Analysis of Discontinuous Deformation New Developments and Applications.pdf

Preface

The first International Conference on Analysis of Discontinuous Conference (ICADD) was held in Taiwan in 1995. Since then, it has been hosted in sequence in Japan (1997), USA (1999), UK (2001), Israel (2002), Norway (2003), USA (2005), China (2007) and Singapore (2009). The conference series aims to exchange ideas and new developments in the various modeling methods for discontinuous deformation. It also promotes the application of the developed methods to rock engineering problems, including but not limited to stability analysis of rock slope, storage carvern, rock tunnels, and underground power stations; and more recently in stress wave propagation in rock mass; rock support design; failure of heterogeneous rock materials; and mining.

This book contains the technical papers presented at the 9th International Conference on Analysis of Discontinuous Deformation (ICADD9) held at the Nanyang Technological University (NTU), Singapore on 25‐27 November 2009. These papers represent the most recent advances and developments in the increasingly important field of discontinuous deformation analysis in rock mechanics and engineering. Following the tradition of the conference series, the main schemes of ICADD9 are on the key block theory and its engineering applications, the discontinuous deformation analysis (DDA) method and the numerical manifold method (NMM). However, other relevant topics, for example, the distinct/discrete element method, the meshless method, the finite element method in rock mechanics and engineering applications are also included.

ICADD9 received more than 140 abstracts from 17 countries. After a vigorous selection and review process, just over 80 papers were accepted for oral presentation at the conference and for inclusion in this book. Still, these represent the largest number of papers presented at the ICADD Conference series.

We would like to acknowledge the authors and speakers for their contribution to ICADD9. Special thanks are due to Dr Genhua Shi for his continuous support and encouragement. Thanks are also due to CMA International Consultant Pte Ltd for their secretariat support. The Underground Technology & Rock Engineering team of NTU and the Defence Science and Technology Agency also provided technical support for the conference.

The successful organization of ICADD9 would not have been possible without the untiring efforts of the organizing committee. Their volunteering but efficient works helped the planning and organizing stages in a great success. We would also like to extend our gratitude to the following sponsors for their support: Tritech Group Pte Ltd, Singapore; Basissoft, LLP, Korea; The Yangtze River Scientific Research Institute, China; Advance Contech (S) Pte Ltd, Singapore; SK E&C Singapore Branch; Hyundai Engineering & Construction Co. Ltd, Korea; Knights Synergy (S) Pte Ltd, Singapore; Dongah Geological Engineering Co Ltd, Singapore.

Guowei MA and Yingxin ZHOU

Editors

About the Book

This book contains 73 technical papers and 7 keynote papers from the industry experts were presented at the 9th

International Conference on Analysis of Discontinuous Deformation — New Development and Applications held in the

Nanyang Technological University, Singapore on 25–27 November 2009. It represents the latest advances in numerical and

analytical methods related to rock mechanics and engineering.

Following the tradition of the conference series, the main schemes of the conference papers are on the key block theory

and its engineering applications, the discontinuous deformation analysis (DDA) method and the numerical manifold method

(NMM). However, other relevant topics, for example, the distinct/discrete element method, the meshless method, the

finite element method in rock mechanics and engineering applications are also included.

For rock masses where the discontinuities dominate their behaviour in a rock engineering system such as rock slopes and

underground rock caverns, the analysis of discontinuous deformation is of critical importance in the stability analysis and

engineering design. It is hoped that this book serves as a useful reference for both researchers and engineers on the new

developments and applications of the various available modeling techniques and tools in this increasingly important field of

rock mechanics and engineering.

Table of Contents

Keynote Papers Rock Stability Analysis and Three Convergences of Discontinuous Deformation Analysis (DDA) Gen‐Hua Shi

Modelling Dynamic Deformation in Natural Rock Slopes and Underground Openings With DDA Y. H. Hatzor

Study on the Formation Mechanism of Tanjiashan Landslide Triggered by Wenchuan Earthquake Using DDA Simulation Wu Aiqing, Yang Qigui, Ma Guisheng, Lu Bo and Li Xiaojun

A G Space Theory with Discontinuous Functions for Weakened Weak (W2) Formulation of Numerical Methods G. R. Liu

Concerning the Influenced of Velocity Ratio and Topography Model on the Result of Rockfall Simulation T. Shimauchi, K. Nakamura, S. Nishiyama and Y. Ohnishi

Development of Numerical Manifold Method and Its Application in Rock Engineering Guowei Ma, Lei He and Xinmei An

Tensorial Approach to Rock Mass Strength and Deformability in Three Dimensions P. H. S. W. Kulatilake

Technical Papers Contact Algorithm Modification of DDA and Its Verification Y. J. Ning, J. Yang, G. W. Ma and P. W. Chen

DDA for Dynamic Failure Problems and Its Application in Rock Blasting Simulation Y. J. Ning, J. Yang, G. W. Ma and P. W. Chen

Study on Roof Caving Problem with DDA Method Liu Yong‐Qian, Yang Jun, Chen Peng‐Wan and Ning You‐Jun

Indeterminacy of the Vertex‐vertex Contact in the 2D Discontinuous Deformation Analysis H. R. Bao and Z. Y. Zhao

Complementary Formulation of Discontinuous Deformation Analysis W. Jiang and H. Zheng

Accelerated Block Sectioning Algorithm Based on Half‐Edge Data Structure Jian Xue

A New Contact Method using Inscribed Sphere for 3D Discontinuous Deformation Analysis Tae‐Young Ahn, Sung‐Hoon Ryu, Jae‐Joon Song and Chung‐In Lee

Study on Failure Characteristics and Support Measure of Layer Structure_Cataclasm Rock Mass Guang Bin Shi, Junguang Bai,Minjiang Wang,Baoping Sun, Ying Wang and Genhua Shi

Stability Analysis of Expansive Soil Slope Using DDA Lin Shaozhong and Qiu Kuanhong

DDA Simulations for Huge Landslides in Aratozawa Area, Miyagi, Japan Caused by Iwate‐Miyagi Nairiku Earthquake K. Irie, T Koyama, E Hamasaki, S Nishiyama, Kshimaoka and Y Ohnishi

Modelling Crack Propagation with Nodal‐based Discontinuous Deformation Analysis H. R. Bao and Z. Y. Zhao

Discontinuous Deformation Analysis for Parallel Hole Cut Blasting in Rock Mass Zhiye Zhao, Yun Zhang and Xueying Wei

The Analysis of Structure Deformation Using DDA with Third Order Displacement Function T. Huang, G. X. Zhang and X. C. Peng

Application of DDA to Evaluate the Dynamic Behaviour of Submarine Landslides Which Generated Tsunamis in the Marmara Sea G. C. Ma, F. Kaneko and S. Hori

3D DDA vs. Analytical Solutions for Dynamic Sliding of a Tetrahedral Wedge D. Bakun‐Mazor, Y. H. Hatzor and S. D. Glaser

Application of Strength Reduction DDA Method in Stability Analysis of Road Tunnels Xia Caichu, Xu Chongbang and Zhao Xu

Micromechanical Simulation of the Damage and Fracture Behavior of a Highly Particle‐filled Composite Material Using Manifold Method Huai Haoju, Chen Pengwan and Dai Kaida

The Application of Discontinuous Deformation Analysis in the Slope Stability of the Expansive Soil Lin Yuliang and Wei Lingjing

Extension of Distinct Element Method and Its Application in Fracture Analysis of Quasi‐brittle Materials Y. L. Hou, G. Q. Chen and C. H. Zhang

A Comparison Between the NMM and the XFEM in Discontinuity Modelling X. M. An and G. W. Ma

Initial Stress Formulae for High‐Order Numerical Manifold Method and High‐Order DDA Haidong Su and Xiaoling Xie

Development of Coupled Discontinuous Deformation Analysis and Numerical Manifold Method (NMM‐DDA) and Its Application to Dynamic Problems S. Miki, T. Sasaki, T. Koyama, S. Nishiyama and Y. Ohnishi

Stability Analysis of Ancient Block Structures by Using DDA and Manifold Method

T. Sasaki, I. Hagiwara, K. Sasaki, R. Yoshinaka, Y. Ohnishi, S. Nishiyama and T. Koyama

Application of Manifold Method (MM) to the Stability Problems for Cut Slopes along the National Roads Y. Ohnishi, T. Koyama, Kazuya Yagi, Tadashi Kobayashi, Shigeru Miki, Takumi Nakai and Yoshifumi Maruki

Boundary Deformability and Convergence in the Higher‐Order Numerical Manifold Method D. Kourepinis, C. J. Pearce and N. Bicanic

The Numerical Manifold Method and Extended Finite Element Method — A Comparison from the Perspective of Discontinuous Deformation Analysis D. Kourepinis, C. J. Pearce and N. Bicanic

Accuracy Comparison of Rectangular and Triangular Mathematical Elements in the Numerical Manifold Method H. H. Zhang, Y. L. Chen, L. X. Li, X. M. An and G. W. Ma

Development of 3‐D Numerical Manifold Method G. W. Ma and L. He

Application of the Optimization for Rock Tunnel's Axis Trend by Block Theory Yang Wenjun, Hong Baoning, Sun Shaorui and Zhu Lei

Quarry Wall Stability Analysis Using Key Block Theory — a Case Study Lu Bo, Ding Xiuli and Dong Zhihong

Probabilistic Key Block Analysis of a Mine Ventilation Shaft Stability — a Case Study Gang Chen

The Support Design for Slope and Tunnel Engineering Based on Block Theory Jiao Liqing, Ma Guowei, He Lei and Fu Guoyang

Hereditary Problems in Long‐Wall Mining by Free Hexagons P. P. Prochazka and Kamila Weiglova

Analysis of Large Rock Deformation Under High in situ Stress S. G. Chen, Y. B. Zhao and H. Zhang

Gotthard Base Tunnel: UDEC Simulations of Micro Tremors Encountered during Construction H. Hagedorn and R. Stadelmann

Discrete Modeling of Fluid Flow in Fractured Sedimentary Rocks Wu Wei, Li Yong and Ma Guowei

An Investigation of Numerical Damping for Modeling of Impact T. Nishimura

Development of Modified RBSM for Rock Mechanics Using Principle of Hybrid‐type Virtual Work N. Takeuchi, Y. Tajiri and E. Hamasaki

High Rock Slope Stability Analysis Using the Meshless Shepard and Least Squares Method X. Zhuang, H. H. Zhu and Y. C. Cai

Numerical Modelling of Laboratory Behaviour of Single Laterally Loaded Piles Socketed into Jointed Rocks W. L. Chong, A. Haque, P. G. Ranjith and A. Shahinuzzaman

Distinct Element Analysis on the Stability of a Stone Pagoda at Mireuk Temple Site in Korea H. Kim and S. Jeon

Distinct Element Analysis of Staged Constructed Underground Cavern in the Vicinity of a Fault H. C. Chua, A. T. C. Goh and Z. Y. Zhao

Numerical Experiment on Thermo‐Mechanical Behavior of Jointed Rock Masses under Cryogenic Conditions S. K. Chung, E. S. Park, Y. B. Jung and T. K. Kim

UDEC Simulation of Block Stability Analysis Around a Large Cavern A. Sookhak, A. Baghbanan, H. Hashemalhosseini and M. Bagheri

The Application of Meshless Methods in Analysis of Discontinuous Deformation M. Hajiazizi

The Optimum Distance of Roof Umbrella Method for Soft Ground by Using PFC Yusuke Doi, Tatsuhiko Otani and Masato Shinji

3DEC Investigation on Slope Stability at Norwich Part Mine S. G. Chen and B. Shen

Evaluation of Deformations Around a Tunnel by Using FEM, FEBEM, UDEC, UDEC‐BE and CFS Rajbal Singh

Numerical Modeling of Undrained Cyclic Behaviour of Granular Media Using Discrete Element Method B. Ferdowsi, A. Soroush and R. Shafipour

A Fundamental Study on the SPH Method Application for Impact Response of RC Structural Members J. Fukazawa and Y. Sonoda

2‐D FEM Analysis of the Rock Fragmentation by Two Drill Bits S. Y. Wang, Z. Z. Liang, M. L. Huang and C. A. Tang

Determination Method of Rock Mass Hydraulic Conductivity Tensor Based on Back‐Analysis of Fracture Transmissivity and Fracture Network Model Li Xiaozhao, Ji Chengliang, Wang Ju, Zhao Xiaobao, Wang Zhitao, Shao Guanhui and Wang Yizhuang

Numerical Simulation of Scale Effect of Jointed Rock Masses Z. Z. Liang, L. C. Li, C. A. Tang and S. Y. Wang

Influence of Cobblestone Geometrical Property on Equivalent Elastic Modulus of Cobblestone‐Soil Matrix M. Z. Gao, H. S. Ma, and J. Zhao

Comparative Studies of Physical and Numerical Modeling on Regular Discontinuities Abbas Majdi, Hessam Moghaddam Ali and Kayumars Emad

Probabilistic Assessment of a Railway Steel Bridge B. Culek, V. Dolezel and P. P. Prochazka

An Analysis of Dynamic Tensile Fracture in Concrete Under High Strain Rate M. Kurumatani, S. Iwata, K. Terada, S. Okazawa and K. Kashiyama

A New Equivalent Medium Model for P‐wave Propagation Through Rock Mass with Parallel Joints G. W. Ma, L. F. Fan and J. C. Li

Stability Analysis of Transformer Cavern and the Corresponding Bus Duct System at Siyah Bishe Pumped Storage Power Plant Abbas Majdi, Kayumars Emad and Hessam Moghaddam Ali

Process Zone Development Associated with Cracking Processes in Carrara Marble L. N. Y. Wong and H. H. Einstein

Simulation of Stress Singularity Around the Crack Tips for LEFM Problems Using a New Numerical Method G. R. Liu and N. Nourbakhsh Nia

Modeling of Three‐dimensional Hydrofracture in Permeable Rocks Subjected to Differential Far‐field Stresses L. C. Li, C. A. Tang, G. Li and Z. Z. Liang

Crack Propagation Analysis using Wavelet Galerkin Method S. Tanaka, S. Okazawa and H. Okada

Simulation of Multiphase Fluid Motion in Pore‐scale Fractures M. B. Liu and J. Z. Chang

An Analysis of Model Tests on Rock Cavern Damage Induced by Underground Explosion Zhang Xingui, Ma Guowei, Wu Wei, Yan Lie, Li Mangyuan and Cheng Qingsheng

Microscopic Numerical Modelling of the Dynamic Strength of Brittle Rock G. ‐F. Zhao and J. Zhao

Fault Studies and Coal‐gas‐outburst Forecast in Coal Mines H. Q. Cui, X. L. Jia, Z. P. Xue and F. L. Yang

Suggestion of Equations to Determine the Elastic Constants of a Transversely Isotropic Rock Specimen Chulwhan Park, Chan Park, E. S. Park, Y. B. Jung and J. W. Kim

Numerical Analysis of Deep Excavation Affected by Tectonic Discontinuity L. Mica, V. Racansky and J. Grepl

The Finite Element Analysis for Concrete Filled Steel Tubular Columns under Blast Load J. H. Zhao, X. Y. Wei and S. F. Ma

Numerical Simulation of Performance of Concrete‐filled FRP Tubes Under Impact Loading C. Wu, T. Ozbakkloglu , G. Ma, Z. Y. Huang

Estimating Hydraulic Permeability of Fractured Crystalline Rocks Using Geometrical Parameters R. Vesipa, Z. Zhao and L. Jing

Mutual Effect of Tectonic Dislocations and Tunnel Linings during Tunnelling K. Weiglová and J. Boštík

Rock Stability Analysis and Three Convergences of DiscontinuousDeformation Analysis (DDA)

GEN-HUA SHI

DDA Company, 1746 Terrace Drive, Belmont, CA 94002, USA

1. Introduction

In the field of practical rock engineering, there are two independent computations: continuouscomputation and limit equilibrium computation. Limit equilibrium is still the fundamentalmethod for global stability analysis. For any numerical method, reaching limit equilibriumrequires large displacements, discontinuous contacts, precise friction law, multi-step com-putation and stabilized time-step dynamic computation. Therefore three convergences areunavoidable: convergence of equilibrium equations, convergence of open-close iterations forall contacts and convergence of the maximum displacement for static computations. Thispaper focuses mainly on applications of two-dimensional DDA. The applications show DDAhas the ability to reach limit equilibrium of block systems. For slope or tunnel stability anal-yses, this paper works on rock block sliding and rotation. For dam foundation stabilityanalysis, this paper presents dam foundation damage computation, where the block slidingis a main issue.

2. Rock Stability Computation

2.1. About discontinuous deformation analysis (DDA)

DDA works on block systems. Each block has linear displacements or constant stresses andstrains.

The current version of 2d-DDA bas 6 unknowns per block:

x direction movement dx,y direction movement dy,rotation rxy,x direction strain εx,y direction strain εy,shear strain τxy.

DDA uses multi-time steps. Both static and dynamic cases use dynamic computation. Staticcomputation is the stabilized dynamic computation by applying small mount of damping.Therefore DDA can perform discontinuous and large deformation computation for bothstatic and dynamic cases.

For each time steps, DDA usually has several open-close iterations. DDA readjust open,close or sliding modes until every contact position has the same contact mode before andafter the equation solving then going to next time step. For each open-close iteration of eachtime step, DDA solves global equilibrium equations. The friction law is ensured in DDAcomputation. This law is the principle law of stability analysis. Also, the friction law isinequality equations in mathematics.

Corresponding author. E-mail: [email protected]

Analysis of Discontinuous Deformation: New Developments and Applications.Edited by Guowei MA and Yingxin ZHOU. Published by Research Publishing Services.Copyright c© 2009 by Society for Rock Mechanics & Engineering Geology (Singapore).ISBN: 978-981-08-4455-4doi:10.3850/9789810844554-keynote-Shi-Genhua 1

Analysis of Discontinuous Deformation: New Developments and Applications

Every single block of 2-d DDA can be a generally shaped convex or concave two-dimen-sional polygon. Each block can have any number of edges. Based on simplex integration, thestiffness matrices, the inertia matrices, the matrices of initial stresses, the loading matricesand all other matrices of DDA are analytical solutions.

DDA has complete linear contact modes. If the time step is small enough and the totalstep number is large enough, DDA can simulate any possible complex movements of blocksystems.

DDA serves as a bridge between FEM and limit equilibrium method. DDA has strict equi-librium at each time step. After certain time step, DDA reaches dynamic or static limit equi-librium for whole simply deformable block systems.

DDA also served as implicit version of DEM method. DDA has all advantages of dynamicrelaxation yet the convergence is strict and the result is accurate.

More important, DDA is a very well examined method by analytical solutions, physicalmodel tests and large engineering projects.

2.2. Five different factor of safety for gravity dam foundation stabilityanalysis

Table 1 shows the input data of three cases of dam foundation stability computation usingtwo-dimensional DDA. Figure 1 shows the mode of failure by increasing total water pressureand reducing the friction angle. Based on the mode of failure, the sliding blocks are chosen.Based on the assumed sliding blocks, different factors of safety are computed. In Table 1,three cases are included:

Case 1 is limit equilibrium method. Here normal loads are applied, the factor of safety ofthe chosen sliding blocks are computed. The factor of safety is 1.94 as shown in Table 1.

Case 2 is the fictitious force method. Keeping the stability, increase the water pressure asmuch as possible. The factor of safety is the ratio of applied total water pressure and thetotal normal water pressure. The factor of safety is 2.80 as shown in Table 1.

Case 3 is the strength reduction method. Keeping the stability, reducing the friction angleas much as possible. The factor of safety is the ratio of the tangent of real friction angle andthe tangent of reduced friction angle. The factor of safety is 2.91 as shown in Table 1.

Table 1. Physical data of rock mass of gravity dam foundation.

Material parameters 1. Limit equilibrium 2. Fictitious force 3. Strength reduction

Unit weight 2.4 2.4 2.4Elastic Modulus 2600000 2600000 2600000Poisson’s ratio 0.25 0.25 0.25Friction angle 17.0 17.0 6.0Cohesion 0.0 0.0 0.0Additional water pressure 0.0 7290 0.0Dynamic ratio 0.9999 0.999 0.999Contact stiffness 10000000 5000000 10000000Time interval 0.05 0.05 0.05Total time step 1500 1500 1500Ending max. displ. ratio <0.0000005 <0.0000005 <0.0000005Factor of safety 1.94 2.80 2.91Margin factor of safety +0.0 +0.14 +0.39

2


Figure 1. Dam foundation failure with increased water pressure and reduced friction angle.

Case 4 is the limit equilibrium method under the fictitious force. All parameters are thesame as case 2. The computation is also identical with case 2. After increasing the waterload, the computed factor of safety FS is still greater than 1.0. The resulting factor of safetyis the factor of safety of case 2 plus the marginal factor of safety FS− 1.0 = 0.14.

Case 5 is the limit equilibrium method under the reduced friction angle. All parameters arethe same as case 3. The computation is also identical with case 3. After reducing the frictionangle, the computed factor of safety FS is still greater than 1.0. The resulting factor of safetyis the factor of safety of case 3 plus the marginal factor of safety FS− 1.0 = 0.39.

The dynamic ratio is 0.9999 in case 1 for example. It means the next time step inherent99.99% of the velocity from the previous time step.

The maximum displacement ratio is the allowed maximum step displacement divided bythe half height of the whole mesh. In Table 1, the maximum displacement ratio is less than0.0000005.

The unified units on weight, length, time and angle are the following:

Weight unit: TonLength unit: meterTime unit: secondAngle unit: degree

The computation results of all case 1 to case 5 are stable. All case 1 to case 5 didn’t reachlimit equilibrium. Except case 1, all case 2 to case 5 are close to limit equilibrium. Figure 2shows the results of all case 1 to case 5. The computation requirement of case 2 to case 5 ishighly difficult because these cases are very close to limit equilibrium or very close to failure.The computation of case 2 to case 5 also requires three convergences.

2.3. Rock falling of underground power chambers

In the following, sections of a given underground powerhouse are analyzed by two-dimen-sional DDA. The underground powerhouse section has two cases: without bolt support andwith bolt support. All two cases are based on the same geometric data of joint sets shown byTable 2. Table 3 shows the physical data of the underground chamber rock mass.

The rock falling computation of underground chamber (Figures 3–4) uses two-dimensionalDDA. Total 1800 time steps are used. In the cases of Figure 2 and 3, the time interval was

3


Figure 2. Result of dam foundation of case 1 to case 5.

automatic chosen and controlled by maximum displacement ratio 0.001. In the cases ofFigures 3 and 4, the dynamic ratio is 1.0. It means in the beginning of the next time step100% of velocity is inherited from the present time step. Therefore rock falling computationis fully dynamic without damping.

The geometric data of joint sets of the underground power chamber for Figure 3–6 areshown in Table 2.

The physical data of the rock mass of Figures 3–4 are shown on Table 3.Figures 3 and 4 show rock fall process and final state of this underground powerhouse.

Table 2. Geometric data of joint sets of the underground powerchamber.

Joint set 1 2 3

Dip angle 35 degree 70 degree 85 degreeDip direction 315 degree 150 degree 25 degreeAverage spacing 6.0 meter 4.0 meter 3.0 meterAverage length 100 meter 20 meter 30 meterAverage bridge 0.2 meter 1.0 meter 0.5 meterDegree of random 0.0–1.0 0.5 0.5 0.5

Table 3. Physical data of underground chamber rock mass.

Material parameters Parameters Computation data Data

Unit weight 2.7 Dynamic ratio 1.0Elastic Modulus 3000000 Contact stiffness 1000000Poisson’s ratio 0.25 Time interval AutomaticFriction angle 25 Total time step 1800Cohesion 0 Ending max. displ. ratio 0.48

4


Figure 3. Rock falling process of the underground powerhouse computed by 2-d DDA without boltsupport.

Figure 4. Rock falling final condition of the underground powerhouse computed by 2-d DDA withoutbolt support.

5


Figure 5. Bolting computation of underground powerhouse where the location of each bolt is drawn.

Figure 6. Bolting forces of the underground powerhouse where the force of each bolt is drawn as thelength along this bolt.

2.4. Rock bolting of an underground power chamber

The rock bolting computation of the underground chamber (Figures 5–6) uses two-dimen-sional DDA. Total 2000 time steps are used. The time interval was not controlled by maxi-mum displacement ratio. The step time interval is 0.002 seconds. The dynamic ratio is 0.97.

6


Table 4. Physical data of underground chamber rock mass.

Material parameters Parameter Computation data Data

Unit weight 2.7 Dynamic ratio 0.97Elastic modulus 3000000 Contact stiffness 3000000Poisson’s ratio 0.25 Time interval 0.002Friction angle 25 Total time step 2000Cohesion 0 Ending max. displ. ratio 0.000026Bolt stiffness 16889 Max. bolting force error <0.0000005

It means in the beginning of the next time step 97% of velocity is inherited from the presenttime step. This miner volume damping can obtain stabilized contact forces. The forces of allbolts are highly stabilized. The bolting forces were not changed in 6 digits after the dismalpoint under large enough time steps.

The physical data of the rock mass of Figures 5–6 are shown on Table 4.Figure 5 shows the location of each rock bolts. The length of the rock bolts are 10 meter

and 12 meters alternatively. The bolt spacing is one meter. The diameter of the rock bolts is32 mm.

Figure 6 shows the bolting forces of the underground powerhouse where the force of eachbolt is drawn as the length along this bolt.

Figure 7 is the time depending resulting bolting forces (unit ton) of bolt number 1 to 30counting from top centre down and right to left in each level of the underground powerhouse section. From bolt 1 to bolt 30, there are two bolts where the tension forces exceed 25tons as shown by Figure 7.

Figure 8 is time depending resulting bolting forces (unit ton) of bolt number 31 to 60counting from top centre down and right to left in each level of the underground powerhouse section. It can be seen that, there are two bolts where the tension forces are greaterthan 25 tons.

bd

tst

ress

time

60

50

10

0

0

1 2 3 4 5 6 7

40

30

20

bol t 3bol t 4bol t 5bol t 6bol t 7bol t 8bol t 9bol t 10bol t 11bol t 12bol t 13bol t 14bol t 15bol t 16bol t 17bol t 18bol t 19bol t 20bol t 21bol t 22bol t 23bol t 24bol t 25bol t 26bol t 27bol t 28bol t 29bol t 30

bol t 2bol t 1

Figure 7. Time depending resulting bolting forces of bolt number 1 to 30 counting from top centredown and right to left in each level of the underground power house section.

7


bd

tst

ress

time

40

35

30

25

20

15

10

5

00

1 2 3 4 5 6 7


bol t 32bol t 31


Figure 9 is time depending resulting bolting forces (unit ton) of bolt number 61 to 90counting from top centre down and right to left in each level of the underground powerhouse section. It can be seen from Figure 9 that, there is no blot where the tension force isgreater than 25 tons.

Figure 10 is the time depending resulting bolting forces (unit ton) of bolt number 91 to115 counting from top centre down and right to left in each level of the underground powerhouse section. It can be seen that, there is only one bolt where the tension force is greaterthan 25 tons.

bd

tst

ress

25

20

15

10

5

0

time

0 1 2 3 4 5 6 7


bol t 62bol t 61


8


25

30

0

20

15

10

5

-50 1 2 3 4 5 6 7

bol t 93bol t 94bol t 95bol t 96bol t 97bol t 98bol t 99bol t 100bol t 101bol t 102bol t 103bol t 104bol t 105bol t 106bol t 107bol t 108bol t 109bol t 110bol t 111bol t 112bol t 113bol t 114bol t 115

bol t 92bol t 91


Table 5. Geometry and physical data of joint set.

Joint set Dip angle Average spacing Average length Cohesion Friction angle

1 (foliation) 70 degree 7 ft 2000 ft 0 psf 10 degree2 20 degree 30 ft 30 ft 0 psf 30 degree

Weight unit: TonLength unit: meterTime unit: secondAngle unit: degree

2.5. Stability computation of toppling slopes: foliation planes have 70degrees dip angle

From the drilling data, the original dip angle of the foliation planes is about 70 degrees. Ifafter toppling, the second joint set has 40 degrees dip angle, the original dip angle of thesecond joint set is about 20 degrees. The following Table 5 includes the geometric data andphysical data of the joint sets and rock masses. The following computation is to simulate orback calculate the past toppling of the slope.

Figure 11 shows the result of the slope, the dip angle of the foliation planes is 70 degrees.Figure 12 is the time depending movements of the measured points under the anchor block

of the slope for the previous case of Figure 11.Computation of two-dimensional DDA uses 30000 time steps, 0.002 second per step. The

dynamic ratio is 0.99. It means the next time step inherent 0.99 of the velocity from theprevious time step. Under the 70 degrees dip angle of the foliation planes, the toppling ofthe slope is very large. It can be seen; the slope is convex which gives more room for blockrotation or toppling.

9


Figure 11. Toppling of the slope where the dip angle of foliation plane is 70 degrees.

AB5 dip=70 with weak zone

0

1

2

3

4

5

6

7

8

9

10

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60

10 ft20 ft30 ft40 ft50 ft60 ft70 ft80 ft90 ft100 ft110 ft

dis

pla

ce

me

nt

ft

time seconds

Figure 12. Displacements of points under the slope where the dip angle of foliation planes is 70degrees.

Figure 11 shows that, after large mount of rotation, the toppling reached a stable state.Therefore the toppling mode is different from the sliding mode. Generally speaking, thetoppling mode can be stabilized after the rotation. For sliding mode, as soon as the slopestart to slide, the sliding can hardly to stop.

10


Acknowledgements

The project of “Discontinuous Deformation Analysis (DDA) of Block Systems” (XDS2007-10) provided financial support to this research.

References

1. Shi, Gen-hua, Block System Modelling by Discontinuous Deformation Analysis, ComputationalMechanics Publications, New Southampton, UK and Boston, USA, 1993.

2. Shi, Gen-hua, “Single and Multiple Block Limit Equilibrium of Key Block Method and Discon-tinuous Deformation Analysis”, Stability of Rock Structures, ICADD-5, Beer Sheva, Israel. 3–46,2002.

3. Shi, Gen-hua, “Applications of Discontinuous Deformation Analysis (DDA) and ManifoldMethod”, The Third International Conference on Analysis of Discontinuous Deformation,ICADD-3, Vail, Colorado, USA, 1999, 3–15.

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Modelling Dynamic Deformation in Natural Rock Slopes andUnderground Openings With DDA

Y.H. HATZOR

Dept. of Geological and Environmental Sciences, and Dept. of Structural Engineering,Ben-Gurion University of the Negev, Beer – Sheva, Israel 84601

1. Introduction

This paper presents recent developments in the validation and application of DiscontinuousDeformation Analysis (DDA), originally developed by Shi.1 We begin with a brief review ofrecently published 2D — DDA validations for cases of dynamic loading,2,3 along with new3D — DDA validations for single and double face sliding.4 Following these validations wepresent dynamic DDA applications in natural rock slopes and underground openings.

The north face of Masada world heritage site is used to demonstrate dynamic rock slopestability analysis with DDA where a highly fractured rock slope is subjected to dynamic inputusing modified earthquake records that take into consideration the measured topographicsite effect at the site. We thus obtain the peak ground acceleration (PGA) that would berequired for damage, present the expected failure modes, and show how the anticipateddeformations may be restrained using rock bolting. The issue of rock bolt reinforcement isfurther explored via an example from an overhanging cliff where the most effective boltingpattern for restraining overturning is obtained.

Finally, we show how DDA can be utilized to predict the minimum overburden required toensure underground opening stability as a function of opening span in horizontally layeredand vertically jointed rock masses.

2. Dynamic DDA Validations

2.1. Single and double face sliding

A displacement based sliding block model was first proposed by Newmark5 and Good-man and Seed,6 is now largely referred to as “Newmark” type analysis. Determination ofthe amount of displacement during an earthquake involves two steps:6 (1) Determinationof horizontal acceleration required to initiate down slope motion, also known as “yieldacceleration” (ay), which can be found by pseudo-static analysis, and (2) Evaluation ofthe displacement developed during time intervals when yield acceleration is exceeded, bydouble-integration of the acceleration time-history, with the yield acceleration used as refer-ence datum. Goodman and Seed6 showed that for the case of a block on an inclined planethe yield acceleration is given by:

ay = tan (φ − α)g (1)

For an acceleration record of the form a = kg sin (ω t), where ω corresponds to the frequencyof the function and k calibrates the proportion between a and g, the corresponding time


Analysis of Discontinuous Deformation: New Developments and Applications.Edited by Guowei MA and Yingxin ZHOU. Published by Research Publishing Services.Copyright c© 2009 by Society for Rock Mechanics & Engineering Geology (Singapore).ISBN: 978-981-08-4455-4doi:10.3850/9789810844554-keynote-Hatzor 13


interval θ until yield acceleration is attained is given by:

θ =sin−1

(ay

kg

)ω

(2)

The down slope acceleration of the sliding block can be determined by subtracting the resist-ing forces from the driving forces:

at =[kg sin (ωt) cosα + g sin α

] − [g cosα − kg sin (ωt) sinα

]tanφ (3)

Similarly, the displacement of the block at any time is determined by double integration onthe acceleration, with θ as reference datum, in a conditional manner, namely:

If a > ay or v > 0:

dt =t∫θ

v =∫∫θ

a = g

[(sin α − cosα tanφ)

(t2

2− θ

)]+

+ agω2 [(cosα + sin α tanφ) (ω cos (ωθ) (t− θ)− sin (ωt)+ sin (ωθ)]

(4)

Otherwise dt = dt−1

Equation 4 provides the analytical solution for the dynamic displacement of a block on aninclined plane with inclination α and friction angle φ, starting from rest and subjected to asinusoidal loading function with frequency ω. Figure 1 displays a comparison between the

Figure 1. DDA vs. analytical solution for the dynamic sliding of a block on an inclined plane (after2).

14


0

0.4

0.8

1.2

1.6D

ispl

acem

ent(

m)

0.010.1

110

1001000

Rel

ativ

eEr

ror(

%)

0 2 4 6Time (sec)

-6

-4

-2

0

2

4

6

Hor

izon

talI

nput

mot

ion

(m/s

2 )0.010.11101001000

Erel, VAErel, DDA

Newmark SolutionVector Analysis3D-DDAInput Motion

A

0

0.5

1

1.5

2

2.5

Dis

plac

emen

t(m

)0.1

1

10

100

Rel

ativ

eEr

ror(

%)

0 0.4 0.8 1.2 1.6 2Time (sec)

-6

-4

-2

0

2

4

6

Hor

izon

talI

nput

mot

ion

(m/s

2 )

0.1

1

10

100

Vector Analysis3D-DDAInput Motion (y)

A

Figure 2. Expansion of the 2D solution to 3D using vector algebra (VA) and comparison between 3D— DDA, VA, and the classic Newmark solution for single face (left panel) and double face (right panel)sliding (after Ref. 4).

analytical and DDA solutions for three friction angles (φ =22◦, 30◦, 35◦) with θ = 0.035,0.18 and 0.28 seconds, respectively. The accumulated displacements are computed for a planeinclination of 20◦. Other than for the first second, in which the relative error reaches 50%,due to a very small absolute error of only 2.6E10−5 meters, the relative error is less than 1%for all three friction angles for all accumulated displacements. Time-step size is kept constantin all DDA runs and is 0.002 sec.

The two dimensional formulation shown in Eq. (4) has recently been expanded by Bakun-Mazor et al.4 to three dimensions for a single block on an inclined plane and for a wedgesliding simultaneously on two planes. Figure 2 presents results obtained with 3D DDA forsingle face (left panel) and double face (right panel) sliding under a sinusoidal input motionacting in the horizontal direction. The relative error is plotted in the lower panels of Fig. 2.As can be seen from inspection of Fig. 2 numerical results obtained with 3D-DDA agree verywell with analytical solutions. For further details see Ref. 4.

2.2. Block response to induced displacements at foundation

The above validations were performed with the loading function applied to the cenroid ofthe block. It is also interesting to study the response of a block to vibrations at the founda-tions where the forces are transmitted through frictional forces along the interface betweenthe “induced” and “responding” blocks. An analytical solution for the response of a singleblock resting on a block that is subjected to time-dependent displacement input function wasdeveloped by Kamai and Hatzor2 for the block system shown in Fig. 3 below.

The displacement function for block 1 is in a form of a cosine function, starting from 0:dt = D (1− cos(2πωt)) where D is the amplitude of the harmonic wave, and the correspond-ing response of block 2 is investigated.

Results of sensitivity analyses for input amplitude and interface friction are presented inFig. 4 below. In the left panel the response of Block 2 to changing amplitudes of motion

15


x

y

1

2

0

x

y

1

2

0

Figure 3. The modeled DDA block system. Block 0 is the foundation block, Block 1 receives thedynamic input motion (horizontal — cyclic), and Block 2 responds (after Ref. 2).

0

0.5

1

1.5

Dis

plac

emen

tof

uppe

rbl

ock

(m)

0 1 2 3 4 5Time (sec)

AnalyticAnalyticAnalytic

DDADDADDA

InputMotion

=0.1=0.6=1

A B

Figure 4. Analytical (line) vs. DDA (symbols) solutions for the response of a block to cyclic displace-ments at the foundation (after2): (A) influence of input amplitude (f = 1 Hz and μ = 0.6), (B) influenceof friction coefficient along interface (D = 0.5 m, f = 1 Hz).

(D) under a constant input frequency of 1 Hz and friction coefficient of 0.6 is presented.The cumulative displacement is in direct proportion to the amplitude, as expected. Note thatthe three displacement curves follow the periodic behavior of the input displacement function(T = 1 sec.), and that divergence between curves starts after 0.25 sec. where the displacementfunction has an inflection point. In the left panel the response of Block 2 to changing frictioncoefficients (μ) along the interface under constant displacement amplitude of 0.5 m and inputfrequency of 1 Hz is presented. The accumulating displacement is in direct proportion to thefriction coefficient up to 0.5 sec., where the input displacement function changes direction.After that point the accumulating displacement for μ = 0.6 is larger than for μ = 1, since thehigh friction works in both directions: forward and backward. Note that curves for μ = 0.1and μ = 0.6 follow the periodic behavior of the displacement function, whereas the curvefor μ = 1.0 is in delay of about 0.25 sec.

2.3. Dynamic rocking

So far we have looked at validations concerning frictional sliding. Another important failuremode in rock mechanics is block rotation. Makris and Roussos7 studied in 2D the dynamicrocking of a column subjected to a sinusoidal input acceleration function for the free bodydiagram shown in Fig. 5.

The solution for the dynamic rocking of a column subjected to an input loading functionof a half-sine pulse is obtained in two stages:

16


2b

2h

R

c.m.

h

b0’ 0

ü(t)

2b

2h

R

c.m.

h

b0’ 0

ü(t)

Figure 5. Free body diagram and sign convention for the rocking column analysis (after7).

(1) Instantaneous response — dynamic motion which takes place simultaneously with appli-cation of the input acceleration function: ug (t) = ap sin

(ωpt + ψ) from t = 0 to t = 0.5

sec., where here ωgis2π(f = 1 Hz) and the phase angle (ψ) is ψ = sin−1(αgap

),

(2) Consequent motion — rocking oscillations after pulse termination from t = 0.5 sec. andonwards.

Naturally when the pulse terminates the input acceleration diminishes (ug (t) = 0, henceap = 0) and the coefficients of integration are updated for changing rotation angle and angu-lar velocity. Furthermore, following each impact (@ θ = 0), the angular velocity and thecoefficients of integration are recalculated as well. The analytical and DDA solutions for col-umn width and height of b = 0.2 m and h = 0.6 m are presented in Fig. 6, following thework of Yagoda-Biran and Hatzor.3 In the top panel results obtained for amplitude ap = 5.43m/s2 (0.5535 g), a value slightly lower than required for overturning according to the analyt-ical solution, are shown, hence only column rocking is obtained. In the lower panel results

t=1.28 sect=1.28 sec



t=0.44 sect=0.44 sect = 0.44 sec

t = 1.28 sec

t = 2.36 sec

t = 4.88 sec




t=0.44 sect=0.44 sect = 0.44 sec

t = 1.28 sec

t = 2.36 sec

t = 4.88 sec




t=0.44 sect=0.44 sect = 0.44 sec

t = 1.28 sec

t = 2.36 sec

t = 4.88 sec

1111111

0 1 2 3 4 5 6-1

0

1

0 1 2 3 4 5 6

-1

0

1

0 1 2 3 4 5 6

0

0.2

0.4

0 1 2 3 4 5 610

-210

-110

010

110

210

3

time (sec)

apeak=5.43 m/sec2

ü(g

)θ

/αan

gula

r ve

loci

ty(r

ad/s

ec)

erro

r (%

)

1111111

0 1 2 3 4 5 6-1

0

1

0 1 2 3 4 5 6

-1

0

1

0 1 2 3 4 5 6

0

0.2

0.4

0 1 2 3 4 5 610

-210

-110

010

110

210

3

time (sec)

apeak=5.43 m/sec2

ü(g

)θ

/αan

gula

r ve

loci

ty(r

ad/s

ec)

erro

r (%

)




t=0.44 sect=0.44 sect = 0.44 sec

t = 1.28 sec

t = 2.36 sec

t = 4.88 sec




t=0.44 sect=0.44 sect = 0.44 sec

t = 1.28 sec

t = 2.36 sec

t = 4.88 sec




t=0.44 sect=0.44 sect = 0.44 sec

t = 1.28 sec

t = 2.36 sec

t = 4.88 sec

1111111

0 1 2 3 4 5 6-1

0

1

0 1 2 3 4 5 6

-1

0

1

0 1 2 3 4 5 6

0

0.2

0.4

0 1 2 3 4 5 610

-210

-110

010

110

210

3

time (sec)

apeak=5.43 m/sec2

ü(g

)θ

/αan

gula

r ve

loci

ty(r

ad/s

ec)

erro

r (%

)

1111111

0 1 2 3 4 5 6-1

0

1

0 1 2 3 4 5 6

-1

0

1

0 1 2 3 4 5 6

0

0.2

0.4

0 1 2 3 4 5 610

-210

-110

010

110

210

3

time (sec)

apeak=5.43 m/sec2

ü(g

)θ

/αan

gula

r ve

loci

ty(r

ad/s

ec)

erro

r (%

)

t = 0.56 sec

t = 1.76 sec

t = 3.32 sec

t = 3.92sec

t = 0.56 sec

t = 1.76 sec

t = 3.32 sec

t = 3.92sec

t = 0.56 sec

t = 1.76 sec

t = 3.32 sec

t = 3.92sec

0 1 2 3 4 5 6-1

0

1

0 1 2 3 4 5 6

-1

0

1

0 1 2 3 4 5 6

0

0.2

0.4

0 1 2 3 4 5 610-310-210-110010

110210

3

6

6

6

6

time (sec)

apeak=5.44 m/sec2

0 1 2 3 4 5 6-1

0

1

0 1 2 3 4 5 6

-1

0

1

0 1 2 3 4 5 6

0

0.2

0.4

0 1 2 3 4 5 610-310-210-110010

110210

3

6

6

6

6

time (sec)

apeak=5.44 m/sec2

ü(g

)θ/

αan

gula

r ve

loci

ty(r

ad/s

ec)

erro

r (%

)ü

(g)

θ/α

angu

lar

velo

city

(rad

/sec

)er

ror

(%)

t = 0.56 sec

t = 1.76 sec

t = 3.32 sec

t = 3.92sec

t = 0.56 sec

t = 1.76 sec

t = 3.32 sec

t = 3.92sec

t = 0.56 sec

t = 1.76 sec

t = 3.32 sec

t = 3.92sec

0 1 2 3 4 5 6-1

0

1

0 1 2 3 4 5 6

-1

0

1

0 1 2 3 4 5 6

0

0.2

0.4

0 1 2 3 4 5 610-310-210-110010

110210

3

6

6

6

6

time (sec)

apeak=5.44 m/sec2

0 1 2 3 4 5 6-1

0

1

0 1 2 3 4 5 6

-1

0

1

0 1 2 3 4 5 6

0

0.2

0.4

0 1 2 3 4 5 610-310-210-110010

110210

3

6

6

6

6

time (sec)

apeak=5.44 m/sec2

ü(g

)θ/

αan

gula

r ve

loci

ty(r

ad/s

ec)

erro

r (%

)ü

(g)

θ/α

angu

lar

velo

city

(rad

/sec

)er

ror

(%)

0 1 2 3 4 5 6-1

0

1

0 1 2 3 4 5 6

-1

0

1

0 1 2 3 4 5 6

0

0.2

0.4

0 1 2 3 4 5 610-310-210-110010

110210

3

6

6

6

6

time (sec)

apeak=5.44 m/sec2

0 1 2 3 4 5 6-1

0

1

0 1 2 3 4 5 6

-1

0

1

0 1 2 3 4 5 6

0

0.2

0.4

0 1 2 3 4 5 610-310-210-110010

110210

3

6

6

6

6

time (sec)

apeak=5.44 m/sec2

ü(g

)θ/

αan

gula

r ve

loci

ty(r

ad/s

ec)

erro

r (%

)ü

(g)

θ/α

angu

lar

velo

city

(rad

/sec

)er

ror

(%)

Figure 6. Solution for dynamic column rotation (b = 0.2 m, h = 0.6 m). Left) ap lower than requiredfor toppling, Right) ap sufficient for column toppling. Solid line: analytical solution, Open circles: DDAresults (after Ref. 3).

17


obtained with an amplitude of ap = 5.44 m/s2 (0.5545g), the minimum value required foroverturning according to the analytical solution, are shown, and indeed column overturningis obtained ∼ 1.5 sec. after pulse termination, exactly as in the analytical solution. The errorremains very small until the first impact occurs, after which the error begins to increase. Nat-urally from the definition of relative error (see3), which depends on the actual value of θ ateach time step, greater error is expected for very small values of θ , and vice versa.

2.4. Block slumping

In a segment along the East face of the Masada mountain, locally known as the ‘snake path’cliff, a prismatic block resting on an easterly dipping bedding plane and separated fromthe cliff by two orthogonal “tension cracks”. The tall and slender geometry of the blockmakes it susceptible to the “block slumping”’ failure mode, initially proposed by Wittke8

and extended to multiple blocks by Kieffer.9 The block, 15 m high, 10 m wide, and weighs13.7MN (1400 ton), has separated from the cliff over geologic and historic times by an accu-mulated displacement of up to 20 cm. Because the resultant weight vector trajectory of theblock acts on the steeply inclined plane (see Figure 7), sliding will commence by mobilizingshear strength along both the steep and the shallow inclined planes simultaneously. Thus,rotation around a center located outside of the block may take place – a failure mode definedas “Block Slumping” by Goodman and Kieffer.10 It is intuitively clear that once slumping isinitiated joint water pressures will rapidly dissipate as an “A type joint”, characterised by awide base and pointing sharp edge at the top, will form behind the block at onset of motionallowing free drainage.

The forces acting on a block that undergoes block slumping are shown in Fig. 7A. Assum-ing the friction angles on the two sliding planes are equal (φ1 = φ2) three equilibrium equa-tions are necessary for solution of the contact forces N1 and N2 and the mobilized frictionangle φmobilized:

∑Fv = 0:W = N1 cosα1 +N1 tanφ1 sinα1 +N2 cosα2 +N2 tanφ2 sinα2 (5)∑

M0 = 0:Wdw +N2 tanφ2d′2 = N2d2 (6)∑Mc = 0:Wx = N2 tanφ2AC+N1 tanφ1OC (7)

A

O

C

W

N1

T1

N2

T2

xdw

d2

d2’

A

O

C

W

N1

T1

N2

T2

xdw

d2

d2’

H = 15m

A B

DATUM Zw = 0

Zw

H = 15m

A B

DATUM Zw = 0

Zw

A B

Figure 7. A — Free body diagram of the block at the snake path cliff, Masada, B — Actual geometryof the studied block (after Hatzor11).

18


Figure 8. DDA results for the studied block with interface friction angle of 20◦ and gravitationalloading for t = 0, 0.8, 1.6, and 2.5 sec.

where α1 and α2 are the inclinations of the sliding plane and the “tension crack” respectively.Simultaneous solution of the three equations for the geometry of the block (Figure 7B) yieldsa mobilized friction angle value of φmobilized = 22◦.

To test the validity of this solution 2D — DDA is employed. The exact two — dimensionalgeometry of the block is studied under gravitational loading with different values of interfacefriction angle as the only varied parameter between simulations. The actual friction angle ofMasada discontinuities was studied experimentally12 using tilt tests, tri-axial tests, and directshear tests. The peak friction angle obtained from direct shear tests on rough surfaces is 41◦.The residual friction angle, obtained from tri-axial tests performed on filled saw-cut planes is23◦. The analytical solution for block slumping indicates that for friction angle values lowerthan 22◦ the studied block will exhibit back slumping by simultaneous shear along bothinterfaces. Therefore, for rough interfaces with available friction angle of 43◦ the block maybe assumed to be stable. However, for interfaces possessing residual friction angle value of23◦ the block may be considered at limit equilibrium considering the block slumping mode.

The original modelled configuration of the studied block with DDA is shown in the leftpanel of Fig. 8. The block remains static until the input friction angle on the interfaces isreduced to 21◦ after which sliding ensues along both interfaces simultaneously, exactly aspredicted by the analytical solution. The dynamic deformation progress for interface frictionangle of 20◦ is shown in Fig. 8 where clearly the block slumping mode is obtained, confirm-ing the analytical solution that requires a minimum friction angle of 22◦ for stability. It isimportant to note here that in DDA the failure mode is a result of the analysis and not a preassumption.

3. Rock Slope Stability

3.1. The significance of correct mesh generation

The validity and accuracy of DDA has been demonstrated in the previous section. In rockengineering analysis however it is the simulated structure that will govern the deformation.Therefore, every effort should be made to represent the rock mass structure correctly onthe basis of field measurements. Consider for example the discontinuous nature of the rockfoundations at King Herod’s palace in Masada, where the rock mass structure consists of

19


0

10

20

30

40

50

60

0.15

2.78

5.42

8.05

10.6

9M

ore

Joint Length (m)

Fre

quen

cy N = 100Mean = 2.7m

0

5

10

15

20

25

0.15 0.61 1.08 1.54

Bed Spacing (m)

Fre

quen

cy

N = 59Mean = 60cm

0

10

20

30

2.82 11.11 19.41 27.70 More

J2 Spacing (cm)

Fre

quen

cy

N = 80Mean = 14 cm

0

5

10

15

3.00 10.93 18.87 26.80 More

J3 Spacing (cm)

Fre

quen

cy N = 69Mean = 16.8 cm

Figure 9. Rock mass structure at the Northern Palace — Masada (after Hatzor et al.12): (Left) Jointlength and spacing distribution, (Right) Joint orientation (upper hemisphere projection of poles).

two orthogonal, sub vertical, joint sets striking roughly parallel and normal to the NE trend-ing axis of the mountain, and a set of well developed bedding planes gently dipping to thenorth (Figure 9). The joints are persistent, with mean length of 2.7 m. The bedding planes,designated here as J1, dip gently to the north with mean spacing of 60 cm. The two joint sets,J2 and J3, are closely spaced with mean spacing of 14 cm and 17 cm respectively.

An E-W cross section of the upper terrace is shown in Fig. 10A, computed using the sta-tistical joint trace generation code (DL) of Shi.1 It can be seen intuitively that while the Eastface of the rock terrace is prone to sliding of wedges, the West face is more likely to fail bytoppling of individual blocks. Block theory mode and removability analyses13 confirm theseintuitive expectations.

While it is convenient to use mean joint set attitude and spacing to generate statisticallya synthetic mesh, the resulting product (Fig. 10A) is not always realistic and in the casehere bears little resemblance to the actual slope. The contact between blocks obtained thisway is planar, thus interlocking between blocks is not modelled. Consequently the resultsof dynamic forward analysis may be overly conservative and the computed displacementsunnecessarily exaggerated.

To analyze the dynamic response of the slope realistically, a photo-geological trace map ofthe face was prepared using aerial photographs and the joint trace lines were digitized. Then,the block-cutting (DC code) algorithm of Shi1 was utilized in order to generate a trace mapthat represents more closely the reality in the field (Fig. 10B). Inspection of Fig. 10B revealsthat block interlocking within the slope is much higher and therefore results of forwardanalysis are expected to be less conservative and more realistic. The deterministic mesh shownin Fig. 10B is used therefore in the forward modelling discussed below.

The results of forward modelling is shown in Fig. 11 for an input motion adopted from thestrong M = 7.1 Nuweiba earthquake from 1995.14 The record, originally measured on a 50m thick layer of Pleistocene fill, was first deconvoluted to obtain proper rock response, andusing results from a topographic site effect study at Masada the rock response record wasconvoluted to take into consideration topographic site effect (for details see12). The resultinginput motion was up scaled in forward DDA simulations until damage was detected. InFig. 11A the response of the fractured slope to a modified earthquake record scaled up to

20


- E - - W -- E - - W -

- -- E - - W -

A B

Figure 10. Synthetic (A) and deterministic (B)joint trace maps for the upper rock terrace of Herod’sPalace (after Ref. 12).

A B

Figure 11. Dynamic forward modelling results at Masada:A) DDA prediction of fractured rock sloperesponse to modified Nuweiba record scaled to PGA= 0.2 g, B) Predicted performance of the reinforcedslope for a modified Nuweiba record up scaled to 0.6 g (after12) with a sparse rock bolting pattern.

PGA = 0.2 g is shown. As expected, with a PGA = 0.2 g at the site block sliding and blocktoppling are obtained in the East and West slopes, respectively. Application of a very sparserock bolting pattern (s = 4 m, L = 6 m) proves to be sufficient to reinforce the fracturedslope even when the record is up scaled to a PGA of 0.6 g (Fig. 11B).

3.2. Overhanging cliffs

Overhanging cliffs pose great risk in rock excavations as possible overturning may ensuewithout prior notice, depending upon the location and extent of the tension crack behindthe excavation surface. Consider the overhanging rock slope shown for example in Fig. 12A.The critical question here is the minimum distance between the toe and the vertical tensioncrack behind the face (B) required to ensure no overturning, so that the minimum requiredanchorage length may be constrained (Fig. 12B). This problem has recently been studied ingreat detail by Tsesarsky and Hatzor15 where an analytical expression for B is provided on

21


0m 13m0m

35m

(0,0) B(0,0)(0,0) B

0 4 8 12 16 20time (sec)

0

0.2

0.4

0.6

0.8

1

u (

m)

joint distance from toe5m

10m

15m

20m

25m (no joint)

A B C

Figure 12. Stability of overhanging slopes: (A) The geometry of the modeled slope, (B) DDA meshshowing a hypothetical tension crack at a distance B behind the toe, (C) The influence of tension crackdistance (B) on slope stability.

the basis of the geometrical properties of the slope and the stress distribution at the base.The dynamic evolution of the horizontal displacement component (u) of the upper tip of theslope as a function of B is plotted in Fig. 12C for the analyzed slope shown in Fig. 12A.Clearly, when the tension crack distance from the toe is equal to or less than 5 m the slopewill overturn, the analytical rationale is provided by Tsesarsky and Hatzor.15

Once the critical distance to the tension crack is determined, rock bolt reinforcement maybe designed such that the free anchor length must be greater than B. Using the bolt elementin DDA it is possible to track the tensile forces that develop in each bolt, as shown in Fig. 13.

1

2

3

4

5

6

7

8

9

1

2

3

4

5

6

7

8

9

1

2

3

4

5

6

7

8

9

0 4 8 12 16 20

time (sec)

0

50

100

150

200

250

bo

lt fo

rce

(kN

)

1

2

3

4

9

8

6

bolts are numbered sequentially from bottomto top (from toe, vertical spacing is 4m)

7

5φ φ φ φ = 2”

0 4 8 12 16 20time (sec)

0

50

100

150

200

250

bo

lt fo

rce

(kN

)

1

2

3

4

9

8

6

bolts are numbered sequentially from bottomto top (from toe, vertical spacing is 4m)

7

5φ φ φ φ = 2”

A B

Figure 13. Rock bolt reinforcement for overhanging slopes with DDA: (A) DDA mesh includinganchor location, (B) developed tension forces in each rock bolt according to elevation in the slopefor 2" diameter rods.

22


4. Stability of Shallow Karstic Caverns Below Open Pit Mines

The limiting relationship between underground cavern span and the minimum required depthof cover for limiting stability is explored here for jointed sedimentary rock masses character-ized by horizontal beds that are transected by vertical joint sets. Understanding this relation-ship is crucial for the design of mining excavations in karstic terrain.

The main challenge in performing rock engineering works in ground susceptible to sink-hole collapse is the lack of basic guidelines for prediction of shallow cavern stability forgiven span width and cover depth. For deep excavations, with rock cover much greater thanthe excavation span, preliminary assessment of the height of the loosened zone above theimmediate roof of the opening may be obtained on the basis of the empirical Terzaghi’srock load classification.16 Indeed, we found Terzaghi’s predictions valid when compared withfield observations17,18 and numerical analyses.19,20 The problems begin when the rock coverdepth is nearly as large as the excavation span, or even smaller. In such cases the karsticcaverns may be of precarious stability: they may hold for many years18 or collapse withoutpreliminary warning with the failure zone breaking through to the ground surface.21

Consider for example open pit mining operations in a ground prone to sinkhole collapse.Assuming the existing karst caverns are explored before mining operations begin, either byemploying geophysical methods or simply by drilling exploration boreholes, the minimumrequired rock cover (h) for a given cavern span (B) must be known in advance for safe min-ing or exploration operations. This relationship will also help determine three economicallysignificant mining parameters: (1) the maximum safe bench height (H), and in the case ofcavern exploration by drilling( 2) the minimum required distance between the explorationboreholes (d) and (3) the minimum required drilling depth (D) (see Fig. 14A).

To determine the critical relationship between opening span and minimum required rockcover for stability we model the deformation of theoretical caverns with a horse-shoe cross-sectional geometry and the following variable parameters (see Fig. 14B): cavern span (B),height (H = 1/2B), roof curvature (c = 1/2H) and rock cover h. The modelled domain isextended laterally to its boundaries by a distance of b = 2B. The infinite lateral continuityof the rock mass beyond the analyzed domain is modelled by fixed boundaries as shownin Fig. 14B. The ground surface is modelled as a horizontal plain. A total of 19 caverngeometries of varied spans and rock covers are modelled. The limiting relationship between

d

D

Bh

d

D

Bh

H

d

D

Bh

d

D

Bh

H

A B

Figure 14. (A) Schematic illustration of a shallow cavern below active open pit mining operation, (B)Geoletrical layout of modeled meshes.

23


Table 1. Simulated rock mass structure for DDA modelling of cavern stability.

Joint Dip/Direction Trace Mean Degree of Rock BridgeSet Length Spacing Randomness Length

1 0/0 ∞ 0.70 m 1.0 0 m2 88/182 5 m 0.96 m 0.5 2.5 m3 88/102 5 m 0.78 m 0.5 2.5 m

0

5

10

15

20

25

30

35

0 10 20 30 40 50

h, m

B, m

1(6) 2(6)

3(11)

4(6) 5(6)

6(10)

7(14) 8(15)

9(20)

10(25)

11(15)

12(22)

13(30)

14(12.5)

16(19)

15(25)

17(30)

19(22)

18(11)

h

B

Legend: Stable Marginal Unstable

Stable

Marginal

Unstable

0

5

10

15

20

25

30

35

0 10 20 30 40 50

h, m

B, m

1(6) 2(6)

3(11)

4(6) 5(6)

6(10)

7(14) 8(15)

9(20)

10(25)

11(15)

12(22)

13(30)

14(12.5)

16(19)

15(25)

17(30)

19(22)

18(11)

h

B

Legend: Stable Marginal Unstable

Stable

Marginal

Unstable

0

5

10

15

20

25

30

35

0 10 20 30 40 50

h, m

B, m

1(6) 2(6)

3(11)

4(6) 5(6)

6(10)

7(14) 8(15)

9(20)

10(25)

11(15)

12(22)

13(30)

14(12.5)

16(19)

15(25)

17(30)

19(22)

18(11)

h

B

Legend: Stable Marginal UnstableLegend: Stable Marginal Unstable

Stable

Marginal

Unstable

Figure 15. Limiting relationship between cavern span and required overburden height for stability forthe representative rock mass shown in Table 1, as obtained with 2D-DDA. The numbers are model #and in brackets the overburden height in each model.

cavern span and rock cover is thus obtained for a representative sedimentary rock mass withgeometrical parameters as shown in Table 1.

Composite results of 19 cases are presented in Fig. 15.

Acknowledgements

Partial funding for this research has been provided by the US – Israel Binational ScienceFoundation (BSF) through contracts 98-399 and 2004-122, Israel Nature and Parks Author-ity, Ministry of National Infrastructure, and Ministry of Housing and Construction. Dr. ShiGen-Hua is thanked for sharing his DDA codes. Dr. Michael Tsesarsky, Ronnie Kamai,Gony Yagoda-Biran, Dagan Bakun-Mazor and Dr. Ilia Wainshtein are thanked for theircollaboration.

24


References

1. Shi G., “Discontinuous Deformation Analysis — A New Numerical Method for the Statics andDynamics of Block System”, Berkeley, University of California, 1988.

2. Kamai R. and Hatzor Y. H., “Numerical analysis of block stone displacements in ancient masonrystructures: a new method to estimate historic ground motions”, Int J Numer Anal Met, 32, 2008,pp. 1321–1340.

3. Yagoda-Biran G. and Hatzor Y. H. “Constraining Paleo PGA Values by Numerical Analysis ofOverturned Columns”, Earthquake Eng Struct Dyn, In Press, 2009, pp. DOI: 10.1002/eqe.950.

4. Bakun-Mazor D., Hatzor Y. H. and Glaser S. D. “3D DDA vs. analytical solutions for dynamicsliding of a tetrahedral wedge ”, ICADD9 Nanyang Technological University, Singapore, 25–27November 2009, 2009.

5. Newmark N. “Effects of earthquakes on dams and embankments”, Geothecnique, 15, 2, 1965,pp 139–160.

6. Goodman R. E. and Seed H. B. “Earthquake-induced displacements in sand embankments”, J SoilMech Foundation Div, ASCE, 90, SM2, 1966, pp 125–146.

7. Makris N. and Roussos Y. S. “Rocking response of rigid blocks under near-source groundmotions”, Geotechnique, 50, 3, 2000, pp 243–262.

8. Wittke W. “Methods to analyze the stability of rock slopes with and without additional loading(in German)”, Rock Mech and Eng Geol, Supp. II., 1965, pp 52.

9. Kieffer S. D. “Rock Slumping — A Compound Failure Mode of Jointed Hard Rock Slopes.”,Berkeley., U. C. Berkeley., 1998.

10. Goodman R. E. and Kieffer D. S. “Behavior of rock in slopes”, J Geotech Geoenviron, 126, 8,2000, pp 675–684.

11. Hatzor Y. H. “Keyblock stability in seismically active rock slopes — Snake Path Cliff, Masada”, JGeotech Geoenviron, 129, 11, 2003, pp 1069–1069.

12. Hatzor Y. H., Arzi A. A., Zaslavsky Y. and Shapira A. “Dynamic stability analysis of jointed rockslopes using the DDA method: King Herod’s Palace, Masada, Israel”, Int J Rock Mech Min, 41,5, 2004, pp 813–832.

13. Goodman R. E. and Shi. G.-H. “Block Theory and its Application to Rock Engineering, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1985pp.

14. Hofstetter A., Thio H. K. and Shamir G. “Source mechanism of the 22/11/1995 Gulf of Aqabaearthquake and its aftershock sequence”, J Seismol, 7, 1, 2003, pp 99–114.

15. Tsesarsky M. and Hatzor Y. H. “Kinematics of Overhanging Slopes in Discontinuous Rock”, JGeotech Geoenviron, 135, 8, 2009, pp 1122–1129.

16. Terzaghi K. Load on tunnel supports. In: Proctor R.V., White T.L., editors. Rock Tunneling withSteel Supports. Ohio: Commercial Shearing Inc. ; 1946. p. 47–86.

17. Hatzor Y. H. and Benary R. “The stability of a laminated Voussoir beam: Back analysis of ahistoric roof collapse using DDA”, Int J Rock Mech Min, 35, 2, 1998, pp 165–181.

18. Hatzor Y. H., Tsesarsky M. and Eimermacher R. C. Structural stability of historic undergroundopenings in rocks: two case studies from Israel. In: Kouroulis S.K., editor. Fracture and Failure ofNatural Building Stones: Springer; 2006. p. 215–237.

19. Bakun-Mazor D., Hatzor Y. H. and Dershowitz W. S. “Modeling mechanical layering effects onstability of underground openings in jointed sedimentary rocks”, Int J Rock Mech Min, 46, 2,2009, pp 262–271.

20. Tsesarsky M. and Hatzor Y. H., “Tunnel roof deflection in blocky rock masses as a function ofjoint spacing and friction — A parametric study using discontinuous deformation analysis (DDA)”,Tunn Undergr Sp Tech, 21, 1, 2006, pp 29–45.

21. Murphy P., Westerman A. R. Clark R., Booth A., et al., “Enhancing understanding of breakdownand collapse in the Yorkshire Dales using ground penetrating radar on cave sediments”, 1st Gen-eral Meeting of the European-Geosciences-Union, Nice, FRANCE, 2004, pp 160–168.

25

Study on the Formation Mechanism of Tanjiashan LandslideTriggered by Wenchuan Earthquake Using DDA Simulation

WU AIQING1,∗, YANG QIGUI2, MA GUISHENG2, LU BO1 AND LI XIAOJUN3

1Yangtze River Scientific Research Institute, Wuhna, 430010, P.R. China2Changjiang Institute of Survey, Planning, Designing and Research, Wuhna, 430010, P.R. China3 Institute of Engineering Mechanics, China Earthquake Administration, Harbin 150080, P.R. China

1. Introduction

The 5.12 Wenchuan earthquake, which occurred at 14:28 on May 12, 2008 in SichuanProvince, was the biggest one for its effects in China since the foundation of the People’sRepublic of China. The surface wave magnitude was Ms 8.0, with the focal depth at 14 kmbelow the surface. The damage suffered area was about 100,000 km2 around the epicenter,and the earthquake caused nearly 90,000 fatalities.

Wenchuan earthquake triggered landslides, rock collapses and debris flow at more than15000 sites. Among them, the Tangjiashan landslide was the biggest and the most notableone. The landslide, with its volume 20.37 million m3and located at 3.2 km north of BeichuanCounty, slipped down at the right bank of Tongkou river. The original river was blockedby the barrage with its height of 82–124 m, and a dammed lake had been formed with itsmaximum reservoir capacity about 316 million m3. The collapse of the barrage, at anytime,would cause another disaster to its downstream, and more than 1.3 million people in thedownstream region were threatened by the hanging dammed lake. In order to remove thepotential menace of the dammed lake to the downstream at a possible short time, a greatdeal efforts, which consists of engineering and non-engineering measures, had been carriedout by Chinese government, army and engineers at an extremely difficult condition.1−3

It is in our knowledge that Discontinuous Deformation Analysis (DDA) possesses theadvantages of real time variable and large discontinuous deformation, and has the abilityto simulate the startup and the whole kinematical process of a landslide.4−8 In order toreproduce the kinematical process of the Tangjiashan landslide and to study the inducedstresses characteristics in the barrage body, DDA is employed here, and a lot of case-trialcalculations have been carried out with the final shape and location of the barrage as theobjective, and the representative acceleration records obtained by strong-motion observationin this earthquake as the basic conditions.

It is shown that the calculation results are quite acceptable and some new knowledgeabout the characteristics related to the high speed landslide triggered by the strong Wenchuanearthquake have been revealed by DDA.

2. Spatial Features and Geological Condition

2.1. Location and morphology of the landslide

Tanjiashan landslide was triggered by the earthquake on the right bank of Tongkou river,a branch of the Fujiang River. It was located at 3.2 km north of Beichuan county where

Analysis of Discontinuous Deformation: New Developments and Applications.Edited by Guowei MA and Yingxin ZHOU. Published by Research Publishing Services.Copyright c© 2009 by Society for Rock Mechanics & Engineering Geology (Singapore).ISBN: 978-981-08-4455-4doi:10.3850/9789810844554-keynote-Wu-AiQing 27


N

Legend

XIXIXVIIIVIVI

LangZhou

Xian

Chong Qing

WenChuan

QingChuan

TangjiashanBei Chuan

MianZhu

ChengDu

IXVII

VIVIII

XI

XIX

Figure 1. Seismic intensity map of Wenchuan earthquake.

Figure 2. Original scenery of Tangjiashan site (from upstream view).

the seismic intensity was in the grade of ten to eleven. Figure 1 shows the seismic intensitydistribution map of the Wenchuan earthquake.

∗Corresponding author. E-mail: [email protected]

Figure 3. Spatial morphology of landslide (from downstream view).

28


origi

nal s

urfa

ce

791.4mCataclastic rockstrongly weathered

cataclastic rock weakly weathered

river level 1664.7mFoundation rock of Qingping groupin the lower Cambrian period

gravel and soil mixture

Foundation rock of Qingping groupin the lower Cambrian

muddy siltstone by deposit

elevation(m)

elevation(m)

N60 E/NW 60

Figure 4. Geological section of the landslide.

Before the landslide occurred, the top elevation of the hill at the right bank of the riverwas about 1580 m, and the slope height was 900 m with the dip angle of 40◦ in its upperpart and 30◦ in the lower part. During the earthquake period, the landslide slipped down,creating a barrage. The original river was blocked, and a dammed lake was formed. Figure 2and 3 show separately the Tangjiashan scenery before and after the landslide.

The main features of the barrage for its morphology can be seen in Fig. 3 and 4 while thelater is a cross section map of the barrage which will be attached in the following part. Thebarrage shows a rectangle shape in the horizontal plane with the width of 611 m across theriver, and the length of 803 m along the river. There is an undulate topography on the barragesurface where the surface elevation at the left part, the front of the barrage body, is higherthan that at the right. The maximum elevation of the left is 793.9 m while the maximumelevation at the right is 775 m. The height of the barrage is 82∼124 m, and its total volumeis 320.37 million m3.

2.2. Geological condition

Figure 4 shows the cross section of the landslide where the original slope surface is estimated.Except of the weathered slope debris in the surface layer, the foundation rock appearedin the slope consists of silty sandstone, siliceous slate, marlite and mudstone in the lowerCambrian period. The hard and soft rock layers, with their orientation of N60◦E/NW� 60◦,exist alternatively, and their thickness is from thin to medium. The landslide occurred in theweakly weathered rock mass, and had a bedding slip in the main sliding direction of N10◦W.

The barrage body formed by the landslide consists mainly of cataclastic rock with stronglyand weakly weathered. A small part of gravel and soil mixture, with layer thickness of 2∼4 m,appeared locally in the rear part of the landslide. Figure 5 shows structure features of thecataclastic rock. It could be seen that features of the original rock structure are still remained.

3. DDA Simulation Model

3.1. Block system

According to the geological condition of the landslide, the block system, formed by differentsliding surfaces and joints, has been obtained as in Fig. 6. In order to reflect adequately theearthquake effects to the deformation and failure features of the landslide, the geometrical

29


Figure 5. Cataclastic rock with its structure remained.

438#

893#

443#

Figure 6. Block system in DDA model.

dimensions of blocks in the block system are normally characterized with uniformity. Here,the length of each block is in the range of 5∼8 m.

In Fig. 6, the maximum thickness above the lowest sliding surface is about 80 m, and thetotal height of the landslide body is 634 m.

Some particular blocks in sliding body at different positions, i.e. upper part, middle part,and lower part of the landslide, are chosen as monitoring points. If all monitoring points haveobvious displacement and velocity at particular time, then the landslide may occur. In theprocess of movement, the DDA kinematical output versus time may reflect the correspondingkinematical process of the landslide. In this paper, three blocks, 438#, 893#, and 443# inFig. 6, have been chosen as monitoring points.

3.2. Mechanical parameters

In the simulation model, the landslide body consists of four types of rock material, whichinclude residual deposits in the surface layer, strongly weathered rock mass, weakly weath-ered rock mass and foundation rock which is located below the lowest sliding surface. Thethickness of the residual deposit and the strongly weathered rock mass is about 15∼20 m. Inaddition, at the bottom of the slope, a layer of silt sand, which has been formed by down-streem reservoir sedimentation, and in thickness of 20 m, is considered.

In order to conduct flexibly different simulations involved in case-trial calculation, differ-ent material regions and block boundaries have been pre-determined individually prior to

30


Table 1. Parameters for case-trial calculation.

itemstrength deformability

φ/(◦) C/kPa E/GPa μ

Residual deposit 29∼40 20∼70 0.5 0.3Strongly weathered 31∼41 30∼80 1.0 0.28Weakly weathered 33∼42 40∼100 3.0 0.25Foundation rock / / 5.0 0.25Sliding surface 15∼42 0∼80 / /Vertical boundaries 33∼42 40∼100 / /

property assignment of the block system. For deformation properties, there are 5 regions intotal, and for shear strength in boundaries, 22 types in total. According to the material con-stitutes and the geological conditions of the landslide, the possible ranges of parameters usedin calculation are listed in Table 1, where parameters in upper limit means that the slope isin the state of equilibrium in the condition of self weight.

3.3. Seismic acceleration records

During the Wenchuan earthquake of May 12,2008, the NSMONS in China obtained recordsfrom 460 stations in 17 provinces, municipalities and autonomous regions and three arraysin Sichuan and Yunnan provinces. Among the acceleration records from the main shock,the largest PGA was recorded at Wolong station in Wenchuan Country, and the recordsin the EW, Ns, and UD directions are 957.7 Gal, 652 Gal and 948.1 Gal, respectively. ForTangjiashan landslide DDA simulation, acceleration records obtained from Qingping stationin Mianzhu county, Sichuan province are accepted as the representative input seismic curve,in which the PGA at directions mentioned above are 824.1 Gal, 802.7 Gal, and 622.9 Gal,respectively. The seismic curve in EW direction is shown in Fig. 7.9−10

Figure 7. Acceleration records used in DDA model.

31


Table 2. Actual parameters of the landslide.

ItemStrength Deformability

φ/(◦) C/kPa E/GPa μ

Residual deposit 29 20 0.5 0.3Strongly weathered 31 30 1.0 0.28Weakly weathered 33 40 3.0 0.25

Foundation / / 5.0 0.25Sliding surface 16.5 0 / /

Vertical boundaries 30 20 / /

4. Landslide Kinematical Process Simulation

4.1. Parameters consistent to the landslide

Based on the front location of the landslide and the morphology of the barrage formed bythe landslide, the objective determining actual parameters of the block system can be estab-lished. By case-trial calculation, parameters consistent to the objective are determined inTable 2. The mechanical parameters listed in the table can be understood to be comprehen-sive, which means that the parameters represent the whole slipping process of the landslide,and the possible changes of parameters, in the whole process of its slipping, are not con-sidered. In DDA calculations, loads acting on the landslide body, within the process of itsslipping, consist of weight and earthquake loads transformed by the earthquake accelera-tion records in Fig. 7. The time step used in DDA model is 0.001 s. It is shown that the shearstrength on the sliding surface, compared to the static status, is decreased in much degree. Theratio of friction coefficient on sliding surface in kinematical and static conditions is no morethan 0.35.

4.2. Kinematical process simulation

(1) Deformation features

The velocity and slipping distance versus time of some blocks, here with blocks of443#, 893#, and 438# as the representative, are chosen to reflect kinematical features ofthe landslide.

Figure 8. Velocity versus time. Figure 9. Displacement versus time.

32


g p

(a) At ti me 10s

(b) At ti me 15s

(c) At ti me 40s

Figure 10. Deformation and failure pattern.

Figures 8 and 9 show respectively curves with velocity and slipping distance versus timeof above blocks, which are carried out by DDA simulation. It has been revealed that theTangjiasha landslide was a high speed landslide, and in the whole kinematical process, non-linear features, from beginning to the end, were presented especially in aspects of slippingvelocity, etc.

The whole slipping time was about 35 s while nearly all of their slipping displacementswere carried out in the beginning 25 s. The average slipping velocity in the beginning 25 swas about 15 m/s to 17 m/s.

Figure 10 shows the deformation and failure patterns of block system at some particulartime steps.

(2) Stresses Distribution

Besides of the deformation features, the stresses distribution in block system, induced by thehigh speed landslide, has also been obtained by DDA.

It is shown that the stresses in the landslide, after the slipping process finished, present adiscontinuous feature. In some areas, i.e. the front, the rear, and the surface of the landslide,

33


material there presents a broken state, the stresses are normally small. In the lower part of thelandslide, especially in the area near to the bottom of the river bed, the crash of rock blocksinduced a much higher stresses with the maximum value of 6–7 MPa, and the principal stressorientation parallel to the bottom boundary in majority.

4.3. Effects of two parameters to landslide behavior

(1) Shear strength of the sliding surface

When a landslide is triggered, much of its potential energy will be absorbed by friction move-ment along the sliding surface and among block boundaries. The friction angle on the slid-ing surface is much more sensitive to the landslide behavior. In the case when there is ahigher friction angle, potential energy in the landslide will be dissipated soon, and a muchless kinematical behavior will be transformed. In other case, lower friction will cause muchmore kinematical behavior. Figure 11 shows quantitatively the effects of friction angle to thevelocity of the landslide, where the block 893# is selected, and three cases of friction angle,φ = 16.5◦, 20◦, and 23◦ are analyzed. The reasonable result is achieved by DDA.

Velo

city

/m/s

35

30

25

20

15

10

105 35 40 45302520150

Time/s

893# 16.5893# 20893# 23

Figure 11. Effects of friction angle.

(2) Earthquake loads

It is shown that the decrease of the shear strength of the sliding surface, from stable stateto unstable state in the condition of earthquake loads, was the main reason to have theTangjiashan landslide triggered as a high speed landslide. In other hand, field investigationsshows that there are still a lot of high rock slopes and potential landslides which have notbeen triggered by this strong earthquake, but failures in terms of loose rock in surface layer,local rock collapse, and rock cracks, etc., can be found generally in field.

In order to reveal the deformation mechanism of the totally stable slopes on the conditionof earthquake loads, the DDA simulation for the Tangjiashan landslide, with the condition ofno parameter reducing, has been conducted while parameters of the upper limits in Table 1are used, and the earthquake load represented by the acceleration records in Fig. 7 is con-sidered in calculation. Figure 12 shows the velocity and displacement versus time of block893#, which is located in the middle bottom of the landslide.

Corresponding to the earthquake processing, it is notable that there is a continuous defor-mation tendency and a fluctuated velocity in rock mass. If reducing of parameters, especiallyfor shear strength on the sliding surface, is not considered, the landslide will not occur. Infact, continuous deformation of rock mass is accompanied usually with decreasing of itsstrength. If the strength is not enough to hold the rock mass stable, the landslide will occur

34


0 5 10 20 25 30 35150 0

8

12

16

1

2

3

4

4

Time/sVe

loci

ty/m

/s

VelocityDisplacement

Dis

plac

emen

t/m

Figure 12. Velocity and displacement versus time under strong earthquake load.

along the weakest path in the slope. DDA simulation considering the decreasing process ofrock strength versus its deformation is still under consideration in further.

5. Conclusions

Wenchuan earthquake, Ms 8.0 in magnitude and occurred on May 12, 2008 in SichuanProvince, China, triggered landslides, rock collapses and debris flow at more than15000 sites, and the Tangjiashan landslide, with its total volume 20.37 million m3, wasthe biggest and the most notable one for its effects in China.

Based on the field geological investigation and the typical acceleration records of the mainshock obtained in the period of the earthquake, numerical simulation of the whole slidingprocess of Tangjiashan landslide has been carried out by use of DDA method. It is shownthat the Tangjiasha landslide was a high speed landslide, behaved with non-linear features inthe whole sliding process. According to DDA simulation, the whole slipping time was about35 s while nearly all of their slipping displacements were carried out in the beginning 25 s,with the maximum sliding velocity about 30 m/s, and the average 15 m/s to 17 m/s in thebeginning 25 s.

The stresses in the landslide behave a discontinuous feature. In areas of the front, the rear,and the surface layer, the stresses are normally small, and in the lower part of the landslide,the crash of rock blocks induced a much higher stresses with the maximum value of 6–7 MPa.

The dynamic earthquake load caused an incessant deformation of the landslide, resultingin reducing of mechanical parameters, especially for the shear strength on the sliding surface.The friction angle on the sliding surface is much more sensitive to the landslide behavior.In the case of Tangjiashan landslide, the ratio of friction coefficient on sliding surface inkinematical and static conditions is no more than 0.35.

In further study, DDA simulation considering the decreasing process of rock strengthversus its deformation will be under consideration.

Acknowledgements

This research work is supported by the National Science Foundation of China with contractNo. 50639090, and the Project of Xdc2007-10 founded by Ministry of Water and Resourcesof People’s Republic of China. The authors thank to Dr. Gen-hua Shi for his valuable advicein the field of DDA engineering applications.

35


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4. Shi G.H. Discontinuous deformation analysis: a new numerical model for the statics and dynamicsof block systems, Ph.D. Dissertation, Department of Civil Engineering, University of California,Berkeley, 1988.

5. Maclaughlin, M.M., Sitar, N., Doolin, D.M., Abbot, T.S. Investigation of slope stability kinematicsusing discontinuous deformation analysis. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 2001,38:753–762.

6. Maclaughlin, M.M., Doolin, D.M., Berger E.A. A decade of DDA validation. In Ming Lu (ed.),Development and application of discontinuous modelling for rock engineering, Proceedings of the6th International Conference on Analysis of Discontinuous Deformation, Balkema, 2003, 13–3117.

7. Hatzor,Y.H. & Feintuch, A.The validity of dynamic block displacement prediction using DDA.Int.J. Rock Mech. Min. Sci. & Geomech. Abstr. 2001, 38: 599–606.

8. Aiqing Wu, Xiuli Ding, Huizhong Li, Gen-hua Shi. Numerical simulation of startup and thewhole process characteristics of Qianjiangping Landslied with DDA method [C]. Proce. of SeventhInternational Conference on the Analysis of Discontinuous Deformation(ICADD-7). December10–12, 2005, Honolulu, Hawaii, 167–174.

9. Li Xiaojun, Zhou Zhenghua, Yu Haiyin, et al. Strong motion observations and recordings fromthe great Wenchuan Eearthquake [J]. Earthquake Engineering and Engineering Vibration, 2008,7(3):235–246.

10. Zhou Zhaohui. The strong ground motion redording of the Ms8.0 Wenchuan Earthquake inSichuan Province [J]. Earthquake Research in Sichuan, 2008, No.4 (129 in total): 25–28 (inChinese).

36

A G Space Theory with Discontinuous Functions for WeakenedWeak (W2) Formulation of Numerical Methods

G.R. LIU1,2

1Centre for Advanced Computations in Engineering Science, Department of Mechanical Engineering,National University of Singapore, 9 Engineering Drive 1, Singapore 117576, http://www.nus.edu.sg/ACES/,2Singapore-MIT Alliance (SMA), E4-04-10, 4 Engineering Drive 3, Singapore, 117576

This paper introduces first a G space theory using the generalized gradient smoothingtechnique for a unified formulation of a wide class of meshfree methods of special propertiesincluding the upper bound properties. The G space is first defined to include discontinuousfunctions allowing the use of much more types of methods/techniques to create shape func-tions for numerical methods. It is also shown that W2 formulation can be used to constructmany meshfree methods, and methods based on the finite element settings can also be for-mulated in the similar manner. Properties and a set of important inequalities for G spaces arethen proven in theory and analyzed in detail. We prove that the numerical methods developedbased on the W2 formulation will be spatially stable, and convergent to exact solutions. Wethen present examples of some of the possible W2 models, and show the major propertiesof these models: (1) it is variationally consistent in a conventional sense, if the solution issought from a proper H space (compatible cases); (2) it passes the standard patch test whenthe solution is sought in a G space with discontinuous functions (incompatible cases); (3) thestiffness of the discretized model is reduced compared to the FEM model and even the exactmodel, allowing us to obtain upper bound solutions with respect to both the FEM and theexact solutions; (4) the W2 models are less sensitive to the quality of the mesh, and triangularmeshes can be used without any accuracy problems. These properties and theories have beenconfirmed numerically via examples solved using a number of W2 models including compat-ible and incompatible cases. A number of W2 models, such as NS-PIM, NS-FEM, ES-PIM,ES-FEM, CS-PIM and CS-FEM, are then presented. The NS- models are used for real-timecomputation based on the reduced basis approximation. The real-time computation modelis then used to inversely identify the interface property of a dental implant system.

Keywords: Numerical methods, meshfree methods, FEM, real-time computation, solutionbound, inverse analysis.

References

1. G.R. Liu, and Quek, S.S., Finite Element Method: a practical course, BH, Burlington, MA, 2003.2. J.S. Chen, C.T. Wu, S. Yoon, Y. You, A stabilized conforming nodal integration for Galerkin mesh-

free methods. Int. J. Numer. Mech. Engrg., 50: 435-466, 2001.3. Dai, K. Y., Liu, G. R. and Nguyen TT. (2007). An n-sided polygonal smoothed finite element

method (nSFEM) for solid mechanics. Finite elements in analysis and design; 43: 847-860.4. Dai, K. Y. and Liu, G. R. (2007). Free and forced analysis using the smoothed finite element

method (SFEM). Journal of Sound and Vibration; 301: 803-820.5. Liu, G. R., Dai, K. Y. and Nguyen, T. T. (2007). A smoothed finite element method for mechanics

problems. Computational Mechanics; 39: 859-877.6. G.R. Liu, T. Nguyen-Thoi, H. Nguyen-Xuan, K.Y. Lam. A node-based smoothed finite element

method (NS-FEM) for upper bound solution to solid mechanics problems. Computers and Struc-tures, 87: 14-26, 2009.


Analysis of Discontinuous Deformation: New Developments and Applications.Edited by Guowei MA and Yingxin ZHOU. Published by Research Publishing Services.Copyright c© 2009 by Society for Rock Mechanics & Engineering Geology (Singapore).ISBN: 978-981-08-4455-4doi:10.3850/9789810844554-keynote-Guirong-Liu 37


7. G.R. Liu, X. Xu, G.Y. Zhang, Y. T. Gu, An extended Galerkin weak form and a point interpolationmethod with continuous strain field and superconvergence using triangular mesh. ComputationalMechanics, 43: 651-673, 2009.

8. G.R. Liu, X. Xu, G.Y. Zhang, T. Nguyen-Thoi, A superconvergent point interpolation method (SC-PIM) with piecewise linear strain field using triangular mesh. International Journal for NumericalMethods in Engineering, 77: 1439-1467, 2009.

9. Liu, G.R., Li, Y., Dai, K.Y., Luan, M.T. and Xue, W. (2006). A Linearly conforming radial pointinterpolation method for solid mechanics problems, International Journal of Computational Meth-ods, 3: 401-428.

10. Zhang, G. Y., Liu, G. R. and Li, Y (2008). An efficient adaptive analysis procedure for certifiedsolutions with exact bounds of strain energy for elasticity problems. Finite Elements in Analysisand Design, 44: 831-841.

11. Zhang, G. Y., Liu, G. R., Nguyen, T. T., Song, C. X., Han, X., Zhong, Z. H. and Li, G. Y. (2007a).The upper bound property for solid mechanics of the linearly conforming radial point interpolationmethod (LC-RPIM). International Journal of Computational Methods, 4(3): 521-541.

12. G. R. Liu, T. T. Nguyen, K. Y. Dai and K. Y. Lam, Theoretical aspects of the smoothed finiteelement method (SFEM), Int. J. Numer. Mech. Engrg., 71: 902-930, 2007.

13. G.R. Liu, G.Y. Zhang, etc., A linearly conforming point interpolation method (LC-PIM) for 2Dmechanics problems, International Journal for Computational methods, 2, 645-665, 2005.

14. G.R. Liu, G.Y. Zhang, Upper bound solution to elasticity problems: a unique property of thelinearly conforming point interpolation method (LC-PIM), Int. J. Numer. Mech. Engrg. 74, 1128-1161, 2008.

15. G.R. Liu, A generalized gradient smoothing technique and the smoothed bilinear form for galerkinformulation of a wide class of computational methods, International Journal of ComputationalMethods, Vol. 5, No. 2 (2008) 199–236.

16. Liu, G. R. (2008). A G space theory and weakened weak (W2) form for a unified formulation ofcompatible and incompatible methods, Part I: Theory and Part II: Applications to solid mechanicsproblems International Journal for Numerical Methods in Engineering, published online, DOI:10.1002/nme.2719, DOI: 10.1002/nme.2720, 2009.

17. Liu, G. R. and Zhang, G. Y. (2008) Edge-based smoothed point interpolation methods. Interna-tional Journal of Computational Methods, 5(4): 621-646.

18. Liu, G. R. and Zhang, G. Y. (2009) A normed g space and weakened weak (W2) formulationof a cell-based smoothed point interpolation method. International Journal of ComputationalMethods, 6(1): 147-179.

19. Liu, G. R., Nguyen, T. T. and Lam, K. Y. (2008). An edge-based smoothed finite element method(ES-FEM) for static and dynamic problems of solid mechanics. Journal of Sound and Vibration,320: 1100-1130, 2009.

20. Liu GR (2009) On the G space theory. International Journal of Computational Methods; 6(2):257-289.

21. Z. Khin, G.R. Liu, B. Deng, K.B.C. Tan, Rapid identification of elastic modulus of the interfacetissue on dental implants surfaces using reduced-basis method and a neural network, Journal ofBiomechanics, 42: 634-641, 2009.

22. G.R. Liu, G.R. Zhang. A novel scheme of strain-constructed point interpolation method for staticand dynamic mechanics problems. International Journal of Applied Mechanics, 1(1): 233-258,2009.

38

Concerning the Influenced of Velocity Ratio and TopographyModel on the Result of Rockfall Simulation

T. SHIMAUCHI1,∗, K. NAKAMURA2, S. NISHIYAMA3 AND Y. OHNISHI3

1Meiji-Consultants Co .Ltd Saitama 333-0801, Japan2A Depertment of Civil Engineering, Factory of Engineering, Tottori University, Tottori, Japan3Depertment of Urban and Environmental Engineering, Kyoto University, Kyoto, Japan

1. Introduction

Rockfall is one of the major hazards in rock slope cut for highways and railways in the moun-tainous area of Japan. In order to prevent the rockfall disasters, it is necessary to evaluatethe influences on the preservation object in advance. The rockfall simulation is thus appliedfor the evaluation. However the behaviours of rockfall are complicated and uncertain phe-nomena, exhibiting a great deviation due to the influences of the scale and shape of fallingrocks, the geographic feature of falling route and the roughness of the slope surface. There-fore input parameters and the deviation of the topographic model are given to take accountof these uncertain factors. However at the present stage, the relationship between the givendeviation and the rockfall behaviours is still not clear.

On the other hand, we have investigated a method to predict the velocity ratio(Rv) dur-ing the rockfall restitution time by using the DDAball.1,2 However the rockfall behavioursas a whole, particularly the reproducibility of the trajectory, the reproducibility of rockfallvelocity and the improvement of prediction accuracy have still not been realized.

In this paper, based upon the previous results, the essential factors for the reproducibilityof velocity and trajectory were investigated through the reproductive analysis of the previousfield experimental results.

2. Summary of Previous Field Experiments for Investigation

The field used in this investigation is located in Takamatsu of Japan where the former Min-istry of Construction has performed the field experiment in 1980.3 (it is designated as theTakamatsu Experiment in the following parts of this paper).

In the Takamatsu Experiment as shown in Fig. 1, one rock was allowed to drop from aheight of 9 m , i. e., from two points with the inclination of 60◦ (A) and that of 30◦(B). Thetrajectory and velocity of the falling rock on the slope were measured. The slope surface ofthe falling point was that of the shotcrete and the slope surfaces from the middle to lowerpart were the rock plate. The diameters of the falling rock were from 0.09 to 1.15 m andtheir average weight was 170 kg of the granite rock.

Figure 1 is for the B point, i.e., the trajectory recorded figure of the falling rock on theslope surface with the inclination of 30◦. The velocity shows the right side with this figureFurthermore, in Table 1, the velocity ratio of the normal direction measured in the A and Bpoints, and the conversed velocity ratio based on this value were shown in Table 1. In thispaper, the velocity and trajectory concerning the measured example in the point of B wereinvestigated.


Analysis of Discontinuous Deformation: New Developments and Applications.Edited by Guowei MA and Yingxin ZHOU. Published by Research Publishing Services.Copyright c© 2009 by Society for Rock Mechanics & Engineering Geology (Singapore).ISBN: 978-981-08-4455-4doi:10.3850/9789810844554-keynote-Ohnishi 39


Figure 1. The Trajectory section and velocity of Takamatsu Experiment.

Table 1. The measurement result in Takamatsu Experiment.

Specimen Incident Angle (◦) Rn Vout Rv(conversion)

A 60 0.49 7.8 ∼ 9.4 0.59 ∼ 0.73B 30 0.26 3.6 ∼ 5.2 0.27 ∼ 0.29

Furthermore, the DDAball was used in this analysis. Oddball is the DDA taking part in thethree dimensional rigid body elements, and is the program code characterizing the analysisof falling rock and rock block.4

3. Reproductive Simulation of Velocity

In the rockfall analysis, the velocity ratio (Rv) and the velocity ratio of the normal direction(Rn) are used as input parameters (Fig. 2 and Eqs. (1), (2)). However, these parameters havethe property which decreases when the incident velocity of the normal direction increases.

Rv = Vout

Vin(1)

Figure 2. Properties of bouncing.

40


Figure 3. Scale Factor Curve was proposed considering to the velocity-dependency.

Rn = Vout cosβVin cos α

(2)

Pfeiffer et al. (1989) proposed the Scale Factor Curve as shown in Fig. 3 and Eq. (3), withthe purpose to consider the dependency of the incident velocity of the normal direction onthe velocity ratio of the normal direction.5 This curve may express the state that the velocityratio of the normal direction will decrease greatly when the incident velocity of the normaldirection increases. Its characteristic is its high adaptability in comparison with the curveof exponential function or other regression curves. This value is actually 0.0 ∼ 1.0. K ofSF in Eq. (3) is designated as the revised velocity, expressing the gradient of the curve. Thereflective velocity may be obtained from Eq. (4) by using the SF curve.

SF = 1

1+(

Vn inK

) (3)

Vn out = Rn(scaled)× SF×Vn in (4)

Vout = RV(scaled)× SF×Vin (5)

On the other hand, we have clarified that the velocity ratio (Rv) has also the dependentproperty on the incident velocity of the normal direction, based on the laboratory experi-mental results of dropping a quartz ball on the reflective plate of rock and wood. We alsoshowed that the Scale Factor Curve could be applied to Rv. In this case, the reflective velocitymay be determined by Eq. (5).

Equations (4) and (5) show the reflective velocity of the normal direction and that theScale Factor can be applied to either prediction method of the reflective velocity. Howeverduring the application, the velocity ratio of the normal direction and each property andcharacteristic of the velocity ratio should be taken into account.

Concerning the velocity ratio and the incident velocity of the normal direction, only themain points are mentioned in this paper.2

• The slower the incident velocity along the normal direction, the closer the velocity ratioapproaches 1.0. The quicker it became, the velocity ratio reduced.• The changing of velocity ratio is mainly due to the incident angle. The smaller the incident

angle becomes, the closer Rv approaches 1.0. Rv will fall as the incident angle becomeslarger.• When the incident angle is the same, the quicker the incident velocity, the lower the

velocity ratio will fall. Its influence is smaller than that of the incident angle.

41


• When the reflective plate is the rock plate (granite), the gradient of the SF curve is rathersmall, but for the soft wooden reflective plate, it becomes large (for the rock reflectiveplate, K = 10 ∼ 12, for the wooden reflective plate, K = 7 ∼ 10).

In the field, because the influences of the shape and state of rockfall as well as and thedestruction of the ground, there exists a great deviation in the measured data. It is not simplelike the laboratory experiment. However based on the above mentioned fundamental prop-erty, it can be considered that the velocity ratio is changing. Taking account of this propertyof the velocity ratio, we first test the reproducibility analysis of the previous TakamatsuExperiment.

Figure 4 is the plotting of the Rv (conversion) value of the Takamatsu Experiment as shownin Table1. This figure shows the SF curve connecting its minima In the coordinate, Rv (scaled)is designated in order to distinguish the measured velocity ratio and the velocity ratio of theScale Factor Curve. The Rv (conversion) value shown here expressed the limits of the max-imum and minimum caused by the reading errors of the incident angle against the incidentvelocity of the normal direction.

Figure 4. Takamatsu Experiment result and the SF curve.

Therefore, it can be considered that the SF curve itself may express the dependency ofthe incident velocity of the normal direction on the velocity ratio, taking account of thecharacteristic of the slope surface. Furthermore, it may also be considered that the two curvesmay express the range of roughness deviations due to the incident and reflective angles. Basedupon this consideration, the iterative calculations have been performed in this investigation,taking Rv(scaled) = 0.95 and K = 9.0 as the mean value, giving the range of Rv(scaled) =±0.05,, K = ±1.0 as the standard deviation. The iteration number was 100.

The results were shown in Fig. 5. In this figure, the trajectory obtained by the simulationwere shown. This figure showed the envelope of the trajectory of Takamatsu Experiment(Fig. 1). At the first falling place immediately after the falling, as the falling rock reached theflat part in the lower side of the slope surface, the reason why the rock would not springhighly up was the effect considering the dependency of incident velocity of normal direction.Thus it may be considered that the velocity has been reproduced. However the trajectoryhave been focused on the narrow range of the restitution area, monotonously rising and theenvelope of the Takamatsu Experiment greatly moved down. The main reason is consideredto be the modelling of the even inclination region with a long straight line.

42


Figure 5. Analysis results considering the dependency of incident velocity of normal direction onvelocity ratio.

4. Reproducibility Simulation of Trajectory

From the reproducibility analysis results of velocity, it is difficult to reproduce the trajectoryby only considering the dependency of the incident velocity of the normal direction on thevelocity ratio. In order to solve such problems of the trajectory, several methods have beenproposed: one is to give deviations to the incident angle; the other is to give deviations to thetopographic model itself. In this paper the second method was investigated.

The investigated model was the three dimensional topographic model as shown in theright side of Fig. 5. It is the model which had been pushed out 30 m in the y direction anddivided the ground surface with one side being approximately 1.5 m of the triangle mesh.In this investigation, the normal distribution with mean value and standard deviation σ wasgiven to each vertex of each triangle mesh with the height z (m). The topographic modelwith randomness was established. The number of these established model was 100. By usingthese models, the analysis had been performed by applying the scale factor curve as shownin Fig. 4.

In the Takamatsu Experiment, Komura or Ushiro et al. reported that the standard devia-tion of the incident angle on the concrete slope surface was σ = ±11.22◦ by assuming thatall the causes of deviation in the velocity ratio of the normal direction Rn was the readingerrors of the inclination angle.6,7 Based on this report, the average side length of the triangleelements composing the investigated model was determined as 1.8 m, and the variation inthe upper and lower directions of the vertex σ = ±0.18 (m) (Fig. 6).

Figure 6. Input of height-deviation to triangle elements.

43


Figure 7. Input of height-deviation to triangle elements.

The analysis results were shown in Fig. 7. In this figure, we can see that there appeared agreat variation in the trajectory and velocity, when the roughness of the topographic modelwas taken into consideration, in comparison with the case where only the velocity ratiodeviations were taken into account. These effects were quite evident. Not mention the fallingpoint, there appeared a new restitution section on the way of the long linear slope surface inthe 2 ∼ 11 m height or 13 ∼ 20 m height. Thus the appearance of the cross-section of thetrajectory and the velocity distribution were close to the experimental results.

On the other hand, it can be seen that the trajectory extended from the falling point whenthe roughness of the topographic model was taken into account in comparison with the threedimensional diagrams of Figs. 5 and 7. The relationship between the extension of the fallingtrajectory and the deviation given to the vertex height of the mesh were shown in Fig. 8, Thedeviation of the vertex height σ was increased from ±0.04 m to ±0.09 m, ±0.18 m, ±0.24 m,that is, the larger the roughness of the slope surface becomes, the farther extended the rockfalltrajectory. Komura et al. reported that the extension from the falling point was about 30◦in the Takamatsu Experiment.6 In addition, Komura et al., determined the roughness basedupon the measured results of the deviation in the incident angle, the extension angle fromthe falling point was also 30◦. It was fairly in agreement with the Takamatsu Experimental

Figure 8. Relationship between deviation of vertex height considering triangle element and extendingangle of rockfall.

44


results. From these results, we may consider that the method of this investigation, whichgives deviations to the small roughness of the topographic model and the method whichgives deviation to the incident angle and the mass point analysis performed by Komura et al.and Ushiro et al. have approximately the equivalent effects of improvement on the trajectory.

5. Conclusions

Concerning the rockfall experimental results performed by the former Mimistry of Con-struction in Takamatsu of Japan (1989), a reproducing experiment was carried out by usingDDAball.

The reproducibility experiment was performed by two steps. At the first, the deviationeffects of the dependency on the incident velocity of the normal direction was investigated byusing the model expressing the even inclination region with a long straight line. The resultswere shown as follows:

(1) There is a dependency of the incident velocity of the normal direction on the velocityratio. It is important to consider this property for the reproducibility of velocity.

(2) In order to consider the deviation of the trajectory, it is important to take account of thesmall roughness in the topographic model, the more roughness in the model, the greaterthe scattering of the trajectory will become.

Naturally, in the practical rockfall simulation, it is necessary to consider both (1) and (2).However, in order to elevate the reliability of the rockfall simulation, it is necessary to clarifyhereafter the determination method of the factor corresponding to the field conditions suchas the bare rock or the soil and sand, the measuring method of roughness and the methodreflecting the analysis mesh etc.

Moreover, although it does not mention in this paper, the occurring place of rockfall, theinfluence of the initial velocity on the results are unexpectedly very important. In the future,it is necessary to investigate the shape of rockfall, the influence of trees and the adequacy ofthe probability distribution.

Acknowledgements

The authors would like to thank Dr. Shigeru Miki, Kiso-Giban Consultants Co. Ltd. formany suggestions.

References

1. Shimauchi, T., Wei, Z., Nakamura, N., Sakai. N., Nishiyama, S and Ohnishi, Y. 2006. Funda-mental Study of Velocity Energy Ratio by Using DDA-Rockfall Simulation. Proc. of the JRMS,No.36.Japan, pp. 473–481.

2. Shimauchi, T., Ohnishi, Y., Nishiyama, S and Sakai, N, 2006, Study on Verification of Rockfall Sim-ulation Using Improved DDA considering the Characteristics of Velocity Energy Ratio at Impact,Journals of Civil Engineers C ,Vol. 62, No.3, pp. 707–721.

3. Japan Road Association. Reference Works for Rockfall Countermeasure Manual. Tokyo: MaruzenPublisher, 1983.

4. Fukawa, Y., Ohnishi, Y., Nishiyama. S., Fukuroi. H., Yonezu. K and Miki. S. 2004. The applica-tion of 3-dimentional DDA with spherical rigid block to rockfall simulation. Proc. of the ISRMInternational Symposium 3rd ARMS, Kyoto, Japan. pp. 1243–1248.

5. Pfeiffer, T, J., Bowen, T. D, 1989, Computer Simulation of Rockfalls; Bulletin of the Association ofEngineering Geologists Vol.XXVI No.1, pp. 135–146.

45


6. Komura, T., et al. 2000. A view of parameter and roughness on field slop used in rock-fall simulationmethod. Proc. of the 5th Symposium on Impact Problem in Civil Engineering. Tokyo, Japan. pp.63–68.

7. Ushiro, T., et al. 2000. A study on the motion of rockfalls on slopes. Proc. of the 5th Symposiumon Impact Problem in Civil Engineering. Tokyo, Japan. pp. 91–96.

46

Development of Numerical Manifold Method and its Applicationin Rock Engineering

GUOWEI MA,∗ LEI HE AND XINMEI AN

School of Civil and Environmental Engineering, Nanyang Technological University, Singapore 639798.

1. Introduction

Rock mass is a natural geological material consisting of both continuous rock medium anddiscontinuous components such as joints, fractures, faults etc. To characterize the mechanicalbehaviors of such discontinuities in a computer model, either explicitly or implicitly, variousnumerical methods have been developed.

For problems where their characteristic length (defined by the smallest dimension of theproblem) is much larger than their representative volume (i.e. the smallest volume over whicha measurement can be made that will yield a value representative of the whole), the discon-tinuities can be implicitly modeled with a homogenization model to obtain the equivalentproperties of the material which is heterogeneous and/or fractured. Continuum-based numer-ical methods such as the finite difference method (FDM), the finite element method (FEM),and the boundary element method (BEM) can be adopted to describe such problems.

However, for problems where the representative volume is either much larger than or of asimilar order of the physical problem, the continuum hypothesis is violated and the explicitrepresentation of discontinuities is desired. In FEM, various joint element or interface ele-ment models like ‘Goodman joint element’,1 six-node fracture element,2 joint element basedon the theory of plasticity,3 thin-layer element,4 and interface element in contact mechanics5

have been implemented. Despite these efforts, the treatment of fractures and fracture growthremains limited in the FEM. The FEM requires the finite element mesh conforming to thecracks, which complicates the meshing task. When fracture growth involved, remeshing isinevitable, which makes the simulation complicated and time-consuming. To overcome theinconveniences in meshing and remeshing processes, a variety of modifications to the conven-tional FEM have been made based on the partition of unity (PU). The extended finite elementmethod (XFEM),6,7 in which discontinuities and discontinuities in derivatives are directlyrepresented by incorporating enrichment functions, and the generalized finite element mesh(GFEM),8 which augments the finite element approximation space with high-order termsor handbook functions of boundary value problems to tackle some typical problems withmultiple reentrant corners, voids, and cracks, are two representative examples.

Other continuum-based numerical methods for fracturing analysis include the BEM andthe meshless methods. In general, the BEM is not efficient as the FEM in dealing with materialheterogeneity and non-linearity. Meshless methods look promising, but are not sufficient toreplace the FEM because of their difficulties in numerical integration of weak forms andimposition of essential boundary conditions.

The continuum-based numerical methods can deal with the fractures to some extent. Blockrotations, complete detachment and large-scale opening can not be treated.

Discontinuum-based numerical methods including the distinct element method (DEM)originated by Cundall9 and the discontinuous deformation analysis (DDA) method pio-neered by Shi10 are also used in modeling the rock mass behaviors. In these methods, the


Analysis of Discontinuous Deformation: New Developments and Applications.Edited by Guowei MA and Yingxin ZHOU. Published by Research Publishing Services.Copyright c© 2009 by Society for Rock Mechanics & Engineering Geology (Singapore).ISBN: 978-981-08-4455-4doi:10.3850/9789810844554-keynote-Ma-Guowei 47


problem domain is treated as an assemblage of rigid or deformable blocks with the contactsbetween them identified and continuously updated during the entire deformation/motionprocess. This kind of methods is especially suitable for the simulation of large-scale displace-ments of individual blocks, block rotations, and complete detachment.

It is often the case that individual discrete blocks can also fracture or fragment, which isin essence a process of transition from continua to discontinua. Such problems can be wellrepresented by the combined continuum-based and discontinuum-based numerical methods,such as the combined finite-discrete element method11 and the numerical manifold method(NMM).12

The NMM was initially proposed by Shi in 1991. It gains her name after the mathematicalnotion of manifold.

Different from other numerical approaches, the NMM adopts a finite number of smallpatches called covers to discretize the problem domain, defines local approximations calledcover functions on each cover, and uses weight functions (or partition of unity functions) topaste the local approximations together to give a global approximation.

Compared with other numerical methods, the NMM has the following three distinct fea-tures: (1) because of the introduction of covers, the discontinuities are treated in a straightfor-ward manner; continuous body, fractured body, and assemblage of discrete bodies are treatedin a unified form; (2) the covers do not need to conform to neither the external boundariesnor the internal features such as cracks and material interfaces, which makes its prepro-cessing easy and discontinuity evolutions be modeled without remeshing; (3) because of thepartition of unity property of its weight functions, high-order terms or special functions canbe easily incorporated into its local approximations to improve the accuracy.

In this paper, firstly, the basic theories of the NMM are briefly introduced Then some appli-cations of the 2D-NMM, including the modeling of multiple discrete blocks, the modeling ofstrong discontinuities, and the 3D-NMM, including 3D block cutting, the displacement anddeformation modeling, are presented, respectively.

2. Basic Theory of the NMM

2.1. NMM components

With reference to an example in Fig. 1, the definitions of three basic components in theNMM, namely the mathematical cover (MC), the physical cover (PC) and the cover basedmanifold element (CE), are presented.

The portrait of the physical problem including the problem domain in which the physicalproblem is defined, and all the physical features such as the internal discontinuities (e.g.cracks, joints, material interfaces, holes, etc.) and the external geometries on which boundaryconditions are prescribed is referred to as a physical domain (Fig. 1(a)), whereas a domainwhich is independent but completely cover the physical domain is called the mathematicaldomain (Fig. 1(b)).

The mathematical domain can be constructed as a union of a finite number of smallpatches, called mathematical covers, denoted as MCI. The mathematical covers are user-defined, can be of arbitrary shape and can overlap each other partially or completely. Theyare defined completely independent of the physical domain. However, their union must belarge enough to cover the entire physical domain. See the example in Fig. 1, there are twomathematical covers in total, denoted as MC1 and MC2 (Fig. 1(c)).

The physical covers are the intersection of mathematical covers and the physical domain.If completely cut by the physical features, a mathematical cover MCI will be partitioned into

48


MC1MC2

11PC 1

2PC

21PC 2

2PC

(a) (b) (c) (d)

1CE 2CE 3CE 4CE 5CE

(e)

Figure 1. NMM components in 2D-NMM: (a) physical domain; (b) mathematical domain; (c) math-ematical covers; (d) physical covers; (e) cover-based manifold elements.

several physical covers, denoted as PCjI(j = 1 ∼ mI). See the example in Fig. 1, mathematical

cover MC1 is completely cut into three isolated regions by the physical features and twoof them are within the problem domain, so two physical covers, denoted as PC1

1 and PC21

are formed (Fig. 1(d)). Similarly, mathematical cover MC2 also forms two physical covers,denoted as PC1

2 and PC22.

The cover-based manifold element is defined as the common region shared by several phys-ical covers. The four physical covers in Fig. 1(d) finally form five cover-based manifold ele-ments, shown in Fig. 1(e).

Figure 2(a) illustrates the three basic components of the 3D-NMM. There are two math-ematical covers in total, i.e. a sphere mathematical cover MC1 and a hexahedron mathe-matical cover MC2. The pyramid defines the physical domain. Intersected with the physicaldomain, two physical covers i.e. PC1

1 and PC12 as shown in Fig. 2(b) are generated. These two

physical covers finally form three cover-based manifold elements, as shown in Fig. 2(c).

2.2. NMM approximations

On each mathematical cover MCI, a weight function ϕI(x) satisfies

ϕI(x) ∈ C0(MCI)

ϕI(x) = 0, x /∈MCI(1a)

∑K

if x ∈MCK

ϕK(x) = 1. (1b)

Equation (1a) indicates that a weight function ϕI(x) is continuous over the mathematicalcover MCI, and has non-zero value only on its corresponding mathematical cover MCI, butzero elsewhere, whereas Eq. (1b) is known as the partition of unity (PU) for the continuityof approximation.

49


(a) (b)

(c)

12PC

PC

Figure 2. NMM components in 3D-NMM: (a) physical domain and mathematical covers; (b) physicalcovers; (c) cover-based manifold elements.

On each physical cover PCjI, a local approximation function called cover function denoted

as ujI(x) is defined. Weight functions defined on each mathematical cover transfer to the

physical covers as

ϕjI(x) = δj

I · ϕI(x) (2)

where δjI is a modifier, with its value to 1 within the physical cover PCj

I and 0 elsewhere. Here,each physical cover has two indices, I and j. To simplify the implementation, we reallocate asingle index i to each physical cover with i gained by

i(I,j) =I−1∑l=1

ml + j (3)

Thus, physical cover PCjI, cover function uj

I(x) and weight function ϕjI(x) are re-denoted

as PCi, ui(x) and ϕi(x), respectively. Then, we use the weight functions ϕi(x) to paste all thecover functions ui(x) together to give a global approximation over each cover-based manifoldelement as

uh(x) =∑

iif CE ⊂ Pi

ϕi(x) · ui(x) (4)

Because of the partition of unity property of the weight functions, any high-order termsor special functions can be incorporated into the cover functions to give a high-accuracyapproximation.

50


2.3. Imposition of essential boundary condition

In the NMM, the mathematical covers are constructed totally independent of the boundaries.The essential boundary condition thus can not be accomplished by directly enforcing thedegrees of freedom like in the FEM, but is usually prescribed by the Lagrange multipliermethod, the penalty method or the augmented Lagrange multiplier method in the weak formof governing equations.13

2.4. Contact problems in the NMM

The NMM aims at solving the discontinuous problems, even with rigid movements. Whenintersected with physical features like weak discontinuities and strong discontinuities, eachmathematical cover forms several independent physical covers associated with individualcover functions. So, the adjacent cover-based manifold elements formed by these physicalcovers are independent on each other in the framework of the NMM.

However, the fact is that the cover-based manifold elements at the two sides of the discon-tinuities are not totally independent, but have some relations. For example, for a problemdomain containing a strong discontinuity, the displacement field across the crack surface isdiscontinuous; however, the two sides, i.e. upper side and lower side, of a crack surface cannot penetrate each other in geometry even under a complex stress state. Another exampleis the problem with discrete bodies. The displacement field across each body boundary isdiscontinuous; however, one body can not penetrate into another body Such constraints arenormally termed non-penetration or unilateral condition, and attributed to a contact prob-lem in physics.

Since the frictional contact problems are inherently nonlinear and irreversible, for the sakeof generality, an incremental approach is adopted in the NMM. The contact state at thebeginning of the current time step is known and the contact state at the end of the currentstep after a time interval of step time is to be solved, while the time incremental for each timestep is chosen small enough so that the displacements of all the points within the problemdomain are less than a predefined maximum displacement limit ρ. With an open-close itera-tion procedure, the contact constraint of no penetration and no tension of the two sides ofdiscontinuous entities are fulfilled. Detailed contact detection and modeling in the 2D-NMMcan refer to Shi.12

0s

0.1s

0.5 s

0.9s

1.3 s

1.7 s

Figure 3. Toppling process of a series of blocks modeled by the NMM.

51


3. Applications of the 2D-NMM

3.1. Modeling of multiple discrete blocks

The problems with multiple discrete bodies which are described well with the DDA can alsobe modeled with the NMM. A typical domino run problem is numerically investigated here.The numerical model consists totally 41 rectangular blocks with the size of 20 mm×150 mmand the spacing of 30 mm on a horizontal surface. The material parameters for the blocks are:Young’s modulus E = 200 GPa, Poisson’s ration ν = 0.3. The fiction coefficient μ betweenthe blocks and the horizontal surface is 0.3. An initial pulse force is applied to the first blockto make it topple to the second block, which induce a toppling process shown in Fig. 3. Thenumerically obtained result is consistent with the experimentally observed phenomenon.

Another example is the NMM modeling of mineral separation process using the vibratingscreen. The simplified model includes a set of coal blocks, a vibrating screen and a container.The screen moves in the horizontal direction with a frequency of 1 Hz and amplitude of5.5 cm.

In the study, an oversize triangle is used to cover the whole problem domain, which treatseach block as a single cover-based manifold element with constant stress field and lineardisplacement field. It is reasonable because the coal blocks generally undergo small deforma-tion during the separation process. The numerically obtained separation process is shown inFig. 4.

(a) time step = 0 (b) time step = 6000

(c) time step = 12000 (d) time step = 18000

Figure 4. Size separation process with the vibrating screen modeled by the NMM.

3.2. Modeling strong discontinuities

The NMM models the strong discontinuities by splitting mathematical covers completely cutby the crack surfaces into several physical covers attached with independent cover functionsand enriching singular physical covers containing the crack tips with asymptotic crack tipfunctions.14,15 The resulting displacement field across discontinuities is naturally discontinu-ous. The NMM models complex cracks in an exactly same way with the modeling of a singlecrack.

52


Numerical examples in Refs. 14 and 15 have demonstrated the efficiency and robustness ofthe NMM in modeling complex cracks and their growth. For illustration purpose, a problemwith a tree-shaped crack under bi-axial tension in a finite plate shown in Fig. 5(a) is examinedhere. The detailed description of the parameters used in the computation can refer to Ref.[14]. The regular mathematical covers for the central part of the problem are depicted inFig. 5(b). The convergence of the stress intensity factors (SIFs) at crack tip D can be easilyobserved when the mathematical covers are gradually refined (Fig. 5(c)).

s

W

s

ss

W

H

H

A

B

CD

6.0 12.0 18.0 24.0 30.01.54

1.62

1.70

1.78

1.86

FD I

W/h

6.0 12.0 18.0 24.0 30.0-0.10

-0.07

-0.04

-0.01

0.02

FD II

W/h

σ

σ

σ

σ

(a) (b)

(c) (d)

Figure 5. Modeling a tree-shaped crack with the NMM: (a) a problem with a tree-shaped crack; (b)part of the mathematical covers; (c) SIFs at crack tip D.

4. Applications of the 3D-NMM

4.1. Block cutting

A cutting algorithm16 is proposed to generate blocks from joints and free faces in a three-dimensional space.

If the joints are long compared with the dimension of the target block, cutting algorithmis simple. Only convex blocks are generated. However, if the joints are shorter than thedimension of the target block, the cutting algorithm becomes complicated because: (1) theblocks may be concave; (2) the faces of the blocks may be concave; (3) a block may containsub-blocks inside; (4) a block may contains holes inside; (5) the faces of blocks may alsocontains sub-faces or holes inside.

Most existing block cutting codes only deal with the cases in which the joints are longenough to cut through the target block and thus produce only convex blocks. However, our

53


proposed algorithm can describe not only convex blocks but also concave blocks, not onlysimply connected blocks but also multiply connected blocks.

Figure 6(a) depicts the Great Pyramid of Khufu with internal chambers, while Fig. 6(b)illustrates our 3D cutting result, which is a convex block with complex internal faces. Figure 7illustrates the cutting of a complex tunnel system in a jointed rock mass. Figure 8 shows thecutting of a slope in a jointed rock mass.

(a) (b)

Figure 6. Cutting the Great Pyramid of Khufu: (a) real structure; (b) 3D cutting result.

Figure 7. Cutting a tunnel system in a jointed rock mass.

Figure 8. Cutting a slope in a jointed rock mass.

54


4.2. Modeling of block free fall

In Fig. 9(a), the geometry of a single block in the 3-D space can be defined by using theexplanation of physical covers and mathematical covers mentioned in the previous sections.The cube falls under the pure influence of the gravity. The acceleration of gravity is 10 m/s2.The time step used is 0.05 s.

Figure 9(b) gives the displacement time history. The error of the numerical solution is lessthan 0.1% in the maximum absolute difference, which illustrates the accuracy of the NMMcalculation.

4.3. Effect of cover size and orientation

A 2 m×2 m×0.1 m plate is subjected to the gravity load with g = 10 and fixed at four corners,with material properties of E = 10 GPa, v = 0.3, ρ = 1200 kg/m3. Its geometry of typicalmesh design is shown in Fig. 8(a) in which faces of the plate conform to the axis planes.

Six mathematical cover size of s = 0.52, 0.32 0.22, 0.12, 0.08, 0.05 are used to examinethe cover size effect. After intersection with the physical plate, 190, 684, 1104, 3706, 8100,19200 cover-based manifold elements are generated respectively. The dynamical ratio is set

(a) (b)

90.0

80.0

70.0

60.0

50.0

40.0

30.0

20.0

10.0

000.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Time (s)

Dis

plac

emen

t (m

)

NMMExact

1670 1

Figure 9. Block free fall simulation: (a) the model; (b) the displacement time history versus theoreticalsolution.

(a) Model (b) Convergence of deformatiom

Tota

l Def

orm

atio

n in

the

Cen

ter

0.40

0.35

0.30

0.25

0.20

0.15

1 2 3 4 5 6

MC size Level

NMM

Figure 10. Deformation convergence with decrease of cover size.

55


to be 0 to clarify the quasi-static responses of the 6 models. Hence, no velocity of the cover-based manifold element is transferred to the next time step. Results are convergent and stablewith the increase of the mesh density shown in Fig. 10(b).

Then, the 2 m×2 m×0.1 m plate is subjected to a constant point load L = 5 at the cen-ter with fixed four corners of the same material properties above. The dynamical ratio isset as 0.999 to investigate dynamical response fully. Two different orientations of the plate(orientation 2 and 3) are shown in Fig. 11 where the orientation 1 is same as the previousexample.

Z direction displacement histories of the center point (Fig. 12) shows that the orientationof the mathematical covers has negligible effect on the plate maximum displacement at theplate center. The maximum displacement is accurate and stable after converging, and theconvergence time is also stable.

This ensures the accuracy and modeling efficiency of the 3-D NMM and decreases themesh division complexity in FEM. It also supports the validity of the 3-D NMM dynamicalgorithm.

Figure 11. Geometry of mesh designs for mesh orientation effect study.

Figure 12. Z direction displacement histories of center point.

56


Figure 13. two tetrahedron contact with flexible slab.

4.4. Contact algorithm

As shown in the Fig. 13 a flexible slab has been fixed on three corners, and one corner is free.Two tetrahedron blocks are suspending above the slab. The whole system is under gravityenvironment and equivalent 0.4 gravity force along minus y direction

In recent developing NMM 3D code, contact system built upon contact hierarchies’ con-cept (HTC), which has been used in the commercial software LSDYNA. It is one of most con-fidential contact algorithm recently. Furthermore, NMM3D formula can simulate the wholedynamical process, such as fell down, contact interaction, frictional sliding, and separation

This algorithm is still under optimization. Up until now, the advantage of this algorithm indiscontinuous description and contact interaction, has facilitated and provided foundationfor its wide usage in complex underground engineering.

5. Conclusions and Remarks

In the NMM, two separate cover systems i.e. the mathematical covers and the physical coversare employed to describe a physical problem with displacement approximations defined oneach physical cover. The displacement approximation on a coverbased manifold element isobtained by the combination of the approximations of the related physical covers using thepartition of unity. As elements across discontinuous entities may be related with differentphysical covers, the NMM can describe continuous problems and discontinuous problemssimultaneously in a unified form. By increasing physical covers, the process from continuato discontinua can be easily actualized without any difficulties that are encountered in othernumerical methods Obviously, the NMM is a promising numerical method in many researchand application areas, and the rock mechanics and rock engineering area is certainly andimportantly included

In this paper, focus is put on the development and verifications of both the 2D-NMMand 3D-NMM. As the basic theory of the 2D-NMM has developed relatively well, it is usedto simulate discrete block systems and developed for crack simulation. Results show thatthe 2D-NMM can deal with discrete and strong discontinuous problems well. Fundamentalstudies are carried out for the 3D-NMM, including block cutting and the application of 3D-NMM in both continuous and discontinuous problems. The satisfied verification results ofthe 3D-MM program put a solid stage for its further development.

57


References

1. Goodman R. E., Taylor R. L., Brekke T. L. A model for the mechanics of jointed rock. Journal ofthe Soil Mechanics and Foundations Division, ASCE, 1968, 94: 637–659.

2. Zienkiewicz O. C., Best B., Dullage C., Stagg K. Analysis of nonlinear problems in rock mechanicswith particular reference to jointed rock systems. Proceedings of the Second International Congresson Rock Mechanics, Belgrade, 1970, pp. 8–14.

3. Ghaboussi J., Wilson E.L, Isenberg J. Finite element for rock joints and interfaces. Journal of theSoil Mechanics and Foundations, ASCE, 1973, 99(10): 849–862.

4. Desai C. S., Zamman M. M., Lightner J. G., Siriwardane H. J. Thinlayer element for interfacesand joints. International Journal for Numerical and Analytical Methods in Geomechanics, 1984,8: 19–43.

5. Katona M. G. A simple contact–friction interface element with applications to buried culverts.International Journal for Numerical and Analytical Methods in Geomechanics, 1983, 7: 371–384.

6. Moes N., Dolbow J., Belytschko T. A finite element method for crack growth without remeshing.International Journal for Numerical Methods in Engineering, 1999, 46: 131–150.

7. Sukumar N., Prevost J. H. Modeling quasi-static crack growth with the extended finite elementmethod Part I: computer implementation. International Journal of Solids and Structures, 2003,40: 7513–7537.

8. Strouboulis T., Babuska I., Copps K. The design and analysis of the generalized finite elementmethod, Computer Methods in Applied Mechanics and Engineering, 2000, 181: 43–69.

9. Cundall P. A. A computer model for simulating progressive, large scale movements in blocky rocksystems, Proceedings, International Symposium on Rock Fracture, Nancy, France, II-8, 1971

10. Shi G. H. Discontinuous Deformation Analysis — A new numerical model for the static anddynamics of block systems, PhD Dissertation, Department of Civil Engineering, U.C. Berkeley,1988

11. Munjiza A. The combined finite-discrete element method, Wiley, Chichester, 200412. Shi G. H. Manifold method of material analysis, Trans. 9th Army Conf. on Applied Mathematics

and Computing, Minneapolis, Minnesota, 1991, pp. 57–76.13. Ma G. W., An X. M., He L. The numerical manifold method-A reivew. Submitted to International

Journal of Computational Methods, 2009.14. Ma G. W., An X. M., Zhang H. H., Li L. X. Modeling complex crack problems with numerical

manifold method. International Journal of Fracture, 2009 156 (1): 21–35.15. Zhang H. H., Li L. X., An X. M., Ma G. W. Numerical analysis of 2-D crack propagation prob-

lems using the numerical manifold method. Submitted to Engineering Analysis with BoundaryElements, 2009.

16. Ma G. W., He L. Development of 3-D numerical manifold method. Submitted to InternationalJournal of Computational Methods, 2009

58

Tensorial Approach to Rock Mass Strength and Deformability in

Three Dimensions

P.H.S.W. KULATILAKE∗

Geological Engineering Program, Dept. of Materials Science and Engineering, University of Arizona,

Tucson, Arizona, 85721, USA

1. Introduction

A good understanding of rock mass strength and deformability is vital to arrive at safeand economical designs for structures built in and on rock masses. Rock mass strengthand deformability depend on (a) the discontinuity network, (b) the geomechanical proper-ties of the discontinuities, (c) the geomechanical properties of the intact rock, (d) the in-situ stress system and (e) the loading/unloading stress path. The presence of complicateddiscontinuity patterns, the inherent statistical nature of their geometrical parameters, andthe uncertainties involved in the estimation of their geomechanical and geometrical prop-erties and in-situ stress make accurate prediction of rock mass strength and deformabilitydifficult. It is a known fact that jointed rock mass strength and deformability exhibit bothanisotropy and scale effects in the presence of distinct joint clusters. Rock mass strength anddeformability change with the direction due to the orientation distributions of the distinctjoint clusters. Joint size distributions of the joint clusters lead to the scale effects on rockmass strength and deformability. Procedures currently used in practice to estimate rock massstrength and deformability do not incorporate appropriate procedures to capture the direc-tional and scale dependence of rock mass strength and deformability. At present, in prac-tice, rock mass strength and deformability are estimated based on rock mass classificationindices. All these indices are scalars. Therefore, they do not have the capability of capturingthe anisotropy or the directional changes of rock mass strength and deformability. This papershows how to use tensorial approaches to capture the anisotropy and scale dependency ofrock mass strength and deformability.

2. Capturing of Anisotropy and Scale Effects in Estimating Rock MassDeformability in Three Dimensions

A numerical decomposition technique (Fig. 1), which has emerged from a linking betweenjoint geometry modelling and generation schemes and a distinct element code (3DEC1), isused to evaluate the effects of joint geometry parameters of finite size joints on the deforma-bility properties of jointed rock at the three-dimensional (3-D) level.2−4 In order to use the3-DEC for 3-D stress analysis, the problem domain should be discretized into polyhedra.With only finite size actual joints, it will not be possible to discretize the domain into poly-hedra. In such a situation, in order to use the 3-DEC code for stress analysis, it is necessaryto create some type of fictitious joints so that when they are combined with actual jointsthe problem región is discretized into polyhedra. These fictitious joints then should behaveas intact rock. Kulatilake et al. have given procedures to create polyhedra in working withfinite size joints3 and also have provided guidelines to estímate strength and deformabilityparameter values for fictitious joints.4 Joint geometry networks used for the actual joints in


Analysis of Discontinuous Deformation: New Developments and Applications.

Edited by Guowei MA and Yingxin ZHOU. Published by Research Publishing Services.

Copyright c© 2009 by Society for Rock Mechanics & Engineering Geology (Singapore).

ISBN: 978-981-08-4455-4

doi:10.3850/9789810844554-keynote-Kulatilake 59


Generate non-persistent actual joints in rock blocks in 2D (in 3D)

Create fictitious joints to discretize rock blocks into polygons

in 2D (polyhedra in 3D)

Link 2D joint generator (3D joint

generator) to the 2D distinct element

code (3D distinct element code) to

generate rock blocks with actual and

fictitious joints

Obtain representative values for mechanical properties of fictitious joints in 2D (in 3D) to simulate the intact rock behaviour

Perform stress analysis for different actual joint configurations under different stress paths using the distinct element menthod in 2D (in 3D)

Evaluate the effect of joint geometry parameters on the stength and deformability of

rock blocks

Figure 1. Flow chart of the procedure used to study the effect of joint geometry on rock massdeformability.2

the study are given in Table 1. The reader is referred to Ref. 2 for the constitutive modelsused for intact rock, actual joints and fictitious joints. The parameter values used for the con-stitutive models are given in Tables 2 and 3. These values have been obtained from a graniticgneiss rock tested in the laboratory and field.5 Three-dimensional stress paths used for thestudy are shown in Figs. 2 and 3. Variation of deformability parameters of jointed rock withjoint geometry parameters such as joint density, joint size/block size and joint orientation, areshown through 3-D plots in Ref. 2. The relations developed between deformability propertiesof jointed rock and fracture tensor parameters are shown in Figs. 4 and 5. For details on thefracture tensor parameters, the reader is referred to Ref. 2. An incrementally linear elastic,orthotropic constitutive model is suggested to represent the prefailure mechanical behaviorof jointed rock (Figs. 6 and 7). This constitutive model has captured the anisotropic, scaledependent behavior of jointed rock (Fig. 8). In this model, the effect of the joint geometrynetwork in the rock mass is incorporated in terms of the fracture tensor components (Fig. 8).

60


Table 1. Generated joint networks of actual joints in rock blocks in 3-D distinct element stressanalysis.2

No. of joints sets Orientations Joint size/block size No. of joints Jointlocation

60◦/45◦ 0.1–0.9 with step 0.1 5, 10, 20, 30 Uniformdistribution

94.42◦/37.89◦ 0.3, 0.5, 0.6, 0.7, 0.9 5, 10, 20, 30 Uniformdistribution

One joint set 30◦/45◦ 0.3, 0.5, 0.6 5, 10, 20 Uniformdistribution0.7, 0.8, 0.9 5, 10, 20

90◦/45◦ 0.3, 0.5, 0.6, 0.7, 0.8, 0.9 5, 10, 20 Uniformdistribution

68.2◦/72.2◦ 0.3, 0.6, 0.7, 0.8 5, 10, 20, 30 Uniformdistribution

248.9◦/79.8◦ 0.3, 0.6, 0.7, 0.8 5, 10, 20, 30 Uniformdistribution

1st joint set 60◦/45◦ 10Two joint sets Uniform

2nd joint set 240◦/60◦ 0.1, 0.2, 0.3, 0.5, 0.6, 0.7 10 distribution

Table 2. Intact rock and fictitious joints parameter values of the constitutive models for graniticgneiss rock.2

Intact rock Fictitious joints

Parameter Assigned value Parameter Assigned value

Density (d) 2500 (kg/m3) Joint normal stiffness (JKN) DifferentYoung’s modulus (E) 60 (GPa) Joint shear stiffness (JKS) values (MPa/m)Poisson’s ratio (v) 0.25 Joint cohesion (Jc) 50 MPaBulk modulus (K) 40 (GPa) Joint dilation coefficient (Jd) 0Shear modulus (G) 24 (GPa) Joint tensile strength (Jt) 10 (MPa)Cohesion (c) 50 (MPa) Joint friction coefficient ( tanφ) 0.839Tensile strength (t) 10 (MPa)Friction coefficient (tanφ) 0.839

Table 3. Parameter values of the constitutive model fornon-persistent actual joints of granitic gneiss rock.2

Parameter Value

Joint normal stiffness (JKN) 6.72× 104MPa/m

Joint shear stiffness (JKS) 2.7× 103MPa/mJoint cohesion (Jc) 0.4MPaJoint tensile strength (Jt) 0MPaJoint friction coefficient (tanφ) 0.654

61


zz

z

z

x

x

x x

y y

y y

Figure 2. Stress paths of first type used to perform distinct element stress analysis of rock blocks withjoints.2

z z

z

x xx

y

y

y

xy

yx

x

zx

zy

yz

z

Figure 3. Stress paths of second type used to perform distinct element stress analysis of rock blockswith joints.2

Figure 4. Relation between rock block deformation modulus in any direction, Em, and the fracturetensor component in the same direction2 (note: Ei is the intact rock Young’s modulus).

62


Figure 5. Relation between rock block shear modulus on any plane, Gm, and the summation of frac-ture tensor components on that plane2 (note: Gi is the intact rock shear modulus).

1ε11ε21ε31γ121γ131γ23

=

1

E1

−v21E2

−v31E3

0 0 0

−v12E1

1

E2

−v32E3

0 0 0

−v13E1

−v23E3

1

E30 0 0

0 0 0 1G12

0 0

0 0 0 0 1G13

0

0 0 0 0 0 1G23

×

1σ11σ21σ31τ121τ131τ23

Figure 6. General form of the incrementally linear elastic constitutive model suggested to representpre-failure behavior of a jointed rock mass in 3-D2.

3. Rock Mass Strength and Deformability Estimations in 3-D for RockBlocks of Sizes 13.5–30 m at the ASPO Hard Rock Laboratory, Sweden

Rock fracture data provided by Swedish Nuclear Fuel and Waste Management Companywere used to develop and valídate a 3-D stochastic fracture network model for a 30m cubeof Aspo diorite rock mass located at a depth of 485m at Aspo Hard Rock Laboratory,Sweden.6 A new procedure is developed to estimate rock block strength and deformabilityin 3-D allowing for anisotropy and incorporating the fracture geometry for the selected 13.5through 30m cubes.6 In situ stress data and laboratory estimated geomechanical propertiesof intact rock and rock joints were used in estimating the block strength and deformabilityin every 45 degree direction in 3-D (Table 4). The mean rock mass strength was found to

63


1ε11ε21ε31γ121γ131γ23

=

aFn1+1

Ei

−v21(aFn2+1)

Ei

−v31(aFn3+1)

Ei0 0 0

−v12(aFn1+1)

Ei

aFn2+1

Ei

−v32(aFn3+1)

Ei0 0 0

−v13(aFn1+1)

Ei

−v23(aFn2+1)

Ei

aFn3+1

Ei0 0 0

0 0 0 b(F1+F2)m+1

Gi0 0

0 0 0 0 b(F1+F3)m+1

Gi0

0 0 0 0 0 b(F2+F3)m+1

Gi

×

1σ11σ21σ31τ121τ131τ23

Figure 7. Specific form of the incrementally linear elastic constitutive model suggested to representpre-failure behavior of a jointed rock mass in 3-D2.

(b)(a)

(c)

Figure 8. Anisotropy of deformation modulus, Em, in three dimensions for different first invariant ofthe fracture tensor, IF1, values:

2 (a) Em on the y-z plane; (b) Em on the x-y plane; (c) Em on the x-zplane.

64


35.6 cm

17.8

cm

Figure 9. Typical frame used in making the jointed specimens of the model material.7

Table 4. Mean rock mass strength and rock mass modulus in different directions in 3-D (Note:downward plunge +ve).

DirectionMean rock mass strength (MPa) Mean rock mass modulus (GPa)

Trend (degs.) Plunge (degs.)

0 −90 152.5 38.20 −45 113.8 34.20 0 129.8 35.70 45 140.6 41.145 −45 136.0 32.845 0 145.2 36.245 45 164.5 39.990 −45 142.3 40.590 0 153.0 39.090 45 131.5 39.0

135 −45 135.0 34.5135 0 139.5 38.8135 45 137.5 34.2180 −45 140.6 41.1180 0 129.8 35.7180 45 113.8 34.2180 90 152.5 38.2225 −45 164.5 39.9225 0 145.2 36.2225 45 136.0 32.8270 −45 131.5 39.0270 0 153.0 39.0270 45 142.3 40.5315 −45 137.5 34.2315 0 139.5 38.8315 45 135.0 34.5

be 47% of the mean intact rock strength of 297 MPa at 485m depth. The mean rock massmodulus was found to be 51% of the intact rock Young’s modulus of 73 GPa. The rock mass

65


Table 5. Principal rock mass strength and rock mass modulus mean magnitudes and theirdirections.6

Rock mass strength Rock mass modulus

First principal direction

Trend (deg.) 63 51Plunge (deg.) 29 45Magnitude 155.2 MPa 39.9 GPa

Second principal direction


Third principal Direction


(a)

(b)

Horizontal aluminum plate mounted on a ball bearing slide

Load cell

Hydraulic cylinder

Block sample

Two thin galvanized sheets with lubrication in between

35.6 cm ×××× 17.8 cm ×××× 2.5 cm Sample

Firm rubber

Applied load in σ2 direction

Spherical seat

Applied load in σ

1 direction

Steel plates Spherical seat

Figure 10. (a) Equipment and the data acquisition system used in performing uniaxial and biaxialcompression experiments; (b) A detailed view around the sample.7

66


Figure 11. (a) Tensile splitting, (b) Sliding on joints, (c) Failure in intact material and sliding on joints.

Prediction

0.0 MPa0.125 MPa0.25 MPa0.5 MPa1.0 MPa

0.0 1.0 2.0 3.0 4.0 5.0

F22

0

0.2

0.4

0.6

0.8

1

σ1

,b/σ

1,I

Experimental Data

0.0 MPa0.125 MPa0.25 MPa0.5 MPa1.0 MPa

σ2 values

Figure 12. A plot of the normalized block strength, σ1,b/σ1,I, of jointed blocks of glastone model

material against fracture tensor component, F22, for different intermediate principal stresses, σ2.7

Poisson’s ratio was found to be 21% higher than the intact rock Poissson’s ratio of 0.28. Theanisotropy resulted for the rock mass strength and mass modulus are shown in Table 5.

4. A New Rock Mass Strength Criterion Based on Laboratory Test Results

To simulate brittle rocks, a mixture of glastone, sand and water was used as a model mate-rial. Thin galvanized sheets of thickness 0.254 mm were used to create joints in blocks madeout of the model material (Fig. 9). To investigate the failure modes and strength, both theintact material blocks as well as jointed model material blocks of size 35.6 × 17.8 × 2.5 cm

67


0.0

0.5

1.0

1.5

2.0

2.5

0.00 0.01 0.02 0.03 0.04 0.05

σσσσ2/σσσσu,I

ωωωω

07.2! 0 =

0.949R

1)/"("942.43

!!

2

0.6703Iu,2

0

=

+×=

Figure 13. Variation of the decay parameter, ω, with σ2/σu,I.7

Figure 14. Proposed biaxial rock mass failure criterion for glastone jointed blocks on σ1,b/σu,I versus

σ2/σu,I space.7.

68


Table 6. Joint geometry configurations used for jointed model material blocks withpersistent joints.7

Sample Number Dip angles of the two joint sets (degrees) Number of Joints per Set

INTACT —- —–

BSP-10 10–10 4 × 4

BNSP-10-20 10–20 4 × 4

BSP-15 15–15 4 × 4

BNSP-15-25 15–25 3 × 3

BNSP-10-30 10–30 3 × 3

BSP-20 20–20 4 × 4

BSP-25 25–25 3 × 3

BNSP-20-30 20–30 3 × 3

BNSP-15-35 15–35 3 × 3

BSP-30 30–30 3 × 3

BNSP-25-35 25–35 3 × 3

BNSP-10-40 10–40 3 × 3

BNSP-20-40 20–40 3 × 2

BNSP-10-50 10–50 3 × 2

BSP-35 35–35 3 × 3

BNSP-15-45 15–45 3 × 3

BNSP-30-40 30–40 3 × 3

BNSP-20-50 20–50 3 × 2

BNSP-25-45 25–45 3 × 3

BNSP-30-50 30–50 3 × 2

BSP-40 40–40 3 × 3

BNSP-15-55 15–55 3 × 2

BNSP-35-45 35–45 3 × 2

BNSP-25-55 25–55 3 × 2

BNSP-40-50 40–50 3 × 2

BNSP-35-55 35–55 3 × 2

BSP-50 50–50 2 × 2

BSP-45 45–45 3 × 3

BNSP-45-55 45–55 3 × 2

BSP-55 55–55 2 × 2

Note: (a) Each joint configuration used in this research had two joint sets. (b) For thespecimen number: BSP stands for blocks with symmetric persistent joints; BNSP standsfor blocks with non-symmetric persistent joints.

having different joint geometry configurations (Table 6) were subjected to uniaxial and biax-ial compressive loadings using the equipment shown in Fig. 10. Three different failure modeswere observed on the tested samples (Fig. 11) depending on the joint geometry configuration

69


and the level of the minor principal stress.7 A new intact rock failure criterion was proposedat the 3-D level (Eq. (1) in Ref. 7. This criterion was validated for biaxial loading throughlaboratory experimental results obtained on intact model material blocks.7 Results obtainedfrom both the intact and jointed model material blocks were used to develop a strongly non-linear new rock mass failure criterion for biaxial loading (Eqs. 7 & 8 in Ref. 7 and Figs. 12and 13). In this failure criterion, the fracture tensor component (Eqs. 3–6 in Ref. 7) is usedto incorporate the directional effect of fracture geometry system on jointed block strength.The failure criterion shows the important role the intermediate principal stress plays on rockmass strength (Fig. 14).

5. Conclusions

The paper provides procedures to estimate rock mass strength and deformability captur-ing the anisotropy and scale dependency. It also shows how to use discontinuum numericalmodeling techniques to estimate equivalent continuum properties for rock mass strength anddeformability that reflect the combined effects of intact rock and fractures. A new rock massfailure criterion for biaxial loading is presented in the paper. In this failure criterion, thedirectional effect of fracture geometry system on jointed block strength is represented by thefracture tensor component.

Acknowledgements

Research published in paper numbers 2 through 4 was funded by the Swedish NaturalScience Research Council with respect to a fruitful research corporation author had withProfessor Ove Stephansson. The Swedish Nuclear Fuel and Waste Management Companyprovided financial support for the research published in paper number 6. Research publishedin paper number 7 was funded by the U.S. National Science Foundation, grant number CMS-9800407. The author would like to thank his former graduate students S. Wang, J. Um,J. Park, D. Wathugala, B. Malama, H. Ucpirti and G. Radberg for making contributions tothe research reported in this paper.

References

1. ITASCA Consulting Group Inc., 3DEC Version 2 Users Guide, 1998.2. Kulatilake, P.H.S.W., Wang, S. and Stephansson, O., “Effect of Finite Size Joints on Deformability

of Jointed Rock at the Three Dimensional Level,” Int. J. Rock Mech. & Min. Sci., 30, 5, 1993, pp.479–501.

3. Wang, S. and Kulatilake, P.H.S.W. “Linking Between Joint Geometry Models and a DistinctElement Method in Three Dimensions to Perform Stress Analyses in Rock Masses Containing FiniteSize Joints”, Soils and Foundations, 33, 4, 1993, pp. 88–98.

4. Kulatilake, P.H.S.W., Ucpirti, H., Wang, S., Radberg, G. and Stephansson, O., “Use of the DistinctElement Method to Perform Stress Analysis in Rock with Non-Persistent Joints and to Study theEffect of Joint Geometry Parameters on the Strength and Deformability of Rock Masses”, RockMechanics and Rock Engineering, 25, 1992, pp. 253–274.

5. Hardin, E. L. et al., “A. Heated Flat jack Test to Measure the Thermo-mechanical and TransportProperties of Rock Masses”, Office of Nuclear Waste Isolation, Columbus, Ohio, Report ONWI-260P, 1982.

70


6. Kulatilake, P.H.S.W., Park, J. and Um, J., “Estimation of Rock Mass Strength and Deformability in3-D for a 30m Cube at a Depth of 480m at Aspo Hard Rock Laboratory, Sweden”, Int. Jour. ofGeotechnical and Geological Engineering, 22, 3, 2004, pp. 313–330.

7. Kulatilake, P.H.S.W., Park, J. and Malama, B., “A New Rock Mass Strength Criterion for Biax-ial Loading Conditions”, Int. Jour. of Geotechnical and Geological Engineering, 24, 4, 2006,pp. 871–888.

71

Contact Algorithm Modification of DDA and Its Verification

Y.J. NING1,2,∗, J. YANG2, G.W. MA1 AND P.W. CHEN2

1Division of Structures and Mechanics, School of Civil and Environmental Engineering,Nanyang Technological University, Singapore 639798, Singapore2State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology,Beijing 100081, China

1. Introduction

Discontinuous Deformation Analysis (DDA) method is a discrete numerical method devel-oped for the simulation of large deformation and displacement of block systems. As a discretemethod, the treatment of the contact between blocks is of vital importance for the actual-ization and the precision of this method, and much research work has been done in thisarea.1–6 Block large deformation and large displacement require the constraint of no pene-tration and no tension between each two blocks, and this constraint is fulfilled using penaltymethod by Dr Gen-hua Shi in the original DDA.1 In penalty method, stiff springs are set innormal and/or shear directions between blocks to transfer the inequality problem of contactconstraint into equality problem of computing contact displacements and contact forces.

Penalty method is proved to be effective in many numerical areas and has been widelydiscussed and applied.7–9 This method is simple in implementation and may not increase thedimension of the system equilibrium equations. The shortage of penalty method is that itcan only fulfil the contact constraint approximately, and the contact treatment precision isaffected by different penalty number selections.5, 6 Study shows that with relatively largerpenalty number, penalty method can treat the contact of blocks well, but too large value ofpenalty number may possibly lead to the ill condition of the system equilibrium equations,and then even error result may be gotten. Meanwhile, for a particular problem, a properpenalty number is not easy to select, so the DDA simulation result may easily be affectedartificially and the result can be unreliable.

To solve the above mentioned problem of penalty method, Augmented LagrangianMethod is adopted to modify this method in the current paper, and two examples are carriedout to verify the improved contact method in contact force computation and block systemdeformation analysis respectively.

2. Modification of Contact Algorithm

2.1. Contact displacement in DDA

In two dimensional DDA, all kinds of contact between blocks can be transformed as thecontact between an angle and an edge.1 As shown in Fig. 1, P1 is a vertex of block i, P2P3is an edge of block j. After a step displacement, P1 moves to P0. Assume (xk,yk) and (uk,vk)are the coordinates and displacements of Pk(k = 0 ∼ 3) respectively. The normal and shearcontact displacements dN and dS of P1 to P2P3 in time step can be expressed as

dN = S0

l+ EDi +GDj, dS =

S′0l+ E′Di +G′Dj (1)


Analysis of Discontinuous Deformation: New Developments and Applications.Edited by Guowei MA and Yingxin ZHOU. Published by Research Publishing Services.Copyright c© 2009 by Society for Rock Mechanics & Engineering Geology (Singapore).ISBN: 978-981-08-4455-4doi:10.3850/9789810844554-0009 73


iblock

iblock

Np

Sp

2P 3P

Sd Nd

jblock

1P

0P

Figure 1. Angle-edge contact.

where

S0 =∣∣∣∣∣∣1 x0 y01 x2 y21 x3 y3

∣∣∣∣∣∣ , S′0 =[x3 − x2 y3 − y2

] {x1 − x0y1 − y0

}, l =

√(x2 − x3)2 + (y2 − y3)2

E, G, E′, G′ are 1× 6 matrices. Where (r = 1 ∼ 6)

er = 1l[(y2 − y3)t1r(x0,y0)+ (x3 − x2)t2r(x0,y0)] (2)

gr = 1l[(y3 − y0)t1r(x2,y2)+ (x0 − x3)t2r(x2,y2)]

(3)+ 1

l[(y0 − y2)t1r(x3,y3)+ (x2 − x0)t2r(x3,y3)]

e′r =1l[(y2 − y3)t1r(x1,y1)+ (x3 − x2)t2r(x1,y1)] (4)

g′r =1l[(x2 − x3)t1r(x0,y0)+ (y2 − y3)t2r(x0,y0)] (5)

In penalty method, two springs with stiffness of pN and pS are used in normal and sheardirections to constrain the contact displacements as zero respectively. Then the strain energyof normal and shear springs can be respectively expressed as

�N = 12

pNd2N, �S = 1

2pSd2

S (6)

2.2. Augmented Lagrangian contact method in DDA

The classical Lagrange Multiplier Method, which is one of the early methods to deal withblock contact problems computing contact forces explicitly, expresses contact strain energyas the product of the contact force λ and contact displacement d. As the introduction ofthe unknown λ, the number of the system equilibrium equations will be increased andthen additional computational effort is needed to do the solution.4 Based on the classicalLagrange Multiplier Method, the developed Augmented Lagrangian Method.5, 6, 10, 11 notonly keeps the briefness of penalty method, but also keeps the number of the system equilib-rium equations unchanged. In the Augmented Lagrangian Method, a penalty number p anda Lagrangian multiplier λ∗ (represents contact force) are used to compute the contact force

74


iteratively as

λ∗m+1 = λ∗m + pd (7)

where λ∗m and λ∗m+1 are the Lagrangian multipliers of time step m and m + 1 respectively,d is the contact displacement. Then at the mth time step, the contact train energy can beexpressed as

�s = �λ +�p = λ∗md + 12

pd2 (8)

where �p = 12pd2 is the same as the spring strain energy in penalty method, so only �λ =

λ∗md needs to be taken into account additionally based on penalty method. Respectively innormal and shear directions, �λ can be deduced as

�λ = λ∗mdN = λ∗m(

S0

l+ EDi +GDj

)(9)

�′λ = λ∗′mdS = λ∗′m(

S′0l+ E′Di +G′Dj

)(10)

According to the minimum potential energy principle, four 6× 1 sub-matrices are added tothe system sub-matrices Fi or Fj. As the normal contact force may become different from thatin penalty method, the friction force sub-matrices should also be adjusted.12

3. Verification

3.1. Contact force computation

As shown in Fig. 2, there is a rectangle block on a fixed incline with a dip angle of 45◦. Thefriction angle between the rectangle block and the incline is set as 70◦, and the cohesion andtensile strength between which are set as 30MPa and 15MPa respectively, so the rectangleblock may keep still on the incline. The Young’s modulus of the two blocks is both set asE = 35GPa. In DDA, the contact between the rectangle block and the incline is transformed

Fixed

UP

LP

hb

m.70 m.30

°45

Figure 2. A block on an incline.

75


as two angle-edge contact between the two vertices (PL and PU) of the rectangle block andthe incline. The theoretical solutions2 of the two normal contact forces respectively at PL andPU are

NL =W cosϕ(

0.5+ h2b

tan θ)= 5197.259N (11)

NU =W cosϕ(

0.5− h2b

tan θ)= 2078.753N (12)

The theoretical solution of the resultant force of the two shear contact forces SL (at PL) andSU (at PU) is

S =W sin θ = 7276.758N (13)

When the penalty number (p) varies from 10 × E to 500 × E (E is the Young’s modulus,and so in the following), using both penalty contact method (shorten as P method) andAugmented Lagrangian contact method (shorten as A-L method), NL and NU can be gottenas 5197.235N and 2078.894N respectively, while S is 7276.124N or 7276.125N, which allare very close to the corresponding theoretical solutions. Table 1 gives some of the contactforce result with several relatively small penalty numbers using P method or A-L method.With different penalty numbers, both SL and SU vary a lot while using P method, but thoseare quite stable while using A-L method, except when p is relatively small (10 × E), eventhough, the variety is smaller than 1N, which is very small compared with the shear contactforces themselves.

Figure 3 shows the variety of SL and SU with the penalty number varying from 10 × E to500× E respectively using P method and A-L method. Using A-L method, SL and SU almostdon’t change with the variety of the penalty number, while using P method, which apparentlyvary with different penalty number selections, and the varying speed is obviously larger whenthe penalty number is relatively small. Meanwhile, as the penalty number value increases, SLand SU of P method go close to that of A-L method, which certificates the correctness of theA-L method for contact force computation.

Figure 4 shows the variety of NL and SL with the increase of time steps respectively using Pmethod and A-L method. With both contact methods, NL and SL come stable at the secondstep, and the NL curves duplicate each other, while the SL curves are quite discrepant. WhenA-L method is used, the difference between the final stable value and the first step value ofSL is more apparent than that when P method is used. This is because the A-L method is an

Table 1. Contact force result with different penalty numbers.

p Contact method NL/N NU/N SL/N SU/N S/N

10× EP 5197.235 2078.894 7143.561 132.564 7276.125

A-L 5197.235 2078.894 7777.840 −501.716 7276.124

20× EP 5197.235 2078.894 7434.782 −158.657 7276.125

A-L 5197.235 2078.894 7778.272 −502.147 7276.125

50× EP 5197.235 2078.894 7633.927 −357.802 7276.125

A-L 5197.235 2078.894 7778.531 −502.406 7276.125

100× EP 5197.235 2078.894 7705.029 −428.904 7276.125

A-L 5197.235 2078.894 7778.617 −502.493 7276.124Theoretical solutions 5197.259 2078.753 − − 7276.758

76


0 50 100 150 200 250 300 350 400 450 500 550

7100

7200

7300

7400

7500

7600

7700

7800

7900S L

/N

Penalty number/ ×E

A-L P

0 50 100 150 200 250 300 350 400 450 500 550

-500

-400

-300

-200

-100

0

100

200

S U/N

Penalty number/ ×E

A-L P

Figure 3. SL and SU variety with different penalty numbers.

0 2 4 6 8 10 12 14 16 18 20 225194.5

5195.0

5195.5

5196.0

5196.5

5197.0

5197.5

NL/N

Time step

P A-L

0 2 4 6 8 10 12 14 16 18 20 22

7600

7620

7640

7660

7680

7700

7720

7740

7760

7780

7800

S L/N

Time step

P A-L

Figure 4. NL and SL variety with increase of time steps.

iterative method to compute contact force, and the contact forces need a longer process tobecome stable.

From the above example, it can be found that using both P method and A-L method, thenormal contact force and the resultant force in shear direction can be computed preciselybetween blocks, but the A-L method eliminates the influence of the penalty number selec-tion on shear contact force result at each contact point. In penalty method, the increase ofpenalty number value is beneficial for improving contact force computation precision, andwith a relatively small penalty number, precise shear contact force can be obtained using A-Lmethod.

3.2. Block system deformation analysis

Contact force computation precision directly affects the deformation analysis of block sys-tems, so a cantilever model is used to observe the influence of the modification of contactmethod on block system deformation analysis. As shown in Fig. 5(a), the cantilever with asize of l × h = 8.0m × 1.0m, is fixed at x = 0.01m and a vertical downward concentratedforce F is loaded at point A (7.9m, 0.99m). Under the effect of load F, σx, the normal stress atx direction, τ , the shear stress at y direction and f , the deflection between x = 0.01m ∼ 7.9m

77


(a) Configuration (b) sub-block cutting

F

m8

m1

)0,0(O x

y

A

Figure 5. Cantilever model.

in the cantilever can be calculated respectively as follows

σx = FyI

(l′ − x), τ = 32

Fbh

, f = Fx2

6EI(3l′ − x) (14)

where l′ = 7.89m, y is the distance from the concerned point to the neutral axis (y = 0.5m),b = 1m, I = bh3/12, E = 35GPa is the Young’s modulus. Some theoretical solutions of σx(at y = 1m, and so in the following), |τ | and |f | are shown in Table 2.

As in DDA with first order displacement approximation, each block has a constant stressand strain state. To use DDA to compute the deformation of the cantilever, the model is cutinto 1340 triangular sub-blocks as shown in Fig. 5(b) and plane stress assumption is used inthe simulation. Joint parameters between blocks are set as: friction angle ϕ = 70◦, cohesionc = 30MPa, tensile strength σt = 15MPa. Respectively using P method and A-L method, thedeformation of the cantilever is computed with several different penalty number p as 10×E,20× E, and 50× E.

Table 3 gives the simulation result of σx, |τ | at x = 4.0m and |f | at x = 7.9m, and theirerror (%, positive means the simulation result is larger than the theoretical solution and neg-ative means opposite, and so in the following) to corresponding theoretical solution. UsingP method, different selections of penalty numbers have small influence on the stress result,but the influence on deformation result is obvious. This phenomenon can be easily under-stood: no matter how much the deformation is, if only the block system is not broken, the

Table 2. Theoretical solutions of σx, |τ | and |f |.x(m) 1.0 2.0 3.0 4.0 5.0 6.0 7.0 7.9

σx/(× 105Pa) 2.070 1.770 1.470 1.170 0.870 0.570 0.270 −|τ |/(× 103Pa) 7.50 7.50 7.50 7.50 7.50 7.50 7.50 −|f |/mm 0.006 0.025 0.053 0.895 0.133 0.181 0.233 0.281

Table 3. DDA result of stress and deflection and the corresponding errors.

p 10× E 20× E 50× E

Contactmethod - σx |τ | |f | σx |τ | |f | σx |τ | |f |

PDDA result 1.143 7.353 0.309 1.144 7.356 0.293 1.144 7.358 0.283

error/% −2.31 −1.96 9.96 −2.22 −1.92 4.27 −2.22 −1.89 0.71

A-LDDA result 1.144 7.359 0.277 1.144 7.359 0.277 1.144 7.359 0.277

error/% −2.22 −1.88 −1.42 −2.22 −1.88 −1.42 −2.22 −1.88 −1.42

Note: σx/(× 105Pa), |τ |/(× 103Pa), |f |/mm

78


Table 4. DDA result of stress and deflection with p=10 ×E.

x 1.0m 2.0m 3.0m 4.0m 5.0m 6.0m 7.0m 7.9m

σx/(× 105Pa)P 2.018 1.738 1.423 1.143 0.858 0.541 0.269 −

A-L 2.021 1.740 1.425 1.144 0.864 0.549 0.272 −|τ |/ × 103Pa)

P 7.31 7.35 7.35 7.35 7.39 7.32 7.45 −A-L 7.30 7.36 7.36 7.36 7.36 7.35 7.45 −

|f |/mmP 0.007 0.027 0.059 0.099 0.147 0.20 0.256 0.309

A-L 0.006 0.024 0.042 0.089 0.131 0.179 0.230 0.277

whole block system is in mechanical balance, so the stress can be stable when the deforma-tion varies. With p = 10 × E, the error of deflection is as large as 10%, and which becomesmaller as penalty number value increases. Using A-L method, the stress and deflection errorsto theoretical solutions are quite small, as −2.22%, −1.88% and −1.42% respectively forσx, |τ | and |f |, and stable with the variety of the penalty number. So A-L method eliminatesthe influence of penalty number selection on the analysis of block system deformation, whichis obvious when P method is used.

Table 4 gives the stress and deflection result at some points in the cantilever respectivelyusing P method and A-L method with p = 10 × E. Fig. 6 is the σx and f curves of thecantilever. σx curves using P method and A-L method are close to each other and both havea little error to the theoretical curve. Using both contact methods, the deflection curves haveobvious error to the theoretical curve, but that using A-L method is much smaller than thatusing P method.

1 2 3 4 5 6 7 80.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

x/M

Pa

x/m

P method A-L method Theoretical

1 2 3 4 5 6 7 8-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

f/m

m

x/m


Figure 6. σx and f curve comparison.

Figure 7 is the time histories of σx at x = 4.0m and f at x = 7.9m, with comparison tocorresponding theoretical solutions. In Fig. 7(a), σx at y = −0.5m is also plotted, and it canbe found that the stress at the up side (y = 0.5m) and the down side (y = −0.5m) of thecantilever has good symmetry. It also can be found that the DDA stress and deflection resultfluctuate during early time steps, and which is more violent while using A-L method. Thisis also because A-L method is an iterative method for contact treatment, as mentioned inprevious contact force verification example.

The cantilever model result indicates that, using P method, different penalty number selec-tions have small influence on the simulation of block system stress distribution, but the block

79


0 600 1200 1800 2400 3000-0.20

-0.16

-0.12

-0.08

-0.04

0.00

0.04

0.08

0.12

0.16

0.20

x/M

Pa

time step


x = 0

0 600 1200 1800 2400 3000-0.50

-0.45

-0.40

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

f /m

m

time step


Figure 7. Time histories of σx and f .

system deformation analysis result will vary with different penalty number selections obvi-ously, and the increase of penalty number value is beneficial for the improvement of defor-mation analysis precision. Using A-L method, block system deformation analysis result willnot be influenced by different penalty number selections, and with a small penalty number,relatively high precise deformation result can be obtained.

4. Conclusions

Considering the influence of penalty number selection on DDA simulation result, AugmentedLagrangian Method is used to modify the penalty contact treatment method which is used inoriginal DDA. Verification examples indicate that, the modified contact algorithm eliminatesthe influence of penalty number selection on the computation of the shear contact force ateach contact position, and also the influence on the analysis of block system deformation.Using the new contact algorithm, quite precise contact force and block system deformationresult can be obtained with relatively small penalty number. The elimination of the influenceof penalty number selection on simulation result is beneficial for the improvement of theprecision of DDA result and its reliability in the application of engineering practice. Mean-while, as relatively small penalty number can be used, the efficiency of DDA computing canbe improved, and the possible ill condition of the system equilibrium equations caused bylarge penalty number can be avoided.

References

1. Gen-hua Shi. Discontinuous deformation analysis – a new numerical model for the statics anddynamics of block systems[D]. Berkeley: University of California, Berkeley, 1988.

2. Yeung, M.R. Application of Shi’s discontinuous deformation analysis to the study of rock behav-ior[D]. Berkeley: University of California, Berkeley, 1991.

3. Mary Magdalen MacLaughlin. Discontinuous deformation analysis of kinematics of landslides[D].Berkeley: University of California, Berkeley, 1997.

4. Lin C.T. Extensions to the discontinuous deformation analysis for jointed rock masses and otherblocky systems[D]. Boulder: University of Colorado, Boulder, 1995.

5. Chihsen T. Lin, Berbard Amadei, Joseph Jung, Jerry Dwyer. Extensions of discontinuous defor-mation analysis for jointed rock masses[J]. International Journal of Rock Mechanics and MiningSciences & Geomechanis, 1996, 33(7): 671–694.

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6. Amadei B., Lin C.T., Dwyer J. Recent extensions to the DDA method[A]. In: Proceedings of theFirst International Forum on Discontinuous Deformation Analysis (DDA) and Simulations ofDiscontinuous Media[C]. California, USA, 1996, 1–30.

7. Felippa C.A. Interactive procedures for improving penalty function solutions of algebraic sys-tems[J]. International Journal of Numerical Methods in Engineering, 1978(12): 821–836.

8. Felippa C.A. Penalty-function interactive procedures for mixed finite element formulations[J].International Journal of Numerical Methods in Engineering, 1986(22): 267–279.

9. Y.B. Bayram, H.F. Nied. Enriched fnite element-penalty function method for modeling interfacecracks with contact[J]. Engineering Fracture Mechanics, 2000(65): 541–557.

10. Harnau M., Konyukhov A., Schweizerhof K. Algorithmic aspects in large deformation contactanalysis using ‘solid-shell’ elements[J]. Computers and Structures, 2005, 83: 1804–1823.

11. Simo J.C, Laursen T.A. An augmented Lagrangian treatment of contact problems involving fric-tion. Computers and Structures[J], 1992, 42(1): 97–116.

12. Ning Youjun. Study on Dynamic and Failure Problems in DDA Method and its application[D].Beijing: Beijing Institute of Technology, 2008.

81

DDA for Dynamic Failure Problems and Its Application in RockBlasting Simulation

Y.J. NING1, 2, ∗, J. YANG2, G.W. MA1 AND P.W. CHEN2

1School of Civil and Environmental Engineering, Nanyang Technological University2State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081,China

1. Introduction

Discontinuous Deformation Analysis (DDA), put forward by Dr Genhua Shi,1 computesthe static and dynamic behaviours of discrete block systems. It gives a unique solution forlarge displacement and failure computations of block structures. Blocks of any shapes can beinvolved in a DDA block system, and block system kinematics is developed to make blocksfulfil the constraints of no tension and no penetration between each other. Meanwhile, theMohr-Coulomb criterion can also be applied between blocks.

Rock mass is a kind of natural geological material consists of continuous rock mediumand discontinuous joints, faults etc. As mathematical models, rock mass can be simplifiedas discrete block systems with joint strength considered between blocks. Therefore, the DDAmethod is applicable in the simulation of rock mass behaviours. Since its birth, DDA has beenwidely used in rock mechanics and rock engineering. However, in dynamic rock mechanicalproblems, for example, rock blasting, DDA hasn’t been applied too much. Mortazavi2 andNing et al.3 simulated rock blasting with DDA, but problems, which are of vital importancefor dynamics, such as stress wave propagation characteristics and non-reflecting boundaryconditions, joint strength reduction in block system failures, etc., were not taken into con-sideration. Due to that the numerical model did not exactly conform to the realities, thesimulated results should be further improved.

In this paper, the dynamic parameter in the DDA computation is adjusted to make it morereasonable. A viscous non-reflecting boundary condition is then implemented into DDA.Besides, sub-block DDA method to deal with crack problems considering both tensile andshear fracture is proposed. Based on the modified DDA, the blasting process in jointed rockmass is carried out by acting detonation pressure on expanding borehole wall and connectedcrack surfaces around. A case study of a horizontal column borehole cast blasting is con-ducted and discussed.

2. Two Problems in Dynamic DDA

2.1. Dynamic parameter adjustment

The only difference between static and dynamic computations in DDA is whether the blockvelocities of the previous time step are inherited to the next time step, proportionally orcompletely. A dynamic parameter pd (0 ≤ pd ≤ 1), defined as an inherited block velocityproportion, is used to control the attenuation of the block velocities in each time step. Forthe effect of system damping such as the mass damping and the modal damping, etc., resis-tances such as the third direction shear resistance,1 the air resistance, etc., and the impactforce between two blocks, the energy of block systems will be consumed in the process of




deformation and movement. In the current modified DDA method, frictions between twoblocks is the only way of energy consumption, so it is reasonable to use this dynamic para-meter to assume the block system energy artificially, except that the value of this parameteris hard to select reasonably.

In current DDA, pd is set as a constant in each computation instance for all the blocks inthe system without the consideration of the variety of the step time and the velocity differencebetween blocks. As the step time is adjusted according to the actual step displacement ratioand the convergence condition of block contact treatment in DDA computation, meanwhile,velocities of blocks in a system can be of a wide range, to attenuate the block velocities witha same proportion is unreasonable obviously. It is a fact that the energy consumed for blockimpact and air resistance is proportional to the velocities of the blocks.4, 5 Therefore, pd isadjusted to be decided using the following formula

pm+1di = 1− �

�0× vm

i

v0(1− pd0) (1)

where pd0 (0 < pd0 ≤ 1), �0 and v0 are constants given by users, pm+1di is the dynamic

parameter for block i in the time step m + 1, � is step time of the time step m + 1, vmi is

the final velocity of block i in the time step m. When vmi = 0, pm+1

di = 1 is obtained. When��0× vm

iv0

is large enough, pm+1di will become unreasonable as negative, so a minimum value

pdmin (0 < pdmin ≤ pd0) is given for pm+1di , when pm+1

di < pdmin, let pm+1di = pdmin.

The physical meaning of the above formula can be understood as when the current steptime is �0 and the velocity of block i is v0 at the end of the previous time step, the inheritedvelocity proportion for this block is pd0. In other words, pd for each block will interpolatebetween pdmin and 1 according to the step time change and the block own velocity. In com-plete static problems, pd is evaluated as zero directly for each block.

Moreover, for different block materials or blocks in different regions in a system, pd0, �0,v0 and pdmin can be evaluated with different values to represent the effect of different systemdamping or the third direction resistance levels.

2.2. Non-reflecting boundary condition

Although stress waves attenuate very rapidly while propagating in DDA block systems, stresswaves reflected at artificially cut boundaries can still be visible in dynamic problems in infi-nite or half-infinite regions, especially when the selected computation domain is relativelysmall.6 To reduce this unwanted reflection of stress waves, which may possibly influencethe correctness of the simulation, non-reflecting boundary conditions are considered in thecurrent DDA method.

In the viscous boundary condition, dampers are used to absorb the energy released fromthe system. In the DDA method, two pairs of perpendicular dampers7 are used at two endsof each non-reflecting block boundary. Assuming there exists a pair of dampers respectivelyin normal and shear directions of a boundary of block i at point (x,y), the damping forces inthese two directions are

fn = −ρCPvn, fs = −ρCsvs (2)

respectively, where ρ is the block density, vn and vs are normal and shear particle velocitiesat point (x,y) respectively, Cp and Cs are particle longitudinal wave velocity and transverse

84


wave velocity respectively. Then, the strain energy of the dampers is

� = − (un us) (fn

fs

)= − (ux uy

) ( nx ny−ny nx

)(fnfs

)

= ρl(ux uy

)( Cpn2x + Csn2

y (Cp − Cs)nxny

(Cp − Cs)nxny Csn2x + Cpn2

y

)(vxvy

)

= ρlDTi TT

i CiTiDi (3)

where l is length of the block non-reflecting boundary, (nx,ny) is the direction cosines of thenon-reflecting boundary, vx and vy are particle velocities at point (x,y) in x and y directionsrespectively, ux and uy are particle displacements at point (x,y) in x and y directions respec-tively, Di is the block deformation parameter matrix, Ti is the displacement transformation

matrix, and C =(

Cpn2x + Csn2

y (Cp − Cs)nxny

(Cp − Cs)nxny Csn2x + Cpn2

y

)is called the wave velocity sub-matrix. As

Di = Di−Di0�= Di

�

1, then the strain energy can be expressed as

� = ρlDTi TT

i CTiDi

�(4)

where� is step time. According to the minimum potential energy principle, a 6×6 sub-matrixis obtained and added to the system coefficient matrix,

2ρlTTi CTi

�→ Kii (5)

3. Sub-Block DDA Method for Failure Problems

In the DDA method, rigid body movement and deformation occur in each single blockwhile opening and sliding occur between blocks. By cutting a continuous domain into manysub-blocks with strong joint strength used between the sub-blocks, a method called sub-block DDA can compute the stress distribution and deformation of continuous mediumprecisely.6 This sub-block method can be used to simulate crack problems while reasonablejoint strength is applied to the artificial joints between sub-blocks. In this method, cracksinitiate and propagate along pre-set artificial joints, and the crack coalescence process canbe naturally considered and actualized without any difficulties encountered. Although thecrack route will definitely be affected by the distribution of the pre-set artificial joints, bycutting the continuous domain into smaller sub-blocks, this unwanted influence can becomeinvisible.

For crack problems in which the actual crack route is not strictly required, the sub-blockDDA method can be used. Both tensile and shear failures are considered in this method, withthe maximum tensile strength criterion and Mohr-Coulomb criterion used respectively,

Maximum tensile strength criterion,

σ = σt (6)

Mohr-Coulomb criterion,

τ = c+ σ tanϕ (7)

where σ and τ are the normal and shear stress on the artificial joint surface respectively, σt , c,ϕ are the tensile strength, cohesion and friction angle of the block material respectively. Once

85


any of the two criterions is fulfilled for an artificial joint, crack will initiate or propagatealong the joint, and the joint strength is to be reduced as real joint strength.

Failure simulation of rock samples indicates that the sub-block DDA method can simulatecrack problems reasonably well. Besides providing a new technique for crack simulation,by comparing the simulation results of rock sample failure with experimental results, theability and correctness of DDA method in failure treatment of block systems have also beenverified.6

4. Simulation of Rock Blasting Using DDA

4.1. Modified DDA method

Rock blasting is a complex dynamic failure process, in which the rock mass may be fracturedand cast under the effect of explosion shock waves and explosion product pressure. In jointedrock mass, stress waves attenuate rapidly for being reflected by joint planes, so the explosionproduct pressure plays a dominating role in rock mass fragmentation and rock block cast.For rock blasting simulation, continuous numerical methods are usually used to simulate thedamage evolution of rock mass so as to study the expansion of boreholes and the blastingfragment size.8, 9 For the simulation of rock mass fracture, rock block cast and the formationof the blasting pile, discontinuous numerical methods are applicable.

For the rapid attenuation of stress waves while propagating in DDA block systems, DDAmethod can only be used in the simulation of jointed rock mass blasting, but not in contin-uous rock mass blasting. As jointed rock mass is simulated, the effect of explosion productpressure should be considered primarily. Jointed rock mass mentioned here, not only includesnatural jointed rock mass, in which the existing joints will be relaxed by explosion shockwaves, but also includes damaged continuous rock mass, in which joints are newly gener-ated under the effect of explosion shock waves. Therefore, joints defined in a DDA blastingmodel can be natural joints, and also can be newly generated joints after the effect of explo-sion shock waves, or both included. The mechanical parameters of these joints are selectedvariably according to actual conditions. In the model, joints are defined as artificial jointsbefore the failure criterion is fulfilled. After failure, they will be transformed as real joints (orcracks) and corresponding strength is reduced. Joints with strength below failure criterion inthe initial model are set as cracks at the beginning.

Assuming that explosion product will propagate along joints with a constant velocity VPonly after the joints have become cracks, by tracking borehole expansion and the propaga-tion and coalescence of cracks in surrounding rock mass, and loading the explosion productpressure on the borehole wall and the connected crack surfaces, a modified DDA methodto simulate the blasting process only considering the effect of explosion product pressure injointed rock mass is developed.

At the beginning of each time step, starting from all the ends of the block boundaries onthe borehole wall, all connected cracks can be searched round by round according to crackconnectivity.6 The instant explosion product pressure in the chamber is then calculated basedon the detonation pressure equation of state10:

P = P0

(V0

V

)γ(8)

where P0 and V0 are initial chamber pressure and chamber volume respectively, P and V arethe chamber pressure and chamber volume at time t respectively, γ is a constant relative tothe properties of the charged explosive and the rock mass. The chamber volume is calculated

86


approximately as expressed in the following section, while the chamber area in plane iscalculated using simplex integration1 precisely.

4.2. Horizontal column borehole cast blasting simulation

Figure 1 is an explosion chamber expanding sketch of a column borehole looked in the axisdirection in rock blasting, where R0 is the borehole initial radius, RP is the propagatingradius of the explosion product at time t. The explosion chamber volume consists of theexpanded borehole volume and the connected crack volume within RP. Taking the planestress assumption, the chamber has a unit length depth in the axis direction, so the chambervolume is equal to the chamber planar area in quantity.

The computed rock blasting model with one column borehole considered is shown inFig. 2. The whole model has a size of 15m × 5m. The left, right and lower boundaries arenon-reflection boundaries and the upper boundary is free surface. The borehole radius isR0 = 0.1m. The distance from the borehole centre to the upper free surface is w = 3m.The rock medium is cut into triangular rock blocks while subdivision is done around theborehole. The subdivision can be understood as the fragmentation of rock mass under theeffect of the explosion waves previously. Taking the rock mass gravity into consideration,this model can be seen as a horizontal column borehole cast blasting.

The initial chamber pressure is assumed as P0 = 1GPa, and select VP = 100m/s, γ =1.4.The density, Young’s modulus, Poisson’s ratio of the rock mass are selected as

Chamber

massRock

massRock

cracksdUnconnecte

cracksConnected0R

PR

Figure 1. Chamber expanding sketch.

B

B orehole

Figure 2. Horizontal column borehole blasting model.

87


Table 1. Joint mechanical parameters.

Joint type Friction angle ϕ/◦ Cohesion c/MPa Tensile strength σt/MPa

Artificial joint 45 2 0.5Real joint 45 0.05 0

0 5 10 15 20 25 30

0

1

2

3

4

5

Cha

mbe

r vol

ume

/ m3

Time / ms 0 5 10 15 20 25 30

0.0

0.2

0.4

0.6

0.8

1.0

Cha

mbe

r pre

ssur

e / G

Pa

Time / ms(a) Chamber volume (b) Chamber pressure

Figure 3. Chamber volume and chamber pressure histories.

ρ = 2600Kg/m3, E = 55GPa, μ = 0.3 respectively. The joint mechanical parameters areshown in Table 1.

Figure 3 gives the chamber volume expanding history and the chamber pressure atten-uating history. Under the effect of the explosion product pressure, the chamber begins toexpand, and the failure of the rock mass around the chamber makes it easier for the chamberto expand further, so the chamber seems expanding faster and faster as time increases. Thechamber pressure attenuates very quickly at the beginning. After about 5 ms, the attenuat-ing speed becomes very much smaller. At 26.57 ms, the explosion product expands to theupper free surface and leaks, so the pressure drops to zero and its loading stops. At thattime, the chamber volume is 4.073 m3, which is 131.4 times of the initial borehole volume(0.0031 m3), and the chamber pressure decreases from 1 GPa to 1.04 MPa correspondingly.

Figure 4 gives the vertical velocity history of the block located right beyond the initialborehole centre near the free surface (indicated as B in Fig. 2) and the corresponding dis-placement history of this block. Under the effect of the chamber pressure, the block is thrownup quickly and reaches a velocity of 31.23m/s at 26.99ms, and then the velocity begins todrop and becomes negative gradually. At 2.065s, the block falls back to the block pile, andfinally it gets still. At 0.740s, the block gets the peak height as 5.21m.

Figure 5 shows the simulated blasting process (in which the red joints represent wherethe explosion product has propagated, block colours represent the block maximum principalstress levels, and the large blocks at both sides are used to keep the blasting pile). Underthe effect of the explosion product pressure, cracks initiate in the borehole surrounding rockmass. The propagation of the explosion product along the cracks then leads to the furtherfailure. Figure 5(a) is the explosion product propagation situation at 0.021s. There are manycracks generated in the rock mass, and a visible bulge can be seen from the upper free surface.The rock mass continues to be fractured and thrown up under inertia effect after the leak ofthe explosion product. Figure 5(c) is the rock mass failure and block flying state when the

88


-0.5 0. 0 0.5 1. 0 1.5 2. 0 2.5 3. 0 3.5 4. 0-10

-5

0

5

10

15

20

25

30

35V

elo

city

/(m

/s)

Ti me/ s

-0.5 0. 0 0.5 1. 0 1.5 2.0 2.5 3.0 3.5 4.0-1

0

1

2

3

4

5

6

Dis

pla

cem

ent

/m

Time / s

(a) Block velocity (b) Block displacement

Figure 4. Block velocity and displacement histories.

(a) t =0.021s (b) t =0.152s

(c) t =0.740s (d) t =1.326s

(e) t =2.006s (f) t =3.618s

Figure 5. Simulated cast blasting process.

89


blocks get the peak cast height. After that, blocks begin to fall back under gravity effect, andalmost get still at 3.618s. The peak height of the block pile is 1.51m, distributed in a rangeof 16.71m in horizontal direction. Visible crater shape can be seen in the blasting pile. Itsradius is r = 5.48m, height is H = 4.08m, and visible depth is h = 1.63m. The blastingacting index10 is

n = rw= 1.827 > 1 (9)

According to the classification of crater blasting, this is a strong cast blasting.

5. Remarks

In this paper, for a better application of DDA in dynamic failure problems such as rock blast-ing, problems including dynamic parameter selection, non-reflecting boundary condition,crack and fragmentation treatment, are studied firstly. Based on these fundamental works,DDA method to simulate rock blasting process is studied and verified thereafter.

The adjustment of the dynamic parameter selection makes the energy assumption indynamic DDA more properly. With non-reflecting boundary condition, the reflected stresswaves at artificially cut boundaries can be absorbed to some extent, so a relative small com-putation domain can be involved in the simulation of dynamic problems in infinite or half-infinite regions. Using the failure computation way of sub-block DDA method, by actingdetonation pressure on borehole walls and connected crack surfaces around, DDA methodto simulate the blasting process in jointed rock mass is developed. The simulation result of ablasting case indicates that the blasting process in jointed rock mass can be duplicated wellwith this newly developed DDA method.

References

1. Gen-hua Shi. Discontinuous Deformation Analysis: A New Numerical Model for the Statics andDynamics of Block Systems[D]. Berkeley: University of California, Berkeley, 1988.

2. A. Mortazavi, P. D. Katsabanis. Modelling burden size and strata dip effects on the surface blastingprocess[J]. International Journal of Rock Mechanics and Mining Sciences, 2001, (38): 481–498.

3. Ning You-jun, Yang Jun. Numerical Simulation of the Blasting Process in Bedded and JointedRock Mass with 2D-DDA Method[A]. In: Proceedings of the 8th International Symposia on RockFragmentation by Blasting[C]. Santiago, Chile, 2006, 119–123.

4. C. Y. Koo, J. C. Chern. Modification of the DDA method for rigid block problems[J]. InternationalJournal of Rock Mechanics and Mining Sciences, 1998, 35(6): 683–693.

5. Chen Guangqi. Numerical Modeling of Rock Fall Using Extended DDA[J], Chinese Journal ofRock Mechanics and Engineering, 2003, 22(6), 926–931.

6. Ning Youjun. Study on Dynamic and Failure Problems in DDA Method and its application[D].Beijing: Beijing Institute of Technology, 2008.

7. Y. Y. Jiao, X. L. Zhang, J. Zhao, Q. S. Liu. Viscous boundary of DDA for modeling stress wavepropagation in jointed rock[J]. International Journal of Rock Mechanics and Mining Sciences,2007, 47: 1070–1076.

8. Chengqing Wu, Hong Hao. Numerical prediction of rock mass damage due to accidental explo-sions in an underground ammunition storage chamber[J]. Shock Waves, 2006, 15(1): 43–54.

9. Zheming Zhu, Heping Xie, Bibhu Mohanty. Numerical investigation of blasting-induced damagein cylindrical rocks[J]. International Journal of Rock Mechanics and Mining Sciences, 2007, doi:10. 1016/j. ijrmms. 2007.04.012.

10. J. Henrych. The Dynamics of explosion and its use[M]. New York: Elsevier Scientific PublishingCompany, 1979.

90

Study on Roof Caving Problem with DDA Method

LIU YONG-QIAN, YANG JUN∗, CHEN PENG-WAN AND NING YOU-JUN

Stake Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081,China

1. Introduction

Discontinuous Deformation Analysis(DDA) is one of the most effective numerical algorithmsfor rock mass behaviours, both static and dynamic analysis of a block system. With the helpof DDA, one could get solution of large deformation and large displacement. It is good atanalyzing block system interactive behaviours, no matter the blocks’ shape and scale, eventhe hollow ones.1, 2 There are lots of cracks in rock stratums and coal seams, when coal seamsexcavated, the rock stratums produced more cracks which coupled with the original ones,rebuilt a new fracture network continuously. In the excavating process, the roof above thegoaf bended, separated, broke and collapsed inevitably at last.3 DDA method is a predom-inant algorithm for dealing with coal seam excavation. In China, some scholars have usedDDA method studying coal mine exploitation problems, for example, WU Hong-ci4 and JUYang5 simulated the roof caving and analyzed the effect of faults in coal seams. Some peoplehave renewed DDA program for mine engineering and hydraulic engineering. This is a tryto simulate coal roof caving synchronously with excavating in the first weighting. Discussedthe crushing phenomena and a comparative study on displacement, velocity and stress in theroof caving.

2. Basic Theory of DDA Method1

In Ref. 1, SHI introduced the DDA algorithm and its theoretical foundation, principle ofminimum potential energy in classical mechanics, study the block contact, displacement,deformation and breakage in a block system.

2.1. Displacements and deformations of blocks

The displacement function for each block is equivalent to the complete first order approxi-mations of displacements. For each step, a block has constant stresses and constant strains,the displacement (u,v) of any point P(x,y) of a block can be represented by six displacementvariables

Di =(u0 v0 r0 εx εy γxy

)T (1)

(u0,v0) is the rigid body translation of a specific point P0(x0,y0) of within the block; r0 isthe rotation angle of the block with the rotation centre at P0(x0,y0). The unit of angle r0 isgiven in radians. εx, εy and γxy are the normal and shear strains of the block. Summing the




displacement matrices and simplifying, the block deformation matrix is obtained

(uv

)=

⎛⎜⎜⎝

1 0 −(y− y0) (x− x0) 012

(y− y0)

0 1 (x− x0) 0 (y− y0)12

(x− x0)

⎞⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎝

u0v0r0εxεyγxy

⎞⎟⎟⎟⎟⎟⎟⎠

(2)

2.2. Simultaneous equilibrium equations of block system

Individual blocks are connected and form a block system by contacts between blocks and bydisplacement constraints on single blocks. Assuming there are n blocks in the defined blocksystem, the simultaneous equilibrium equations are obtained⎛

⎜⎜⎜⎜⎜⎝

K11 K12 K13 · · · K1nK21 K22 K23 · · · K2nK31 K32 K33 · · · K3n

......

.... . .

...Kn1 Kn2 Kn3 · · · Knn

⎞⎟⎟⎟⎟⎟⎠

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

D1D2D3...

Dn

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

F1F2F3...

Fn

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

(3)

Each element Kij in the coefficient matrix given by Equation (3) is a 6 × 6 suBmatrix,Diψ and Fiψ are 6 × 1 submatrices, where Di is the deformation variable of block i; andFi represents the loading on block i distributed to the six deformation variables. SuBmatrix[Kii] depends on the material properties of block i and [Kij], where iψ �= ψj is defined by thecontacts between block i and block j.ψ

2.3. Blocks contact and its judgments

When DDA program running, there are no tension and no penetration can be seen betweenblocks even when the displacements and the deformations are large. Judge the contactbetween blocks by the penalty function, put the constrained inequality into the global simul-taneous equations, correct the contact positions though adding or deleting springs and judg-ing by the reference lines step by step.6 In a plan, there are three types of contact (see Fig. 1),angle- angle, angle-edge and edge-edge.

Gen-hua SHI used the penalty function algorithm judging the contact conditions, this algo-rithm has its own strongpoint, for example, the global simultaneous equations are constant,and it is easy to get the result though Summing the contact stiffness matrices simply. However,

Block i

block jblock jblock j

Block iBlock i

Figure 1. Three types of contact in a plan.

92


the penalty function algorithm has some shortcomings.7 Firstly, the accuracy of solution forcontact computing rests with penalty value, but it’s barely getting the appropriate value fore-handed. Secondly, for the contact constraint conditions, the penalty function algorithm canonly satisfy the accuracy approximately. Finally, contact forces have to compute separately.Because of the reasons above, Reference 7 used Augmented Lagrangian Method (ALM) tojudge the blocks contact conditions, while Reference 8 used joint element method.

3. Roof Breakages Analyzing in Excavation

It is a general rule for roof caving and appearance of the mine pressure in different stages,from starting cut mining to the first weighting, to the periodic weighting.9 Before the firstweighting, controlled by the gravity of rock stratums above, the roof above the goaf wasbecoming curve in vertical, the deformation of roof increased with the coal seam excavating.From starting cut to workface, when the length of unsupported roof came to the limited span,the roof would break down and lost its natural continuity. Because of compressive stressturning to tensile stress in parts, fractures in rock stratums increased. When rock stratumscollapsed, the blocks formed a interpenetrating skeleton structure, it’s a equilibrium structurein mechanics.10, 11 Once the workface advanced, the mine pressure appeared, and the roofcollapsed instability, it is the first weighting of roof.

It was the compound effect of overlying strata pressure function for overburden failure.The collapsed height, fracture dip angle and span of roof were induced by the occurrence stateof joints in rock. It was agreed by the verifying experiment,12 the original joints in horizontalinduced spalling and slipping layer, and the original joints in vertical induced rock stratumsslipping. Moreover, References 13 and 14 insisted that, the mechanic property parameter ofjoints controlled the distribution of new cracks and rock caving, even the broken fragmentof the roof.

4. Simulation of Roof Caving in Excavating

4.1. Numerical model and the postulate

This was a simplified model from F16−17 coal seam excavation of 5th coal mine of Pingding-shan mining district, Henan province, China. The buried depth of F16−17 coal seam wasD = 550m, and the average mining height was h = 4m. Assuming the length of the modelwas L = 200m, and the height was H = 90m; The uniform loading was applied on the topof the model, the strength of the uniform loading was σy = 13MPa; The initial horizontalstress σx = 3MPa, and the shear stress τ = 0MPa; There were two groups of joints in therock stratums, the horizontal and the vertical, the parameters of joints in stratums was listedon Table 1; The parameters of rock and coal was listed on Table 2, and the model was fixedby 7 fixed points, point A, B, C, D, E, F and point G (Fig. 2). Set 9 measured points in therock stratums, point 1st, point 2nd, . . . point 9th (see Fig. 2). The velocity of excavating was2 m/s.

Table 1. Parameters of joints in strata.

Friction Angle/◦ Poisson ratio Cohesive force/MPa Tensile strength/MPa

Horizontal joints 12.00 0.23 1.80 2.00Vertical joints 15.00 0.21 1.20 1.40

93


Table 2. Parameters of rock and coal.

Density/kg/m3 Friction angle/◦ Elastic modulus/MPa Poisson ratio

Rock 2.6× 103 23 4.5× 103 0.22Coal 1.4× 103 21 1.2× 103 0.24

G

F

EDC

B

Uniformly distributed load Q

rock stratum

Figure 2. Analysis model for overlying stratums.

4.2. Simulated result and analysis

4.2.1. Displacement analysis

In the course of excavating, the development trends of the roof changes were rock bodycaving and the micro-rotation of some bocks along the axis in vertical (Figs. 3b and c). Thedistinct differences in horizontal displacement were not turned up until the groups of bed sep-arations increased (Fig. 3d, e). Compared with the vertical, the horizontal displacement wasremarkable (Fig. 4). In the vertical, the longer distance from the coal seam, the smaller dis-placement (Fig. 5). The displacement of measured point 3rd was 4m, while the displacementof point 5th was 0.41m. Because of excavating, the caving-rock-head was not a symmetricalbanded structure, and the axis in vertical was moving from time to time (Fig. 3d). So it wasforming an inverted arch structure, the displacements were declining from the axis to bothsides (Figs. 3f, g and h). The layer separation zone was formed progressively.

4.2.2. Velocity analysis

More excavating, more layer crannies increasing. When the distance from staring cut to theworkface long enough, coming with the excavation, the layer crannies became compacted,the velocities of measured points changed obviously (Fig. 6). The velocities have increasedrapidly since excavation went though the points. In general, the velocities in vertical compo-nent were declining from the bottom rock stratum to the top (Fig. 7)

4.2.3. Stress analysis

During the excavating time, the stress both in horizontal and in vertical increased gradually,more excavating, the stronger. However, in the collapse area, the stress in vertical increaseddeclined more quickly than the horizontal stress once the excavation went though. Whenrock stratums collapsed, the blocks formed a interpenetrating skeleton structure, the kinetic

94


a. t= 1.0s b. t=14.4s

c. t=28.2s d. t=35.5s

e. t=46.3s f. t= 60.4s

g. t=90.5s h. t=116.7s

Figure 3. The process of roof caving.

energy was changed into elastic potential energy, which increased the horizontal stress in rock(Fig. 8). According to the theory of displacement-shear stress in mining field, once excavatingin stress concentration region, the stress in vertical would increased times,15 and when themeasured points coming to the goaf, the stress would declined at once, that’s the reason forthe change of stress in vertical.

The roof caving caused the stress declining in the middle part (Figs. 3d–h), and then theoverlying strata pressure rebuilt the mechanical equilibrium, the stress recovered gradually.In horizontal, the stress steadily increased from the middle to both sides (Fig. 9).

95


-10 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

Velo

city

/m/s

time/s

horizontalvertical

Figure 4. Displacement comparing between in horizontal (x) and vertical component (y) (point 7th).

0 2 0 4 0 60 80 100 120 1 4 0

-4

-3

-2

-1

0

disp

lace

men

t/m

point 7point 6point 9

point 3

time/s

Figure 5. Displacement comparing graph of some measuring points in different level courses.

-10 0 10 20 30 40 50 60 70 80 90 100 110 120

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

horizontalvertical

Velo

city

/m/s

time/s

Figure 6. Velocity and histories curve of measured point 2nd.

5. Conlusions

• DDA is a good method to deal with the underground excavating problems.• Both the displacements and velocities in vertical are declined from the caving-rock-

head to both sides during the first weighting.• In the collapse area, the stress in vertical decreased more quickly than in horizontal,

and the horizontal stress steadily increased from the middle to both sides.• The mechanics Parameters of joints in rock controlled roof caving and subsidence

area.

96


-10 0 10 20 30 40 50 60 70 80 90 100

-0.4

-0.3

-0.2

-0.1

0.0

disp

lace

men

t/m/s

time/s


Figure 7. Comparing graph of the vertical components of measuring point velocities.

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

-50

-40

-30

-20

-10

0

time/s

stre

ss/M

Pa

horizontalvertical

Figure 8. The curve of stress and histories of measured point 2nd.

-10 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150-16

-14

-12

-10

-8

-6

-4

-2

0


point 1

time/s

stre

ss/M

Pa

Figure 9. Comparing graph of the horizontal components of measured points stress.

• For DDA computing efficiency and precision, some questions have not been resolved,for example, spring stiffness selecting, it need to optimize the source routine of DDA.

Acknowledgements

This research is supported by Project 985 funds of BIT and Fundamental Research funds ofBIT (BIT_RDC_20060242004). The authors are grateful for the financial support.

97


References

1. SHI Gen-hua. Numerical Manifold method and Discontinuous Deformation Analysis[M]. PEI Jue-min Translator. Beijing: Tsinghua University Press, 1997, 92–167. (in Chinese).

2. SHI G H. Discontinuous Deformation Analysis: A New Numerical Model for The Statics andDynamics of Block Systems [D]. Berkeley: Department of Civil Engineering, University of Califor-nia, 1988.

3. LIU Yong-qian. Study on Gas Migration for Close Quarters Upper Protective Coal Seam Exploita-tion in No.5 Mine, Pingdingshan [D]. Jiaozuo: Henan Polytechnic University, 2007, 23–70 (inChinese).

4. WU Hong-ci, ZHANG Xiao-bin, BAO Tai, et al. Dynamic Modeling for Movement Behavior ofMined Overlying Strata Using Discontinuous Deformation Analysis Method [J]. Journal of CoalScience & Engineering (China), 2001, 26(5): 486–491 (in Chinese).

5. JU Yang, ZUO Jian-ping, SONG Zhen-duo, et al. Numerical Simulation of Stress Distribution andDisplacement of Rock Strata of Coal Mines by Means of DDA Method [J]. Chinese Journal ofGeotechnical Engineering, 2007, 29(2): 268–273 (in Chinese).

6. LIU Jun, LI Zhong-kui. Current Situation and Development of DDA Method [J]. Chinese Journalof Rock Mechanics and Engineering, 2004, 23(5): 839–845 (in Chinese).

7. Amadei B Lin Chihsen Jerry Dwyer. Recent extensions to the DDA method [A]. In: Proc. of theFirst International Forum on Discontinuous Deformation Analysis (DDA) and Simulations of Dis-continuous Media [C]. Albuquerque TSI Press 1996, 5–18.

8. Cheng Y M. Advances and Improvements in Discontinuous Deformation Analysis [J]. Computersand Geotechnics, 1998, 22(2), 153–163.

9. MENG Zhao-ping, SU Yong-hua. Theory and Method of Sedimentary Rock Mass Mechanics [M].Beijing: Science Press, 2006, 188–227 (in Chinese).

10. QIAN Ming-gao, XU Jia-lin. Study on the “O-shape” Circle Distribution Characteristics ofMining-induced Fractures in the Overlaying Strata. Journal of Coal Science & Engineering(China), 1998, 23(5): 466–469 (in Chinese).

11. LI Shu-gang. Movement of the Surrounding Rock and Gas Delivery in Fully-Mechanized TopCoal Caving[M]. Xuzhou: China University of Mining and Technology Press, 2000, 74–183 (inChinese).

12. YU Guang-ming, LI Bao. Study on the Effect of Joints on the Boundary of Mining Covered RockBreak[J]. Journal of Liaoning Technical University (nature science edition), 1998, 17(1): 22–26 (inChinese).

13. CAI Mei-feng, HE Man-chao, LIU Dong-yan. Rock Mechanics and Engineering[M]. Beijing: Sci-ence Press, 2002, 79–125 (in Chinese).

14. XIE He-ping. Damage Mechanics of Rocks and Concrete[M]. Xuzhou: China University of Miningand Technology Press, 1990, 105–181 (in Chinese).

15. ZHANG Jing-cai, ZHANG Yu-zhuo, LIU Tian-quan. Rock Seepage and Water Inrush in MineStratum[M].Beijing: Geological Publishing House, 1997, 6–14 (in Chinese).

98

Indeterminacy of the Vertex-vertex Contact in the 2DDiscontinuous Deformation Analysis

H.R. BAO AND Z.Y. ZHAO∗

School of Civil and Environmental Engineering, Nanyang Technological University, Singapore 639798,Singapore

1. Introduction

In the past two decades, many developments have been carried out to improve the perfor-mance of the discontinuous deformation analysis (DDA).1 Most of the improvements arefocused on reducing the rigid body rotation error2–4 and refining the stress distributioninside the blocks.5–11 However, little work had been done on the contact problem in the2-dimensional DDA. Among them, Lin8 improved block contact by using the augmentedLagrangian method instead of the penalty method employed in the DDA.

In the DDA, polygons are basic geometric entities and they will cause nonsmooth con-tact. For two-dimensional polygons, only three types of contacts exist: vertex-vertex (V-V),vertex-edge (V-E), and edge-edge (E-E). The vertex-edge contact is the basic type and theedge-edge contact can be decomposed as the combination of two vertex-edge contacts whilethe vertex-vertex contact can degenerate into a vertex-edge contact. But the procedure ofthe degeneration is indeterminate in numerical modelling. The indeterminacy is due to thediscontinuity of the normal direction of the contact boundary, i.e. the non-unique directionof the contact force at the corner.

There are two types of indeterminacy exist in the vertex-vertex contact model. The firsttype of indeterminacy is referred as the genuine indeterminacy (GI) in this paper becauseit originates from the real vertex-vertex contact. When the trajectory of the vertex of thecontact pair passes its target vertex, the genuine indeterminacy will arise. The second typeof indeterminacy is referred as the pseudo indeterminacy (PI) in this paper because it is aphenomenon belongs to particular numerical modelling methods when dealing with the quasivertex-vertex contact (i.e. the contact detected to be a vertex-vertex contact at the beginningof the time step).

In the DDA, if two possible reference edges are passed by the corresponding vertex simul-taneously, invasion takes place, as shown in Fig. 1b, where the normal penetration distancesare d1 and d2, respectively. The choice of contact edge is controlled by the shortest pathmethod, which picks the one with smaller penetration among two potential reference edges.If d1 < d2, the entrance edge is DE and the final vertex-edge contact is A-DE, and a contactspring is attached between vertex A and the projecting point A1 on the reference edge, other-wise, the entrance edge is DF and contact spring is attached between A and A2. The physicalmeaning of the shortest path method is pushing the invaded vertex A out of the block alongthe shortest path.12 However, the shortest path method has two obvious shortcomings: (1)it cannot work when the initial state of a quasi vertex-vertex contact does not have penetra-tion; (2) it cannot resolve the case when the penetration distances from both reference edgesare equal.

The vertex-vertex contact indeterminacy is a well-known problem in numerical modellingand many heuristic works have been proposed to solve it. Oden13 and Chaudhary14 used the




A1 A2

d2d1

A

C B

E

F

DD D

A A

(a) (b) (c)

Figure 1. Working principle of the shortest path method: (a) quasi V-V contact case 1; (b) overlappedat the end of time interval; (c) quasi V-V contact case 2.

minimum principle to select a unique trajectory for the vertex, excepting the case in whichthe possible scattered trajectories are indistinguishable by symmetry. Pandolfi15 and Kane16

provided the variational principle for selection of trajectories in problems where multiple tra-jectories of vertex are possible, but with the same limitation as the minimum principle. Fengand Owen17 used an energy-based normal contact model in which the normal and tangentialdirections, magnitude and reference contact position of the normal contact force are uniquelydefined. Cundall18 employed a corner rounding procedure so that blocks can smoothly slidepast one another when two opposing corners interact. Krishnasamy and Jakiela19 provided asimple scheme to resolve the ambiguity of vertex-vertex contact in the penalty based model.Most of the above works have their own scheme dealing with the indeterminacy of vertex-vertex contact in the framework of specific contact interaction dealing method.

To overcome the limitations of the shortest path method, the original DDA code providesa simple scheme. For the quasi vertex-vertex contact without initial penetration at the begin-ning of time interval, the code specifies the initial entrance edge which is the steepest amongthe two potential reference edges. This scheme is proper in some cases but quite sensitive tothe choice of stiffness of the contact spring and time interval size. Sometimes, the artificialchoice of an entrance edge will affect the result of the analysis significantly. The original DDAcode will not be able to deal with the case when two penetration distances from the potentialreference edges are equal. To make the shortest path method employed in the DDA moreprecise and more applicable, an enhancement to it is provided in this paper. More detailsabout this enhancement are introduced in the following sections.

2. Solution to the Genuine Indeterminacy

In the genuine indeterminacy, two blocks collide at the vertexes but the direction of contactforce cannot be obtained by the normal vector of the boundary. The new scheme proposedin this section is based on the fact that there must have contact forces interact betweentwo vertexes when they collide into each other and the contact forces will direct the furthermotion of them. Hence, a temporary contact spring between vertexes is necessary for eachvertex-vertex contact pair to reflect the contact interaction. The temporary contact springwill connect two vertexes and act as a hinge between them. The ‘temporary’ means it willbe removed right after the further motions are detected. This procedure will not affect thecomputational efficiency much because it is done in a normal open-close iteration.

100


r1 r2

u

¦ È1 ¦ È2

P1

P2

P1'

block i

block j

r1

r2

u

P1 ¦ È2¦ È1'

P2

P1'

block i block j

¦ È1

(a) (b)

Figure 2. Relative displacement vector and potential reference edge vectors: (a) two potential referenceedge on the same block (b) two potential reference edge on different block.

Consider the motion of a vertex-vertex contact pair, vertex P1(x1, y1) on block i and vertexP2(x2, y2) on block j, with a temporary contact spring attached between them. At the end ofthe time interval, there will have a relative displacement between point P1 and P2, as shownin Fig. 2. The relative displacements of P1 with respect to P2 is

u = u1 − u2 = Ti(1)di − Tj(2)dj (1)

here, for simplicity, use Ti(1) for Ti(x1,y1) and Tj(2) for Tj(x2,y2). Assume that the vectors forthe potential reference edge are r1 and r2, respectively, with origin at point P2. Denote theangle between r1 and u as θ1, the angle between r2 and u as θ2, then

sin θ1 = |r1 × u||r1||u| ; sin θ2 = |r2 × u|

|r2||u| (2)

By comparing the value of sin θ1 and sin θ2, it is easy to find the proper reference edge ontowhich the vertex will slide when removing the vertex-vertex contact spring. If sin θ1 < sin θ2,the proper reference edge is r1, otherwise it is r2. If sin θ1 = sin θ2, the vertex-vertex contactwill not degenerate into corresponding vertex-edge contact and the temporary vertex-vertexcontact spring is kept in this step until the symmetry is destroyed in the following steps.Indeed, the above selection procedure is another form of the shortest path method because|u| sin θ is the penetration distance.

3. Solution to the Pseudo Indeterminacy

For the pseudo indeterminacy, since the shortest path method is sensitive to the penaltyparameter and the time step size, an alternative scheme is proposed here by considering thetrajectories of vertexes during the time interval. For any vertex-vertex contact which does nothave any penetration at the start of time interval, the new scheme can work out the properentrance edge for it unless it falls into the genuine indeterminacy.

The new scheme is based on the fact that no block can jump to the final positionfrom the start position. Hence, each vertex must have its own trajectory which must intersectwith the boundary of the target block if the vertex invades into the target block. It is easyto find the proper entrance edge of this vertex from the intersection point. If the intersection

101


P2

P3

P1

P3'P1'

P2'

Figure 3. Vertex-edge contact process.

point is also a vertex on the opposite block, this vertex-vertex indeterminacy falls into thegenuine indeterminacy which can be resolved by the enhanced shortest path method providedin the previous section.

Figure 3 shows a vertex-edge case. Points P1(x1, y1), P2(x2, y2), and P3(x3, y3) correspondto the start positions of time interval, and points P′1, P′2 and P′3 are corresponding to theend positions of that time interval. The displacement vectors of P1, P2, and P3 at this timeinterval are {u1,v1}T, {u2,v2}T, and {u3,v3}T, respectively.

Then the coordinates of P1, P2, and P3 at any time instant within current time step, can bewritten out by a linear interpolation as follows:

x1(t) = u1

Tt+ x1

y1(t) = v1

Tt+ y1

(3)

with the other coefficients obtained by a cyclic permutation of subscripts in the order 1, 2, 3.Here, T is the time step size which must be small enough to ensure the linear displacementassumption. Denote �(t) as two times of the area of �P1P2P3 at time instant t, obtained by

�(t) =∣∣∣∣∣∣1 x1(t) y1(t)1 x2(t) y2(t)1 x3(t) y3(t)

∣∣∣∣∣∣ =∣∣∣∣x2(t)− x1(t) y2(t)− y1(t)x3(t)− x1(t) y3(t)− y1(t)

∣∣∣∣ (4)

102


substituting Eq. (3), then

�(t) =

∣∣∣∣∣∣∣u2

Tt+ x2 − u1

Tt − x1

v2

Tt+ y2 − v1

Tt− y1

u3

Tt+ x3 − u1

Tt − x1

v3

Tt+ y3 − v1

Tt− y1

∣∣∣∣∣∣∣=∣∣∣∣u2 − u1 v2 − v1u3 − u1 v3 − v1

∣∣∣∣ t2

T2+∣∣∣∣u2 − u1 y2 − y1u3 − u1 y3 − y1

∣∣∣∣ tT

+∣∣∣∣x2 − x1 v2 − v1x3 − x1 v3 − v1

∣∣∣∣ tT+∣∣∣∣x2 − x1 y2 − y1x3 − x1 y3 − y1

∣∣∣∣≈∣∣∣∣u2 − u1 y2 − y1u3 − u1 y3 − y1

∣∣∣∣ tT+∣∣∣∣x2 − x1 v2 − v1x3 − x1 v3 − v1

∣∣∣∣ tT+∣∣∣∣x2 − x1 y2 − y1x3 − x1 y3 − y1

∣∣∣∣ (5)

At the time instant that vertex P1 contacts edge P2P3, we have

�(t) = 0 (6)

Solve Eq. (6) with respect to t will obtain the time instant when the contact occurs. Equa-tion (6) is solved for both potential reference edges. If the root of Eq. (6) is negative or biggerthan T, vertex P1 will not collide with edge P2P3 in current time step. Even a nonnegativeroot is found for Eq. (6), it is not necessarily the right one because vertex P1 may drop onthe extension line of segment P2P3. Assume the root of Eq. (6) is t0, and the coordinatesof P1, P2, and P3 at that time instant are denoted as P1(x0

1,y01), P2(x0

2,y02), and P3(x0

3,y03),

respectively. If the following inequality holds, vertex P1 is colliding with edgeP2P3 and it isthe entrance edge for vertex-vertex contact. Otherwise, the reference edge must be the otherpotential reference edge. ⎧⎨

⎩(x0

1 − x02

) (x0

1 − x03

) ≤ 0(y0

1 − y02

) (y0

1 − y03

) ≤ 0(7)

After obtaining the proper entrance edge, the vertex-vertex contact is transformed into avertex-edge contact and the open-close iteration will continue.

4. Applications

4.1. Example 1

A two-block system under vertical body force is at rest at the beginning of the analysis (seeFig. 4). The bottom block is fixed by two points at the bottom. There will be a vertex-vertexcontact between the bottom vertex of upper block and the top vertex of the bottom blockafter upper block starts to move. At the initial position, there is a small gap between them,which can assure the vertex of upper block will land on the right edge of the bottom block.

The material properties for both blocks are as follows: mass density = 2.8 × 103 kg/m3;Young’s Modulus = 50MPa, Poisson’s ratio = 0.25, body force is fx = 0N and fy = −2.8×105N. Friction angle and cohesion of joint material is zero. The allowed displacement ratiois 0.1, and the maximum time increment for each time step is 0.05s. It is assumed that thereis no friction between the interfaces.

In the test, the time step is 0.05s and the penalty value is 40 times the value of Young’sModulus. The results from the original DDA code are shown in Fig. 5 and the results fromthe new DDA code is shown in Fig. 6. Because the upper block is land on the right edge of

103


1

3 5

1

43

0.10.02

Figure 4. Geometry of example IV (unit: m).

(a) step=0 (b) step=1 (c) step=2 (d) step=3

Figure 5. Results from original DDA code (p = 40E, T = 0.05s).

Figure 6. Results from new DDA code (p = 40E, T = 0.05s).

the bottom block, the contact edge should be the right edge, and the contact vertex of theupper block will slide along this edge. The enhanced DDA code can show this phenomenonwhile the old DDA code cannot.

4.2. Example 2

The second example is a four-block system on a fixed table with gravity neglected, and allblocks are at rest at the beginning of the analysis, as shown in Fig. 7. There is a vertex-vertexcontact between block a and block c.

104


1525

1525

2020

20 20

F

10

10

ab

c d

Figure 7. Configuration (unit: m).

The material properties for all blocks are as follows: mass density = 2300kg/m3; Young’sModulus = 10GPa, and Poisson’s ratio = 0.3. Friction angle of joint material is 15◦ andcohesion is zero. The allowed maximum displacement ratio is 0.05, and the maximumtime increment for each time step is 0.01s. The stiffness of the normal contact spring is4×1011N/m, which is 40 times of the value of Young’s Modulus of block’s material. A pointforce, F(−4× 107KN, −3× 107KN), is applied on block a. The dynamic factor is set to zerofor the static analysis.

For such a static problem, since Fx/Fy = 4/3, i.e. |Fx| > |Fy|, the deformation along xdirection should be larger than the deformation along y direction. Therefore, the vertex ofblock a should move forward along the top edge of block c. The revised DDA code shows thecorrect result while the original DDA cannot in this special case. The results from both codesare shown in Figs. 8 and 9, respectively. Only the revised DDA code obtained the correctresult.

Figure 8. Original DDA.

105


Figure 9. Revised DDA.

5. Conclusions

This paper provided an enhancement to the shortest path method for resolving the first classindeterminacy of the vertex-vertex in the DDA model. For resolving the second class inde-terminacy, this paper proposed a scheme based on the vertex trajectory to determine theentrance edge of a vertex-vertex contact. Two examples showed the problems caused by theimproper choice of initial contact edge in the vertex-vertex contact. In the examples, theenhanced DDA code obtained the correct results while the original DDA code could not.Although these examples used in the demonstration are simple, they show that the enhancedmethod offers a solid basis and potential for its further application to complicated cases. Thenew scheme can also be applied to the numerical manifold method and other penalty basedcontact model for resolving the indeterminacy of vertex-vertex contact.

References

1. MacLaughlin, M.M. and D.M. Doolin, Review of validation of the discontinuous deformationanalysis (DDA) method. International Journal for Numerical and Analytical Methods in Geome-chanics, 2006. 30: pp. 271–305.

2. Ke, T.C., Modification of DDA with respect to rigid body rotation, in Proceedings of the FirstInternational Conference on Analysis of Discontinuous Deformation, J.C. Li, C.Y. Wang, andJ. Sheng, Editors. 1995, National Central University, Chungli, Taiwan, ROC: Chungli, Taiwan.p. 260–273.

3. MacLaughlin, M.M. and N. Sitar, Rigid body rotations in DDA, in Proceedings of the First Inter-national Forum on DDA and Simulations of Discontinuous Media, M.R. Salami and D. Banks,Editors. 1996, TSI Press, Albuquerque, New Mexico: Berkeley, California. pp. 620–636.

4. Cheng, Y.M. and Y.H. Zhang, Rigid body rotation and block internal discretization in DDA anal-ysis. International Journal for Numerical and Analytical Methods in Geomechanics, 2000. 24:p. 567–578.

5. Ke, T.C., Simulated testing of two-dimensional heterogeneous and discontinuous rock massesusing discontinuous deformation analysis, in Civil Engineering. 1993, University of California:Berkeley.

6. Chang, C.T., Nonlinear dynamic discontinuous deformation analysis with finite element meshedblock system. 1994, University of California, Berkeley: United States — California.

106


7. Shyu, K., Nodal-based discontinuous deformation analysis, in Civil Engineering. 1993, Universityof California Berkeley.

8. Lin, C.T., Extensions to the discontinuous deformation analysis for jointed rock masses and otherblocky systems, in Civil Engineering. 1995, University of California: Berkeley.

9. Koo, C.Y., J.C. Chern, and S. Chen, Development of second order displacement function for DDA,in Proceedings of the First International Conference on Analysis of Discontinuous Deformation,J.C. Li, C.Y. Wang, and J. Sheng, Editors. 1995, National Central University: Chungli, TaiwanROC. pp. 91–108.

10. Hsiung, S.M., Discontinuous deformation analysis (DDA) with nth order polynomial displace-ment functions, in Rock Mechanics in the National Interest, Proceedings of the 38th U.S. RockMechanics Symposium, D. Elsworth, J.P. Tinucci, and K.A. Heasley, Editors. 2001, American RockMechanics Association, Balkema: Rotterdam, Washington DC. pp. 1437–1444.

11. Clatworthy, D. and F. Scheele, A method of sub-meshing in discontinuous deformation analysis(DDA), in Proceedings of the Third International Conference on Analysis of Discontinuous Defor-mation, B. Amadei, Editor. 1999, American Rock Mechanics Association, Balkema: Rotterdam,Washington DC: Vail, Colorado. pp. 85–96.

12. Shi, G., Discontinuous deformation analysis – A new numerical model for the statics and dynamicsof block systems, in Civil Engineering. 1988, University of California: Berkeley.

13. Oden, J.T. and E.B. Pires, Algorithms and numerical results for finite element approximationsof contact problems with non-classical friction laws. Computers & Structures, 1984. 19(1–2):pp. 137–147.

14. Chaudhary, A.B. and K.-J. Bathe, A solution method for static and dynamic analysis of three-dimensional contact problems with friction. Computers & Structures, 1986. 24(6): pp. 855–873.

15. Pandolfi, A., et al., Time-discretized variational formulation of non-smooth frictional contact.International Journal for Numerical Methods in Engineering, 2002. 53: pp. 1801–1829.

16. Kane, C., et al., Finite element analysis of nonsmooth contact. Computer Methods in AppliedMechanics and Engineering, 1999. 180(1–2): pp. 1–26.

17. Feng, Y.T. and D.R.J. Owen, A 2D polygon/polygon contact model: algorithmic aspects. Engineer-ing Computations 2004. 21(2/3/4): pp. 265–277.

18. Cundall, P.A., UDEC 4.0 Manual — Theory and Background. 2004, ITASCA Consulting Group,Inc.

19. Krishnasamy, J. and M.J. Jakiela, A method to resolve ambiguities in corner-corner interactionsbetween polygons in the context of motion simulations. Engineering Computations, 1995. 12:pp. 135–144.

107

Complementary Formulation of Discontinuous DeformationAnalysis

W. JIANG1 AND H. ZHENG1,2,∗1China Three Gorges University, Key Laboratory of Geological Hazards on Three Gorges Reservoir Area Ministryof Education, Yi Chang 443002, China.2State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics,Chinese Academy of Science, Wuhan 430071, China

1. Introduction

Existence of various discontinuities creates difficulties in numerically modeling geotechnicalproblems. DDA1 proposed by Shi is recognized as an efficient method for analyzing highlydiscretized models in the condition of large displacement, and has been serving diverse prob-lems in geotechnical engineering, see Refs. 2–4 and many others.

Since the object analyzed by the DDA is usually a system of blocks, the treatment of con-tact between blocks is the major task of the DDA. Now the penalty function method and theLagrange multiplier method or its variants are generally utilized to solve contact problems.Each method has the merits and demerits. In the penalty function method, adopted in theoriginal DDA and its improved version,5 small penetration between blocks is allowed andeven necessary. Although the penetration is inappreciable, it is not easy to select reasonablepenalty factors (stiffness parameters). Too small factors lead to undue deviation from thecontact conditions; while too large factors result in an ill-conditioned stiffness matrix whichcauses numerical problems. The Lagrange multiplier method, like the LDDA proposed byCai and Liang,6 enforces the contact conditions, but might fail because of rank deficiencyof the stiffness matrix if some improper constraints are involved during the open-close itera-tion. The Augmented Lagrange Multiplier method (ALM) proposed by Amadei and Lin7 canalleviate difficulties of the conventional DDA in selecting penalty factors and overcome rankdeficiency of the stiffness matrix in the LDDA.

All existing methods use a process called the open-close iteration to enforce the contactconditions. Although simple and intuitive, the scheme cannot assure the iteration is alwaysconvergent. During iteration the change in the contact mode of any contact-pair will causean abrupt change in the stiffness matrix, so the change in degrees of freedom correspondingto two consecutive iterations is usually abrupt.

To avoid the penalty factors and the open-close iteration, we reconfigure the DDA. Fromthe variational formulation of momentum conservation instead of minimizing the potentialfunction, we derive a system of equations for momentum conservation, which facilitates con-sidering non-linearity of block materials in future. Then we utilize C-functions to transformthe inequalities reflecting contact conditions on all contact-pairs into the equivalent equa-tions called the contact equations. Then we combine the momentum conservation equationswith the contact equations and thus obtain a system of nonlinear equations some of whichare continuous but non-smooth. Finally the Path Newton Method (PNM) designed for thenonlinear complementary problems8 is utilized to solve the system derived in the study. Twoclassical numerical examples are tested, which demonstrate the feasibility of the proposedprocedure.




2. Complementary Formulation of Discontinuous Deformation Analysis

2.1. Introduction of complementary theory

Complementary Problems (CP) were proposed by Dantzig and Cottle in 1963.8 The CPshave forms of variety. Here, we only recapitulate some concepts related to this study. Thedetailed complementary theories can be referred the monograph by Facchinei and Pang.8

Definition (MiCP) Let three continuous vector-valued functions g:Rn1 ×Rn2 → Rn1 , h:Rn1 ×Rn2 → Rn2 and f :Rn1×Rn2 → Rn2 , with n1 and n2 being two positive integers. A generalizedmixed complementary problem, denoted by MiCP

(g,h,f

), is to find a pair of vectors (w,v) ∈

Rn1 × Rn2 such that ⎧⎨⎩

g (w,v) = 0f (w,v) � 0, h (w,v) � 0f (w,v)T h (w,v) = 0

(1)

We will apply system (1) to reformulate the DDA. The first equations will stand for themomentum conservation equations; the second inequalities for the constraints on the contactpairs; and the third equations will indicate the complementary conditions on the contactforces and the relative displacements of the contact-pairs. We can rewrite (1) as a system ofequations of the form {

g (w,v) = 0min

(f (w,v) , h (w,v)

) = 0(1.1)

The second equations in (1.1) can be look as a vector-valued function whose componentfunctions are an identical min function, i.e., min

(fi (w,v) ,hi (w,v)

). Function min(x, y) is an

easiest C-function8 that is continuous over R2 and smooth everywhere except on the liney = x. Of course min function can be instead of by other C-functions. The theories on thefinite dimensional variational inequalities, e.g. Ref. 8, have had some practical algorithmsfor the solution of systems of non-smooth equations such as those continuous and piecewisesmooth functions that will be involved in the study.

2.2. Equations of momentum conservationof a block system in discrete form

Considering the DDA always reduces contact between blocks to angle-edge contact, we canclassify the forces acted on a typical block �i shown in Fig. 1 as follows.

1) Unknown contact force p, a point load that is divided into two types:

(i) Master force pm acted at a vertex. It occurs when an angle of block �i is in contactwith an edge of �j adjacent to �i. In this case, we call �i as “master” and �j as“slave”. pm has the following decomposition

pm = pτm + pnm (2)

with

pτm = pτmτ , pnm = pn

mn (2.1)

where n is the unit exterior normal vector of �j and τ is perpendicularn to and alongcounterclockwise boundary of �i; pτm and pn

m are components of pm alongτ andn,respectively.

110


i

ps

p-

pm

mp n

pm

n

sp

ps

sp

-f

Figure 1. Force diagram of a single block.

(ii) Slave force ps acted on an edge. It occurs when an angle of a block �j (master)adjacent to �i (slave) is in contact with the edge of �i adjacent to �j. Similarly wehave the decomposition:

ps = pτs + pns (3)

with

pτs = pτs τ , pns = pn

s n (3.1)

where n is the unit inner normal vector of �i and τ is tangential to �i and alongcounterclockwise boundary of �i; pτs and pn

s are components of ps along τ and n,respectively.

2) Known surface traction, p acted on the boundary segment Sp of �i.3) Known point load f , acted on some point of boundary of �i.4) Unknown volume load b− ρ u, with b being the volume force and ρ the density.

We first discuss a typical block �i subject to the forces mentioned above. For this purpose,we start with the weak form of momentum conservation:∫

�i

(δ ε)T σ d� =∫�i

(δu)T(b− ρ u

)d�+

∫Sp

(δu)T pd�+∑

(δu)T f +∑

(δu)T p (4)

where σ and u represent the stress vector and the displacement vector of a point in �i,respectively; δu and δ ε denote the virtual displacement and the virtual strain. For ease ofpresentation, the approximation mode for displacements of blocks and the temporal dis-cretization of uare taken the same as the original DDA; similarly we have a system of linearequations for momentum conservation in discrete form

Kd− Cp = q (5)

where dT = (u0,v0,r0,εx,εy,γxy), Kis the stiffness matrix, C is a 6× 2cn matrix of the form

C6×2cn =[TT

c1Lc1 · · ·TT

cnLcn

](6)

Here, cn is the number of contact force vectors on �i, Tci represents matrix T at contactpointci which transforms the vector of the degree of freedom into the displacement vector,1

111


and Lci is the local frame at contact point ci defined as

Lci =[τci nci

](7)

pin (3) is contact force vector of dimension-2cn with the form

pT = (pτc1,pn

c1, . . . ,pτcn

,pncn

)(8)

At the last, q in (3) is a 6-dimensional vector called the generalized force vector.Suppose that the system of blocks has nb blocks and nc contact-pairs within the time step.

Shi1 has proposed a set of rigorous criteria for constituting the nc contact-pairs. The criteriacan assure that the number of contact-pairs will arrive at the minimum. For any time step ,each block has a system in the form (3). We collect all the systems of the blocks into a systemof 6nb equations in 6nb + 2nc unknowns

(d,p

)as follows

Kd − Cp = q (9)

in which K = diag(K1, . . . ,Knb), with Ki being the matrix of block �i relevant to (19.1); C,a 6nb × 2nc matrix, is formed by assembling Ci of all blocks; d is a 6nb-dimensional vectormade up of all di; q is the generalized force vector composed of all qi; p is the contact forcevector of all contact pairs with a dimension of 2nc

p = (pτ1,pn1, . . . ,pτnc

,pnnc

)TTo solve for d and p from system (9), we have to complement other 2nc equations. This willconstitute the contents of the next section.

2.3. Contact equations

Now we will complement 2nc equations from the nc contact-pairs and call them the contactequations. For simplicity of notations, we will omit the subscripts indicating contact-pairs,and specify that all relations will refer to the moment at the end of a time step.

Normal contact equation

As shown in Fig 2, the fact that the two blocks corresponding to the contact-pair of interestcannot be penetrated into each other requires that the normal relative distance be nonnega-tive, that is,

gn � 0 (10)

Here,

gn = nT (xi − xj) = nT

(xi − xj + Tidi − Tjdj

)

where xi and xj represents the positions of the two nearest points of the contact- pair at thestart of this step; subscripts i and j refer to �i and �j, respectively.

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ττττgpττττng

pn

j

iΩ

Ω

xj

ix

Figure 2. Contact conditions.

At the same time, the normal contact force pn cannot be tensile

pn � 0 (11)

In addition, the two variables cannot be nonzero simultaneously,

pngn = 0 (12)

Equations (10), (11) and (12) can be expressed equivalently with a min function

min(pn,Egn) = 0 (13)

where E is a constant with the same dimension as Young’s modulus, ensuring that pn andEgn have a nearly equal order of magnitude.

Tangential contact equation

In tangential direction, a contact pair might have two contact states: the sticky state andthe sliding state. For the sticky state, the contact pair has no relative movement in tangentialdirection and the friction has not arrived at the maximum specified by the Colombian frictionlaw. In this case, we have

gτ = 0,∣∣pτ ∣∣ � C+ μpn (14)

where gτ = the velocity of relative movement in direction τ ; μ = the friction factor andC = the inner cohesion. For the sliding state, the contact pair has relative movement intangential direction and the friction will arrive at the maximum. In this case, we have

gτ �= 0,∣∣pτ ∣∣ = C+ μpn (15)

In any way, however, we always have for the two states

gτ(C+ μpn − ∣∣pτ ∣∣) = 0 (16)

Considering that the direction of tangential relative movement is in agreement with thefriction pτ acted on the slave block �i, see Fig. 2, we have

Ggτpτ � 0 (17)

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where G is introduced for the same purpose as E in (13). Still Equations (11)–(14) can beexpressed equivalently with a min function identical to (10)

min(Ggτpτ ,C+ μpn − ∣∣pτ ∣∣) = 0 (18)

In this study, gτ is approximated by the Euler-backward difference, that is

gτ = 1τT (uj − ui

) = 1τT(Tjdj − Tidi

)where τ is the unit tangential vector of slave block �i.

2.4. System of equations from the MiCP in DDA

Now that for each contact-pair two contact equations (13) and (18) are deduced, the nccontact-pairs with the current time step will supply 2nc contact equations. Combining sys-tems (9) (13) and (18), we have a system of equations in unknowns d and p

H(d,pn,pτ

) =⎛⎜⎝ Kd − Cp− f(

min (pn,Egn)

min (Ggτpτ ,μpn + C− |pτ |))i=n2

i=1

⎞⎟⎠ = 0 (19)

2.5. Solution procedure

The fact that H includes min function indicates that H is nonsmooth. For ease of presenta-

tion, let m = 6nb + 2nc and z =(

dp

)∈ Rm, and system (16) reduces to H(z) = 0. To solve

it, we introduce the merit function of H

(z) ≡ 12

H (z)T H (z) (20)

Based on the fact that the minimal vector z0 of (z) impliesh (z0) = 0, the Path NewtonMethod (PNM) is utilized to minimize(z), which is of global convergence.8. We first listthe algorithm and then give an explanation for its application to the solution of H(z) = 0.

Algorithm Path Newton Method

Data: Let z0 ∈ Rm be given and ε be given scalars, ε > 0 small.

Step 1: Set k = 0.

Step 2: If (zk)

� ε, terminate withzkas an approximate zero of H (z).

Step 3: Solve the directional Newton equation to obtain the direction dzk ∈ Rm:

H(zk)+H′

(zk)

dzk = 0 (21)

Step 4: Determine step size. Letωk = 2−ik , whereikis the smallest non-negative integer forwhich the following decrease criterion holds:

(zk + 2−ikdzk

)�(1− 2−ik

)(zk)

(22)

Step 5: Set zk+1 = zk + ωkdzk, return to Step 2 with k← k+ 1.

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In Step 1, the toleranceε = 10−4 ‖q‖2, q is the generalized force vector in equation (6).H′(zk) in Step 3 is an arbitrary element in the Clarke generalized Jacobian T(zk)8 of H(z)at zk. In Step 4, starting with i = 1, one can find the smallest i satisfying inequality (22),and thus the largest step length along the search direction dzk can be determined. For easeof presentation, we will call the proposed procedure as the Complementary DiscontinuousDeformation Analysis (CDDA).

3. Numerical Examples

3.1. Sliding problem

Using this example, we will compare the precision given by the CDDA and the classical DDArespectively. Shown in Fig. 3 is a sliding rectangle block of dimension 2m×1m on a rampthat has a slope angle of 30˚. The block and the ramp have the same material properties:the density ρ = 2.75×103 kg/m3, Young’s modulus E = 2.0 MPa, Poisson’s ratio μ = 0.25.The rigid body displacement components of the ramp are fixed, i.e., u0 = v0 = r0 = 0.0.Suppose no frictional force between the block and the slope. So, the block has an exactsliding displacement

S = 14

gt2 (23)

where S(m) = the sliding distance of the block center point , t (s) = the time elapsed.

Figure 3. A sliding rectangle block on a ramp.

Let the time step length = 0.01s and 100 time steps be calculated. Table 1 lists the resultsfrom the analytical solution, the DDA and the CDDA. It is shown that CDDA’s precision ishigher than DDA’s.

From this and other examples we have conducted, the DDA usually gives rise to displace-ments slightly greater than the CDDA. It is explainable — the CDDA can be viewed as thelimit of the DDA in approaching infinite spring stiffness.

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Table 1. Compare CDDA’s results with DDA and analytic theory.

time(s) 0.20 0.40 0.60 0.80 1.000analytic results (m) 0.10 0.40 0.90 1.60 2.50DDA’s results (m) 0.1066 0.4084 0.9087 1.6088 2.5092DDA’s error (mm) +6.6 +8.4 +8.7 +8.8 +9.2DDA’s relative error (%) 6.60 2.10 0.97 0.55 0.37CDDA’s results (m) 0.1023 0.4019 0.9009 1.5993 2.4978CDDA’s error (mm) +2.3 +1.9 +0.86 −0.71 −2.43CDDA’s relative error (%) 2.30 0.50 0.10 0.04 0.09

3.2. The surrounding rock deformation of roadways

The surrounding rock deformation of roadways is a common issue frequently encountered.Figure 4 shows a typical roadway, the rigid body displacement components of outermost

rock are fixed, initial ground stress(σ 0

x ,σ 0y ,τ0

xy

)= (−2,− 1,0), the self weight is ignored. All

blocks have the same material properties: the density ρ = 0.3×103 kg/m3, Young’s modulusE = 5.0 kPa, Poisson’s ratio μ = 0.2. Suppose no frictional force between blocks. Let thetime step length = 0.01 s, the max step displacement radio δ = 0.005, we conduct thestatic analysis to the example by the CDDA.

Figure 5 illustrates the surrounding rock deformation and stress of roadways after 500steps. We can see the surrounding rock is basically stable. The stress of outermost and middlerock changes little and the main stress is still horizontal. The clear change of stress occursin the innermost rock, main direction varies with the position of block. The main direction

Figure 4. A typical roadways.

Figure 5. Deformation and stress of roadway.

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slants gradually, it is horizontal in the roof and the floor but is vertical in the left and theright.

4. Conclusions

This study reformulates the DDA as a mixed nonlinear complementary problem and developsa procedure called the CDDA. The following conclusions are deduced from the experimentalresults:

• The CDDA has a more compact computational scheme. No artificial stiffness parametersare introduced and the “open-close” iteration is avoided.• In general, the CDDA has better convergence.• The CDDA is easier to implement the nonlinearity of blocks.

Acknowledgements

The authors are sincerely grateful to Dr. G. H. Shi, inventor of the DDA, for his valuablesuggestions and guidance, and are looking forward to his great breakthroughs in the 3-dimensional DDA.

References

1. Shi, G.H., Discontinuous Deformation Analysis — A New Numerical Model for the Statics andDynamics of Block Systems (Ph. D. Thesis), Department of Civil Engineering University of Cali-fornia, Berkeley, 1988

2. Moosavi M. and Grayeli, R. “A model for cable bolt-rock mass interaction: Integration withdiscontinuous deformation analysis (DDA) algorithm” International Journal of Rock Mechanics& Mining Sciences 432006, pp. 661–670

3. Hatzor, Y.H., Arzi, A.A., Zaslavsky, Y. and Shapira, A., “Dynamic stability analysis of jointedrock slopes using the DDA method: King Herod’s Palace, Masada, Israel”, International Journalof Rock Mechanics & Mining Sciences 41,2004, pp. 813–832

4. Tsesarsky, M. and Talesnick, M.L., “Mechanical response of a jointed rock beam-Numerical studyof centrifuge models”, International Journal for Numerical and Analytical Methods in Geome-chanics, 31,8,2007, pp. 977–1006

5. Cheng, Y.M., “Advances and improvements in discontinuous deformation analysis”, Computersand Geotechnics, 22, 2, 1998, pp. l53–l63.

6. Cai Y.E., Liang G.P., Shi G.H. etc., “Studying impact problem by LDDA method”, DiscontinuousDeformation analysis (DDA) and Simulations of discontinuous media TSI Press, 1996.

7. Amadei, B. and Lin, C., etc., “Modelling fracture of rock masses with the DDA method”, RockMechanics (Nelson L ed), Balkema, 1994, pp. 583–590.

8. Facchinei, F. and Pang, J.S., Finite-dimensional Variational Inequalities and Complementary Prob-lems. New York, Springer, 2003

117

Accelerated Block Sectioning Algorithm Based on Half-edge DataStructure

JIAN XUE

College of Computing & Communication Engineering, Graduate University of the Chinese Academy of Sciences

1. Introduction

Block sectioning is an important pre-processing step for Discontinuous Deformation Anal-ysis, Numerical Manifold Method and other similar methods, for their computations relyon the block system with correct geometrical and topological structure. The blocks of thecomputed block system are the result of sectioning by the existing discontinuities. Thereforeblocks with any shapes and any number of vertices can be produced: convex blocks, concaveblocks, the union of convex blocks or block containing holes.1 Thus, the block sectioningstep is also a very time-consuming process and the acceleration of such process is helpful toaccelerate the entire analysis procedure.

In order to accelerate the speed of this pre-processing step, a new block sectioning algo-rithm based on half-edge data structure is proposed in this paper. The optimized implemen-tation of this data structure can accelerate the block searching process markedly, and theexperimental results show that the new algorithm is almost two times faster than the originalalgorithm.

2. Related Work

2.1. Block sectioning by discontinuities

The original algorithm for block sectioning by discontinuities was proposed by G. H. Shi etal.2 and was also introduced in Ref. 1. It mainly includes following steps.

1. Compute the intersection points of any two discontinuities;2. Connect adjacent intersection points and generate line segments (possible block edges);3. Delete tree like segments (whose branches with their nodes go nowhere and can not form

block boundaries or loops);4. Trace blocks (whose boundary is a loop consisting of edges);5. Identify the outside and inside boundaries (outside boundary should be deleted).

Among these steps, Steps 2 and 4 are crucial. In order to find all possible block edges andtrace valid blocks, the following three items of information are necessary and sufficient1:

1. The intersection points and their order on each curved line;2. All of the curved lines passing through an intersection point;3. The direction angles of the line segments passing through a node.

Several matrices are used in the original algorithm to record these items of informationand trace the blocks. These matrices are actually two dimensional arrays in implementation.




For example, as mentioned in Chapter 6 of Ref. 1, matrix Q = [Qij] is used for recordingthe intersection points along each curved line. Each row of matrix Q

qi1,qi2, . . . ,qij, . . .

contains the intersection point numbers on the i-th curved line.1 In such a manner, blocks canbe found out properly, but some inefficient operations also are introduced to the algorithm.When locating the next vertex of a block, for instance, a linear search with time complexityO(n) is performed on one row of matrix Q (n is the number of intersection points recorded inthis row). Although the length of row is usually short (n is small), a lot of such linear searchoperations will slow down the algorithm greatly.

2.2. The half-edge data structure

The half-edge data structure is very popular in geometric modeling, which was introducedin Refs. 3 and 4 early. Its main purpose is to perform topological queries for polygon mesh(e.g. “which edges use this vertex?”) in constant time. The structure of half-edge mesh isillustrated with Figure 1.

Half-Edge

Edge

Vertex

Pointers

Figure 1. Half-Edge structure.

The half-edge data structure is called that because instead of storing the edges of the mesh,half-edges are stored. A half-edge is a half of an edge and is constructed by splitting an edgedown its length. The two half-edges that make up an edge are called a pair. Half-edges aredirected and the two edges of a pair have opposite directions.5 In Figure 1, the black dots arethe vertices of the mesh and the gray bars are the half-edges. The short lines with arrows inthe figure represent pointers (links).

The common implementation for representing a polygon mesh with half-edge data struc-ture can be described in C as follows5:

struct HE_edge { // represent a half-edge in meshHE_vert* vert; // vertex at the end of the half-edgeHE_edge* pair; // oppositely oriented adjacent half-edgeHE_face* face; // face the half-edge bordersHE_edge* next; // next half-edge around the face

};

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struct HE_vert { // represent a vertex in meshfloat x, y, z;HE_edge* edge; // one of the half-edges emantating

// from the vertex};struct HE_face { // represent a face in mesh

HE_edge* edge; // one of the half-edges bordering the face};

With these data structures, the topological queries of the polygon mesh can be done easilyand quickly. The time complexity of these queries is linear in the amount of informationgathered and independent of global complexity. For example, iterating over the half-edgesadjacent to a face is done like this5:

HE_edge* edge = face->edge;do {

// do something with edgeedge = edge->next;

} while (edge != face->edge);

3. New Algorithm

3.1. The data structure

The half-edge data structure for the new algorithm is implemented in C++ language asfollows:

struct halfedge {typedef unsigned int index_type;typedef double value_type;enum tag_type { READY, INIT, DELETED, VISITED };

halfedge(): line(INVALID_INDEX), end_node(INVALID_INDEX), pair(INVALID_INDEX), next(INVALID_INDEX), t(0), angle(0), tag(READY)

{ }

index_type line;index_type end_node;index_type pair;index_type next;

value_type t;value_type angle;

121


tag_type tag;};

The ‘line’ field indicates on which line the half-edge lies. The ‘end_node’ field recordsthe index of the node at the end of this half-edge. The ‘pair’ and ‘next’ fields are the indicesof its opposite half-edge and its next half-edge around the same face. The ‘t’ field stores thelocation of this half-edge on the line (i.e. length parameter of the node at the start of thishalf-edge), which is used for sorting the half-edges along the line. The ‘angle’ field is usedfor two purposes, which will be explained in details in Section 3.2. Finally, the ‘tag’ field isused for recording the status of this half-edge when tracing blocks.

3.2. Overall Algorithm

Based on above data structure, the new block sectioning algorithm can be described as fol-lows:

1. Compute the intersection points of any two discontinuities (lines) and set up a node struc-ture for each intersection point;

2. Connect adjacent nodes to form possible block edges and create two half-edges for eachedge;

3. Establish the half-edge mesh (i.e. fill each half-edge structure with proper information);4. Delete tree like edges;5. Trace the valid blocks using topological queries based on half-edge structure;6. Check the correctness of the topological and geometrical structure for the generated block

system.

Obviously, Steps 3 and 5 are the vital steps of this algorithm, which will be explained indetails by following sub-sections.

Based on the half-edge mesh established in Step 3, the time complexity of the topologicalsearch is reduced. Therefore, the block sectioning process is speeded up.

3.3. The establishment of half-edge mesh

The establishment of half-edge mesh can be divided into two steps (as shown in Figures 2(a)and 2(b)).

out boundary

(a) (b) (c)

Figure 2. Establishment of the half-edge mesh.

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In the first step, the ‘pair’ and ‘end_node’ links are set up when the new half-edges arecreated for the possible block edges, as shown in Figure 2(a). The length parameter ‘t’ is alsoset according to the node position at the start of each half-edge. Further more, a temporaryarray is attached to each node for recording all the half-edges started from this node and thedirection angle of each half-edge is also computed and stored in its ‘angle’ field.

In the second step, all the outward half-edges of each node obtained in the last step aresorted according to their direction angles, as shown in Figure 2(b). If the array ‘segs’ storesthe indices of the outward half-edges of current node and the array ‘halfedges’ stores allthe created half-edges, the ‘next’ links of the opposite of the outward half-edges can be setup as follows:

for (int i=0; i<segs.size(); ++i) {halfedges[halfedges[segs[i]].pair].next= segs[(i+1)%segs.size()];

}

Besides, the ‘angle’ field is updated to the angle between two adjacent outward half-edges,which is used to identify the outside and inside boundaries later.

After all the nodes have been processed, the half-edge mesh is finally established. Figure 2(c)shows a simple example of the established half-edge mesh.

3.4. Block tracing

Based on the half-edge mesh, block tracing can be done as following procedure:

1. for each half-edge index hei do {2. if (halfedges[hei].tag �= READY) continue;3. curhei← hei;4. accangle← 0;5. sidenum← 0;6. do {7. curhe← halfedges[curhei];8. add nodes[curhe.end_node] to the new block;9. curhe.tag← VISITED;10. accangle← accangle + curhe.angle;11. sidenum← sidenum + 1;12. curhei← curhe.next;13. } while (curhei �= hei);14. if (accangle > sidenum * π) discard this new block (outside boundary);15. else add this new block to the block set;16. }

From above description, we can see that the time complexity of locating the next vertex ofa block is reduced from O(n) of the original algorithm to O(1).

3.5. Experimental results

The proposed new block sectioning algorithm has been implemented in C++ language.Table 1 gives the comparison between the new algorithm and the original algorithm. Allthe tests are run on a Windows-PC with an Intel Core 2 Duo 1.83GHz processor and 2GB

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Table 1. Comparison between the new algorithm and the original algorithm.

Data Size Time (ms)

Input Lines Output Blocks Original Algorithm New Algorithm

Data 1 88 197 55.86 33.17Data 2 1186 1769 2606.03 1322.04Data 3 1810 1739 3062.42 1592.84

(a) (b)

(c) (d)

Figure 3. Some experimental results.

physical memory. The results indicate that the new algorithm is 68% ∼ 97% faster than theoriginal algorithm. Figure 3 shows the output blocks of the three experimental examples.Figures 3(a) and 3(b) show the result of Data 1 generated by the original algorithm and newalgorithm respectively along with the software GUI. Figures 3(c) and 3(d) show the resultsof Data 2 and Data 3. The blocks are rendered with random colors.

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4. Conclusions

We have developed an efficient block sectioning algorithm based on half-edge data structure,which can accelerate the process of generating blocks from arbitrary input discontinuities forblock system analysis algorithms, such as Discontinuous Deformation Analysis, NumericalManifold Method, etc. The experimental results indicate that the new algorithm is effectiveand efficient.

Acknowledgements

This work is supported in part by the President Fund of GUCAS.

References

1. Gen-hua Shi, Discontinuous Deformation Analysis: A New Numerical Model for the Statics andDynamics of Block Systems, Ph.D. Thesis, 1988.

2. G. H. Shi, R. E. Goodman and J. P. Tinucci, “Application of Block Theory to Simulated Joint TraceMaps”, Fundamentals of Rock Joints, O. Stephansson (ed.), Lulea: Centak Publishers, 1985, pp.367–383.

3. Martti Mantyla, An Introduction to Solid Modeling, Computer Science Press, Inc., New York, NY,USA, 1987.

4. Swen Campagna, Leif Kobbelt and Hans-Peter Seidel, “Directed Edges — A Scalable Representationfor Triangle Meshes”, Journal of Graphics Tools: JGT, 3(4), 1998, pp 1–12.

5. Max McGuire, “The Half-Edge Data Structure”, flipcode: http://www.flipcode.com/archives/The_Half-Edge_Data_Structure.shtml, 2000.

125

A New Contact Method Using Inscribed Sphere for 3DDiscontinuous Deformation Analysis

TAE-YOUNG AHN1, ∗, SUNG-HOON RYU1, JAE-JOON SONG1 AND CHUNG-IN LEE2

1Department of Energy Resources Engineering, Seoul National University, Seoul, Korea2Geogeny Consultants Group Inc., Seoul, Korea

1. Introduction

The discontinuous deformation analysis (DDA) is widely recommended numerical methodfor blocky system.1 In DDA, the penalty method which adds contact spring and friction towhere the contact occurs is used to represent the kinematics of a block system. It shouldbe noted that the contact model using penalty springs is very effective but has shown someinstabilities on the densely populated contacts. In addition, it becomes more complicatedwhen all the contacts are simplified to vertex-to-face contacts in 3D DDA.

It is very difficult to build a perfect contact theory and its detection algorithm because thecontact should be consistently defined between any two blocks with various combinationsof vertices, edges and faces in 3D space. Recently, many researches, therefore, have beenconducted about 3D contact models since Shi2 presented the contact algorithm using thepenalty method. Jiang and Yeung3 developed a model of 3D vertex-to-face contact as apart of the contact theory and Yeung4 presented the details of an edge-to-edge contact in3D DDA. Wu5 developed an algorithm to find the vertex-to-face contact as the first steptowards a more comprehensive 3D contact model and Keneti6 developed a new algorithmfor identifying contact points and types between convex blocks using the concept of mainplanes and dominant contacts.

2. Contact Models in Present 3D DDA

Contacts between polyhedrons can be defined as following 6 types; vertex-to-vertex, vertex-to-edge, vertex-to-face, edge-to-edge, edge-to-face and face-to-face. Figure 1 shows the typi-cal examples of each contact type.

Among these contact types, edge-to-face contact and face-to-face contact can be consideredas combination of the other four contact types and therefore converted to one of the firstfour types in Fig. 1. These four contacts are finally brought under vertex-to-face contact in acontact treatment stage which represents for kinematics of contacting blocks. After detectingall possible contacts precisely, the contacts should be treated by contact penalty springs andfriction force by determining a penetrating point and penetrated plane (i.e. reference face).The penetrating point is one of the vertices of the penetrating block and the penetrated planeis a plane selected among the faces of the penetrated block.

Selecting a proper penetrated plane from candidate faces is important task in the contacttreatment stage. A vertex-to-vertex contact, for example, has three or more candidates anda penetrated plane selected in the present step can be identified as a non-penetrated plane inthe next step by only a small displacement. Wrong selection of the penetrated plane occursoccasionally and it increases the instability of contact treatment algorithm.




Figure 1. Possible contact types between two polyhedrons.

In this paper, a new method using an inscribed sphere to define the penetrating point andpenetrated plane is proposed.

3. Contact Method Using an Inscribed Sphere

Ohnishi7 developed an advanced 2D DDA algorithm using elliptic elements in addition topolygons. The basic concept of his algorithm could be also adopted in 3D cases by usingellipsoids. In blocky systems, however, there are few cases using ellipsoidal elements insteadof polygons. While some DEM codes like PFC3D use spherical particles to construct a blockysystem, DDA can hardly employ the ellipsoidal or spherical elements to present a blockysystem because it would require a huge size of simultaneous equilibrium equations and along time of Open-Closed Iteration (OCI) process.

Using spherical elements can be very effective approach when they are used for the contacttreatment algorithm. The present contact treatment algorithm, however, includes a processof selecting a proper penetrated plane, and therefore, this possibly causes the increase ofcalculation time as well as decrease of stability of contact treatment by selection of improperplanes.

Figure 2. A tetrahedron and its inscribed sphere.

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3.1. Inscribed sphere of a tetrahedron

Figure 2 shows a tetrahedron and its inscribed sphere. Fiagbedzi8 proved that a tetrahedronhas the one and only inscribed sphere as following:

For the tetrahedron, the equations of 4 planes constituting the tetrahedron,�i, are given as

�i:aix+ biy+ ciz+ di = 0 (i = 1,2,3,4) (1)

Then the radius and the center of an inscribed sphere are obtained as (2) and (3) respec-tively.

R = d4 −(a4 b4 c4

) · A−10 d0√

a24 + b2

4 + c24sgn (�4(x))− (a4 b4 c4

) ·A−10 b0

(2)

C0 = A−10 {Rb0 − d0} (3)

where the variables used in Equation (2) and (3) are denoted as

A0 =⎛⎝a1 b1 c1

a2 b2 c2a3 b3 c3

⎞⎠ , b0 =

⎛⎜⎜⎜⎝√

a21 + b2

1 + c21sgn (�1 (x))√

a22 + b2

2 + c22sgn (�2 (x))√

a23 + b2

3 + c23sgn (�4 (x))

⎞⎟⎟⎟⎠ and d0 =

⎛⎝d1

d2d3

⎞⎠

and x is center of the tetrahedron.

3.2. Contact using the inscribed sphere

Figure 3 shows the progress of installing an inscribed sphere into a vertex. When a contactis detected at the vertex, firstly, 3 edge vectors are used to generate a tetrahedron as shownin Fig. 3(a) and 3(b). The length of the edge vectors are important for determining the sizeof the inscribed sphere and it has to be smaller than allowable maximum displacement ofthe each time step. Using the combinations of 4 points which are start-points and end-pointsof the vectors, the equations of 4 planes can be obtained as the form of Eq. (1). Then aninscribed sphere is inserted into the vertex using Eqs. (2) and (3).

Differently with the vertex-to-face contact whose candidate for proper penetrated plane isonly one, the vertex-to-vertex contact has 3 or more candidates for proper plane in existingcontact treatment algorithm. In the new algorithm, the proper penetrated plane is not selectedamong the candidates but generated between two inscribed spheres in contacting vertices.

(a) getting 3 edge vectors (b) defining a tetrahedron (c) obtaining the inscribed sphere

Figure 3. Install-progress of the inscribed sphere into a vertex.

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Figure 4. The inscribed spheres and penetrated plane.

Figure 5. Generation of penetrated plane.

Figure 4 shows the generated ‘imaginary’ penetrated plane. The imaginary plane can begenerated as shown in Fig. 5. A normal vector of the plane,−→n , is determined by two inscribedspheres. Both start point and end point of −→n are the centers of two spheres respectively. Thethree points on the plane can be obtained as follows: P(1) is the tangential point of the sphereinscribed in penetrated block. P(2) is the projection of P(tmp) to the plane after generating a

vector randomly whose direction is different from −→n . P(3) is the end point of−→b , which is

obtained by the cross product of −→n and −→a as shown in Fig. 4. The 3 points are to be usedto install contact springs and friction force to the contact as Shi.2

Table 1 show the difference between existing contact method and newly proposed method.

4. Verification Example

4.1. Tetrahedron free drop test

To verify the accuracy of the new contact method, a tetrahedron subjected to gravity forcewas dropped to a flat bottom as shown in Fig. 6. The tetrahedron had 1m long bottom sides

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Table 1. The difference between two contact methods.

Contact typeExisting contact method Proposed contact method

Penetrating point Penetrated plane Penetrating point Penetratedplane

Vertex-to-vertex contact Vertex Selected faceamong 3 or more

Sphere Imaginaryplane

Vertex-to-face contact Vertex face Sphere FaceVertex-to-edge contact Vertex Selected face

between 2Sphere Selected face

between 2

Figure 6. Exaggerated free drop test progress.

and 2m height. Since the penetrating point of the dropped block was on the surface of theinscribed sphere, the bigger the sphere size was, the deeper penetration occurred. In the test,the size of sphere was varied by changing R-value, which meant the ratio of the length of agenerated edge vector to the length of an edge itself. Three different spheres in Fig. 6 showsthe example spheres when R values are 1.0, 0.5 and 0.1 respectively. In the test, however,R-values was 0.02, 0.01 and 0.001.

The penetration depth of a dropped block was plotted as shown in Fig. 7. In Fig. 7,z-coordinate of vertices where sphere was inscribed is plotted. The vertices reached a bottomblock at 20th step and the penetrations were occurred with different depth. The maximumpenetration depth with 0.001 R-value was 0.0033 m while the maximum penetration depthwith 0.02 R-value was 0.0650. The penetration depth was gradually decreased as the tetra-hedron fell down gradually and it was converged to zero after 175 steps. The 0.0033 mpenetration error from a tested block was reasonably small and the error can be smaller byreducing R-value.

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Figure 7. Penetration depth of the dropped blocks.

4.2. Block stack test

Because the vertex-to-vertex contacts are sensitive with block movement, small displacementof the block can change the proper referential plane. Furthermore, some contacts are negligi-ble in some cases. As shown in Fig. 8(a), for example, the contact between block 1 and block4 can be neglected while the contacts between block 1 and block 2 or 3 should be consid-ered. In existing contact method, the contact between block 1 and block 4 is not negligiblebecause very small overlapping occurs between vertices. Using new method, however, thecontacts can be negligible one because two inscribed spheres hardly overlap to each other.The block stack test was conducted to verify the efficiency of the new method by skippingthe negligible contacts.

(a) 2 2 2 block stack (b) 3 3 3 block stack (c) 4 4 4 block stack

Figure 8. Install-process inscribed sphere into a vertex.

Table 2. Contact states of the block stack cases.

Cases 2× 2× 2 block stack 3× 3× 3 block stack 4× 4× 4 block stack

Contacts open closed total open closed total Open closed Total

Existing contact method 4 88 92 4 424 428 20 1160 1180New contact method 28 64 92 216 212 428 540 640 1180

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As shown in Fig. 8, three kinds of block stacks were analyzed; 2 × 2 × 2, 3 × 3 × 3 and4×4×4 stacks. There were only vertex-to-vertex contacts between blocks in the stacks. Afteranalysing 10 steps, every contacts and their contact state was checked as shown in Table 2.Open state means that two blocks are very near but not in contact so the contacts do notneed contact penalty using contact springs or friction force. Closed state, on the other hand,means that two blocks are in contact so they need contact springs or friction force. Table 2shows that the new contact methods set much more open contact though total contacts aresame with existing one. The result means the efficiency of the analysis is enhanced by usingnew contact method.

4.3. Analysis Examples

An artificial rock slope model was analyzed to test the workability of new algorithm as shownin Fig. 9. 18 blocks on the left slope and 20 blocks on the right slope were generated. Table 3shows the block properties used in the test.

Figure 9 shows the result of the analysis using new contact method. As shown in Fig. 9,the new contact method was worked well in 3D DDA algorithm.

5. Conclusions

In this paper, 3D DDA with new contact treatment method using inscribed sphere is pro-posed. The inscribed spheres are inserted into the contacting vertices. In addition, the imag-inary penetrated planes are adopted for the vertex-to-vertex contacts. The result of verifica-tion tests shows that:

(1) The penetration errors occurred by inscribed spheres were reasonably small so that thenew method can be used to analyse the blocky system.

(a) Before the analysis (b) after 25,000 steps (c) after 50,000 step

Figure 9. Artificial rock slope analysis example.

Table 3. Input parameters for the artificial rock slope analysis.

Blocks on the left slope(18 blocks)

Blocks on the right slope 1(15 blocks)

Blocks on the right slope 2(5 blocks)

Elastic modulus 0.7 GPa 1.0 GPa 0.5 GPaPoisson Ratio 0.20 0.10 0.25

Unit mass 2.8 t/m3 2.5 t/m3 2.7 t/m3

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(2) The new method enhances contact calculation efficiency by reducing the number ofclosed contacts. In addition, it increases the stability of contact treatment algorithm anddecreases calculation time by generating imaginary referential plane without selecting theproper plane among the real block faces.

(3) The behaviour of an artificial slope was coincided with the real state. Therefore, it isthought that the new contact method was worked well in 3D DDA algorithm.

The new algorithm was focused on the vertices to which the inscribed spheres wereinserted. However, edges can also make the contact algorithm complicated. So, inserting thecylinders into the edges can be the new approach of the existing contact method. Therefore,the contact method using spheres and cylinders for vertices and edge can be interesting topicfor further study.

Acknowledgements

Authors thank the Rock Fall and Landslide Prevention Research Center for financial support.

References

1. Shi G.H., Block system modelling by discontinuous deformation analysis. Boston: ComputationalMechanics Publications, 1993.

2. Shi G.H., Theory and examples of three dimensional discontinuous deformation analyses, Proceed-ings of the second Asian rock mechanics symposium, Beijing, China, 2001, p. 27–32.

3. Jiang Q.H. and Yeung M.R., A model of point-to-face contact for three-dimensional discontinuousdeformation analysis. Rock Mech Rock Eng. 37, 2004, p. 95–116.

4. Yeung M.R., Jiang Q.H., Sun N., A model of edge-to-edge contact for three-dimensional discontin-uous deformation analysis, Computers and Geotechnics 34, 2007, p. 175–186.

5. Wu J.H, Juang C.H., Lin H.M., Vertex-to-face contact searching algorithm for three-dimensionalfrictionless contact problem, Int J Numer Methods Eng 63, 2005, p. 199–207.

6. Keneti A.R, Jafari A., Wu J.H., A new algorithm to identify contact patterns between con-vex blocks for three-dimensional discontinuous deformation analysis, Comput Geotech (2008),doi:10.1016/j.compgeo.2007.12.002.

7. Y. Ohnishi, S. Nishiyama, and S. Akao, Proceedings of the 7th International Conference on Analysisof Discontinuous Deformation, 2005.

8. Y. A. Fiagbedzi, M. El-Gebeily, Classroom note: The inscribed sphere of an n-simplex, InternationalJournal of Mathemetical Education in Science and Technology 35(2), 2004, p. 261–268.

134

Study on Failure Characteristics and Support Measure of LayerStructure-Cataclasm Rock Mass

GUANG BIN SHI1,2,3,∗, JUNGUANG BAI2, MINJIANG WANG2, BAOPING SUN2,YING WANG1 AND GENHUA SHI1

1Graduate University of the Chinese Academy of sciences, BeiJin 100049, China2Northwest Investigation Design & Research Institute, Xi’an 710065, China3Xi’Aa university of Architecture and Technology, Xi’an 710054, China

1. Introduction

A underground powerhouse group from the underground powerhouse cavern, the volt-age changer cavern, tailwater surge-chamber, composed of three main cavern, three cavernarranged in parallel, the axis direction of chamber NE50◦, spaces were 40.0 m, 39.5 m; theoutline of the underground plant size 175.0 m × 27.4 m × 74.0 m (length × with ×height),the main voltage changer cavern size of 134.8 m × 16.5 m × 46.6 m, tailwater surge chamber130.0 m × 25.0 m × 67.0 m. Mountain strong base hole from the outer side-wall slope forthe shortest distance between the level of 146 m, a thickness of rock covering 80 m ∼ 275 m.Axis with the three major rock cavern was a wide-angle to the intersection(angle of 55 ∼ 65◦).

Surrounding the main underground powerhouse of green sandstone and gray metamorphicmetamorphic quartz sandstone composed of gray between black sandy slate folder. Maxi-mum principal stress value measured roughly 10 ∼ 13 MPa, the value of intermediate prin-cipal stress is generally 5 ∼ 8 MPa, minimum principal stress value of about 4 ∼ 6MPaabout the maximum principal stress direction NE25◦. Rock mass deformation modulusE = 10 ∼ 15MPa, cohesion c = 1.0 ∼ 1.5MPa, friction angle = 45 ∼ 50◦; structuresurface cohesion c = 0.4 ∼ 0.7MPa, friction angle = 22 ∼ 35◦.

Since the formation of the underground powerhouse and the special conditions of rockmass structure, chamber rock deformation and failure of the plant primarily by high andsteep inclination of the rock surface under the control of the structure, the main advantageof the structure outside inclined surface, a secondary structure tend to inside edge. Surround-ing Rock deformation is usually characterized by the performance of non-continuous, andthe dumping of certain rock structures or cutting load conditions, the underground power-house rock occurs in some local damage due to rock deformation or instability caused bylocal. Application of discontinuous deformation analysis (DDA) of the main plant of the sur-rounding rock deformation and failure characteristics of the numerical simulation, combinedwith rock deformation monitoring and on-site excavation of rock destruction, focusing onan analysis of the regional structure of plant spacing and yield distribution shape, stress level,measures such as anchoring rock cavern deformation and stability.

2. Distribution of Characteristics of Rock Mass Structure Analysis

Figure 1 is a layered excavation of underground powerhouse. To April, 2009 , powerhouse1st to 4th floor excavation is end, Geological engineers carried out the excavation _exposed

∗Corresponding author.



Figure 1. Cavern excavation stages.

Figure 2. Step 1–Step 4 excavation _exposed geological discontinuous plane logging Figure.

geological discontinuous plane for logging in Figure, such as Figures 2 and 3 is the upstreamwall rock photos (part). Figure 2 based on statistical analysis of surface structure, As shownin Figures 4–6.

From the photos (Figure 3) can clearly see that the advantages of structure tend to chamber,the secondary structure and the cutting surface to form a block of different sizes. Through theunderground powerhouse excavation step 1 ∼ step 4 surface geological structure(Figure 2) statistics and analysis (see Figures 4–6), one can clearly see the structure of themain advantages of the top surface in the cavern arch, upstream and downstream wall of thedistribution of wall as follows: Group 1 occurrence for the 335◦ ∼ 350◦ SW � 55◦ ∼ 65◦edge structure and the No. 2 group occurrence for the 12◦ ∼ 54◦ NW � 70◦ ∼ 80◦ advantage

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Figure 3. Upstream rock wall photos (part). Figure 4. Crown structural plane statistics.

of the structure of the surface in the chamber Top arch, upstream and downstream wallsboth wall; occurrence for the 340◦ NE � 72◦ appear in the top arch; occurrence for the 57◦NW � 77◦ appear in the upper reaches of the wall; occurrence for the 67◦ SE � 54◦ appear inthe downstream wall. The structure of the tendency for the NW face, in the upper reaches ofthe outside wall is a tendency in the lower reaches of the wall is a hole in the wall inside thedump; the structure of the tendency for the SE side of the wall is inclined in the downstreamoutside, in the upper reaches of the wall is inside the hole in the wall dump; structure catalogside wall maps and excavation can be clearly seen in the upstream side of the wall inside thecave, there tend to, and with the chamber of less than 30◦ angle between the axis of the struc-ture are not many, and in the downstream side of the wall or more, the distribution of cavitywall and the scene is characterized by destruction of the surrounding rock as the tendency tocave, and with the chamber of less than 30◦ angle between the axis of the structure facing thechamber wall is detrimental to the stability of surrounding rock. In rock excavation Taiwanrock anchor beam, the upper reaches of the wall rock of the rock-anchored beam, about10% rock bench did not succeed (rock bench excavation cross-section in Figure 7), while thedownstream side-wall rock of the rock anchor beam about 70% rock bench did not succeed.In the analysis of block stability, certainty and half_ certainty block on the downstream wallwas significantly higher than the upstream wall.

Figure 5. Upstream wall structural Figure 6. Downstream wall structuralplane statistics. plane statistics.

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Figure 7. Rock bench excavation.

Table 1. In-situ stress measurement result.

Measuring points Principal stress Stress value (MPa) Dip direction (◦) Dip (◦)

σ1 10.56 23 18ZK248 σ2 7.20 122 26

σ3 5.89 262 58σ1 10.05 42 14

ZK208 σ2 7.44 136 16σ3 4.64 274 69σ1 10.24 30 37

ZK206σ2 6.08 174 47σ3 5.01 285 19

3. DDA Analysis Scheme and Model

Hydraulic fracturing method used in the main plant axis stress measured results shown inTable 1. Of course, in DDA2D analysis, must use formula (1) to transform the three dimen-sional initial stress field to the analysis section plane in the stress field, its resultis 5.7–7.7MPa,5.2–7.5MPa, 0.1–0.5Mpa. Sprayed concrete on the role of rock resistance by formula (3) cal-culation to the form of cloth are imposed on the rock; prestressed force to focus on the formof restrictions on the rocks. The geological structural plane provides which according togeologic engineer extends the length and the distribution spacing, jointing statistical lengthl = 25m which the computation analysis uses, jointing average spacing d = 1.5m, 2.0m. Theestablishment of the DDA analysis model in Figure 8. 1, 2, 3 in Figure 8 are the calculationobservation points, 1 and 3 which is located in the wall centre, 2 located in the vault.

{σ ′} = [T]{σ } (1)

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[T] =

⎡⎢⎢⎢⎢⎢⎢⎣

l21 m21 n2

1 2l1m1 2m1n1 2l1n1l22 m2

2 n22 2l2m2 2m2n2 2l2n2

l23 m23 n2

3 2l3m3 2m3n3 2l3n3l1l2 m1m2 n1n2 l1m2 + l2M1 m1n2 +m2n1 l1n2 + l2n1l2l3 m2m3 n2n3 l2m3 + l3M1 m2n3 +m3n2 l2n3 + l3n2l1l3 m1m3 n1n3 l1m3 + l3M1 m1n3 +m3n1 l1n3 + l3n1

⎤⎥⎥⎥⎥⎥⎥⎦

(2)

where: {σ } — New coordinate system (x′, y′, z′) stress: [T] — Stress alternation matrix: {σ }— Original coordinate system (x, y, z) stress: li, mi, ni—Respectively is the new coordinatesystem (x′, y′, z′) to original corresponding system (x, y, z) corresponding direction cosine,i = 1, 2, 3.

pc = δ · τb + As · fyv (3)

where: pc — shotcrete force on a single block of the resistance; δ — shotcrete thickness, if δ isgreater than 100mm, δ = 100mm: τb — shotcrete shear strength: As — steel cross-sectionalarea: fyv — steel shear strength.

4. Characteristics of Rock Deformation and Damage Analysis

4.1. Characteristics of rock mass damage

Rock in the absence of anchor bolt-shotcrete support, due to excavation unloading theimpact of typical rock mass damage occurs is characterized by the dumping of shear slid-ing and extrusion toppling damage, Figures 9 and 10. Spacing distribution of rock massstructure of the rock form of damage has obvious impact on the distribution of joint spacingvarying hours, mainly for the destruction of rock dumping extrusion damage; joints becomelarger when the distribution of spacing, rock damage is mainly expressed in shear slidingdamage is the destruction of these two phenomena occurs at the scene, as travel in someplaces the wall, there is a chamber group and less than 20◦ angle between the axis and thesteep inclination angle outside the existence of joints, on the occurrence of rock excavationslide block, Figure 11, in this article Figures 9 and 10 also reflects the characteristics of the

Figure 8. DDA model.

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a Steps= 400 (b) steps= 1000

Figure 9. Surrounding rock collapes caused by joint intersections (d = 1.5m).

(a) Steps= 400 (b) steps= 1000

Figure 10. Surrounding rock collapes caused by joint intersections (d = 2.0m).

damage; Also in the upper reaches of the wall, in the excavation of rock under unloading, theoccurrence of shear sliding or extrusion toppling damage, leading to sprayed concrete cracksin Figure 12.

When imposed on the rock hanging nets sprayed concrete, prestressed anchor and cable,can effectively suppress the shear or extrusion damage, Figure 13, the observation pointssafety factor K = 1.53. No anchor anchoring conditions, the safety factor K = 0.37.

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(a) Bad (b) Good

Figure 11. Excavation current situation (photos).

4.2. Characteristics of rock mass deformation

(1) Non-anchorage conditions, different levels of initial stress, deformation of the under-ground rock cavern there was a clear distinction, in Figure 14, σ0 = [7.7MPa, 7.5MPa,0.2MPa]. From Figure 14 it is clear that, with the initial stress levels increase, the radial dis-placement of the basic linear increase in side-wall radial deformation from 17mm to 48mm,the radial deformation of the vault from 22mm to 61mm. The Observation points safetyfactor K changed from 2.85 to 0.37.

(2) Under the conditions in the anchorage, when σ0 = [7.7MPa, 7.5MPa, 0.2MPa], thedeformation of 1# point observation on the upstream wall 33.5mm, 2# point for the top archis 47.6mm, 3# point for the downstream side-wall is 35.6mm, in Figure 15. As can be seenfrom Figure 15, calculated values are basically the same values with the scene monitoring.

(3) The surrounding rock deformation Under the anchor bolt-shotcrete support is obvi-ously smaller than the non-anchor supports, the former is approximately the latter 70%.

shotcrete crack shotcrete protuberanceFigure 12. Damage of shotcrete (photos). Figure 13. Stable status of rock block under support.

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Figure 14. Deformation of observation points under non- support.

Figure 15. Deformation of observation points under support.

The anchor bolt stress of the upstream wall is 45–168MPa; The crown anchor bolt stress is33–184MPa; The downstream wall anchor bolt stress is 50–147MPa.

5. Conclusion

This paper analyzed the use of DDA layered structure of rock fragmentation under the largeunderground chamber and deformation characteristics of rock destruction. The calculation

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results show that the contrast with the scene, DDA numerical simulation can be a good rockshow cavern large deformation and destruction of the entire process to help engineers under-stand the failure mechanism of rock; to stress the level of the surrounding rock deformationplays a major role in; linked network of spray concrete, prestressed anchor and cable toeffectively suppress the deformation and destruction of layer structure_cataclasm rock mass.This research is still under continuation.

References

1. Shi Genhua. Manifold Method and Discontinuous Deformation Analysis[M]. Translated by PeiJuemin. Beijing: Tsinghua University Press,1997.

2. Shi Gen-hua, Discontinuous deformation analysis, a new numerical model for the statics anddynamics of block systems, PhD thesis, university of california, berkeley (1988).

3. Sheng Zhenzhong, Ni zhibin, Zhao jian. Stability analysis of rock mass for underground hydraulicpowerhouse by dda. Chinese Journal of Rock Mechanics and Engineering, 2003, 22(Supper1):2299–2303.

4. Sun Dongya, Peng Yijiang, Wang Xingzhen. Application of dda methodin stability analysis oftopplerock slope. Chinese Journal of RockMechanics and Engineering, 2002, 21(1): 39–42.

5. Dong Zhi-hong, WUAi-qing, Ding Xiu-li. Rockbolts simulation bynumerical manifold methodandits preliminary application. Chinese Journal of Rock Mechanics and Engineering, 2005, 24(20):3754–3759.

6. Wu Aiqing, Ren Fang, Dong Xuecheng. A study on the numerical model of DDA and its preliminaryapplication to rock engineering[J]. Chinese Journal of Rock Mechanics and Engineering, 1997,16(5): 411–417.

143

Stability Analysis of Expansive Soil Slope Using DDA

LIN SHAOZHONG∗ AND QIU KUANHONG

DDA Center, Yangtze River Scientific Research Institute, Wuhan 430010, China

1. Introduction

Expansive soil is a special type of soil; it swells and softens when wet, but shrinks and crackswhen dry. Thus, fissures often occur in the shallow layer of expansive soil slopes during theprocess of weathering, wetting and drying. The existence of fissures can disrupt the unity ofsoil and reduce its strength. On the other hand, it also provides a channel for rainwater fil-tration and evaporation, and as a result, climate changes further affect the nature of interiorsoil. Rainfall infiltration reduces the shear strength of soil in the shallow layer, and due tomoisture absorption, the expansion deformation, which is restrained differently from differ-ent directions, will produce shear stress that may lead to shallow landslide along the slope.According to cases of expansive soil slope failures, landslide could happen even in where theexpansive soil slope is gentle.

Currently, the stability analysis of expansive soil slopes primarily adopts the Limit Equi-librium Method (LEM) and the Finite Element Method (FEM). LEM can not simulate theexpansive force and the failure process of slopes while FEM can simulate the expansiveforce but not suitable for analyzing discontinuous deformation and failure process. DDA,1

which has been widely used in geotechnical engineering, can analyze discontinuous deforma-tion, failure process and stability, but rarely used in expansive soil slope analysis. Althoughthere has only been a few application research,2 the calculation model was too simple as itonly simulated macro fissures and the unsaturated soil suction without the consideration ofexpansion effect.

Based on the characteristics of shallow landslide of expansive soil slopes and the Mohr-Coulomb criterion, a DDA calculation model was established to analyze the stability ofexpansive soil slopes with the consideration of both expansion and strength reduction due tothe moisture absorption.

2. Computation Model

2.1. Block division

To analyze the slope stability by DDA, it is necessary to divide blocks according to joint dis-tributions of the slope. Fissures in expansive soil include original stretching and secondaryweathering fissures. The number of original stretching fissures is small. They are discretelydistributed in the slope and generally do not cut each other. Therefore, they do not affectthe stability of the slope significantly. The secondary weathering fissures, which are causedby climate effects, normally occur in the shallow layer where the thickness is less than 3 m.These fissures are densely distributed where only the distribution of fissures on the surfacecan be collected while the distribution of interior fissures is unknown. As a result, it is dif-ficult to complete the statistics and establish a geologic model for secondary fissures at thescene. In this research, three groups of potential shear failure planes were set based on the




characteristics of shallow landslide of expansive soil slopes and the Mohr-Coulomb criterion(The known original fissures can be added into the model).

The direction of potential shear failure planes of one group is parallel to the slope sur-face. The direction of maximum principal stress in shallow soil is approximately parallel tothe slope surface. According to the Mohr-Coulomb criterion, the angles between the slopesurface and the potential shear failure planes of another two groups are:

θ = ±(π/4− ϕ/2) (1)

ϕ is the internal friction angle of soil.

2.2. Relationships between material parameters of expansive soil andwater content

During the process of wetting, shear strength (including cohesion c and internal friction angleϕ) will decrease, and deformation modulus E, Poisson ratio μ and expansion coefficient αwill change as well. As water content w is becoming higher, E will become smaller. Therelationships between the physical and mechanical parameters of expansive soil and w aredetermined according to the experiment. Referring to relevant researches, it is assumed thatc, ϕ, E, μ and α change linearly with w in the research.

2.3. Stress-strain relationship of expansive soil

The expansion strain is treated as initial strain. It is assumed that material parameters areconstant in each time step calculation, and the stress-strain relationship in incremental formfor plane stress condition is:⎧⎨

⎩�σx�σy�τxy

⎫⎬⎭ = E

1− μ2

⎡⎢⎣

1 μ 0μ 1 0

0 01− μ

2

⎤⎥⎦⎛⎝⎧⎨⎩�εx�εy�γxy

⎫⎬⎭−

⎧⎨⎩α�wα�w

0

⎫⎬⎭⎞⎠ (2)

For plane strain condition, E,μ,α should be transformed accordingly.

3. Numerical Example

This example simulated the failure process of an experimental slope which is located in themiddle route of a south-to-north water diversion project in China. The slope is 9.01 m highand mainly formed by marlstone with weak expansion. When cumulative time of artificialrainfall was about six hours, widespread landslide occurred as shown in Figure 1.

The water content of natural soil is 9%, and the water content of saturated soil is 20%.The loading process of moisture is shown in Figure 2. The first 0.5 s is the calculating timefor the gravity load. It is assumed that the moisture of the soil within 2m below the surfaceis the same; the moisture below 3 m of the surface is unchanged; and the moisture between2 m and 3 m under the surface changes linearly.

It is assumed that the weathered depth is 3 m. According to the soil test, c = 84.9 kPa,ϕ = 45◦ in the unweathered soil; c = 48.7 kPa, ϕ = 27.2◦ in natural state and c=13.7kPa,ϕ = 22.2◦ in saturated state of the weathered soil. The density of natural soil is 2300 kg/m3. αvaries with overburden pressure P: α = 0.147 when P = 0 kPa, α = 0.050 when P = 15 kPaand α = 0.032 when P = 25 kPa. The overburden pressure of the slope soil is determinedapproximately according to its depth and unit weight. Due to lack of test data, it is assumed

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Slope before slide

Slope after slide

Sliding surface

1;1,5

Figure 1. Landslide shape in field.

8

12

16

20

0 5 10 15 20 25

Time (s)

Wa

ter c

on

ten

t (%

)

Figure 2. Loading process of water content.

that E, μ and α do not change with water content, E and μ are seen as constant: E = 23 MPa,μ = 0.33.

Block division is shown in Figure 3. Time step for DDA computation is 0.005 s. Landslidehappens when the water content of soil increases to a certain level. The calculated failureshape (Figure 4) is similar to the failure shape in the testing site (Figure 1).

Figure 3. Block system.

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Figure 4. Landslide shape by DDA calculation.

0

0.2

0.4

0.6

0.8

1

1.2

10 12 14 16 18 20

Water content (%)

Ho

riz

on

tal

dis

pla

cem

en

t (m

) Point A

Point B

-0.8

-0.6

-0.4

-0.2

0

10 12 14 16 18 20

Water content (%)

Verti

ca

l d

isp

lacem

en

t (m

)

Point A

Point B

(a) Horizontal displacement (b) Vertical displacement

Figure 5. Displacement curves at measured points.

In order to analyze the failure process of the expansive soil slope, some measured pointsare chosen as shown in Figure 3. The measured points move with the increase of the watercontent as shown in Figure 5. When the water content of the surface soil is less than 16%,the displacements of measured points are small. However, when the water content reaches16%, the displacements speed up and the landslide starts. Landslide depth is less than 3 m.

4. Analysis of Influencial Factors of Slope Stability

It primarily analyzes the effects from slope ratio, strength reduction and expansion on thestability of expansive soil slopes.

4.1. Influence of inclination

It is important to choose an appropriate slope ratio as steep slope may lead to landslide whilegentle slope requires large scale excavation.

This research compared the stability of three expansive soil slopes with the same height of10m but different slope ratios: 1:1.5, 1:2.0 and 1:2.5. The material properties are the same

148


0.0

0.2

0.4

0.6

0.8

1.0

1.2

10 12 14 16 18 20

Water content (%)

Ho

riz

on

tal

dis

pla

cem

en

t (m

)

Slope ratio= 1:1.5

Slope ratio= 1:2.0

Slope ratio= 1:2.5

-0.8

-0.6

-0.4

-0.2

0.0

0.2

10 12 14 16 18 20

Water content (%)

Verti

ca

l d

isp

lacem

en

t (m

)

Slope ratio= 1:1.5

Slope ratio= 1:2.0

Slope ratio= 1:2.5

Figure 6. Displacement curves of slopes with different slope ratios.

(a) Slope ratio=2.0 (b) Slope ratio =2.5

Figure 7. Displacement vectors of slopes with different slope ratios at water content of 20%.

as in Chapter 3. As seen in Figure 6, the inflection point appears earlier on the displacementcurves and the displacements of measured points are larger if the slope is steeper. This meansthat the steeper the slope is, the more unstable it will be. The slope with slope ratio 1:1.5fails when the water content reaches 16%.

As seen in Figure 7, the slopes with ratio of 1:2.0 and 1:2.5 approximate their limit equi-librium state when the surface soil reaches the saturated state. This implies that landslidecan also happen on a gentler slopes and further inclination reduction might not make anyobvious contribution to the stability of expansive soil slopes. Therefore, it is important toensure stability and avoid over-excavation at the same time.

4.2. Influence of strength reduction

Expansive soil in unsaturated state possesses higher strength, however, it will reduce signifi-cantly in saturated state. In order to analyze the effect of strength reduction to the stability,an expansive soil slope with the height of 10 m and the slope ratio of 1:2.0 was researched.The shear strength parameters c and ϕ of saturated soil in the weathered layer are reducedto 1/F of those of natural soil. Without considering the expansion effect, other calculationparameters are the same as in Chapter 3.

As shown in Figure 8, if the reduction coefficient F is 2.8, landslide starts when the watercontent reaches 19.5%. However, landslide does not occur when F is 2.6. It means shallowlandslide would likely happen if F is between 2.6 and 2.8 without considering expansioneffect.

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0.0

0.4

0.8

1.2

1.6

10 12 14 16 18 20

Water content (%)

Ho

riz

on

tal

dis

pla

cem

en

t (m

)

F=2.6

F=2.8

-0.8

-0.6

-0.4

-0.2

0.0

10 12 14 16 18 20

Water content (%)

Verti

ca

l d

isp

lacem

en

t (m

)

F=2.6

F=2.8

Figure 8. Displacement curves of slopes with different strength reduction degrees.

4.3. Influence of expansion

Swelling is an important characteristic of expansive soil. Soil swells when wet. Swellingchanges its stress state and may lead to shear failure. Expansion also depends on mineralcomponent of soil. Using an expansive soil slope with the height of 10 m and the slope ratioof 1:2.0 as research object, the expansion coefficient α in Chapter 3 is multiplied by K. Theshear strength parameters c and ϕ of the weathered soil keep the values of natural state dur-ing wetting (Namely the strength reduction is ignored). Other parameters are the same as inChapter 3.

As shown in Figure 9, higher expansion results in earlier inflection point on the curve andlarger displacement. It means that the slope will easily be damaged in high expansive soils.Although strength reduction during wetting is ignored, local shearing failure still happensunder the action of expansion as shown in Figure 10. Combining the results with Section 4.2,it is concluded that swelling is the primary factor for the failure of expansive soil slopes.

0.00

0.02

0.04

0.06

10 12 14 16 18 20

Water content (%)

Ho

riz

on

tal

dis

pla

cem

en

t (m

)

K=1.0

K=2.0

0.00

0.02

0.04

0.06

0.08

0.10

10 12 14 16 18 20

Water content (%)

Verti

ca

l d

isp

lacem

en

t (m

)

K=1.0

K=2.0

(a) Horizontal displacement at point B (b) Vertical displacement at point A

Figure 9. Displacement curves of slopes with different expansion coefficients.

150


(a) K=1.0 (b) K=2.0

Figure 10. Displacement vectors of slopes with different expansion coefficients at water contentof 20%.

5. Conclusions

This research adopted DDA to analyze the stability of expansive soil slopes under the effectsof both expansion and strength reduction due to moisture absorption, and established calcu-lation model for DDA. The failure process of an expansive soil slope in an artificial rainfalltest was simulated by DDA. Several influence factors of slope stability were analyzed. Theresults are as follows:

(1) The calculation result of DDA is close to the failure feature from artificial field rain-fall test. This means that it is feasible to analyze deformation, stability and failure processthrough the use of DDA and the established calculation model for expansive soil slopes issuitable.

(2) Under the effect of swelling, landslide can also happen in gentle expansive soil slope.Thus, attempting to ensure the stability by reducing inclination may not be the best solution.

(3) Swelling is the primary factor that causes the failure of expansive soil slope, and thestrength reduction as a result of moisture absorption can be seen as a stimulus. Therefore,taking measures, such as waterproofing; modifying expansive soil; or covering with non-expansive soil can help to avoid shallow failure of expansive soil slopes.

References

1. Shi G.H. and R.E. Goodman. Two dimensional discontinuous deformation analysis. InternationalJouurnal for numerical and analytical methods in geomechanics. Vol. 9: 541–556, 1985.

2. An Yanyong. Study of discontinuous deformation analysis to the slope stability analysis of theexpansive soil. Master Thesis of Guangxi University. 27 May 2006.

3. Qiu Kuanhong, Lin Shaozhong, Huang Bin. Failure Simulation of Expansive Soil Slope based onDDA. J. Yangtze River Scientific Research Institute (to be published).

151

DDA Simulations for Huge Landslides in Aratozawa Area, Miyagi,Japan Caused by Iwate-Miyagi Nairiku Earthquake

K. IRIE1,∗, T. KOYAMA1, E. HAMASAKI2, S. NISHIYAMA1, K. SHIMAOKA1 AND Y. OHNISHI3

1Department of Urban and Environmental Engineering, Kyoto University2Advantechnology, Inc., Sendai, Japan3Excutive, Vice President, Kyoto University

1. Introduction

In the mountainous area of Japan, landslides due to earthquakes are one of the major haz-ards which cause serious damages not only to human lives but also to the various importantstructures and infrastructures such as roads and/or railway. To investigate the mechanicalbehavior as well as its mechanisms for landslides caused by earthquakes, numerical simula-tions, especially discontinuum based approaches such as discontinuous deformation analysis(DDA)1 and/or distinct element method (DEM), which can simulate large displacement ofrock/soil masses, will be useful. The finite element method (FEM) has been commonly usedto investigate the earthquake response of the ground. However large displacement can not betreated by FEM properly. On the other hand, a few researchers applied DDA to simulate largelandslides caused by earthquakes so far because there are still some difficulties to determinethe stiffness of contact springs and to give the seismic boundary conditions.2 In this study,one of the largest landslides occurred in Aratozawa area, Miyagi, Japan, on June 14, 2008,which was caused by the Iwate-Miyagi Nairiku Earthquake3 (the damage cost was about38.5 billion yen) was investigated. The 2D DDA model for the landslide was created basedon the geological survey and selected one of the cross section was selected. The mechanicalparameters were determined by laboratory mechanical tests using soil/rock samples. For theseismic boundary conditions, the observed acceleration for the earthquake was converted tothe displacement and given to the bottom basement. The failure process of this landslide wassimulated by DDA and its mechanism was investigated. The simulation results by DDA werecompared with the one obtained from geological survey and the applicability of DDA to thelandslide simulation with large scale was also discussed.

2. Basic Concept for DDA

In the DDA, the fractured rock masses are treated as an assemblage of many independentblocks separated by discontinuities and mechanical behavior of each block are represented bysix deformation parameters in the 2-dimensional problems. Six parameters are displacementsof a rigid body at the center of gravity of blocks, angle of rotation around the gravity centerof blocks and normal/shear strain of the blocks.

The DDA is formulated by the kinematic equations based on Hamilton’s principle. Theequation of motion can be expressed as

Mu+ Cu+ Ku = F (1)

where, M; mass matrix, C; viscosity matrix, K; stiffness matrix, F; external force vector, u;displacements, u; velocity, and u; acceleration of blocks at the gravity center.




Equation (1) is discretized in time using Newmark’s β method and solved the followingthree equations including the block contact in each time step.

K ·�u = F (2)

with

K = 2�t2

M+ 2η�t

M+ Ke + Kf (3)

F = 2�t

M · u+ (�F− f ) (4)

where, �t; time increment, �u; incremental displacement, Ke; elastic matrix for linear term,Kf ; displacement constraint and contact matrix, f ; initial stress vector, and �F; body forceand point road vector. In addition, the DDA applies the penalty method to contacts andintroduces the contact force by setting contact springs. In this study, seismic vibration (in bothvertical and horizontal directions) was given by inputting the time history of the displacement(which was calculated by integrating the observed accelerations)to displacement constraintpoint.

3. Landslide in Aratozawa Area

The studied landslide area was located in Aratozawa area, Miyagi Prefecture, Japan. On the14th of Jun, 2008, Miyagi-Iwate Nairiku Earthquake was occurred and caused this landslide.The size of this huge landslide was 1.3km in length, 900 m in width and 150m (maximum)in depth and the total amount of the collapsed rock/soil masses was about 70 million m3

(Fig. 1).3

Geological survey was carried out along several survey line and geological cross sectionswere obtained. In this study, one of the cross sections called the D-survey line (see Fig. 2) wasselected to create the 2-D DDA model. Because the amount (volume) of collapsed rock/soil

Figure 1. The landslide occurred in the Aratozawa Area.

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Figure 2. D-survey line.

Figure 3. Geological models (a) before and (b) after the earthquake.

masses did not change much before and after the earthquake. The geological cross sectionsand their geological map along the D-survey line before and after the earthquake is shownin Fig. 3. According to the geological survey, geological map in this area consists of thefollowing five different layers: welded tuff (purple), pumice tuff (yellow), sandstone and/orsiltstone (light blue), tuff (green), and old caving zone (orange) respectively (see Fig. 3).

The angle of sliding plane is low (0 ∼ 2◦). This is one of the geological features in thisarea.

155


4. Numerical Modeling with DDA

4.1. DDA model for the Aratozawa Landslide

The 2-D numerical model with DDA was created based on the geological map and fracturemapping results. Considering the geological maps before and after the earthquake and frac-ture directions, the upper stream region was divided using relatively large blocks. On theother hand, the since lower stream region and old caving zone were highly fractured areaand moved fluidly and spread widely after the earthquake was divided using small blocks.Finally, the collapsed rock/soil masses were divided into 177 blocks (Fig. 4).

4.2. Parameters for simulations and the seismic boundary conditions

The parameters for the simulations are summarized in Table 1. Each parameter was deter-mined empirically from previous research works and literature.4 One of the most importantparameters in DDA is the stiffness of contact springs. In this study, the stiffness of 720,000

Figure 4. Block model on this site for DDA.

Table 1. Parameters for the DDA simulations.

Item Value

Welded TuffUnit Weight (kN/m3) 19.0

Young’s Modules (MPa) 1,000Poisson’s Ratio 0.35

Pumice TuffUnit Weight (kN/m3) 16.5

Young’s Modules (MPa) 80Poisson’s Ratio 0.4

Sandstone and/or SiltstoneUnit Weight (kN/m3) 17.5


TuffUnit Weight (kN/m3) 22.5

Young’s Modules (MPa) 2,000Poisson’s Ratio 0.3

Old Caving ZoneUnit Weight (kN/m3) 17.0


JointFrictional Angle (degree) 10

Cohesion(kPa) 0Tensile Strength (Mpa) 0

156


(kN/m) was given.5 The internal friction angle for discontinuities (with static condition) rep-resenting the shearing resistance was measured (φ = 10◦). According to the geological sur-vey, groundwater level was relatively high (there is a dam close to this area) and groundwaterplays important role for the landslide. However, in this study, the effect of groundwater wasnot considered directly (consider only seismic forces).

As for the seismic boundary conditions, the observed acceleration data obtained fromstations was converted to the displacement (see Fig. 5) and given to the bottom basement inthe DDA model as a compulsory displacement. Other parameters for DDA are summarizedin Table 2.

Figure 5. Seismic boundary conditions converting acceleration data to displacement history (a) hori-zontal direction, (b) vertical direction.

Table 2. Parameters for the DDA simulations.

Item Value

Time interval (sec) 0.0005Maximum permissible displacement ratio 0.001Penalty coefficient(Kn) (kN/m) 720,000Penalty coefficient(Ks) (kN/m) 720,000Velocity/Energy Ratio 1

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5. DDA Simulation Results

As mentioned before, one of the features of this landslide was the angle of sliding plane wasextremely small (0 ∼ 2◦). To investigate the mechanism of landslide with very small slidingangle, the parametric study in terms of internal friction angle of discontinuities was carriedout. One of the possible scenarios will be the internal friction angle which represents theresistance for shearing becomes smaller and finally close to zero during the earthquake dueto the cyclic loading and/or the groundwater.

5.1. Simulation results only considering seismic loading

There are still difficulties to consider the effect of the pore water pressure directly in DDA(combining with flow simulations). Hence, the effect of groundwater was considered as thedecrease of the internal friction angle for discontinuities (sliding plane). When the pore waterpressure head of 51m was applied to the sliding plane, the internal friction angle will decreaseup to 5◦. The simulation result with the internal friction angle of 5◦ is shown in Fig. 6.

Figure 6. The result analyzed without compulsion sliding force.

From this figure, the calculated sliding distance (in horizontal direction) after the earth-quake was about 100m and much smaller than the observed sliding distance at the center ofthe collapsed rock masses (300m) (see Fig. 3b). This result indicates that the internal frictionangle in dynamic condition was smaller than 5◦, and consistent with the fact that the angleof slinging plane was 0 ∼ 2◦.

5.2. Simulation results with additional sliding force

In the previous section, the simulated sliding distance was smaller than the observed one(especially, the movement of two ridges marked in Fig. 7). Hence, in this section, addi-tional sliding force was considered and applied to the two ridges (additional sliding forceof 197,000 kN/m and 141,000kN/m were applied to the ridges A and B, respectively). Theadditional sliding force of 338,000kN/m will be necessary to reproduce the landslide (ridgemovement of 300m). The simulation result is shown in Fig. 8. The movement upward of

A B

Figure 7. Two ridges which are added the compulsion sliding force.

158


Figure 8. The result analyzed with compulsion sliding force.

highly fractured region (downstream) and the toppling failure (upstream) can be reproducedby using DDA.

The amount of additional sliding force was determined by trial and error to reproduce thesliding distance of 300 m. Assuming the total length of sliding plane is 800m, the additionalhydraulic pressure (head) can be calculated by 338,000 [kN/m] ÷ 800 [m] ÷ 10[kN/m3] =42m. Hence, to reproduce the large displacement of 300m, the decrease of internal frictionangle of 5◦ (the internal friction angle of 10◦ was decreased by the effect of pressure head of51m) wan not sufficient and more pressure head of 42m was required.

In this case, the internal friction angle was significantly decreased up to less that 1◦, whichagrees with the fact that the angle of sliding plane is 0 ∼ 2◦ and the internal friction angelwas close to zero during sliding. The DDA simulations can evaluate the mechanism of landslide qualitatively.

6. Conclusions

In this study, the mechanism and processes of the landslide in the Aratozawa area was investi-gated using the observation and measuring results such as geological feature (maps), seismicwave and physical properties value of the rock masses. The 2-D DDA model was createdbased on the geological maps before and after the earthquake. The sensitivity analysis interms of the internal friction angle of sliding plane was carried out to investigate the failuremechanism and processes quantitatively. To consider the effect of groundwater, the additionalsliding force was also considered. The findings obtained from this study can be summarizedas follows.

• The simulation results clearly showed that the large displacement of rock/soil masseswas caused not only seismic loading but also other factors such as groundwater.• By considering the additional sliding force caused by the high groundwater level,

the DDA simulation can reproduce the landslide with large movement of the ridges(about 300m).• The DDA simulations can reproduce the toppling failure in the upstream region and

fluidly movement in the highly fractured area in the downstream region, which wasobserved at the Aratozawa landslide area.• This result indicates that the internal friction angle in dynamic condition was smaller

than 5◦, and consistent with the fact that the angle of slinging plane was 0 ∼ 2◦.

From the simulation results, the fracture geometry, the stiffness of contact springs andmechanical properties for fractures play important roles for the failure processes of rock/soilmasses. Further study will be necessary to consider the effect of groundwater. Since thislandslide occurred close to the dam site, the effect of the groundwater may not be neglected.

159


Acknowledgements

The authors would like to thank to Dr. Takeshi Sasaki, Suncoh Consultants Co., Ltd. and Dr.Shigeru Miki, Kiso-Ziban Consultants Co., Ltd. for their valuable comments and suggestions.

References

1. Shi, G.H. and Goodman, R.E., “Discontinuous Deformation Analysis”, Proceedings of the 25thU.S. Symposium on Rock Mechanics, 1984, pp. 269–277.

2. Jian-Hong Wu, Jeen-Shang Lin and Chao-Shi Chen, “Dynamic discrete analysis of an earthquake-induced large-scale landslide”, International Journal of Rock Mechanics and Mining Sciences Vol-ume 46, Issue 2, 2009, pp. 397–407.

3. The farm promotion section of Miyagi Prefecture, “Iwate-Miyagi Nairiku Earthquake, the infor-mation of the damage in Aratozawa Area, 2008” Web site: http://www.pref.miyagi.jp/nosonshin/kouikisuirityousei/pdf/20080614,aratosawadamu.pdf

4. Akao, S., Ohnishi, Y., Nishiyama, S. and Nishimura, T., “Comprehending DDA for a block behav-ior under dynamic condition”, Proceedings of The 8th International Conference on Analysis ofDiscontinuous Deformation, 2007, pp. 135–140.

5. Shimaoka, K., Koyama, T., Nishiyama, S. and Ohnishi, Y., “Earthquake response analysis ofrock slopes by Discontinuous Deformation Analysis (DDA)”, Proceedings of International Mini-Symposium for Numerical Analyses, 2008, pp. 31–40.

160

Modelling Crack Propagation with Nodal-Based DiscontinuousDeformation Analysis

H.R. BAO AND Z.Y. ZHAO∗

School of Civil and Environmental Engineering, Nanyang Technological University, Singapore 639798, Singa-pore

1. Introduction

The discontinuous deformation analysis (DDA)1 is a discontinuum-based method. It choosesthe displacements and strains of blocks as variables and solves the equilibrium equations inthe same way as the finite element method (FEM) does. Since it is also an implicit methodas FEM, one attractive advantage of the DDA is that an existing FEM code can be readilytransformed into a DDA code while retaining all the advantageous features of the FEM.The DDA has emerged as a more attractive model than the continuum-based methods forgeomechanical problems, due to its intrinsic feature of block discontinuity at the contactboundaries.

The DDA employs the complete first order polynomial as the displacement function fora two-dimensional block, which restricts the block to constant stresses and limits the defor-mation abilities of the block. If a more complicated stress field and more deformable blockboundary are desired, enhancement to the original DDA method is necessary. Three kinds ofenhancement have been developed by previous researchers. The first one incorporates higherorder displacement functions or two-dimensional Fourier series.2, 3 The second enhancementis done by introducing artificial joints into the real block.4, 5 The last but most promisingenhancement is to couple the finite element mesh into a block.6, 7 In the writer’s opinion, thecoupled DDA/FEM perhaps is the most promising enhancement because it largely improvedthe deformation ability of a single block without inducing any unnecessary contact. How-ever, previous work on coupling the finite element mesh into the DDA cannot consider thefragmentation of a block or crack initiation and propagation inside a block. The nodal-basedDDA provided in this paper is based on the valuable work of previous researchers with anenhanced ability of treating with crack initiation and propagation.

2. Theory of the Nodal-Based DDA

2.1. Basic concepts

In the nodal-based DDA (NDDA), the triangular element is the basic analysis object and thenodal displacements become the unknowns of simultaneous equation. A group of triangularelements build up a block. The grid lines can be referred as virtual joints, which will fracturewhen certain failure criterion is triggered. The topology of a nodal-based DDA model isillustrated in Fig. 1.

In the original DDA method, basic unknowns are six displacements of a block, {u0, v0, r0,εx, εy, γxy}, which are independent of the block shape. It is very rough to use six variables todescribe the deformation of a big block under complex loading condition. In the nodal-basedDDA, the basic analysis object is triangular element and nodal displacements are the basic




element

block i

real joint

virtual joint

block j

Figure 1. Illustration of a nodal-based DDA model.

unknowns. Hence, the degree of freedom of a block is now depending on the number ofnodes it has. The more nodes a block has, the better deformation ability it will gain.

For a triangular element, three nodes provide six unknown displacements, {ui, vi, uj, vj, um,vm}T. In a special case where the block is triangular and include only one triangular element,the six unknowns of NDDA is equivalent to the unknowns of original DDA. The advantageof using {ui, vi, uj, vj, um, vm}T as element unknowns makes it possible to unify continuousand discontinuous region in the analysis.

2.2. Displacement functions

The displacement of any point (x, y) inside the element can be obtained by the followingdisplacement function:

{uv

}=[Ni 0 Nj 0 Nm 00 Ni 0 Nj 0 Nm

]⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

uiviujvjumvm

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

(1)

where

Ni = ai + bix+ ciy2�

; Nj =aj + bjx+ cjy

2�; Nm = am + bmx+ cmy

2�

in which

ai = xjym − xmyj; bi = yj − ym; ci = xm − xj

with the other coefficients obtained by a cyclic permutation of subscripts in the order i, j, m,and where � is the area of �ijm.

2.3. Simultaneous equations

Individual triangular elements are connected by nodes to form a block, and blocks are con-nected by joints and contact springs to build a system. Assuming there are n nodes in the

162


system, the global equilibrium equations will have the following form:

KD = F (2)

where K is the global stiffness matrix, a 2n × 2n matrix. D is the unknown vector and F isthe equivalent force vector. The global equilibrium equation can be written in a submatrixform as follows: ⎡

⎢⎢⎢⎢⎢⎣

k11 k12 k13 · · · k1nk21 k22 k23 · · · k2nk31 k32 k33 · · · k3n

......

.... . .

...kn1 kn2 kn3 · · · knn

⎤⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

d1d2d3...

dn

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

f1f2f3...fn

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

(3)

in which, each element kij in the coefficient matrix is a 2×2 submatrix. di is a 2×1 submatrixand denotes the displacements unknowns {ui, vi} of node i. fi is a 2×1 submatrix and denotesthe equivalent nodal forces {fix, fiy} applied on node i.

2.4. Fracture criterion

In this paper, a fracture mechanism is introduced to make it possible for the fracture initiationand propagation in a block along the virtual joint. The Mohr-Coulomb criterion is employedas the failure criterion.

According to the Mohr-Coulomb law, the shear strength on a grid line is

τ = c− σn tanφ (4)

where c is the cohesion and φ is the angle of internal friction. It is noted that the normalstress σn acting on an inclined plane is defined here to be negative in tension. The stresses onthe grid line are the average value of the two elements sharing the grid line. If σn > σt, tensilefailure will happen; if τ > c− σn tanφ, shear failure will happen.

2.5. Mesh update

A crack is introduced where the normal stress or shear stress reaches the ultimate strength. Inthis case, the topology of the system needs to be updated so that this crack can be consideredin the next time step. In a continuum media, cracks are not independent of each other. Once acrack happen somewhere, it will cause the stress redistribute in the area around it and releasesome energy. And finally affect the behaviour of the crack propagation in the nearby field.

The numerical model in the DDA is a multiple crack model with many cracks appearingin a time step. Traditional crack inserting procedure of DDA just simply separates the jointsamong blocks according to the Mohr-Coulomb criterion without taking any considerationof the fracture sequence among them. When the time interval of a step is small enough andthe parameters is appropriate, the original DDA can provide a proper result.

In the nodal-based DDA, a fracture sequence is considered by a special crack insertingprocedure inside each time step. The procedure includes four steps:

Step 1. Compute the weighted-average normal stress and shear stress on the grid lines amongall elements.

Step 2. Look for the grid line which will crack first according to their failure time and markthis grid line as a crack, then insert new nodes if need.

Step 3. Update the mesh and all related information of the nodes and elements.

163


Step 4. Compute the updated system again and go back to step 1, unless the last time stephas been reached.

During the Step 2, the most important task is to check the failure time sequence, whichis based on the failure time of each mesh grid. The failure time of each grid line needs toconsider two cases: tensile failure time and shear failure time. The smaller one of them is thefailure time of that grid line.

3. Applications

The above algorithm is applied into a nodal-based DDA program called NDDA. Two numer-ical examples are introduced here to show the enhanced ability of the nodal-based DDA.

3.1. Example 1

This example is designed to validate the ability of the nodal-based DDA in the analysis ofP-wave propagation in an elastic rock bar with free ends. The elastic rock bar is 1 meter longand 0.03 meter in height, as shown in Fig. 2. The material properties and analysis parametersare shown in Table 1.

In this example, the P-wave pulse is generated by a pressure applied on the left boundaryof the bar. The P-wave pulse is shown in Fig. 3. The results obtained by the NDDA are shownin Fig. 4. The first crack appeared at step 256 (time instant is 0.0002560s), and ran throughthe whole section at step 261 (time instant is 0.0002562 s).

The horizontal particle velocity time histories at three different measure points: start point,midpoint, end point (i.e. nodes 343, 679, 171) are presented in Fig. 5. The calculated P-wave

Figure 2. Configuration of the bar (1m × 0.03m).

Table 1. Analysis parameters.

Rock sample Unit mass (kg/m3) 2600Young’s modulus (GPa) 50Poisson ratio 0.25Friction angle 30◦Cohesion strength (MPa) 24Tensile strength (MPa) 18

Joint/crack Friction angle 25◦Cohesion strength (MPa) 0Tensile strength (MPa) 0

Control parameter Penalty stiffness (GN/m) 500Time step size (s) 1× 10−6

Max displacement ratio 0.1SOR factor 1.0Total analysis time (s) 0.002

164


0.0 4.0x10-50

1x107

2x107

Pre

ssur

e(pa

)

Time(s)

Figure 3. Input P-wave pulse.

(a) step 200 (0.0002s)

(b) step 256 (0.000256s)

(c) step 261 (0.0002562s)

(d) step 1016 (0.001s)

Figure 4. Analysis results from NDDA.

0.0000 0.0004 0.0008

0

1

2

3

4

5

Vel

ocity

(m/s

)

Time(s)

Node 343

0.0000 0.0004 0.0008

0

1

2

3

4

5

Vel

ocity

(m/s

)

Time(s)

Node 679

0.0000 0.0004 0.0008

0

1

2

3

4

5

Vel

ocity

(m/s

)

Time(s)

Node 171

(a) start point (b) midpoint (c) end point

Figure 5. Horizontal particle velocity of different measure points.

165


velocity propagating through the bar is 4386 m/s and the theoretical one calculated usingthe elastic constants is 4385 m/s. The velocity obtained by the NDDA program for P-wavepropagation agrees well with the theoretical value.

3.2. Example 2

A numerical test on the Brazilian disc of rock materials was executed to validate the probabil-ity of the NDDA on dealing with crack initiation and propagation. Assume the test materialis a continuous, isotropic and homogeneous elastic body. The diameter of the rock disc is50 mm. The dimension for the rectangular loading plate is 50×10 mm. The thickness of discand rigid plate are 25 mm, namely the thickness-to-diameter (T/D) ratio is 0.5. The materialproperties are shown in Table 2. The load strain rate is applied by the displacement of tworigid plates with a speed of 0.2 m/s.

Table 2. Analysis parameters.

Rock sample Unit mass (kg/m3) 2600Young’s modulus (GPa) 10Poisson ratio 0.25Friction angle 25◦Cohesion strength (MPa) 25Tensile strength (MPa) 12

Rigid plate Unit mass (kg/m3) 7800Young’s modulus (GPa) 2000Poisson ratio 0.25Friction angle 25◦Cohesion strength (MPa) 2500Tensile strength (MPa) 2500

Joint/crack Friction angle 20◦Cohesion strength (MPa) 0Tensile strength (MPa) 0

Control parameter Penalty stiffness (GN/m) 4000Time step size (s) 1× 10−5

Max displacement ratio 0.01SOR factor 1.0Total analysis time (s) 0.003

The results from the NDDA are shown in Fig. 6. The first crack appeared at around0.00214 s and propagated toward the contact point with rigid plate very quickly. The numer-ical experiment results agree well with the laboratory observations.

4. Conclusions

With the addition of a fine element discretization in each block, the nodal-based DDA canprovide more realistic deformation ability in each block and consequently more precise stressdistribution filed for crack initiation in it. In the first example, it is found that the nodal-basedDDA is able to model the wave propagation accurately because it is a continuum-basedmethod when no crack occurs. In the second example, while cracks occur among elements,it becomes a discontinuum-based method and the kinematics of blocks come into work. Theresults agree well with the laboratory observation.

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(a) first crack appear (b) cracks propagation (c) after failure

Figure 6. Results from NDDA.

References

1. Shi, G., Discontinuous deformation analysis — a new numerical model for the statics and dynamicsof block systems, in Civil Engineering. 1988, University of California: Berkeley.

2. Koo, C.Y., J.C. Chern, and S. Chen, Development of second order displacement function for DDA,in Proceedings of the First International Conference on Analysis of Discontinuous Deformation,J.C. Li, C.Y. Wang, and J. Sheng, Editors. 1995, National Central University: Chungli, TaiwanROC., pp. 91–108.

3. Hsiung, S.M., Discontinuous deformation analysis (DDA) with nth order polynomial displacementfunctions, in Rock Mechanics in the National Interest, Proceedings of the 38th U.S. Rock MechanicsSymposium, D. Elsworth, J.P. Tinucci, and K.A. Heasley, Editors. 2001, American Rock MechanicsAssociation, Balkema: Rotterdam, Washington DC., pp. 1437–1444.

4. Ke, T.C., Simulated testing of two-dimensional heterogeneous and discontinuous rock masses usingdiscontinuous deformation analysis, in Civil Engineering. 1993, University of California: Berkeley.

5. Lin, C.T., Extensions to the discontinuous deformation analysis for jointed rock masses and otherblocky systems, in Civil Engineering. 1995, University of California: Berkeley.

6. Shyu, K., Nodal-based discontinuous deformation analysis, in Civil Engineering. 1993, Universityof California Berkeley.

7. Chang, C.T., Nonlinear dynamic discontinuous deformation analysis with finite element meshedblock system. 1994, University of California, Berkeley: United States — California.

167

Discontinuous Deformation Analysis for Parallel Hole Cut Blastingin Rock Mass

ZHIYE ZHAO1, ∗, YUN ZHANG1 AND XUEYING WEI2

1School of Civil and Environment Engineering, Nanyang Technological University, Singapore 639798,Singapore2School of Civil Engineering, Chang An University, Xi’an 710061, China

1. Introduction

Rock blasting is a rock excavation technique most widely used in the mining and constructionindustry due to its reliability, economy and safety. The explosive is loaded in boreholes andblasted in a prearranged sequence to fracture, fragment and displace a well defined portion ofthe rock from its natural position. The goal of blast design is to attain the expected technicaltarget (advance and good contour) at an economical cost. In a well designed blast, most ofthe explosive energy is spent in breaking rock (to achieve high blasting efficiency and obtaindesired fragmentation level), but some is converted into vibrations, either ground motionor air overpressure. In a badly designed blast, where poor breakage is obtained, or heavydamage in the surround rock is caused; or where boreholes are over or under loaded, it can bethat much of the energy is converted into vibrations, since it is not expended in fragmentingthe rock as it should be. The cost saving, small vibration and little damage in the surroundingrock may be achieved by optimising parameters of the blast design. On the other hand, suchexperiments are very expensive and hard to catch the results. Thus the numerical methodson simulating such problems become feasible and important.

The discontinuous deformation analysis (DDA) proposed by Shi1 is a discrete elementprogram to investigate the two-dimensional behaviour of the fractured rock mass, and hasbeen widely used to model the motions of blocky system in rock engineering recently. Due toits inherent discontinuous mechanics formulation, the DDA has been widely used to modelboth static and dynamic problems of rock mass containing multiple parallel or intersect-ing fractures on stability analysis of rock slope and underground structures. For continuousanalyses where the rock mass are composed of one intact block without fractures, the DDAcan be used as an FEM program by gluing all the subblocks together with very strong jointproperties.2 For discontinuous analysis where the rock mass are composed of multiple blocksseparated by fractures, all the blocks can be assembled to interact with each other with thereal joint properties. The ability to solve both the continuous and the discontinuous prob-lems makes the DDA a potential method to model the one-dimensional or two-dimensionaldynamic analysis of fractured rock mass.

2. Blast Design Models in Rock Mass

Drill and blast is one kind of methods to detach the rock in excavation. It was first applied in1627 by the Tyrolean Kaspar Weindl in a silver mine in Banská Štiavnica (former Schemm-nitz, Slovakia). Since it is suitable for hard rock (e.g. granite, gneiss, basalt, quartz) as well asfor soft rock (e.g. marl, loam, clay, chalk), drill and blast is applicable for rocks with varyingproperties. Moreover, drill and blast is advantageous for: (a) relatively short tunnels, where




a tunnel boring machines (TBM) does not be suitable; (b) very hard rock; (c) non-circularcross section.

Tunnel blasting is a much more complicated operation than bench blasting because theonly free surface that initial breakage can take place towards is the tunnel face.3 Blast designhas direct influence on the time consumption and construction cost of drill and blast tunnels.The most important operation in the tunnel blasting procedure is to create an opening in theface in order to develop another free surface in the rock. This is the function of the cut holes.Cuts can be classified in two groups: parallel hole cuts and angle hole cuts. The parallel cutconsists of one or more larger diameter unloaded boreholes. All holes are drilled at a rightangle to the face and parallel to the tunnel direction. The breakage is against the opening orvoid formed by these unloaded holes of diameter 76-150 mm. The parallel hole cut is mostused in operations with mechanized drilling. As this group tends to easier to drill and do notrequire a change in the feed angle, it is widely used in practice and the second one has lost itsappeal due to the difficulty in drilling.

2.1. NTNU models

The NTNU blast design model4 developed by the Department of Civil and Transport Engi-neering at NTNU is an empirical blast design model based on the parallel hole cut. Thetunnel face is usually divided into cut, stoping (easers), lifters (invert) and contour as shownin Fig. 1.

The blasting starts against an opening that is established by drilling one or more empty(large) holes. Three standard parallel hole cuts are shown in Fig. 2. The numbers indicatesmillisecond detonators interval. The empty hole is drilled first to provide sufficient space forexpansion of the rock.

There are several parameters governing the drill pattern: (1) drill hole diameter; (2) drillhole length; (3) rock mass blast ability; (4) tunnel cross-section; (5) look-out angle; and (6)skill level of the tunnel crew. Start the drill pattern design by positioning the cut and thecontour holes first. Secondly, the row nearest the contour and the lifter holes should bedetermined. The stoping holes are finally designed.

Figure 1. Different tunnel sections.6

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Figure 2. Large hole cut for 45 mm drill holes.4

2.2. Swedish models

The tunnel face is divided into five separate sections as shown in Fig. 1. Similarly, the designcalculation includes the following parameters: (1) length of drillhole; (2) diameter of drill-hole; (3) linear charge concentration; (4) maximum burden; (5) type of explosive; (6) rockconstant; (7) fixation factor.5

The four-section cut is shown in Fig. 3(a). The length of drilling hole depends on thediameter of the empty hole as shown in Fig. 3(b). The distance between the empty holeand the blastholes in the first quadrangle should not be more than 1.7 times the diameterof the empty hole to obtain breakage and a satisfactory movement of the rock. Breakageconditions differ very much depending upon the explosive type, structure of the rock anddistance between the charge hole and the empty hole.

The burden for the lifters and stopping holes is in principle calculated with the same for-mula as for bench blasting. The bench height is just exchanged for the advance, and a higher

(a) (b)

0.00 0.05 0.10 0.15 0.200

2

4

6

Dril

l hol

e le

ngth

(m)

at 9

5% a

dvan

ce

Diameter of the empty hole(m)

Figure 3. (a) Four-section cut; (b) Drill hole length as a function of empty hole diameter for four-section cut.6

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fixation factor is used due to the gravitational effect and to a greater time interval betweenthe holes.6

The burden depends on the linear charge concentration, fixation factor, rock constant andexplosive type. A condition that must be fulfilled is B ≤ 0.6H where H is drillhole length. Thesame fixation factor (f = 1.45) is used for lifters and stopping holes in section B (breakagedirection horizontally and upwards, Fig. 1). The fixation factor for stopping holes in SectionC (breakage direction downwards, Fig. 1) is reduced to f = 1.20.

The spacing value of the lifter holes are equal to burden value (S/B = 1) and for bothtypes of stopping holes the spacing is 1.25 times the burden values (S/B = 1.25). Like the cutholes, the uncharged length of the lifter and stopping holes is 10 times the drillhole diameter.The linear charge concentration in the column and the bottom charge (1.25B) may differ; thecolumn charge can be reduced to 70% (of the bottom charge) for the lifter holes and 50% forthe stopping holes. This is, however, not always common since it is time-consuming chargingwork. Usually the same concentration is used both in the bottom and in the column.

3. Numerical Simulation Based on DDA

The DDA1 is a discrete element program to investigate the two-dimensional behaviour of thefractured rock mass which has been widely used to model the motions of blocky system inrock engineering recently. During the process of blasting, the rock mass will experience theprocess from continuous to discontinuous state. A developed DDA program special for thecrack formation analysis is constructed. In every time step, the program will automaticallycheck all joint boundaries with the Mohr’s and cut off (tensile) criteria. With the gray backcolour specimen, all the joint boundary lines are drawn gray at first, if the strength boundariesreach the criteria of failure, it means rock mass occurs failure in certain region, cracks willinitiate indicated by red lines in the next time step. The fractured block will thus detach withthe major part and move independently.

In this section, the DDA method is selected as the developed discontinuous method forsimulating the fracture generation of rock mass induced by the explosives. The objectiveis to study whether the DDA method could be applied as an useful numerical method forsuch practical problem. Considerations are mainly concentrated on the fracture generationprocesses on different loading history.

3.1. DDA model description

A DDA model is constructed based on the charge design in the experiment as shown inFig. 4(a). 5516 blocks have been used and each charge hole is modeled by six loading pointaround. The block configuration is generated by the input data from another software visu-alFEA which makes the mesh work on the DDA more efficient and accurate. The outsidefour boundaries are free and the rock mass could be moved outwardly free. Dynamic modeis applied and no damping effect is considered. The explosive is simulated by the load historycurve of each loading point. The simplified curve used in this model is shown in Fig. 4(b).The t in the figure is a unit which could be changed to any detail value for the considerationof different blast duration.

For rock mass, laboratory tests were conducted on the Bukit Timah granite at the field site.It was found that the granite at the site was of good quality. The average density of the rockwas 2650kg/m3, Possion’s ratio was 0.16,7 Young’s modulus of intact granite was 73.9GPa,uniaxial compressive strength was 186MPa, tensile strength was 16.1MPa.8 These data areadopted in this study.

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(a) (b)

0

10

20

3040

50

60

70

80

90

100

110120

130

140

150

160

0 1 2 3 4 5 6 7 8 9 10 11

Time (t-ms)

Pre

ssu

re (

MP

a)

Figure 4. (a) Block configuration of the model by the DDA method; (b) Loading history of the explo-sive in the DDA model.

3.2. Results discussions

3.2.1. Simultaneous blasting

After the numerical simulation, the next step is to conduct the crack configuration analysis.In this section, the basic introduction on how to read and understand the simulation resultsby the DDA method is given. Fig. 5(a) shows the cracks initiated around the charge holesafter explosion. The cracks are starting to generate and beginning to propagate in the radialdirection. During the radial cracks development, the cracks between the radial cracks couldmeet each other and form a closed loop. The others are continuing to propagate outward.Fig. 5(b) shows the formation of the detached body as highlight in yellow lines. More andmore cracks from the adjacent charge hole meet and form the closed district. When thecracks are enough, the closed loop will form a separate body detached with the other part.The cracks will also stop to propagate for the energy assumption during the form of thisdetached body. Fig. 5(c) displays two cracks modes in detail: cracks connected together andnon-connected.

In this blast design, the control of the detached body is an important technique. The pur-pose of the design is to produce the detached bodies sufficient with suitable size and lessimpact on the surrounding rock mass. Therefore, many factors e.g. the charge magnitude,coupled type, delay time into the blast design need to be considered.

3.2.2. Comparison of the final crack configuration of Cases 1, 2 and 3

The numerical simulation is conducted using the DDA method to study the effect of delaytime on the rock fragmentation.

In the Case 1, 12 charge holes are considered to blast at the same time. The t in Fig. 4(b)is taken as 0.1 which means the rising time to the peak pressure 150 MPa is equal to 0.1ms.The time interval of each time step is taken as 2.5 µs. The maximum displacement ratio inthe program is set as 0.001. This basic set is used in all the cases.

Same as the experiment designed, the charge holes will blast in sequence. The inside fourholes will initiate firstly, then after an interval, the outside four follow and finally the mid

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(c)

(a) (b)

Figure 5. (a) Crack initiation; (b) Detached body of the rock mass; (c) Connection and non-connectionof the cracks.

four will start at the third stage. See the sequence in Fig. 6. The time interval here is 1ms forCase 2 and 2 ms for Case 3, respectively. All the other sets are same as the Case 1.

The cracks surrounding the charge holes will start to occur soon after the initiation of theexplosives. The final crack generation will be stable after all the charge holes are blasted. Theprocess for Case 2 and Case 3 is similar only the start of the crack occurrence is different.

Here the results are put together as shown in Fig. 7. The yellow lines are sketched for thedetached body of rock mass. It could be clearly found that the Case 1 has the least amount ofdetached body and Case 2 generates most. This means for this experiment case, the sequenceblast with time interval delay is better than the simultaneous explosion. For loading timeinterval, the case of 1 ms delay will give the best blast effect.

4. Conclusions

Rock blasting is a rock excavation technique most widely used in the mining and construc-tion industry due to its reliability, economy and safety. In this blast design, the control of thedetached body is an important technique. The purpose of the design is to produce enoughfragmentation with suitable size and less impact on the surrounding rock mass. The numer-ical simulation could be carried out by the discontinuous method based on the concept offailure of the rock mass strength. The DDA method considered several different influence

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Figure 6. Blast sequence of the rock mass model.

Figure 7. Final Cracks configuration of Case 1, 2 and 3.

effects. The results show that the numerical method could be a good support for the futureblasting design for its efficiency, economy and feasibility.

References

1. Shi, G., Discontinuous deformation analysis — a new numerical model for the statics and dynamicsof block systems, in Civil Engineering. 1988. University of California: Berkeley.

2. Lin, C.T., Amadei, B., Joseph, J., Jerry, D. Extensions of discontinuous deformation analysis forjointed rock masses, International Journal of Rock Mechanics and Mining Sciences & Geome-chanics, 1996. 33(7), p. 671–694.

3. Zare, S., Bruland, A., Comparison of tunnel blast design models, tunneling and UndergroundSpace Technology, 2006. 21, p. 533–541.

4. NTNU, Project Report 2A-95 Tunnelling — Blast Design, NTNU, 1995. Department of Civil andTransport Engineering, Trondheim.

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5. Langefors, U., Kihlstrom, B., The Modern Technique of Rock Blasting, 1978. Third Ed. Almqvist& Wiksell Forlag AB, Stockholm.

6. Persson, P. A., Holmberg, R., Lee, J., Rock Blasting and Explosives Engineering, 2001. Sixthprinting. CTC Press, USA.

7. Hao, H., Wu, C., Zhou, Y., Numerical analysis of blast-induced stress waves in a rock masswith anisotropic continuum damage models. Part 1: equivalent material property approach. RockMechanics and Rock Engineering, 2002. 35(2), p. 79–94.

8. Hao, H., Wu, Y.K., Ma, G.W., Zhou, Y.X., Characteristics of surface ground motions induced byblasts in jointed rock mass. Soil Dynamics and Earthquake Engineering, 2001. 21(2), p. 85–98.

176

The Analysis of Structure Deformation Using DDA with ThirdOrder Displacement Function

T. HUANG∗, G.X. ZHANG AND X.C. PENG

China Institute of Water Resources and Hydropower Research, Beijing, China 100038

1. Introduction

Since the concept of Discontinuous Deformation Analysis (DDA) was proposed by Shi1 inlate 1980’s, DDA has become a rapidly developing new numerical modeling technique, andhas been more and more widely applied to engineering problems, such as underground tun-nels and caverns, analysis of slope stability, engineering blast etc.

In Current version of DDA program, the first order displacement functions were usedto approximate the block movement and deformation. Therefore, the stress and strain of ablock is constant, which degrades the calculating ability of DDA. To overcome the limitation,traditional methods are adding the artificial joints into the blocks2 or using FEM mesh todiscrete the blocks.3 A new mesh-free displacement approximation mode for DDA is alsoproposed,4 namely the MLS approximation, which never re-mesh blocks but uses mesh-freeinterpolative nodes scattering into blocks or along the borders. The third order displacementfunction is also implemented into the DDA code by some researchers.5–8 But there are alsotwo problems need to be solved for the DDA with third order displacement function. Onone hand, when it comes to nonlinear deformation, the straight edge block will change intoa curved block, followed by two questions: (1) Precision of curved edge simplex integrationproblem; (2) contact detect and simulation of curved edge block. On the other hand, it is thathow to transfer to the next step of initial stress of each step.

This paper will presents analysis of derivation and solve two problems above mentioned.

2. The Third Order Displacement Functions for Block Element

The first order displacement function is used to approximate the displacement field of theblock. The first order Taylor series can be expressed as follows:

u = a1 + a3x+ a5y

v = a2 + a4x+ a6y(1)

The six variables (u0,v0,r0,εx,εy,εxy) were used to describe the movement and deformationof individual block. Substituted the coefficients of displacement function (a1,a2,a3,a4,a5,a6)by six variables, formula (1) can be expressed in the following matrix form:[

uv

]= [T]2×6 [D]6×1 (2)

Where matrix [T]2×6 is the transformation matrix, and [D]6×1 is the displacement matrix.




X

Y P(x, y)

Figure 1. Definition of DDA block.

Based on the simplex integration, DDA algorithms are simple, efficient yet accurate. Theintegrations are simply represented by the coordinates of boundary vertices. In the case ofDDA with high order displacement functions, considering the deformation and contact ofblocks, one random shape block is formed by some small segments on the material boundary.Thus, DDA program will obtain accurately simplex integration of curved blocks and alsodeal with contact detection and contact simulation of curved blocks. Shown as Figure 1, thethird order displacement function of one block has the following form:

u = a1 + a3x+ a5y+ a7x2 + a9xy+ a11y2 + a13x3 + a15x2y+ a17xy2 + a19y3

v = a2 + a4x+ a6y+ a8x2 + a10xy+ a12y2 + a14x3 + a16x2y+ a18xy2 + a20y3

Rewrite above formula in matrix form, the displacement function can be expressed as:

[uv

]= [T]2×20 [D]20×1 (3)

in which [T]2×20 may be expressed as:

[T]2×20 =[1 0 x 0 y 0 x2 0 xy 0 y2 0 x3 0 x2y 0 xy2 0 y3 00 1 0 x 0 y 0 x2 0 xy 0 y2 0 x3 0 x2y 0 xy2 0 y3

](4)

the displacement variable matrix [D]T is represented:

[D]T = [d1 d2 d3 · · · d20]1×20 (5)

3. The Equilibrium Equations and Sub-Matrices Base on the Third OrderDisplacement Function

3.1. Coefficient matrix of simultaneous equations for block system

For a system with N blocks, the simultaneous equations may be derived by minimizing thetotal potential energy produced by the forces, stresses, and boundary constraints. And total

178


potential energy can be expressed in matrix form as follows:

∏= 1

2

[DT

1 DT2 DT

3 · · · DTN

]⎡⎢⎢⎢⎢⎢⎣

K11 K12 K13 · · · K1NK21 K22 K23 · · · K2NK31 K32 K33 · · · K3N

......

......

...KN1 KN2 KN3 · · · KNN

⎤⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎣

D1D2D3...

DN

⎤⎥⎥⎥⎥⎥⎦

+ [DT1 DT

2 DT3 · · · DT

N

]⎡⎢⎢⎢⎢⎢⎣

F1F2F3...

FN

⎤⎥⎥⎥⎥⎥⎦+ C (6)

where [Kii]20×20 is the stiffness matrix; [Fi]20×1 is the force matrix, [Di]20×1 is the unknownmatrix which is the displacement variables [di1 di2 di3 · · · di20]T

1×20 of each block andC is energy produced by friction force.

By minimizing the total potential energy, the simultaneous equations can be expressed inmatrix form as follows:

⎡⎢⎢⎢⎢⎢⎣

K11 K12 K13 · · · K1NK21 K22 K23 · · · K2NK31 K32 K33 · · · K3N

......

......

...KN1 KN2 KN3 · · · KNN

⎤⎥⎥⎥⎥⎥⎦

(20×N)×(20×N)

⎡⎢⎢⎢⎢⎢⎣

D1D2D3...

DN

⎤⎥⎥⎥⎥⎥⎦

(20×N)×1

=

⎡⎢⎢⎢⎢⎢⎣

F1F2F3...

FN

⎤⎥⎥⎥⎥⎥⎦

(20×N)×1

(7)

Following, initial stress in the formulation of DDA with third order displacement functionwas derived.

3.2. Initial stress matrix

Following the time sequence, the DDA computes step by step. Current version of DDA withthe first order displacement function compute stresses of previous time step will be settled asthe initial stress transferred to the next time step. Therefore the initial stresses are essentialfor DDA computation step by step. Whereas the initial stress within block are not necessarilyconstant in third order displacement function. The stress of a random point (x, y) is givenby:

⎛⎝σx0σy0τxy0

⎞⎠ = [E]

⎛⎝ εx0εy0γxy0

⎞⎠ = [E][Bi]3×20

[D0

i

]20×1= [E]

⎧⎪⎪⎨⎪⎪⎩

Bi1Bi2Bi3Bi4

⎫⎪⎪⎬⎪⎪⎭ [Di] (8)

179


[Bi1] =

⎡⎢⎢⎢⎢⎣

0 0 00 0 01 0 00 0 10 0 1

⎤⎥⎥⎥⎥⎦ ; [Bi2] =

⎡⎢⎢⎢⎢⎣

0 1 02x 0 00 0 2xy 0 x0 x y

⎤⎥⎥⎥⎥⎦ ;

[Bi3] =

⎡⎢⎢⎢⎢⎣

0 0 2y0 2y 0

3x2 0 00 0 3x2

2xy 0 x2

⎤⎥⎥⎥⎥⎦ ; [Bi4] =

⎡⎢⎢⎢⎢⎣

0 x2 2xyy2 0 2xy0 2xy y2

0 0 3y2

0 3y2 0

⎤⎥⎥⎥⎥⎦ (9)

The matrix [Bi]3×20 of third order displacement function has the form as formula (9),which is not same as matrix [B]3×6 of first order displacement function. The latter has theform as follows:

[B]3×6 =⎡⎣0 0 0 1 0 0

0 0 0 0 1 00 0 0 0 0 1

⎤⎦ (10)

In this case, transfer the initial stress to next step is same as transfer previous displacement.But in the case of DDA with third order displacement function, it is not convenient to transferthe stress of previous time step. It is difficult to integrate stress with simplex integrationbecause matrix [B]3×20 of third order displacement function includes unknown variables x,y. So transfer the displacement of previous time step to the next step is right and reasonable.

Therefore, the potential energy∏σ of the initial stress is given by:

∏σ

=∫∫

(εxσx0 + εyσy0 + γxyτxy0)dxdy =∫∫ [

εx εy γxy] ⎡⎣σx

σyτxy

⎤⎦dxdy

= [Di]T(∫

[Bi]T [E] [Bi] dxdy)[

D0i

](11)

Then the derivatives are computed to minimize the energy∏σ :

[f ]20×1 =∫∫

[Bi]T[E][Bi]dxdy[D0

i ] (12)

Sub-matrix [f ]20×1 is added to the sub-matrix [Fi] in the global Equation (7).

3.3. Simulation of contact

All DDA blocks are nonlinear deformation under plane stress condition. In this study, whentwo blocks come in contact,9 two directional springs, including normal contact spring andtangential contact spring, are setting at contact points. Contact mechanism is shown inFigure 2. Value of contact spring is related with the contact length, which is defined bybelow formulas (13): ⎧⎪⎨

⎪⎩Kn = L

2 kE (edge− edge)orKn = kE (corner− edge)Ks = Kn

2(1+υ)

(13)

where k is penalty coefficient, and its value is 10 to 100. In current program k = 20 is chosen.

180


Case: corner-edge Case: edge-edge

Ks

KN

L

i block

j block

Ks

KN

i block

j block

Figure 2. Contact mechanism of DDA with third displacement function.

4. Verification Sample

4.1. General stress and deformation problem

A cantilever beam is presented here for the purpose of validation and demonstration of theDDA3O. Figure 1 shows the analytical model. The left end is fixed, and right end is appliedvertical point load P, where P = 1T. The cantilever has a span of 1m, a height of 0.1m.Material properties of the beam were assumed to be E = 105T/m2, v = 0.2, a densityρ = 2.4T/m3, and the gravity acceleration is ignored.

The closed form solution for the beam deflection v along its axis can be written as below:

v = px2

6EI(x− 3L) (14)

where L is the length of the beam; I is the moment of inertia of beam cross section.The stress σx of analytical solution of the beam is:

σx = MyI

(15)

where M is moment of concentrated P force; I is the moment of inertia of beam cross section.As shown in Figures 4 and 5, solution of the deflection and stress of DDA3O is close to

analytical solution, and the maximum relative error is no more than 1.5%. It is shown theDDA3O can accurately describe the deformation and stress of a single block.

Figure 3. Cantilever beam configuration and calculating mesh of DDA3O.

181


0.0 0.2 0.4 0.6 0.8 1.0

-0.008

-0.006

-0.004

-0.002

0.000

Def

lect

ion

in y

dire

ctio

n/m

Distance from the fixed end(m)

DDA3O's Result Theoretical Result

Figure 4. Deflection of cantilever beam calculated by analytical solution and DDA3O.

0.0 0.2 0.4 0.6 0.8 1.0-700

-600

-500

-400

-300

-200

-100

0

xal

ong

the

botto

mof

cant

ileve

r/104

Pa

Distance from the fixed end

Theoretical SolutionSimulation Result by DDA3O

Figure 5. Stress of cantilever beam calculated by analytical solution and DDA3O.

5. Conclusions

The DDA with a simple linear displacement function is only suitable for simple blocks sys-tem calculation. Without adding artificial joints and FEM mesh, the numerical model andformulations of DDA with third order displacement function is presented in this paper. Thestress field and deformation field are greatly improved. Some conclusions can be obtain asfollows:

• Initial stress transfer and Simplex Integration issues are successively solved.• The deformation and stress and strain in one block can be obtained, which can

greatly enhance the application range of DDA.

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• The large deformation, material nonlinear mechanical response can be simulatedwith DDA3O. In small deformation case, DDA3O can give as well as analyticalresult within one block. DDA3O has great potential ability to simulate and analyzestatic and dynamics problem.

Acknowledgements

The authors are very grateful to Dr. Genhua SHI for his valuable guidance and encour-agement. The work has been supported by National Natural Science Foundation of China(Contact Numbers: 50539010, 50539120).

References

1. Shi, Gen-Hua. Discontinuous Deformation Analysis-A New Numerical Model for the statics andDynamics of Block Systems, Ph.D Dissertation, Department of Civil Engineering, University ofCalifornia at Berkeley (1988).

2. Ke, Te-chih. Artificial Joint-Based DDA, proceedings of the First International Forum on Discon-tinuous Deformation Analysis (DDA) and Simulation of Discontinuous Media, June 12-14, TSIPress: USA, 1996:326–333.

3. CHUNG-YUE WANG, CHING-CHIANG CHUANG, JOPAN SHENG. Time Integration The-ories for the DDA Method With Finite Element Meshes, proceedings of the First InternationalForum on Discontinuous Deformation Analysis (DDA) and Simulation of Discontinuous Media,June 12–14, TSI Press: USA, 1996:263–287.

4. Ma Yong Zheng, Jiang Wei, Huang zhe Chong, Zheng Hong. A new meshfree displacementapproximation mode for DDA method and its application, Proceedings of the Eighth internationalconference on analysis of Discontinuous Deformation Fundamentals & Applications to Mining &Civil Engineering, August 14–19, Beijing, 2007:81–88.

5. Koo, C.Y. and Chern, J.C. The development of DDA with third order displacement function,proceedings of the First International Forum on Discontinuous Deformation Analysis (DDA) andSimulation of Discontinuous Media, June 12–14, TSI Press: USA, 1996:342–349.

6. WANG Xiao-Bo, DING Xiu-Li, LU Bo, Wu Ai-Qing. DDA with high order polynomial displace-ment functions for large elastic deformation problems, proceedings of the 8th International Con-ference on Analysis of Discontinuous Deformation, August 14–19, Beijing, 2007:89–94.

7. Max Y. Ma, M. Zaman, J.H. Zhu. Discontinuous Deformation Analysis Using the Third OrderDisplacement Function, proceedings of the First International Forum on Discontinuous Defor-mation Analysis(DDA) and Simulation of Discontinuous Media, June 12–14, TSI Press: USA,1996:383–394.

8. S.A.Beyabanaki, A.Jafari, M.R.Yeung and S.O. Biabanaki. Implementations of a trilinear hexa-hedron mesh into Three-Dimensional Discontinuous Deformation Analysis. Proceedings of theEighth international conference on analysis of Discontinuous Deformation Fundamentals & Appli-cations to Mining & Civil Engineering, August 14–19, Beijing, 2007:51–55.

9. G.X Zhang, X.F Wu. Influence of Seepage on the Stability of Rock Slope – Coupling of Seepage andDeformation By DDA Method[J], Chinese Journal of Rock Mechanical and Engineering, 2003,22(8):1269–1275.

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Application of DDA to Evaluate the Dynamic Behaviour ofSubmarine Landslides Which Generated Tsunamis In theMarmara Sea

G.C. MA1,∗, F. KANEKO2 AND S. HORI3

1Gifu University, Japan2OYO International Corporation, Japan3DPT corporation, Japan

1. Introduction

According to the catalogues of historical earthquakes and tsunamis in the Marmara Sea,their sources would be located along the North Anatolian Fault (NAF). The dates of pasttsunamis are mostly correlating to those of past earthquakes with magnitude M > 7. Also,a big tsunami traces has been recently discovered at the construction point of the railwaystation for the new lines under the Bosporus strait. That was the past port in the southerncoast of old Istanbul city. This makes clear that the past big tsunamis attacked Istanbul coastat around every half a millennium.1

However, since NAF is a strike slip type, only small tsunamis can be thought to haveoccurred. Further, the estimated tsunami heights due to the NAF segments obtained bytsunami simulation show small vertical values that cannot reach those of the past tsunamiheights. Therefore, not only the movement due to the active fault segment of the NAF, butalso the landslides along the cliffs generated and followed by the movement of NAF, have tobe considered as the possible tsunami sources.

Thus, in order to estimate the tsunami effects exactly, the evaluation of the dynamicbehaviour due to submarine landslides becomes very important. For this purpose, the threedimensional landslide simulations should generally be adopted. But, since most of the meth-ods were developed for ground landslide simulation, these methods cannot directly considerthe resistance of the seawater. In order to solve this problem, a two dimensional simulation byusing DDA was attempted. The resistance of seawater was considered by using the viscositycoefficient, which was used in the rock fall simulation to express the arborous resistance.2–4

In this study, firstly, the development of DDA due to simulate the behaviour of the sub-marine landslide were summarized. Next, an application to evaluate the dynamic behaviourof submarine landslide was executed. In the application, firstly, the topographic analysis wascarried out to identify vulnerable slope slipped or may slip by using the existing detailedbathymetry data; Next, identification simulation was performed to decide the viscosity coef-ficient by simulating the past landside trace which was estimated by topographic analysis;Finally, estimating simulation was performed to simulate the vulnerable slope when theystate to slip. This paper reports the outline and results of the topographic analysis, identifi-cation simulation and the estimating simulation.

2. Development of DDA

2.1. Modeling of discontinuities

Recently, DDA has been popularly applied to Rock-fall analysis2–4 and DDA can be appliedto estimate mass movement of landslide, blasting5 etc.




By DDA, discontinuities were handled as a boundary surface of a rock mass block. Inother words, rock mass block is surrounded with the discontinuities, but blocks were dealtwith as homogeneous rock mass without discontinuities. Therefore, the strength evaluationof discontinuities expressed mechanical relations in contact or separation between rock massblocks, and it has to be using different mechanics models for the discontinuities existing oroccurring with a failure. Both discontinuities would need to obey Mohr-Coulomb criterion,but failure standards will need to be viewed differently.

We have supposed the failure criterion of new discontinuity obeys Coulomb criterion.In addition, when a crack was judged to form, the shear strength and tension strength ofthe potential surface would become zero together, but supposing the friction angle does notchange. And the potential surface would be has behaviour as a new crack distinguished froman existing crack after failure.

2.2. Estimate of rock mass strength

By weathering, erosion and scale effect, the strength of rock mass around the potential surfaceshould be much smaller than the strength provided from the rock examination using theintact rock. About this, we have supposed that compressive strength and tension strengthof a rock mass would fall off simply with an agreed ratio from its of a rock materials, andcalls the coefficient k as “strength reduction rate”. And the strength reduction rate have beendiscussed by reproduced an actual rock mass failure.6, 7

2.3. Energy loss by the collision between blocks

Using DDA, we can simulate the response of rock block systems under general loading andboundary conditions considering rigid body movements and deformations simultaneously.This method is suitable because large deformations, such as sliding, jumping and rotatingmotions of rock fall can be simulated. But, the actual motions of rock mass have not beensimulated adequately so that energy loss of rock mass has not been calculated by the originalDDA.

The energy loss with collision caused by plastic deformation of rock mass is very impor-tant to simulate rock mass. Hence, DDA should be improved to consider energy loss duringcollision between blocks. Finally, we have developed the improved DDA to simulate energyloss of rock mass caused by slope absorbability.

Energy loss of rock falls caused by collision is given by a factor β as follows:

F′ = (1− β)× F (1)

where, β is an energy loss factor, which means collision damping coefficient between blocks.F′ and F are the reaction forces with and without an energy loss, respectively (Fig. 1). Thereaction force F is given as:

F = m× a (2)

where, a is the acceleration of rock fall, m is the mass of rock fall.Equation (2) indicates that the reaction force depends on the acceleration and mass of

blocks. The coefficient β can be used to improve the original DDA by modifying the acceler-ation of rock fall reduced by collision as follows.

[v′] = �t[a′]+ [v0] (3)

[a′] = (1− β)[a] (4)

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FF’

Rock fall

Ground

F : Reaction forces withoutan energy loss F’ : Reaction forces with an energy loss

Figure 1. Energy loss model considering collision.2,4

where, [a] is the acceleration calculated at current step (Equation (5)), [v0] is the initial veloc-ity of block for the current step calculation, and �t is the time interval. [a’] and [v’] are themodified acceleration and initial velocity for the next step calculation.

[a] = ∂2[D(t)]∂t2

= 2�t2

[Di]− 2�t

[v0] (5)

where [Di] is the deformation calculated at current step.

2.4. Resistance of Seawater

The resistance of seawater was considered by using the viscosity coefficient, which was usedin the rock fall simulation to express the arborous resistance.2, 3, 8 In the rock fall simulationusing DDA to analyze arborous resistance force such as energy loss when the block goesdown a slope, we utilize the viscous force to represent the effects. In specific, the resistanceforce corresponds to the volume and falling speed of block. As well as rock fall simulation,the resistance of seawater can be considered as viscous force. We assumed the followingequation as viscous force.

(fxfy

)= μd

�t

(uv

)(6)

where fx, fy — Viscous force per unit volume, �t — Time step, u and v — Increment ofdisplacement, μd: Viscosity coefficient.

The potential energy accumulated with viscous force resistance can be expressed with thefollowing equation.

∏v=∫s

(u v)(

fxfy

)ds (7)

where s — Area of rock mass block.It is possible to analyze the effects of viscous force by differentiating the displacement

variables of potential energy and applying them in the conventional DDA formulation.

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3. Application to Evaluate the Dynamic Behaviour of Submarine Landslide

3.1. Topographical analysis

Fortunately the precise bathymetry data (20m grids) in Marmara Sea bottom is available,shown in Fig. 2.9 Using the data, the topographic analysis was conducted for identificationof vulnerable slopes that may generate disastrous tsunamis. Some criteria were given by exist-ing information, for example scale of slopes,10 past activities of active faults11 etc. Thoughthe analysis was based on land slopes, the situation at submarine level was taken into consid-eration in so much as we included sea water resistance, weathering by sea water and activemovements by the North Anatolian Fault etc. Soil samplings with dating tests at sea bottomsurface certificate the trace of the past events of slope failure, and past tsunami records mightsuggest the possibility of generating tsunami by submarine landslides during or just after theearthquakes.12 Consequently, the 10 vulnerable slopes are identified, including 3 past eventsdebris sites for verification of methodology.

Bathymetry (m)

Figure 2. Bathymetry and Topography around the Marmara Sea including Istanbul.11

3.2. Identification simulation for the seawater resistance

Identification simulation of past event was performed to decide the viscosity coefficient bysimulating the past landside traces which identified by topographic analysis. Figure 3 showsthe selected two vulnerable slope locations to be examined for identification simulation.Figure 4 is the cross section of the vulnerable slopes for “Identification Simulation East”.From the shapes, the simulation model was established as shown in Fig. 5. Soil propertieswere given from the results of the seismic microzonation study at Istanbul municipality.11

After try and error analysis for the sensitivity of key properties like μd, β and angle offriction etc, are examined. Figure 6 shows the simulation results of Identification simulationEast. In this simulation, μd was 0.12, β was 0.7, angle of friction between slide body and bedwas 8 degrees are thought to be reasonable to reproduce the past landslide event. The shapeof the surface simulated reproduced generally the current geography shown in Fig. 7. Andby applying the shape variation along time and the velocity simulated (The average velocityshown in Fig. 6) into tsunami simulation as input data, the results of tsunami simulations

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Identification simulation West Identification

simulation East

Used for Identification simulation

Used for Estimating simulation

Estimating simulation East

Figure 3. Vulnerable slopes identified by topographic analysis.

EL

(m)

0

-200

-400

-600

-800

-1000

-1200

-14002500 3000 3500 4000 4500 5000 5500 6000 6500 7000

Distance (m)

Section 07Landslide LineOld geographyCurrent geography

Figure 4. Section of Vulnerable slopes identified by topographic analysis (Identification simulationEast).

6900 2500 -1550

-95

Unit : m

Figure 5. Simulation model of Identification simulation East shown in Fig. 6 and Fig. 7.

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d : 0.12, :0.7, Angle of friction between slide body and bed: 8 degree

6900 2500 -1550

-95

Unit : m

Figure 6. Simulation Results of Identification simulation East.

have been compared with historical records of tsunamis. The estimated tsunami height dueto submarine landslides could approach the level of the past tsunami heights of the records.

3.3. Estimating simulation for the vulnerable slope

Estimating simulation was performed to simulate the vulnerable slope when they start to slip.Figure 7 and 8 shows the simulation results of estimating simulation. By using the estimatedvelocity and the shape of surface into tsunami simulation, we have estimated tsunami heightdue to submarine landslides.

4. Conclusions

In this paper, in order to simulate the dynamic behaviour of submarine landslide, the DDAwere developed by adopting the energy loss model which was explained the energy losscaused by collision between blocks, and considered the seawater resistance as viscous force.

By the application, we mostly reproduced the submarine landslide and estimated the move-ment behaviour of the vulnerable slope. It was cleared that the simulated results was closelydepended to the shape of slide line, shape of surface and the seawater resistance, but not soclosely to the strength, shape of the discontinuity. If the slide line and the geography beforeslip of the past submarine landslides, the past submarine landslide can be reproduced accu-rately, and the seawater resistance would be identified accurately. Further, if the possible slideline of the vulnerable slope can be truly presumed, the behaviour of the vulnerable slope canbe estimated exactly.

190


step: 9000 time: 107s



step: 60500 time: 1006s

6451 0-1508

-78

Unit : m

Figure 7. Simulation results of estimating simulation (Estimating simulation East).

Velo

city

[m/s

]

Maximum average velocity:3.6m/s

6

5

4

3

2

1

00 500 1000 1500 2000 2500 3000

Time [s]

Maximum velocityAverage velocity

Figure 8. Simulated velocity of estimating simulation (Estimating simulation East).

191


References

1. Kanako, F., 2009. A Simulation Analysis of Possible Tsunami affecting the Istanbul Coast, Turkey,Proc. Of the International Workshop on Tsunami and Storm Surge Hazard Assessment and Man-agement in Bangladesh, Dhaka, 41–50.

2. Ma, G.C., Matsuyama, H., Nishiyama, S. and Ohnishi, Y., 2007. Study on analytical method forrock fall simulation, Journals of the Japan Society of Civil Engineers, Vol. 63, No. 3, 913–922 (inJapanese).

3. MA, G.C., Yashima, A., Nishiyama, S., Ohnishi, Y., Monma, K. & Matsuyama, H., 2008. Applica-tion of DDA to evaluate the mechanism of rock slope instability/rock fall, The Third InternationalSymposium on Modern Mining and Safety Technology (ICSSE), Fuxin, 67–78, 2008.

4. Ma, G.C., Nakanishi, A., Ueno, S., Mishima, S., Nishiyama, S. and Ohnishi, Y., 2004. Study onthe applicability of rock fall simulation by using DDA, In Proc. of the ISRM Symp. 3rd Asia RockMechanics Symp., Kyoto, 1233–1238.

5. Ma, G.C., Miyake, A., Ogawa, T., Wada, Y., Ogata, Y. and Katsuyama, K., 1995. Numericalsimulations on the blasting demolition by DDA method, Proc. of 2nd International Conference onEngineering Blasting Technique (ICEBT), Kunming, 79–84.

6. Ma, G.C., Ohnishi, Y. and Monma, K., 2007. Application of DDA to Evaluate the Mechanism ofRock Slope Instability, Proc. Of ICADD-8, Beijing, 285–295.

7. Monma, K., Chida, Y., Ma, G.C., Shinji, M. and Ohnishi, Y., 2004. Study on the application ofdiscontinuous deformation analysis to evaluate the mechanism of rock slope instability, Journalsof the Japan Society of Civil Engineers, No. 757/III-66, 45–55 (in Japanese).

8. Shinji, M., Ohno, H., Otsuka, Y. and Ma, G.C., 1997, Viscosity coefficient of the rock-fall simu-lation, Proc. of ICADD-2, Kyoto, 201–210.

9. TUBITAK/IFREMER, 2005. Bathymetrical Map for the Marmara Sea.10. Hebert, H., F. Schindele, Y. Altinok, B. Alpar and C. Gazioglu, 2005. Tsunami hazard in the

Marmara Sea (Turkey) : a numerical approach to discuss active faulting and impact on the Istanbulcoastal areas. Marine Geology, 215, 23–43.

11. IMM (Istanbul Metropolitan Municipality), 2007. Seismic Microzonation for Istanbul Metropoli-tan Municipality, European side.

12. McHugh, C.M.G., L.Seeber, M.H. Cormier, J. Dutton, N. Cagatay, A. Polonia, W. B. F. Ryan, andN. Gorur, 2006. Submarine earthquake geology along the North Anatolia Fault in the MarmaraSea, Turkey: A model for transform basin sedimentation, Earth and Planetary Science Letters, 248,661–684.

192

3D DDA vs. Analytical Solutions for Dynamic Sliding of aTetrahedral Wedge

D. BAKUN-MAZOR1,∗, Y.H. HATZOR1,2 AND S.D. GLASER3

1Department of Geological and Environmental Sciences, Ben-Gurion University of the Negev, Beer-Sheva, Israel2Department of Structural Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel3Department of Civil and Environmental Engineering, University of California Berkeley, CA. USA

1. Introduction

In this research, the validity of the three dimensional Discontinuous Deformation Analysis(3D-DDA)1 is examined using analytical solutions for three dimensional problems involvingtwo different failure modes: 1) dynamic sliding of a single block on an inclined plane, and 2)dynamic sliding of a tetrahedral wedge simultaneously on two faces.

From the early nineties, researchers in the DDA community have documented the accuracyof the original two-dimensional method (2D-DDA) by performing validation studies withrespect to analytical solutions, by comparison with results of other numerical techniques, andfrom laboratory and field data. A paper by MacLaughlin et al.2 contains a summary of nearly100 published quantitative validation studies. With respect to dynamic loading and response,several works [e.g. Refs. 3, 4] have calibrated 2D-DDA results with respect to the Newmarkmethod5 and the Goodman and Seed6 solution. Wartman et al.7 investigated the analyticalimplementation of the Newmark method and Goodman and Seed solution with laboratorytests, using physical tests of a block sliding on a tilting and shaking table. Tsesarsky et al.8

used Wartman’s data to explore the validity of the 2D-DDA results for dynamic loading.As expected from numerical forward modelling analysis, the input parameters, such as thecontact spring stiffness, the boundary conditions, and time interval, have a decisive influenceon the accuracy of the output results.

Recently the validity and accuracy of 3D-DDA has been explored, yet only preliminary orpartial work on this subject has been published to date.9–14 The reason may be due to thedifficulty in developing a complete contact theory that governs the interaction of many 3Dblocks.14 Considering 3D validations, Shi1 reports very high accuracy for two examples ofblock sliding modelled with 3D-DDA, subjected to gravitational load only. Moosavi et al.11

compare 3D-DDA results for dynamic block displacement with an analytical solution. Yeunget al.13 validate the wedge stability analysis method using physical models and field casehistories, and report a good agreement between physical and numerical results in terms ofboth the effective failure mode and the block displacement history, although no quantitativecomparison between 3D-DDA and lab test results is reported.

In this study, an independent mathematical solution for dynamic block sliding in 3D isdeveloped based on the vector analysis (VA) formulation presented by Goodman and Shi.15

The developed 3D solution employs static formulation of the force balance on the block ateach time-step, according to the assumed sliding mode. The incremental sliding force or accel-eration thus calculated is integrated numerically twice to yield the three displacement com-ponents (x,y,z) versus time t.16 We first compare the developed VA and existing Newmarksolutions, and then proceed with the developed 3D VA solution in validation of 3D-DDA.1




2. Analytical Formulations of Vector Analysis Solution

2.1. Limit equilibrium equations

The static limit equilibrium equations formulated for each time step are discussed in thissection for both single face and double face sliding. Note that the expected failure modemust be known in advance to formulate these equations. Furthermore, in all cases studiedhere the static and dynamic resultant forces are applied to the centroid of the sliding block,this is slightly in contrast to the physical reality where the input motion is applied to thefoundation upon which the block rests.

2.1.1. Single face sliding

A typical model of a block on an incline is illustrated in Fig. 1(a). The dip and dip directionangles are, α = 20◦ and β = 90◦, respectively. Although it is a simple 2D problem, themodel is plotted as if it were 3D to demonstrate the robust VA solution. For this purpose, aCartesian coordinate system (x,y,z) is defined where X is horizontal and points to east, Y ishorizontal and points to north, and Z is vertical and points upward. The normal vector ofthe inclined plane is: n = [nx,ny,nz], where:

nx = sin (α) sin (β)

ny = sin (α) cos (β)

nz = cos (α)

(1)

The force equations presented below refer to a block with a unit mass. Hence, these equa-tions can be discussed in terms of accelerations. The resultant force vector that acts on thesystem at each time-step is r = [rx, ry, rz]. The driving force vector that acts on the block (m),namely the projection of the resultant force vector on the sliding plane, at each time step is:

m = (n× r)× n (2)

The normal force vector that acts on the block at each time step is:

p = (n · r)n (3)

Figure 1. Typical models of sliding blocks in 3D coordinate system. (a) Block on single face inclinedto 20/090, (b) a tetrahedral wedge sliding on two faces, the line of intersection is inclined 30◦ belowNorth and inclination of the boundary faces are 52/063 and 52/296.

194


At the beginning of a time step, if the velocity of the block is zero then the resisting forcevector due to the interface friction angle φ is:

f ={− tan (φ)

∣∣p∣∣ m, tan (φ)∣∣p∣∣ < |m|

−m, else(4)

where m is a unit vector in direction m.If, at the beginning of a time step the velocity of the block is not zero, then:

f = − tan (φ)∣∣p∣∣ v (5)

where v is the direction of the velocity vector.In an unpublished report,16 the author refers only to the case of a block subjected to

gravitational load, where the block velocity and the driving force have always the same sign.However, in dynamic input loading cases, momentary driving force could be opposite to theblock velocity.

2.1.2. Double face sliding

Double face sliding, or wedge analysis stability is a classic problem in rock mechanics thathas been studied by many authors.17–19 A typical model of a wedge is shown in Fig. 1(b). Thenormal to plane 1 is n1 = [nx1,ny1,nz1] and the normal to plane 2 is n2 = [nx2,ny2,nz2]. Con-sider a block sliding simultaneously on two boundary planes along their line of intersectionI12, where:

I12 = n1 × n2 (6)

The resultant force in each time step is as before r = [rx, ry, rz], and the driving force ineach time step is:

m = (r · I12)I12 (7)

The normal force acting on plane 1 in each time step is p = [px,py,pz], and the normalforce acting on plane 2 in each time step is q = [qx,qy,qz], where:

p = ((r× n2) · I12)n1 (8)

q = ((r× n1) · I12)n2 (9)

As in the case of single face sliding, the direction of the resisting force (f ) depends uponthe direction of the velocity of the block. Therefore, as before, in each time step:

f =

⎧⎪⎨⎪⎩−(tan (φ1)|p| + tan (φ2)|q|)m, V = 0 and (tan (φ1)|p| + tan (φ2)|q|) < |m|−m, V = 0 and (tan (φ1)|p| + tan (φ2)|q|) ≥ |m|−(tan (φ1)|p| + tan (φ2)|q|)v, V �= 0

(10)

2.2. Dynamic equations of motion

The sliding force, namely the block acceleration during each time step, is s = [sx sy sz] and iscalculated as the force balance between the driving and the frictional resisting forces:

s = m+ f (11)

The block velocity and displacement vectors are V = [Vx, Vy, Vz] and D = [Dx, Dy, Dz],respectively. At t = 0, the velocity and displacement are zero. The average acceleration for

195


time step i is:

Si = 12

(si−1 + si) (12)

The velocity for time step i is therefore:

Vi = Vi−1 + Si�t (13)

It follows that the displacement for time step i is:

Di = Di−1 + Vi−1�t+ 12

Si�t2 (14)

Due to the discrete nature of the VA algorithm, sensitivity analyses were performed todiscover the maximum value of the time increment for the trapezoidal integration methodwithout compromising accuracy. The results are found to be sensitive to the time intervalsize as long as the friction angle is greater than the slope inclination. We find that the timeincrement can not be larger than 0.001 sec to obtain accurate results.

3. Results

The validity of the 3D VA formulation presented in Section 2 is tested using the classicalNewmark solution for the dynamics of a block on an inclined plane. Once the validity ofthe VA approach is confirmed we proceed to check the validity of 3D-DDA using the VAapproach.

3.1. Single face sliding

The typical Newmark solution requires condition statements and is solved using a numer-ical time steps algorithm as discussed for example by Kamai and Hatzor.4 We relate hereto the Newmark’s procedure as the ’analytical solution’, to distinguish between the analyt-ical approach and the VA and DDA solutions. Fig. 2(A) shows a comparison between theanalytical (Newmark), VA, and 3D-DDA solutions for a plane with dip and dip direction ofα = 20◦ and β = 90◦, respectively, and friction angle of φg 30◦. We use for dynamic loadinga sinusoidal motion in the horizontal X axis, so the resultant input acceleration vector isr = [rx ry rz] = [0.5g sin (10t)0 − g]. The accumulated displacements are calculated up to10 cycles (tf = 2π sec). The input horizontal acceleration is plotted as a shaded line and theacceleration values are shown on the right hand-side axis. The theoretical mechanical prop-erties as well as the numerical parameters for the 3D DDA simulations are listed in Table 1.For both the Newmark and VA methods the numerical integration is calculated using a timeincrement of �t = 0.001 sec. For the 3D methods (VA and 3D-DDA), the calculated dis-placement vector is normalized to one dimension along the sliding direction.

An excellent agreement is obtained between the VA and analytical solutions throughoutthe first two cycles of motion. There is a small discrepancy at the end of the second cyclewhich depends on the numerical procedures and will decrease whenever the time incrementdecreases. The relative error of the VA and 3D-DDA methods with respect to the exist-ing Newmark solution is shown in the lower panel of Fig. 2A, where the relative error isdefined as:

Erel = |DNewmarkl −Dnumerical||DNewmarkl| · 100% (15)

The relative errors for both VA and 3D DDA are found to be less the 3% in the finalposition.

196


0

0.4

0.8

1.2

1.6

Dis

plac

emen

t (m

)

0.01

0.1

1

10

100

1000

Rel

ativ

e E

rror

(%

)

0 2 4 6Time (sec)

-6

-4

-2

0

2

4

6

Hor

izon

tal I

nput

mot

ion

(m/s

2 )

0.01

0.1

1

10

100

1000Erel, VAErel, DDA

Newmark SolutionVector Analysis3D-DDAInput Motion

A

0

0.5

1

1.5

2

2.5

Dis

plac

emen

t (m

)

0.1

1

10

100

Rel

ativ

e E

rror

(%

)

0 2 4 6Time (sec)

-6

-4

-2

0

2

4

6

Hor

izon

tal I

nput

mot

ion

(m/s

2 )

0.1

1

10

100

Vector Analysis3D-DDAInput Motion (x)Input Motion (y)

B

Figure 2. Block displacement vs. time for the case of a block on an incline subjected to gravitationaland cycles of horizontal sinusoidal loading. (A) Comparison between the Newmark solution, VA and3D-DDA for 1D horizontal input motion along the X axis. The relative error for the numerical solutionsis plotted in the lower panel where the Newmark solution is used as a reference. (B) Comparisonbetween VA and 3D-DDA for 2D horizontal input motion along the X and Y axes simultaneously. Therelative error is plotted in the lower panel where the VA is used as a reference.

After the verification procedure with respect to the existing Newmark solution has beensuccessfully completed, the VA algorithm is found to be suitable to serve as a reference solu-tion for 3D dynamic problems that are examined using 3D DDA. Fig. 2(B) shows a compar-ison between the VA solution and 3D DDA results for block sliding on an inclined plane aspresented in Fig. 1(a) and subjected to two components of dynamic, horizontal, input load-ing. The resultant input acceleration vector is r = [rx ry rz] = [0.5g sin (10t) 0.5g sin (5t)−g],and the friction angle is again φg 30◦. The two components of the input horizontal accelera-tion are plotted as shaded lines and the acceleration values are shown on the right- hand sideaxis. Note that the relative error presented in the lower panel now refers to the VA solutionand defined here as:

Erel =∣∣DVectorAnalysis −D3D DDA

∣∣∣∣DVectorAnalysis∣∣ · 100% (16)

The relative error in the final position in this simulation is approximately 8%.

3.2. Double faces sliding

A comparison between VA and 3D-DDA for the dynamic sliding of a wedge is shown inFig. 3(A) using cumulative displacement versus time. The input acceleration is now definedby a sinusoidal curve on the horizontal Y axis, and gravitational load on the vertical Z axis,so the resultant dynamic load vector is r = [rx ry rz] = [0 0.5g sin (10t)more− g]. The lineof intersection between the two planes is inclined 30◦ below the Y axis, and the orientationsof the bounding planes are 52/063 and 53/296, as illustrated in Fig. 1(b). The studied frictionangles of the planes are φ1 = φg = 20◦; all other numerical control parameters are listed inTable 1. The relative error is calculated using equation 16 and plotted in the lower panel ofFig. 3(A).

197


p p

0

0.5

1

1.5

2

2.5

Dis

plac

emen

t (m

)

0.1

1

10

100

Rel

ativ

e E

rror

(%

)

0 0.4 0.8 1.2 1.6 2Time (sec)

-6

-4

-2

0

2

4

6

Hor

izon

tal I

nput

mot

ion

(m/s

2 )

0.1

1

10

100

Vector Analysis3D-DDAInput Motion (y)

A

0

0.4

0.8

1.2

1.6

Dis

plac

emen

t (m

)

-15-55

15

a x

0 1 2 3 4 5Time (sec)

Vector Analysis3D-DDA ; k=10 MN/m3D-DDA ; k=1 MN/m

-15-55

15

a y

-25-15

-55

a z

0

0.4

0.8

1.2

1.6

Inpu

t Mot

ion

(m/s

2 )

B

Figure 3. Dynamic sliding of a wedge: comparison between 3D-DDA and VA solutions. (A) Wedgeresponse to one component of horizontal sinusoidal input motion and self weight. lower panel presentsthe relative error calculated according to equation 16. (B) Wedge response to 3D loading using datafrom the Imperial Valley earthquake (the three components, multiplied by a factor of 5, are shown inthe lower panel).

The response of the modelled wedge to the Imperial Valley earthquake recorded as mea-sured in El-Centro, CA, is studied and presented in Fig. 3(B). The lower panel presents thethree components of the recorded signal multiplied by a factor of 5 to obtain meaningfuldisplacements. The modelled friction angles of the boundary planes here are φ1 = φ2g = 30◦so that the block is at rest under gravity load only. Two different numerical spring stiffnessvalues are studied and the obtained results are plotted with comparison to the VA solution.

Table 1. Numerical parameters for all 3D DDA forward modelling simulations andVA algorithm.

Model Type Single plane Wedge, sine curve Wedge, El Centro(Figure) (Fig. 2A,B) (Fig. 3A) (Fig. 3B)

Mechanical Properties:Elastic Modulus, MPa 20000 20000 20000Poisson’s Ratio 0.25 0.25 0.25Density, kg/m3 1000 1000 1000Friction angle, Degrees 30 20 30Numerical Parameters:Dynamic control parameter 1 1 1Number of time steps 628 800 2500Time interval, Sec 0.01 0.0025 0.002Assumed max. disp. Ratio, m 0.01 0.005 0.002Penalty stiffness, MN/m 10 50 1 and 10Max. time step for VA, Sec 0.001 0.01 0.001

198


4. Conclusions

• The newly developed VA algorithm can be utilized to validate 3D numerical solu-tions, for example dynamic sliding of a block on a single plane and along two planessimultaneously. Since the resisting frictional force direction depends upon the slidingdirection a set of condition statements must be implemented in the VA solution toobtain the correct solution at each time step.• We report here very good agreement between VA solution and results obtained with

the existing 2D Newmark solution for dynamic sliding of a block on an incline.• 3D DDA is validated here using the VA solution for cases of dynamic sliding and

an excellent agreement is found using both synthetic and real earthquake records asdynamic input for both single and double plane sliding.• We want to note here that the resultant input acceleration in this study is always

applied to the centre of mass of the sliding block, in all Newmark, VA, and 3D-DDAtypes of analyses. In the physical reality, however, the dynamic input is applied tothe foundations and the block responds dynamically to the induced vibrations at thefoundations. Further physical tests must be conducted, for example using carefullymonitored shaking table experiments, to explore wave propagation behaviour fromthe shaking foundation to the responding block.

Acknowledgements

Financial support from the U.S. – Israel Bin-national Science Foundation (BSF) through con-tract 2004122 is gratefully acknowledged.

References

1. Shi G.H., "Three dimensional discontinuous deformation analysis", 38th US Rock MechanicsSymposium, Washington, DC, 2001, pp. 1421–1428.

2. MacLaughlin M.M., Doolin D.M., "Review of validation of the discontinuous deformation anal-ysis (DDA) method", Int J Numer Anal Met, 30, 4, 2006, pp 271–305.

3. Hatzor Y.H., Feintuch A., "The validity of dynamic block displacement prediction using DDA",Int J Rock Mech Min, 38, 4, 2001, pp. 599–606.

4. Kamai R., Hatzor Y.H., "Numerical analysis of block stone displacements in ancient masonrystructures: A new method to estimate historic ground motions", Int J Numer Anal Met, 32, 11,2008, pp. 1321–1340.

5. Newmark N.M., "Effects of earthquakes on dams embankments", Geotechnique, 15, 1965, pp.60–139.

6. Goodman R.E., Seed H.B., "Earthquake induced displacements in sands and embankments", JSoil Mech Foundation Div ASCE, 92(SM2), 1966, pp. 125–146.

7. Wartman J., Bray J.D., Seed R.B., "Inclined plane studies of the Newmark sliding block proce-dure", J Geotech Geoenviron, 129, 8, 2003, pp. 673–684.

8. Tsesarsky M., Hatzor Y.H., Sitar N., "Dynamic displacement of a block on an inclined plane:Analytical, experimental and DDA results", Rock Mech Rock Eng, 38, 2, 2005, pp. 153–167.

9. Jiang Q.H., Yeung M.R., "A model of point-to-face contact for three-dimensional discontinuousdeformation analysis", Rock Mech Rock Eng, 37, 2, 2004, pp. 95–116.

10. Liu J., Kong X.J., Lin G., "Formulations of the three-dimensional discontinuous deformation anal-ysis method", Acta Mechanica Sinica, 20, 3, 2004, pp. 270–282.

11. Moosavi M., Jafari A., Beyabanaki S., "Dynamic three-dimensional discontinuous deformationanalysis (3-D DDA) validation using analytical solution", The Seventh international conferenceon the analysis of discontinuous deformation (ICADD-7), Honolulu, Hawaii, 2005, pp 37–48.

199


12. Wang J., Lin G., Liu J., "Static and dynamic stability analysis using 3D-DDA with incision bodyscheme", Earthquake Engineering and Engineering Vibration, 5, 2, 2006, pp 273–283.

13. Yeung M.R., Jiang Q.H., Sun N., "Validation of block theory and three-dimensional discontinuousdeformation analysis as wedge stability analysis methods", Int J Rock Mech Min, 40, 2, 2003, pp.265–275.

14. Yeung M.R., Jiang Q.H., Sun N., "A model of edge-to-edge contact for three-dimensional discon-tinuous deformation analysis", Computers and Geotechnics, 34, 3, 2007, pp. 175–186.

15. Goodman R., Shi G., "Block theory and its application to rock engineering, Prentice-Hall Engle-wood Cliffs, NJ, 1985, 338 pp.

16. Shi G.H., "Technical manual and verification for Keyblock codes of dynamic Newmark Method",Unpublished technical report, DDA Company, Belmont, CA, 1999.

17. Goodman R.E., "Methods of Geological Engineering in Discontinuous Rocks, West PublishingCompany, San Francisco, 1976, 472 pp.

18. Hatzor Y.H., Goodman R.E., "Three-dimensional back-analysis of saturated rock slopes in dis-continuous rock-a case study", Geotechnique, 47, 4, 1997, pp. 817–839.

19. Hoek E., Bray J.W., "Rock slope engineering, Institution of Mining and Metallurgy, London, 1981,358 pp.

200

Application of Strength Reduction DDA Method in StabilityAnalysis of Road Tunnels

XIA CAICHU1,2, XU CHONGBANG1,2,∗ AND ZHAO XU3

1Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China2Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education,Tongji University, Shanghai 200092, China3The Key Laboratory of Urban Security and Disaster Engineering, Ministry of Education,Beijing University of Technology, Beijing 100124, China

1. Introduction

Surrounding rock of mountain tunnel is formed in nature under complex geological andtectonic forces in its evolution and formation progress, has numerous discontinuous inter-faces such as joints, fractures, faults, folds and fractured rock zones with different sizes andgeometric shapes. Rock mass integrity and uniformity are damaged by these interfaces androck strength is reduced. The rock mass is cut into a non-continuum of rock blocks withvarious volume and shapes. Obviously, the discontinuity is of very important significance tothe mechanical property of rock mass. To understand such mechanical property is a primaryproblem in design tunnels economically and constructs them safely.

Two major means in studying tunnel stability considering the rock mass discontinuity aremodel test and numerical analysis.

Many researchers have carried out physical scaled model tests to study tunnel stability.Everling1 (1964) analyzed deformation property of supported jointed rock mass by physicalscaled model tests; Hobbs2 (1969) proved that a mixture of sand, plaster and water can sat-isfy the requirements of material for physical model tests and studied the effects on tunnelstability made by orientation of joints and tunnel span. Goodman3 (1972) studied the defor-mation of tunnel in jointed rocks by model tests. Jiang4 (1984) studied tunnel stability underdifferent joint occurrence. Zhu5 (1997) studied results of bolts strengthening of jointed rocktunnels in excavation. Song6 (2002) researched the effects on tunnel stability of joint spacing,orientation and the angle between fractures and tunnels.

Though physical scaled model tests can be used in researching tunnel stability in jointedrocks under certain conditions, but are impossible to be adopted in studying all influencingfactors as they are hard to be repeated and costly. To overcome such disadvantages, numericalsimulation can be used, which provides researchers a fast and convenient method. There aretwo main methods in discontinuous rock mass analysis, namely, the distinct element method(Cundall,7 1971) and discontinuous deformation analysis (Shi,8 1988). The later is adoptedin this paper.

MacLaughlin9 (1997) studied calculation accuracy of the discontinuous deformation anal-ysis by modelling sliding of a block on a slope, whose results showed that by DDA compu-tation, sliding displacement of the block could be accurately forecasted. McBride10 (2001)also proved the efficiency of DDA by compare numerical computation with physical modeltests. Michael11 (2006) carried out sensitivity analysis of parameters in DDA computation.Zhang12 (2008) studied the failure progress of semi-penetrative joints rock of tunnel.




2. Basic Principals of DDA Theory

In DDA, each block of arbitrary geometry has six degrees of freedom, among which threecomponents are rigid body motion terms and the other three are constant strain terms. There-fore, the deformation variable of block i can be written as

(u0,v0,r0,εx,εy,γxy

)T , where u0,v0are the rigid body translations along the x, y direction; r0 is the rigid body rotation aroundcetroid (x0,y0) of the block; εx,εy,γxy are the strain components. By the minimum energyprinciple, the general function of the system can be solved, the general function of the sys-tem is:

KD = F

Assuming that this system is composed of n blocks, we can have the function as follows:

K =

⎡⎢⎢⎢⎣

K11 K12 · · · K1nK21 K21 · · · K2n

......

......

Kn1 Kn2 · · · Knn

⎤⎥⎥⎥⎦ , D =

⎡⎢⎢⎢⎣

D1D2...

Dn

⎤⎥⎥⎥⎦ , F =

⎡⎢⎢⎢⎣

F1F2...

Fn

⎤⎥⎥⎥⎦

Where, Kij(i,j = 1,2, · · · ,n) is a submatrix with 6× 6 components and related to materialproperties of block i, Kij (i �= j) is related to the interaction of block i and j; Di is thedisplacement vector of block i; Fj is the load on block i.

3. Strenght Reduction Method Combined with Catastrophe Theory

3.1. Catastrophe function of surrounding rock displacement from strengthreduction method

At first, divide the value of rock strength parameters, cohesion C and internal friction angleϕ, by a same factor F, according to the strength reduction method.13 Then a set of newparameter are obtained as C′, ϕ′. This new set of parameters will be used in computationagain and displacement ζ0 can be obtained.

C′ = CF

ϕ′ = arctan (tanϕ

F) (1)

Repeat this progress for n times. Displacement ζ of these n times computation can begained. For critical points of a tunnel, the relationship curve of their displacements andstrength reduction factor can be fitted by least square method, by which we can get a functionbetween ζ and F which is defined as the safety factor of the tunnel:

ζ = f (F)

This function can be described as a quintic polynomial with sufficient precision, so that wecan get

ζ = a5F5 + a4F4 + a3F3 + a2F2 + a1F+ a0 (2)

Where a0, a1, a2, a3, a4, a5 are fitting coefficients to be calculated out.The first derivatives of Eq. (2) with respect to F can be written as Eq. (3), this is a cusp

catastrophe potential function.

V = 5a5F4 + 4a4F3 + 3a3F2 + 2a2F+ a1 (3)

Let q = a45a5

, F = p − q and substitute them into Eq. (3), terms with degree 3 can beeliminated and ignore constant term as the constant term dose not change the property of

202


Eq. (4)

V = b4p4 + b2p2 + b1p (4)

Then let p =(

1b4

) 14

x while b4 > 0, or let p =(− 1

b4

) 14

x while b4 < 0, substitute theminto Eq. (4), we get

V = x4 + ux2 + vx (5)

V = −x4 + ux2 + vx (6)

Where u = b2/

√∣∣b4∣∣, v = b1/

4√∣∣b4

∣∣.Equations. (5) and (6) are standard potential function14 of canonical cusp catastrophe

model and dual cusp catastrophe model whose control variables are state variables u and v.Their equilibrium surface and bifurcation graph are shown in Figures 1 and 2.

3.2. Failure criteria of tunnel surrounding rock stability

According to theory of cusp catastrophe model, let the first derivative of function V equalsto 0, the following equations can be got:

4x3 + 2ux+ v = 0 (canonical cusp catastrophe) (7)

− 4x3 + 2ux+ v = 0 (dual cusp catastrophe) (8)

The critical points set defined by Eqs. (7) or (8) composed the equilibrium surface whoseshape in space (x,u,v) are shown in Figures 1 or 2. The shape includes 3 parts includingupper lobe, lower lobe and middle lobe, among which the former two are stable while thelatter one is unstable. Phase points would change on the upper lobe (or the lower lobe) withequilibrium regardless their path. When the point reaches the edge of the upper lobe (or thelower lobe), it will jump abruptly across the middle lobe to the lower lobe (the upper lobe).

Upper leaf

Middle leaf

Lower leaf

Contro

l plan

e

Balanc

e curv

ed su

rface

x

u

u

v

v

x

>0

<0

Δ

Δ

Figure 1. Canonical cusp catastrophe model.14

203


Figure 2. Dual cusp catastrophe model.14

Thus, point with vertical tangent line at equilibrium lobes composes a catastrophe points set(singular points set) of certain status, whose function can be written as

∂2V∂x2= 12x2 + 2u = 0 (canonical cusp catastrophe) (9)

∂2V

∂x2= −12x2 + 2u = 0 (dual cusp catastrophe) (10)

The projection of singular points set on plane defined by control variables (u,v) formeda bifurcation set which is a points set makes status variables catastrophe. Combine Eqs. (7)and (9), Eqs. (8) and (10) separately, eliminate x, we can get the following questions:

� = 8u3 + 27v2 and � = −8u3 + 27v2 (11)

For displacement analysis based on strength reduction method of tunnel surrounding rock,a failure criterion can be defined as follows: while � > 0, the tunnel surrounding rock isstable under the strength reduction factor; while � = 0, the rock is at the critical status ofstable to unstable; while � < 0, the rock is failed. Thus the rock mass stability status can bejudged according to the stable status of displacement data.

4. Application

4.1. Engineering background

Jinjishan multiple-arch tunnel of the second phase project of Fuzhou city airport is a bigspan tunnel with eight traffic lanes, locates between milestone K22 + 235 to K22 + 630,the total length of which is 295m (Fig. 3). Span of a single tunnel is 18.2m and the crosssection area of inner profile of a tunnel is 171.06 m2. The whole span of the multiple-archtunnel is 41.9m and the height is 14.2m (Fig. 4). Middle wall of the tunnel is designed with3 composite layers.

The tunnel locates in an area of low mountains and hills with a complex ground sur-face. Surrounding rock of the main tunnel is composed of weak weathered granite, granite-porphyry and dioritic porphyrite, which is of a rather high strength but a poor integrity,can be classified as lever IV according to “code for design of road tunnel” in China (JTG

204


Figure 3. Geological profile of Jinjishan tunnel (Left tunnel).

Figure 4. Cross section of Jinjishan tunnel.

D70-2004). The surrounding rock at tunnel portal is composed of residual hillside waste andhighly weathered rock, which is rather loose and classified as lever V. The maximum coveringlayer of rock IV is 46.7m and 36.0m of rock 36.0.

4.2. Parameters in computation and strength reduction factor

There are two sets of joints in the surrounding rock which are oriented as 62◦� 75◦ and63◦� 169◦ according to joint data collected during construction. Average intervals of these 2sets of joint are 1.6m and 1.8m. Dimension of DDA numerical simulation model is 120.0m× 80.0m. The covering layer is 30.0m. Mechanical parameters in computation are listedin Table 1. Mechanical computation Parameters of joints (internal friction angle, cohesiveparameter and tensile strength of joints) are reduced according to the strength reductionmethod. Increscent of reduce factor for each computation in sequence is 0.01. The originalreducing factor is 1.00 and the last factor after 31 computations is 1.30. Position of 9 criticalpoints in the tunnel surrounding rock in computation was shown in Fig. 6.

Table 1. Surrounding rock parameters of Jinjishan tunnel.

Rock Rock Elastic Poisson’s Internal Friction Cohesive TensileDensity/kN Modulus /GPa ratio Angle of Joints /◦ Parameter of Strength of

Joints/kPa Joints/kPa

20.0 1.5 0.28 34.0 160.0 80.0

205


Figure 5. DDA numerical simulation model.

Figure 6. Critical points.

Horizontal, vertical and total displacement of the critical points in Jinjishan tunnel sur-rounding rock under different strength reduction factor of joints are shown in Figures 7–9.According to these figures, the value of vertical displacements is about twice as much as thatof horizontal displacement, so that vertical displacement should be the main object to be ana-lyzed. Further, of all critical points, the vertical displacements at arch top are more sensitiveto strength reduction factors. So, stability evaluation of Jinjishan tunnel will be made mainlybased on analysis of arch top stability of surrounding rock.

4.3. Tunnel stability analyses

The relationship between displacement of critical points and strength reduction method ofJinjishan tunnel arch top is shown in Figure 10. According to strength reduction methodcombined with catastrophe theory, this relation curve is fitted and analyzed by the trinomialwith degree 5, ζi = fi(F) (where i is the computation number and is bigger than 5). The resultsare shown is Table 2.

As shown in Table 4, Safety factor of Jinjishan left tunnel judged by the strength reductionmethod combined with catastrophe theory is 1.10, and 1.16 of right tunnel. This means thatthe stable status of left tunnel is obviously smaller that the right tunnel under the 2 sets ofjoints with the orientation of 62◦� 75◦ and 63◦� 169◦.

206


Figure 7. Horizontal displacement of critical points.

Figure 8. Vertical displacement of critical points.

Figure 9. Total displacement of critical points.

207


Figure 10. Relation between ζ and F of critical point 3 and 7.

4.4. Discussion

Two problems should be mentioned here in adopting the strength reduction method com-bined with strength reduction method:

(1) Number of data samples.In fitting the trinomial ζ = a5F5+ a4F4+ a3F3+ a2F2+ a1F+ a0, number of data samples

should be at least 5. This means at least 6 computations of strength reduction method shouldbe carried out to use the catastrophe theory. Or to say, stable displacements results should beperformed in the first 5 computations in strength reduction method.

(2) Selection of critical pints in tunnel surrounding rock according to the orientation ofjoints.

In discontinuous deformation analysis, deformation of a block is effected by the interactionof sides and numbers of surrounding blocks. Block chosen with critical points should belocated at the tunnel free face and with the smallest number of surrounding contact blocks.For example (Fig. 11), there are 5 blocks can be chosen among which block 2 is hindered byblock 1 and block 3 while moving downward and block 4 is hindered by block 3 and block5. In all these 5 blocks, only block 3 and block 5 are subjected to the smallest resistance.Compare block 3 with block 5, the most satisfactory block to be adopted is block 3.

Figure 11. Possible blocks with critical points.

208


Table 2. Judgment on displacement catastrophe of critical point 3 and 7.

No. Strength reduction factorCritical point 3 Critical point 7

Judging value� Stable status Judging value� Stable status

1 1.00

Stable

Stable

2 1.013 1.024 1.035 1.046 1.05 1.25 9385.797 1.06 0.04 19757.718 1.07 0.14 73368.699 1.08 0.24 86245.5810 1.09 0.01 59386.5411 1.1 0.01 76176.7312 1.11 −0.24

Unstable

136365.8913 1.12 −0.58 138751.5914 1.13 −21.97 0.0815 1.14 −28.34 0.0216 1.15 −2.73 0.0417 1.16 −26.74 0.0818 1.17 −16.25 −0.12

Unstable

19 1.18 −16.49 −0.1420 1.19 −29.12 −0.1121 1.2 −87.13 −0.0522 1.21 −41.52 −0.0423 1.22 −581.67 −0.3224 1.23 −7704.77 −0.4525 1.24 −758.02 −0.4726 1.25 −3656.04 −0.5127 1.26 −560.56 −0.2128 1.27 −2939.17 −0.8229 1.28 −8934.00 −2.1330 1.29 −5988.02 −5.0331 1.30 −5475.79 −12.17

5. Conclusions

The strength reduction method is used in discontinuous deformation analysis, and by dis-placement catastrophe theory, the catastrophe point is judged. Thus the safety factor of tun-nel surrounding rock is decided. The following conclusions are obtained in the research:

• Cusp catastrophe theory provided a theoretical basis to judge the displacement catas-trophe point of tunnel surrounding rock, which avoids the subjectivity in judging thepoint of sudden change.• The strength reduction method combined with displacement catastrophe theory is

feasible to evaluate the stability of tunnel surrounding rock. This method provideddiscontinuous deformation analysis and tunnel stability evaluation a new researchway.

209


• In studying tunnel surrounding rock stability by the strength reduction method com-bined with displacement catastrophe theory, at least 6 computations should be made.Blocks with critical points should be delicately chosen as those blocks with leastnumber of contacting blocks at free face.

Acknowledgements

This study is sponsored by the Chinese National Natural Science Foundation 50579088,50639090. And thank Mr. Peng Yuwen and Mr. Guo Rui for their help in completing thispaper.

References

1. Everling, G. Model Study of rock-joint deformation. Int. J. Rock. Mech. Min. Sci & Geomech.Abstr. 1: 1964, pp. 319–326.

2. Hobbs, D.W. Scale model study of strata movement around mine roadways–Roadway shape andsize. Int. J. Rock. Mech. Min. Sci &Geomech. Abstr. 6, 1969, pp. 305–404.

3. Goodman, R.E., Heuze, H.E and Bureau, G.J. On modeling techniques for the study of tunnels injointed rock. Fourteenth Symposium on Rock Mechanics: 1972, pp. 441–479.

4. Jiang Jueguang, Li Sonpeng, Qian Huiguo,et al. Scale model study of stability of tunnels in differ-ent occurrence joined rock. Hydrogeology & Engineering Geology, 5, 1984, pp. 13–19.

5. Zhu Weishen, Ren Weizhong, Zhang Yujun, et al. Scale model study of the anchorage effect injointed rock under the excavation condition. Rock and Soil Mechanics, 18, 1, 1997, pp. 1–7.

6. Song Xuanmin, Gu Tiefeng, Liu Chongwei. Experimental study on roadway stability in rock-mass with connected fissures. Chinese Journal of Rock Mechanics and Engineering. 21, 12, 2002,pp. 1781–1785.

7. Cundall, P.A. Acomputermodel for simulating progressive, large scalemovements in blocky rocksystem. Symposium of International Society of Rock Mechanics, Nancy, France, 1971, pp. 11–18.

8. Shi,G-H. Discontinuous deformation analysis — a newmodel for the statics and dynamics of blocksystems. Ph.D. thesis, Department of Civil Engineering, University of California, Berkley. 1988.

9. MacLaughlin, M.: Discontinuous deformation analysis of the kinematics of rock slopes. Ph.D.thesis. Department of Civil Engineering, University of California, Berkley. 1997.

10. McBride, A., Scheele, F. Investigation of discontinuous deformation analysis using physical lab-oratory models. In: Bicanic,N. (ed.), Proc. Fourth International Conference on DiscontinuousDeformation Analysis, 2001,pp 73–82.

11. Michael Tsesarsky, Yossef H. Hatzor. Tunnel roof deflection in blocky rock masses as a function ofjoint spacing and friction — A parametric study using discontinuous deformation analysis (DDA)Tunnelling and Underground Space Technology 21, 2006, pp. 29–45.

12. Zhang Xiuli, Jiao Yuyong, Zhao Jian Simulation of failure process of jointed rock. J. Cent. SouthUniv. Technol.15, 2008, pp. 888–894.

13. Zienkiewicz O.C., Humpheson C & Lewis R. W. Associated and non-associated viso-plasticityand plasticity in soil mechanics. Geotechnique, 25, 4, 1975, pp. 671–689.

14. Saunders P.T. Introduction of Catastrophe Theory. Translated by Ling Fuhua. Shanghai: ShanghaiScientific and Technical Documents Publishing House, 1983.

210

Micromechanical Simulation of the Damage and FractureBehavior of a Highly Particle-filled Composite Material UsingManifold Method

HUAI HAOJU, CHEN PENGWAN∗ AND DAI KAIDA

State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology,Beijing PR China, 100081

1. Introduction

Polymer bonded explosives (PBXs) are highly particle filled composite materials comprisedof 90–95% by weight of powerful secondary explosive particles and 5–10% by weight ofbinder. Understanding and modeling the mechanical responses of PBXs is of great interest tothe defense industry and commercial applications to enable predictive constitutive models.Previous research has shown that the properties of PBXs are a strong function of crystallinefracture, interfacial strength and initial damage. For the PBXs and solid propellant materials,there is a little of mesoscopic simulation data in the literature. Hubner et al.1 developed athree-dimensional FEM model of an elementary cell to study the effect of detachments of thematrix material from the filler surface on Poission’s Ratio. In his work, debonding leads to adecrease in Poisson’s Ratio. A cohesive law is developed by Tan2 for modeling the mechanicalresponse of hydrostatic and uniaxial tension loaded PBX9501 based on Mori-Tanaka’s effec-tive medium theory, the bulk modulus of PBX9501 is more than 40% lower than that of thesame material with perfect interfaces without any debonding. Wu3 developed a viscoelasticcohesive zone model, which was implemented into an implicit finite element code based on aslightly modified version of Yoon’s model, to predict combined damage of particles and inter-face debonding in PBXs materials. The interactions can be made responsible for asymmetricmechanical behavior in the tension and the compression range. The internal deformation andgrain interaction of granular explosives were studied by Bardenhagen using a particle-in-cellmethod4. With more computing power now available, it enables the development of detailedmesoscopic models, which assess the evolution of internal microstructure, and analyze thebehavior of interfacial debonding.

Manifold Method proposed by Shi5 is a new numerical method, which provides a unifiedframework for solving problems with both continuous and discontinuous media. By employ-ing the concept of cover and two sets of meshes, manifold method combines the advantagesof FEM and Discontinuous Deformation Analysis. It can not only deal with discontinuities,contact, large deformation and block movement as DDA, but also provide the stress distri-bution inside each block accurately as FEM can. The numerical model of the original MMpossesses only the first-order accuracy, leading to dissatisfaction in simulating problems thatneed high accuracy in displacement and stress distribution. To overcome this and expand theapplicability of MM, Zhang et al.6 developed the second order manifold method with sixnode triangle mesh. In this paper, MM method is used to study the process of damage andfracture of the PBX material under tensile and compressive loading basing on a micromechan-ical numerical model. The influences of initial microcrack and microvoid on the deformation,fracture and stress-strain curves of the material are also analyzed.




Table 1. Material parameters of PBX

Youngmodulus(GPa)

Poisson’sratio

Tensilestrength(MPa)

Cohesivestrength(MPa)

Frictionangle(◦)

Particle 31 0.3 4.75 2 30Matrix 0.12 0.48 2 2 25Interface / / 0.5 0.5 15

2. Micromechanical Simulation of PBX Under Tensile Loading

The MM models of some irregular particles without and with initial damage are shown inFigures 1(a) and 2(a). The entire computational domain size is 1×1 unit square and containsexplosive particle and matrix, and the volume fraction of explosive particle is 53%. Threemicrocracks and one microvoid are prefabricated as initial damage. The tensile displacementloading is applied to vertical direction. The explosive particle and matrix are modeled aselastic materials. The material parameters of PBX are listed in Table 1.

Figures 1 and 2 show the simulation results without and with initial damage. The resultsshow that interfacial debonding first occurs in the horizontal direction under the tensilestresses. And then debonding surface develops gradually in the model without initial damage,while crack develops along the prefabricated crack direction in the model with initial dam-age. With the development of loading, the crack initiation also appears in the matrix due tothe smaller strength of matrix. The stress concentration forms around particles because of the

(a) (b) (c)

(d) (e) ( f)

Particle

Matrix

1

1

F

Figure 1. Simulation results of irregular particle without initial damage.

212


(a) (b) (c)

(d) (e) ( f)

Matrix

1

1

F

Void

Particle

Cr ack

Figure 2. Simulation results of irregular particle with initial damage.

(a) Without initial damage (b) With initial damage

Strain Strain

Stre

ss

Stre

ss

Figure 3. The stress-strain curves.

redistribution of stress, which also causes the initiation and development of other cracks. Thelong continuous trans-binder crack dominates the development process, which penetrates thewhole material model and induces the fracture of material. Interfacial debonding and matrixcracking are key damage modes.

Figure 3 shows the relation of stress and strain without and with initial damage respec-tively, demonstrating that the stress-strain curves change linear to nonlinear due to interfacialdebonding. The nonlinearity of material is more obvious with development, connection andperforation of crack.

213


3. Micromechanical Simulation of PBX Under Compressive Loading

The numerical model and material parameters are the same with that of Section 2. The com-pressive displacement loading is applied to vertical direction, see in Figures 4(a) and 5(a).Figures 4 and 5 show the development process of crack without and with initial damageunder compressive loading respectively. The interfacial debonding first occurs in the load-ing direction, and then crack initiation appears in matrix at many different locations. Thesecracks propagate and develop due to stress concentration. The fracture of explosive particleoccurs when the stress exceeds the tensile strength of particle, and develops along prefabri-cated crack direction. Connected crack causes the rupture of material in the end. Comparedwith Section 2, the particle occur fracture besides interfacial debonding and matrix crack-ing in simulation results. In the model with initial damage, the initial damage reduces themechanical property, and causes explosive particle to damage easily. It is in good agreementwith microscopic test results of reference.7

4. Asymmetry of Tension and Compression of PBX

In order to only consider the influence of interfacial debonding, we increase the strength ofexplosive particle and matrix, and decrease the strength of interface in numerical simulation.Figure 6 shows the result of asymmetry of tension and compression of PBX without and withinitial damage respectively. The range of asymmetry is controlled by tensile strength of inter-face. The phenomenon exists due to different influence factors of tension and compressioncondition. The interfacial friction effect determines compression mechanical property, whilethe bonding strength determines tensile mechanical property. The simulation results are agreewith the result of test,8 see in Figure 7.

(a) (b) (c)

(d) (e) ( f)

Matrix

F

F

Particle

Figure 4. Simulation results of irregular particle without initial damage.

214


(a) (b) (c)

(d) (e) ( f)

Particle

Matrix

F

F

CrackVoid

Figure 5. Simulation results of irregular particle with initial damage.

(a) Without initial damage (b) With initial damage

TensileCompressive Compressive

Tensile

Strain Strain

Stre

ss

Stre

ss

Figure 6. Asymmetry of tension and compression of PBX.

5. Conclusions

The process of damage and fracture of PBX under tensile and compressive loading was stud-ied by the manifold method. The simulation results show that interfacial debonding andmatrix cracking are key damage modes, while transgranular fracture is only exist undercompressive loading. Initial damage decreases the mechanical property of PBX, and makesfracture easily. The results also demonstrate asymmetry of tension and compression of PBX.The results show that manifold method is effective to simulate the micromechanical damageand fracture of particle-filled composite materials, and can be used to predict the mechanicalbehavior of the composite materials.

215


Compressive

Tensile

Figure 7. Stress-strain curve of tensile and compressive loading of PBX.

Acknowledgements

The authors of this paper acknowledge the support from The National Basic Research Pro-gram of China (No. 613830202), The National Natural Science Foundation of China (No.10832003), and New Century Excellent Talents in University of China.

References

1. Hubner, C., Geibler, E., et al, “The Importance of Micromechanical Phenomena in Energetic Mate-rial”, Propellants, Explosives, Pyrotechnics, 24, 1999, pp. 119-125.

2. Tan, H., Huang, Y., Liu, C., et al, “The Mori-Tanaka Method for Composite Materials withNonlinear Interface Debonding”, Int J Plasticity, 21, 10, 2005, pp. 1890–1918.

3. Wu, Y.Q. and Huang, F.L., “A Micromechanical Model for Predicting Combined Damage of Par-ticles and Interface Debonding in PBX Explosives”, Mechanics of Materials, 41, 2009, pp. 27–47.

4. Bardenhagen, S.G., Brackbill, J.U., Sulsky, D., “The Material-point Method for Granular Materi-als”, Comput. Methods Appl. Mech. Engrg., 187, 2000, pp. 529–541.

5. Shi, G.H, and Goodman, R.E., “Discontinuous Deformation Analysis”, Proceedings of the 25thUS Symposium of Rock Mechanics, Published by Society of Mining Engineers ,1984, pp. 269–277.

6. Zhang G.X., Sugiura, Y., Hasegawa, H. and Wang, G.L., “The Second Order Manifold Methodwith Six Node Triangle Mesh”, Structural Eng./Earthquake Eng. JSCE, 19, 1, 2002, pp. 1-9.

7. Chen, P.W., Damage Theory of Energetic Material and Its Application, Beijing: Beijing Instituteand technology Publishing Company, 2006.

8. Li, M., Wen, M.P., Huang, M., et al, “Evaluation of Coherence Strength of Energetic CrystallineGranules by Compressive Stiffness Method”, Chinese Journal of Energetic Materials, 15, 3, 2007,pp 243–247

216

The Application of Discontinuous Deformation Analysis in theSlope Stability of the Expansive Soil

LIN YULIANG∗ AND WEI LINGJING

College of Civil & Architectural Engineering, Guangxi University, Nanning, Guangxi, 530004, China

1. Introduction

The expansive soil expands while moisture content increases and contracts while moisturecontent decreases. According to the relative information about geological prospecting inChina, especially the information about geological prospecting of expansive soil slopes inNayou expressway in Guangxi, it is shown that under the effects of geology and naturalweather of dry and wet cycles, fissures are caused and developed in many slopes of expan-sive soil, the strength of the soil continuously weaken, the unloaded joints on the slope footaccelerate the joint system development, the slope finally become a block system. Thus thereare following features of the expansive soil slope: (1) Block structure. That is said, undersome certain conditions, many slopes of expansive soil are finally become a block system.The continuous medium mechanics is not suitable for these slopes. (2) Traction-type failure.The slope slides again and again for many times. After the first slide, the strength of thesoil continuously weaken, which could form the second, or the third slide again. (3) Step bystep failure. The strength of the soil continuously weakens step by step. (4) Expansibility.The expansive forces between the blocks and the volume of the blocks increase while themoisture content increases.

Based on the fact that slope of expansive soil had became block system on the later stage,and considering the other features of the slope of expansive soil, we put forward a newmethod of discontinuous deformation analysis for the slope stability of the expansive soil inpapers.1−2 We will proceed to study further and to discuss the relationship between the slopestability and the moisture content of the expansive soil especially in this paper.

2. DDA Method for the Slope Stability of the Expansive Soil

2.1. The calculating model of slope block system

There are two component mediums in the expansive soil slopes in NANYOU expressway inGuangxi: the layer of expansive soil and the layer of mudstone (sometimes there is a layerof surface soil above). The expansive soil is classified into two layers: the upper layer isgreyish white expansive soil; the under layer is greyish black expansive soil. There are somehorizontal or incline discontinuous planes in expansive soil. Generally, these discontinuousplanes are surface of soil layer, but some probably are joint plane or crack planes causing bydry condition.

There are also some vertical discontinuous planes which are caused by dry condition: someare parallel to the top line of slope; another is normal to the top line of slop. The mudstoneis subdivided into several layers in general.

According to the features of slopes of the expansive soil and Discontinuous DeformationAnalysis, we put forward the calculating model of expansive soil slope as Fig. 1. If the vertical




Figure 1. 2-D calculating model of slope.

discontinuous planes which are normal to the top line of slop are well developed, there is aplane stress model; if not, there is a plane strain model. This model has bellow features: Underthe effects of geology and natural weather of dry and wet cycles, many slopes of expansivesoil become a block system; each block can be considered elastic; There are the weight ofblock, the frictional force, the water pressure, the expansive force, etc. between the blocks inthe block system; the lower layers are mudstone layers.

2.2. Basic calculating equation of slope block system

According to Dr. Gen-Hua Shi’s Discontinuous Deformation Analysis 3, we put forward anew method of discontinuous deformation analysis for the slope stability analysis of theexpansive soil.

Suppose there are constant stress and constant strain in every block at any time, the dis-placements (u,v) of any point (x,y) in the block can be expressed by six displacement con-stants:

(u0 ,v0 ,r0 ,εx,εy ,γxy)

Where (u0,v0) are the rigid displacements of the special point (x0,y0) in the block; r0 is theturning angle round about turning center(x0,y0); (εx ,εy ,γxy) are the block strains.

The blocks are connected to form a block system by the contact between the blocks andthe displacement restraints for blocks. Suppose there are n blocks in the block system, thebasic equations are as follows:

⎡⎢⎢⎢⎢⎢⎣

K11 K12 K13 · · · K1nK21 K22 K23 · · · K2nK31 K32 K33 · · · K3n...

......

. . ....

Kn1 Kn2 Kn3 · · · Knn

⎤⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

D1D2D3...Dn

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

F1F2F3...Fn

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

(1)

Because of every block has six degrees of freedom (u0 ,v0 ,r0 ,εx,εy ,γxy), so every matrixelement Kij in above equations is a 6× 6 submatrix. [Di]and [Fi] are 6× 1 submatrixs where[Di]represents the deformed variations (d1i ,d2i ,d3i ,d4i ,d5i ,d6i) and [Fi] represents theloads which are distributed to six variations. Submatrix [Kii] is relative with the properties ofblock i; Submatrix [Kij] (i �= j) represents the contact relationship between block i and block j.

218


2.3. The constitutive law of a block and handle of parameters

2.3.1. The shape of a block

For convenient, suppose there are two types of a block shape: one is quadrilateral; anotheris triangle.

2.3.2. The physical and mechanical properties of the blocks

Every block can be consisted of different materials, such as normal soil, expansive soil, nor-mal rock and expansive rock. So we can input different material parameters for each block.

2.3.3. The force analysis of a block

Every block is acted upon by elastic stresses, dead weight of a block, expansive force, initialstresses and inertial force etc. The dead weight and expansive force can be considered asbody forces. The dead weight is constant. The expansive force is a function of block moisturecontent.

2.3.4. The constitutive equation of a block for expansive soil

Expansive soil belongs to the type of unsaturated soil. We consider that the Fredlund modelcan better represents the constitutive model of a bock. In the plane strain condition, theconstitutive equation of a bock is as follows:

εx = 1− μ2

E

[(σx − ua)− μ

1− μ (σy − ua)]+ us

Eus

εy = 1− μ2

E

[(σy − ua)− μ

1− μ (σx − ua)]+ us

Eus

γxy = 2(1+ μ2)E

τxy (2)

where E — elastic modulus of expansive soil;μ — Poisson’s ratio of expansive soil;ua — pore air pressure;us — Absorption forceEus — Elastic modulus of soil structure relating with the variation of absorption force.

2.3.5. The choice of the calculating time step

The calculating time step can not be determined accurately in terms of theory, but in practice,we adjust the calculating time step by the method of trial and error gradually to determinethe optimum value in the case of the calculating time step is not very large.

2.4. Mechanics of block system

2.4.1. The geometrical contact relation between blocks

The geometrical contact relation between blocks is simplified as the contacts of the edges andthe angle points between adjacent blocks as shown in Fig. 2.

219


��

��

� �

��

� �

��

�

�

�

�

� �

��

� �

��

��

� �

��

��

� �

��

Figure 2. Geometric Contacts between blocks.

2.4.2. The criterions of to plus or to subtract springs between blocks

There are two criterions that must be satisfied while two blocks contact:

(1) Intrusion to each other between the blocks is not allowed;(2) There are not tensile forces between the edges of blocks.

For satisfying these two criterions, it is necessary to plus or to subtract springs betweenblocks in the processing.

2.4.3. The determination of boundary displacement conditions

The points are called fixed point which displacements are known (including the point whichdisplacements are equal to zero). Stiff springs are needed to plus to block system in the direc-tions of displacement variety to insure fixed point has appointing displacements. Accordingto the precision of displacement of fixed point, we can adopt different restraints in varietydegrees that is said to adjust the stiffness of spring for to achieve an optimum restraint state.

3. DDA Program and Calculating Examples

DDA program are compiled by using Visual C++ as development platform of Windows.The program is two dimensions (plane strain and plane stress). We adopt some slopes whichis composed of expansive soil and mudstone in Nanyou highway of Guangxi, China as cal-culating examples.

3.1. The relating mechanical parameters of the slopes

The mechanical parameters of expansive soil and mudstone are derived by experiments asfollows.

Table 1. The deformation parameters of greyish white expansive soil.

Deformation parameters in different moisture content

moisture content 5.27% 9.59% 14.39% 17.24% 21.43% 24.90% 28.39% 54.40%Poisson’s ratio μ 0.20 0.23 0.24 0.35 0.37 0.41 0.48 0.50E(mPa) 54.26 41.62 33.35 30.85 17.77 13.82 5.52

220


Table 2. The strengths parameters of greyish white expansive soil.

Shear strengths in different moisture content

Saturation level 100.00% 92.38% 85.16% 80.74% 75.01%moisture content 29.62% 26.89% 24.82% 24.17% 22.01%Friction angle (˚) 10.77 12.63 15.49 17.4 21.27Cohesion (kPa) 71.28 86.03 98.28 113.71 156.27

Table 3. The mechanical parameters of grayish black expansive soil.

Deformation parameters Shear strengths

moisture content 10.70% 18.23% 32.05% Moisture content 35% 32.70% 27.41%Poisson’s ratio 0.26 0.28 0.46 Friction angle (˚) 10.4 14.3 23.5E(MPa) 45.96 23.04 12.57 Cohesion (kPa) 53.31 71.4 76.4

3.1.1. The mechanical parameters of expansive soil

There are two layers for expansive soil: one is greyish white; another is greyish black. Themechanical parameters of expansive soil are largely depended on the moisture content asshown in Tables 1–3 and Figs. 3–6.

According to above information, we can get the mechanical parameters of expansive soilas shown in Table 4 by the interpolation method.

The unit weights of these two expansive soils are 17.5 and 19.4 kN/m3 respectively.

��

��

��

�

��

��

��

�

Figure 3. Poisson’s ratio μ for grayish white soil. Figure 4. Elastic modulus for grayish white soil.

��

��

��

��

��

��

��

��

�

��

��

��

��

�

Figure 5. Poisson’s ratioμ for grayish black soil. Figure 6. Elastic modulus for grayish black soil.

221


Table 4. The values of μ and E in different moisture content.

Grayish white expansive soil

moisture content 10% 15% 20% 25% 30%Poisson’s ratio 0.23 0.26 0.36 0.41 0.48Elastic modulus (mPa) 40.91 32.81 22.23 13.58 4.14

Grayish black expansive soil

moisture content 10% 15% 20% 25% 30%Poisson’s ratio 0.26 0.27 0.30 0.37 0.43Elastic modulus (mPa) 47.12 32.87 21.70 17.91 14.12

3.1.2. The mechanical parameters of mudstone

The elastic modulus E is 169(mPa), the Poisson’s ratio μ is 0.27, the unit weight is 21.1kN/m3.

3.2. The calculating examples

Two cases about the block displacement and failure situation of expansive soil slopes arecalculated: one is in the case that has the same moisture content; another is in the case thathas different moisture content. Then we analyze these examples according to the results ofcalculating.

3.2.1. The displacement patterns of an expansive soil slope in the case of 20%moisture content

The displacement and failure patterns of an expansive soil slope in the case of 20% moisturecontent are as Figs. 7–12 in different calculating time steps.

We can see that, the displacement patterns of an expansive soil slope can be derived byusing the discontinuous deformation analysis step by step. These results are consistent withpractical situation.

Figure 7. Displacement pattern at time step 2. Figure 8. Displacement pattern at time step 5.

222




Figure 13. Displacement pattern for 10% moisture. Figure 14. Displacement pattern for 15% moisture.

3.2.2. The displacement patterns of an expansive soil slope in different moisturecontents

From Sec. 3.1.1, we can see that the mechanical parameters of expansive soil are largelydepended on the moisture contents of expansive soil: while moisture content of expansive soilincrease, the poisson’s ratio μ increase, the elastic modulus and shear strengths decrease. Sothe slope stability of the expansive soil must be largely depended on the moisture contents of

223


Figure 15. Displacement pattern for 20% moisture. Figure 16. Displacement pattern for 25% moisture.

Figure 17. Displacement pattern for 30% moisture.

expansive soil too. The relationship between the slope stability of expansive soil and moisturecontent is studied by using the DDA program as follows.

The displacement patterns of an expansive soil slope in different moisture content whencalculating time step is 25 (Figs. 13–17).

From Figs. 13–14, we can see that the slope displacement pattern of the expansive soil islargely relating with its moisture content, the higher the moisture content of expansive soil,the larger the displacements of slope blocks if the other conditions are the same. These resultsare also consistent with practical situation.

4. Conclusions

From above calculating results and the analysis, we can get the following conclusions:

(1) Under the effects of geology and natural weather of dry and wet cycles, many slopesof expansive soil are caused the development of the fissures and finally become a blocksystem;

(2) Calculating the block displacement and analyzing the slope stability of expansive soilare feasible by using Discontinues Deformation Analysis (DDA) while expansive soilbecomes a block system;

224


(3) The slope stability of the expansive soil depends on the moisture content. In general, thehigher the moisture content, the lower the stability of slope if the other conditions arethe same.

References

1. Liu Longwu, Lin Yuliang, An Yanyong, Yan Lie. The Research of Calculating Models and Meth-ods for the Stability of Swelling Soil Slopes. Theory and Practice of Expansive Soil TreatmentTechnology. China Communications Press. 2005. pp. 124–131

2. LIN Yu-liang, CHEN Xiao-liang, YANG Yang. A new method of discontinuous deformation anal-ysis of the slope stability of expansive soil. Rock and Soil Mechanics. 2007, Vol. 28. pp. 255–258.

3. Shi Genhua. Numerical Manifold Method and Discontinuous Deformation Analysis. QinghuaUniversity Press. 1997.

225

Extension of Distinct Element Method and Its Application inFracture Analysis of Quasi-Brittle Materials

Y.L. HOU1, G.Q. CHEN1,∗ AND C.H. ZHANG2

1Dept. of Civil Engineering, Kyushu University, Fukuoka, Japan2Dept.of Hydraulic and Hydro-power Engineering, Tsinghua University, Beijing, China

1. Introduction

Fracture and collapse analysis of a system comprised of quasi-brittle materials such as rockand concrete is an important research topic in the field of computational solid mechanics.Development of a numerical model that is capable of treating the failure of continuous mediaas well as the simulation of transition process from the continuum to discontinuum is a quitechallenging problem. Continuum-based finite element method (FEM) has achieved great suc-cess in simulating the initiation and propagation of crack by nonlinear fracture mechanicsmodels such as smeared crack model1 and discrete crack model.2 But the method is notsuitable for the simulation of post-failure processes, especially when large deformation andtopological changes are involved.

In recent years, the distinct element method (DEM) has been applied in the fracture analysisof quasi-brittle materials due to its inherent advantage in modelling discontinuous media.In particular, rigid particle DEM3 has attracted much attention nowadays which explainsthe macro-scale fracture phenomena through the simulation of the material meso-structure.Rigid particle DEM is conceptually simple and the material failure as well as the fracturepropagation appears naturally in the simulation process. Unfortunately, approaches based ontrial-and-error are usually needed to determine the microscopic parameters yielding requiredmacroscopic parameters, which has significantly limited the application of the particle DEM.Other methods such as the combined finite/distinct element methods4−5 are also available,which adopt triangular or quadrilateral finite elements by discretizing each block. Usuallythe fracture of material is limited to inter-element boundaries if a finite element is adoptedin each block. Obviously, it is necessary to arrange a large number of blocks in order tosolve practical engineering problem, which results in exhaustive computational cost for thecontact detection. When many finite elements are adopted in each block, it is necessary tointroduce new physical cracks through previously intact mass and the finite element meshof blocks must be changed adaptively to explicitly capture the propagation of the crack6−7,which becomes a bottleneck of the application of this method.

In order to overcome the above shortcomings, a new method is proposed to deal with theproblem of fracture of quasi-brittle materials in this paper. A discrete crack model is intro-duced into three-dimensional deformable discrete element code (3DEC)8 to simulate modeI fracture and I/II mixed mode fracture. In the method, discrete blocks are discretized intotetrahedral elements to simulate the deformation of the blocks. Moreover, the fictitious inter-faces between discrete blocks are created along the potential crack paths from the beginningof the analysis when potential crack paths are known in advance or predicted by other meth-ods. Along the fictitious interfaces, a discrete crack model is adopted, accounting for thefailure and softening property of the quasi-brittle materials. To illustrate the performance ofthe presented method, the numerical model is applied to analyze a concrete beam in mode I




fracture experiment and an asymmetric three-point bending beam with a single-edge notch inI/II mixed mode fracture experiment. The comparison between experimental and numericalresults illustrates that the method is capable of quantitatively predicting material failure andfracture propagation. It is also concluded that the method lays a theoretical foundation forthe numerical simulation from damage initiation in meso-scale (even micro-scale) to completeprocess of rupture in macro-scale.

2. Discrete Crack Model

Discrete crack model based on the fictitious crack model concept proposed by Hillerborg andco-workers9 has been widely and successfully used in the analysis of the fracture of quasi-brittle materials. In order to simulate the crack initiation and its propagation, it is necessaryto establish some reasonable failure criteria and constitutive models, governing the mate-rial failure and crack propagation. Different types of failure criteria have been put forwardin the past several decades. In early research works, the criterion for crack occurrence wasexclusively maximum tensile stress criterion, i.e. the crack opens if the normal stress alongthe crack plane reaches the tensile strength of the material. Later, Margolin10 suggests thatcrack starts if an effective stress intensity factor for mixed-mode fracture is larger than thefracture stress. Some recent literatures provide new criterion by introducing a failure surfacein the stress space, similar to the yield surface in the classic plasticity theory, which is usedto determine the crack initiation under pure tension, shear-tension and shear-compressionloading.11−12 Once crack occurs, crack propagation mechanism is described with a consti-tutive model that explains the relationship between the normal and shear stresses on thecrack plane and the corresponding normal opening and shear sliding displacements. In theproposed approach, the failure surface concept and the shear retention factor in smearedfixed crack model1 are followed and the approach used by Camocho et al.13 to deal with thetraction-displacement relationship is extended. As to the discrete crack model, a failure cri-terion featuring tensile-shear zoning is developed based on the classic Mohr-Coulomb jointfailure criterion. The mechanism of micro-cracks initiation and propagation is taken intoaccount by incorporating the softening behaviour of the material.

2.1. Failure criterion of tensile-shear zoning

The critical stress for crack initiation is estimated from the stress point position in the failuresurface as illustrated in Figure 1 according to the failure criterion of tensile-shear zoning. Forthe designated fictitious interface of discrete blocks, the failure surface (positive stands fortension) F, which is a function that specifies the limiting stress combination for which cracktakes place, is defined by the following expression:

F (σn,τ ) = |τ | + tanφfσn − c (1a)

F (σn) = σn − ft (1b)

where σn and τ are the normal and shear stresses on the crack plane (fictitious interface),respectively. c, φf and ft are cohesion, friction angle and tensile strength, respectively. Failuresurface consists of tension failure surface and tensile-shear failure surface and all the stresspoints below the surface are characterized by elastic behaviour. Supposing the stress pointat time t is below the surface and lies in zone ©1 of Figure 1, an elastic prediction for thecontact stresses at time t+�t is given by:

σn = −knun (2a)

228


nσ

τ

( ) 0nF σ =

( , ) 0nF σ τ =

fφ

Figure 1. Crack surface of tensile-shear zoning.

τ = −ksus (2b)

where kn and ks are the crack contact stiffness in the normal and tangential directions, respec-tively, and un and us are the normal and tangential relative displacements, respectively. If theelastic prediction violates the failure function, i.e. F(σn,τ ) � 0 or F(σn) � 0, then micro-cracks take place and a softening contact constitutive model is adopted (to be presented inSection 2.2) in the computation of contact stresses. If the elastic prediction stress point lies inzone©2 , tension failure criterion of the Equation (1b) is adopted and crack starts mainly dueto the normal stress of contact exceeding the tensile strength. If the elastic prediction lies inzone©3 , crack starts due to both tensile damage and shear damage. If the elastic predictionstress point lies in zone©4 , Mohr-Coulomb failure criterion of the Equation (1a) is adoptedand crack starts mainly due to shear damage. The stresses at crack initiation as marked bythe arrows in Figure 1 are given by:

σ0 = ft, τ0 = −ksus(zone©2 ) (3a)

σ0 = ft, τ0 = −sign(us)τmax(zone©3 ) (3b)

σ0 = −knun, τ0 = −sign(us)τmax(zone©4 ) (3c)

where σ0 and τ0 are the normal and tangential stresses at crack initiation, respectively;sign(x) is the sign function and τmax = −tanφfσ0 + c denotes maximum shear stress. Theadoption of the above proposed failure criterion featuring tensile-shear zoning allows thedeterminations of the stress conditions at which micro-cracks initiate in the quasi-brittlematerials considered.

2.2. Simulation of crack propagation

Quasi-brittle materials do not totally exhaust their strength after micro-cracks initiate andpropagate, as the material within the fracture process zone is able to transfer stresses acrossthe crack interface. Hence, the definition of the traction-displacement relationship along the

229


interface can affect the overall response to a great extent. Camacho et al.13 considered thatthe tractions which resist the opening and sliding of the crack decrease with the increase of thecrack opening. Well and Sluys14 also argued that the softening behaviour at the interface wasdriven only by the normal separation at the interface. Moreover, many experiments show thatcrack is initiated with mixed mode I/II effect, but propagates with mode I dominant undertensile-shear loading. In this paper, the following rules are applied to model the interfacialsoftening behaviour:

(1) Crack initiation is determined by the aforementioned failure criterion of tensile-shearzoning. The normal stress is a function of the crack opening and is independent of therelative sliding displacement. A bilinear tension softening model for the tensile strengthproposed by Petersson15 as shown in Figure 2 is adopted and the maximum crack open-ing wf is determined by fracture energy Gf and normal stress σ0 at crack initiation:

wf =3.6Gf

σ0. (4)

The normal stress transferred across the discontinuity interface is calculated from:

σn ={σ0 − w

wf(σ0 − fm), (0 ≤ w < wm)

fmwf−wm

(wf −w), (wm ≤ w ≤ wf )(5)

The turning point (wm, fm) of the curve is (2wf/

9, σ0/

3). Unloading behaviour issimulated using the secant stiffness.

(2) The softening curve for the cohesion is the same as the softening curve of the tensilestrength and the cohesion reduces to zero when the normal stress transmitted across theinterface becomes zero. When crack propagates, the cohesion c′ is given by:

c′ = σncσ0

(6)

(3) The shear stiffness reduces with the crack opening. A linear constitutive law is adoptedbetween tangential stress and relative sliding displacement. Meanwhile the Mohr-Coulomb

w

0σ

fwmw

mf

nσ

Figure 2. Bilinear tension-softening model.

230


failure criterion must be satisfied:

β = σn

σ0(7a)

τ = −βksus, when |τ | ≤ tanφfσn + c′ (7b)

τ = −sign (us) τ′max when |τ | > tanφfσn + c′ (7c)

where β is the reduction coefficient of shear stiffness and τ ′max = tanφfσn+ c′ denotes themaximum shear stress.

3. Numerical Implementation of Discrete Crack Model in 3DEC

The above discrete crack model is implemented into the 3DEC program through modify-ing the failure criterion and constitutive model of contact definition with FISH language,which enables the user to define new variables and functions or to modify the intermediatedata computed by 3DEC according to the user’s request during computation. DEM uses anexplicit time-marching scheme to solve the equations of motion directly. At each time step,the contact relative displacements calculated by 3DEC in normal and shear directions canbe obtained, and the contact stresses are calculated with FISH language in accordance withthe discrete crack model described above. Based on the extended 3D deformable DEM withdiscrete crack model along the fictitious interface, progressive damage and fracture withinquasi-brittle materials can be simulated. Some verification analyses results are shown in thefollowing section.

4. Verification Analyses

4.1. Analysis of mode I fracture in notched beam

The experimental results published by Petersson15 are commonly used for the verificationanalysis of mode I fracture of quasi-brittle materials. The geometry of the notched beam was2000 × 200 × 50mm3 (length × height × thickness). The depth of the notch was 100 mm.The loading F is applied by imposing displacement of the loading point on the top of thenotched beam at a velocity of 2.0e-4m/s. The discretization of the test beam is shown inFigure 3, in which a fictitious interface is created at the top of the notch.

The materials properties in the simulation are shown in Table 1. As mode I fracture isdominant in the test specimens, the shear stresses transferred across the crack plane shouldbe nearly zero, hence the cohesion and friction coefficient are set to zero in the numericalsimulations.

The predicted and experimental load-displacement curves of the notched beam are shownin Figure 4. It can be seen that the computational results mostly fall in the range of those of

notch fictitious interfaceF

Figure 3. Discretization of the notched beam.

231


Table 1. Material properties of the test of Petersson beam.15

Young’s modulus (GPa) Tensile strength (MPa) Fracture energy (N/m) Poisson’s ratio

30 3.33 124 0.2

DEM Simulation

Exp. Upper

Exp. Lower

D (mm)

F (

N)

0.0 0.2 0.4 0.6 0.80

300

600

900

Figure 4. Load-displacement curves.

(a) (b)

Figure 5. Deformed configuration at different displacements (scaled by 100) (a) D = 0.3 mm,(b) D = 1.0 mm.

the experiments The calculated peak load is very close to the test results, which approves thatthe failure criterion adopted in the paper is quite suitable for modelling mode I fracture. It isalso shown that the computed post-peak load-displacement curve compare favourably withthat of the experiment, which is greatly influenced by the definition of the softening curve.

Further results are shown in Figure 5 which presents the deformed configuration scaledby 100 at two typical loading stages. It is shown that a true crack forms along the fictitiousinterface and stability of system would be lost if the displacement at the top side of the beamabove the notch continues to increase. During the loading phase, the modelled system isprogressively damaged and subsequently fractured. The proposed method allows a successivenumerical simulation of the whole loading process from elastic stage to local damage and tothe final overall collapse of the system analyzed.

232


D

7D/4 D/2 2D D/4

D

P

notch

Figure 6. Geometry, force and boundary conditions of the notched beam.

Table 2. Material properties of the test of by Gálvez.16

Young’s modulus (GPa) Tensile strength (MPa) Fracture energy (N/m) Poisson’s ratio

38 3.0 69 0.2

4.2. Analysis of I/II mixed mode fracture test

A mixed-mode fracture experiment with a single-edge notched beam tested byGálvez et al.16 is selected to test the performance of the presented method in this paper, whichhas also been analyzed by Gálvez et al.12 using a cohesive crack finite element approach. Thegeometry, force and boundary conditions of the test are shown in Figure 6. The beam wasdimensioned as 300mm in height, 1350mm in length and 50 mm in thickness. The notchdepth was a half of the beam height. The fictitious interface is predicted by the smearedcrack model developed into 3DEC by Hou et al.17. Table 2 gives a summary of the materialproperties. The fictitious interface had a friction coefficient of 1.2, cohesion of 3.6MPa andthe initial normal and shear stiffnesses of the fictitious interface are 2.0e + 4GPa/m. Thecalculation is performed under displacement control at the loading point using displacementincrements corresponding to a velocity of 2.0e − 4m/s. The discretization of the beam isshown in Figure 7.

The load-displacement (F−D) response of the notched beam is shown in Figure 8, also acomparison of the load-crack mouth opening displacement (F-CMOD) is given in Figure 9.It can be seen that the computed peak load is a little smaller than that of the experiments,but quite close to that calculated by Gálvez et al.12 The cause of lower prediction may be dueto an earlier crack initiation in the computation according to the proposed failure criterionas compared to the crack occurrence during the experimental test. It can also be observed

fictitous interface

Figure 7. Discretization of the beam.

233


0.0 0.1 0.2 0.30.0

3.0

6.0

9.0

12.0

Exp. Upper Exp. Lower DEM SimulationGálvez et al. [12]

D (mm)

F (

KN

)

Figure 8. Load-displacement curves.

0.0 0.2 0.4 0.60.0

2.0

4.0

6.0

8.0

10.0

12.0

Exp. Upper Exp.Lower DEM SimulationGálvez et al. [12]

CMOD (mm)

F (

KN

)

Figure 9. Load-CMOD curves.

that the calculated post-peak response is quite close to the experimental results, which illus-trates that the proposed relationship of traction-displacements after crack initiates can wellsimulate the crack propagation under tensil-shear loading.

Figure 10 shows the deformed configuration of the beam in the vicinity of the notch whenthe displacement of load point is 0.1mm and 0.3mm, respectively. It can be seen that eventhough noticeable relative sliding displacement can be mobilized as a result of deep crackformation, its magnitude is much smaller than the crack mouth opening displacement. Theobservation from the simulation results illustrates that crack starts under mixed mode underthe loading conditions but propagates in a mode I dominant state.

234


(a) (b)

Figure 10. Deformed configuration at different displacements (scaled by 100) (a) D = 0.1 mm,(b) D = 0.3 mm.

5. Conclusions

A new method for modelling fracture of quasi-brittle materials under tensile and tensile-shearstress conditions has been developed based on conventional 3D deformable DEM. A newdiscrete crack model with failure criterion featuring tensile-shear zoning is introduced into3DEC program by the FISH language. Crack propagation is simulated through the gradualreduction of stiffness and strength which are driven only by the normal opening of the crack,in which only some ordinary material properties, including the tensile strength, the frictioncoefficient and the cohesion, are taken into account. Through verification analyses of bothmode I and I/II mixed mode fracture notched beam experiments, it is shown that the newapproach of incorporating discrete crack model into DEM is capable of simulating the entirefracture process within quasi-brittle materials.

Acknowledgements

The presented research work and the preparation of this paper have received financialsupport from the Global Environment Research Found of Japan (S-4), grants-in-Aid for Sci-entific Research (Scientific Research(B), C19310124, G. Chen) from JSPS (Japan Society forthe Promotion of Science). These financial supports are gratefully acknowledged.

References

1. De Borst, R. and Nauta P., “Smeared Crack Analysis of Reinforced Concrete Beams and SlabsFailing in Shear”, Proceedings of the International Conference on Computer Aided Analysis andDesign of Concrete Structures, Swansea: Pineridge Press, 1984, pp. 261–273.

2. Gálvez, J.C, Èervenka, J., Cendón, D.A. and Saouma, V.A., “A Discrete Crack Approach toNormal/shear Cracking of Concrete”, Cement and concrete research, 32, 1, 2002, pp. 1567–1585.

3. Azevedo, N.M. and Lemos, J.V., “A Generalized Rigid Particle Contact Model for Fracture Anal-ysis”, International Journal for Numerical and Analytical Methods in Geomechanics, 29, 3, 2005,pp. 269–285.

4. Ghaboussi, J., “Fully Deformable Discrete Element Analysis Using a Finite Element Approach”,Computers and Geotechnics, 5, 3, 1988, pp. 175–195.

5. Munjiza, A. and John N.W.M., “Mesh Size Sensitivity of the Combined FEM/DEM Fracture andFragmentation Algorithms”, Engineering Fracture Mechanics, 69, 2, 2002, pp. 282–295.

6. Owen, D.R.J., Feng, Y.T., de Souza Neto, E.A., Cottrell, M.G., Wang, F., Andrade Pires F.M. andYu, J., “The Modelling of Multi-fracturing Solids and Particular Media”, International Journalfor Numerical Methods in Engineering, 60, 1, 2004, pp. 317–339.

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7. Klerck, P.A., Sellers, E.J. and Owen, D.R.J., “Discrete Fracture in Quasi-brittle Materials underCompressive and Tensile Stress States”, Computation Methods in Application Mechanics Engi-neering, 193, 27–29, 2004, pp. 3035–3056.

8. ITASCA Consulting Group, Inc. USA, “3DEC, 3 Dimensional Distinct Element Code, Version 3.0,User’s manual”, 2005, Web site: http://www.itascacg.com.

9. Hillerborg, A., Modéer, M. and Petersson, P.E„ “Analysis of Crack Formation and Crack Growthin Concrete by means of Fracture Mechanics and Finite Elements”, Cement and ConcreteResearch, 6, 6, 1976, pp. 773–782.

10. Margolin, L.G., “A Generalized Griffith Criterion for Crack Propagation”, Engineering FractureMechanics, 19, 33, 1984, pp. 539–543.

11. Carol, I., Prat, P.C. and López, C.M., “Normal/Shear Cracking Model: Application to DiscreteCrack Analysis”, Journal of Engineering Mechanics, 123, 8, 1997, pp 765–773.

12. Gálvez, J.C., Cendón, D.A. and Planas, J., “Influence of Shear Parameters on Mixed-mode fractureof Concrete”, International Journal of Fracture, 118, 2, 2002, pp . 163–189.

13. Camacho, G.T. and Ortiz, M., “Computational Modelling of Impact Damage in Brittle Materials”,International Journal of Solids and Structures, 33, 20–22, 1996, pp. 2899–2938.

14. Wells, G.N. and Sluys, L.J., “A New Method for Modelling Cohesive Cracks Using Finite Ele-ments”, International Journal for Numerical Methods in Engineering, 50, 12, 2001, pp. 2667–2682.

15. Petersson, P.E., “Crack Growth and Development of Fracture Zones in Plain Concrete and SimilarMaterials”, TVBM-1006, Division of Building Materials, Lund Institute of Technology, 1981.

16. Gálvez, J.C., Elices, M., Guinea, G.V. and Planas, J., “Mixed Mode Fracture of Concrete underProportional and Nonproportional Loading”, International Journal of Fracture, 94, 3, 1998, pp.267–284.

17. Hou, Y.L., Zhou, Y.D. and Zhang, C.H., “I/II Tensile-shear Mixed Mode Fracture Simulation by3D Discrete Element Method”, Engineering Mechanics, 24, 3, 2007, pp. 1–7.

236

A Comparison Between the NMM and the XFEM in DiscontinuityModelling

X.M. AN AND G.W. MA∗

School of Civil and Environmental Engineering, Nanyang Technological University, Singapore

1. Introduction

Discontinuities such as voids, cracks, material interfaces widely exist in nature. In order tocharacterize such discontinuities explicitly, various numerical methods have been developed.

The finite element method (FEM) models the cracks by incorporating joint elements orinterface elements. However, such methods require the finite element mesh to coincide withthe cracks, which often complicates meshing task. When crack propagation involved, remesh-ing is inevitable, making the simulation tedious and time-consuming. In addition, variablessuch as displacement, stress and strain need to map to a new set of nodes and quadraturepoints. In order to overcome such inconveniences, various modifications to the conventionalFEM has been made within the framework of partition of unity.

Belytschko and Black1 enriched finite elements near the crack tips and along the cracksurfaces with the asymptotic crack tip functions, which allows the cracks to arbitrarily alignwithin the finite element mesh.

Moes et al.2 introduced the generalized Heaviside function H(x) to describe the discontin-uous field across a single crack. The finite element mesh is allowed to be independent of thecracks and remeshing is completely avoided for crack growth problems. In order to repre-sent branched and intersecting cracks, Daux et al.3 introduced the junction function J(x) andnamed their method the extended finite element method (XFEM). The XFEM has been suc-cessfully applied to 2D static and quasi-static crack growth problems,2 3D crack problems,4

with its extension to modelling voids,3,5 material interfaces,5,6 tangential discontinuities,6

and so on.Another example of modifications to the conventional FEM is the generalized finite ele-

ment method (GFEM). The mesh in the GFEM can be totally independent of the prob-lem domain. Recently, Simone et al.7 applied the GFEM for polycrystals with discontinuousaggregate boundaries by incorporating discontinuous functions Hα(x) corresponding to eachaggregate. Later, Duarte et al.8 extended it for branched cracks.

The numerical manifold method (NMM) can also be viewed as an extension or generaliza-tion to the conventional FEM. However, different from the XFEM and the GFEM, the NMMapproximation is based on covers. The NMM describes the discontinuities by splitting coverstogether with their cover functions. The covers in the NMM are usually generated from afinite element mesh. Similar to the GFEM, the NMM does not require the mesh conformingto neither the external boundaries nor the internal discontinuities, therefore the meshing taskin the NMM is easy and remeshing is totally avoided. The NMM has been successfully usedto describe both strong discontinuities9 and weak discontinuities10,11.

In this paper, the NMM and the XFEM are compared in modelling arbitrary discontinuitiessuch as voids, cracks, material interfaces. How the covers in the NMM favours the modellingof discontinuities is fully discussed. Incorporating the concept of covers into the XFEM issuggested to improve its efficiency.




2. Fundamentals of the NMM

With reference to an example in Fig. 1, the basic concepts in the NMM are introduced.The NMM adopts a mathematical domain (Fig. 1(b)) to cover the physical domain (Fig.

1(a)). Mathematical domain can be completely independent of but must be large enough tocover the physical domain.

The mathematical domain is constructed as a union of a finite number of mathematicalcovers, denoted as Mi (i = 1 ∼ nM). In the example in Fig. 1, there are totally two mathe-matical covers, M1 and M2 (Fig. 1(c)).

The physical covers are the intersection of mathematical covers and the physical domain.For example, if it is completely cut into several pieces and mi of them are within the problemdomain, a mathematical cover, say Mi will form mi physical covers, denoted as Pj

i(j = 1∼mi).In the example in Fig. 1, the M1 is completely cut by the physical features into three piecesand two of them are within the problem domain, and thus forms two physical covers P1

1 andP2

1 (Fig. 1d). Similarly, the M2 also forms two physical covers, P12 and P2

2 (Fig. 1(d)).The manifold element is defined as the common region of several physical covers. For

example, the four physical covers in Fig. 1(d) finally form five manifold elements, shown inFig. 1(e).

On each mathematical cover Mi, a weight function ϕi(x) satisfying

ϕi(x) ∈ C0 (Mi)

ϕi(x) = 0, x /∈ Mi∑k

if x ∈Mk

ϕk(x) = 1 (1)

is defined. The last term of Eq. (1) is known as partition of unity to guarantee the continuityof approximation.

On each physical cover Pji, a cover function uj

i(x) is defined.Weight functions defined on each mathematical cover transfer to physical covers as

ϕji (x) = δj

i · ϕi(x) (2)

where δji is a modifier, with its value equal to 1 within Pj

i and 0 elsewhere. It can be provedthat the weight functions in Eq. (2) also have the partition of unity property.

The global displacement field is approximated as

uh(x) =nM∑i=1

mi∑j=1

ϕjiu

ji =

nM∑i=1

ϕi

mi∑j=1

δjiu

ji. (3)

(a) (b) (c) (d) (e)

1M 2M1

1P 12P

21P 2

2P

11( )E P 1

2( )E P

21( )E P 2 2

1 2( , )E P P 22( )E P

Figure 1. Basic concepts in the NMM: (a) physical domain; (b) mathematical domain; (c) mathemat-ical covers; (d) physical covers; (e) manifold elements.

238


Theoretically, any shape of mathematical covers can be used in the NMM. However, con-struction of weight functions of mathematical covers and integration of stiffness matricesover manifold elements are related to the cover shape, thus a reasonable choice of covers isvery important. The most convenient and commonly used way is to adopt a finite elementmesh to generate mathematical covers and construct weight functions. The so-called finiteelement mesh here can be completely independent of the problem domain, therefore actuallydifferent from the one used in the FEM. Regarding the node in the finite element mesh as astar, the union of the finite elements sharing a common star forms a mathematical cover. Thefinite element shape functions naturally form the weight functions of mathematical covers.

3. Comparison Between the NMM and the XFEMin Discontinuity Modeling

3.1. Modelling voids

Modelling a single void with the NMM is illustrated in Fig. 2(a), where each mathematicalcover with squared star intersects the boundary of the void and forms one physical coverwithin its material fraction, while each mathematical cover with circled star is completelyinside the void and does not form any physical covers. The displacement field in the interiorof the void is naturally zero since no physical covers and thus no manifold elements areformed there.

Modelling a complex case with three voids is depicted in Fig. 3a, where each mathemati-cal cover with squared star forms one physical cover within its material fraction, while eachmathematical cover with circled star does not form any physical covers. Again, the displace-ment field in the interior of the voids is naturally zero.

Modelling the same single void with the XFEM is illustrated in Fig. 2(b), where eachsquared node whose support intersects the boundary of the void is enriched with the V(x)(V(x) = 1 when x outside the void, V(x) = 0 when x inside the void) through replacingclassical nodal shape function ϕi(x) by ψi(x) = ϕi(x)V(x), while deleting all the DOFs for thecircled nodes whose support is completely inside the void.

Modelling the complex case with three voids using the XFEM is depicted in Fig. 3(b).In this case, we need to define three discontinuous enrichment functions Vj(x), j = 1,2,3corresponding to three voids, where Vj(x) = 1 when x outside the void j, and Vj(x) = 0when x inside the void j. The enrichment to the nodes is also given in Fig. 3b.

The NMM approximation is based on covers. The displacement field within the voids arenaturally zero since no physical covers are generated there. Multiple voids are modelled in an

(a) (b)

Node whose support cut by boundary of void, enriched with V(x)

Node whose support completely inside the void, all DOFs deleted

MC cut by boundary of void, forms one PC within its material fraction

MC completely inside the void, forms no PC

Figure 2. Modelling a single void with: (a) NMM (MC=mathematical cover; PC=physical cover);(b) XFEM.

239


(a) (b)

Node enriched with 1( )V x

Node enriched with 2 ( )V x

Node enriched with 3( )V x

Node enriched with 1 2( ) ( )V Vx x



Node enriched with

1 2 3( ) ( ) ( )V V Vx x x

Node, all DOFs deleted

Void 1 Void 2

Void 3

MC cut by boundaries of voids, forms one PC within its material fraction

MC completely inside the void, forms no PC

Void 1 Void 2

Void 3

Figure 3. Modelling a complex case with three voids with: (a) NMM; (b) XFEM.

exactly same way with a single void. In contrast, the XFEM models the voids by introducingdiscontinuous enrichment functions into the standard finite element space. Different enrich-ment functions need to be defined corresponding to different voids. Treatment of a singlevoid is easy in the XFEM. However, when hundreds of voids involved, defining the enrich-ment functions and enriching the nodes will be tedious. In the following part, the XFEMwill be re-examined, and the modelling strategy of the NMM will be incorporated into theXFEM to make it more efficient.

Careful observation on Figs. 2(b) and 3(b) reveals that the XFEM initially defines thesupport of each node as the union of all the finite element sharing the node, then introducesenrichment functions to each squared node and deletes all the DOFs of each circled node torestrict the real support of each node to its material fraction.

To make it easy, without introducing any enrichment functions or deleting any DOFs, wecan directly define the material fraction of the original support of a node as its real support,define the material fraction of each original finite element as a real element, and do theintegrations only in the real elements. This modification makes the implementation mucheasier than before. It is inspired the modelling strategy of the NMM.

3.2. Modelling cracks

Modelling a single crack with the NMM is illustrated in Fig. 4(a), where each mathematicalcover with squared star is completely cut by the crack surface into two isolated regionsand thus forms two physical covers attached with independent cover functions, while eachmathematical cover with circled star is partially cut by the crack surface and thus forms onesingular physical cover, enriched with the asymptotic crack tip functions.

Modelling a branched crack with the NMM is depicted in Fig. 5(a), where each mathemat-ical cover cut by the crack surface into two isolated regions forms two physical covers, each

(a) (b)

Node whose support cucompletely by crack, enriched with ( )H x

Node whose support contains crack tips, enriched with crack tip functions

MC completely cut by crack, each forms two PCs

MC contains crack tip, each form one singular PC, enriched with crack tip functions

Figure 4. Modelling a single crack with: (a) NMM; (b) XFEM.

240


(a) (b)

Node enriched with I ( )H x

Node enriched with II ( )H x

Node enriched with ( )J x

Node enriched with crack tip functions

+1 -1

I ( )H x

+1 -1

II ( )H x

( )J x

+1 -1

0

Crack I Crack II

MC completely cut by crack, each forms several PCs

MC contains crack tip, each form one singular PC, enriched with crack tip functions

Figure 5. Modelling a branched crack with: (a) NMM; (b) XFEM.

mathematical cover cut by the crack surface into three isolated regions forms three physicalcovers, and each mathematical cover with circled star partially cut by the cracks forms onesingular physical cover. Each physical cover has an independent cover function. The singularphysical covers are enriched with the asymptotic crack tip functions.

Modelling the same single crack with the XFEM is illustrated in Fig. 4(b), where eachsquared node whose support is completely cut by the crack is enriched with the general-ized Heaviside function H(x), while each circled node whose support is partially cut by thecrack is enriched with the asymptotic crack tip functions. The corresponding displacementapproximation is

uh(x) =∑

i

ϕi(x)

⎛⎜⎜⎜⎜⎜⎝ui +H(x)bi︸︷︷︸

i∈n�

+4∑α=1

�αaαi︸︷︷︸i∈n�

⎞⎟⎟⎟⎟⎟⎠ (4)

where n� is the set of squared nodes, n� is the set of circled nodes.In order to represent branched or intersecting cracks, additional enrichment functions need

to be introduced. Modelling a branched crack with the XFEM is shown in Fig. 5(b). Thebranched crack is treated as the intersection of a main crack, crack I and a secondary crack,crack II. Three discontinuous functions HI(x), HII(x), and J(x) are defined corresponding tocrack I, crack II, and the junction of crack I and crack II, given in Fig. 5b. The enrichment ofthe nodes is as follows: nodes whose support is completely cut by crack I are enriched withHI(x), nodes whose support is completely cut by crack II are enriched with HII(x), nodeswhose support contains the junction are enriched with J(x), and nodes whose support ispartially cut by the cracks are enriched with the asymptotic crack tip functions.

Similarly, an intersecting crack needs to be treated as the intersection of a main crack,crack I and two secondary cracks, crack II and crack III. Five discontinuous functions, HI(x),HII(x), HIII(x), JI,II(x), JI,III(x) need to be defined corresponding to crack I, crack II, crack III,junction of crack I and II, junction of crack I and crack III. The nodes are then enrichedaccordingly.

The NMM approximation is based on covers. Splitting mathematical covers completelycut by the cracks into several physical covers attached with independent cover functions andenriching the singular physical covers make the displacement jump across arbitrarily com-plex cracks be modelled in a straightforward manner. In contrast, the XFEM describes thecracks by introducing discontinuous enrichment functions. Modelling a single crack is easyin the XFEM. However, when multiple arbitrarily branched or intersecting cracks involved,

241


defining the enrichment functions and enriching the nodes become tedious. In the followingpart, the XFEM is re-examined, and incorporating the modelling strategy of the NMM intothe XFEM is suggested.

The XFEM initially defines the support of a node as the union of all the finite elementssharing the node. Each node, say xi, originally has a unique unknown ui. If the support ofa node is completely cut by a single crack into two pieces, the H(x) is used to enrich thenode, which results in two independent unknowns ui + bi, ui − bi corresponding to twopieces of its support. Similarly, if its support is completely cut by a branched crack into threepieces, the node will be enriched with the J(x), which results in three independent unknownscorresponding to three pieces of the support. Again, if its support is completely cut by anintersecting crack into four pieces, the node will be enriched with two junction functions,resulting in four independent unknowns corresponding to four pieces of its support.

To make it easy, without using any enrichment functions, we can directly assign severalindependent unknowns to a node. To be more specific, if its support is completely cut into mi

pieces, the node xi will be assigned mi independent unknowns, denoted as uji (j = 1 ∼ mi).

One node is usually associated with only one unknown. Here, we totally have mi independentunknowns, it is equivalent to that we totally have mi nodes at the same position of node xi.We denote these nodes as xj

i (j = 1 ∼ mi), each of them takes one piece of the support as itssupport and uj

i (j = 1 ∼ mi) as its unknown. This modification makes the implementationmuch easier. It is also inspired by the modelling strategy of the NMM.

3.3. Modeling material interfaces

Modelling a material interface with the NMM is illustrated in Fig. 6a. The whole domainis divided into two distinct sub-domains, [1] and [2], corresponding to two materials,respectively. The material interface between[1] and[2] is denoted as �[1−2], which consistsof two coincident surfaces, �[1] and �[2], corresponding to two sub-domains [1] and [2],respectively. We define two outward unit normal vectors n[1] and n[2], associated with thesetwo distinct surfaces �[1] and �[2]. The numerically obtained displacement field should satisfythe following interface compatibility condition along the material interface �[1−2]:

u[1] = u[2]

t[1] = −t[2]. . (5)

Each mathematical cover with squared star in Fig. 6a forms two individual physical coversattached with independent cover functions. From Eq. (3), the displacement field is discon-tinuous across the material interface. In order to satisfy the interface compatibility conditiongiven in Eq. (5), the Lagrange multiplier method, the penalty method or the augmentedLagrange method can be adopted.

If the Lagrange multiplier method is adopted, the corresponding weak form of governingequation is ∫

δ εTσdV −∫�1−2

δ(u[1] − u[2]

)T · λd� =∫

δuTbdV +∫�t

δuTtd� (6a)∫�[1−2]

δλT ·(u[1] − u[2]

)d� +

∫�u

δλT · (u− u) d� = 0 (6b)

where ε is the strain tensor, σ is the stress tensor, u is the displacement vector, b is the bodyforce per unit volume, t is the traction prescribed on the traction boundary �t, λ is the vectorof Lagrange multiplier, which is actually the traction vector at the material interface.

242


(a) (b)

Node whose support completely cut by material interface, enriched with

( )ϕ x

[1]Ω [2]Ω

[1 2]−Γ

MC completely cut by material interface, each forms two PCs

[1]Ω [2]Ω

[1 2]−Γ

Figure 6. Modelling a material interface with: (a) NMM; (b) XFEM.

The second term of the left-hand side of Eq. (6a) is the enforcement of the interface com-patibility condition. The discrete equations can be derived accordingly, see Ref. 10.

If the penalty method is adopted, the Lagrange multiplier is approximated to be the corre-sponding displacement multiplied by a penalty parameter p such that

λ = p(u[1] − u[2]

)on �[1−2]. (7)

Substituting Eq. (7) into Eq. (6) gives the penalized weak form.In augmented Lagrange multiplier method, it is assumed that

λ = λ+ p [u] (8)

where λ is a given algorithmic multiplier, [u] is the error given by

[u] = u[1] − u[2]on�[1−2]. (9)

Substituting Eq. (8) into Eq. (6) yields the corresponding weak form, see Ref. [11].With the augmented Lagrange multiplier method, based on the parameter at step k, the

iterative procedure is followed by setting the multiplier at step k+1 as

λk+1 = λk + p [u]k . (10)

The augmented Lagrange method results in an accurate evaluation of the traction vectorat the material interface, which favours the modelling of debonding at the material interface.

Modelling the same material interface with the XFEM is illustrated in Fig. 6b, where eachsquared node whose support is completely cut by the material interface is enriched withthe absolute value of signed distance function, denoted as |ϕ(x)|. The corresponding XFEMapproximation is

uh(x) =∑

i

ϕi(x)

⎛⎜⎝ui + |ϕ(x)| ai︸︷︷︸

i∈n�

⎞⎟⎠ (11)

where n� is the set of nodes whose support is completely cut by the material interface.Since |ϕ(x)| is continuous, but |ϕ(x)|,n is discontinuous, therefore, the obtained displace-

ment field from Eq. (11) is continuous, but its derivative is discontinuous across the materialinterface.

The NMM initially results in a discontinuous displacement field across the material inter-face, and then adopts additional techniques to enforce the interface compatibility condition.The debonding at the material interface is ready to be modelled. The XFEM presents an

243


easier way to account for the material interface by introducing an enrichment function.However, similar to void problems and crack problems, when multiple material interfacesinvolved, defining the enrichment functions and enriching the nodes will become tedious.When debonding is desired, additional techniques also need to be incorporated.

4. Conclusions

In this paper, detailed comparison between the NMM and the XFEM in discontinuity mod-elling is presented. The NMM approximation is based on covers. It adopts a set of mathemat-ical covers to discretize the physical domain. Intersection between the mathematical coversand the physical domain forms physical covers attached with independent cover functions.The displacement field within the voids is naturally zero since no physical covers are gen-erated there. The displacement jump across arbitrarily complex cracks is straightforwardlycaptured by splitting mathematical covers into physical covers and enriching singular phys-ical covers. Interface compatibility condition is enforced by adopting additional techniques,which seems inconvenient, but makes the debonding ready to be modelled. In contrast, theXFEM introduces enrichment functions to account for the voids, cracks, and material inter-faces. When multiple voids, material interfaces, arbitrarily branched cracks involved, definingenrichment functions and enriching the nodes become tedious. Modification to the XFEMis then suggested to make it more efficient. For problems with voids, without using V(x),we can directly define the material fraction of the original support of each node as its realsupport. For problems with cracks, without using H(x) and J(x), we can directly assign miindependent unknowns to node xi whose support is completely cut by cracks into mi isolatedpieces. These modifications are inspired by the modelling strategy of the NMM. They makecomplex cases with multiple cracks and arbitrarily branched cracks to be modelled mucheasier than before.

References

1. Belytschko, T., Black, T., “Elastic crack growth in finite elements with minimal remeshing”, Inter-national Journal for Numerical Methods in Engineering, 45, 5, 1999, pp. 601–620.

2. Moes, N., Dolbow, J., Belytschko, T., “A finite element method for crack growth without remesh-ing”. International Journal for Numerical Methods in Engineering, 46, 1999, pp. 131–150.

3. Daux, C., Moes, N., Dolbow, J., Sukumar, N., Belytschko, T., “Arbitrary branched and intersect-ing cracks with the extended finite element method”, International Journal for Numerical Methodsin Engineering, 48, 2000, pp. 1741–1760.

4. Sukumar, N., Moes, N., Moran, B., Belytschko, T., “Extended finite element method for three-dimensional crack modeling”, International Journal for Numerical Methods in Engineering, 48,11, 2001, pp. 549–1570.

5. Sukumar, N., Chopp, D.L., Moes, N., Belytschko, T., “Modeling holes and inclusions by level setsin the extended finite element method”, Computer Methods in Applied Mechanics and Engineer-ing, 190, 46–47, 2001, pp. 6183–6200.

6. Belytschko, T., Moes, N., Usui, S., Parimi, C., “Arbitrary discontinuities in finite elements”, Inter-national Journal for Numerical Methods in Engineering, 50, 4, 2001, pp. 993–1013.

7. Simone, C.A., Duarte, E, Van, der Giessen, “A generalized finite element method for polycrystalswith discontinuous grain boundaries”, International Journal for Numerical Methods in Engineer-ing, 67, 8, 2006, pp. 1122–1145.

8. Duarte, C.A., Reno L.G., Simone A. “A high-order generalized FEM for through-the-thicknessbranched cracks” International Journal for Numerical Methods in Engineering, 72, 2007, pp.325–351.

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245

Initial Stress Formulae for High-Order Numerical ManifoldMethod and High-Order DDA

HAIDONG SU1,2,∗ AND XIAOLING XIE2

1DDA Center, Yangtze River Scientific Research Institute, Wuhan, China2Department of Material and Structure, Yangtze River Scientific Research Institute, Wuhan, China

1. Introduction

Based on mathematical manifold of modern mathematics, numerical manifold method(NMM) is invented by Dr. Shi.1 In NMM, the entire material volume is divided into manyfinite covers overlapped each other. These covers are called physical covers. They are formedby two independent cover systems: one is mathematical mesh system defining only the fineor rough approximations; the other is physical mesh system defining the boundaries of thematerial volume and the interfaces of different material zones. On each cover, an independentlocal cover function is defined, which can be constants, polynomials or other series. Thesefunctions are connected together to form a global function on the entire material volumeby means of weighted average via weight functions. The intersection areas of these coversare named manifold elements that can be in arbitrary shapes. Similar to the element in finiteelement method (FEM), the manifold element is the fundamental computation unit in NMM.

At present, finite element meshes are often employed as mathematical meshes to definefinite covers of NMM. All elements attached to any a FEM node form a mathematical cover.Here, the FEM shape function is the weight function for weighted average. Due to the arbi-trary shape of manifold elements, conventional integration methods, i.e., Gauss quadrature,are difficult to give precise results for element integrations. Therefore, simplex integrationformulae1 given by Dr. Shi are usually adopted to obtain exact integration results, but it isrequired that integrands should be polynomials.

Before the appearance of NMM, Dr. Shi invents Discontinue Deformation Analysis (DDA)method to investigate the motions and deformations of blocks. On the base of the entrancetheory of contacts between blocks, DDA has been widely utilized in Geotechnical Engineer-ing. In fact, DDA is a special case of NMM in the point that an independent physical coverrepresents a block. That is to say, covers or blocks do not overlap each other. Same as thegeneral NMM, displacements and deformations of the block are usually approximated viapolynomials in DDA, with the exception that the weight function is always equal to 1.

Some researches show that constant cover functions in NMM or first-order displacementapproximation in DDA usually bring about inaccurate results of displacements and stresses,unless very fine mathematical meshes are used in NMM. So high-order NMM or high-orderDDA is widely studied. 2–11 Computational accuracy is improved greatly in the case of smalldeformations of material, when high-order polynomials are employed as cover functionsor displacement functions. Whereas poor accuracy is obtained and even computation isnot convergent while solving large deformation problems (also called geometric nonlinearproblems),7–10 with the exception of Ref. 11. Although good results are achieved in 11, theapproach is not a typical high-order NMM because of the adoption of six constant coversin a triangular mesh. It has been pointed out in our previous papers10, 12 that inaccurate orincorrect treatments for high-order initial stresses, such as applying lower-order initial stress




expressions, or not considering the variation of structural configurations when accumulatinginitial stresses, are accounted for the unsuccessful cases. The difficulties of solving this prob-lem come from not only the correct way to accumulate initial stresses, but also the demandof being expressed explicitly in the form of polynomials in order to use simplex integrations.

This paper makes an attempt to obtain the correct initial stress formulae for high-orderNMM and high-order DDA. The contents are organized as follows: first, a brief introductionof the equations for large deformations in NMM or DDA is given; then high-order initialstresses formulae are developed; at last results of large deformations of a cantilever beam arepresented in order to demonstrate the validity of the formulae. For the sake of simplicity, thefollowing restricts the study to the case of isotropic linear-elastic material of continuum, notinvolving the material nonlinear problems due to large deformations.

2. Inertia Dominant Equilibrium Equation

The computations in NMM or DDA follow the time-step sequence. A so-called inertia domi-nant equilibrium equation is presented in Ref. 1. For linear-elastic material, at each load step(or time) the equations are given (from t = n− 1 to t = n) as:

([K]+ [Kg]){�dn} = {F} + {Fg} −∫V

[B]T{σn−1}dV (1)

{dn} = {dn−1} + {�dn} (2)

{σn} = {σn−1} + [D][B]{�dn} (3)

where [K], [B], [D], {F}, {d } and {σ } denote stiffness matrix, strain matrix, elasticity matrix,load vector (F is the total load which is already applied by the time of n), displacement vectorand stress vector, respectively. [Kg] and {Fg} are stiffness matrix and load vector due to inertiaforces, respectively. Expressions of these matrices and vectors can be referred to Ref. 1.

It can be seen that the above equations differ greatly from classical FEM formulae forlarge deformation computation. The governing Equation (1) involves neither complicatednonlinear terms nor equilibrium iterations usually appearing in FEM processes. All matrices,such as [K] and [B], are identical to those of small deformation problems. Hence, programcodes for small deformations can be directly utilized to implement the computation conve-niently. Equation (2) means coordinates of mathematical meshes and physical meshes arerenewed according to new incremental displacements obtained in each step. Thus all matri-ces are computed in the new structural configuration. After a number of steps are computed,small deformations of each step are accumulated to a large deformation. Equation (3) showsstresses are accumulated as initial stresses for the next step.

Dr. Shi has the following explanations about the rationality of the above equations.14

First, the nonlinear equation for the geometrical nonlinear problems can be transferred to aseries of linear equations via linear approximation, with the only demand that incrementaldisplacements of each step should be small enough to neglect the second order of the strain.Second, Eq. (1) is the equilibrium equation that material volume must satisfy at any time.It is an implicit equation that has a certain ability to eliminate unbalanced forces due toignorance of the nonlinear factors. Finally, Cauchy stresses are obtained by means of Eq. (3).On the other hand, as an important factor to the iteration solver, introduction of inertialeffect can improve the condition of the linear equations to be solved finally, because thequadratic term of �t in the denominator of coefficients of [Kg] has a strong effect to amplify

248


the principal diagonal terms when �t is very small. This is why they are called “inertiadominant equation”, and why taking account of inertial effect is recommended even in staticanalysis. Considering that the statics is the ultimate stabilized state of the dynamics, energydissipation should be introduced to solve static problems, such as velocity multiplying anappropriate constant that is 0.95 in this paper.

In the case of large deformations, it is still a problem to accurately compute the initialstress load term

∫v [B]T{σ }dV, on which this paper focuses.

3. Initial Stress Formulae for High-Order NMM

The approach is illustrated using one-order polynomial cover functions with triangular math-ematical meshes of two-dimensional problems. In this case the displacement cover functionsof the i-th node (i = 1,2,3) of the mesh are{

ui(x,y)vi(x,y)

}=[d1 d2 d3d4 d5 d6

]i{t} (4)

where d1 to d6 are coefficients of the series, defining generalized degree of freedoms to besolved, and {t} = [1 x y]T is the vector of monomial functions. Number of such termsincreases along with the rise of the order of polynomial functions.

The displacement field in the triangular mesh is expressed by using weighted average ofcover functions of three nodes, as given below{

u(x,y)v(x,y)

}=∑

i=1,3

Li{

ui(x,y)vi(x,y)

}=∑

i=1,3

(ai + bix+ ciy)[d1 d2 d3d4 d5 d6

]i{t} (5)

where Li = ai + bix + ciy is the shape function of triangular meshes, and ai, bi, ci arecoefficients. It can be seen that the displacement functions are two-order polynomials. Asstrains are one-order partial derivative of displacements, stresses in the triangular mesh aredistributed as linear functions.

At present, a method given by Refs. 8 and 9 is widely used to handle the initial stressproblem. In this method, incremental stresses are expressed as:

{�σ } = [D]{�ε} = [D][B]{�d} = [D][�S]{t} (6)

where [B]{�d} is written in the form of a strain coefficient matrix [�S] multiplying by thevector {t}. Then accumulation of σ in Eq. (3) is represented by superposition of coefficientsof [S] for linear-elastic material in which [D] is a constant matrix. This method is efficientto the problems of relative small deformations. However, it brings about great calculationerrors and even computation failure when large deformations occur.10

In Ref. 12 we find out the reason for the unsuccessful method is that variation of thestructural configuration is not considered during the initial stress accumulation procedure.According to Eq. (2), stress accumulation and integration of initial stress load should beimplemented in the present configuration. However, {x, y} coordinates of {t} remain unchangedin this method, implying that material points always stay in their original positions.

We have presented a so-called stress point method to handle the initial stress problem ofhigh-order NMM,12 in which three nodes of the triangular mesh are usually used as stresspoints. After each step is completed, stresses of each point are accumulated according toEq. (3). Stresses of any points in the mesh are linear interpolated, similar to the way ofdisplacement interpolation via Eq. (5), then the stress distribution formula in the mesh isdeduced. It has to be emphasized here that the coordinates of these stress points should be

249


renewed to reflect the variation of the configuration. This method is successfully applied tocompute large deformations of a cantilever beam, and computation errors comparing withthe analytical solutions are very small.

Following the idea, this paper introduces variations of coordinates to make an improve-ment to the original method given by 8 and 9. When the n step is to be computed, let theaccumulated strain coefficient matrix of the n− 1 step be [Sn−1]. Superposition of the initialstrain is written as:

[Sn] = [Sn−1]{tn−1} + [�Sn]{tn} (7)

In order to transfer the coordinates of the n−1 configuration, [tn−1] = [1 xn−1 yn−1]T,to those of the n configuration, [tn] = [1 xn yn]T, the relationship of them is deduced asfollows:

After the n−1 step is completed, the incremental displacements of the i-th node are �uin−1

and�vin−1, where subscript n−1 represents the computation step and superscript i represents

the i-th node. The displacements can be obtained according to Eq. (4) in an incremental form.Based on Eq. (5), coordinates of any a point in the triangular mesh are interpolated as

xn = xn−1 +�un−1 = xn−1 +∑

i

Lin−1�ui

n−1

=∑

i

ain−1�ui

n−1 +(∑

i

bin−1�ui

n−1 + 1

)xn−1 +

(∑i

cin−1�ui

n−1

)yn−1 (8)

yn = yn−1 +�vn−1 = yn−1 +∑

i

Lin−1�vi

n−1

=∑

i

ain−1�vi

n−1+(∑

i

bin−1�vi

n−1

)xn−1 +

(∑i

cin−1�vi

n−1 + 1

)yn−1 (9)

Substitute Eq. (8) and (9) into {t} and write it in the form of a matrix

{tn} =⎧⎨⎩

1xnyn

⎫⎬⎭ =

⎡⎢⎢⎣

1 0 0∑i

ain−1�ui

n−1∑i

bin−1�ui

n−1 + 1∑i

cin−1�ui

n−1∑i

ain−1�vi

n−1∑i

bin−1�vi

n−1∑i

cin−1�vi

n−1 + 1

⎤⎥⎥⎦⎧⎨⎩

1xn−1yn−1

⎫⎬⎭

= [Cn−1]{tn−1} (10)

As displacements of each step are very small, the principal diagonal terms of the squarematrix [Cn−1] are dominated. So

{tn−1} = [Cn−1]−1{tn} (11)

Substituting Eq. (11) into Eq. (7), we have the coefficient matrix of initial strain written as

[Sn] = [Sn−1][Cn−1]−1 + [�Sn] (12)

then the initial stress load∫

v [B]T[D]{Sn}{t}dV can be computed for the next step.When cover functions with more than one order are used, Eqs. (8) and (9) remain the same.

What is only to be done is substituting them into a more complicated vector {t} of monomialfunctions. With regard to tetrahedron meshes in three-dimensional problems, z coordinate isintroduced into the above equations.

Since strain matrix [B] is also the matrix of polynomials, integrands of initial stress loadterm

∫V [B]T{σ }dV can be expressed as polynomials. Thus simplex integration can be used,

250


but expressions of the integrands for high order terms are usually much complicated. Asa foundation of this paper, approaches are achieved to develop expressions of high-orderNMM matrices and to automatically form program codes by using mathematical softwarein Ref. 3.

4. Initial Stresses of High-Order DDA

DDA makes use of polynomials to depict displacements of a block as:{u(x,y)v(x,y)

}=[du1 du2 · · ·dv1 dv2 · · ·

]{t} (13)

where du1 and so on are coefficients, and {t} is also the vector of monomial functions, suchas

{t} = [1 x y x2 xy y2 x3 x2y xy2 y3]T (14)

in the case from one order to three order monomials.Since a block can be regarded as one cover of NMM, all matrices and vectors of high-order

DDA are similar to those of high-order NMM except that weight function always remains1. Accumulation of initial strain is also the same as Eq. (7). However, if we follow the sameidea of NMM for considering variation of coordinates as

xn = xn−1 +�un−1 = xn−1 + [du1 du2 · · · ]{tn−1} (15)

yn = yn−1 +�vn−1 = yn−1 + [dv1 dv2 · · · ]{tn−1} (16)

the maximal power of xn and yn in {tn} gets up to the square of the original order, forexample, 9th power in the case of three-order polynomials. This leads to not only morecomplicated derivation of equations but also a large computation amount of integration ofinitial stress load.

So an alternate way is chosen. Only for computing initial stress load, the block of DDAis divided into several triangular meshes, or in an easier way that triangular meshes areintroduced to cover the block, as what is done in NMM in the preceding chapter. Hence, theintegration of initial stress load in the entire block is changed to the sum of integrations ofthe small blocks.

For each small block, incremental displacements of the nodes of the triangular mesh towhich the block belongs are computed according to Eq. (13). Coordinates of xn and yn ofany points in the block are obtained according to Eqs. (8) and (9). It can be seen that themaximal order of xn and yn in {tn} remains the same. Then compute [Cn−1] and [Cn−1]−1 ofEqs. (11) and (12) for every small block.


The two-dimensional program written by Dr. Shi is adapted to the above procedures throughimplementation of Fortran codes. We consider a cantilever beam shown in Figure 1. Thelength of the beam is 10m, with both height and width are 1m. The Young’s modulus E is3 × 105 kN/m2, and Poisson’s ratio v is 0.2. The beam is subjected to a concentrated forceP at the midpoint of the section of the free end, which is always downward vertically. Thenumber of the triangular meshes is 38 in high-order NMM with one-order polynomial coverfunctions.

Table 1 gives the results of the displacements of the free end at several steps, and corre-sponding deformation when P = 1250 kN is shown in Figure 2. Comparing with analytical

251


P

Figure 1. Manifold meshes of a cantilever beam.

Table 1. Displacements of the mid-point at the free end of the cantilever beam (by high-orderNMM).

P Horizontal displacements — u (m) Vertical displacements — v (m)

(kN) Numericalsolutions

Analyticalsolutions13

Relativeerror (%)

Numericalsolutions


Relativeerror (%)

125 0.15 0.16 −6.25 1.60 1.62 −1.23250 0.56 0.56

0.003.02 3.02

0.00500 1.61 1.60

0.634.96 4.94

0.40750 2.56 2.55

0.396.08 6.03

0.831000 3.32 3.29

0.916.76 6.70

0.901250 3.91 3.88

0.777.21 7.14

0.98

Figure 2. Deformation of the cantilever beam when P = 1250 kN.

solutions,13 good results of numerical solutions demonstrate the validity of high-order initialstress formulae. The results are close to those of stress point method in Ref. 12, implyingthese two methods are equivalent.

With regards to high-order DDA, 18 triangular meshes cover the cantilever beam in orderto calculate initial stress load. To obtain a precise result in a one-step static analysis, three-order polynomials for displacements are needed, while four-order functions are required tocompute large deformations of several steps. Results also show the validity of the proposedmethod, as given in Table 2.

252


Table 2. Displacements of the mid-point at the free end of the cantilever beam (by high-orderDDA).

P Horizontal displacements—u (m) Vertical displacements—v (m)

(kN) Numericalsolutions


Relativeerror (%)

Numericalsolutions


Relativeerror (%)

125 0.15 0.16 −6.25 1.61 1.62 −0.62250 0.51 0.56 −8.92 2.90 3.02 −3.97500 1.59 1.60 −0.63 4.92 4.94 −0.40750 2.52 2.55 −1.18 6.00 6.03 −0.50

1000 3.27 3.29 −0.61 6.66 6.70 −0.601250 3.84 3.88 −1.03 7.08 7.14 −0.84

It is found that if inertia matrix is not considered, computation is not convergent when verylarge deformations occur (when p is greater than 1000 kN). However, if inertia is introduced,a period of time is required to eliminate the effect of inertia for the static computations. So theresults of some early steps are smaller than the analytical solutions. For example in Table 2,when p = 250 kN, the horizontal and the vertical displacements are 0.51m and 2.90m, andrelative errors comparing with analytical solutions are 9% and 4%, respectively. If we ignoreinertial effect, the above results change to 0.56 m and 3.05 m, and relative errors are all under1%.

6. Conclusions

This paper presents initial stress formulae for high-order NMM and high-order DDA, sat-isfactorily settling the problem of poor accuracy or no convergence for large deformationcomputation. This method lay a foundation of discontinue deformation analysis using high-order NMM and DDA, and fixed-mesh NMM in the future.

NMM is well known for its independence of mathematical meshes from physical meshes.When mathematical meshes satisfy boundaries of material, and constant cover function areused, NMM is degraded to the conventional FEM. Hence the equations for large defor-mations including the initial stress formulae presented in this paper are suitable for FEM.Since matrices of NMM are the same as those of small deformations, programming is quiteconvenient. Therefore, it can be regarded as the improvement of the conventional FEM forgeometrical nonlinear problems.

Acknowledgement

The project is supported by the National Science Foundation of China (10772034).The research is done under the guidance of Dr. Gen-hua Shi. Thanks for his great helps.

References

1. Gen-hua Shi, Numerical Manifold Method (NMM) and Discontinuous Deformation Analysis(DDA), Qinghua University Press, Beijing, China, 1997. (in Chinese)

2. Rong Tian, Maotian Ruan, etc, “Fundametals and applications of high-order manifold method”,Engineering Mechanics, 18, 2, 2001, pp 21–26. (in Chinese)

3. Haidong Su, Xiaoling Xie, Qin Chen, “Application of high-order numerical manifold method instatic analysis”, Journal of Yangtze River Scientific Research Institute, 22, 5, 2005, pp. 74–77 (inChinese).

253


4. Shaozhong Lin, Yongfeng Qi, Haidong Su, “Element analysis of high-order numerical manifoldmethod based on special matrix operations”, Journal of Yangtze River Scientific Research Institute,23, 3, 2006, pp. 36–39. (in Chinese)

5. S.A. Beyabanaki, A. Jafari, M.R. Yeung and S.O. Biabanaki, “Implementation of a trilinear hexa-hedron mesh into three-dimensional discontinuous deformation analysis (3-D DDA)”, Proceedingsof the Eighth International Conference on the Analysis of Discontinuous Deformation, Beijing,China, 2007, pp. 51–56.

6. D. Kourepinis, N. Bicanic, C.J. Pearce, “A higher-order variational numerical manifold methodformulation and simplex integration strategy”, Proceedings of the Sixth International Conferenceon the Analysis of Discontinuous Deformation, Trondheim, Norway: A.A Balkema, 2003, pp.145–151.

7. Xiaobo Wang, Xiuli Ding, Bo Lu, Aiqing Wu, “DDA with higher order polynomial displacementfunctions for large elastic deformation problems”, Proceedings of the Eighth International Con-ference on the Analysis of Discontinuous Deformation, Beijing, China, 2007, pp. 89–94.

8. Ming Lu, “High-order manifold method with simplex integration”, Proceedings of the Fifth Inter-national Conference on the Analysis of Discontinuous Deformation, A.A Balkema, 2002, pp. 75–83.

9. Ming Lu, “Complete n-order cover function for numerical manifold method”, STINTEF report,STF22 F01139, 2001.

10. Haidong Su, Xiaoling Xie, Qin Chen, “Soving large deformation problems using numerical mani-fold method”, Acta Mechanica Solida SINICA, 25, S. Issue, 2004, pp. 88–92. (in Chinese)

11. Guoxin Zhang, Jing Peng, “Second-order manifold method in structure failure analysis”, ActaMechanica Sinica, 34, 2, 2002, pp 261–269. (in Chinese)

12. Haidong Su, “Study on new methods for solving fluid-solid coupling vibration and their appli-cations”, PHD thesis, Huazhong University of Science and Technology, Wuhan, China, 2006. (inChinese)

13. Zhenxing Liu, Yan Sun, Guoqing Wang, Computational Mechanics, Shanghai Jiaotong UniversityPress, Shanghai, China, 2000, 270 p. (in Chinese).

14. Gen-hua Shi, 2006, Private communication.

254

Development of Coupled Discontinuous Deformation Analysis andNumerical Manifold Method (NMM-DDA) and Its Application toDynamic Problems

S. MIKI1,∗, T. SASAKI2, T. KOYAMA3, S. NISHIYAMA3 AND Y. OHNISHI4

1Kiso-Jiban Consultants Co., Ltd. 1-5-7, Kameido, Koto-ku, Tokyo, 136-8577, Japan2SUNCOH Consultants Co., Ltd. 1-8-9, Kameido, Koto-ku, Tokyo, 136-8522, Japan3Kyoto University, Kyoto Univ.-Katsura 4, Saikyo-ku, Kyoto, 615-8540, Japan4Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto, 606-8501, Japan

1. Introduction

Earthquakes and heavy rainfall, which repeatedly occurred in Japan, can be a major triggerfor rock slope failures and rock falls and these disasters often caused serious damages tonot only human lives but also infrastructures. The greater part of the Japanese territory isoccupied by mountainous area, and there are many dangerous slopes along the national roadswhich are the basis of our social activities. However, it is difficult to reinforce all dangerousslopes from technical and economical points of views. Hence, it is necessary to establish themethods to evaluate the damages and/or risks for social stocks and structures/infrastructuressuch as roads, railways, houses etc against collapsed rock masses.

In order to predict and estimate the traveling distance and velocities of collapsed rockblocks, the discontinuum-based numerical approaches such as Distinct Element Method(DEM)1 and Discontinuous Deformation Analysis (DDA),2, 3 which can introduce fracturesexplicitly in the model, will be effective tools. So far, DEM and DDA have been widely usedto analyze the slope stability, excavation of underground cavern, etc in discontinuous rockmasses. Recently, these discontinuum-based numerical methods were applied to the seismicproblems such as slope failures due to earthquakes, where one of the key issues is the esti-mation of traveling velocities and distances for the collapsed rock blocks and the safety ofthe structures adjacent to the slopes.4, 5 For the dynamic response analysis of discontinuousrock slopes, seismic forces are commonly applied to the basement block modeled using asingle DDA block. However, it is necessary to consider the local variation of seismic forcesand stress conditions, especially when the size of slopes is large and/or the slope geometrybecomes complicated. There is difficulty in DDA to consider the local displacements of thesingle block for the basement due to the fact that the strain in the single block is uniformand displacement function is defined at the gravity center. On the other hand, the NumericalManifold Method (NMM)6 can simulate both continuous and discontinuous deformationof blocks with contact and separation. However, the rigid body rotation of blocks, which isone of the typical behaviors for rock slope failure, cannot be treated properly because NMMdoes not deal with the rigid body rotation in explicit form.

For the numerical simulations of the dynamic behavior of slopes during earthquakes, itis necessary and preferable to consider both continuous and discontinuous deformations offractured rock masses appropriately. According to the above mentioned features and draw-backs, it is reasonable to combine these DDA and NMM for the slope stability problemsfrom practical point of view. The mechanical behavior of falling rock blocks is simulated byDDA with the basement block covered by the NMM mesh, where seismic forces are given.In this paper, the formulation for the coupled NMM and DDA (NMM-DDA) is presented




with the programming code developments. For the formulation, NMM and DDA can beeasily combined by choosing displacements of the DDA blocks and NMM cover nodes asunknowns, because the total potential energy is minimized to establish the equilibrium equa-tions and block system kinematics are same between DDA and NMM. In this paper, theapplication of the NMM-DDA was also presented with the discussion of the applicability forthe dynamic response analysis.

Some technical terms used in this paper are defined as follows (see Figure 1):‘Block’ means a closed area surrounded by one ‘joint loop’. In DDA, a block is a base

unit to solve displacements. On the other hand, in NMM, a block consists of ‘elements’.‘Element’ shows an area divided by the numerical ‘cover’ in a ‘joint loop’. In NMM, anelement is a base unit to solve displacements. ‘Cover’ implies numerical mesh (with triangleand/or rectangular shape) dividing the area surrounded by ‘joint loop’. The ‘cover’ is used forcalculating displacement of the elements. ‘Joint loop’ means the peripheral boundary of the‘block’. ‘Joint’ means the boundary between ‘blocks’ in contact, and common line segmentsof two ‘joint loops’.

2. Development of Coupled NMM and DDA (NMM-DDA)

2.1. Basic concept of NMM-DDA and its displacement function

The combined analyses of NMM and DDA should be performed in the joint loop defined asDDA blocks and/or NMM elements. Therefore, the joint loop defined as DDA and NMMwill be contact through the joint, which is a segment of common line between two joint loopsas shown in Figure 1. It should be noted that the DDA block and NMM element cannot sharethe common area in the same joint loop.

There are two possible methods to combine DDA with NMM theoretically. Chen et al.proposed one possible method as the extension of NMM, which treats a single DDA blockas a single NMM element overlapped one triangle cover mesh.7 In their method, the displace-ments of the DDA block are obtained by connecting the displacements for NMM elementand DDA block after NMM calculations and the rigid body rotation is applied separatelyto DDA blocks. The other possibility, which is applied in this study, is to combine DDAand NMM directly, where the coupled term for NMM elements and DDA blocks appearsin the equilibrium equation. In this method, it is also easy to combine NMM and/or DDAwith rigid block system because the equilibrium equation for both DDA and NMM can beobtained from the same principle so-called the minimization of potential energy.

Joint loop modeled as DDA Material boundary

Element

Cover

Joint loop modeled as NMM Joint boundary (Joint) Block

Figure 1. Notations of NMM-DDA block system.

256


The total potential energy�sys of the block system, which includes DDA blocks and NMMelements, can be expressed as the following equation:

�sys = �msys +�d

sys +∑B,i

∑E,j

�i,j (1)

where �dsys and �m

sys are the potential energy for DDA part and NMM part, respectively. Thelast term on the right side of Eq. (1) represents the potential energy for the contacts betweenDDA block i and NMM element j. The NMM-DDA is formulated from Eq. (1) with thekinematic equations based on Hamilton’s principle expressed as:

MD+ CD+ KD = F (2)

where M is mass matrix, C is viscosity matrix, K is stiffness matrix, and F is external forcevector. D, D and D are displacement, velocity and acceleration of DDA block and NMMelement, respectively. The matrices and vector in kinematic equations based on Hamilton’sprinciple can be also obtained by minimizing the potential energy expressed as Eq. (1). How-ever, the potential energy for DDA part�d

sys and NMM part�msys are minimized with respect

to the displacement of the DDA block and the NMM element, respectively. These processesare similar to those in original DDA and NMM. Consequently, the formulation of NMM-DDA is necessary only for the potential energy of the contacts between DDA blocks andNMM elements. In NMM-DDA, the unknowns are six displacement variables in Eq. (3) andnodal displacements of the cover in Eq. (4) for DDA and NMM parts, respectively.(

uv

)= [Td

i (x,y)][Ddi ], [Dd

i ] = (u0 v0 r0 εx εy γxy)T (3)(

uv

)= [Tm

i (x,y)][Dmi ], [Dm

i ] = (u1 v1 u2 v2 u3 v3)T (4)

In Eq. (3), [Tdi ] is the block deformation matrix (displacement function) for i-th DDA block,

(u0, v0) is the rigid body transformation, r0 is the rigid body rotation of the block at thegravity center, and εx, εy, γxy are the normal (in the x- and y- directions) and shear strainsof the block, respectively. In Eq. (4), [Tm

i ] is the element deformation matrix for i-th NMMelement and (uj, vj)(j = 1,2,3) means the displacements at the triangle nodes of the cover.

2.2. Contact sub-matrices

For the NMM-DDA codes, the contacts between DDA block and NMM element should beconsidered and newly formulated. Assuming that the contact between corner P1 on the DDAblock i and edge P2P3 on the NMM element j as shown in Figure 2, the distance d from pointP1 to edge P2P3, which should be zero after some displacements to prevent the penetrationbetween DDA block and NMM element, is expressed as follows:

d = �

l= 1

l

∣∣∣∣∣∣1 x1 + u1 y1 + v11 x2 + u2 y2 + v21 x3 + u3 y3 + v3

∣∣∣∣∣∣ , l =√

(x2 − x3)2 + (y2 − y3)2 (5)

� = S0 +(y2 − y3 x3 − x2

) (u1v1

)+ (y3 − y1 x1 − x3

) (u2v2

)+ (y1 − y2 x2 − x1

) (u3v3

),

S0 =∣∣∣∣∣∣1 x1 y11 x2 y21 x3 y2

∣∣∣∣∣∣257


d

P1(x1, y1)

P3(x3, y3) P2(x2, y2)

DDA block i

MM element j

P0(x0, y0)

Figure 2. Contact between DDA block and NMM element.

where, (xi, yi) and (ui, vi) are the coordinates and displacements of point Pi (i = 1, 2, 3),respectively, l is the length of edge P2P3. Using deformation matrices T(xi, yi) (i = 1, 2, 3)defined at the points P1, P2 and P3, the distance d can be calculated as follows:

d = S0

l+ [Hd]T[Dd

i ]+ [Gm]T[Dmj ]

[Hd] = 1l[Td

i (x1,y1)]T(

y2 − y3x3 − x2

), (6)

[Gm] = 1l[Tm

j (x2,y2)]T(

y3 − y1x1 − x3

)+ 1

l[Tm

j (x3,y3)]T(

y1 − y2x2 − x1

)

When the DDA block contacts with the NMM elements, the contact spring with the stiff-ness of kp is introduced to prevent the penetration between the DDA blocks and the NMMelements. The potential energy �p for the contact spring is:

�p = kp

2d2 = kp

2

(S0

l+ [Hd]T[Dd

i ]+ [Gm]T[Dmj ])2

(7)

Minimizing potential energy expressed as Eq. (7) by taking the derivatives in terms of[Di], the four 6 × 6 sub-matrices and two 6 × 1 sub-matrices can be obtained. These sub-matrices are assembled to the global stiffness matrix. The processes for deriving the contactsub-matrices mentioned above are same as the ones in original DDA and/or NMM. The sub-matrices for rock bolts, which connect DDA block and NMM element, can be also derivedeasily in the similar way.

2.3. Kinematics of DDA block and NMM element

The NMM and DDA originally developed by Shi2, 3, 6 use the same kinematics for block sys-tem, and the criteria for penetration and separation in DDA is also same as NMM. The con-tacts between blocks and/or elements are searched along joint loop in both DDA and NMM.Therefore, the same kinematics for block system in DDA and/or NMM can be adopted inNMM-DDA. When the DDA blocks contact with the NMM elements, sub-matrices for thecontacts can be easily derived as shown in the previous section.

258


2.4. Programming codes

The developed NMM-DDA code consists of the following four different programs: (1) pre-processing program to generate the NMM mesh based on the joint geometry data, (2) pre-processing program to define the joint loops as NMM elements and convert into the jointloops defined as DDA blocks, (3) main calculation program for NMM-DDA, and (4) post-processing program to illustrate the results of NMM-DDA calculations.

When all joint loops are defined as NMM, the main calculation program is exactly sameas original NMM code. On the other hand, the main calculation program is also same asoriginal DDA code when all joint loops are defined as DDA blocks. All programs are writtenin C and C++ language.

3. Application to Earthquake Response Analysis of Rock Slope

Figure 3 shows the application of newly developed NMM-DDA to the earthquake responseanalysis of rock slopes. The length and height of the rock slope are 250m and 120m, respec-tively. The basement of the slope was divided by NMM elements, and the rock slope consistsof rectangular DDA blocks. The material properties and analytical conditions are summa-rized in Table 1. A series of simulations were carried out with different joint strength (cohe-sion of joint) called Case 1, 2 and 3 to investigate the effect of joint strength on the mechan-ical behavior of fractured rock masses. The largest cohesion of joint was given for the Case1 and smallest for Case 3. For Case 3, the larger viscosity coefficient of blocks and elements,which worked as damper, was given comparing the one for Case 1 and 2. The seismic forces,which were calculated from the acceleration records of the actual earthquake obtained from

250m

120m

Acc.(m/s2)

Time(s)

Acc.(m/s2)

Time(s)

Time(s)

Time(s)

Time(s) Time(s)

Time(s) Time(s)

Time(s) Time(s)

Disp.(cm)

Disp.(cm)

Disp.(cm) Disp.(cm) Disp.(cm)

Disp.(cm) Disp.(cm) Disp.(cm)

Vertical

Loading acceleration

Horizontal

Response displacement (horizontal)

Response displacement (horizontal)

Figure 3. Loading acceleration and seismic response of displacements at different parts of the slope(Case 1).

259


Table 1. Physical properties and parameters.

Case 1 Case 2 Case 3

Unit mass (kg/m3) 2000 2000 2000Unit weight (kN/m3) 20 20 20Young’s modulus (GN/m2) 1.0 1.0 1.0Poisson’s ratio 0.3 0.3 0.3Viscosity coefficient for blocks and elements 100 100 1000Penalty stiffness (GN/m) 10 10 10Friction angle of joint (deg) 45 45 45Cohesion of joint (kN/m2) 10.0 1.0 1.0Tensile strength of joint (kN/m2) 0.0 0.0 0.0Max. time step(s) 0.01 0.01 0.01

the seismic observation station at the ground surface, were given to the basement (NMMelements) as a dynamic body force. This will be the simplest method to apply seismic forcesto the basement elements because dynamic body forces are proportional to the input seismicaccelerations and can be calculated directly.

The simulation results for Case 1 are shown in Figure 3. From this figure, the waveformof displacement responses in the basement was similar to the one for the input accelerationsand the displacements in the basement increase toward right hand side because the rightside boundary of the model was free. The DDA blocks moved together with the basementin the initial stage until the slipping of DDA blocks became dominant. When the slippingand/or separation between NMM basement and DDA blocks occurred with increasing seis-mic forces, the displacements of the DDA blocks increased rapidly.

Figure 4 also shows the distribution of displacements for DDA blocks and NMM elements(for Case 1). From this figure, the DDA blocks fell down along the basement modeled byNMM and the DDA blocks moved toward the toe of the slope, and the failure mode wasregarded as sliding. This figure also clearly shows that the problem with large displacementand contact/separation of blocks can be simulated correctly by newly developed NMM-DDAwith the local distribution of the stress and deformation. The NMM-DDA can be success-fully applied to the dynamic response analysis for the model including both continuous anddiscontinuous media.

60

5652

4844

4036

32

28

2420

16 12

8 4

Disp.(cm)

after 1100 step (6.00s) after 1400 step (8.26s)after 1300 step (7.67s)

after 3000 step (13.34s) after 4000 step (14.14s) after 6000 step (16.33s)

Figure 4. Displacement distribution for each block after applying seismic loads (Case 1).

260


60 565248444036 32 282420 16 1284

Disp.(cm)




Figure 5 shows the distribution of displacement for DDA blocks and NMM elements forCase 2. The DDA blocks moved toward the toe of the slope along the basement and felldown from the slope surface. The failure mode of the slope for Case 2 is toppling, which isdifferent failure mode from Case 1. The DDA blocks started slipping and were separated inthe early stage of the seismic loads, and the separation lines clearly observed in the verticaldirection and grew along the slope surface. However, comparing with the results of Case 1,the traveling distance of the blocks in Case 2 was shorter.

Figure 6 shows the distribution of displacement for DDA blocks and the basement modeledas NMM elements for Case 3 with heavily damped case. In this case, the collapse of the DDAblocks was not observed even though the large displacement occurred during the seismicloads. However, slipping toward left side and separation between DDA blocks appearedalong the slope surface and these displacements of the blocks were stored as the cumulativedeformation of the slope. This simulation result also indicates that the joints and/or fracturesin the slope will open by earthquake especially near slope surface and these open joints turnto a defect for following earthquake.

From Figures 4 and 5, the joint strength between blocks plays significant roles for thefailure modes of the slopes during earthquake and traveling distances of falling blocks alsodepend on the failure modes. From the simulation results presented in this paper, the travelingdistance of DDA blocks for Case 1 shows larger than one for Case 2 and the failure modes forCase 1 and 2 were ‘sliding’ and ‘toppling’, respectively. Therefore, it is possible to estimate

60 565248444036 32 282420 16 12 84

Disp.(cm)




261


that the slope failure by sliding tends to show large traveling distance of falling blocks. Infact, the failure modes of slopes during earthquakes is affected by many other factors such asjoint distributions, joint strength, slope geometry, intensity and frequency characteristics ofseismic waves, and so on. The effect of these factors on the slope failure modes are key issueand will be investigated in the future.

4. Conclusions

In this paper, the NMM-DDA analysis method was newly developed and applied to earth-quake response analysis of the rock slope. Both NMM and DDA originally developed byShi 2, 3 and 6 share the common mathematical principles so-called the minimization of thepotential energy to establish the equilibrium equations for kinematics of block system. Thisenables to combine NMM and DDA easily and DDA blocks and NMM elements can betreated at the same time. The findings obtained from this study are summarized as follows:

• In NMM-DDA, the blocks modeled as NMM elements are analyzed by NMM, andthe blocks modeled as DDA blocks are analyzed by DDA.• The only difference between NMM-DDA and original NMM (DDA) is the treatment

for the contact between NMM elements and DDA blocks.• The contact sub-matrices between NMM element and DDA block are easily derived

by using the deformation matrices for NMM element [Tmj ] and DDA block [Td

i ],and the proposed NMM-DDA satisfy the principle of minimum potential energy.• The NMM-DDA compensates the drawbacks of original NMM and DDA, and

NMM-DDA can be applied to many geotechnical problems including both continu-ous and discontinuous media.• The results of the earthquake response analysis indicate that the failure modes and

traveling distance of the collapsed rock blocks are affected by the joint strengthbetween blocks significantly.

The NMM-DDA was applied to the simulation of the discontinuous rock slope behaviorduring earthquake successfully, as shown in Figure 3. However, when NMM-DDA is used tosolve the actual geotechnical problems, how to determine the specific parameters such as stiff-ness of contact springs and displacement control parameters is still key issue. Moreover, veryfew systematic methods to determine these parameters have been proposed so far even fororiginal DDA and NMM. The selections of these parameter values are closely related to theconvergence speed of the calculation and the performance of contact behaviors. Therefore,the quantitative approaches as well as sensitivity analysis are required in order to determineparameter values systematically and appropriately in the future.

References

1. Cundall P.A., “A Computer Model for Simulation Progress, Large Scale Movement in Block Sys-tem”, ISRM Symp., Nancy, France, 1971, pp. 11–18.

2. Shi G.H. and Goodman R.E., “Two dimensional discontinuous deformation analysis”, Int. J.Numer. Anal. Meth. Geomech., 9, 1985, pp. 541–556.

3. Shi G.H and Goodman R.E., “Generalization of two-dimensional discontinuous deformation anal-ysis for forward modelling”, Int. J. Numer. Anal. Meth. Geomech, 13, 1989, pp. 359–380.

4. Sasaki T., Hagiwara I., Sasaki K., Horikawa S., Ohnishi Y., Nishiyama S. and Yoshinaka R.,“Earthquake response analysis of a rock-fall by discontinuous deformation analysis”, Proc.ICADD-7, Hawaii, 2005, pp. 137–146.

262


5. Hamasaki E. and Sasaki A., “Study on landslide due to earthquake by using Discontinuous Defor-mation Analysis”, Proc. 3rd ARMS, Kyoto, Japan, 2004, pp. 1253–1256.

6. Shi G.H., “Manifold Method of Material Analysis”, Transactions of the 9th Army Conference onApplied Mathematics and Computing, Report No.92-1, U.S. Army Research Office, 1991.

7. Chen G.Q., Zen K., Ohnishi Y. and Kasama K., “Extension of Manifold Method and Its Applica-tion”, Proc. ICADD-4, Glasgow, UK, 2001, pp. 439–450.

263

Stability Analysis of Ancient Block Structures by Using DDA andManifold Method

T. SASAKI1,∗, I. HAGIWARA1, K. SASAKI1, R. YOSHINAKA2, Y. OHNISHI3,S. NISHIYAMA3 AND T. KOYAMA3

1Rock Engineering Lab., Suncoh Consultants Co., Ltd., Tokyo, Japan2Saitama University, Saitama, Japan3School of Urban & Environment Engineering, Kyoto University, Kyoto, Japan

1. Introduction

Kerisel, J. (1985) have studied long-term stability and stress concentration of many ancientmasonry structures and evaluates settlement of those foundations by using empirical theoryand pointed out factors of collapse these are the load concentration, the ground conditionand the settlement of foundation.1

Since, in order to evaluate of sedimentary rock in the long–term stability behaviour ofdeformation, the authors are analyzed the masonry structure of the Pharaoh Khufu’s Pyra-mid located in Giza area constructed B.C. 2551 and the Pont of Gard located in Frenchconstructed A.D. 17 by Discontinuous Deformation Analysis (DDA) and Manifold Method(MM). Those ancient structures still standing and carry us a message. The Khufu’s Pyramidwas constructed on the sedimentary limestone and the formation is not horizontally but littleinclined on the hill. The Khufu’s Pyramid was measured by F. Pytory in 1880 and Ministry ofmeasurement of Egypt in 1925. And the precise measurement involving the settlement wasdone by T. Nakagawa in 1978.2 The width of foundation is about 230m and the height isabout 147m. In the Khufu’s pyramid, there are three chambers as King’s chamber, Queen’schamber and underground unfinished chamber at the 30m depth from ground level.3

Those chambers are still stable from after 4500 years in construction. Therefore, theauthors expected to get the knowledge of the long-term stability of sedimentary rock toanalyze those ancient masonry structures. The objective of the studies are evaluates of rockmaterials to elucidation of the stress concentrations and the deformation of those ancientstructures by using DDA and MM.

2. Analyses of Khfu’s Pyramid

2.1. Outline of the geology at Giza area

Figure 1 shows the geological map around Giza area as three pyramids constructed.4 Thosepyramids are constructed on the tertiary deposit limestone along a flood plain field of riverNail. The tertiary deposit limestone called Mokattam formation and the area is about 1.5Km square and surrounded perpendicular two faults.

Figure 2 shows the geological section near Giza area and the thickness of Mokattam for-mation is 120m and it is not horizontally with little inclined of South-East direction.




Figure 1. Geological map around Giza area.

Figure 2. Geological section near Giza area.

2.2. Analysis of the upper masonry structure of the Pyramid

The total number of blocks of the Pharaoh Khufu pyramid is estimated about 2300 thousand.The height of the Pyramid is 147m and the width of the foundation is 230m. The total storyof the masonry is 203. The total volume is estimated about 2 million and 595 thousand andthe total weight is estimated about 5.62 MN. The height of each level, the lower level isabout 1.2m and the upper level is about 75 cm which those are smaller towards the upperlevel. The unevenness of each level is 2 to 3 cm and those are surprisingly horizontal. Theblock length of horizontal direction, the lower level is 1 to 1.5m and the upper level is 10cm to 1m at random of the places.2 Figure 3 shows two dimensional DDA model of the

Figure 3. DDA model of the Pharaoh Khufu pyramid.

266


Table 1. Material properties and parameters.

Items ParametersMaterial Upper block FoundationTime increment 0.001 secondElastic modulus 1GPa 1GPaUnit mass 20kN/m3 18kN/m3

Poisson’s ratio 0.30Friction angle, Cohesion φ = 35◦, c = 0.0 MPaPenalty coefficient (Normal)Penalty coefficient (Shear)

10GN/m3

4GN/m3

Viscosity (Block body) 10%× (unit mass)× secViscosity (Normal contact)Viscosity (Shear contact)

10%× (Penalty Pn)× sec10%× (Penalty Ps)× sec

Velocity – energy ratio Rock VS Rock: 0.80

Pharaoh Khufu pyramid. The total story of DDA model is 48 and the average height of stepis 2.5m and the average length of horizontal direction is 5m for simplicity. The total numberof blocks is about 1200.

The material property of limestone blocks was determined by experiments considering sizeeffect of the block as shown Table 1.5

2.3. Results of the DDA model of Pharaoh Khufu Pyramid

Figure 4 shows the principal stress distribution of the static mode calculation. The maximumprincipal stress distribution around King’s chamber is 6 to 7 MPa and this value is almostequivalent what Kerisel (1985) was pointed out by empirical method. The maximum prin-cipal stresses of near outside part of the pyramid is distributed along pyramid shape andthe inside part is distributed toward perpendicular to the foundation. Figure 5 shows the

Figure 4. Principal stress distribution of DDA (Static analysis).

267


Figure 5. The vertical displacement by the iso-parametric Manifold Method.

principal stress distribution of the dynamic mode calculation. The stress distribution of nearoutside part of the pyramid is distributed confused and not smoothed comparison with thestatic mode case. In the dynamic mode calculation, once the unbalance force cause numeri-cally the displacement of the pyramid structure towards development to the collapsed. Thisphenomenon shows actual physical behaviour.6

2.4. Analysis of the foundation of the Pyramid by Manifold Method

In order to estimates settlement of the pyramid foundation, the same model as DDA is ana-lyzed by the triangle original and the 4-node iso-parametric Manifold Methods.7–9

Figure 5 shows the vertical displacement and the maximum settlement is about 60 cmat centre of pyramid bottom part and relative settlement between centre and side part ofthe pyramid is about 10 to 20 cm. The principal stress distribution of the pyramid and thefoundation, no stress concentration caused in the pyramid and the tensile stresses causedparallel along the bottom part near the foundation. The maximum shear strain distributionand maximum value is located central part of the structure.

3. Analysis of the Pont DU Gard Ancient Roman Arch Structure

In order to analyze the ancient Roman arch structure of the stress distribution and the set-tlement, the Pont of Gard is analyzed by Discontinuous Deformation Analysis and the iso-parametric manifold method. The structure was constructed for aqueduct in ancient Romanage B.C. 17. The foundation of the structure and the blocks is made of limestone as sameas Khufu’s pyramid. The bridge comprised three levels and build with large blocks with outcement (dry joints). The first and second level features six and eleven superimposed archeswhose spans vary from 15.5 to 24.5m. And the height is 21.87 and 19.5m respectively. Thethird level, which is 490m long, comprises forty seven arches (12 of which were destroyedin the middle Ages), supports the aqueduct proper. These thirty-five arches are 4.8m wideand height is 7.4m. To render them waterproof, the walls of the canal are covered by areddish-brown plaster-work, a Roman secret call the ‘malthe’.

268


490m

21.87m

7.4m

19.5m

Figure 6. The DDA model of the Pont of Gard.

Figure 7. The principal stress distribution by DDA.

3.1. Analysis of the foundation of the Pon du Gard by DDA

Figure 8. The principal stress distribution by the iso-parametric Manifold Method.

Figure 6 shows the DDA model and the total block is about 2500. The material propertiesand the parameter of calculation are same as the pyramid model as shown Table 1.

269


Figure 9. The vertical displacement by the iso-parametric Manifold Method.

Figure 7 shows the principal stress distribution of around main arch and the major stressesare distributed along each arch shapes. The maximum stresses distribute at the bottom ofpillars which is 1.5 MPa to 2 MPa.

3.2. Analysis of the foundation of the Pon du Gard by Manifold Method

Figure 8 shows the principal stress distribution by the iso-parametric Manifold Method. Andthe stresses are disturbed compare with the DDA result.

Figure 9 shows the vertical displacement by the iso-parametric Manifold Method. Thesettlements around main arch at upper level are 20 to 27cm. And the first level is 10 to 15cmexperimentally.

4. Conclusions

In order to analyze for long term stability of block structures the ancient two internationalheritages in which the Khuhu’s pyramid and the Pont of Gard are analyzed by using Dis-continuous Deformation Analysis, the triangle and the iso-parametric manifold method. Thereasons of long-term stability of these structures for more than 2000 to 4500 years, theauthors focused the settlements of foundation of the structures and both structures are notcollapsed and still keep standing.

The first reason is to keep horizontal level of foundations of the structures depends on themechanical property of the foundation deformability.

The second reason that the masonry structure is released surplus shear forces betweenblocks in its self each others because of compare with the distribution of the principal stressesbetween DDA and Manifold Method as shown before. And this reason is effective for earth-quake motions of the masonry structures also.

The third reason, the contemporary design method of structures is estimate the settlementof foundation or the members of beam before construction and finally constructed themconsidering to keeps horizontal position of the structures.

The second bridge of the Gard constructed in 1743 alongside the old aqueduct concealingall the arches on the first level, which the centre part of bridge is height about 20 centimetresthan the side parts. According to the measurement by T. Nakagawa in 1978, the difference of

270


horizontal relative level between the South-East corner and the North-West corner was 45 to30 mm of the first story masonry.2 In fact, the constructing works of masonry of the Pyramidwas very précised. But it is impossible to measure the absolute settlement of the foundationfrom the initial conditions.

Therefore, we imagine that the ancient designers are also considering in this circumstancesof their experiments and the presented methods of DDA and Manifold Method are havegood applicable to explain those phenomena in static analyses. Both structures have samecondition of the foundation as sedimentary limestone which is stable in chemically for rainfalland water flow.

However, this paper studied only for the static condition, the earthquake motion of thedynamic condition is another important factor for the stability of the structure. We willanalyze those structures of the dynamic condition of the earthquake motion in the future aswe recommended model as considering frequency characteristics of the structure.10, 11

The following conclusions are deduced from the analytical results:

• The principle stresses distributions of the upper structure of Pyramid and Pon duGard are good agreement by using DDA.• The settlements of the foundations of Pyramid and Pon du Gard are good agreement

by using Manifold Method.• The stability of masonry structures are strongly depends the settlements of their

foundations, therefore, the combined analysis of DDA and NMM12 could be effec-tive for the models.

Acknowledgements

The authors thank Dr. Gen Hua Shi for many informative discussions.

References

1. Kerisel, J., The history of geotechnical engineering up until 1700, Proceedings of the eleventhinternational conference on soil mechanics and foundation engineering, San Francisco, 1985,pp. 12–16.

2. Yoshimura, S., The mystery of pyramid, Kodansha (in Japanese), 1979.3. Guardian’s Egypt, Web site: http://www.guardians.net/egypt/pyramids.htm4. Egyptian Geological Survey and Mining Authority, Geological map of grater Cairo area, 1983.5. Aboushook, M.I., Wkizaka, Y. and Shinagawa, S., Environmental impact on the durability of some

Egyptian and Japanese lime stones, Proc. of 9th ISRM Congress Paris, Vol. 2, 1999, pp. 991–996.6. Sasaki, T. and Yashinaka, R., Studies of masonry structures by using Discontinuous Deformation

Analysis and Manifold Method, Proceedings 7th Japan Society for Computational Engineeringand Science, 2002, pp. 431–434.

7. Shi, G.H., Block system modeling by discontinuous deformation analysis, Univ. of California,Berkeley, Dept. of Civil Eng., 1989.

8. Shi, G.H., Manifold method of material analysis, Trans. 9th Army Conf. on Appl. Math. andComp., Rep. No. 92-1. U.S. Army Res. Office, 1991.

9. Sasaki, T. and Ohnishi, Y., Analysis of the discontinuous rock mass by four node iso-parametricManifold method, Fourth International Conference for Analysis of Discontinuous Deformation,Glasgow, Scotland, UK, 2001, pp. 369–378.

10. Yoshinaka, R., Sasaki, T., Sasaki, K. and Horikawa, S., Consideration on stability and collapse atearthquake of soft rock slope based on an example, 11th ISRM Congress Lisbon, Portugal, 2007,pp. 1109–1112.

271


11. Sasaki, T., Hagiwara, I., Sasaki, K., Ohnishi, Y. and Ito H., Fundamental studies for dynamicresponse of simple block structures by DDA, Eighth International Conference on the Analysis ofDiscontinuous Deformation, 2007, pp. 141–146.

12. Miki, S., Sasaki, T., Koyama, T., Nishiyama, S. and Ohnishi, Y., Combined analysis of DDA andNMM (NMM-DDA), and its application to dynamic response models, ICADD9-Singapole, 2009.(to be appear)

272

Application of Manifold Method (MM) to the Stability Problemsfor Cut Slopes along the National Roads

YUZO OHNISHI1, TOMOFUMI KOYAMA2,∗, KAZUYA YAGI3, TADASHI KOBAYASHI3, SHIGERU MIKI4,TAKUMI NAKAI5 AND YOSHIFUMI MARUKI5

1Exsective, vice president, Kyoto University2Dept. of Urban and Environmental Engineering, Kyoto University3Hanshin Kokudo, Kinki Regional Development Bureau, Ministry of Land, Infrastructure and Transport4Kiso-Jiban Consultants Co., Ltd5Earthtech Toyo, Co, Ltd

1. Introduction

Recently, because of the limitation of the available lands, one of the greatest challengingproblems for civil engineers is how to construct the new structures/infrastructures adjacentpre-existing ones and how to evaluate the effect of the new construction on the pre-existingstructures. The area studied in this paper (one of the national road, Route 176, in Kinkiregion, Japan) is not the exception. The rapid increase of the population in this area causesthe heavy traffic jams and the extension of the national road has been required for longtime. The main planed road extension processes are excavation of the bypass tunnel and cut-ting the rock slope. One of the difficulties for these constructions will be pre-existing struc-tures/infrastructures such as old railway tunnel, railway line and pylon (for electric powersupply) on the top of the slope. Therefore, to investigate the effect of the newly constructedstructures and their construction processes on the pre-existing structure/infrastructure is nec-essary and important because rock slope failure causes serious damage to not only the humanlives but also structures and infrastructures, as well as serious economical losses.

The other challenging issue is how to design the structures in the fractured rock masses.So far, finite element method (FEM) has been widely used to design the structures. However,since pre-existing fractures play an important role for the mechanical behavior of fracturedrock masses, it is necessary to consider and evaluate the effect of fractures on the mechanicalbehavior of rock slopes properly for the design of cut slopes in the fractured rock masses. TheManifold Method (MM)1 is one of the discontinuum based numerical approaches to simulatethe mechanical behavior of fractured rock masses including large deformation/displacementalong fractures as well as stress/strain conditions of the rock blocks/masses. In this study, theMM was used to investigate the stability of the cut slope in the highly fractured rock massesalong one of the national roads in Japan focusing on the effects of new construction onthe pre-existing structures/infrastructures. The effect of reinforcement during the construc-tion such as rock bolts, anchors etc. was also investigated and evaluated quantitatively bynumerical simulations using MM.

2. Outline for the Constructions and Geology

2.1. Outline of the constructions (bypass tunnel and slope cutting)

Figure 1 shows the outline of the planned constructions (bypass tunnel and slope cutting)to extend the national road, Route 176 (currently 2 lanes for two directions). The planed




constructions to extend the national road and their processes are as follows: (1) filling theold railway tunnel (which has not been used any more), (2) excavating the bypass tunnel (fornational road) with the length of 293m and (3) cutting the slope and extend the road from2 to 4 lanes. The railway line also passes along the slope in the opposite side of the nationalroad and the pylon for electric power supply is situated on the top of the slope. Hence, inthis study, the following issues were investigated carefully including the slope stability: (1)the effect of old tunnel on the excavation of the new bypass tunnel and excavation disturbedzone (EDZ), (2) the effect of cutting slope on the newly constructed bypass tunnel and (3)the effect of the excavation of bypass tunnel and cutting slope on the pylon located on thetop of the slope (cross section No. 420, in Fig. 1).

(a) (b)

Bypass tunnel

Old railway tunnel

Slope cutting

National road, Route 176 (2 lanes)

National road, Route 176 (4 lanes)

Bypass tunnel

Old railway tunnel

Figure 1. The construction processes, (a) filling the old railway tunnel and excavating the new bypasstunnel, and b) cutting slope.

2.2. Geological investigation results (fracture mapping)

The hard rocks distribute widely in the slope (granite, porphyry, granite porphyry, welded tuffand rhyolite) and two faults passes close to the construction area as well as the rock massesare highly fractured. Hence, for the design of the construction, the effect of fractures on themechanical behavior of slope should be investigated. For the numerical simulations usingthe Manifold Method (MM), the fracture geometry and mechanical properties of fracturesare required. The fracture distributions were obtained from the borehole TV image and theon-site geological survey (see Fig. 2). Three different fracture sets were observed and used tocreate the 2-D models for the slope (see Table 1). The fracture geometry data was analyzedby Schmidt net and shown in Fig. 3.

2.3. Geological investigation results (fracture mapping)

The hard rocks distribute widely in the slope (granite, porphyry, granite porphyry, welded tuffand rhyolite) and two faults passes close to the construction area as well as the rock massesare highly fractured. Hence, for the design of the construction, the effect of fractures on themechanical behavior of slope should be investigated. For the numerical simulations using

274


Southern edge, railway line

Dominant fractures

(a) (b)

Figure 2. The fracture geometry obtained from a) on-site geological survey and b) borehole TV image.

Table 1. Three fracture sets obtained from borehole TV image and on-site geological survey.

Fracture set Direction Interval No. 420cross

section

No. 422cross

section

Features

I EW20N 10–15m J1–J4 J1–J3 20◦ inclined against thecut slope (dip slope)

II N86E64N 5–10m J5–J7 J4–J6 60◦ inclined against the cutslope (dip slope)

II N70W70SN70W80S

10–30m J8–J12 J7–J8 70–75◦ inclined against thecut slope (reverse dip slope)

EW20N

N N

N32E39NN41E44N

N86E64N EW59NNo. 420 No. 422

Figure 3. The statistical analysis for fractures using Schmidt net.

the Manifold Method (MM), the fracture geometry and mechanical properties of fracturesare required. The fracture distributions were obtained from the borehole TV image and theon-site geological survey (see Fig. 2). Three different fracture sets were observed and used to

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create the 2-D models for the slope (see Table 1). The fracture geometry data was analyzedby Schmidt net and shown in Fig. 3.

3. Numerical Simulation for Slope Stability Problem Using ManifoldMethod (MM)

3.1. Outline of MM

The Manifold Method (MM) originally developed by Shi1 is one of the discontinuum basednumerical methods and can simulate the mechanical behaviors including the displacement/deformationfor both continuous and discontinuous media. The detail explanation, formulation and deriva-tion of equations can be seen in the literature.1 The main features of MM can be summarizedas follows: (1) MM divides blocks into small regions (elements) and the nodal displacementsare unknown parameters like FEM and can calculate the stress/strain with good accuracy, (2)MM can treat the block contacts and/or separations easily like DDA by using the overlappedcovers (mathematical mesh), (3) MM uses coves (mathematical mesh) where the displace-ment function (shape function) is defined and can over the physical boundaries unlike FEM(a kind of SmeshlessT method) and (4) MM can solve both static and dynamic problems.

3.2. Analytical domain and conditions for the simulations

Figure 4 shows the 2-D models of the slope with fractures for two different cross sectionsnamed No. 420 and No. 422. Based on the statistical analysis for fractures, three fracturesets were introduced in the model (J1–J9, see Table 1). The slope was divided into fourdifferent regions: (1) highly weathered, (2) weathered, (3) weakly weathered zone and (4)excavation disturbed zone (EDZ) around old railway tunnel. The material properties for eachregion are summarized in Table 2. Since laboratory mechanical tests for fractures were notperformed, the cohesion for fractures was assumed to be zero. Then the sensitivity analysis interms of internal friction angle for fractures was performed (details will be explained later).As for the boundary conditions, the displacements in both x- and y-directions were fixedalong the bottom boundary and the displacement in the x-direction was fixed along the sideboundaries. The first order triangle elements with linier elastic constitutive law were used.The rock bolts introduced around the newly constructed bypass tunnel were not modeled

J1 J2 J3

J4

J5 J6 J7

J8 J10J9

J12

J11

J4 J5 J6 J9 J8

J3 J2

J1

J7

No420 No422

Cross section No.420 Cross section No.422

Bypass tunnel

Old railway tunnel (EDZ)

Old railway tunnel (EDZ)

Bypass tunnel

Weathered zone

Highly weathered zone

Weathered zone

Highly weathered zone

Weakly weathered zone

Cutting slope

Cutting slope

Figure 4. 2-D models of the slope with fractures for different cross sections.

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Table 2. The material properties for each region.

Zone Hightly Weathered Weakly EZD aroundweathered weathered old tunnel

Rock classification D-CL CL-CM CM −Density (kg/m3) 2200 2600 2600 2600Young’ modulus (MN/m2) 30 220 410 290Poisson’s ratio 0.4 0.3 0.3 0.35Cohesion (MN/m2) 0.1 0.4 4.5 3.2Internal friction angle (◦) 30 32 34 32others − − − include the filling materialFriction angle for fractures (◦) Parametric study (see Table 3)Cohesion of fractures(MN/m2) 0 0 0 0

Table 3. The material properties for fractures in each region.

CasesInternal friction angle for fractures (◦)

Highly weathered Weathered Weakly weathered

Case 1 30 35 40Case 2 35 40 45Case 3 No fractures

directly, but consider the stress relief ratio of 70% was introduced to consider the effectof rock bolts’ support. In the simulations, slope was cut at the same time in the whole cutarea. The simulation processes followed exactly same as the abovementioned constructionprocesses. As for the parameters for fractures, according to the literature,2–3 the basic frictionangles of fractures for granite, porphyry, granite porphyry, welded tuff, rhyolite (observed inthe studied area) distribute from 30–40◦. The parameters for fractures used in this study aresummarized in Table 3.

3.3. Simulation results

Because of the page limitation, only simulation results for the cross section No. 422 werepresented in this paper. Figure 5 shows the distribution of horizontal and/or vertical dis-placements for each simulation case. From this figure, the relatively larger displacements aredistributed in the highly weathered and weathered regions, and the displacement along thefracture J5 is significant. The amount of the displacement becomes smaller with increase ofthe friction angle of the fractures. This figure also clearly shows that the fractures play impor-tant roles for mechanical behavior of fractured rock masses. However, the continuum basedapproaches such as FEM can not treat separation of the blocks properly. On the other hand,MM can simulate both continuous and discontinuous media. Figure 6 shows the displace-ment vectors and final displacement for Case 2 (after cutting the slope). From this figure,the blocks above and below the fracture J8 move downward (subsidence) and upward (ele-vation), respectively. The maximum displacement of the blocks was less than 4cm and thisdisplacement can be controlled by installing the reinforcement such as anchors and rockbolts. Figure 7 shows the stress distributions and maximum shear strain. Some stress concen-tration can be observed along the fractures, around the old railway tunnel and both sidewall

277



Case 1 Case 2 Case 30.15 0.14 0.13 0.12 0.11 0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01

(m)

Figure 5. Simulation results by MM, horizontal (upper) and vertical (lower) displacements for eachcase.

Disp = 0.1600m Disp = 0.400m

Figure 6. Simulation results by MM, the displacement vectors and final displacements for Case 2.

of the bypass tunnel. The large shear strain of 0.6% occurs around the old railway tunnel.However, significant shear strain cannot be observed at the toe of the slope.

3.4. Design of the reinforcement

Based on the simulation results presented in the previous section, the reinforcement such asanchors and rock bolts was designed and their effect was investigated quantitatively. Theselection of the method for reinforcement depends on the volume of the rock masses whichshould be fixed as well as the depth location of the sliding planes and the condition of the

278



0.60 0.56 0.52 0.48 0.44 0.40 0.36 0.32 0.28 0.24 0.20 0.16 0.12 0.08 0.04

(%)


Stress = 4e+006 Stress = 4e+006 Stress = 4e+006

Figure 7. Simulation results by MM, the stress distributions (upper) and the maximum shear strain(lower) for each case.

Table 4. Results of slope stability analysis.

Slope stability analysis

Safety factor Planed safety factor Restraint force (kN/m)

Current situation 1.139Aftercuttingslope 1.026 1.200 630.16

fractures. For the slope investigated in this study, the combination of ground anchors withrock bolts and/or iron reinforcing rods will be suitable. Figure 8 shows the suggested rein-forcement for the slope to control the displacement along the possible sliding plane, J5 (seeFigs. 4 and 6).

To investigate the effect of the suggested reinforcement, the numerical simulations werecarried out using MM with rock bolt element and distributed loads (represents restraintforce for anchors). The material properties for the rock bolt element can be seen in Fig. 8.The distributed load of 630.162kN was introduced as the restraint force of the anchors. Thesame material properties and boundary conditions as Case 2 (see Tables 2 and 3) were usedfor each region excpt the stress relief rate of 100% for cutting slope.

The distribution of displacments (in horizontal and vertical directions) and maximumshear stress are compared between cases without and with reinforcement and shown in Fig. 9.From this figure, the displacements of blocks in both horizontal and vertical directions weresignificantly decreased along the fracture J1 and J2 by the reinforcement. This means that thesuggested reinforce system will work properly. However, the suggested reinforcement could

279


Fig. 8 Suggested reinforcement.

Iron reinforcing rods (D22, 4 m) → rock bolt element

Restraint force for anchors → distributed loads

Iron reinforcing rods (D22, 4 m) → rock bolt element

llAEF /Δ⋅⋅= where F: axial force, E: Young’s modulus, A cross section area, l: length, Δl: increment of length. Now, E = 105(kN/mm2), A = (22.2 mm-1.0 mm)2·π/4 = 353 mm2 (1 mm for corrosion) l = 4 (m)

Figure 8. Simulation results by MM, the stress distributions (upper) and the maximum shear strain(lower) for each case.

1.20 1.12 1.04 0.96 0.88 0.80 0.72 0.64 0.56 0.48 0.40 0.32 0.24 0.16 0.08

(%)

0.15 0.14 0.13 0.12 0.11 0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01

(m)

Without reinforcement

With reinforcement

Horizontal disp.

Horizontal disp. Vertical disp.

Vertical disp. Max shear stress

Max shear stress

Figure 9. Simulation results by MM for the cases without (upper) and with installing the reinforce-ment (lower), horizontal and vertical displacements as well as the maximum shear stress.

not improve the displacement of the block srounded by J2, J3 and J5 along the J3 and morerock bolts need to be intoduced. On theothr hand, the distribution of maximum shear stressdid not change much after installing the reinforcement.

4. Concluding Remarks

In this study the MM was applied to real slope stability problems along one of the nationalroads in Japan with tunnel excavation and slope cutting. Especially, in this constructionsite, the effect of new construction on the pre-existing structures/infrastructures such as old

280


railway tunnel, railway line pylon should be carefully investigated. The effect of the rein-forcement such as anchors, rock bolts and etc was also investigated. In this construction site,since the mechanical behavior along the fractures is significant, the numerical method basedon the discontinuum approaches will be suitable. The MM can introduce fractures explicitlyin the model and fracture information such as geometry and material properties are requitedto make a model. In this study, the distribution of fractures was obtained from the on-sitesurvey/observation using borehole camera etc. The simulation results show that displacementof less than 4cm was observed along the fractures in the stress relaxation region and appro-priate reinforcement such as rock bolts and/or anchor will be necessary for the stability ofthe cut slope. The MM will be applicable to the design of the cut slope in the fractured rockmasses.

References

1. Shi, G.H., “Manifold Method of Material Analysis”, Transactions of 9th Army Conference onApplied Mathematics and Computing, Report No. 92-1, U.S. Army Research Office, 1991.

2. Nakai, T. and Shimauchi T., “Investigation, analyses and evaluation for fractured rock masses (1)”,Soil Mechanics and Foundation Engineering, 48(1), 2000, pp. 57–62 (in Japanese).

3. Shimauchi T. and Nakai, T., “Investigation, analyses and evaluation for fractured rock masses (2)”,Soil Mechanics and Foundation Engineering, 48(3), 2000, pp. 55–60 (in Japanese).

281

Boundary Deformability and Convergence in the Higher-OrderNumerical Manifold Method

D. KOUREPINIS1,∗, C.J. PEARCE2 AND N. BICANIC2

1Halcrow Group Ltd2University of Glasgow

1. Introduction

Safety critical structures such as nuclear containment vessels and dams, as well as mostcivil engineering structures are normally designed to perform in the linear elastic regimeunder normal operating conditions. However, there is a wide range of problems that requiresknowledge of how structures behave beyond the linear elastic range; for example, when itis required to allow for acceptable levels of inelastic deformation, to design for structuralperformance under extreme loads induced by earthquakes, blast or impact, or to conductforensic studies and assess the effectiveness of retrofit. Furthermore, there are several prob-lems, such as slope stability, tunnelling and mining, which are inherently discontinuous.

The deformation analysis of problems that exhibit inelastic behaviour through strain local-ization, followed by development of micro-fractures and coalescence into discrete discontinu-ities such as cracks has posed significant challenges to the mechanics community for severaldecades. This is due to difficulties to resolve numerically and in an accurate and efficientmanner the gradual transition from continuum to discontinuum and, potentially, interactionof fragmented parts.

One of the difficulties to describe this behaviour numerically stems from requirements toadapt the approximation locally in order to capture accurately the stress field around evolv-ing discontinuities. Traditionally, in finite element methods this class of problems has beenapproached using remeshing, while in discrete element methods it was attempted using arti-ficial connection of discrete bodies which are identified a priori to act as continua. However,neither of these attempts comprises a diritta via for modelling the transition from continuumto discontinuum efficiently.

The enrichment of finite element methods with higher-order shape functions (p-enrichment)appears to be appealing for improvement of the numerical approximation without remesh-ing. However, to capture the evolution of discontinuities, in traditional finite element meth-ods it is necessary to couple enrichment with remeshing. The Numerical Manifold Method(NMM) is an attractive alternative for modelling discontinuous deformation problems, advo-cating the ability to adapt the level of approximation locally,1 coupled with the ability tointroduce discrete discontinuities without remeshing.

Although higher-order enrichment can potentially improve the numerical approximation,it can also lead to convergence difficulties and insufficiently constrained boundaries. Fur-thermore, the enforcement of essential boundary conditions, which in NMM is undertakentraditionally using penalty constraints, is not straightforward and can result to incompleteenforcement. Similar problems manifest if distributed loads are not applied consistently.

This paper examines convergence issues in higher-order NMM and presents a strategyfor direct and complete enforcement of essential boundary constraints, which is achieved ingeneral terms using a point collocation method, whereby boundary conditions are enforcedexactly at specified points on the boundary without the use of penalty constraints.




2. Higher-Order Numerical Manifold Method

The problems of improvement and convergence are intrinsic and universal in numericalapproximation techniques. In principle, a numerical solution can be improved by increasingthe number of unknowns employed to define the approximation field, so that the trial fieldbecomes a closer representation of the actual field. Assuming that the trial field is capable toreproduce the displacement form of the continuum (or discontinuum), then with refinementor enrichment of the approximation with additional unknowns, the exact solution may beobtained.

NMM, which was introduced by Shi,2 is based on the partition of unity concept3 and ideassimilar to those utilized in meshless methods. It integrates aspects of traditional and hierar-chical finite element methods, and exhibits strong parallels with the more recently developedExtended Finite Element Method (XFEM) as noted by Kourepinis.1 NMM can be viewed asa more generalized formulation of Discontinuous Deformation Analysis (DDA),4 wherebyblocks are substituted by assemblages of elements formed by overlapping covers, or domainsof influence. Similar to DDA, the NMM approximation can be enhanced using higher-orderpolynomial basis functions, to achieve a variable and potentially improved strain field withinelements without altering the mesh. In NMM it is possible to improve the numerical approx-imation in two principal ways, similar to finite elements:

• By adapting the level of discretization via remeshing or by modifying the existingmesh without changing the number of elements, nodes or connectivity (h- and r-refinement respectively).• By using higher-order basis functions (p-refinement) without introducing new nodes

and hence without undertaking remeshing. This is typically undertaken in a hierar-chical manner.

The latter approach appears to be particularly attractive for adaptivity, as it entails astraightforward and similar implementation for problems of any spatial dimension and meshstructure.

To date, the original NMM has been extended in attempts to exploit its potential toimprove the level of approximation with higher-order basis functions,1, 5 while preservingthe ability to undertake integration analytically. Furthermore, it has been demonstrated thatlocal higher-order enhancement can be undertaken by adapting the order of displacementpolynomials that define the approximation field for any arbitrary level, without explicitlyderiving the matrix system.1 A distinct advantage of the latter case is that it is possible toimprove the solution by adapting locally the order of nodal displacement functions usingerrors estimators or indicators, without undertaking remeshing. However, to date there hasbeen limited evidence of potential issues associated with convergence or enforcement of con-straints in higher-order formulations.1

3. Boundary Deformability in Higher-Order NMM

In traditional finite elements, due to the interpolation of nodal displacements, constraints onelement boundaries between nodes are enforced naturally. This is not the case when higher-order displacement functions are employed in NMM, since the deformation between nodesassociated with higher-order functions is not anymore an interpolation of nodal displace-ments. In addition, the unknowns corresponding to nodes associated with higher-order func-tions are not simply displacements and the unknowns of higher-order functions associatedwith constrained nodes are not all necessarily zero. In physical terms the result is that, if

284


the boundary conditions require an entire edge to be constrained and only nodes are utilizedfor this purpose, the increased deformability associated with higher-order NMM leads toincomplete enforcement (Fig. 1). A similar problem manifests itself if distributed loads arenot applied consistently.

Based on the local enrichment concept discussed earlier, this issue can be resolved byenforcing zero-order displacement functions on the boundary, without necessarily changingthe order of displacement functions elsewhere. By doing this, displacement function coeffi-cients associated with boundary nodes are always nodal displacements interpolated betweenboundary nodes, thereby enforcing the correct boundary conditions along the entire bound-ary and not only at nodes. This can be demonstrated using the following single element test(Fig. 2), where zero displacement functions are employed at nodes 2 and 3 and first orderdisplacement functions at node 1.

The only variables at nodes 2 and 3 are the (already known) prescribed displacements:

a2 =[0 0

]T and a3 =[0 0

]T (1)

At point A between these nodes it can be shown that the weight functions of both nodes 2and 3 are equal to 0.5, therefore the associated shape matrices are equal to:

T2 = 0.5[1 00 1

]and T3 = 0.5

[1 00 1

](2)

whereas the weight function of node 1 at point A is zero, since this point lies on the boundary.Therefore, the displacement at point A equals zero as shown in Eq. (3), although the resulting

Figure 1. Deformed shape of a two-element test. (a) FEM/NMM with constant-strain triangles(b) NMM with higher-order displacement functions.

Figure 2. Single element test.

285


Figure 3. Single element test with zero-order displacement functions at nodes 2 and 3 and 1st-orderdisplacement functions at node 1. (a) Contours of displacement along the horizontal axis (b) Contoursof stress in the horizontal axis.

stress field is not constant as illustrated by Fig. 3.[uAxuAy

]= 0+ T2a2 + T3a3 = 0 (3)

4. Convergence

In principle, higher-order enhancement may improve the approximation but there are casesin which convergence is bound to fail. For example, where singularities exist or where pointloads are applied. Furthermore, the use of interpolation with high-order polynomials atequidistant points can introduce errors as the solution tends to oscillate with increasing poly-nomial orders at interpolation intervals (Fig. 4). Potential remedies are spline curves andChebyshev nodes that become increasingly closer near boundaries.

To illustrate this the problem of Fig. 5 is considered. The problem consists of a 400 ×200mm membrane of 1mm thickness subject to a uniform axial pressure of 100MPa. Atraction-free circular hole of 20mm radius is situated in the middle of the membrane toweaken the section. The elastic modulus and Poisson’s ratio are taken as 100,000 MPa and

Figure 4. Runge’s phenomenon: interpolation of Runge’s function with high-order polynomials.

286


Figure 5. Definition of quadrant idealisation of semi-infinite perforated plate (left); problem discreti-sation (right).

0.3 respectively and the problem is idealised in plane-stress. Since it is desired to enhancenodes in the vicinity of the hole, it is necessary to discretise the full problem.

The mesh is deliberately coarse in order to examine the effect of enhancement viap-refinement using zero, first, second, third and fourth-order displacement polynomials. Com-parison of the energy error in the case of h-refinement alone around the hole (Fig. 6), andp-refinement (Fig. 7) illustrates that the solution diverges when displacement polynomials oforder higher than two are employed.

Another significant issue is that the imposition of essential boundary conditions associatedwith higher-order displacement functions can lead to rank deficiency of the system matrix.Duarte7 and subsequently Lin8 noted this phenomenon in the context of h-p Clouds andNMM respectively and recognized that this was due to the fact that the polynomials used toconstruct the displacement function included monomials that are reproduced by the weight-ing functions. Lin proposed that the linear term should be omitted from the displacementfunctions although in fact this only partially tackles the problem1 and a potential resolutioncan be obtained with appropriately modified displacement functions for nodes on restrained

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 100 200 300 400 500 600 700 800

Mesh density (number of nodes)

En

erg

y n

orm

err

or

Figure 6. Energy norm error at A versus mesh density; h-refinement study using remeshing and zero-order displacement functions.

287


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 1 2 3 4 5

Order of displacement polynomials

En

erg

y n

orm

err

or

Figure 7. Energy norm error at A versus the order of global p-refinement.

boundaries. The most straightforward way to achieve this is by enforcing boundary nodeswith zero-order displacement functions.

5. Enforcement of Essential Boundary Conditions Using ProjectionMatrices

Due to the non-interpolating nature of the NMM shape functions with respect to nodaldisplacements, the enforcement of essential boundary conditions can be rather more compli-cated than it is in FEM. Traditionally, essential boundary conditions in NMM are enforcedusing the penalty method. Although enforcement with penalty constraints is straightforwardfrom an implementation point of view, the solution is susceptible to artificially high stiffnessconstraints which can lead to ill-conditioning of the matrix system.

Alternatively, the enforcement of essential boundary conditions can be resolved in a morerobust way by means of Lagrange multipliers. In paradox, whereas this approach elimi-nates the requirement for artificial constraining forces, it introduces additional unknowns.An alternative Lagrange multiplier technique which restores the problem to its original num-ber of unknowns is employed here in NMM as presented by Ainsworth6 within the contextof finite elements. It can be shown that the discrete system of equations can be derived fromminimization of the discrete version of the modified (constrained) energy functional as:

mina,λ

� = 12

aTKa− fTa+ λT(Aa− q) (4)

where a is a vector which contains the deformation coefficients of the displacement functionvector, λ is the vector of Lagrange multipliers, K is the structural stiffness matrix, A is amatrix which couples displacement degrees of freedom to Lagrange multipliers, whereas fand q are the vectors of external forces and applied displacements respectively. Assumingthis problem is well-posed, the following matrices are well-defined:

Q = I− R A (5)

R = AT(AAT)−1 (6)

288


where R is an auxiliary matrix and Q is a projection matrix. Thus, there is a unique solutionto the following modified problem:

Ka = f (7)

where the modified stiffness matrix and force vector are defined as:

K = QTKQ+ ATA (8)

f = ATq+QT(f− KRq) (9)

and the corresponding Lagrange multipliers can be recovered from:

λ = RT(f− Ka) (10)

The constrained system has the same number of unknowns as the original problem. Forcomputational efficiency, it is also possible to implement a sequential approach6 rather thana single step using a global constraint matrix. Thus the approach offers clear advantages overboth the Lagrange multiplier method and the penalty method.

6. Conclusions

Although higher-order enrichment of NMM can potentially improve the numerical approx-imation, it requires a special level of attention and engineering judgment in order to attainmeaningful results. In particular, care must be given to apply loads and boundary condi-tions consistently. This paper discussed issues such as incomplete enforcement of boundaryconditions and rank deficiency of the system matrix, and it was advocated that these issuesmay be resolved using a local enhancement strategy with zero-order displacement functionson restrained boundary nodes. Furthermore, a technique for enforcing essential boundaryconditions exactly was presented for any general higher-order case using projection matrices,without the use of penalty constraints and without increasing the number of unknowns. It isworthwhile to note that the numerical issues and considerations discussed here are potentiallyapplicable to other techniques which are conceptually similar to NMM, such as Discontinu-ous Deformation Analysis (DDA) and XFEM.

Acknowledgements

Halcrow Group Ltd and the Engineering and Physical Sciences Research Council (UK) areacknowledged gratefully for providing financial support for this research.

References

1. Kourepinis, D. (2008), Higher-Order Discontinuous Analysis of Fracturing in Quasi-Brittle Materi-als, Doctoral Thesis, University of Glasgow, Glasgow University Library.

2. Shi, G.H. (1995). Simplex integration for Manifold Method and Discontinuous Deformation Anal-ysis. In Proceedings of Working Forum on the Manifold Method of Material Analysis, pp. 129–164.

3. Babuška, I. and Melenk, J.M. (1996). The partition of unity finite element method: Basic theoryand applications. In Computer Methods in Applied Mechanics and Engineering, 139: 289–314.

4. Shi, G.H. (1988). Discontinuous Deformation Analysis — A new numerical model for statics anddynamics of block systems. PhD Thesis. University of California, Berkeley.

5. Lu, M. (2002). Numerical Manifold Method with complete N-order cover function. STF22-F02121to STF22-F02124, SINTEF.

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6. Ainsworth, M. (2001). Essential boundary conditions and multi-point constraints in finite elementanalysis. In Computer Methods in Applied Mechanics and Engineering, 190, 6323–6339.

7. Duarte, C.A. and Oden, J.T. (1996). H-p Clouds – an h-p Meshless Method. In Numerical Methodsfor Partial Differential Equations, 12, 673–705, John Wiley and Sons, Inc.

8. Lin, J.S. (2003). A mesh-based partition of unity method for discontinuity modelling. In ComputerMethods in Applied Mechanics and Engineering, 192, 1515–1532.

290

The Numerical Manifold Method and Extended Finite ElementMethod — a Comparison from the Perspective of DiscontinuousDeformation Analysis

D. KOUREPINIS1,∗, C. J. PEARCE2 AND N. BICANIC2

1Halcrow Group Ltd2University of Glasgow

1. Introduction

The computational description of discontinuous phenomena has presented significant chal-lenges to the mechanics community over the past few decades, driven by rapid technologicaldevelopments, increasing social and economical constraints for safer and more complicatedengineering designs, and consequently by increasing requirements for more accurate under-standing of macro- and micro-structural processes.

Finite element methods have been pushed to their limits in an attempt to resolve strainlocalization and ultimately fracturing in a unified and objective manner, while discrete meth-ods have been utilized by artificial connection of discrete bodies which are identified a priorito act as continua. Neither of these attempts comprises a diritta via for modelling the tran-sition from continuum to discontinuum efficiently and this has led to the investigation ofalternative techniques.

The Numerical Manifold Method (NMM) and Extended Finite Element Method (XFEM)are two relatively recent numerical techniques which advocate distinct advantages for mod-elling evolving discontinuities, such as cracks, due to their ability to resolve jumps in theapproximation field without the requirement for a priori assumptions and remeshing, butwithin a continuum setting.

This paper presents a discussion of the similarities and differences of the two techniquesfrom the perspective of discontinuous deformation analysis.

2. The Partition of Unity

The essence of finite element techniques lies in approximating the unknown function usingan expansion which contains the unknown parameters (e.g. displacements) and some form ofshape functions, which are typically expressed in terms of independent variables. In the finiteelement method (FEM), its extension, XFEM, and NMM, the approximating displacementfunction can be expressed in a form equivalent to:

u =n∑

i=1

Ni ai = Na (1)

where n is the total number of nodes, Ni are the shape functions and ai are the unknownparameters. Ni and ai can be non-scalar; for example, in NMM when higher-order dis-placement functions are used Ni are matrices and ai are vectors containing the unknowncoefficients of the approximating displacement polynomials.

It is worthwhile to note that in their simplest form FEM, XFEM and NMM are equiv-alent. For example, in continuum mechanics when linear shape functions are employed in




NMM, combined with constant displacement functions and simplex elements (i.e. elementsthat adopt the form of the simplest possible shape in any given space) the NMM approxi-mating function is identical to FEM and XFEM.1

The shape functions satisfy the partition of unity condition:

n∑i=1

Ni(x) = 1 ∀x ∈ �e (2)

0 ≤ Ni(x) ≤ 1 ∀x ∈ �i (3)

Ni(x) = 0 ∀x /∈ �i (4)

where �e is the element domain and �i is the influence area of a shape function. The aboverelationships state that (1) the sum of shape functions at any position within the discretizeddomain must equal to one (hence each function is a “partition of unity”), (2) each shapefunction takes values from 0 to 1 within its influence area and (3) every shape functionequals to zero at any position which does not lie within its influence area.

From the perspective of discontinuous deformation analysis, the partition of unity is signif-icant since it can be used to incorporate discontinuous shape functions in the approximation,and thereby resolve jumps in the approximation field without the requirement to undertakeremeshing and without a priori assumptions with respect to the paths followed by disconti-nuities. Consequently, methods based on the partition of unity approach appear particularlyattractive for problems that involve simulation of moving boundaries, such as fracturing andcrack propagation.

For example, using Eqs. (2) to (4) it can be observed that once an element is intersectedby a discontinuity, for Eq. (4) to hold, affected shape functions must be set to 0 over theboundary of the discontinuity. As a result, the shape functions become discontinuous and theelement is partitioned into two sub-domains, one on either side of the discontinuity. However,this invalidates Eqs. (2) and (3). The partition of unity can be restored by partitioning theoriginal element it into two elements with boundaries that lie on the discontinuity (henceundertaking remeshing). Alternatively, additional nodes associated with discontinuous shapefunctions that restore the partition of unity may be added so that they overlap the originalnodes. The procedure is illustrated in Figs. 1 and 2.

Figure 1. One-dimensional element intersected by a discontinuity at x.

292


Figure 2. Additional overlapping nodes are introduced with discontinuous shape functions to restorethe partition of unity.

3. Modelling of Evolving Discontinuities

Traditional finite element techniques for modelling strong discontinuities (where the displace-ment and strain fields in the vicinity of displacement jumps are fully discontinuous), are nor-mally associated with interface models and remeshing. The use of interface models withoutremeshing implies a priori assumptions with regard to the location and trajectory of potentialdiscontinuities, and therefore may yield results that are not objective due to mesh alignmentand issues associated with integration and fictitious elastic stiffness of interface elements. Ifinterface elements are introduced only when required, then the use of remeshing techniquesis entailed. This implies that where the continuum is intersected by discontinuities remeshingtakes place and interface elements are introduced aligned to boundaries of discontinuities.However, whenever a localization zone is remeshed, the neighbouring region may also beremeshed in order to obtain a smooth transition of the approximation field. This process canresult in a significant increase of unknowns.

A more robust approach can be accomplished with NMM and XFEM, which exhibitstrong parallels since they are both based on the partition of unity framework and exploitthat ability to employ discontinuous shape functions. Jirásek2 notes that in chronologicalterms, the concept of modelling discontinuities using the partition of unity concept is tracedback to NMM, which appears in literature almost the same time as the identification of thepartition of unity framework.3

In both NMM and XFEM discontinuous shape functions can be introduced in order torestore the partition of unity when the domain is intersected by discontinuities. The dis-continuities introduced in this manner are not limited to element boundaries, but can belocated anywhere in the mesh as displacement jumps are represented by additional degreesof freedom that overlap the existing mesh. This type of enrichment of the continuum followsdirectly from the original topology, so that the additional degrees of freedom are introducedonly on existing nodes intersected by discontinuities. Surrounding elements are unaffected

293


Figure 3. Arbitrary domain with discontinuity introduced using additional (overlapping) nodes.

and therefore remeshing does not take place in the traditional sense. Consequently, the pro-cedure is similar for any type of element and problem geometry.

The key difference between NMM and XFEM is that NMM captures the jump in the dis-placement field using discontinuous displacement functions to the ‘left’ and to the ‘right’ of adiscontinuity in order to restore the partition of unity. XFEM on the other hand, introducesan additional degree of freedom at each node by enriching the trial function. In both tech-niques the resulting stiffness matrix is symmetric and there is an increase of its size due to theintroduction of additional degrees of freedom.

Another distinctive divergence between the two methods arises from the way the integra-tion process is carried out. In XFEM integration is carried out numerically at Gauss points,while in NMM it is undertaken analytically using simplex integration as discussed in the nextsection.

Furthermore, it is worth noting that in any case the introduction of displacement discon-tinuities can be associated with algorithmical difficulties, particularly in three-dimensionaldomains. For example, tracking of discontinuities in 3D represents a significant geometricalchallenge although developments in the area of level sets4 employed in XFEM may also beof use in NMM.

4. Integration

Although NMM and XFEM have several conceptual similarities, where discontinuitiesemerge NMM traditionally constructs the approximation space as an enriched product ofstandard basis functions, while XFEM constructs the approximation space as a product ofstandard basis and special enrichment functions.5

This key difference affects the integration approach adopted in each case as the shape func-tions and their derivatives can be conceptually different. As a result, in NMM the stiffnessmatrix can consist only of integrals of monomial terms, whereas in XFEM it can consist ofintegrals of monomial terms and other special functions. In situations where fracture prob-lems are considered within the framework of linear elasticity and zero traction boundaryconditions on crack surfaces, the XFEM special functions are typically singular linear elasticnear-tip field functions,6, 7 in order to enrich the crack-tip.

In linear elastic cohesive crack models the situation is slightly different, as tractionsbetween either side of a discontinuity lead to a reduction of stress at the tip. This is desirable

294


since it reduces the non-physical singular stress field at the tip.8 However, singular enrich-ment functions are not valid. In this case, enrichments at the tip have been undertaken usingnon-singular asymptotic functions8 in XFEM, or enrichment functions based on higher-orderpolynomial bases.9

Consequently, integration in XFEM is not always as straightforward as it is in NMMand it is undertaken numerically. This implies that the integration domain must conform tothe boundary of the discontinuity. If the discontinuity is not taken into account, then thesolution can lead to poor results or a non-invertible set of equations if integration pointsdo not track the discontinuity.10 Therefore, integration of elements that are intersected bydiscontinuities requires a form of partitioning of elements into triangular sub-domains. Inaddition, in cohesive models the variational principle involves integration over the domainand integration over segments of the cohesive zone.7, 8

XFEM partitioning is undertaken without introducing additional unknowns since basisfunctions are only associated with nodes tied to parent elements.11 However, the computa-tional cost can increase as the number of integration points increases. Furthermore, integra-tion by partitioning around a singularity can yield poor results if the integration rule is notadequate or if the mesh in the proximity of the singularity is coarse.

In NMM with simplex integration non-simplex domains resulting by the intersectionof (simplex) elements and discontinuities are also in essence partitioned into simplex sub-domains1 similar to the XFEM approach. Also similar to XFEM, no additional unknownsare introduced. However, in contrast to XFEM, integration in this case is exact and thereforethe additional computational cost of integration of parts of the domain intersected by discon-tinuities depends only on the order of the displacement functions associated with nodes tiedto parent elements. Furthermore, the NMM approach is guaranteed to yield precise resultsas it is exact.

It is worthwhile to note that although simplex integration constrains the shape of theapproximating basis functions, it is likely to avoid potential issues of zero energy modesassociated with inadequate or reduced numerical integration rules12 of non-smooth as wellas smooth problems without the requirement for additional considerations when the order ofthe basis functions is increased (for example in adaptive enrichment). However, to date appli-cation of simplex integration has been restricted to linear elastic problems whereas numericalintegration has been applied extensively to both linear and nonlinear mechanics.

Depending on the form of the special enrichment functions employed, simplex integra-tion can also be used in XFEM. Similarly, NMM can benefit from the work undertaken inXFEM in situations where a simplex approach is not desirable, whether this is due to useof non-simplex elements or enrichment of the approximation field with non-standard basisfunctions.

5. Conclusions

From the viewpoint of discontinuous deformation analysis, the principal differences betweenNMM and XFEM appear to be restricted to the way the discontinuous approximation spaceis constructed and the way integration is carried out (traditionally). Although distinct, it canbe postulated that these divergence points are subtle and as a result there is strong poten-tial to integrate aspects of XFEM in the NMM framework and vice versa. For example,the extensive amount of research undertaken with regard to modelling curved or branchedcracks and tracking discontinuities in XFEM using level sets may be potentially utilized forfurther developments of NMM. Similarly, the straightforward approach of NMM for mod-elling discontinuities can be used in XFEM where the use of special enrichment functions is

295


not necessary, or simplex integration may be used in cases where it is more desirable thannumerical integration.

Acknowledgements

Halcrow Group Ltd and the Engineering and Physical Sciences Research Council (UK) areacknowledged gratefully.

References

1. Kourepinis, D. (2008), Higher-Order Discontinuous Analysis of Fracturing in Quasi-Brittle Mate-rials, Doctoral Thesis, University of Glasgow, Glasgow University Library.

2. Jirásek, M. and Belytschko, T. (2002), Computational resolution of strong discontinuities, WCCMV — Fifth World Congress on Computational Mechanics, Mang, H., Rammerstorfer, F.G. andEberhardsteiner, J.

3. Babuška, I. and Melenk, J.M. (1996). The partition of unity finite element method: Basic theoryand applications. In Computer methods in applied mechanics and engineering, 139: 289–314.

4. Moes, N. (2003), A computational approach to handle complex microstructure geometries. InComputer Methods in Applied Mechanics and Engineering, 192: 3163–3177.

5. Belytschko T., Moes N., Usui S. and Parimi C. (2001), Arbitrary discontinuities in finite elements.International Journal for Numerical Methods in Engineering, 50: 993–1013.

6. Fleming, M., Chu, Y.A., Moran, B., Belytschko, T. (1997), Enriched element-free Galerkin meth-ods for crack tip fields, In International Journal for Numerical Methods in Engineering, 40,pp. 1483–1504.

7. Moes, N., Dolbow, J. and Belytschko, T. (1999), A Finite Element Method for Crack Growthwithout Remeshing, In International Journal of Numerical Methods in Engineering, Vol. 43, 1:131–1500

8. Moes, N. and Belytschko, T. (2002), Extended finite element method for cohesive crack growth.In Engineering Fracture Mechanics, 69, pp. 813–833.

9. Mariani, S. and Perego, U. (2003), Extended finite element method for quasi-brittle fracture. InInternational Journal for Numerical Methods in Engineering, 58, pp. 103–126.

10. Laborde, P., Pommier, J., Renard, Y. and Salaun, M. (2005), High-order extended finite elementmethod for cracked domains. In International Journal for Numerical Methods in Engineering, 64,pp. 354–381.

11. Sukumar, N. and Prevost, J.H. (2003), Modelling quasi-static crack growth with the extendedfinite element method, Part I: Computer implementation. In International Journal of Solids andStructures, 40, pp. 7513–7537.

12. Zienkiewicz, O.C. and Taylor, R.L. (2000), The Finite Element Method. Volume 1. The Basis.Butterworth-Heinemann.

296

Accuracy Comparison of Rectangular and TriangularMathematical Elements in the Numerical Manifold Method

H.H. ZHANG1, Y.L. CHEN1, L.X. LI1,∗, X.M. AN2 AND G.W. MA2

1MOE Key Laboratory for Strength and Vibration, Xi’an Jiaotong University, Xi’an, Shaanxi, PR China, 7100492School of Civil and Environmental Engineering, Nanyang Technological University, Singapore, 639798

1. Introduction

The numerical manifold method (NMM), originally proposed by Shi1,2 is a combinationof the finite element method (FEM) and the discontinuous deformation analysis (DDA).3 Itprovides a unified framework for both continuous and discontinuous problems using math-ematical covers that are independent of the physical domain of problem.

Since the advent of the NMM, it has been extensively investigated either in theoreticalstudy or practical applications. For example, Shyu and Salami4 implemented the mappedquadrilateral mathematical elements in the NMM, which were more efficient than triangularmathematical elements originally suggested in the NMM. However, mapped elements sufferfrom the element shape. Moreover, lower order elements can give rise to shear locking forbending problems. Chen et al.5 derived formulations of the high-order NMM using high-order cover functions and verified its necessity for complicated deformation problems. Chenget al.6 incorporated Wilson non-conforming elements in the NMM and verified its efficiencyand accuracy for a cantilever slab bending problem. On the other hand, applications of theNMM to strong discontinuity problems with cracks are successful. Tsay et al.7 combined theNMM with the crack opening displacement method to predict the crack growth path. Chiouet al.8 applied the NMM to model crack growths by means of the virtual crack extensionmethod. However, in their work, local remeshing was required to accurately capture thecrack path, which increased the burden of meshing to some extent. By incorporating thesingular physical covers in the original NMM, Ma et al.9 investigated the complex crackproblems involving multiple, branched and intersecting cracks. Meanwhile, Zhang et al.10

simulated the propagation of complex cracks without remeshing using the same strategyas in Ref. 9.

Mathematical covers play a most important role in the NMM because they determine theapproximation accuracy and the physical covers. The prominent feature of mathematicalcovers in the NMM is that they are independent of physical features, and therefore regu-lar mathematical elements are possible to be used in building mathematical covers. In thispaper, rectangular, right-angled (RA) triangular and equilateral (EL) triangular elements arerespectively examined to study their difference in accuracy.

2. Basic Theory

2.1. A brief of the NMM

For ease of reading, the NMM is briefly introduced here.10

For a problem, the mathematical cover system, i.e. the union of mathematical covers, mustbe first built to completely cover the physical domain of problem ignoring all the physical




features such as joints, material interfaces, cracks and boundaries. On each mathematicalcover MI, a weight function is defined such that

φI(x) ∈ C0 (MI) (1)

φI(x) = 0, x /∈MI (2)

which satisfy the partition property of unity∑J

if x ∈ MJ

φJ(x) = 1 (3)

With well-defined mathematical covers, physical covers can be obtained by their inter-sections with the physical domain, and then the manifold elements are generated as the com-mon regions of physical covers.

In the NMM, the displacement on an element E is approximated by

uh(x) =NP∑i=1

φi(x) · ui(x) (4)

where NPis the number of physical covers sharing element E and φi(x)is the weight functioncorresponding to the physical cover Pi, which is the same as that defined on the mathematicalcover MI ⊃ Pi. ui(x) is the cover function defined on Pi as

ui(x) = PT(x) · ai (5)

where PT(x) is the matrix of polynomial bases which may have constant, linear or higherorder terms. ai is the vector of unknowns to be determined.

As we can see, use of displacement approximation in Eq. (4) is adequate to model con-tinuous problems, however, when strong discontinuity problems with cracks are consid-ered, this approximation cannot effectively represent the stress singularity around the cracktip10. Hence, for a more general problem, the displacement approximation in Eq. (4) can beenriched as Ref. 10

uh(x) =NP∑i=1

φi(x) · ui(x)+NP

S∑j=1

φj(x) · uj(x) (6)

where the additional cover functions uj(x) are

uj = � · cj (7)

for singular physical covers. cj is the array of additional unknowns, and NPS is the number of

singular physical covers associated with the element E ·� is the matrix of singular bases as

� =[�1 0 �2 0 �3 0 �4 00 �1 0 �2 0 �3 0 �4

](8)

with

[�1,�2,�3,�4] =[√

r sinθ

2,√

r cosθ

2,√

r sin θ sinθ

2,√

r sin θ cosθ

2

](9)

where (r,θ) are the polar coordinates in the local coordinates system with the origin at thecrack tip.

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2.2. Remarks on mathematical elements

The NMM starts with building mathematical covers. In geometry, mathematical covers canbe of any shape, and overlapping of mathematical covers may generate any shape of math-ematical elements. However, for ease of constructing the weight functions, mathematicalcovers are always composed of several elements with a simple shape such as triangles orquadrilaterals which will be mathematical elements through overlapping of the mathematicalcovers.

A straightforward trick to generate the mathematical cover system is to borrow the finiteelement mesh, despite that the mesh is conceptually different from that used in the finiteelement method (FEM) because the latter requires consistence with the physical domain.Based on the finite element mesh, a mathematical cover consists of the elements sharinga common node, which is termed the star of this cover in the NMM. Hence, the wholemathematical cover system is eventually formed by all mathematical covers starred by everynode in the finite element mesh.

The advantage to employ a finite element mesh is evident. That is, the shape function atthe star for each element will directly transfer to the weight function through being piece-wise pasted together. Thus, for a mathematical cover with triangular elements, the weightfunction is in the form of global coordinates via the area coordinates, while for a mathemat-ical cover with quadrilateral elements, the weight functions will be obtained via a mappingtechnique. Under this circumstance, the approximation accuracy is dependent on the shapeof quadrilateral element. For some extreme cases such as concave quadrilateral elements, thefinite element mesh together with the shape functions is disabled at all.

From the geometric viewpoint, the RA triangle and the EL triangle are simple and perfect inshape as a triangle. However, for most of previous numerical methods, it is impossible to usesuch kinds of elements to discretize the real but complex domain due to the requirement thatthe mesh must be consistent with the physical domain. On the other hand, as a quadrilateral,rectangular elements are the simplest and the perfect one, especially that the shape functioncan be expressed in terms of global coordinates as a triangular element. Unfortunately, for acomplex domain in reality, a rectangular shape cannot be guaranteed for each element.

The situation changed since the NMM was born. Due to the independence of the phys-ical domain, a mathematical cover system with rectangular elements as shown in Fig. 1(a)becomes feasible. Thus, the weight functions on a rectangular element in the NMM can

Mathematical cover

x

y

o

(a) A mathematical cover system with rectangular elements

(b) The rectangular element and the coordinate system

1 2

3 4 a

b

Figure 1. A mathematical cover system with rectangular elements.

299


be obtained directly from the shape functions on such an element. See Fig. 1(b), they areexpressed by

⎧⎪⎪⎨⎪⎪⎩φ1 (x,y) = 1

4

(1+ 2x′

a

)(1+ 2y′

b

); φ2 (x,y) = 1

4

(1− 2x′

a

)(1+ 2y′

b

)

φ3 (x,y) = 14

(1− 2x′

a

)(1− 2y′

b

); φ4 (x,y) = 1

4

(1+ 2x′

a

)(1− 2y′

b

) (10)

with relative coordinates to the center of the rectangle being

⎧⎪⎨⎪⎩

x′ = x− x1 + x2

2y′ = y− y1 + y4

2

(11)

It is noted that the rectangular elements are assumed to be right oriented in the presentpaper to avoid unexpected troubles.11

Next, we consider a mathematical cover system with RA triangular elements shown inFig. 2 or EL triangular elements shown in Fig. 3. Following the guideline of area coordinatesξ1, ξ2 and ξ3, the weight functions on the two kinds of triangular elements are

⎧⎨⎩ϕ1 (x,y) = ξ1ϕ2 (x,y) = ξ2ϕ3 (x,y) = ξ3

(12)

Mathematical cover

Figure 2. A mathematical cover system with right-angle triangular elements.

Mathematical cover

Figure 3. A mathematical cover system with equilateral triangular elements.

300



To compare the accuracy of different mathematical elements, two numerical examples areexamined, including a cantilever beam bending problem and a mixed mode crack problem.

3.1. Bending of a cantilever beam

A cantilever beam is subjected to a uniform lateral loading at the free end in plane strainstate. The specifications are illustrated in Fig. 4.

Three mathematical cover systems with different mathematical elements are used, in whichthe element h, defined as the diameter of the circumscribed circle of the mathematical ele-ment, manifesting the resolution of the cover systems. Figs. 5 (a)–(c) illustrate the discretiza-tion when h = 5.10. The vertical displacements at the point (100, 0) normalized by theanalytical solution of 4.03 (see Ref. 12) are summarized in Table 1. It is seen that givingcover size h, the results by system (a) with rectangular elements are the best and not sensitiveto h. The results by systems (b) and (c) with two kinds of triangular elements are almost sameand become better with h.

3.2. Mixed mode crack problem

A finite plate with an edge crack in the plane strain state is considered, as shown in Fig. 6.In calculation, the dimensions are H = 8.0,W = 7.0, a = 3.5, and the uniform shear force τis taken to be unity. The reference mixed mode stress intensity factors (SIFs) are KI = 34.0

(0,10)

(0,0) (100,0)

(100,10)

y

x

P/2

P/2

P=104

E=1.0×107, ν=0.3

Figure 4. A cantilever beam subjected to a lateral loading.

(a) An illustrative cover system with rectangular elements

(b) An illustrative cover system with right-angled triangular l

(c) An illustrative cover system with equilateral triangular elements

Figure 5. Three different cover systems when h = 5.10.

301


and KII = 4.55, respectively13. Accuracy tests are also performed on three different kindsof mathematical elements and the normalized SIFs by the reference solutions are listed inTable 2, from which we can tell that given the cover size, the cover system with rectangularelements is the best, while the cover system with equilateral triangular elements is apparentlybetter than that with right-angled triangular elements.

a

W

H

H

E=1.0×107

ν=0.3

τ

Figure 6. A finite plate with an edge crack under shear loading.

4. Conclusions

In the present paper, the difference of three mathematical cover systems, respectively withrectangular, right-angled triangular and equilateral triangular elements is comparatively stud-ied in accuracy through typical numerical experiments. The results show that, given the ele-ment size, the cover system with rectangular elements is the best and therefore recommended.The cover system with equilateral triangular elements is apparently better than that withright-angled triangular elements for complex problems (e.g. a crack problem), but almostidentical for simple problems (e.g. a bending problem). Due to the difference of inscribed cir-cles for the three elements, these results can be reasonably interpreted with the mathematicaltheory of finite element methods.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (10472090,10572109, 10972172) and the New Century Excellent Talents of Ministry of Education(NCET-04-0930).

Table 1. Normalized vertical displacement of the point (100, 0).

Mathematical cover system Mathematical Element Size h

5.10 3.15 1.57

(a) Rectangular 0.892 0.902 0.907(b) RA triangular 0.793 0.855 0.885(c) EL triangular 0.802 0.859 0.890

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Table 2. Normalized SIFs for different mathematical elements.

Mathematical cover system Normalized SIFsMathemetical Element Size h

1.18 0.61 0.31

RectangularKI 0.916 0.966 0.984KII 0.974 0.985 0.993

RA triangularKI 0.828 0.916 0.970KII 0.908 0.956 0.958

EL triangularKI 0.864 0.937 0.973KII 0.945 0.974 0.989

References

1. Shi, G.H., “Manifold method of material analysis”, Transactions of 9th Army Conference onApplied Mathematics and Computing, Minneapolis, Minnesota, 1991, pp. 57–76.

2. Shi, G.H., “Modeling rock joints and blocks by manifold method”, Proceedings of the 33rd USRock Mechanics Symposium, San Ta Fe, New Mexico, 1992, pp. 639–648.

3. Shi, G.H. and Goodman, R.E., “Generalization of two-dimensional discontinuous deformationanalysis for forward modeling”, International Journal for Numerical and Analytical Methods inGeomechanics, 13, 1989, pp. 359–380.

4. Shyu, K. and Salami, M.R., “Manifold with four-node isoparametric finite element method”, 1stWorking Forum on the Manifold Method of Material Analysis, California, USA, 1995, pp. 165–182.

5. Chen, G., Ohnishi, Y. and Ito, T., “Development of higher-order manifold method”, InternationalJournal for Numerical Methods in Engineering, 43, 1998, pp. 685–712.

6. Cheng, Y.M., Zhang, Y.H. and Chen, W.S., “Wilson non-conforming element in numerical mani-fold method”, Communications in Numerical Methods and Engineering, 18, 2002, pp. 877–884.

7. Tsay, R.J., Chiou, Y.J. and Chuang, W.L., “Crack growth prediction by manifold method”, Journalof Engineering Mechanics, 125, 1999, pp. 884–890.

8. Chiou, Y.J., Lee, Y.M. and Tsay, R.J., “Mixed mode fracture propagation by manifold method”,International of Fracture, 114, 2002, pp. 327–347

9. Ma G.W., An X.M., Zhang, H.H. and Li, L.X., “Modeling complex crack problems with numeri-cal manifold method” International of Fracture, 156, 2009, pp. 21–35

10. Zhang, H.H., Li, L.X., An, X.M. and Ma, G.W., “Numerical analysis of 2-D crack propagationproblems using the numerical manifold method”, Engineering Analysis with Boundary Elements,34, 2010, pp. 41–50.

11. Zhang, H.H., Liu, S.J. and Li, L.X., “On the smoothed finite element method”, InternationalJournal for Numerical Methods in Engineering, 76, 2008, pp. 1285–1295.

12. Li, L.X., Han, X.P. and Xu, S.Q., “The analysis of interpolation precision of quadrilateral ele-ments” Finite Elements in Analysis and Design 41, 2004, pp. 91–108.

13. Yau, J.F., Wang, S.S. and Corten, H.T., “A mixed-mode crack analysis of isotropic solids usingconservation laws of elasticity”, Transactions of the ASME, Journal of applied mechanics, 47,1980, pp. 335–341.

303

Development of 3-D Numerical Manifold Method

G.W. MA∗ AND L. HE

School of Civil & Environmental Engineering, Nanyang Technological University, Singapore

1. Introduction

Finite element method (FEM) is inconvenient in meshing process, and limitation of small dis-placement/deformation, while mesh-freed analysis is limited by boundary treatments, espe-cially for contact and multi-physical problems. Heterogeneous structure and jointed solidsenable researchers to model a discontinuous material; however, it is restricted in areas suchas multi-intersecting interface cases. Even though Distinct element method (DEM) or Discon-tinuous deformation analysis (DDA) [Shi, 1988] enable the analysis in discontinuous domain,their relatively inflexible description of block deformation and insufficiently accurate descrip-tion on stress field matters.

Numerical manifold method (NMM), proposed by Shi [1991], incorporates these meth-ods’ benefits and can be viewed as a transition and combination of FEM and DDA, throughits important concept of manifold covers (mathematical and physical covers). Its manifold ismathematically defined when a function is continuous and differentiable at each independentcover in the description domain. These overlapping mathematical covers create various man-ifold elements through intersecting with the physical domain, which generates the continuousand differentiable function description in the whole domain. Geometrical shape can be user-defined and each mathematical cover does not require conforming to the boundaries of itsstructure, which also reduce the workload in processing the meshes remarkably.Increasingunderground construction and geotechnical works surge a high demand for 3-D discon-tinuous deformation for more reliable 3-D models. Cheng and Zhang (2008) derived onerelatively basic theoretical formation of 3-D NMM without implementations.

This paper mainly describes the implementation of 3D-NMM. Examples illustrate thatthe developed 2D numerical manifold program is effective and applicable to 3D continuumsolids.

2. Geometric Configuration of 3-d Mathematical Cover

2.1. Fundamentals of the NMM

The NMM is based on three important concepts: mathematical cover (MC), physical cover(PC) and cover-based element (CE). MCs are user-defined overlapping patches. One signifi-cant advantage of the NMM is that arbitrary geometric shapes (e.g. polyhedron in 3-D) canbe the basic MC of the mathematical domain. Each different shaped MC has its own math-ematical description inside. These polyhedrons are overlapping in space, and those MCs areonly required to be able to fulfil the whole space completely. Physical domain is used to rep-resent the portrait of target physical objects (TPO) in a general sense. TPO includes all thephysical features such as internal discontinuities (e.g. joints, material interfaces and cracks)and external geometries on which boundary conditions are prescribed. MCs intersect andpaste seamlessly to re-divide the TPOs, during which PCs are generated. Further, elements(CEs) in the NMM framework can be considered as the common part of the overlapping




(a)Physical domain and mathematical covers (b) Physical covers (c) Cover based elements

Figure 1. Illustration of finite cover system in the 3D NMM.

PCs. In this way, the NMM can be easily understood and extended to the three dimensionalcase.

Fig. (1a) illustrates three basic concepts of the NMM in a 3-D view. There are two MCsin total, a sphere MC1 and a hexahedron MC2. The pyramid defines the physical domain.Intersected with the physical domain, two PCs (i.e. PC1 and PC2) (shown in Fig. (1b)) aregenerated. These two PCs finally form three CEs, which are CE1, CE2 and CE3, as shown inFig. (1c).

On an MC denoted as MI, a weight function is defined, which satisfies

ϕI(x) ∈ C0(M1) (1a)

ϕI(x) = 0, x � MI

with ∑I

if x ∈MI

ϕI(x) = 1 (1b)

Equation (1a) indicates that the weight function has non-zero value only on its correspond-ing MC, but zero otherwise, whereas Equation (1b) is just the partition of unity property toassure a conforming approximation. The weight function ϕI(x) associated with MI will betransferred to any of the PCs in MI accordingly. The interpolation approximation can beconstructed. First, a cover function ui(x) is defined individually as a local approximation onthe PC Pi for the displacement field, which can be constant, linear, high order polynomialsor other functions with unknowns (also termed DOFs) to be determined. Then, the globaldisplacement u(x) on a certain CE e is approximated to be

uh(x) =∑

iif e ⊂ Pi

ϕi(x) · ui(x). (2)

2.2. Geometric patterns

A Geometric Pattern (GP) is a type of theme of the recurring objects, sometimes referred to aselements of a set, to generate space or parts of an object [Nooshin et al. (1997)]. The first stepto extend to 3D NMM is to select a basic GP to fill the entire 3-D space, as also required in the2-D NMM. Most often GP geometry is based on platonic solids. The NMM creates a possibleplatform to extend the applications of different platonic solids. However, for computationalconvenience, simple and regular geometric patterns are suggested. The equilateral triangleand rectangle are excellent choice for the 2-D NMM, as their high identity and uniformity.

306


Unfortunately, regular tetrahedron is not able to fill the entire 3-D space. Hexahedron is thebest choice geometrically because it has relatively simple topological structure, and equallysized hexahedrons can fill the space completely. The generalized octahedron is another wisechoice. It is worth noting that MCs can be any shapes in 5 platonic solids and even some otherpossible irregular solids, if proper weight functions are implemented. In addition, the NMMis based on the incremental method, suitable for large deformation and large displacementanalysis. The simplex integration method can gain the accurate integration for any arbitraryshape [Li et al. (2005)]. If the interpolation function in a global Cartesian system for anyarbitrary shape is able to be built, the governing equation with the NMM incremental stepis viable. In other words, the hexahedron/octahedron is not the best MC choice in the 3-DNMM. For integration accuracy, the general tetrahedron is the ideal unit cell to form themanifold element in 3-D.

Similar to the 2-D NMM, one background node is a star of a MC, which can be anygeometry significance points in the cover (e.g. center of gravity). By considering a tetrahedronmanifold element, four MCs are required to overlap to create one manifold element. Onthe other hand, it is well known that any shaped convex polygon can be decomposed intoseveral triangles. Any convex polyhedron in the 3-D space can also be decomposed intoseveral tetrahedrons. The decomposition plans are innumerable (e.g. choosing different blocksize). The division plan determines the shape of the MCs. It can be seen in Fig. (2) that ahexahedron can be

As a matter of fact, it is not the only of 6 Tetrahedrons plan (as shown in Fig. (2a)), theadvantage is that the cube divided by the figure shown, has the property of center-symmetry,and it can build up the entire 3D space without rotation. Two schemes of 5 Tetra-plan (Fig.(2b) and (2c)) are usually coupled and applied as a mixed-discretization (M-D) zone [ItascaConsulting Group, Inc (2003)]. As these two kinds of mesh are symmetric to each other, theycan overlap at the same position to decrease “the strain instability disturbed by choosingmesh direction”. One cubic domain and cover is generated by two “anisotropy" mathemat-ical covers, which is another extension of the manifold concept. When using the centers ofcube and 6 faces as auxiliary points, a cube can be decomposed into 24 different tetrahe-drons. The mathematical mesh has extremely symmetrical property by this 24-tetrahedrondecomposition scheme.

2.3. MC and its formulation

The corresponding MC belongs to the field of geometrical topology. When the decompositionin Fig. (2b) is applied, their geometry structure is shown in Bronstein et al. (2007). Similarly,other decompositions can also be found to gain a cover structure.

(a) (b) (c) (d)

Figure 2. Decomposition plan from hexahedron to tetrahedron.

307


• Weight Function: Here, four overlapping MCs (one type-I and three type-II covers) havea common area E (complete tetrahedron element), the displacement functions u(x,y,z),v(x,y,z) and w(x,y,z) in this common area E can be obtained by taking the weightedaverage of the four cover functions, which satisfy Equation (1b). In a tetrahedral commonarea (tetrahedron cornered by four nodes), four weight functions of the four covers exist.• Cover Function: The cover displacement functions can be constant, linear, and even

higher order polynomials or locally defined series. The cover function is expressed as

Ui =m∑

j=1

⎡⎣ fij(x,y,z)

fij(x,y,z)fij(x,y,z)

⎤⎦ (3)

Then the displacement for the common area is written as:

U =q∑

i=1

ϕi(x,y,z)Ui (4)

where q is the sum of the mathematical covers for an element(CE).On substituting Equation (3) into (4), the following displacement function is obtained:

U = ϕFd = Nd. (5)

3. General Formulations of NMM Approximation

3.1. Weak form of governing equations

According to the principle of virtual work, the virtual work done by external forces is equalto the virtual strain energy of the system, which leads to the corresponding weak form ofgoverning equation∫

�

δεTσdV =∫�

δuTbdV +∫�t

δuTtd� −∫�

uTρudV (6)

where ε is the strain tensor; u is the stress tensor; u is the displacement vector; b is the bodyforce per unit volume; t is the traction prescribed on the corresponding boundary; ρ is thedensity; u is the acceleration vector.

An increment approach is adopted in the dynamic analysis. The total-time is divided intoa finite number of time steps. Variables such as displacement, velocity and acceleration areknown at the beginning time tn of the current step, and the increments of several variablesduring the current time step tn ∼ tn+1 are as unknowns to be determined. Therefore, theincremental form of the principle of virtual work is required. Energy status at time tn andtn+1 are both satisfied under Equation (6). The increment form of the principle is obtained bysubtracting two energy equations at time tn and tn+1 , and ignoring the higher order terms.∫

VδεTσdV −

∫�

δuTbdV −∫�t

δuTtd� +∫�

uTρudV = 0. (7)

3.2. NMM interpolations

In a cover-based element, the NMM interpolation follows Equation (5). The strain is expressedas

ε = LNd = Bd (8)

where B = LN is the strain matrix; L is the differential operator matrix.

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The constitutive relation is

σ = DLNd = Sd (9)

where S = DLN is the stress matrix.Equation (5) can be rewritten in an incremental form

u = Nd (10)

Hence, the incremental strain is

ε = LNd = Bd (11)

The constitutive relation in the incremental form is

σ = DLNd = Sd (12)

4. Illustration Example

4.1. Movement of a free falling block

Since the primary aim of the 3-D NMM is to predict the stability and motion of a discontin-uous system, the developed program is first validated by simulating the free-falling processof a single block.

In Fig. (3), the geometry of a single block in the 3-D space can be defined by using theexplanation of physical covers and mathematical covers mentioned in the previous sections.The cube falls under the pure

Fig. (4) gives the displacement and velocity time history. The error of the numerical solu-tion is less than 0.1% in the maximum absolute difference, which illustrates the credibilityof the NMM calculation accuracy.

4.2. Effect of MC size and orientation

The NMM consists of independent MCs and PCs and thus the generation of the cover basedmanifold elements is not restricted to the MC configurations, instead, the concerned physical

Figure 3. Falling cube under gravity.

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domain can be obtained by intersection with MCs. The following examples are to investigatethe effect of the MC size and the cutting plane orientations. A plate structure is generated bycutting a cube and the density of the MC is variable. The orientation of the cutting planes toproduce the same geometry plate can be different.

• MC size effect (quasi-static condition).

A 2 m×2 m×0.1 m plate is subjected to the gravity load with g = 10 and fixed at fourcorners. The material properties are assumed to be E = 10000, v = 0.3, ρ = 1.2. Its geometryof the typical mesh design is shown in Fig. (5) in which the faces of the plate conform to theaxis planes.

Six mathematical cover size of s = 0.52, 0.32 0.22, 0.12, 0.08, 0.05 are used to examinethe MC size effect. After intersection with the physical plate, 190, 684, 1104, 3706, 8100,19200 manifold elements are respectively generated. In order to clarify the quasi-staticresponses of the 6 models, the dynamical ratio is set to be 0. Hence, no velocity of themanifold element will be transferred to the next time step. Totally five measurement pointsare set at the middle of the panel for every edge for each case. More details are listed inTable 1.

The deformation of case 6 (refinement 5) at the maximum displacement is illustrated inFig. (6). It is found that they converge well and all the 4 edges are adequately symmetric, eg.

Figure 4. Comparison between NMM result and exact value of displacement and velocity history.

Figure 5. Geometry of typical mesh design for mesh density effect study.

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Table 1. Solution history at the center of panel.

Name Base Refinement Refinement Refinement Refinement RefinementSolution 1 2 3 4 5

“Total 0.16404 m 0.25027 m 0.34334 m 0.37919 m 0.39694 m 0.39973 mDeformation” [+34.45%] [+27.11%] [+9.45%] [+4.47%] [+0.76%]

Mesh Spacing: Spacing: Spacing: Spacing: Spacing: Spacing:properties 0.52 0.32 0.22 0.12 0.08 0.05

Elements: Elements: Elements: Elements: Elements: Elements:190 684 1104 3706 8100 19200

Vertexes: Vertexes: Vertexes: Vertexes: Vertexes: Vertexes:1156 3812 6116 20184 44282 76800

Figure 6. Deformation of case (refinement 5) at the maximum displacement time.

M1∼4 curves in Fig. (6) are presented the deformation history of four middle points fromup, down, left and right side. The result converges to the closed-form solution when the meshdensity increases. This property is consistent with the result of the FEM, and it supports thevalidity of the 3-D NMM calculation.

• MC orientation effect (dynamic condition).

The 2 m×2 m×0.1 m plate is subjected to a constant point load L = 5 at the center and itsfour corners are fixed. The material properties of the plate are same as the previous example.The dynamical ratio is set as 0.999 to investigate the dynamical response fully. Two differentorientations of the plate (Orientation 2 and 3) are shown in Fig. (7) while the orientation 1is same as the previous example.

The Z displacement histories of the center point are shown in Fig. (8), from which it can beseen that the orientation of the MCs has negligible effect on the plate maximum displacementat the plate center. The maximum displacement is accurate and stable after converging, andthe convergence time is about the same when the total element number is almost same.

The results show that the orientation of the MCs has little effect to the simulation result.This ensures the accuracy and modeling efficiency of the 3-D NMM decreases the mesh divi-sion complexity in FEM, and increases the modeling efficiency. It also supports the validityof the 3-D NMM dynamic algorithm.

311


p p

Figure 7. Geometry of mesh designs for mesh orientation effect study.

Figure 8. Z displacement histories of center point.

5. Conclusions

The present study aimed to extend the 2-D NMM to the three dimensional application.The fundamentals of the NMM are briefly outlined in the 3-D space, and the potential3-D manifold cover geometry configurations are discussed. The three basic concepts, i.e., themathematical cover, physical cover and the manifold element are defined in the 3-D frame-work and the general formulation of 3-D NMM derived based on the incremental form ofthe principle of virtual work has been given. The three numerical examples demonstratedin section 5 all support the accuracy of the developed 3-D NMM code. Results are credibleboth under quasi-static and dynamic analysis. It highlights the algorithm in the aspect ofindependence between physical modeling and mathematic cover orientation. Application ofthe 3-D NMM to engineering problems can thus be possible especially when a block systemis under concern. Currently we are still seeking for the most suitable contact model to the3-D NMM and the results will be reported in the next stage.

Acknowledgements

The Ministry of Technology provided financial support to this research.

312


References

1. Cheng, Y.M., and Zhang, Y.H., “Formulation of a three-dimensional numerical manifold methodwith tetrahedron and hexahedron elements”, Rock Mechanics and Rock Engineering, 41, 4, 2008,pp. 601–628.

2. Bronstein, A.M. Bronstein, M.M. Kimmel, R., “Weighted distance maps computation on paramet-ric three-dimensional manifolds”, Journal of Computational Physics, 225, 2007, pp. 771–784.

3. Li, S.C. Li, S.C. Cheng, Y.M., “Enriched meshless manifold method for two-dimensional crackmodelling”, Theoretical and Applied Fracture Mechanics, 44, 2005, pp. 234–248.

4. Itasca Conculting Group, Inc., 3 Dimensional Distinct Element Code-Theory and Background,Version 3.0. Minneapolis Press, Minnesota,2003.

5. Nooshin, H., Disney, P. and Champion, O., “Computer Aided Processing of Polyhedric Configu-rations”, Beyond the Cube: The Architecture of Space Frames and Polyhedra, Chapter 12, 1997,pp. 343–384.

6. Shi, G.H., “Discontinuous Deformation Analysis: A New Numerical Model for the Statics andDynamics of Block Systems”. PhD Thesis. University of California, Berkeley. 1988.

7. Shi, G.H., “Manifold method of material analysis”, Transaction of the 9th Army Conference onApplied Mathematics and Computing, Minneapolis, Minnesota, 1991, pp. 57–76.

313

Application of the Optimization for Rock Tunnel’s Axis Trend byBlock Theory

YANG WENJUN1,2,∗, HONG BAONING1,2, SUN SHAORUI1,2, AND ZHU LEI1,2

1Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering,Nanjing 210098 China2Geotechnical Research Institute, Hohai University, Nanjing 210098 China

1. Introduction

It is one of the key factors to optimize tunnel’s axis both rock cranny’s structural planesand excavated planes of the tunnel, and it affects awfully engineer if the route of the tunnelis not circumspect. After known the generating mechanism of weak structural planes whenchoosing routes of tunnel, we use key block theory and stereographic projection method,adjust the trend of excavated planes, and make the area both of stereographic projection andexcavation cone have the same point so that the weight of joints cone is the smallest. In otherwords, after known the generating mechanism of structural planes, mechanical propertiesand geometric parameter of excavation planes when optimizing and design of tunnel axisand section, we can adjust the tunnel axis linking the rock maximal orientation of rockinitial stress field, so that the potential arisen key blocks are the smallest, also can analyseeach part stabilization status of the tunnel and anchorage design.

Using the statistic of characteristic parameter of wall rock cranny’s structural planes ofthree tunnels, also using key block theory and stereographic projection method, this papersimulates and analyses the change of the position, size and stability in differ trend of thetunnel, and it proves simulative correctness and necessity in optimization in contrast withpractical instance, lastly it programmes and reckons the best trend of the tunnel by using thereformatory measure of the Genetic Algorithm. The results obtained in this paper show thatit is necessary to optimize tunnel’s axis by using key block theory and the modified GeneticAlgorithm, and it has very important referenced worthiness in traffic programming, designand construction of the tunnel.

2. Block Theory Basic Principles and Rock Joints Statistic

2.1. Block theory rationale and characteristic

Block Theory1, 2 is brought forward by Gen-hua Shi in middle of 1970s, it adopts the Numer-ical Manifold Method, and adopts absolute math cover and physics network to solve theinterface of material boundary, crack, block and different materials fields, and it is a goodkind of analytic method in rock mass stability analysis, and has been widely used in ourcountry.

Block Theory is a new method of stability analysis to block crack rock by means of analy-sissitus, set theory, geometry and vector algebra. Its basal assumptions: (1) structural planesare plane and pierce rock mass on research; (2) Structure is rigid body; (3) rock mass instabil-ity begins with engendering shear displacement along structural planes in the different kindsof load. Three theorems: (1) block bounded-ness theorem; (2) the removability theorem ofbounded convex block; (3) the removability theorem of combinatorial block.




The existent structural planes and its intensity of rock mass often control the intensityand stabilization of rock mass. Key block theory analyses and differentiate the configuration,size, location and the movement direction of the key block though the known geologicalinvestigation measure about the generating mechanism of structural planes, space coordinate,and so on, and it bases on set theory, vector analysis and stereographic projection method,and then it confirms all the key block and calculates its downslide strength and safety factorby physical mechanics properties of slip surface, so as to offer decision both of economicrational reinforcing rock mass and choosing excavation orientation.

2.2. Statistics and analytical method of random block structural planesparameters

Because the combination of structural planes is random, rock block which are incised bystructural planes is random block. Random block nearby the planes of tunnel excavationmaybe become key slippage block. The main approach on research of random block is:

1. On the basis of on-the-spot geological investigation, we should hold statistical groupingof cranny’s plane of disquisitive region, and analyses their range of generating mecha-nism. Statistical grouping of cranny’s plane need to be reshuffled in distinctness of orien-tation and excavation planes according to the orthonormal theory, and research on theoutcropped characteristic of random block which are incised by structural planes andexcavation planes.

2. On the basis of each combined generating mechanism of structural planes, we shouldplat stereographic projection maps of structural planes and each excavated planes, stere-ographic projection can show easily the points, lines, planes and solids of space in theplanes, and distinguish the types of key blocks in the excavated planes by key blocktheory.

3. Statistical analysis includes modality, size, space distribution characteristics character-istic, and so on. We can calculate both block’s stability and block’s safety factor of atdifferent operation points by the method of limit equilibrium, so as to offer decisionboth of systemic anchorage design and the choosing of classic tunnel orientation. Safetyfactor of key block is:

➀ Removable block straightway drop, its safety factor is η = 0.➁ Sliding block in single plane, its safety factor is:

η = G cos δ tanϕ +�scG sin δ

(1)

Formula: G is the weight of removable block, δ is single of slip plane, c and φ is cohesivestrengths and internal friction angle of slip plane, and�s is area of slip plane. The weightof block: G =∑ 1

3γh�S, γ is density of rock mass, �S and h is bottom area and heightof block respectively.

➂ Sliding block in double planes, its safety factor is:

η = G cos δ( sin δ2 tanφ1 + sin δ1 tanφ2)G sin δ

+ (s1c1 + s2c2) sin (1800 − δ1 − δ2)G sin δ

(2)

Formula: δ is the single of intersectant lines of slip plane, δ1 is the single of intersectantlines normal and slip plane s1, δ2 is the single of intersectant lines normal and slip planes2, else parameters are as described above.

316


3. Genetic Algorithm and Reformatroy Measure

3.1. The model structure and principle of genetic algorithm

Genetic Algorithms (GA) is a high-effective randomly searching algorithms, which bases onthe natural evolution and Darwinism. A simple and efficient global optimization method thatit make individual into a certain length of chromosome, and only one chromosome should beincluded in each fitness function, so that it may have better “next generation” by inheritance.

Reformulating the possible potential solutions of the problems as a certain length of indi-vidual form population by genetic code, inter-population evolves the highest disturbed solu-tion by generation following the principle of genetic selection and nature elimination. Witheach generation according of individual fitness value to choose individual in crossover andmutation probability, they generate new population of new solutions. Through genetic selec-tion next generation is more easily adapt to environment than the previous one, and theclassic individual of the last generation can get the classic solution by genetic code. BasalGenetic Algorithms mainly steps include coding, producing initial population, confirming offitness function and estimate function (fitness account), choice operation, search operation,decode, and son. The programme drawing of Genetic Algorithms is shown in Figure 1.

3.2. Proper reformatory measure of genetic algorithms

Basal Genetic Algorithms may find the global optimal solution to solve easy problems, whilebasal Genetic Algorithms seems a little spirit is willing but the flesh is weak because the

Calculate fitness(object function W)

NO

NO

YES

Output result

Over

I=I+1

I=I+1

Use GA, select unit, mating unit, aberrance unit to optimize

Basing on estimationterms, estimate if end to optimize?

Start

Input initial data

I 1

Random create variable in range

Generate next population

I>maxg?

Figure 1. The programme drawing of GA.

317


problems are very complicated and the amount of form population is much. The algorithmof BP neural network is BP algorithm, its ability to seek the topical superior is strong whileits ability to seek the global superior is weak, and easily gets into the topical superior, soas to BP neural network becomes anamorphic in mapping complicated nonlinear functioncoupling.

In view of the advantage and disadvantage of them, we improve on fitness selection,crossover and mutation operator of basal Genetic Algorithms, and import BP operator toameliorate the defect that the ability to seek topical superior is weak, and get easily theglobal superior solution because of its strong ability of function mapping.3 Many investiga-tions show, firstly to optimize initial weight distributing by Genetic Algorithms, and locate abetter search space, then find out the best solution in the small space by BP algorithm. Thismixed BP algorithm is feasible.4

4. Geological Features of Some Tunnels and Stability Analysis of Blockand Tunnel Axis to Optimize

4.1. Engineering geological features of some tunnels and parametersstatistic of structural planes

Yongjia-lucheng Segment of Jinliwen Freeway is 22 kilometers long, Huayantou lying onthe border of Qingtian and Yongjia Counties at its north, its south joining the constructedfreeway in the suburb of Wenzhou City. The segment goes through mountains, the threeselected tunnels are chosen from five tunnels. The structure abridged drawing of multiplearch tunnel is shown in Figure 2.

There is mainly the quaternary incrustation, the upper Jurassic tuffs and Yanshan forepartmigmatitic granite. The upper Jurassic tuffs and explosive volcanic eruption lie in the projectof the three tunnels. The lithology is compact and hardy, massive structure, and has thehigher weatherproof ability.

The excavation of the three tunnels adopts the means firstly to excavate guided room andmid-board, the excavation method adopts matte surface demolition. The types of structureare joints and fissures in the tunnel region, and there no outcrop of fault and fracture zonein the reconnaissance. Rock joints and fissures are generally developed in the tunnel region.According to field statistics measure, we find that rock mostly developed three groups. Theabove two groups’ structure planes are primarily in the tunnel region, and have small theangle between their trends and tunnel’s axis.

The three tunnels are Hongfeng Tunnel, Muxidai Tunnel and Yangwan Tunnel of Jinli-wen Freeway. On-the-spot data acquirement of wall rock joints structural planes are done

Figure 2. The structure abridged drawing of multiple arch tunnel.

318


by Statistics analysis, and past through model and practice distributing by approach andχ2 tests, then models tests meet the requirements. The generating mechanisms of structuralplanes of the three tunnels’ wall rock basically follow the Gaussian distribution, and spacingfollow the Weibull Distribution. According to the characteristic parameters of the models,calculate the generating mechanisms of structural planes and spacing by confidence level of95% and 99% respectively, which are list in Tables 1–3.

Table 1. Geometric features space Statistic of Hongfeng Tunnel’s structural planes.

Structural planes Structural planes 1 Structural planes 2 Structural planes 3

95% dip direction (◦) 149.87∼157.77 54.12∼61.72 60∼80(confidence dip angle (◦) 81.22∼83.15 81.78∼83.75 8∼15level) Spacing (cm) 39.96∼53.58 29.36∼62.18

Reach (cm) 53.20∼82.80 49.26∼98.7499% dip direction (◦) 142.75∼159.01 47.43∼62.92(confidence dip angle (◦) 76.71∼83.46 77.43∼84.06level) Spacing (cm) 29.87∼56.20 26.79∼67.33

Reach (cm) 52.17∼87.45 53.33∼106.51

Table 2. Geometric features space Statistic of Muxidai Tunnel’s structural plane.


95% dip direction (◦) 60.74∼71.72 243.73∼256.91 324.49∼333.98(confidence dip angle (◦) 82.39∼86.72 71.35∼75.50 83.52∼85.23level) Spacing (cm) 21.36∼30.50 32.92∼56.23 28.54∼39.86

Reach (cm) 53.31∼85.00 62.91∼95.14 48.27∼85.9199% dip direction (◦) 53.06∼73.45 238.92∼258.66 318.13∼335.47(confidence dip angle (◦) 76.29∼87.40 65.87∼76.15 79.66∼85.50level) Spacing (cm) 13.71∼31.94 26.47∼59.89 21.63∼41.64

Reach (cm) 52.61∼89.98 61.06∼100.21 48.56∼91.82

Table 3. Geometric features space Statistic of Yangwan Tunnel’s structural planes.


95% dip direction (◦) 29.11∼33.22 239.95∼253.79 329.42∼336.62(confidence dip angle (◦) 80.11∼82.85 73.28∼77.69 71.60∼76.59Level) Spacing (cm) 33.87∼50.36 47.64∼87.46 28.59∼43.30

Reach (cm) 71.28∼93.79 43.58∼78.65 34.39∼52.3899% dip direction (◦) 24.01∼33.87 234.78∼255.97 323.65∼337.75(confidence dip angle (◦) 75.64∼83.28 68.65∼78.39 66.29∼77.38Level) Spacing (cm) 27.35∼52.96 46.05∼93.72 22.42∼45.61

Reach (cm) 68.39∼96.86 45.71∼84.16 29.10∼55.21

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Figure 3. Distributing diagram of some Figure 4. Key block distributing diagram ofrock joints. some rock joints.

4.2. Simulation methods of tunnel axis

On the known that tunnel figures and tunnel sizes, we adopt stereographic projection methodto select three typical structural planes and excavation slope pattern which form the bound-ary conditions of tetrahedrons, and automatic search and calculate potential wrought blockshape and size by given structure planes in different position of chambers, and calculate safetyfactor of each part block. Distributing diagram of some rock joints is shown in Figure 3, andkey block diagram distributing of some rock joints is shown in Figure 4. Because structureplanes are no filled joints rigid structure planes, take parameters φ = 23.8◦, c = 0.2 MPa.

4.3. The computing results along tunnels’ axis trend

The quantity of structural planes combination is large, but their basal uptrend is rule-based,next take typical structural planes of every tunnel for an example to analyze their uptrend.

Three groups of structural planes of Hongfeng Tunnel: α1 = 153.8◦, θ1 = 82.2◦, α2 =57.9◦, θ2 = 82.8◦, α3 = 70.0◦, θ3 = 11.5◦, and their distributing diagram are shown inFigure 3, and key block distributing diagram are shown in Figure 4 in a trend of the tunnel.Under their conditions relations diagram of curves of tunnel trends and stability of key blockare shown in Figure 5.

Three groups of structural planes of Muxidai Tunnel: α1 = 66.2◦, θ1 = 84.6◦, α2 =250.3◦, θ2 = 73.4◦, α3 = 329.2◦, θ3 = 84.4◦. Under their conditions relations diagram ofcurves of tunnel trends and stability of key block are shown in Figure 6.

Three groups of structural planes of Yangwan Tunnel: α1 = 31.2◦, θ1 = 81.5◦, α2 =246.9◦, θ2 = 75.5◦, α3 = 330.0◦, θ3 = 71.4◦. Under their conditions delaminating withtunnel trends and stability of key block relations diagram of curves are shown that theirsafety factors of reliability are all over three, so the stability of Yangwan Tunnel wall rock isbasically steady-going.

320


right tunnel block weight

020406080

100

120140160180200

0 40 80 120 160 200 240 280 320 360tunnel trend (°)

bloc

k w

eigh

t (kp

a)

η<1 η<1.5 η<2

left tunnel block weight

02040

6080

100120140

160180200

0 40 80 120 160 200 240 280 320 360tunnel trend (°)

bloc

k w

eigh

t (kp

a)

η<1 η<1.5 η<2

Figure 7. Tunnel trends and stability of key block relations diagram of curves of Hongfeng Tunnel.

right tunnel block weight

020

4060

80100120

140160

180200

0 40 80 120 160 200 240 280 320 360tunnel trend (°)

bloc

k w

eigh

t (kp

a)

η<1 η<1.5 η<2

left tunnel block weight

020

406080

100

120140160

180200

0 40 80 120 160 200 240 280 320 360tunnel trend (°)

bloc

k w

eigh

t (kp

a)

η<1 η<1.5 η<2

Figure 8. Tunnel trends and stability of key block relations diagram of curves of Muxidai Tunnel.

4.4. The contrast analysis of between simulation result of tunnel axis’trend and practical trend

1. The orientation of three tunnels cross the mountains chain is the NE direction, and thetunnels trends (90◦ ∼ 110◦) approximately perpendicularly intersect the mountains. Inthe eyes of simulation result of the safety factors and weight of tunnels key block, thesafety factor of Hongfeng Tunnel wall rock is not big, but basic steady, and the quantityof immediate fallen block is small and weight is not big; the safety factor of MuxidaiTunnel wall rock is big, but always has block to fall firsthand in any case — their safetyfactors of key block η = 0; the stability of Yangwan Tunnel is the best of all, and basicallyno immediate fallen block.

2. By the Figure 5 ∼ 6 knowable, when the trends of tunnels is near the joints dip directionof tunnels wall rock, the safety factors and weight of tunnels key block fall sharply, whileblock weight increase sharply. So we should avoid the instance that the trends of tunnelsand the joints dip direction of tunnel wall rock are basically accordant in selecting tunnelaxis.

3. Because of two groups of joints dip direction and dip angles of Hongfeng Tunnel arenear, it results in the safety factors of wall rock are not large as a whole, and the weightof key block is not large when the tunnel trends of and the joints dip direction of tunnelwall rock are at Larger Angles. Two groups of joints dip direction of Muxidai Tunnel

321


wall rock are opposite and their angels are near, which result in rock mass is incisedreciprocally by joints planes and excavation planes, so a lot of key block firsthand fall andlead to exist overbreak phenomenon. Three groups of joints dip direction of YangwanTunnel have definite dip angles reciprocally, so the stability of wall rock is very good. Inthe whole, we should avoid the rock mass which two groups of joints dip direction ofare opposite while their angels are near in selecting tunnel axis, and it is very significantto select the best trend of tunnel.

4.5. Tunnel trend to optimize and simulation result

According to geometric features space Statistic parameters of structural planes, we calculateevery different combined structural planes, get relevant the stability of key block, also useMatlab software and reformatory measure of the Genetic Algorithm to edit programs, simu-late and analyse to filtrate large numbers of data, at last gained optimization result of tunnelaxis trend.

Whereas the gradient of tunnel is small and to predigest filtration large numbers of data,we simulate in the case that tunnels gradient is 0◦ and tunnels orientation is in the range of 0◦to 180◦. Adopting safety factor criterion η to analyse, we search the distribution and weightof corresponding block in the different case, η < 1, η < 1.5, η < 2, η < 3, and make η > 2as one of the judgment basises. The best trend of tunnel will be selected by filtrating data.

As selecting data in the 95% confidence Level of Hongfeng Tunnel, its dip direction rangeis 149.87◦ to 157.77◦ and dip angles range are 81.22◦ to 83.15◦ of structural planes 1, andits dip direction range is 54.120 to 61.72◦ and dip angles range are 81.78◦ to 83.75◦ ofstructural planes 2, and its dip direction range is 60◦ to 80◦ and dip angles range are 8◦to 15◦ of structural planes 3. In the same way the data to select are listed in Table 1 as inthe 99% confidence Level of Hongfeng Tunnel. safety factor criterion η > 2 and weightG < 100 KN are chosen as judgment basises, use Matlab software and reformatory measureof the Genetic Algorithm to optimize tunnel axis trend of Hongfeng Tunnel, and the result islisted in Table 4.

Be based on same argument, structural planes parameters of Muxidai Tunnel and YangwanTunnel can be selected respectively in Table 2 and 3, relevant optimization results of tunneltrend are listed in Table 4.

Table 4. Optimization Result of Three Tunnel’s Trend.

tunnel / result 95%(Confidence Level)

99%(Confidence Level)

Hongfeng Tunnel(condition: η > 2.0,G < 100 kN)

31◦ ∼ 55◦75◦ ∼ 133◦

31◦ ∼ 48◦75◦ ∼ 127◦

Muxidai Tunnel(condition: η > 2.0,G < 300 kN)

72◦ ∼ 97◦ (no)

Yangwan Tunnel(condition: η > 2.0,G < 100 kN)

0◦ ∼ 92◦175◦ ∼ 180◦

5◦ ∼ 92◦

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5. Conclusions

Using Block Theory and stereographic projection method built up, and combining the refor-matory measures of the Genetic Algorithm (GA) to optimize tunnel axis trend, we may safelydraw a conclusion:

• According to the instance of excavation construction investigation of the three tunnelswall rock of Jinliwen Freeway, they are basically agree with the simulation result, whichshow that Block Theory has the correctness and the affectivity. In fact, tunnel axis trendcan be optimized further, and find out more economical and reasonable routes. To takeHongfeng Tunnel for an example, the instance of tunnel actual axis trend could be anal-ysed by simulation result, the key block stability of the left room wall is worse than theright, and spot investigation also confirm this instance.5 If tunnel can’t blench, we shouldtake precautionary measures on construction, and sprayed concrete and take bolts toreinforce in time in the first lining time, avoiding result in a chain reaction for blockfalling.• In the time of programming and design, we have no excavation slope pattern, and

can’t find out the actual instance of generating mechanism of structural planes, den-sity and position, but we may statistically analyse according to geological prospectingdata, and get preponderant structural planes combination, average generating mecha-nism and spacing, and then select the best excavation orientation of tunnels axis, andfind the best design scheme of tunnels axis. Also we could forecast possible unstableblock types to face possibly, and potential needful measure to reinforce, which needenough preparation. When choosing tunnels routes, we still need consider ingredients tojudge synthetically on landform, physiognomy, groundwater, and so on, in order to reachthe economical, high effect, safe, reasonable aims.• At the stage of engineering excavation, adverse combination of structural planes and

tunnel excavation methods will have an direct impact on the stability of tunnel wall rockand overbreak questions, so we should ascertain combination forms of structural planesin reason, and forecast arisen probability of structural planes in time.5 In the course ofexcavation actual condition of rock mass structure will be continually exposed, and weshould feed back information and adjust construction scheme in time, even modulatetunnels axis trends if necessary, in order to increase the stability of tunnel wall rock andreduce the construction risk of tunnel.

Acknowledgements

Foundation item: Traffic science and technology project of Zhejiang Province (2004H27).Jinliwen Freeway Corporation of Zhejiang Province provided financial support to this

research.

References

1. Richard E. Goodman, Gen-hua Shi, “Block Theory and Its Application to Rock Engineering”,Englewood Cliffs, New Jersey: Prentice-Hall, Inc, 1985.

2. Liu Jinhua, Lu Zuheng, “Block Theory and Its Application to Stabilization Analysis on the Engi-neering Rock Mass”, Beijing of China, Water Resources and Electric Power Press, 5, 1985.

3. LI Duan-you, Gan Xiaoqing, Zhou Wu, “Back Analysis on Mechanical Parameters of Dams Basedon Uniform Design and Genetic Neural Network”, Chinese Journal of Geotechnical Engineering,1, 2007, pp. 126–127.

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4. Chen Guoliang, “Stereographic Projection Method and Its Applications”, Beijing of China, People’sPostal press, 1999, pp. 89–90.

5. Wei Jihong, Wu Jimin, Chen Xianchun, “Application of Block Theory on Overbreak Forecast inDouble-arch Tunnel”, Hydrogeological and Engineering Geological, 5, 2005, pp. 60–61.

324

Quarry Wall Stability Analysis Using Key Block Theory — aCase Study

LU BO∗, DINGXIULI AND DONGZHIHONG

Yangtze River Scientific Research Institute, Key Laboratory of Geotechnical Mechanics andEngineering of the Ministry of Water Resources, Wuhan, China, 430010

1. Introduction

An abandoned quarry is put into reuse for some reason. The norite rock mass in the regionis heavily jointed and contains a number of major and minor discontinuities. The persistenceof most of these major discontinuities is expected to between 10 m and 40 m. Therefore, theengineering problems of the quarry wall are very complicated. The stability of the quarrywall is obviously controlled by geological structure, i.e. the combination of joints. When therock mass of the quarry is further excavated for engineering construction, some blocks willdevelop sliding along joints, and then a chain reaction may occur. So, the key block theory isthe best suited tool for the stability analysis.

The paper is organized as follows. Section 2 is a brief introduction of the Key Block Theory.Site investigation results are introduced in section 3, and statistical analysis of collected rockjoint data acquired through field survey is performed. We delineated the discontinuities intosubgroups using the density contours of the pole on Schmidt net and identified the dominantjoint sets for later analysis. In section4, the block theory has been applied to identify thepotential unstable rock blocks at the surface of researched area. The orientations of themajor discontinuities that occur in the researched area have been considered in rock blockstability analysis. With the orientations of the dominant joint sets, stereographic projectionmethod is used to determine the maximum safety angle of the rock slope. Section 5 ends witha brief conclusion and some suggestions.

2. Key Block Theory — A Brief Introduction

The block theory is a new method to analyze rock mass stability with graph theory, settheory, and vector algebra, and it has been widely used in the past 30 years for the stabilityanalysis of rock mass. The underlying axiom of block theory is that failure of an excavationbegins at the boundary with the movement of a block into the excavated space. When therock mass is excavated, some blocks will develop sliding along discontinuities, and that maycause a chain reaction, eventually lead to the collapse of the whole rock mass. These initialblocks are called key blocks. Base on above thoughts “Goodman and Shi” proposed “blocktheory” (Goodman & Shi, 1985). In this theory, the principal assumptions follow.

a. All the joint surfaces are perfectly planar.b. Joint surfaces cut through the volume of interest.c. Joint surfaces are rigid.

The main idea behind block theory analysis is that it allows many different combinationsof discontinuities to be passed over and to directly identify and consider critical rock blockknown as "key blocks". Types of blocks can be divided into infinite and finite blocks. An




infinite block is not dangerous as long as it is incapable of internal cracking. Finite blockscan be classified into non-removable and removable blocks. Non-removable tapered block isfinite, but it cannot come out to free space because of its tapered shape.

2.1. Rock blocks defined by system of joints

A block is the region of intersection of half-space formed by the discontinuities that form theblock faces. Each discontinuity is described by two parameters: the dip angle α and the dipdirection β.

A particular block can be created by the intersection of the designated upper or lowerhalf-spaces corresponding to each of the discontinuities. The block corners are calculated asthe intersection points of three different planes. Only a few corners which are real actuallybelong to the considered block. The volume of any type of block can be calculated using thesimplex integration method (Shi Genhua, 1996).

There are five types of blocks in the block theory. An infinite block is of no hazard toan excavation. Finite blocks are divided into non-removable and removable types. A finiteblock may be non-removable because of its tapered shape. The other three are removableblocks. Their stability depends on the orientation of the resultant force, frictional resistanceof discontinuities and support implementation, etc.

2.2. Removability of blocks

The blocks are defined partly by discontinuity and rock slope half-spaces. The discontinuitysubset of the half-spaces determines the joint pyramid (JP). The set of slope half-spaces isdesignated as the excavation pyramid (EP). The block pyramid (BP) is then the intersectionof the JP and the EP for a particular block:

BP = JP ∩ EP. (1)

If the BP is empty (�), the block is infinite.

JP ∩ EP = �. (2)

Whether a finite block is removable or not is based on the following conditions. A blockis removable if its BP= � and JP�= � and becomes non-removable if its BP= � and JP= �.

2.3. Failure modes of removable blocks

Only removable blocks require further analysis. There are three failure modes considered.They are lifting (or falling), sliding on a single plane, and sliding on the intersection of twoplanes.

The lifting or falling mode occurs when there are no discontinuities in contact and thesliding direction is along the resultant force. In the case of sliding in a single plane, therewill be only one discontinuity in contact and the sliding direction is along the orthogonalprojection of the resultant force on that contact plane. As for sliding on the intersection oftwo planes, there are two discontinuities in contact and the sliding direction is along theintersection of those two planes. A fully kinematic analysis used to determine the slidingdirection of the removable blocks has been developed in the block theory.

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2.4. Stability analysis

From the kinematic analysis of failure modes for removable blocks, one can obtain therequired information for identifying the possible sliding conditions of the removable blocks.If the removable blocks for a given rock slope do not have any failure mode, they will bestable and safe.

On the basis of geometric information derived from morphology analysis and with themechanical parameters of the discontinuities, static equilibrium calculations are performed.The factor of safety (FOS) of a defined removable rock block will be given. If the FOS isbelow 1.0, then the removable block is viewed as a key block, and support force needed tokeep the block stable will be calculated.

3. Site Investigation

Site investigation helps to understand the geological condition of the quarry. Comprehensivegeological research for the quarry wall has been performed. The aim of site investigation isto ascertain the rock mass structure, and find out the potential unstable blocks.

3.1. Outline of geological condition

The dip angle of the quarry is about 90 degrees. The lithology of most rock masses is norite.The top of the quarry is covered by vegetations or soil. The thickness of the completelyweathered layer on the top of the quarry is about 3 m, and the thickness of the intensivelyweathered layer is about 5 m.

The quarry wall was formed by blasting. Because of blasting and unloading, the rockmasses were mostly loosened, and many discontinuities are opened. Most of the discontinu-ities are hard joints and others are weak joints that have infillings.

All the discontinuities that longer than 3m are mapped. The persistence of most of thesemajor discontinuities is expected to between 10 m and 40 m.

3.2. Method

The whole site investigation work was divided into 3 phases. The first step is to survey thearea where the height is less than 5m, the second step is to survey the area where the heightis between 5 m and 25 m, and the third step is to survey the area higher than 25m (Fig. 1).

The lower part of the quarry wall where the height is less than 5m is easy to deal with theaid of ladders or other simple facilities. So in the first step, all the joints longer than 3 m onthe bottom quarry were surveyed and recorded in detail.

Boom-lift was used to help to survey the joints on the quarry where the height is between5 m and 25 m. Almost all the joints longer than 5m expose on this area has been surveyedduring this step.

The boom-lift can only elevate the geologists to the height less than 25 m. So, as to thejoints which are on the area higher than 25 m, they can hardly be surveyed by contact mea-surement. Eyeballing and photograph method were used here. Then by comparing with thejoints on lower part, the properties of the joints on higher part were approximately deter-mined.

Scan window method was used to do the joint mapping, the size of each window is10 m×5 m. The quarry wall was divided into dozens of parts according to the windowssize and then the windows have been surveyed one by one. In each window, all the big joints

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estimated mapping

step

3st

ep2

step

1(boom lift)-

(ladder) 5m25

m

Figure 1. Site investigation was divided into 3 steps.

whose midpoints located in the window had been described and recorded. Almost all thejoints longer than 3m on the quarry wall have been surveyed.

3.3. Contents

With scan window method, almost all the discontinuities including joints, cracks, beddingplanes, faults, have been investigated. As to each discontinuity, the following properties havebeen recorded: orientation, aperture, length, infillings, roughness, weathering degree, under-ground water condition, and so on.

Except for discontinuities, the following geological conditions were also been recorded:(a). Height and dip angle of the quarry wall; (b) Depth of the covering layer; (c).Weatheringdegree and zoning; (d)Seepage condition on the quarry; (e) Detailed properties of controllingdiscontinuities (faults or other big joints longer than 25m); (f) Properties of veins.

What’s more, the geo-physical datum and drilling datum were also been collected andanalyzed.

3.4. Dominant joint sets

Statistical analysis of collected rock joint or discontinuity data acquired through field surveyhas been performed. All the joints were projected onto the Schmidt projection net, and thenthe discontinuities were delineated into subgroups using the density contours of the pole onSchmidt net in order to identify the dominant joint sets.

Figure 2 is the rose map of joint treads; it gives the distribution of the joints strike. Figure 3is the statistical histogram of the dip angles of all the joints, apparently, the joints are mainlysteep angled. The predominant strike of most joints are north-west, and the dip angles aremostly bigger than 60◦.

From Fig. 4 we can see that there have 3 group of dominant joint sets on the whole quarry,their dip direction and dip angles are 205◦� 39◦, 64◦� 46◦ and 6◦� 65◦.

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Figure 2. Rose map of the joint trends of all the joints.

Figure 3. Statistical histogram of the dip angles of all the joints.

4. Block Stability Analysis

In this section, we’ll just introduce the analysis result of one segment of the quarry wall.

4.1. Computational condition and method

The mechanical parameters needed are the friction angle and the cohesion of joints; theseparameters are estimated based on filed observations. Table 1 list the mechanical parametersof joints used in the following block stability analysis. For hard joint, the friction angle isabout 44.7 degrees, and the cohesion is 208 kPa. For weak joints, the friction angle variesfrom 14 to 37 degrees, and the cohesion varies from 20 to 130 kPa.

We perform parameter sensitivity analysis, the friction angle varies from 14 to 44.7 degrees,and in order to account the weakening effect of water on the joints, cohesion varies from 0to 208 kPa, for the cohesion is more sensitive to water than the friction angle, especially

Table 1. Mechanical parameters of joints.

Type Friction Angle (Deg) Cohesion (KPa)

Hard Joint 44.69 208.5Weak Joint 14∼37 20∼130

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Figure 4. Density contour map of the joints of all the joints.

the weak joints with infillings. Parameter sensitivity analysis is necessary, for the mechanicalparameters in table 1 are not based on test data. They are evaluated according to the charac-teristics of the joints, such as the JRC, aperture, infillings etc, and there is some engineeringexperience in it too. So, parameter sensitivity analysis will provide some help when determinethe joint mechanical parameters.

Gravity is the basic load case, and overlaying load is also considered if exists. The overlay-ing load will be simplified as uniform pressure exerted on the contact interface between theblock and the overlaying rock layer. The magnitude of pressure is γH, γ is the gravity densityof the overlaying rock layer, and H is vertical distance from the top of the quarry wall to thecontact interface. It is important to note that the stability analysis doesn’t consider the waterpressure that may be caused by raining and the dynamic force caused by earthquake.

According to the parameter sensitivity analysis results, we obtain the parameter combina-tion with which the FOS of each block is just 1 through back analysis. Then, with the newparameters, we calculate the minimum support force needed to ascertain the FOS of eachblock is above 1.3. The support is supposed to be perpendicular to the wall face. In mustbe pointed out that the parameter combinations with which the FOS is below 1.0 are notreasonable. The FOS is at least 1.0, for the rock blocks don’t develop sliding up to now.

4.2. Located rock blocks

Figures 5 and 6 show one segment of the quarry wall. Large joints outcropped here includingJa86, Ja82, Ja76 and Ja77 etc, and they form 8 adverse combinations. The above joints allextend to the top the quarry wall, so there is no overburden on the blocks here.(1) For the joint combination of Ja86, Ja82 and Ja87, the stereographic projection and mor-phology analysis results are shown in Fig. 7.

In Fig. 7(a), the “001” identify the block type, “0” for the upper half space of a joint planeand “1” for the lower half space of a joint plane. So,”001” signify a rock block that confined

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Figure 5. Sketch plot of one part of the quarry wall.

Figure 6. Image of the joints numbered Ja86, Ja76 etc).

by the upper half space of joint Ja86 and joint Ja82 and the lower the lower half space ofthe joint Ja87. The digit “12” below “001” signifies the failure mode of rock block. For thiscase, the block “001” will slide along joint Ja86 and joint Ja82.

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(a) Stereographic projection of Ja86, Ja82 and Ja87 (001, 12)

(b) shape of located rock block defined by Ja86, Ja82 , Ja87and slope surface

Figure 7. Key block confined by Ja86, Ja82 and Ja87 (lower part).

The volume of the located rock block defined by above joints and the slope surface is36.44 m3, and the failure mode of the block is double-face sliding, that’s along the intersec-tion line of Ja86 and Ja82. The areas of the two sliding faces are 41.26 m2 and 23.40 m2

respectively.(2) For the joint combination of Ja86, Ja82 and Ja87, the stereographic projection and mor-phology analysis results are shown in Fig. 8.

The volume of the located rock block defined by above joints and the slope surface (upperpart) is 170.58 m3, and the failure mode of the block is single-face sliding, that’s along jointJa87. The area of the sliding face is 17.80 m2.(3) For the joint combination of Ja76, Ja86, Ja81 and Ja82, the stereographic projection andmorphology analysis results are shown in Fig. 9.

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a. Stereographic projection of Ja86, Ja82, Ja87 (00 0, 3)

b. shape of located rock block defined by Ja86, Ja8 2, Ja87 and slope surface

Figure 8. Key block confined by Ja86, Ja82 and Ja87 (upper part).

The volume of the located rock block defined by above joints and the slope surface is74.27 m3, and the failure mode of the block is single-face sliding, that’s along joint Ja76. Thearea of the sliding face is 15.57 m2.(4) For the joint combination of Ja77, Ja76, Ja86, Ja81, Ja73, the stereographic projectionand morphology analysis results are shown in Fig. 10.

The volume of the located rock block defined by above joints and the slope surface is72.01 m3, and the failure mode of the block is double-face sliding, that along the intersectionline of Ja76 and Ja73. The areas of the two faces are 67.82 m2 and 11.55 m2 respectively.(5) For the joint combination of Ja77, Ja76, Ja81 and Ja73, the stereographic projection andmorphology analysis results are shown in Fig. 11.

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a. Stereographic projection of Ja76, Ja86, Ja81, Ja 82 (0001, 1)

b. shape of located rock block defined by Ja76, Ja86, Ja81, Ja82 and slope surface

Figure 9. Key block confined by Ja76, Ja86, Ja81 and Ja82.

The volume of the located rock block defined by above joints and the slope surface is195.44 m3, and the failure mode of the block is double-face sliding, that along the inter-section line of Ja77 and Ja73. The areas of the two sliding faces are 65.01 m2 and 7.10 m2

respectively.

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((a) Stereographic projection of Ja77, Ja76, Ja86, Ja81, Ja73 (10010, 25)

(b) shape of located rock block defined by Ja77,Ja76, Ja86, Ja81, Ja73 and slope surface

Figure 10. Key block confined by Ja77, Ja76, Ja86, Ja81 and Ja73.

(6) For the joint combination of Ja77, Ja76, Ja73 and Ja91, the stereographic projection andmorphology analysis results are shown in Fig. 12.

The volume of the located rock block defined by above joints and the slope surface is25.04 m3, and the failure mode of the block is double-face sliding, that along the intersectionline of Ja76and Ja91. The areas of the two sliding faces are 20.68 m2 and 8.12 m2 respec-tively.

335


(a) Stereographic projection of Ja77, Ja76, Ja81 and Ja73 (0010, 14)

(b) shape of located rock block defined by Ja77, Ja76, Ja81, Ja73 and slope surface


(7) For the joint combination of Ja77, Ja73 and Ja91, the stereographic projection and mor-phology analysis results are shown in Fig. 13.

The volume of the located rock block defined by above joints and the slope surface is8.97 m3, and the failure mode of the block is double-face sliding, that along the intersection

336


(a) Stereographic projection of Ja77, Ja76, Ja73, Ja91 (1010, 24)

(b) shape of located rock block defined by Ja77, Ja76, Ja73, Ja91 and slope surface

Fi 12 bl k fi d b 6 d


line of Ja77 and Ja91. The areas of the two sliding faces are 11.07 m2 and 1.00 m2 respec-tively.(8) For the joint combination of Ja77, Ja76, Ja91, the stereographic projection and morphol-ogy analysis results are shown in Fig. 14.

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(a) Stereographic projection of Ja77, Ja73, Ja91 (010, 13)

(b) shape of located rock block defined by Ja77, Ja73, Ja91 and slope surface

Figure 13. Key block confined by Ja77, Ja73 and Ja91.

The volume of the located rock block defined by above joints and the slope surface is17.15 m3, and the failure mode of the block is double-face sliding, that along the intersec-tion line of Ja76 and Ja91. The areas of the two sliding faces are 19.78 m2 and 8.12 m2

respectively.

4.3. Back analysis and support force

For convenience, the information of above removable rock blocks is collected in Table 2. TheFOS in this table is calculated by using the strength parameters of hard joints.

338


(a) Stereographic projection of Ja77, Ja76, Ja91 (101, 23)

(b) shape of located rock block defined by Ja77, Ja76, Ja91 and slope surface

Figure 14. Key block confined by Ja77, Ja76, and Ja91.

For each of the rock blocks, back analysis is performed to assess all possible parametercombinations which the FOS is 1.0. Then, with these new parameter combinations, supportforces needed to fit FOS=1.3 is obtained. The results are listed in Table 3.

4.4. Maximum safe slope angle

With the dominant joint sets acquired above, we determine all the unfavourable combina-tions of joints, according to the spatial relationship of the quarry wall surface and the domi-nant joint orientations. The orientations of the discontinuities and the excavation surface forthe segment mentioned above are listed in Table 4.

Figure 15 shows the identification of removable blocks for the segment of the quarry wallcorresponding to Fig. 6 using the stereographic projection. When the dip of the excavation

339


Figure 15. Identification of removable blocks for the quarry wall using stereographic projection.

Table 2. Information of removable rock blocks.

Block No. Block type /sliding mode Sliding plane Volume (m3) Area of sliding faces(m2) FOS

1 001,12 Ja86, Ja82 36.44 41.26 23.40 17.482 000, 3 Ja87 170.58 17.80 2.483 0001, 1 Ja76. 74.27 15.57 3.024 10010, 25 Ja76 , Ja73 72.01 67.82 11.55 14.225 0010, 14 Ja77 and Ja73 195.44 65.01 7.10 6.666 1010, 24 Ja77 and Ja91 25.04 20.68 8.12 14.887 010, 13 Ja77 and Ja91 8.97 11.07 1.00 18.808 101, 23 Ja76 and Ja91 17.15 19.78 8.12 20.63

surface is between 50◦ and 90◦, the JPs 001, and 101 provide removable blocks. For thissegment, the maximum safe slope angle is 50◦.

To have a design slope angle greater than the maximum safe slope angle, it is necessaryto perform a limit equilibrium analysis incorporating all the forces to estimate the requiredsupport system.

5. Conclusions

From the above analysis, we can see that if the mechanical parameters of hard joints areused, the removable blocks are stable and there is still a certain safety margin. As a matter ofexperience, the block defined by Ja86, Ja82 and Ja87 (the upper part) and the block definedby Ja76, Ja86 Ja81 and Ja82 are the most dangerous, the safety margin is relatively small.

Still there are considerable loosened or detached rock blocks on the quarry surface andon the top of wall. So, a thoroughly clearance aiming at above unstable blocks is stronglyrecommended. In addition, systematically bolt- shotcrete support is necessary for engineeringsafety.

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Table 3. Support force needed of rock block based on the parameters from back analysis.

Block No.FOS=1.0

Support force Needed (FOS=1.3) (kN)Friction Angle (Degree) Cohesion (kPa)

1 43.0 0 160

2

40.0 0 84035.3 25 90030.0 50 98024.2 75 100017.5 100 1000

341.3 25 10029.4 50 10014.1 75 100

4 31.1 0 2405 24.8 0 6306 30.7 0 857 24.5 0 308 30.1 0 60

Table 4. Orientation of Dominant of joint sets.

No Dip Direction (Deg) Dip Angle (Deg)

1 205◦ 39◦2 64◦ 46◦3 6◦ 65◦

Quarry Surface 100◦ 50◦

The degree of accuracy of the analysis is largely dependent on the mechanical parametersand the information of discontinuities and on what extent the principal assumptions of blocktheory deviate from the actual situation.

Current mechanical parameters of joints are estimated based on field observation. It’sinsufficient for large or important project. Specific in-situ mechanical testing work shouldbe carried out, especially for the controlling large joints.

Acknowledgements

This research is supported by the Ministry of Water Resources, PRC, under the contractsof No. XDS2007-10. The authors thank to Dr. Gen-hua Shi for his valuable advices in thefield of DDA engineering applications. The authors also give thanks to Dr. ZhangYihu andPangZhengjiang for their help with the site investigation.

References

1. R.E. Goodman, Gen-hua Shi. Block Theory and Its Application to Rock Engineering. Prentice-Hall, Inc, Englewood Cliffs, New Jersey. 1985, 1st edition.

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2. Shi Genhua. Simplex integration for manifold method, FEM, DDA and analytical analysis. In:SalamiMR, Banks D (eds.) Discontinuous Deformation Analysis (DDA) and Simulations of Dis-continuous Media, TSI Press, Albuquerque, New Mexico, USA, 1996, pp. 205–262.

342

Probabilistic Key Block Analysis of a Mine Ventilation ShaftStability — a Case Study

GANG CHEN∗

Department of Mining and Geological Engineering, University of Alaska Fairbanks,Fairbanks, Alaska, USA

1. Introduction

The stability of rock excavations depends on many factors, such as mechanical propertiesof rock masses, orientations and properties of rock discontinuities, hydrological conditions,excavation geometries, and others. In an excavation stability analysis, these parameters areacquired by either in-situ investigations and/or laboratory tests. Because of the complexnature of rock masses, these parameters do not usually appear to be uniquely valued, butvary over a certain range, causing uncertainties in rock excavation stability estimation. Theuncertainty associated with rock excavation stability has been brought to the attention ofmany investigators (Chen, et al, 1997 & 1998; Mauldon, et al, 1997; Tyler, et al, 1991; Hat-zor and Goodman, 1992; Rethati, 1988; Piteau, 1977). It has been recognized that employ-ing probabilistic analysis to determine the stability of a rock excavation may provide a betterunderstanding and lead to more rigorous designs of rock excavations.

A vertical ventilation shaft was developed in an underground mine in Alaska, USA. Theshaft, excavated by a raise borer, is approximately 300 feet in length and 3.5 feet in diam-eter. The surrounding rock is granite with several sets of joints. The discontinuities in therock mass had major impacts on the stability of the shaft excavation. Initial static limiting-equilibrium key block analysis indicated that the shaft should be stable without support.In-depth probabilistic analysis, however, showed that due to random variations in both jointorientations and rock properties, there was a high probability of key block failure with sig-nificant rock volumes.

This paper presents the failure probability study of the ventilation shaft. In the study, theblock theory developed by Goodman and Shi (1985) was applied with the considerationof uncertainties in rock discontinuities and rock mass properties. Probabilistic analysis wasapplied to evaluate the likelihood of a key block occurrence and its failure probability basedon the given distribution of the discontinuity orientations and the mechanical properties ofthe discontinuities. The first-order second-moment approximation of the probabilities wasutilized in the study to simplify the computation. In combination with Monte Carlo simu-lations, analyses were also conducted on the probabilistic safety factor distribution and thepotential rock sliding volume distribution.

2. Probability of Key Block Formation

Mine site investigation revealed that there were 3 major sets of joints in the rock mass sur-rounding the ventilation shaft. The variations in joint orientations may fit in either Betadistribution or the well-known Fisher distribution. In this study, only the Beta distribution




was employed. The density function of Beta distribution is given by:

fX(x) = 1B(q,r)

(x− a)q−1(b− x)r−1

(b− a)q+r−1a � x � b. (1)

In the density function, the parameters a and b define the range of the random variablevariation, and the parameters q and r define the shape of the distribution.

The mean orientations of the joint sets are plotted on a stereographic net as shown in Fig. 1.As can be seen on the plot (without considering those less likely repeated blocks), there are 8block types, namely, 000, 001, 010, 011, 100, 101, 110, and 111. Since the ventilation shaftis vertical, the vector of the upward shaft axis is at the center of the reference circle (dashedline) and the vector of the downward shaft axis is infinitely away outside the reference circle,which can not be plotted. Based on the block theory, block types 000 and 111 are infiniteblocks and the other block types: 001, 010, 011, 100, 101 and 110 are potential key blocks.Preliminary analysis indicated that, block types 010, 011, 100 and 101 would have minimalvolume of caving and should not be of any major concern for the shaft stability. The analysiswas, therefore, focused on the key block types 001 and 110, which would have relativelylarge volume of rock failure.

In order for block 001 or 110 to become a key block, each block should have a down-ward sliding direction. As shown in Fig. 1, the sliding direction for Block 001 is along theintersection line of J1 and J2 (I12). It is projected outside of the reference circle indicatinga downward vector. For Block 110, it is along the intersection line of J2 and J3 (I23), alsoprojected outside of the reference circle to point downward. Assuming the joint plane ori-entations are random variables, I12 and I23 may have a certain probabilities to fall withinthe reference circle and become upward vectors and Blocks 001 and 110 would, therefore,become stable blocks. The probability of this occurs is discussed below.

Figure 1. Great circle plot of the mean orientations of joint sets.

344


A pair of joint planes passing through the origin can be defined as:

A1X+ B1Y+ C1Z = 0 and A1 = sinα1 sinβ1; B1 = sinα1 cosβ1; C1 = cosα1

A2X+ B2Y+ C2Z = 0 and A2 = sinα2 sinβ2; B2 = sinα2 cosβ2; C2 = cosα2

where α1 and α2 are the dip angles for the joint planes 1 and 2; and β1 and β2 are the dipdirections for the two planes respectively. Since the dip angles and the dip directions arerandom variables, the six parameters given by these angles are also random variables thatdefine two randomly distributed planes in the rock mass.

The intersection line of these two planes is a line passing through the origin (0, 0, 0) andcan be defined as:

X = pt; Y = qt; Z = rt (2)

and: p = sinα1 cosβ1 cosα2−cosα1 sinα2 cos β2; q = cosα1 sin α2 sin β2−sinα1 sinβ1 cosα2

r = sinα1 sin β1 sinα2 cosβ2 − sin α1 cosβ1 sin α2 sin β2

where p, q and r are directional numbers defining the orientation of the line. The variablet is a reference variable and varies in the range of 0 < t < ∞. The directional numbers, p,q and r, are random variables defined by the parameters of the two random joint planes asgiven above. Therefore, the probability that the sliding vector pointing upward, representinga stable block, can be defined as:

P[block stable] = P[rt > 0]. (3)

Since the parameters p, q and r are all nonlinear functions of multi random variables, α1,α2, β1 and β2, applying direct integration to compute the probability is formidable. Anapproximation technique with a simplified computation procedure has to be adopted inorder to determine the probability of a block being stable. In this study, the First OrderSecond Moment (FOSM) approach is employed.

3. First-Order-Second-Moment Approach for Probability Estimation

The probability as given in Equation 3 involves a random-variable parameter r, which is afunction of random variables, α1, α2, β1 and β2. The calculation of the probability, therefore,requires multivariable integration, which is in a fairly formidable form and the close-formedsolution may not exist. In order to overcome this difficulty, the First Order Second Moment,i.e. the FOSM approximation technique is employed to estimate the probability of a blockbeing stable. The FOSM method is delineated below.

For practical problems of rock excavation stability, the available information is frequentlylimited to the means and variances of parameters that are regarded as random variables, thatis, the first and second moments of the random variables. Under this condition, the estimationof the probability of a block being removable is limited to a formulation based on the firstand second moments of the random variables, i. e. the second-moment formulation.

Based on rock stable probability as given in Equation 3, the performance function g canbe defined as (let t = 1):

g = r = sinα1 sinβ1 sin α2 cosβ2 − sinα1 cosβ1 sinα2 sinβ2 (4)

If g > 0, the rock block is stable and the rock excavation being studied is in the safe state.Whereas if g < 0, the block is a potential key block. Since r is a function of several random

345


variables, g is also a function of those random variables, i.e.:

g(x) = g(x1,x2, . . . ,xn) (5)

The equation g(x) = 0 defines the boundary between the region of safe state and that offailure state. When g(x) is a linear function, g(x) = 0 is a flat plane in an n-dimensional space.The distance from g(x) = 0 line to the origin is defined as the safety index β, which can beestimated by the following equation (Ang and Tang, 1984):

β = μx1 − μx2√σ 2

x1+ σ 2

x2

(6)

where: μx1 , μx2 — the mean values of the random variables; σx1 , σx2 — the standard devia-tions of the random variables.

If the random variables have normal distributions, the probability can then be estimatedby the standard normal distribution as below (Ang and Tang, 1984):

P = 1−�(β). (7)

For non-normal distributions, such as the beta distribution used in this study, transforma-tions can be performed to obtain equivalent normal distributions. The mean value and thestandard deviation of the equivalent normal distribution for a random variable Xi can beestimated as:

μNXi= x∗i − σN

Xi�−1[FXi(x

∗i )] (8a)

σNXi= φ{�−1[FXi(x

∗i )]}

fxi(x∗i )

(8b)

where: μNXi

, σNXi

- the mean value and standard deviation of the equivalent normal distribu-tion for Xi; fXi(x

∗i ), FXi (x

∗i ) - the original PDF and CDF of the random variable Xi evaluated

at x∗i ; φ(-), �(-) – the PDF and CDF of the standard normal distribution.If the performance function g(x) is nonlinear, as in the case presented in this study, the equa-

tion g(x) = 0 is a curved surface in the space. The curve can be either convex or concave asshown in the figure. To calculate the exact probability of g(x) < 0 for nonlinear performancefunction, it generally requires complex integrations. For practical purposes, approximationto the exact probability is necessary. According to Shinozuka (1983), the point (X∗) on thefailure surface g(x) = 0 with the minimum distance to the origin is the most probable failurepoint. Taking a Taylor series of the function at (X∗) and truncating at the first-order termproduce a tangent plane at (X∗). This tangent plane to the failure surface may then be usedto approximate the actual failure surface and the safety index β may be evaluated as in thelinear case. This first-order approximation of g(x) = 0 at X∗ = (X∗1,X∗2, . . .X∗n) is expressedas (Ang and Tang, 1984):

n∑i=1

(xi −X∗i )(∂g∂xi

)= 0. (9)

The most probable failure point is (Shinozuka, 1983):

X∗i = −α∗i β (10)

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Table 1. Orientation distribution parameters for joint planes.

Mean (dip/dip direction)Beta Distribution Parameters

a b q r

J1 73◦/116◦ 66◦/109◦ 80◦/123◦ 9.5/10.0 9.0/9.0J2 85◦/272◦ 78◦/265◦ 92◦/279◦ 10.5/9.5 10.0/9.0J3 30◦/285◦ 23◦/278◦ 37◦/292◦ 8.0/8.0 9.0/9.5φ 32◦ 0◦ 51◦ 4 4

Note: φ - frictional angle for all the surfaces

where α∗i is evaluated by:

α∗i =(∂g∂xi

)∗√

n∑i=1

(∂g∂xi

)2

∗

(11)

All the derivatives are evaluated at (X∗) and the solution of g(X∗) = 0 yields β. The valueof β can be evaluated by going through several iterations with a simple numerical algorithmproposed by Rackwitz (1976) and the probability of failure is then estimated by Equation (7).Due to the linear approximation of the nonlinear function, the failure probability is eitherslightly overestimated or underestimated depending on convexity of the function, but in mostcases a sufficiently accurate approximation can be obtained.

With the FOSM approximation, probabilistic analyses for more complex and practicalproblems can be performed without much difficulty. By adopting this approach, a FOR-TRAN90 program, RBK.F90 was developed employing the algorithm proposed by Rackwitz(1976) to estimate the stability probability of rock blocks. Running the program with theperformance function in Equation 4 and the parameters provided in Table 1 produced verylow probability for either block 001 or block 110 (0.0 and 0.00001 respectively) to be stablenon-key block. Both blocks, therefore, were treated as potential key blocks with certainty.

4. Failure Probability of Key Blocks and Probabilistic Distribution of BlockVolumes

The potential key blocks of 001 and 110 are approximately tetrahedrons formed by threejoint planes, namely, J1, J2, J3 and the shaft excavation surface. As can be determined on thestereographic plot in Fig. 1, both blocks may potentially slide on two joint planes. Block 001may slide on J1 and J2, and Block 110 may slide on J2 and J3. Based on the block theory(Goodman and Shi, 1985) for double face sliding, the sliding force, T, can be calculated by:

T =∣∣−→w · (−→n i ×−→n j)

∣∣∣∣−→n i ×−→n j∣∣ (12)

and the resistance force, R, due to friction is given by:

R = Ni tanφi +Nj tanφj (13)

where: −→w is the resultant force vector acting on the rock block (weight only in this study),−→ni and −→nj are the normal vectors of joint planes i and j respectively, Ni and Nj are normal

347


forces on planes i and j respectively, and φi and φj are frictional angles for planes i and jrespectively.

Considering only the dry friction of the rock blocks for resistance, the safety factors can,then, be calculated by:

SF = R/T (14)

Using the mean values in Table 1, the safety factors were calculated for the two concernedblocks of 001 and 110 to be:

SF(001) = 1.1 and SF(110) = 6.8.

Both blocks seemed stable. However, all the parameters listed above, including the fric-tional angles are either a random variable itself or a function of several random variables.The failure probability of a key block is, therefore:

Pf = P(T − R > 0). (15)

Employing the FOSM procedure discussed in the last section, the performance functioncan be defined as:

g = T − R =∣∣−→w · (−→n i ×−→n j)

∣∣∣∣−→n i ×−→n j∣∣ −Ni tanφi −Nj tanφj (16)

Running the FORTRAN90 program, RBK.F90 as discussed in the previous section, thefailure probabilities for the two blocks were estimated to be:

Pf (001) = 0.485 and Pf (110) = 0.0012.

The analysis indicated that block 110 was of much less concern in terms of its stability. Ithad a very high safety factor of 6.8 and very low failure probability of 0.0012, and should bestable without support. The most critical block, however, was block 001. Although the safetyfactor is above 1.0, further probabilistic analysis showed a significant failure potential of48.5%. To further verify the probabilistic analysis, Monte Carlo simulations were performedusing a platform package of “R”, a popular statistic analysis package, to check the failureprobabilities and examine the distribution of safety factors. With 100,000 runs, the MonteCarlo simulation produced similar failure probabilities as given by the program RBK.F90:

Pf (001) = 0.475 ∼ 0.49 and Pf (110) = 0.001 ∼ 0.0013.

The distribution of safety factors generated by the simulation are also shown in Figs. 2(a)and 2(b). The results clearly indicated that for Block 001, there is a significant chance for thesafety factor to drop below 1.0, while Block 110 is basically stable with much higher safetyfactor.

Another critical data in excavation stability analysis is the potential volume of rock blockfailure. Following the procedure proposed by Goodman and Shi (1985), it was identified thatthe potential maximum key block for both 001 and 110 would be formed by limiting planesalong the projection lines of edges I13 and I23. The maximum key block can approximatelybe modeled as a tetrahedron, formed by the three joint planes, J1, J2, J3 and the excava-tion surface, approximated to be a flat cut surface. Since J1, J2, J3 and the limiting edgesI13 and I23 are all functions of random variables of the joint orientations, the maximumpotential rock failure volume also follows a random distribution. Following Goodman andShi’s procedure and applying the random variable parameters listed in Table 1. The prob-abilistic distribution of potential rock failure volumes was estimated in the Monte Carlosimulation procedure and was illustrated in Fig. 3. The results revealed that the key block

348


0

2000

4000

6000

8000

10000

12000

00.2

0.4

0.6

0.8 1

1.2

1.4

1.6

1.8 2

2.2

2.4

Safety Factor

Fre

qu

en

cy

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

01.5 3

4.5 6

7.5 9

10.5 12 13

.5 15 16.5 18 19

.5

Safety Factor

Fre

qu

en

cy

(a) (b)

Figure 2. (a) Block 001 safety factor distribution (b) Block 110 safety factor distribution

0

2000

4000

6000

8000

10000

12000

14000

16000

200

240

280

320

360

400

440

480

520

560

600

Max Key Block Volume (cubic ft)

Fre

qu

en

cy

Figure 3. Probabilistic distribution of max key block volume.

had a potentially sizable volume, most likely in the range of 300 to 450 cubic feet. This isa fairly significant volume and special attention has to be paid to the support and stabilitycontrol of the excavation. Without proper control and support of the potential key blocks,the sliding of the blocks may cause sever damage and/or serious personal injury.


In this study, probabilistic analysis was applied to evaluate the stability of a mine ventilationshaft. The FOSM approximation procedure was applied in estimating the probability ofkey block formations and key block failure probabilities. In conjunction with Monte Carlosimulations, the probabilistic distribution of key block safety factors and the probabilisticdistribution of potential maximum key block volumes were also estimated.

The study indicated that taking into account of the uncertainties in rock structure orienta-tions and rock mass properties, the probabilistic analysis provided a more complete under-standing of the mine excavation stability as compared to the conventional deterministic anal-ysis. Although the deterministic analysis in this ventilation shaft stability study resulted in a

349


safety factor of 1.1 for the stability of the most critical key block of 001, the probabilisticanalysis revealed a considerable failure probability of 48.5% with a potentially significantmaximum key block volume of 300 to 450 cubic feet.

The conclusion of the probabilistic study should have prompted special attention to thekey block control and support in the excavation to avoid sever property damage and pre-vent possible personal injuries. However, in-depth probabilistic analysis was not properlycarried out before the development of the ventilation shaft and there was significant delaybetween the completion of the raise boring excavation and the application of the concretelining support. The bottom half of the shaft collapsed two months after the completion ofthe excavation. Although no personal injury occurred at the time of rock failure, the signif-icant caving rock volume forced re-development of the ventilation shaft, causing significantproduction delay and unexpected cost.

References

1. Ang, A.H.S. and W.H. Tang, 1984. Probability Concepts in Engineering Planning and Design, Vol.II, John Willey & Sons, New York, USA.

2. Baecher, G.B. and J.T. Christian, 2003. Reliability and Statistics in Geotechnical Engineering, JohnWilley & Sons, New York, USA.

3. Chen, G., et al 1998. Probabilistic Analysis of Rock Slope Stability with First-Order Approxima-tion, International Journal of Surface Mining and Reclamation, Vol. 12, No. 1, pp. 11-17.

4. Chen, G., et al 1997. Probabilistic Analysis of Underground Excavation Stability, InternationalJournal of Rock Mechanics and Mining Sciences, Vol. 34, No. 3–4, pp. 6.

5. Fisher, R. 1953. Dispersion on a sphere, Proceedings of the Royal Society of London, A217, 295-350.

6. Goodman, R.E. and G.H. Shi, 1985. Block Theory and Its Application to Rock Engineering,Prentice-Hall, Englewood Cliffs, NJ, USA.

7. Hatzor, Y. and R.E. Goodman, 1992. Application of block theory and the critical key block con-cept in tunneling: two case histories. In Proc. Int. Soc. Rock Mech. Conf. on Fractured and JointedRock Masses, Lake Tahoe, California, 632-639.

8. Mauldon, et al, 1997. Limit analysis of 2-D tunnel key-blocks. International Journal of RockMechanics and Mining Sciences, 34(3–4):193.

9. Piteau, D.R. and D.C. Martin, 1977. Slope stability analysis and design based on probabilitytechniques at Cassiar mine. CIM Bulletin, March, pp. 139-150.

10. Rackwitz, R. 1976. Practical probabilistic approach to design. Bulletin 112, Comite European duBeton, Paris, France.

11. Rethati, L. 1988. Probabilistic Solutions in Geotechnics, Elsevier Science Publishing Co., Inc., NewYork, USA.

12. Tyler, D.B., et al 1991. Rockbolt support design using a probabilistic method of key block analysis.In Rock Mechanics as a Multidisciplinary Science, J.C. Roegiers, Editor, Rotterdam: Balkema,1037–104.

350

The Support Design for Slope and Tunnel Engineering Based onBlock Theory

JIAO LIQING, MA GUOWEI∗, HE LEI AND FU GUOYANG

School of Civil & Environmental Engineering, Nanyang Technological University, Singapore

1. Introduction

The support design is of great importance in the fields of underground projects, such as slopeexcavation and tunnel engineering. So it is important to develop rules for the performancechecking of the support design. Till now, many models have been developed to simulatethe rock mass system, and several rules have been developed to check the performance ofthe system. The Q-system and rock mass rating (RMR) are the traditional methods usedin underground projects. The Q-system of rock mass classification was developed by Bar-ton, Lien and Lunde in 1974. In Q-system rock mass classification, six parameters includingDeere’s Rock Quality Designation (RQD), joint set number (Jn), joint roughness number (Jr),joint alteration or filling (Ja), joint water leakage or pressure (Jw) and Stress Reduction Factor(SRF), are used to characterize the rock mass quality. Three key aspects of the rock mass suchas block size (RQD/Jn), inter block shear strength (Jr/Ja) and active stress (Jw/SRF) are com-prehensively considered by the Q-system. The RMR rock mass classification was developedby Bieniawski (1976). In RMR classification, six parameters including uniaxial compressivestrength of intact rock material, Rock Quality Designation (RQD), joint spacing, joint condi-tion, ground water condition, joint orientation, are used to determine the rock mass quality.

The six parameters, in Q-system and RMR rock mass classification, can cover almostall the rock mass classifications. However, it’s really difficult to exclude the value of theseparameters from one specimen, and only geological experts can confirm the value of all thesix parameters. So it’s necessary to develop a theory to express the rock mass system witheasy-to-obtain parameters.

Block theory, developed by Genhua Shi in 1985, can be used to check the performanceof the support design. Using the block theory, five parameters including cohesion, friction,dip and dip direction of joints and rock density, are used to characterize the performanceof the rock mass system. The values of these parameters are easy to get and it is unique toone specimen. Only the instability of the block should be considered. So block theory is apromising method to check the performance of the support design.

In this paper, the block theory was used to study the support design in undergroundprojects. It is easier to check the performance of the support design compared with thetraditional methods, which may be significant in the application of the slope and tunnelengineering in underground projects.

2. Support Design Investigation in Tunnel and Slope Engineering UsingBlock Theory

This present study focuses on the stability analysis of keyblock. The excavations, which canskip over many conceivable combinations of joints and proceed directly to consider certaincritical blocks, are named as keyblocks. The keyblock is potentially critical to the stability




of an excavation because by definition, it is finite, removable, and potentially unstable. Itmust be made up of the joints and free surfaces. A finite, convex block is removable or notremovable according to its shape relative to the excavation. We have previously termed anonremovable finite block as tapered. Necessary and sufficient conditions for the removabil-ity or nonremovability of a finite block are established by the following theorem. A convexblock is removable if it has the free surfaces and its joints plane can not make a block.

We establish relationships connecting the direction of the resultant force, on an incipientlysliding block, and the direction of sliding. Coupled with other kinematic constraints and aspecific direction for the resultant force, these rules will permit us to establish which, if any,sliding mode is applicable to each block. There are three modes of sliding acting on the blockif it can be removed. The direction and the value of the three modes of sliding are definedshown as below by Genhua Shi in 1985.

Lifting:

s = r F = |r|. (1)

Sliding on a single face:

si = (ni × r)× ni/|ni × r| F = |ni × r| − |ni · r| tanφi. (2)

Sliding on two faces:

sij = (ni × nj)sign((ni × nj) · r)/|ni × nj| (3)

F = [|r · (ni × nj)||ni × nj|− |(r× nj) · (ni × nj)| tanφi − |(r× ni) · (ni × nj)| tanφj]]/|ni × nj|2 (4)

s, si, sij are the directions of sliding acting on the block the sliding direction si is the ortho-graphic projection of r on plane i, sij is the direction along the line of intersection of twoplanes i and j that makes an acute angle with the direction of the active resultant r, r is theresultant force of all other forces acting on the block, F is the value of sliding forces acting onthe block, ni, nj are the upward normal vectors to plane i ,j respectively, φi, φj are the frictionangles of joints i, j respectively.

If the normal force from the adjacent block through the joints and the cohesion acting onthe joint planes are considered, the direction and the value of the three modes of sliding areredefined as follow.

Lifting:

s = r F = |r| +∑

Fl

∑Fl ∗ tanφl. (5)

Sliding on a single face:

si = (ni × r)× ni/|ni × r| (6)

F = |ni × r| − |ni · r| tanφi +∑

Fl −∑

Fl ∗ tanφl − ci ∗ Ai. (7)

Sliding on two faces:

sij = (ni × nj)sign((ni × nj) · r)/|ni × nj| (8)

F = [|r · (ni × nj)||ni × nj| − |(r× nj) · (ni × nj)| tanφi

− |(r× ni) · (ni × nj)| tanφj]]/|ni × nj|2 +∑

Fl −∑

Fl ∗ tanφl − ci ∗ Ai − cj ∗ Aj (9)∑Fl is the force from all joint planes which made up the block, the direction is normal of

its joint plane, φl is the friction angle of joint planes which made up the block, ci, cj are

352


the cohesions of joint i, j respectively, Ai, Aj are the areas of joint i, j acting on this blockrespectively.

The merits of Q-system and RMR are that they can almost cover all rock mass conditionsbased on the individual required six parameters of rock mass. Their shortage however isthat determining the values of the parameters requires intensive geological expert and theexclusive value of them from one specimen is difficult. For block theory, the advantage isobvious. To support design based on block theory requires only cohesion, friction, the dipand dip direction of joints and density of rock. The values of these engineering parameters arerelatively easy to obtain, and are unique for an individual specimen. Only the unstable blockidentified by the block theory requires support However, the block theory is not adapted tothe excavation around the weak rock mass. Due to the deficiencies of traditional supportdesigns such as Q-system or RMR, the block theory is also recommended to use in checkingthe traditional rock support and reinforcement design.

3. Discussion About Support Design in Slope and Tunnel Engineering

The slope engineering is shown in Fig. 1. The keyblocks of the slope engineering will slidewithout support after excavation computed as shown in Fig. 2. The all chromatic blocks willbe detached from the main rock mass without the support.

These blocks are called the batch time sliding blocks as shown in Fig. 3. If they are sup-ported by enough support force, the overall slope will be safe.

Each individual potential sliding block identified in Fig. 2 are listed in Fig. 4. The firsteight blocks are the first batch potential sliding blocks from (1) to (8). The next six blocksand the last two blocks are the second batch potential sliding blocks from (9) to (14) and thethird batch respectively, which will slide without any support after the first batch blocks slideWhen each joint plane friction and the cohesion are equal to zero, the volume, the potentialsliding force, sliding direction of blocks, the support pressure and the support direction arecomputed and summarized in Table 1. The slope will be safe if the support design is basedon the block theory

Figure 1. A rock slope model.

353


Figure 2. The potential sliding blocks in the slope.

Figure 3. The first batch potential sliding blocks.

The density of rock mass is 2.0×103kg/m3, the gravity acceleration is 9.8 m/s2 the fric-tion is zero, the cohesion is zero. The direction of support is the vector of (0.995, −0.000,−0.100).

A tunnel model with various joint planes is shown in Fig 5. The potential sliding blocks ofthe tunnel model will slide without support after excavation as shown in Fig. 6 Due to theorientation of the joins, the potential sliding blocks are concentrated in the top and right sideof the tunnel excavation.

In the traditional support analysis, if the support pressure is not enough to support thekeyblocks which are the first batch sliding blocks, the decrease of the spacing of the rockbolts and the increase of the number of rock bolts are usually recommended for the safetyof project. However, it is not necessary for the other blocks, which are inherently stableIn theory, the use of the block theory can facilitate the identification of keyblocks duringunderground excavation. It is also necessary to point out the essence of reinforcing the secondbatch or the following sliding blocks by adjusting the support force acting on the keyblocks.

354


(1) (2)

(3) (4)

(5) (6)

(7) (8)

(9) (10)

355


(13) (14)

(15) (16)

(11) (12)

Figure 4. The potential sliding blocks.

Figure 5. The tunnel model.

356


Table 1. The information of potential sliding blocks.

Thenumberblock

The sliding force (kN) The sliding directionvector

The volume ofblock (m3)

Thesupportpressure(kPa)

1 194328 0.100 0.000 0.995 100 41981.92 61152 0.100 0.000 0.995 31 325276.93 165194 0.100 0.000 0.995 85 45249.44 107312 0.100 0.000 0.995 55 45249.45 32395 0.100 0.000 0.995 17 45249.46 191689 0.100 0.000 0.995 98 113430.17 79093 0.100 0.000 0.995 41 330564.88 93909 0.100 0.000 0.995 48 56599.99 21140 0.100 0.000 0.995 11 56599.910 181233 0.100 0.000 0.995 93 56599.911 106188 0.100 0.000 0.995 54 40848.712 600275 0.707 0.707 0.000 43 40848.713 239486 1.000 0.000 0.000 12 1738683.614 403603 0.100 0.000 0.995 207 1831515 595424 1.000 0.000 0.000 30 40950.916 156731 1.000 0.000 0.000 8 28952.9

Figure 6. The potential sliding block of tunnel model.

4. Conclusions

In the present study, the advantages and shortages of Q-system RMR and block theory havebeen briefly discussed. Based on the block theory, the optimised support design combinedwith the traditional support design is given. The resultant sliding force and direction of forceacting on any potential sliding blocks are computed by the developed code. In the meanwhile,the support pressure and direction can also be obtained. The support system can then bedesigned for each individual potential sliding. To achieve an effective and economical rock

357


support design for slope and tunnel engineering projects, it is recommended to combine thetraditional support based on Q-system and RMR with the block theory.

References

1. N.Barton, R.Lien and J.Lunde. “Engineering Classification of Rock Masses for the Design of TunnelSupport”. Norwegian Geotechnical Insitute. NR.106. 1974

2. N.Barton. Some New Q-value Correlations to Assist in Site Characterisation and Tunnel Design.International Journal of Rock Mechanics and Mining Sciences. p185-216.2002

3. Bhawani Singh. Rock Mass Classification-A Practical Approach in Civil Engineering.19994. Bieniawski Z. T. Rock Mechanics Design in Mining and Tunnelling p272. Balkema, Rotterdam19845. Richard E. Goodman and Shi Genhua. Block Theory and its Application to Rock Engineering.

p295-330.1985

358

Hereditary Problems in Long-wall Mining by Free Hexagons

P.P. PROCHAZKA∗ AND KAMILA WEIGLOVA

Association of Czech Civil Engineers, Prague, Czech Republic, Technical University in Brno

1. Introduction

In the paper time dependent free hexagon DEM is formulated and solved for the stability offaces during the long-wall mining. The time factor is involved in the natural way into themodel of discrete elements created by the boundary element method. Generally, the mainapplication of the approach put forward is found in geomechanics, namely in bumps occur-rence in deep mines. In the deep mines the way of depositing packs and its mechanical prop-erties are decisive. One of the most important phenomena for long-wall mining, for example,is the velocity of excavation. Their mutual coupling (the velocity and the way of deposition ofpacks) can principally influence the safety against bumps and the appearance of the bumpsis mostly caused by this phenomenon. For correct understanding the behavior of the rockaggregate (coal seam vs. overburden) nucleation of cracks finally leading to bumps has to betreated as time depended and certain hereditary problems are to be solved, such as creep andvisco-plasticity mainly in the overburden. According to new experiments and results fromaccessible literature and on scale models those effects will be involved into the formulation.

Contact problems created in terms of the free hexagons describing the bumps occurrencein deep mines have been solved in certain papers of the author for statical and dynamicalcase.1, 2 Either lagrangian multipliers,3 or penalty formulation can be used in formulatingthe interface conditions between particles. New formulation is submitted in terms of penalty,which if high enough (bond effect of adjacent elements), it suppresses the influence of debond-ing forces and debond occurs otherwise. The second case is an impact of rearranged forcesinside of the underground mass due to time dependent behavior. Involving then the inter-face properties, complex nucleation can be studied and improve the information on possiblerock bursts. The mechanical behavior of the rock or coal is very important. A comprehen-sive book involving the visco-plastic properties of rock was issued by Elsevier.4 Results fromobservations of creeping rock in uniaxial compression regime are accessible in Ref. 5. For thepurpose of this paper description of the visco-plastic development is taken from Ref. 6. Theresults are accepted from Kunming location, China. Some examples will show the applicationof the procedure proposed.

2. The Creep Model and Basic Equations

In the classical theory of plastic deformation the following basic assumptions are oftenadopted:

1. A change of a body’s shape or an increment in this change is regarded as being broughtabout by a stress deviator and is not influenced by the spherical stress tensor (classicalassumption in the theory of plastic deformations).

2. On the other hand, the change of a body’s volume or an increment in volume changeresults from the spherical stress tensor and is not influenced by the stress deviator.




The normal compressive stress plays an important role in the visco-plastic shear creepdeformation in rock. Therefore, the effects of normal compressive stress on non-linear visco-plastic shear deformation are further taken into account. At any point located in the sur-rounding visco-plastic rock, the shear viscosity η in the plane of the maximum shear stressis regarded as a non-linear function of the maximum shear stress τ , the normal compressivestress σn acting in the plane of admissible shear stress, and time t, that is:

η = f (τ ,σn,t) (1)

If the initial stress field in the rock mass results from elasticity, then, three stress fieldsin the surrounding rock can be assumed. Around an unsupported underground opening therock mass in which the stress value reaches the yield strength is in a state of visco-plasticity.The rock mass which is far from the visco-plastic region is in an elastic state. The rock massbetween the visco-plastic region and the elastic region is in a state of visco-elasticity. Then,the strain causing the redistribution of stresses in the surrounding rock can be written as:

γ = γe + γep + γvp (2)

where γe,γep,γvp represent the elastic, visco-elastic and visco-plastic shear strains, respec-tively. If the opening is located in a rock mass at great depth, the surrounding rock canapproximately be regarded as being in the elastic and visco-plastic regions. Therefore, thestrain causing the redistribution of the stresses in the surrounding rock can be simplified andthe governing conditions for visco-plastic state are written as:

if τ < τadm − Fσn then γ = γe;

if τ > τadm − Fσn then γ = γe + γvp,τ − τadm + Fσn = f (τadm,σn,t) · γ (3)

where τ is the current shear stress at a point located in the surrounding visco-plastic rock,τadm is the yield limit in the pure shear, parameter F is the equivalent friction coefficientin the plane of the admissible shear stress, and γe obeys Hooke’s law with G being theshear modulus. The value of the parameter F is determined by the shear creep tests of rockmaterial subjected to various normal compressive stresses. In this paper Vyalov’s hypothesis4

pertaining to the similarity of all isochrones including the instantaneous deformation curveis adopted. If the state of plane strain and the associated flow rule is adopted, the constitutiveequation showing the effect of strain rate in the visco-plastic region can be expressed as:

�εij,vp = kP2η(τ ,σn,t)

· ∂P∂σij

�t, P = 12

(σ1 − σ2)2 + Fσn − τadm, (4)

where σ1,σ2 are maximum and minimum principal normal stresses, respectively. The incre-ments of strains appear in the expression of Hooke’s law for eigenstrains.

It remains to express the shear viscosity function η. In the sense of Ref. 6 we obtain:

η(τ ,σn,t) = Ga · T

[T + t

(1− A

τs

)]2 Hs + σn

Hs, (5)

where T is the time to which the process of creep is observed, and the other coefficients areobtained fro the shear test.

3. Free Hexagon Method

Starting with statical equilibrium in the first stage of excavation and elastic state, in therock continuum and in the coal seam creep appears after making opening due to mining

360


Figure 1. Adjacent grains set up.

and time dependent visco-plastic equilibrium has to be considered. Plain strain state is suffi-cient to solve. Under the assumption that the material properties of both rock and coal areknown, hexagon elements are created and linear behavior inside of them is supposed. Sincethe elements are considered to be small enough, isotropic case is taken into account, i.e. theelements are homogeneous and. Classical problem involving generalized Coulomb’s frictionand exclusion of tensile stress exceeding the tensile strength along the interfaces (possibledislocations) is solved. Typical set up of adjacent elements is illustrated in Fig. 1. First thesolution of elastic problem in an element is formulated and the element is put into neigh-borhood of the adjacent elements. Regular distribution of elements is assumed, i.e. only onematrix relating tractions and boundary displacements will be provided.

4. Boundary Element Solution in One Particle

The solution of elasticity in each hexagonal element is approximated by concentration ofDOFs to vertices of the hexagon, and distribution of boundary displacements and tractionsalong edges s,s = 1, . . . ,6 of the hexagon is assumed to be linear. Then, generally, integralequations formulate the problem:

cikuk(ξ ) =6∑

s=1

∫s

pj(x)u∗ij(x,ξ ) dx−6∑

s=1

∫s

uj(x)p∗ij(x,ξ ) dx

+∫

bj(x)u∗ij(x,ξ ) dx+∫

μjk(x)σ ∗ijk(x,ξ ) dx (6)

where i and j run 1, 2, and s = 1, . . . ,6. In case the regular hexagons are used and lineardistribution of both displacements and tractions is used, cik = 1

3δik, δik is Kronecker’s delta.Knowing the form of kernels denoted by asterisk and substituting approximations for

boundary displacements u and tractions p, neglecting the volume weight influence (for deepopenings a symmetric problem with respect to the horizontal axis can be considered), andassume that the eigenstrains are uniform in each particle, matrix equations are obtained:

Au = Bp+ b, Ku = P+ V (7)

where A, B and K are square matrices (12 ∗ 12), u is the vector of displacement approxima-tions at vertices, P that of tractions and b and V are vectors of eigenstrain influences. Thelatter are vectors (1 ∗ 12).

361


Figure 3. Two hexagons in possible contact. Figure 4. Angles and normals with x1- direction.

5. Statical Contact Conditions

Let us consider two hexagons being in possible contact, see Fig. 2. Arrangement of cur-rent and neighboring particles together with their geometry is seen from Fig. 3. Introduce apseudo-cone K, which is defined as:

K ≡ {u ∈ V,[u]n ≥ 0,pn ≤ p+n , if pn ≥ p+n ⇒ pn = 0,

|pt| ≤ c κ(p+n − pn)− pn tanφ, if |pt| ≥ c κ(p+n − pn)− pn tanφ ⇒ pt

= pn tanφ sgn[u]t}(8)

where [u]n = u2n − u1

n, [u]t = u2t − u1

t , u is split into normal un and tangential (shear)ut components, n is the unit outward normal with respect to element 1, V is an admissiblespace of displacements, the traction p has now components {pn,pt}, i.e. projections to normaland tangential directions, p+n is the tensile strength, c is the cohesion or shear strength, and φis the angle of internal friction of the material (rock, coal), κ is the Heaviside function beingequal to one for positive arguments and zero otherwise. Here strict sign convention is used:positive sign is tension, while negative one means compression. The pseudo-cone K becomesa cone for p+n = 0 and frictionless case.

6. Generalized Fischera’s Conditions

Fischera’s conditions have been formerly formulated for K being a cone. In our case theconditions in normal direction can be written as:

p+n κ(p+n − pn)− pn ≥ 0, [u]n ≥ 0, (9)

Generalizing the above conditions to the tangential direction, it holds:

c κ(p+n − pn) − pn tanφ − |pt| ≥ 0, |[u]t| ≥ 0, {c κ(p+n − pn)− pn tanφ − |pt|}|[u]t| = 0,(10)

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Setting pn = kn[u]n,pt = kt[u]t, where kn,kt are normal spring and tangential springstiffnesses, the energy of the system can be stored as:

� = 12

N∑α=1

aα(u,u)−∫

pTu dx+n∑

β=1

∫β

{kβn ([u]βn)2 + kβn [u]βn |[u]βt | + kβt ([u]βt )2} dx

+−n∑

β=1

∫β

{(p+n )βκ(p+n − pβn )[u]βn + cβκ(p+n − pβn |[u]βt |}dx,

aα(u,u) =∫α

(σα)Tεα dx (11)

where α runs over all hexagon elements, α = 1, . . . ,N,β runs all contact edges of possiblecontacts β , β = 1, . . . ,n, is the external boundary where p is prescribed, aα is the internalenergy (bilinear form) inside a hexagon α, σα,εα are respectively stresses and strains in α.

Note that the spring stiffnesses kn,kt play the role of penalty. Since the functional� shouldbe minimized, then for kn being large enough, the jump in normal displacement along thisinterface is suppressed, while the tangential displacements can admit possible jump. On theother hand, if kt is large, the jump in tangential displacement is zero and the normal jump canoccur. The tangential tractions being induced by sticking of elements in tangential directionshave to obey the Mohr-Coulomb law, see (10). If p+n ≤ pn (the tensile strength is exceeded)then pnmust be zero and no contact occurs as also pt must be zero from the generalizedMohr-Coulomb law (using the Heaviside function in specification of the interfacial law).

7. Equilibrium in One Particle Embedded Into Its Neghborhood

Suppose the only element i can move while the others in the neighborhood remain stable atsome time instant t; the denotation t is dropped out in the following formulas. The unknownsin the formulation of the problem will be displacements along the boundaries of elements inx1 and x2 directions, which create interfaces between adjacent elements. Recall that alongthe interfaces debond or slip can occur. On the contrary to the finite elements a priori stickingbetween elements is not assumed and this is why classification of this method leads us to theset of distinct element methods (DEM).

Then in the element i the interfacial forces act, see Fig. 3.In x1- direction:

Fij1 = kij

11[uji1 − uij

1]+ kij12[uji

2 − uij2] = kij

11[u]ji1 + kij

12[u]ji2,

Qij1 = −

∫s

μijkσ∗1jk(x;ξj)dx,j = j1, . . . ,j6 (12)

where [u]ji1,[u]ji

2 are obviously jumps in displacements, kij11,kij

12 are transformed spring stiff-nesses, and in x2-direction:

Fij2 = kij

12[uji1 − uij

1]+ kij22[uji

2 − uij2] = kij

12[u]ji1 + kij

22[u]ji2,

Qij2 = −

∫s

μijkσ∗2jk(x;ξj)]dx, j = j1, . . . ,j6 (13)

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Since the internal equilibrium has to be fulfilled, it inevitably holds:

6∑j=1

[(Ki11kj + kij

11)uij1+(Ki12

kj + kij12)uij

2 − kij11uji

1 − kij12uji

2 −Qij1] = 0,

6∑j=1

[(Ki21kj + kij

21)uij1+(Ki22

kj + kij22)uij

2 − kij21uji

1 − kij22uji

2 −Qij2] = 0

k = 1, . . . ,6(14) (14)

which is a system of 12 equations for 12 unknowns displacements, six in x1 direction andsix in x2 direction. This system is always solved for fixed time and iteration step, i.e. theneighboring elements are considered fixed and the value of displacements for them is taken

Figure 4. Viscosity in the rock. a) at the ceiling b) at the side wall.

Figure 5. Setup of free hexagons: grey is the coal.

Figure 6. Low speed of excavation.

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Figure 7. Fast speed of excavation.

from the previous step. The iteration of position is assumed at a current time instant. Afterthe iteration of position is stable, the time is simply increased by in advance selected timeincrement and the iteration of position repeats. As the movement of the particles is very fast,the time increment has to be very small.

8. Examples

Study on a longwall mining with various speed of excavation is studied. The material coeffi-cients are taken from 6. The depth of the opening is 700 m, the opening is about 8×8 m2 massdensity of the rock is 2 780 kg/m3, the initial stress field is induced by the virgin state. Nextmaterial parameters of the rock mass have the following values: G = 1200 MPa, ν = 0.35,the angle of internal friction is 35 degrees, i.e. F = 0.4, the shear strength τadm = 4.7 MPa,τs = 14.5 MPa, a = 3.4 Hs = 2.9 MPa, T = 2100 hours. The coal seam is characterized asbrittle with G = 150 MPa, and ν = 0.25. The surface forces along the free space are rep-resented by an appropriate horizontal and vertical volume weight substituting its influence.The load due to the volume weight is given by the overburden along the upper elements. Thenumber of particles, which are regularly distributed, is 1532. In Fig. 4 observed distributionof the viscosity coefficient is depicted. The setting of hexagonal elements is seen in Fig. 5,where the shaded part describes the coal seam and the upper part the overburden. In Fig. 6and 7 the movements of the particles due to a low speed of excavation (the increment ofsurface forces is slow, divided into 10 time steps with the final time t = T) and due to a fastspeed of excavation (the forces are applied at the beginig of the opening the free space).

9. Conclusions

Visco-plastic behavior of the surrounding rock of a coal seam at the moment of possiblerock burst and closely after it is studied in this paper. In comparison with classical DEM, forinstant with the PFC, we start with different shape of particles to enable us to get also stressesin the particles and between them and with static equilibrium. Generally, in contradiction tothe PFC static equilibrium is taken into consideration and stability problem is solved basedon time increments. The forces induced along the boundaries of adjacent particles or aftermutual contact of extruded particles are caused by Eshelby’s forces, which come out fromthe virgin state of the massif along the free space boundary, i.e., if the opening is created dueto a mining, these forces are applied at the proper position in opposite directions on the faceof the mine.

Regular distribution of elements is assumed in the examples, i.e. only one stiffness matrixrelating tractions and boundary displacements is provided inside of each element. Thisassumption speeds up the iteration and also provides a better overview at the behavior ofparticles in the pertinent pictures. The presented mathematical and numerical approach can

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consider much more elements, but then the description of movements of particles has to beprobably zoomed.

In order to describe the behavior of the massif no deformation is displayed in the picturesto show the stress concentration zones. Only two cases, which are decisive for assessment ofpossible bumps occurrence, are considered and calculated. Those cases simulate very speedexcavation of the opening (this is not realistic, but for estimate of the effects of the speed ofmining illustrative) on one hand side and on the other side very slow speed of excavation,when the time increments are divided into 10 steps. In each step the iteration is finishedafter moving particles are settled up, i.e. small error between the movements in the old andnew stages is attained. The distribution of time steps is projected also in Eshelby’s forces,which increase accordingly. It appears that the side wall and the ceiling are relatively stable(it depends on a quality of material above envisaged) for low velocity of mining and thestability principally decreases for fast opening.

Acknowledgments

This paper was prepared under financial support of GAÈR, project No. 103/08/0922. Finan-cial support of Ministry of Education and Sport of the Czech Republic, project numbersMSM 6840770001 is also acknowledged.

References

1. Prochazka, P.P., “Application of discrete element methods to fracture mechanics of rock bursts”,Engineering Fracture Mechanics, 71, 2004, pp. 601–618.

2. Prochazka, P.P., “Rock bursts due to gas explosion based on hexagonal and boundary elements”,to appear in Engineering Analysis with Boundary Elements 2009.

3. Prochazka, P.P. and Sejnoha, M., ”Development of debond region in lag model”, Computers &Structures, 55, 2, 1995, pp. 249–260.

4. Vyalov, S.S., Rheological Fundamentals of Soil Mechanics. Elsevier. Amsterdam (1986).5. Okubo, S., Nishimatsu, Y. and Fukui, K., “Complete creep curves under uniaxial compression”,

Int. J. Rock Mechanics and Mining Science & Geomech. Abstracts, 28, 1991, pp. 77–82.6. Song, D., “Non-linear Visco-plastic creep of rock surrounding an underground excavation-

Technical Note”, Int. J. Rock Mechanics and Mining Science & Geomech. Abstracts, 30, 6, 1993,pp. 653–658.

366

Analysis of Large Rock Deformation Under High in situ Stress

S.G. CHEN∗, Y.B. ZHAO AND H. ZHANG

Southwest Jiaotong University, China

1. Introduction

The Jinping Hydropower Project consists of two hydropower stations: namely Jinping GradeI and Jinping Grade II. The Jinping Grade I is at upstream with a capacity of 3.6 MkW.The Jinping Grade II is at downstream with a capacity of 4.8 MkW, which is the biggesthydropower station among 21 stations on the Yalong River, China, with a dam of 305mhigh, the highest dam in the world. The distance of the two stations is 150km along theYalong River but only about 18 km in straight-line distance.

The Jinping Auxiliary Twin Tunnel is built to connect the two hydropower stations witha length of 17.5 m, which is parallel to the 4 diversion tunnels of Jinping Grade II. As it isconstructed prior to the diversion tunnels, the construction of the Jinping Auxiliary TwinTunnel can provide valuable geological information and site trials for the construction ofdiversion tunnels in the next stage. The Jinping Auxiliary Twin Tunnel is deeply buried witha maximum overburden of 2375m and the overburden over 73% of its total length is 1500 m,thus there is no condition to build shafts in between.

The tunnel is laid mainly in Triassic system as shown in Fig. 2, while Carbonate rock(Marbles) occupies about 70–80% including Triassic series (T1), Zagunao formation (T2z),Baishan formation (T2b) and Triassic upper series (T3). Due to the existence of very high insitu stress, this area is heavily compressed with very close composite folds in SN direction.The rock is basically solid with very high UCS of up to 210 MPa and mostly classified asGrades II and III. The tunnel has a very high groundwater head with a maximum waterpressure of up to 10 MPa.1

The rock in the Jinping Auxiliary Twin Tunnel is generally hard, while the rock betweenK3+140∼K4+460 in the tunnel is soft carbon phyllite and silty slate. This section has anoverburden of 2000m and a lateral pressure ratio of 0.8, thus ensuring the tunnel stabilityduring the construction is the main concern. This study is to investigate the stability of thetunnel between K3+140∼K4+460 during the construction by using the discrete elementmethod code UDEC.

2. Computational Model and Material Properties

Figure 1 shows the geological mapping from the site at BK3+237. It illustrates that the rockis heavily jointed by two major joint sets, one (Joint set I) is a vertical joint set with dip angleof 82◦ and joint spacing of 0.8 m, the other (Joint set II) is a horizontal joint set with dipangle of 150◦ and joint spacing of 1–2 m. In addition, one fault (Joint set III) with a dip angleof 30◦ crosses the tunnel. The rock is classified as Grade IV, a soft rock.

The discrete element code UDEC2 is used to simulate the rock deformation due to tunnel-ing. The computational model is shown in Fig. 2. Five types of material are involved in themodel including rock, joint, rockbolt, shotcrete and concrete lining. the material propertiesare listed in Tables 1–5.




Figure 1. The geological mapping on site. Figure 2. The computational model.

Table 1. Rock properties.

Property Value

Density, kg/m3 2840Young’s Modulus, GPa 80Poisson Ratio 0.25

Table 2. Joint properties.

Property Set I Set II Set III

Normal stiffness, MPa/m 1000 10000 800Shear stiffness,MPa/m 4.25 6.3 3.4Cohesion, MPa 0 0 0Friction angle, 35 40 30UTS 0 0 0

3. Numerical Modelling

Two cases are modelled including the case of no rock support and of with rock support.

3.1. Tunnel stability with no rock support

The tunnel stability due to excavation is closely related to the in situ stress existed in rockprior to excavation, and the rock joint geometrical distribution apparently affects the in situstress distribution. Figure 3 shows the influence of rock joints on the maximum principlestress distribution in rock mass. It can be seen that the stress distribution is regular in case of

368


Table 3. Shotcrete properties.

Property Value

Density, kg/m3 2500Young’s modulus, GPa 21Poisson ratio 0.15UTS, MPa 1.29UCS, MPa 27.2Residual UTS, MPa 1.0Interface of rock/shotcreteNormal stiffness, GPa/m 10Shear stiffness, GPa/m 10Cohesion, MPa 0.5UTS, MPa 1.0

Table 4. Rockbolt properties.

Property Value

Density, kg/m3 7500Diameter, mm 22Young’s modulus, GPa 210Maximum tension, tons 15.4Maximum comp., tons 15.4Grout stiffness, GPa 24Grout cohesion, MPa 1.0Rochbolt spacing, m 1.0Rockbolt length, m 3.5

Table 5. Lining properties.

Property Value

Density, kg/m3 2500Young’s modulus, GPa 33.4Poisson ratio 0.15UCS, MPa 29.5UTS, MPa 2.7

no rock joint. When rock joint exists, the in situ stress distribution is much complicated andeven stress concentration appears at the intersection of joints.

Figure 4 shows the rock deformation distribution after tunneling in case of no rock sup-port. It illustrates that local rock fall occurs at right arch waist and bigger rock deformationappears on left and right walls. Particularly, serious rock sliding occurs on left wall, indicatingthat the tunnel stability could be a problem if no rock support is applied.

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(a) no rock joint (b) with rock joints

Figure 3. Influence of rock joint on maximum principle stress distribution.

3.2. Tunnel stability with rock support

To avoid above tunnel stability problem, rock support should be applied. As the in situ stressis very high, the stress in the rock needs to be released at a certain degree before the rocksupport is applied.

Based on the computational result and site observation, a rock support design is made.After the excavation, system rockbolts of 3.5 m in length and 1.0 m in spacing and shotcreteof 0.2 m in thickness are immediately applied. Point rockbolts and reinforcement mesh couldbe applied if necessary. At a certain distance, lining of 60 cm in thickness is applied.

As UDEC is a 2D analysis model, stress release ratios for different rock support applicationneeds to be taken into account, which can be achieved by using the built-in FISH language inUDEC.3 In this study, the stress release ratio before the application of rockbolt and shotcreteadopts 30%. Another 50% stress will be released before the application of the lining. Theleft 20% stress will be released after applying the lining.

Figure 5 show the rock deformation distribution after the excavation but before applyingrock support and shotcrete, in which 30% stress is released. From the figure, it can be seen

Figure 4. Tunnel failure with no rock support.

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Figure 5. The geological mapping on site. Figure 6. The computational model.

that rock is loosen locally at right arch waist and walls, but no rock fall occurs, and thetunnel is basically stable. The maximum displacement is about 5cm where the rock would bemoved away before applying rock support.

Figure 6 shows the rockbolt and shotcrete support design, except the system rockbolt,more point rockbolts are applied to the walls to strength the rock there. Figure 7 is the rockdeformation distribution after applying rockbolts and shotcrete but before applying concretelining. It indicates that maximum displacement of 6.0cm occurs on the left wall. Potentialrock fall appears at right arch waist, but because of the rockbolt, the rock still stays stable.

Figure 8 shows the final rock deformation distribution after the lining is applied. It illus-trates that the maximum displacement of 6.4 cm occurs on left and right walls. The mostunstable part comes from left and right walls. Because of the lining application, the tunnelfinally becomes stable.

Rock deformation monitoring is carried out during the construction by using Baset System.The monitoring data agrees well with the UDEC modelling results.4

Figure 7. Displacement distribution before lining.

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Figure 8. Final displacement distribution.

4. Conclusions

A numerical study on large rock deformation under high in situ stress is carried out by usingthe discrete element code UDEC. From the study, following conclusions can be drawn.

• UDEC is well suitable to simulate the mechanical behaviour of joint rock mass,including the physical discontinuity involvement in the model, and the discontinuousrock deformation distribution at the tunnel outline, which agrees well with the siteobservation.• To obtain reliable computational result by using UDEC, the stress release at various

stages must be properly taken into account. The determination of the stress releaseratios is closely related to the rock type and the rock properties, which needs furtherinvestigation.

References

1. Zhang, J., Ren, S., Jiang, H, Chen, X. and Shu, J., “A study on major engineering geological issuesfor Jinping Auxiliary Twin Tunnels”, Advances in Science and Technology of Water Resources, Vol.26, No. 6, 2006, pp. 66–70.

2. Yan, C., Yang, J. and Chen, S.G., “An integrated geological prediction technology and its applica-tion at Fault F6 of Jinping Auxiliary Twin Tunnel”, Highway Tunnel. Vol. 61, No. 1, 2008, pp.34–38.

3. Cundall, P.A., UDEC — A generalised distinct element program for modelling jointed rock. ReportPCAR-1-80, Peter Cundall Associates, U.S. Army, European Research Office, London, ContractDAJA37-79-C-0548, 1980.

4. Chen, S.G., Ong, H.L. and Tan, K.H., “Main considerations on UDEC modeling of tunnel exca-vation and supports”, IS-Kyoto 2001: Modern tunneling science and technology, Kyoto, Japan,October, 2001, pp. 433–438.

5. Chinease 2nd Railway Engineering Group Co., The construction technology of deep and long tun-nels under high water pressure and rich water. Project Report, April, 2008.

372

Gotthard Base Tunnel: UDEC Simulations of Micro TremorsEncountered During Construction

H. HAGEDORN∗ AND R. STADELMANN

Amberg Engineering Ltd., Switzerland

1. Introduction

The Gotthard Base Tunnel, a 57 km long twin tube single track railway tunnel traversing theSwiss Alps is currently under construction. The aim of the tunnel is to connect the high speedrailway systems of Germany and Italy. In order to attain a reasonable construction time,heading started at the portals and at three intermediate points of attack. During operation ofthe tunnel, two of these intermediate attacks shall serve as multifunction stations (MFS) forrescue in case of fire, operation control and maintenance. Additional tunnels are required forthe special ventilation system of the rescue stations (see Fig. 1). The excavation of the threeintermediate points of attack was drill and blast. During construction of the southern pointof attack at Faido a big unknown fault was encountered. A comprehensive drilling campaignrevealed a big fault zone with a kernel consisting of layers with decomposed rock such asrock debris and rock fragments in a fine-grained matrix (kakirites). During constructionin the fault zone, large displacements in the tunnels occurred giving rise to considerabledifficulties.1 In addition, micro tremors and rock bursts started to develop with increasingintensities, some of them severely damaging the tunnels and its support, mainly in the Easttube (see Fig. 1).

During March 2004 to June 2005 the Swiss Seismological Service (SED) recorded an accu-mulation of seismic activity in the area of the MFS Faido, an area normally exhibiting verylow seismic activity. In July 2005 the tunnel’s owner ATG (AlpTransit Gotthard Base TunnelAG) formed a work group ‘Micro Tremors’ consisting of representatives from the client, theEngineering Joint Ventured ‘GBT South’, the supervision, geologists, the SED (Swiss Seis-mological Service of the Swiss Federal Institute of Technology) and external specialists forstructural dynamics. The aim of this work group was to investigate the following aspects:

• Which were the reasons giving rise for micro tremors and rock bursts?• Did the construction of the MFS Faido initiate the micro tremors?• Where are the locations of the micro tremors hypocenters with respect to the tunnels?• Is the lining’s originally designed bearing capacity capable to resisting additional

seismic loading?• What is a micro tremor’s damage potential regarding the final tunnel lining?• Which investigations are required to assess above mentioned aspects?

The work group decided to install an extended seismic measuring program with twoaccelerometers in the MFS Faido with first priority. Together with geologists, numerical mod-els for dynamic simulations were established. The paper shows results from seismic measure-ments and from numerical discontinuous simulations with UDEC.3

Detailed information regarding the Gotthard Base Tunnel can be found at: http://www.alptransit.ch/pages/e/




East Tube EON

West Tube EWN

Cross cavern

West

East

Figure 1. Left: Initial layout of MFS Faido. Right: Joint system in the fault.

2. The Multifunction Section Faido

The initial layout of the MFS Faido is shown in Fig. 1 (left). The MFS Faido had to beaccessed by a 2.7 km long access tunnel declining at 12.7% from the portal. The overburdenin the MFS Faido is between 1500 and 1800 meters.

The fault strikes at an average angle of about 20◦ to 15◦ to the tunnel axis and dips atabout 80◦ to the East. This unfavorable spatial orientation is shown in Fig. 1, right. Thefault is embedded between hard and brittle Leventina gneiss on the east and folded, moreductile Lucomagno gneiss on the West.1

3. Results from Seismic Measurements

3.1. Epicenters and chronology

The epicenters of registered micro tremors during Oct. 2005 to Feb. 2008 are depicted inFig. 2. The micro tremors’ sources are concentrated in the brittle Leventina gneiss most ofthem at a distance of 50 to 350 meters to the East of the tunnel system. The accuracy of theepicenters’ localization is less than 100 m and less than 250 m in focal depth as determinedby relocation of the calibration shots. Within the error ellipsoid the tremors’ sources were attunnel level.2, 4

Measuring station MFS-AMeasuring station MFS-B

Date

M2.4

East tube, EON

West tube, EWN

East

Figure 2. Epicenters of the micro tremors from Oct. 2005 to Feb. 2008.

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The highest seismic activity took place during December 2005 and May 2006. During theperiod from October 2005 to February 2008 a total of 112 micro tremors were recorded.The highest magnitude of M2.4 (Richter scale) occurred on 25th March 2006.

3.2. Measured particle velocity (V) of the M2.4 micro tremor at measuringstation MFS-A

The measured particle velocities V (Vx, Vy, Vz) of the M2.4 micro tremor, registered at thestation MFS-A (see Fig. 2), are shown in Fig. 3.

-0.03

-0.01

0.00

0.01

0.02

0.03

0.0 0.2 0.4 0.6 0.8 1.0

Start of Computations

Vy

Vx

Vz

Part

icle

Vel

ociti

es V

(m/s

ec)

X

YZ

Time

Figure 3. Measured particle velocities’ components Vx-, Vy- and Vz at measuring station MFS-A.

The orientations of the X-, Y-, and Z — coordinates correspond to those of the modelcoordinates. The total measuring interval was 1 sec. The maximum measured amplitude of0.024 m/s was recorded for Vx, normal to the tunnel’s axis.

The work group ‘Micro Tremors’ decided to consider the M2.4 micro tremor as the ‘design– tremor’. The corresponding seismic load was used to investigate the bearing capacity of theinitially designed tunnel linings.

4. Models for UDEC Analyses

4.1. Specification of the model’s seismic design load

First, the seismic model load (input wave) corresponding to the M2.4 design tremor had to bedetermined. The hypocenter’s distance to the EON tunnel (see Fig. 2) was approximately 250meters. Prior to arrive at measuring station MFS-A, the emitted wave at the tremor’s sourcehad to pass the tunnel system and the fault zone. Therefore, the measured signal at MFS-Acorresponded to a damped wave. To determine the wave’s attenuation, 10 times amplifiedmeasured particle velocities were applied at the eastern boundary of the 3D UDEC modelshown in Fig. 4.

In the model point P41 the 10 times amplified measured wave (input wave) excited particlevelocities similar to those of the measured signal (see Fig. 3). The seismic design load wastherefore determined as the 10 times amplified measured signal of the M2.4 micro tremorregistered in measuring station MFS-A (see Fig. 4).

375


-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.0 0.1 0.2 0.3

Time t (s)

Part

icle

vel

ocity

Vx

(m/s

ec)

Input Vx

P41

0.40

Horizontal cut through 3D model at tunnel level

Input wave

EWN

EON

Fault

P41

t

Figure 4. Left: 3D Model. Right: Horizontal cut through the model with model point P41 simulatingapproximately the measuring station MFS-A. Vx input and Vx at P41.�t is the wave’s travel time fromthe model’s load boundary to Point P41.

4.2. UDEC models for the dynamic simulations

Due to the high computation time parametric studies were not possible with the 3D model.For parametric studies 2D UDEC models were used instead. Special investigations have beenshowing the deviation of seismic waves along weak rock layers. Therefore two different 2Dmodels with one and with two kakirite layers (see Fig. 5), respectively, were investigated.The properties for the rock and the two major joint systems considered in the models werespecified together with the geologists.

A joint spacing of 2 m was used for the fault’s kernel zone. In reality the joint spacing inthe fault’s kernel is much smaller. This was accounted for by setting correspondingly reducedrock’s properties in the model. In addition, weak rock properties were assigned to the layerscontaining kakitrite. For modeling support and final lining block elements and structural barelements respectively were used. According to the applied shotcrete the support thickness inthe model is 1 meter. The lining thicknesses for the tunnels EWN and EON are 0.6 meters and0.4 meters respectively. In a first step the static equilibrium of the supported tunnel systemwas computed. In a second step the final lining was inserted (stress-free) and the design load

Rock

Hard, brittle

Intensively jointed

Faults kernel

Kakirite layer

Medium hard

40 m

EONEWN

Model_2

EWEONEWN

Model_1

EW

XY

Figure 5. Model_1 and Model_2 with 2 and 1 layers of kakirite in the fault’s kernel.

376


applied at the eastern model boundary. The presented results regarding the final lining refersolely to the dynamic load case.

With Model_1 a stress drop was initiated nearby the EON tunnel. The influence of thisstress drop on the tunnel’s liner is discussed in Chapter 5.2.

5. Computational Results

5.1. Static load case, without tunnel support

Numerical modeling without support was carried out to identify possible reasons for themicro tremors. It was presumed that high stress concentrations accumulated during con-struction as a result of the large deformations developing around the tunnels. From overbur-den, rock mass density and gravity a primary principal stress level of σyy = 42 MPa andσxx = 21 MPa was defined at the top of the model.

yy = 70 – 80 MPa yy = 60 – 70 MPa

Stress concentration

EastWest

Figure 6. Stress redistribution without tunnel support. Vertical stress σyy.

Figure 6 shows magnitude-ranges of vertical stresses σyy after excavation of the 3 tunnels.The results in Fig. 6 show a considerable extension of the stress redistribution due to theexcavation of the tunnels. A general stress reduction occurred within the area of the threetunnels, σyy decreasing partially to 10 MPa. To the east of the fault in the hard brittle gneissa stress concentration is extending beneath the tunnel along the fault’s boundary. At tunnellevel the vertical stress’ concentration amounts to σyy = 70 MPa to 80 MPa. Such stressconcentrations were probably the reason for the micro tremors. In general the location of themodel’s stress concentration is in accordance with the micro tremors’ hypocenters determinedfrom seismic measurements (see Fig. 2).

5.2. Stress drop near tunnel EON in Model_1

The development of the stress drop beneath the EON tunnel and its disturbing influence onthe input wave are depicted in Fig. 7. The load wave’s deviation along the fault’s weak rockis obvious. The stress drop (shear stress drop) occurred along a joint.

The stress drop starts prior to the arrival of the input wave’s first Vx – peak and wastriggered by a small disturbance. The drop’s shear direction and its magnitude of 17 MPa instress release are shown in Fig. 8. The particle velocity Vx induced by the stress drop amountsto more than 0.40 m/s (Fig. 8).

377


t = 0.1372s t = 0.1403s t = 0.1532s

Vx - Load wave

EON

Figure 7. Development of a stress drop in the vicinity of tunnel EON. Represented are the particlevelocities Vx. Vx — range = −0.2 to +0.2 m/s.

Vx (m/s) Shear direction

+ VxVxY

X

Tunnel EON

Reference point

-30.0

-25.0

-20.0

-15.0

-10.0

-5.0

0.00.0 0.1 0.2 0.3 0.4 0.5

Time t (sec)

Shea

r st

ress

(MPa

) at c

onta

ct 2

5997

97

Shear stress drop: 17 Mpat = 0.0022 s

0.135289 s

Figure 8. Left: Development of Vx around the stress drop area. Vx — range = −0.4 to +0.4 m/s.Right: Stress drop of 17 MPa. Remaining residual shear stress: −14.5 MPa.

5.3. Influence of the tunnel’s position with respectto the weak (kakirite) zones

The influence of the tunnels position with respect to the weak rock layers on the lining’sdynamic loading is shown in Fig. 9 by comparing the stresses in a liner segment of the EONtunnel for Model_1 and Model_2. The liner segment for comparison is in the eastern invertabutment. In Model_1 the EWN tunnel is sheltered by the eastern kakirite layer. In Model_2the EWN is in front of the kakirite layer and is exposed to the arriving load wave form theEast (see Fig. 5). Negative stress denotes compression.

The liner’s concrete peak stresses occur at high frequencies and do not affect the struc-ture. The computed liner stresses were therefore low-pass (LP) filtered at 60 Hz to obtainthe relevant stress peak for the proof of the liner’s bearing capacity. In case of Model_2 aremaining compression of −1.3 MPa remains after impact. The damping effect of the easternkakirite layer in Model_1 is obvious.

5.4. Loading of the lining due to a stress drop in the vicinity of a tunnel

In addition to stresses and displacements the peak particle velocities (PV) were investigated.This method is used in mines to specify impacts from rock bursts. The method used for

378


-5

-4

-3

-2

-1

0

1

2

0.0 0.1 0.2 0.3 0.4 0.5

Time t (s)

Con

cret

e st

ress

in li

ner (

MPa

)

unfiltered60 Hz LP filter

Kakirite zones

EWN

-5

-4

-3

-2

-1

0

1

2

0.0 0.1 0.2 0.3 0.4 0.5

Time t (s)

Con

cret

e st

ress

in li

ner (

MPa

)

unfiltered60 Hz LP filter

EWN

Kakirite zone

Figure 9. Left: Concrete stresses (MPa) for Mod_1. Right: Concrete stresses (MPa) for Mod_2.

0

200

400

600

800

1000

0.0 0.1 0.2 0.3 0.4 0.5Time t (s)

Part

icle

Vel

ociti

es P

V (m

m/s

)

stress dropno stress drop

Lining segment

EON

0

200

400

600

800

1000

0.0 0.1 0.2 0.3 0.4 0.5Time t (s)

Part

icle

Vel

ociti

es P

V (m

m/s

)

stress dropno stress drop

Point near lining

EON

Figure 10. Left: PV of a point in the rock mass near the lining. Right: PV of the lining segment. Withand without stress drop.

calculation PV’s was:

PV =√

(Vx)2 + (Vy)2 (1)

The effect of the stress drop near the EON tunnel5 is visualized by comparing PV’s for thecases with and without stress drop. The rock’s PV in a reference point (see Fig. 8) near theeastern invert abutment of the EON tunnel is shown in Fig. 10. Figure 10 contains also thePV’s for a liner segment near the reference point.

Between the tunnel’s support and the liner a sealing foil was mounted. This foil was sim-ulated by means of a UDEC-interface. The interface properties specify the transmissivity ofthe foil. The transimissivity affects the liners vibration behavior. This topic was especiallyinvestigated. Figure 10 reveals the sealing foil’s effect on the lining’s vibrations. A stress dropin the tunnel’s vicinity could give rise to partial damage of the final lining.

6. Conclusions

• During tunneling at great depth in geological conditions as encountered in the MFSFaido micro tremors are likely to occur.• The stress redistribution due to tunneling and the stress concentration in hard rock

in combination with a weak fault striking at small angle to the tunnel axis favors theoccurrence of micro tremors.

379


• The micro tremors did correlate with the excavation activities. After completion ofthe excavation of the MFS Faido no more micro tremors occurred.• A seismic wave is deviated by a weak zone. In the MFS Faido the steep dipping fault

caused deviation of the tremors’ seismic wave towards the tunnels.• If a tunnel is located in front of a weak zone the impact of a more or less unhin-

dered seismic wave is considerably higher compared to the impact on a tunnel in the‘shelter’ of a weak zone.• Micro tremors can trigger stress drops.• Excluding the loading due to a spontaneous ‘stress drop’ in the direct vicinity of a

tunnel there has been no need for improving (thickness, additional reinforcing) thelinings designed for the static load case in the MFS Faido.• The assumption for the seismic design load is conservative. The case of a stress drop

in the vicinity of a tunnel has been accepted as a residual risk.

Acknowledgements

Amberg Engineering Ltd. is member of the Engineering JV Gotthard Base Tunnel South,GBTS, consisting of: Poyry Infra Ltd., Lombardi Engineering Ltd. and Amberg EngineeringLtd. The authors thank the AlpTransit Gotthard for the permission to publish the paper.

References

1. Hagedorn, H., Rehbock-Sander, M., Flury, S. “Gotthard Base Tunnel: State of the works and SpecialAspects”, Proc. Int. Symposium on Construction and Operation of Long Tunnels, Taipei, Taiwan,7. 10.11.2005, Vol. 2, pp. 961–973.

2. Rehbock-Sander, M., “Rock Bursts Experience gained in Mines and Deep Tunnels”, Proc. 4th AsianRock Mechanics Symposium (ARMS 2006), Singapore, 8.-10.11.2006, pp. 413.

3. ITASCA, UDEC – Manual, Theory and Background, Itasca Consulting Group, Inc. Minneapolis,Minnesota, USA

4. Kaiser, P.K., Vasak, P., Suorinemi, F.T. and Thibodeau, D. (2005). New dimensions in seismic datainterpretation with 3-D virtual reality visualization in burst-prone ground, RaSiM6, Perth, Aus-tralia, pp. 33–47.

5. Heuze, F.E., Morris, J.P., Insights into ground shock in jointed rocks and the response of structuresthere-in, Int. Journal of Rock Mechanics & Mining Sciences 44 (2007) 647–676.

380

Discrete Modeling of Fluid Flow in Fractured Sedimentary Rocks

WU WEI, LI YONG AND MA GUOWEI∗

School of Civil and Environmental Engineering, Nanyang Technological University,50 Nanyang Avenue, Singapore

1. Introduction

The hydro-mechanical properties of rock masses are sensitive to fluid flow, which has alwaysbeen a significant issue in rock engineering design and construction. Rock masses consist ofintact rock blocks and various kinds of discontinuities, such as joints, cracks, bedding planes,etc. For low permeability rocks, the discontinuities always provide the major pathways forfluid flow while the permeability of the intact rock blocks is generally very low.1 The strengthand deformation of the rocks affected by fluid flow are mainly due to the hydro-mechanicalproperties of discontinuities and these behaviors are more pronounced in fine-grained sedi-mentary rocks.2

The strength and deformation of rocks under the fluid effect was widely investigated inexperimental studies2–5 and numerical simulations.6–9 Euguler and Ulusay,2 Gutierrez et al.,3

Zhang et al.4 and Backstrom et al.5 recognized that the fluid had a remarkable influence onthe mechanical properties of rocks. Generally the presence of fluid reduced the compressivestrength and elastic modulus and affected the stress-strain behavior of rocks. However, rocksfrom different origins may display diverse fluid sensitivity due to their physical propertiesand chemical compositions. Indraratna et al.,6 Zhang and Sanderson,7 Min et al.8 and Bagh-banan and Jing9 simulated the fluid flow through rock fractures using UDEC and particularlyconcentrated on the permeability of rock masses with various flow rates, joint properties andboundary conditions. However, there have been little discrete models generated by UDECconcentrated on the hydro-mechanical properties of the sedimentary rocks and no experi-ences to validate the realism of the model simulation.

A 3D geological model of a sedimentary strata, as shown in Fig. 1(a), presents that threesedimentary formations are recognized for a concerned engineering site: a sand layer of7 meters thickness below the seabed, a mixed sandy and silty clay layer of 42 meters thicknessand a 60-meter mixed layer of sandstone and siltstone at the bottom until 130 meters depth.The concerned engineering site is located at the lower part of the sandstone and siltstonelayer, between 100 meters to 130 meters depth. At this range, the sedimentary rocks displaylow permeability due to high density, low porosity and small apertures. The variable fracturedensity and rock materials are governing factors that affect the mechanical properties of therocks under fluid flow condition. The objective of the study is to build a discrete model basedon the results of the laboratory testing and estimate how fluid flow influences the mechanicalproperties of the sedimentary rocks.

2. Laboratory Testing of the Sedimentary Rocks

2.1. Rock specimen selection and sampling

Ten fully saturated rock specimens were extracted from referenced rock cores 1 and 2, whichwere two parts of a typical rock boreholes located at the depth of concerned engineering site,




Sand

Clay

Sand stone and siltstone

Referenced rockcores

Concerned engineering site

(a) (b)

Figure 1. (a) 3D geological model and concerned engineering site; (b) Sedimentary rock layers andreferenced rock cores.

as shown in Fig. 1(b). The specimens were divided into two groups based on the differentrock materials, sandstone and siltstone. The siltstone specimens appeared a higher fracturedensity on the surfaces than sandstone specimens. The specimens, having a dimension of50-mm-diameter and 100-mm-high, were manufactured and kept the fully saturated stateduring the process.

2.2. Uniaxial compressive tests

The uniaxial compressive tests, based on the ISRM suggested method,10 were conducted onthe specimens by a computer-controlled, servo-hydraulic testing mechine having a maximumloading capacity of 2.67 MN, which emphasize on the uniaxial compressive strength andelastic modulus of the rock specimens. The specimen was applied a measured load in axialdirection continuously at a constant stress rate of 0.53 MPa/s such that the failure occurredwithin 5–10 min of loading. A LVDT was installed near the specimen vertically to recordthe axial deformation of the specimen. The stress-strain curves of four typical sandstone andsiltstone specimens are shown in Fig. 2(a).

3. Numerical Modeling of the Sedimentary Rocks

3.1. The universal distinct element code

The Universal Distinct Element Code is a two-dimensional numerical program based on thedistinct element method to simulate the behavior of discontinuous media under static anddynamic loading.11 The discontinuous medium is represented as an assemblage of discreteblocks, which display either rigid or deformable material, and the discontinuities are treatedas boundary conditions between blocks. The program is based on a ‘Lagrangian’ calcula-tion scheme that is well-suited to perform the large movements and deformations of blockysystem, including the analysis of fluid flow through the fracture network of impermeabilityblocks.

382


(a) (b)

0 0.0014 0.0028 0.0042Strain

0

40

80

120

160

200

240

280

Stre

ss/M

Pa

SA1SA2SI1SI2

0 0.0014 0.0028 0.0042Strain

0

40

80

120

160

200

240

280

Stre

ss/M

Pa

SA1SA2SI1SI2

Figure 2. The stress-strain curves of the sedimentary rocks (1) Laboratory testing (2) Numerical mod-eling.

3.2. The modeling of rock specimens

A preliminary model was firstly established in the same dimension as the specimen used inthe uniaxial compression test with known fractures and materials according to the specimensurface. The mechanical parameters used in the modeling were listed in the Table 1. A seriesof stochastic fractures was then introduced to the model with different fracture length anddensity as listed in the first item of Table 2. The generation of discrete fracture network wasbased on a stochastic representative elementary volume approach using the Monte CarloSimulation technique.1

For the first step of the numerical simulation, a steady fluid flow was applied to the frac-tured rock specimen to make it become a fully saturated state. The left and right boundariesof the model were set as impermeable. The steady flow was injected from the top of themodel, while the bottom was fixed and permeable. With sufficient calculation cycles, thefluid flow filled in the fractures of the model. In the second step, the bottom became imper-meable to simulate the undrained state of the rock specimen under uniaxial compression. Anaxial velocity boundary was then applied on the top of the specimen and the stress-strainrelationship was calculated until rock failure.

4. Discussion

Figure 2(a) reveals that the fully saturated sandstone displays excellent strength and deforma-bility with the increasing compressive loading and explodes into pieces at the failure point.

Table 1. Mechanical parameters of the UDEC model.

Rock

Mass Joint

Density(103kg/m3)

Elasticmodulus

(GPa)

ShearModulus

(GPa)

Initialfrictionangle (◦)

Residualfrictionalangle (◦)

Initialaperture

(mm)

Residualaperture

(mm)

Sandstone 2720 51 30 75 200.5 0.2Siltstone 2742 40 24 65 15

383


Table 2. Numerical simulation results on the aperture and pore pressure of specimens under increasingloading.

1. Effective DFN

2. Variable aperture pattern 3. Pore

pressure

SA1

0MPa 100MPa 200MPa 270MPa

SA2


SI1


SI2


• Note: SA: Sandstone, SI: siltstone.

Siltstone has a lower compressibility than sandstone due to the higher fluid sensitivity of pre-existing fractures that induce shear failure under uniaxial compression. The fracture densityalso has significant influence on the strength of both rock types. Generally the lower fracturedensity, the less surface exposed to fluid, the higher rock material strength.

384


According to the results of numerical simulation, the stress-strain curves, as shown inFig. 2(b), are accordance with the results of laboratory testing. The joints at the center ofthe specimen open, while those at the top and bottom close under the increasing compressiveloading. It is implied for the evolving procedure of joint hydraulic aperture as shown in thesecond item of the Table 2. Moreover, the pore pressure along the joints distributes increas-ingly in the vertical direction mainly due to the gravity effect, as plotted in the third itemof Table 2. It is also found that the strength of the joint rock specimens under the saturatedstate decreases significantly with the increase of fracture density and the differences of rockparameters is slight that can not affect the mechanical properties of the rocks seriously.

5. Conclusions

A discrete model that represented the stress-strain behaviors of saturated rock specimenswith the effect of rock materials and fracture density was established based on the resultsof laboratory testing. It simulated the evolving procedure of joint hydraulic aperture andthe distribution of pore pressure and found fracture density had a more significant effect onthe mechanical properties of the sedimentary rocks than rock materials, because mechanicalparameters of the sandstone and siltstone have slight differences. However, the material effectmay be serious at the interface of sandstone and siltstone. In the future work the mechanicalbehaviors at the interface under fluid flow effect shall be studied to make a better under-standing on the hydro-mechanical properties of the sedimentary rocks.

References

1. Min, K.B., Jing, L.R. and Stephansson, O., “Determining the Equivalent Permeability Tensor forFractured Rock Masses Using a Stochastic REV Approach: Method and Application to the FieldData from Sellafield, UK”, Hydrogeology Journal, 12, 2004, pp. 497–510.

2. Euguler, Z.A. and Ulusay, R., “Water-induced Variations in Mechanical Properties of Clay-bearingRocks”, International Journal of Rock Mechanics and Mining Sciences, 46, 2, 2009, pp. 355–370.

3. Gutierrez, M., Oino, L.E. and Hoeg, K., “The Effect of Fluid Content on the Mechanical Behaviourof Fractures in Chalk”, Rock Mechanics and Rock Engineering, 33, 2, 2000, pp. 93–117.

4. Zhang, J., Standifird, W.B., Roegiers, J.C. and Zhang, Y., “Stress-dependent Fluid Flow and Perme-ability in Fractured Media: from Lab Experiments to Engineering Applications”, Rock Mechanicsand Rock Engineering, 40, 2007, 3–21.

5. Backstrom, A., Antikainen, J., Backers, T., Feng, X.T., Jing, L.R., Kobayashi, A., Koyama, T., Pan,P.Z., Rinne, M., Shen, B.T. and Hudson, J.A., “Numerical Modelling of Uniaxial CompressiveFailure of Granite with and without Saline Porewater”, International Journal of Rock Mechanicsand Mining Sciences, 45, 7, 2008, pp. 1126–1142.

6. Indraratna, B., Ranjith, P.G. and Gale, W., “Single Phase Water Flow through Rock Fractures”,Geotechnical and Geological Engineering, 17, 2009, pp. 211–240.

7. Zhang, X. and Sanderson, D.J., “Anisotropic Features of Geometry and Permeability in FracturedRock Masses”, Engineering Geology, 40, 1995, pp. 65–75.

8. Min, K.B., Rutqvist, J., Tsang, C.F. and Jing, L.R., “Stress-dependent Permeability of FracturedRock Masses: a Numerical Study”, International Journal of Rock Mechanics and Mining Sciences,41, 2004, pp. 1191–1210.

9. Baghbanan, A. and Jing, L.R., “Hydraulic Properties of Fractured Rock Masses with CorrelatedFracture Length and Aperture”, International Journal of Rock Mechanics and Mining Sciences,44, 2007, pp. 704–719.

385


10. Ulusay. R. and Hudson, J.A., “Determining Uniaxial Compressive Strength and Deformability ofRock Materials”, The Complete ISRM Suggested Methods for Rock Characterization, Testing andMonitoring: 1974–2006, pp. 153–156.

11. ITASCA Consulting Group, Inc. (2006) UDEC-Universal Distinct Element Code, Version 4.0,Vol. 1, 2 and 3, User’s Manual.

386

An Investigation of Numerical Damping forModeling of Impact

T. NISHIMURA∗

Tottori University

1. Introduction

When a falling rock block strikes the ground, the block does not bounce much. The blockloses the kinetic energy at impact. This energy loss can be simply represented by the coeffi-cient of restitution for mass point motion. This unconservative phenomenon is often mod-eled with the parallel placed spring-viscous dashpot system at contact point and the damp-ing applies to the mass point or the centroid of the block a force which is proportional tothe velocity but in the opposite direction as well-known in the Distinct Element Method(P.A. Cundall, 1971). This spring-dashpot system is often used for static problems as well asdynamic problems.

For static problems, the critical damping coefficient is often adopted to absorb vibrationsdue to initial or transient force imbalance and to demonstrate a static equilibrium state (ifit exists) with numerical stability. Dynamic problems, such as rockfall or rock avalancheanalysis, generally require less damping than static ones. Based on the equation of motionfor single mass-spring-dashpot system, Omachi et al. (1986) have derived a form of therelation between the coefficient of restitution and the fraction of critical damping in thenormal direction. In the tangential direction, the same system can be introduced but theselection of the damping coefficient can not be performed with certainty because the frictionaldissipation must be considered. The damping force is omitted if sliding occurs and for thenon-sliding, the closely related physical definition in the tangential direction, such as thecoefficient of restitution in the normal, does not exist. The value of the damping coefficientin the tangential direction is often assumed referring to the value in the normal direction.However, the tangential damping coefficient controls not only the energy loss but also thetangential component of velocity when the mass point takes off from the contact plane. Thevalue of the tangential damping coefficient should be carefully selected.

Numerical damping for modeling of impact using the spring-dashpot system, especially inthe tangential direction of contact plane, is investigated. To control the take-off tangentialvelocity component, the relation between the fractions of the critical damping coefficient inthe normal and the tangential is formulated. Numerical simulations on impact motion ofmass point to a plane are conducted using the time marching scheme for the equation ofmotion. The result shows the dependency of the take-off direction on the fractions of criticaldamping in the normal and the tangential. The formulation explains the dependency welland can be used for the damping coefficient determination in the numerical schemes whichintroduce the spring-dashpot system as well as in the distinct element analysis.




2. Impact of a Mass Point to a Plane

2.1. Restitution of reflecting velocity to incoming velocity

The most common used definitions for the bouncing phenomenon are expressed in terms ofrestitution of velocities:

Ren = −vn,r

vn,iRet = vt,r

vt,i(1)

where v is the translational velocity, the subscripts n and t stand for normal and tangential,and the subscripts i and r stand for incoming and reflecting (Fig. 1).

x

x

1

3

1 2

o

n

t

vi vr

Figure 1. Impact of a mass point to a plane.

The change in the momentum of the mass point is given as:

- tangential fttc = m(vt,r − vt,i) (2)

- normal fntc = m(vn,r − vn,i) = −m(1+ Ren)vi cosα1 (3)

where m and tc are the mass and the duration of contact, fn, ft are the components of thereaction force, vi is the impact velocity and α1 is the impact angle.

If the tangential component reaches the frictional resistance:

μ = ft

fn= fttc

fntc= (vt,r − vt,i)−(1+ Ren)vi cosα1

(4)

where μ(= tanφ) is the coefficient of friction. Then Ret is given by:

Ret = vi( sinα1 − μ(1+ Ren) cosα1)vi sin α1

= 1− μ(1+ Ren) cotα1 (5)

For tanα1 = μ(1 + Ren), Ret = 0, i.e. vt,r = 0, this means that the mass reflects in thenormal direction of the plane. For tanα1 < μ(1 + Ren), Ret < 0, i.e. vt, r < 0, the directionof the reflection is in the incoming side to the normal direction. However, this reflectionis unrealistic, the static friction angle φ is not fully mobilized so as to give vt, r > 0. Fortanα1 > μ(1 + Ren), the friction angle must be fully mobilized, and then the value of Retexists in the range of 0 < Ret < 1.

388


2.2. Modeling of impact using spring-dashpot-slider system

The above unconservative contact is often represented by a set of spring-dashpot-slider sys-tem as shown in Figure 2. The contact forces are related to displacements through the fol-lowing equation in incremental manner:

�fn = kn�x3 �ft = kt�x1dn = ηnx3 dt = ηtx1

(6)

where kn, kt are the normal and the tangential stiffness, and ηn, ηt are the damping coeffi-cients in the normal and tangential directions. Here, the following conditions are introducedto the above to explain no tension force and the frictional limit:

If fn < 0 set fn = 0, ft = 0 and dn = 0, dt = 0 (7)

If abs(ft) > μfn set ft = μfn · sign(ft), dt = 0 (8)

Contact

BlockBlock

t

nkn

kt

Contact

BlockBlock

t

nkn

kt

Figure 2. A set of spring-dashpot system with slider at contact point.

x3

contact

Qe- n t

Tdn= 2dn

tcvr

vit

0

Figure 3. The damped response of the mass point in the normal direction.

389


The following differential equations express the motion of the mass for the duration ofcontact:

mx1 + ηtx1 + ktx1 = 0 (9)

mx3 + ηnx3 + knx3 = 0 (10)

The Eq. (10) will give the following response with the condition of x3 = 0 and x3 = −vn, i att = 0:

x3 = −e−βnt vn,i

ωdnsinωdnt (11)

x3 = −(

cosωdnt− βn

ωdnsinωdnt

)vn,ie

−βnt (12)

where ωdn =√ω2

0n − β2n is the damped circular frequency, ω0n =

√kn/m is the natural

circular frequency and 2βn = ηn/m. The duration of impact is given by:

tc = π

ωdn= Tdn

2(13)

where Tdn is the damped period.At t = tc, the mass takes off from the plane, having the velocity in the normal direction:

vn,r = −vn,i exp(−βnπ

ωdn

)(14)

The coefficient of restitution Ren will be given by:

Ren = −vn,r

vn,i= exp

(−βnπ

ωdn

)or Ren = exp

⎛⎝− ζnπ√

1− ζ 2n

⎞⎠ (15)

where ζn is the fraction of critical damping in the normal direction and this fraction is givenby ζn = ηn/η0n in which η0n is the critical damping coefficient and η0n = 2

√mkn.

From the Eq. (9) with x1 = 0, x1 = vt, i at t = 0:

x1 = e−βtt vt,i

ωdtsinωdtt (16)

x1 =(

cosωdtt− βt

ωdtsinωdtt

)vt,ie

−βtt (17)

The take-off velocity in the tangential direction at t = tc:

x1|t= πωdn=(

cosωdt

ωdnπ − βt

ωdtsin

ωdt

ωdnπ

)vt,i exp

(−βtπ

ωdn

)(18)

The restitution of velocity in the tangential for no-sliding contact is given by:

Ret= vt,r

vt,i= exp

(−βtπ

ωdn

)(cos

ωdt

ωdnπ − βt

ωdtsin

ωdt

ωdnπ

)

=Ren exp

⎛⎝ ζtζn

√kt

kn

⎞⎠(cos

ωdt

ωdnπ − βt

ωdtsin

ωdt

ωdnπ

) (19)

390


Once sliding occurs, Ret becomes as shown in (20):

Ret = 1− μ (1+ Ren) cotα1 (20)

As described in the first chapter, the take-off tangential velocity depends on the value of therestitution coefficient when the friction angle is fully mobilized. When the friction angle is notfully mobilized, the spring-dashpot system will explain the energy dissipation at the contactpoint. The following condition is needed for Ret > 0 in Eq. (19):

cos(ωdt

ωdnπ

)− βt

ωdtsin(ωdt

ωdnπ

)> 0 (21)

According to the value of (ωdt/ωdn), this inequality is rewritten as follows:

(a) for 0 ≤ ωdt

ωdn<

12

βt

ωdt> tan

(ωdt

ωdnπ

)(22a)

(b) for12≤ ωdt

ωdn< 1 (Ret ≤ 0) (22b)

(c) for 1 ≤ ωdt

ωdn<

32

βt

ωdt< tan

(ωdt

ωdnπ

)(22c)

(d) for32≤ ωdt

ωdn< 2 Ret > 0 (22d)

The relation between Ren and ζn is derived in Eq. (15). The above conditions are alsoexplained with ζn and ζt as shown below, e.g. for (a):

(a) for 0 ≤√

kt

kn

√1− ζ 2

t

1− ζ 2n<

12

√1− ζ 2

t

ζ 2t

> tan

√kt

kn

√1− ζ 2

t

1− ζ 2nπ (21a’)

where ζt is the fraction of critical damping in the tangential direction.

Table 1. Examination of (ζn, ζ t) for kt/kn=1 to determine the pairs which give Ret > 0.( ) p g

(a)0< ds/ dn<1/2 (b)1/2< ds/ dn<1 (c)1< ds/ dn<3/2 (d)3/2< ds/ dn<2 2 < ds/ dn

tn

391


A pair value of (ζn, ζt) is grouped into the four ranges by the value of√

kt/kn√(1− ζ 2

t )/(1− ζ 2n ) and is examined with the following function:

f (ζn,ζt) =√

1− ζ 2t

ζ 2t− tan

√kt

kn

√1− ζ 2

t

1− ζ 2nπ (22)

Table 1 shows an example of the value of f (ζn, ζt) for kt/kn = 1 with the interval of 0.05of (ζn, ζt). In the table, the figures written in italic mean that the pairs of (ζn, ζt) satisfy theinequality in Eq. (21).

3. Simulation for Impact of a Mass Point to a Plane

Simulations for impact of a mass point were carried out as shown in Figure 1. In the sim-ulation, a mass point impacts with the incident angle α1 = 30◦ and the impact velocityvi = 10.0 m/s. The movement of the mass during the contact is simulated using the timemarching scheme for the equation of motion as described in the Section 2.2. The time step is�t = 1.0× 10−4 s and the gravitation is not considered.

Figure 4 shows the movement profiles during contact for kn = kt = 10 MN/m (kt/kn = 1)and φ = 90◦, i.e. slip never occur during contact and the natural period is the same in boththe normal and tangential direction. The mass point impacts the plane x3 = 0 at x1 = 0.Two movement profiles for the two values of ζn with the value of ζt = 0.15 are displayed inthe figure. For ζn = 0.75, the take-off direction of the mass point is in the opposite side ofthe incoming side, for ζn = 0.70, the direction is in the incoming side.

Figures 5 and 6 show the changes in the velocity components and x3 coordinate (normaldirection of the plane) during the contact shown in Figure 4. For ζn = 0.70 in Figure 5,the mass leaves from the plane at t = 0.033 s with vt,r < 0 and, after taking off, the massflew with the constant velocity. In Figure 6, the mass reflects having vt,r > 0 after the longercontact period than the case of ζn = 0.70. The contact period depends on the damped period2π/ωdn and the ratio of ωdt/ωdn is the governing parameter of the reflecting direction asdescribed in the Chapter 2.

x1 (m)

x 3(m

)

n=0.70 t 0.15, =90° and kt/kn=1.0

n=0.75 t 0.15, =90° and kt/kn=1.0

SurfaceImpact

Flight

Figure 4. Movement profiles during contact for kt/kn = 1 with non-sliding.

392


Time (s)

v n,v

t(m

/s)

n=0.70 t 0.15, =90° and kt /kn=1.0

x 3 (m

)

x3

vt

vn

Figure 5. Changes in velocity components and x3 coordinate for the case of vt,r < 0 as shown inFigure 4.

v n,v

t(m

/s)

Time (s)x 3

(m)

n=0.75 t 0.15, =90° and kt /kn=1.0

x3

vt

vn

Figure 6. Changes in velocity components and x3 coordinate for the case of vt,r > 0 as shown inFigure 4.

Figure 7 shows the case of (ζn, ζt) = (0.36,0.36) for kn = 10 MN/m,kt=1 MN/m (kt/kn =0.1) and φ = 90◦, i.e. the natural period of the tangential direction is longer than the periodof the normal. The mass leaves from the plane at t=0.025s with vt,r > 0. This is the casegrouped (a) in which the tangential velocity decrease with the effect of the dashpot but themass point takes off from the plane before the tangential velocity becomes negative. Thesesimulations support the condition written in the Chapter 2 to evaluate the reflecting directionand to remove the conditions of the reverse-reflection.

4. Conclusions

When we introduce this consideration into the modelling using a rigid body in which theblock has its own shape and volume, the effect of the consideration is sometimes invisible.

393


v n, v

t(m/s)

n=0.36 t 0.36, =90° and kt /kn=0.1

Time (s)

x 3 (m

)

x3

vt

vn

Figure 7. Changes in velocity components and x3 coordinate for the case of vt,r > 0.

This is because not only the translational motion but also the rotational motion must be for-mulated in the rigid body simulation. The equations of motion are formulated at the positionof the geometrical centroid but not at the contact point where the spring-dashpot system isplaced. A reverse reflection could also occur depending on the position of the contact point(s)to the centroid. However, this consideration will not reduce the effect. Each force at the con-tact points must be evaluated with a model which does not reproduce unrealistic responsessuch as the reverse reflection as shown in Figure 4. The reflecting direction of the mass pointor the block depends on the fraction of critical damping. The following conclusions arededuced from the theoretical results:

• The restitution of velocity in the tangential direction for non-sliding contact is for-mulated with using the restitution coefficient.• The restitution of velocity in the tangential direction using the spring-dashpot system

is formulated with the damped circular frequency.• When using the spring-dashpot system for the impact problem of mass point, the sys-

tem produces the reverse-reflecting in the tangential direction depending on the ratioof the damped circular frequency. The condition producing the reflection was inves-tigated and removed with the consideration to the ratio of the tangential dampedcircular frequency to the normal one.

Acknowledgements

This work is partly supported by Japanese Society for the Promotion of Sciences (JSPS),Grant-in-Aid for Science Research, No.20260462.

References

1. Cundall, P.A., A computer model for simulating progressive, large-scale movements in blocky rocksystems”, Symposium on Rock Mechanics, Nancy, Vol. 2, 1971, pp. 129–136.

2. Omachi, T. and Arai, Y: How to determine mechanical properties for Distinct Element Method,Journal of Structural Engineering, JSCE, Vol. 32A, 1986, pp. 715–723.

394

Development of Modified RBSM for Rock Mechanics UsingPrinciple of Hybrid-type Virtual Work

N. TAKEUCHI1,∗, Y. TAJIRI2 AND E. HAMASAKI3

1Department of Mechanical Engineering, Hosei University, Japan2Graduate School of Art and Technology, Hosei University, Japan.3Advantechnology Co.,Ltd., Japan.

1. Introduction

The Rigid-Bodies Spring Model (RBSM) is a generalized model for discrete limit analysis, andassumes a rigid displacement field.1 This model was applied to the non-linear problem suchas the metallic material at first, but it came to be applied to a discrete limit analysis about thesoil and rock foundation. Since this model assumes rigid displacement field, it can evaluateneither the stress in each element, nor element deformation. On the other hand, authorsdeveloped the Hybrid-type Penalty Method (HPM) which assumes the linear displacementfield using the principle of hybrid type virtual work (PHVW).2, 3 The HPM is based on thediscontinuous Galerkin (dG) method,4, 5 and applies the concept of the spring of RBSM inLagrange multiplier.

First, in this paper, we describe the method of computing the stress in each element inRBSM by applying the rigid displacement field to the displacement field of this HPM. Then,we propose new numerical model which improves the rigid displacement of RBSM usingPHVW. Finally, we apply this model to simple problem, and examine the accuracy of thesolution of the proposed method.

2. Hybrid-Type Virtual Work and Discretization Equation

2.1. Hybrid-Type virtual work (weak form)

Let � ⊂ Rndim, with (1 ≤ nndim ≤ 3), be the reference configuration of a continuum body

with smooth boundary �: = ∂� and closure �: = �∪ ∂�. Here Rndim is the ndim dimensional

Euclidean space. The local form of the equilibrium equation for a deformable body is asfollows:

divσ + b = 0 in � (σ = tσ ) (1)

where b is the body force per unit volume, σ is the Cauchy stress tensor respectively. u isa displacement field of particles with reference position x ∈ � and denote the infinitesimalstrain tensor by

ε = ∇su: = 12{∇u+ t(∇u)} (2)

where ∇ := (∂/∂xi)ei is the differential vector operator, ∇s shows the symmetry part of ∇.In what follows, we assume that the boundary � = �u ∪�σ (� = �u ∪�σ ,�u ∩�σ = ∅). Here�u := ∂u�⊂ ∂� where displacement are prescribed and �σ := ∂σ�⊂ ∂� where tractions t




:= σn are prescribed as

n|�u = u (given) (3)

σ |�σ n = t (given) (4)

Here n is the field normal to the boundary �σ . The constitutive equation to the elastic bodyis provided as follows by the use of the elasticity tensor D.

σ = D : ε (5)

Let � consist of M sub-domains �(e)⊂� with the closed boundary �(e): = ∂�(e) as shown inFig. 1(a). That is � = ∪M

e=1�(e), here �(r) ∩�(q) = 0 (r = q).

(a) Sub-domain (b) Common boundary of sub-domain and

Figure 1. Sub-domain and its common boundary.

We denoted by �<ab> the common boundary for two subdomain �(a) and �(b) adjoined asshown in Fig. 1(b), and which is defining as �<ab>: = �(a) ∩�(b). The relation for u(a) andu(b) are following:

u(a) = u(b) on �<ab> (6)

They are the displacements on �<ab> that is intersection boundary in �(a) and �(b) sub-domain. It introduces this subsidiary condition into the framework of the variational equa-tion with Lagrange multipliers λ as follows:

Hab =def δ

∫�<ab>

λ · (u(a) − u(b))dS (7)

where δ(• ) shows the variation of (• ). Physical meaning of the Lagrange multiplier λ is equalto the surface force on the intersection boundary �<ab>.

As described above, the hybrid type virtual work equation is as follows about M subdo-main and N intersection boundary:

M∑e=1

(∫�(e)

σ :grad∂u dV −∫�(e)

f · ∂udV −∫�(e)

t · ∂udS)

−N∑

s=1

(∂

∫�<s>

λ ·(u(a) − u(b)

)dS)= 0 ∀∂u∈V (8)

396


2.2. Discretization equation for HPM

Compatibility of the displacement on the intersection boundary is approximately introducedusing the penalty as a spring constant. Therefore, the displacement filed can be assumed foreach element without restraining by the condition of compatibility. So, we assume the lineardisplacement field u(e) with rigid displacement and rotation d(e) and constant strain ε(e) ineach sub-domain �(e) as follows:

u(e) = N(e)d d(e) +N(e)

ε ε(e) (9)

Since it has the meaning that Lagrange multiplier λ is the surface force on the boundary�<ab> in sub-domain �(a) and �(b), the surface force is defined as follows:

λ<ab> = k · δ<ab> (10)

Here, δ<ab> shows relative displacement on the sub-domain boundary �<ab>, and it is shownin two dimensional problems as follows:{

λn<ab>λt<ab>

}=(

kn 00 kt

){δn<ab>δt<ab>

}(11)

where, δn<ab>, δt<ab> are relative displacement in the normal and the tangential direction tothe sub-domain boundary �<ab>. Similarly, λn<ab>, λt<ab> are Lagrange multipliers in thenormal and tangential direction of the surface forces. The HPM can be described as followsby penalty function p use as coefficient k.

kn = kt = p (12)

The following discretization equations are obtained by substituting Eqs. (9) and (10) for Eq.(8).

M∑e=1

(tδU(e)K(e)U(e)

)+

N∑s=1

(δU<s>K<s>U<s>) =M∑

e=1

(tδU(e)P(e)

)(13)

K(e) =∫�(e)

tB(e)D(e)B(e)d�, P(e) =∫�(e)

tN(e)f d�+∫�σ

tN(e)td�,K<s> =∫�<s>

tB<s>kB<s>d�

ε(e) = B(e)U(e),δ<ab> = B<ab>U<ab>

where U(e) is the degree of freedom concerning the sub-domain �(e) and U<ab> is the degreeof freedom of the sub-domain boundary �<ab>. Since virtual displacement δU is arbitrary,we obtain the following discretized equations.

KU = P (14)

K =M∑

e=1

K(e) +N∑

s=1

K<s>, P =M∑

e=1

P(e)

When U of Eq. (14) is represented in d and ε, the discretization equation is as follows inmatrix form: [

Kdd KdεKεd Kεε +D

]{dε

}={

PdPε

}(15)

The discretization equation of this model becomes a simultaneous linear equation shown inEquation (14).

397


3. HPM and RBSM

Equation (15) shows the discretization equation of HPM with linear displacement field bymatrix form. On the other hand, RBSM assumes rigid displacement field as follows:

u(e) = N(e)d d(e) (16)

If these relations are applied to formulation of above-mentioned HPM, Eq. (15) is as follows:

Kddd = Pd (17)

The detailed descriptions of this equation are as follows:

N∑n=1

[K<sasa>

dd K<sasb>

dd

K<sbsa>

dd K<sbsb>

dd

]{d(sa)

d(sb)

}=

N∑s=1

⎧⎨⎩

P(sa)d

P(sb)d

⎫⎬⎭ (18)

Here, the coefficient matrix is as follows:

k<ab>dd =

∫�<ab>

tN(a)d kB

(b)d d� (19)

In HPM, k of (19) means a penalty function assumed in Equation (10) and (12). On theother hand, in RBSM, k is supposed to be a spring constant as shown in Fig. 2, and, in thetwo dimensional case, it is defined as follows:{

λnλs

}=[kn 00 ks

]{δnδs

}(20)

Here, in the case of a plane strain, kn, ks are as follow:

kn = (1− ν)E(1− 2ν)(1+ ν)(h1 + h2)

, ks = E(1+ ν)(h1 + h2)

(21)

E is elastic modulus, ν is Poisson’s ratio and h1, h2 is height of perpendicular line given to theintersection boundary from the center of figure of each element. When this spring constant isused, Equation (17) accords with a discretization equation of RBSM[1].

kn

u

v

1

2

u

v

3

4

5

6P

ks

P'

P"

s

n

before deformation after deformation

Figure 2. Rigid bodies-spring model.

398


4. Modified RBSM

4.1. Stress within element of RBSM

The relation of Eq. (15) can also be written as follows:

Kddd = Pd − Kdεε (22)

Dε = Pε − (Kεdd+ Kεεε) (23)

The discretization equation of RBSM is the same as the case of ε = 0 of equation (22). Thus,RBSM is the first order approximation of HPM which assumed the rigid displacement field.

On the other hand, although Eqs. (22) and (23) are simultaneous equations, in this paper,Eqs. (22) and (23) are solved iteratively. As a result, the approximate stress within an elementis obtained by using the rigid displacement which is the solution to equation (17).

Dε = Pε − Kεdd (24)

As shown in Fig. 3, Equation (24) is expanded by making into an example element (1) andelement (2)–(4) which adjoins it. At this time, the integration about the intersection boundaryof a focused element (1) is related only to adjoining element (2)–(4), and it becomes indepen-dent on simultaneous equations with other elements. A part of Equation (24) to this exampleis expressed as follows:⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

. . .A(1)D(1) 0

A(2)D(2)

A(3)D(3)

0 A(4)D(4)

. . .

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

...ε(1)

ε(2)

ε(3)

ε(4)

...

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

...P(1)ε

P(2)ε

P(3)ε

P(4)ε...

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭−

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

. . .0 3k(1,1)

ε k(1,2)ε k(1,3)

ε k(1,4)ε 0

k(1,2)ε k(2,2)

ε

k(1,3)ε k(3,3)

ε

k(1,4)ε k(4,4)

ε

. . .

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

...d(1)d(2)d(3)d(4)

...

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭

(25)

(4) (3)

(2)

(1)

Figure 3. Element (1) and adjoining element (2)–(4).

399


Here, A(e) expresses the area of the e-th element. A(e)D(e) in Eq. (25) is independent foreach element, and the equation about the Eq. (1) element is expressed as follows.

A(1)D(1)ε(1) = P(1)ε −

(3k(1,1)

εd d(1) + k(1,2)εd d(2) + k(1,3)

εd d(3) + k(1,4)εd d(4)

)(26)

The right side is the force that multiplied stress in the element by an element area and theleft side means force in the element by the surface force in the intersection boundary. Stressin the element is calculated approximately by these forces being balanced.

4.2. Development of modified RBSM using iterative algorithm

This stress in the element is calculated by a rigid body displacement solved in defiance of astrain in Equation (22). In other words, a rigid body displacement is not improved by thismethod. Therefore a strain solved from Equation (26) is substituted for Equation (22), anda solution is improved iteratively by computing again.

Kdddn+1 = Pd − Kdεεn (27)

Here, n is the iteration number of times and is assumed n = 1 in this paper. However, by thestep of the iteration, a penalty function is used as a spring constant.


5.1. Plate with a circle hole

Figure 4 shows the numerical modelling for plate with a hole acting tensile force.

Figure 4. Numerical model and material properties for plate with a hole.

(a) modified RBSM (b) HPM (linear) (c) HPM (quadratic)

Figure 5. vonMises stress distribution.

400


A

B

RBSM Modified RBSM

(a) displacement at Point A and B (b) displacement mode

Figure 6. Displacement.

Stress distribution of vonMises is shown in Fig. 5. Figure 5(a) is a solution by proposedmethod. Figure 5(b) is a solution by HPM with the linear displacement field, and the Fig. 5(c)assumed quadratic displacement field. The solution of HPM with linear displacement fieldaccords with a solution by the constant strain element in the FEM (CST). The maximumstress occurred in element (a red part) of the hole upper part. The difference of the stress ofpresent method (1.717 kN/mm2) and HPM (quadratic) (1.778 kN/mm2) was about 3.5%.The distribution state of the stress shows an approximately similar tendency in this methodand HPM.

Figure 6 shows the displacement mode and Table 1 shows the displacement value at pointA and B. The difference of modified RBSM and HPM (linear) & FEM (CST) was a littleless than 2%. In the Fig. 6(b), it is comparison in displacement mode of RBSM and modifiedRBSM. Because RBSM is rigid displacement field, an unnatural gap occurs between elements,but it is canceled in modification RBSM.

Table 1. Displacement at Point A and Point B.

Method Point A Point B

Modified RBSM 0.292 0.521RBSM 0.316 0.541HPM (linear) & CST 0.287 0.512HPM (quadratic) 0.310 0.532

P/2P/2

h h h h

h

h/2 h/2

(h=200mm)

150N150N

(a) numerical model (b) mesh division (1487 elements )

Figure 7. Non-reinforced concrete beam.

401


(a) modified RBSM (b) HPM (quadratic)

Figure 8. vonMISE stress distribution.

(a) modified RBSM (b) HPM

Figure 9. Displacement mode.

5.2. Non-reinforce concrete beam

Figure 7 shows the numerical model for non-reinforced concrete beam subjected two-pointloading. Figure 7(a) shows numerical modelling, and a Fig. 7(b) expresses mesh division.

VonMises stress is shown in Fig. 8. Figure 8(a) is a solution by modified RBSM, andFig. 8(b) is a solution by HPM (quadratic). The ranks of the color contour lines are unifiedwith both.

Distribution of a contour differs a little at loading and supporting point because processingof the load and support condition differ by RBSM and HPM. The stress of modified RBSMin the bottom edge of the central part of the span is 2.34MPa, and the stress of HPM is2.29MPa and is about 2.2% of difference.

Table 2. vonMises stress and deflection at center of span.

Method σmises (MPa) δ(mm ×10−3)

Modified RBSM 2.34 4.60RBSM 2.34 4.76HPM (linear) & CST 2.38 4.22HPM (quadratic) 2.29 4.46

About the deflection of the central part of the span, the difference of modified RBSM andHPM (quadratic) is about 3%. These values are shown in Table 2.

402


6. Conclusions

In formulation of RBSM, surface force in the intersection boundary was developed by thedynamics of the spring having a physical meaning. However, when the rigid displacementfield was applied in the formulation of HPM based on the PHVW, it was shown that thecompletely same discretization equation as original RBSM is obtained. That is, RBSM canbe considered with a special case of HPM having rigid displacement field.

In this process, relations of strain in the element and rigid displacement can be obtained.The method to calculate stress in the element with these relations was proposed, and theaccuracy of the solution was verified with a simple numerical examples. As a result, the goodapproximate value was able to be calculated.

Furthermore, modified RBSM which improved the accuracy of the rigid displacement bythe iterative processing was proposed. The displacement by this method became the valuethat was near to HPM with quadratic displacement field.

The proposed method uses a result obtained by discrete limit analysis with RBSM, and theapplication to other technique such as DEM is possible.

References

1. Kawai, T., “New element models in discrete structural analysis”, Journal of the Society of NavalArchitects of Japan, 114, 1977, 1867–193.

2. Takeuchi, N., Ohki, H., Kambayashi, A. and Kusabuka, M., “Material nonlinear analysisby using discrete model applied penalty method in hybrid displacement model”, Transactionsof JSCES, Paper No 20010002, 2001, 52–62. (http://www.jstage.jst.go.jp/article/jsces/2001/0/20010002/_pdf/-char/ja/)

3. Ohki, H. and Takeuchi, N., “Upper and low bound solution with hybrid-type penalty method”,Transactions of JSCES, No. 20060020, 2006, 1–10. (http://www.jstage.jst.go.jp/article/jsces/2006/0/20060020/_pdf/-char/ja/)

4. Amold, D.N., Brezzi, F. Cockburn, B., and Marini, L.D., “Unified analysis of discontinu-ous Galerkin methods for elliptic problems”, SIAM Journal on Numerical Analysis, 39, 5,2002, 1749–1779.

5. Mergheim, J., Kuhl, E. and Steinmann, P., “A hybrid discontinuous Galerkin/interface method forthe computational modeling of failure”, Communications in Numerical Methods in Engineering,20, 2004, 511–519.

403

High Rock Slope Stability Analysis Using the Meshless Shepardand Least Squares Method

X. ZHUANG1,2,∗, H.H. ZHU1 AND Y.C. CAI1

1Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education,Department of Geotechnical Engineering, Tongji University, Shanghai, 200092, China2The School of Engineering and Computing Science, Durham University,South Road, Durham, DH1 3LE, United Kingdom

1. Introduction

The evaluation of safety factors of potential slip surface is always an important but notwell addressed topic in rock slope stability analysis. There are two problems need to beaddressed, namely, how to properly model the influence of discontinuities on stress state theslip surface and how to evaluate the safety factors. Conventional methods usually introduceassumptions on internal forces for specific failure modes, e.g., balances of force or momentin a circular mode for a heavily fractured slope or in a wedge mode for blocky rock system.However, a substantial rock slope is rarely ideal or as simple as assumed due to varying posi-tions, shapes, orientations and complex spatial spreading and cross link of joints. Findingthe proper assumptions of internal forces becomes difficult and an improper characteriza-tion of internal forces will result in predictions deviating potentially dangerously. Numericalmethods fare a little better, for example the strength reduction method has been widely usedand is free of any assumption of internal forces according to the failure mode. This methodis associated with the finite element approaches but also has been applied to discontinu-ous approaches such as the discrete element method (DEM) and discontinuous deformationanalysis (DDA). However, this method changes the mechanical and physical properties of thegeo-material during the analysis can make the final state of the material totally different fromits true and original state. Besides, the need for nonlinear analysis can significantly increasethe computational cost and requires robust software.

A meshless method to analyze rock slope stability based on a meshless model is describedin this paper. The meshless Shepard least squares (MSLS) method is used to model the dis-continuous stress field near the joints and the stress results are used to evaluate the safetyfactor of potential slip surface. There is no assumption on the internal forces as in analyticalmethods and no requirement for non-linear analysis in the strength reduction method. Detailsare provided of meshless simulation of jointed rock and determination of safety factor. In theend, the method described here is used to analyze a high rock slope on the left bank of YalongRiver where the junction of Jinping hydropower station is located. The results obtained bythe present method are compared with analytical methods showing the implementation effec-tively to model the influence joints on potential slip surface and to evaluate the safety factor.They also show the present method outperforms the conventional methods in modelling theeffects of supporting structure on the slope stability.




Figure 1. An arbitrary analysis domain.

2. The Stress Analysis of Jointed Rock by the MSLS Method

Consider a block of rock with some discontinuities as a problem domain� shown in Fig. 1. Ameshless model can be built by discretising the domain into a set of N scattered nodes. Takean arbitrary node from these nodes, node i, for example, its support domain �i is definedby a circle of radius dmi as shown in Figure 1 and any point inside �i is influenced by thenode I, and the superscript or subscript i indicates the node index {i = 1,2, . . . ,N}. The intactpart of rock and the joint are separately modelled in the MSLS method. The intact part ofrock is modelled as continuum and the discontinuous stress field near the joint is captured bythe jump of displacement interpolations. Here, the visibility criterion1 is used for the jumpfunction. For example, the support domain of node j and k are cut by the joint as shown inFigure 1 and only the parts light gray parts are used for displacement interpolation.

To get the displacement interpolation over the domain, the local approximation at eachnode is firstly constructed. If the distance between a node and node i is less than dmi, thenode is selected to construct the local approximation at node i. Suppose M selected for nodei, then the local approximation is given by

uLi(x) =M∑

J=1

�iJ(x)uJ = �

iu (1)

where

�i(x) = [�i

1(x)−�i1(xi), · · · , 1+�i

i(x)−�ii(xi), · · · ,�i

M(x)−�iM(xi)] (2)

and

�iJ(x) =

m∑j=1

pj(x)(A−1B)jJ = (3)

are the shape function of supporting nodes obtained by a standard least squares approach.2

The subscript J is the index of supporting nodes and {J = 1,2, . . . ,M}, pj(x) is a monomial

406


in the polynomial basis function p(x) = [1,x,y,xy]T and m are the number basis, e.g., m = 4for the bilinear basis in two dimensions. A and B are

B = PT = [p(x1) p(x2) · · · p(xM)]

=

⎡⎢⎢⎢⎣

1 1 · · · 1x1 x2 · · · xMy1 y2 · · · xM...

......

...

⎤⎥⎥⎥⎦ (4)

and

A = PTP (5)

respectively. With the local approximation defined, the displacement at any point x inside thedomain is the partition of unity (PU) of local approximations of all supporting nodes at x

u(x) =n∑

i=1

φ0i (x)uLi(x) (6)

Here, the Shepard shape function is used as the PU function that

φ0i (x) = wi(x)

n∑i=1

wi(x)(7)

and n is the number of nodes containing x within the support domains and wi(x) is the weightfunction defined at node i. The weight function used here is a singular function3

wi(x) =

⎧⎪⎨⎪⎩

d2mi

d2i + ε

cos2(πdi

2dmi

), di ≤ dmi

0 di > dmi

(8)

where dmi is the radius of nodal support, di is the distance between the point of interest andnode i and ε is a small value parameter to least the a zero value in denominator in Equation 1and here ε = 1e− 10 is used in analysis. The use of this singular weight functions makes thePU functions satisfying the delta property so that the prescribed displacement can applieddirectly as in the FEM.

The MSLS method is a new kind of meshless method, which removes the difficulty inapplying the essential boundary conditions that appeared in many of the meshless methodsbased on the moving least squares (MLS) approximations. It possesses the delta propertyand preserves the completeness of the field up to the order of the basis. The proof of theseproperties and the choice of support radius dmi can be referred to Ref. 2. In the followingsections, we apply the MSLS method in analyzing rock slope stability.

407


3. Determination of Safety Factors and Implementations

The safety factor is calculated by

F =

∫l

(c+ σn tanφ) dl

∫lτndl

(9)

as a result of the resistant force over sliding forces along the sliding surface of length l in twodimensions, where σn and τn are the normal stress and shear stress, φ and c are the frictionalangle and cohesion of the material.4 The slope stability analysis follows these steps. Firstly,the stress field of a rock slope containing discontinuous stress around joint are analyzed.Secondly, the stress results along the potential sliding surfaces needed to compute the safetyfactor are recovered using a proper stress recovery method. Then, the sliding force and resis-tant force along the sliding surfaces are obtained using a certain integration scheme and thesafety factor is calculated using Equation 8. The Mohr-Coulumn failure criterion is used inthe case study while other failure criterions such as Tresca or tensile failure may also be used.

4. A Case Study on a High Rock Slope at Jinping Hydropower Station

In this section, the method described in this paper is used to analyze a high rock slope onthe left bank of Yalong River, where the junction of Jingping hydropower station is located.The cross section taken for analysis in 2D is shown in Fig. 2. The dashed curve shows thenatural slope before excavation and the solid line shows the slope profile after excavation.Two potential sliding modes, mode A and mode B are predicted in 5 based on the analysis ofspatial spreading and cross link of joints and the most unfavourable combinations of joints.According to the design scheme in 5, anchors and nails as supporting structures need to beinstalled during excavation and are shown as several sets of parallel slant lines in Fig. 2. The

(a) Division of slices of sliding mode A (b) Division of slices of sliding mode B

Figure 2. The cross section of the slope on the left bank of Yalong River taken for analysis.

408


safety factors of the slope with and without supporting structures are evaluated respectivelyfor each of the two modes using the method described here and are also compared with theresults in 5 of two slice methods, namely, Sarma method and unbalanced thrust method.

Figure 3. The meshless model of the slope on the left bank of Yalong River.

Figure 4. Contour plot of maximum principle stress σ1 of the slope by the MSLS method (kPa).

4.1. Sliding mode A

The predicted sliding surface of mode A is shown as a thick black curve in Fig. 2(a). Ameshless model is built accordingly as shown in Fig. 3. Figure 4 shows the stress results by theMSLS method indicating high gradient of stress near the joints. The maximum tensile stress

409


Table 1. The material parameters of slices (mode A).

Index of slices 1 2 3 4 5 6 7 8

Along the cohesion (KPa) 20 20 20 20 20 20 20 20sliding surface frictional angle (◦) 16.7 16.7 16.7 16.7 16.7 16.7 16.7 16.7Interface dohesion (KPa) 1000 1000 1000 1000 1000 893 812 N/Abetween slices frictional angle (◦) 45.6 45.6 45.6 45.6 45.6 43.3 41.4 N/A

Table 2. The material parameters of slices (mode B).

Index of slices 1 2 3 4 5 6 7 8

Along the cohesion (KPa) 1000 1000 1000 1000 1000 20 20 20sliding surface frictional angle (◦) 45.6 45.6 45.6 45.6 45.6 16.7 16.7 16.7Interface dohesion (KPa) 1000 1000 1000 1000 1000 893 812 N/Abetween slices frictional angle (◦) 45.6 45.6 45.6 45.6 45.6 43.3 41.4 N/A

Table 3. The safety factors by different methods and with or without supporting structures.

Mode Case /Method

Excavation with nosupporting structures

Excavation with designedsupporting structures

ASarma 0.724 1.395

Unbalanced thrust 0.412 0.995MSLS based method 0.487 1.344

BSarma 2.160 3.292

Unbalanced thrust 2.041 3.457MSLS based method 2.391 4.330

appears at the crosslink of two controlling joints about 3MPa. Safety factors are evaluated forthe predicted sliding surface by the present method and are compared with the slice methodsin Ref. 5 in Table 3. The division of slices used for the slice methods are marked with number1-8 as shown in Figure 2(a) and the material parameters are listed in Table 1. It can be seenfrom the Table 3 that without supporting structure the safety factor of the present method isclose to the unbalanced thrust method and is lower than the Sarma method. With supportingstructures added, the safety factor is significantly increased by the present method and theincrease in more significant than by the Sarma method or the unbalanced thrust method.

4.2. Sliding mode B

In mode B, the sliding surface initiates between lamprophyre dyke and the boundary ofweakly unloaded area shown as a thick black curve in Figure 2(b). In Table 3, the safetyfactors of the predicted sliding surface are evaluted by the present method and are comparedwith slice methods in Ref. 5. The division of slices used in Ref. 5 is shown in Figure 2(b) andthe material parameters are listed in Table 2. Results show the safety factor of the presentmethod slightly higher than the slice methods without supporting structures and about 30%higher than the slice methods with supporting structures.

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4.3. Remarks on results

The results show the sliding mode A more likely than B after excavation. In mode A, thesafety factor becomes less than 1.0 in all methods after excavation indicating supportingstructures needed. With supporting structure added to the slope, the safety factor is increasedin both modes. However, the increase in the present method is much higher the slice methodsand this is due the different way of modelling structures. In the present method, rock andsupporting structures are deforming compatibly and their stress redistributed. While in theanalytical methods, rigid body assumptions are used so the effects of structure on the slopestability can not be correctly modelled.

5. Discussions

The method described in this paper requires no prior assumptions on balances of internalforce and thus can be used as a general method for rock slope stability analysis. The eval-uation of satey factors here is based on the stress analysis using the MSLS method for elas-tostatics; however other meshless methods are equally viable, e.g., the element-free Galerkinmethod (EFG) and the meshless local-Petrov Galerkin method (MLPG). Results show thepresent method effectively evalute the safety factors along slip surface and outperforms theslice methods in modelling the effects of supporting strtures on rock slope stability. The lim-itation of the present method is that the slope failure is not modelled in a progressive waywhere the severity of sliding may be alleviated by stress redistribution of along sliding surface.The Mohr-Coulumn criterion is used here and other material failure criterion such as Trescacriterion or maximum tensile stress failure may also be applied with the present method. Thepotentials of the present method can be explored in 3D analysis where the assumptions onslip surface can cause results deviating significantly from the substantial cases. We expect tofinish these works in our subsequent papers.

Acknowledgement

The authors gratefully acknowledge the support of the Joint Fund of Yalong RiverHydropower Development, NSFC (50579093). The first author is supported by a DorothyHodgkin Postgraduate Studentship at Durham University.

References

1. Belytschko, T., Liu, Y. and Gu, L., “Element-free Galerkin Methods”, International Journal ofNumerical Methods in Engineering, 37, 1994, pp. 229–256.

2. Cai, Y.C., and Zhu H.H. “A local meshless Shepard and least square interpolation method basedon local weak form”. Computer Modeling in Engineering and Sciences (CMES), 2008, 34(2):179–204.

3. Lancaster, P., and Salkauskas K. “Surfaces generated by moving least squares methods”. Mathe-matics of Computation, 1981, 37: 141–158.

4. Zhuang, X., Cai Y.C. and Zhu, H.H., “Rock slope stability analysis based on meshless methodand shortest path algorithm”. Chinese Quarterly of Mechanics, 2008, 29(4): 537–543.

5. The internal research report on the slope stability and scheme of supporting structure at theleft bank of hydro-junction of Jinping hydropower station, School of Water Conservancy andHydropower Engineering, Wuhan University, China, 2005.

411

Numerical Modelling of Laboratory Behaviour of Single LaterallyLoaded Piles Socketed into Jointed Rocks

W.L. CHONG1,∗, A. HAQUE1, P.G. RANJITH1 AND A. SHAHINUZZAMAN2

1Department of Civil Engineering, Monash University, Australia2Powerlink, Queensland, Australia

1. Introduction

Pile foundations are commonly used to provide support for a wide range of laterally-loadedstructures, such as transmission towers, offshore platforms, bridge foundations and retain-ing structures. These foundations can be constructed in either soil, rock or a combinationof both. Substantial research has been conducted in the past on the performance of laterallyloaded piles founded in soils (Broms, 1964). However, rock mass which is usually a discon-tinuous entity as opposed to soils requires a different design approach. The overall strengthand the stiffness of jointed rock mass are highly variable depending on secondary structuressuch as joint spacing, orientation, persistence, roughness and presence of infill. All these fac-tors may have significant impact on the lateral load capacity of single piles. Chong et al.(2008) carried out a comprehensive review of the published methods for design of laterallyloaded piles socketed in rock concluding that these methods do not physically consider allthese influential factors. Some of the methods assumed the rock to be intact (Carter and Kul-hawy, 1992; Reese, 1997), or employed rock mass classification systems, such as Rock MassRating (RMR) (Bieniawski, 1989) or Geological Strength Index (GSI) (Hoek et al., 1988)to account for the secondary structure in the rock therefore assuming it to be homogeneousand isotropic in their analyses (Zhang et al, 2000; Gabr et al., 2002; Yang, 2006). There hasbeen very few studies conducted using numerical modelling to study laterally loaded pilessocketed into rock. Recently, Yang (2006) used a 3D finite element modelling (FEM) code,ABAQUS, to predict the full-scale lateral pile load behaviour considering the rock mass asa homogenous medium. However, the model was unable to capture the non-linearity of theload-deflection response. It is evident that further research is required to provide a full under-standing of the complex load-deflection behaviour of the pile-rock mass systems under lateralload conditions.

To reliably predict the load-deflection (P–y) behaviour, it is crucial to study the influenceof physical joint sets in rock on the performance of pile-rock systems. Experiments on pilesembedded in jointed rock mass were conducted by Francis et al. (2004). They reported asignificant drop in lateral capacity of the piles socketed into jointed rock mass compared tointact rock and employed 2D UDEC to simulate the laboratory behaviour of the laterallyloaded piles. However, the predictions of the load-deflection (P–y) behaviour of the pile wereinconsistent especially in the estimation of the ultimate lateral capacity. It was hypothesisedthat this inconsistency was due largely to the three-dimensional nature of the problem. Thus,this study investigates the numerical modelling of laboratory tests of laterally loaded singlepiles using a commercial 3D Distinct Element Code (3DEC). The key modelling parameterswere chosen based on trial and error to match the laboratory test results (Francis et al, 2004).In addition, this study also investigates the effect of joint orientations on the load-deflection(P-y) behaviour of laterally loaded piles.




Figure 1. (a) Johnstone rock mass with two joint sets. (b) Conceptual sketch of rock mass fromlaboratory testing (Francis, 2003).

2. Laboratory Experiment

In this study the laboratory test results of Francis (2003) were used. These were based onlaterally loaded single piles socketed into jointed rock masses of various joint set geome-tries and orientations. A synthetic rock known as Johnstone (Johnston and Choi, 1986), theproperties of which resemble the properties of mudstone rock local to Victoria, Australia,was employed. Two joint sets, as shown in Fig. 1, were cut into the cured rock blocks using arock saw. Timber planks were used to clamp the jointed rock mass to ensure tight joint con-tacts. The model pile was a 25mm diameter, solid aluminium bar. A core barrel was drilledinto the rock block to create a cavity for the cylindrical aluminium pile. A customised plasterwhich consisted of Portland cement and plaster was used to fill the gap between the pile-rockinterface. Displacement transducers were positioned to measure the lateral displacement ofthe pile close to the rock surface and a laterally-driven ram was utilised to apply the lateralforce. Figure 1 shows the overall set up of the experiment. Two sets of joint configurations,(45/45) and (30/60) were used. Where (30/60) means the first joint set has a dip angle of 30◦while the second has a dip angle of 60◦.


3.1. Model set up

Considering the jointed rock mass as a discontinuum medium, the discrete element approachwas considered the most suitable option (Eberhardt, 2003). Therefore, numerical modellingpackage 3DEC (Itasca, 2008) was employed to simulate the laboratory behaviour of the lat-erally loaded pile. Laboratory specifications and dimensions were employed in the numericalmodel with a pile diameter (D) of 25mm, embedded pile length (L) of 100mm, and jointspacing (s) of 40mm. Figure 2 shows the overall dimensions and specifications used in thenumerical model. Mesh size was specified with an edge length of 50mm as illustrated inFig. 2. Roller boundary conditions were prescribed to all four sides of the model allowingmovements only along the z-axis. The bottom of the model was fixed in all directions asshown in Fig. 3. Lateral load was applied in the form of uniformly distributed stress overthe same area on the pile head as used in the laboratory experiment. A FISH subroutine waswritten to calculate the total applied force on the pile head.

414


Figure 2. Dimensions and specifications of 3DEC model.

Figure 3. Boundary conditions of model.

Some preliminary trials were conducted to verify the feasibility of the model dimensionsand the extent of joints below the pile tip. The boundary effect was investigated by usingblock sizes of lengths 23D and 45D for a given joint condition as shown in Fig. 2. Theresults showed that the larger block size required a significantly longer run time (about 20hours) compared to a small block size that requires approximately 10 hours. However, itresulted in a difference in lateral deflections of only ±10%. Another trial was carried outusing a fully jointed block as opposed to the partly jointed model shown in Fig. 2. It wasfound that the runtime was significantly longer with a minimal difference in lateral deflec-tions. Moreover, insignificant displacements were encountered beyond the pile tip. Therefore,the model specifications shown in Fig. 2 were considered to be appropriate for subsequentanalyses.

3.2. Material properties and constitutive models

Chiu (1981) proposed a set of equations to calculate the properties of mudstone based onwater content, w. The measured water content for the laboratory models as reported byFrancis (2003) varied between 15% and 19%. Using the median value of 17%, the mudstoneproperties were estimated (Table 1) as a typical input values into the numerical simulation.In this study, previously established pile-rock interface properties by Francis (2003) were

415


Table 1. Properties for numerical modelling input.

Artificial Johnstone Properties Pile-Rock Interface Properties

Water content, w: MPa 17 Normal Stiffness, jkn: GPa 95Density, γ : kg/m3 2200 Shear Stiffness, jks: GPa 95Rock Modulus, E: MPa 153 Friction Angle, φ: degrees 32Poisson’s ratio, v 0.3 Cohesion, c: kPa 55Friction Angle, φ: degrees 32.3 Tensile Strength, kPa 27Cohesion, c: kPa 269.7 Joint PropertiesDilation, ψ : degrees 10 Normal Stiffness, jkn: GPa 95Tensile Strength, σt: kPa 97.7 Shear Stiffness, jks: GPa 95

Uniaxial Compressive Strength, qu: MPa 1

Friction Angle, φ: degrees 28Cohesion, c: kPa 0Tensile Strength, kPa 0Dilation, ψ : degrees 10

used. The pile was modelled as linear elastic and isotropic material. Elastic-plastic Mohr-Coulomb constitutive model was assigned to the rock to capture the non-linear responseof the load-deflection behaviour. Pile-rock interface and rock joints were assigned the elastic-plastic Coulomb slip failure model. Bulk modulus (K) and shear modulus (G) were calculatedbased on rock modulus (E) and Poisson’s ratio (v). Details of all input properties are given inTable 1.

4. Results and Discussion

4.1. Effect of joint normal (kn) and shear stiffness (ks)

Joint normal and shear stiffnesses are fundamental parameters for the numerical modellingof laterally loaded piles socketed into jointed rock mass and they are especially important

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.2 0.4 0.6 0.8 1 1.2

Deflection (mm)

Lo

ad

(kN

)

Laboratory Test

Trial 1: kn=ks=9.5x10^8Pa



Figure 4. Trials using different values of kn and ks.

416


in determining the linear portion of a load-deflection (P-y) curve. These parameters are diffi-cult to measure in the laboratory and a recommended guideline to estimate the values of thestiffnesses is given by Itasca (2008). However, as normal and shear stiffnesses are fictitiousvalues in 3DEC, a trial and error method was employed to obtain a best-fit laboratory pre-diction for an appropriate set of stiffness values. The values tested ranged from 9.5× 108Pato 9.5× 1010Pa.

Figure 4 shows the load-displacement (P–y) behaviour of the piles having various shear(ks) and normal (kn) stiffnesses. Trial 1 (ks = kn = 9.5 × 108Pa) shows a less stiff responsein the linear part of the P-y curve when compared with the laboratory test results. A fur-ther improvement of the predicted results could be achieved by increasing the stiffness val-ues to Trial 2 (ks = kn = 9.5 × 109Pa). However, as also shown in Fig. 4, there are stillsome discrepancies between the laboratory results and Trial 2. Therefore, another trial withhigher stiffness values (Trial 3) was carried out. As can be seen, this higher stiffness val-ues (ks = kn = 9.5 × 1010Pa) most closely predict the linear portion of the laboratory P-ycurve behaviour. Therefore, pile-rock interface and joint normal and shear stiffness values of9.5× 1010Pa were used in all modelling work.

4.2. Effect of Joint Configuration

4.2.1. Joint Configuration (45/45)

Joint configuration of (45/45) was used to investigate the ability of 3DEC to capture theload-deflection behaviour of the laboratory experiment. As it was reported that the watercontent of the rock blocks in the laboratory varied from 15% to 19% (Francis, 2003), amedian value of water content of 17% was used in this study. It was found that for this jointconfiguration of (45/45), water content of 17% for the rock predicted the laboratory resultsmost accurately. Results from the prediction of 3DEC model together with the laboratoryresults are shown in Fig. 5(a). It can be seen from this figure that the pile load behaviouris almost elastic up to point A followed by non-linear response up to point B. The 3DECpredictions agree well with the laboratory trend for a water content of 17%. The numericalmodel predicted plastic zones perpendicular to the loading direction and this behaviour wasobserved during the laboratory testing reported by Francis et al. (2004).

A

B

Figure 5. Comparison of 3DEC predictions with laboratory test results (Francis, 2003) for joint con-figurations: a) (45/45) and b) (30/60).

417


Table 2. Artificial Johnstone rock properties for joint configuration (30/60).

Water content, w: MPa 18 Cohesion, c: kPa 234.7Density, γ : kg/m3 2200 Dilation, ψ : degrees 10Rock Modulus, E: MPa 140 Tensile Strength, σt: kPa 50Poisson’s ratio, v 0.3

Uniaxial Compressive Strength, qu: MPa 0.86Friction Angle, φ: degrees 31.4

4.2.2. Joint configuration (30/60)

A second joint configuration of (30/60) was modelled in 3DEC to predict the load-deflection(P–y) behaviour of the laboratory model pile tests. Different water contents were tested todetermine the water content that best fits the laboratory results. It was found that for thisjoint configuration of (30/60), water content of 18% for the rock predicted the laboratoryresults most accurately (Fig. 5b). Table 2 shows the rock properties corresponding to 18%water content. Pile-rock interface and joint properties were kept the same as for the (45/45)case. Figure 5b shows that a water content of 18%, which is within the reported water con-tent range, generates a matching elastic and post-elastic response of the laboratory behaviour.This implies that the Mohr-Coulomb material model with a Coulomb slip failure for jointsis adequate to simulate the complex P–y behaviour of laterally loaded piles socketed intojointed rock mass. Moreover, this study has shown that 3DEC predicts the lateral capacitymore accurately compared to the 2D UDEC predictions as reported by Francis et al. (2004).

As shown in Fig. 5, steeper joint configuration of (45/45) exhibits stiffer linear behaviourwith a 17% higher ultimate lateral capacity compared to the joint configuration of (30/60).The good predictions of the numerical modelling as evident in Fig. 5 have provided a strongbenchmark model to conduct further investigation on the effect of joint dip angle.

4.2.3. Effect of joint dip angle

The orientations of joints in rocks have substantial impact on the load-deflection (P–y)behaviour of piles socketed into jointed rock mass. In this study, dip angles ranging between30◦ and 60◦ for joint set one were investigated by fixing the dip angle of joint set two at 45◦.Fig. 6 shows the load-deflection (P–y) results for various dip angles of joint set one.

It is evident from Fig. 6 that as the dip angle increases from 30◦ to 60◦, the lateral loadcapacity of the pile increases significantly for a given pile head deflection. This is not sur-prising as the shear strength of the higher dip angled joint would provide greater resistanceagainst shearing under an applied loading condition. This increased shear resistance restrictsthe slip movement along the joint planes and returns a lower lateral deflection of the pile,hence making the pile-rock system behave more rigidly. In contrast, (30/45) and (45/45)joints exhibit almost identical load-deflection (P–y) response (Fig. 6). This suggests that thepile-rock mass system becomes relatively rigid when the dip angle of joint set one increasesbeyond 45◦ for a constant dip angle of joint set two of 45◦. In other words, a thresholdlateral capacity of pile exists up to a certain dip angle beyond which a much higher capacitycan be reached for a given pile geometry.

5. Conclusions

Secondary structures such as jointing and fractures present in a rock mass play an importantrole in determining the lateral capacity of piles socketed in rock. In this study, numericalmodelling has been carried out to simulate the laboratory behaviour of laterally loaded piles

418


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5Deflection (mm)

Lo

ad

(kN

)

3DEC (30/ 45)

3DEC (45/ 45)

3DEC (60/ 45)

Figure 6. 3DEC predictions with different front joint dip angles.

in jointed rock using the 3D distinct element code, 3DEC. This paper has examined the capa-bility of 3DEC to simulate the laboratory behaviour as well as the effect of joint orientationon the load-deflection (P–y) behaviour of piles. In summary, the following conclusions canbe drawn:

• Fictitious modelling parameters such as joint normal (kn) and shear stiffnesses (ks)have significant impact on the prediction of the linear portion of the P-y curve. Inthis study, a joint stiffness value of 9.5× 1010Pa was found to be able to capture thelinear portion of the load-deflection curve most accurately.• 3DEC is proven to be able to predict better the laboratory behaviour of laterally

loaded single piles socketed into jointed rock than UDEC, thus indicating the neces-sity to use 3D modelling when analysing laterally load piles in jointed rock.• Joint orientations dictate the lateral load-deflection (P–y) behaviour of rock socketed

piles. An increase in dip angle beyond a threshold value significantly increases thelateral capacity of piles.

Future research is aimed at understanding the cause of the threshold capacity, establish-ing a set of influential parameters through a parametric study, developing non-dimensionalrelationships between the parameters, understanding the failure mechanisms of the jointedrocks, and developing design guidelines for laterally loaded piles socketed into jointed rock.

Acknowledgements

The authors acknowledge the financial support provided by Monash University, Australiaand Powerlink, Queensland for this research.

References

1. Bieniawski, Z.T. (1976), “Rock mass classifications in rock engineering”, Proceedings of the Sym-posium on Exploration for Rock Engineering, Z.T. Bieniawski (editor), Vol. 1, A.A. Balkema,Rotterdam, Holland, pp. 97–106.

2. Broms, B.B. (1964), “Lateral resistance of piles in cohesionless soils”, Journal of Soil Mechanicsand Foundation Division, ASCE, Vol. 90, SM2, pp. 123–156.

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3. Carter, J.P. and Kulhawy, F.H. (1992), “Analysis of laterally loaded shafts in rock”, Journal ofGeotechnical and Geo-environmental Engineering, ASCE, Vol. 118, No. 6, pp. 839–855.

4. Chiu, H.K. (1981), “Geotechnical properties and numerical analysis for pile design in weak rock”,PhD thesis, Department of Civil Engineering, Monash University.

5. Chong, W.L, Haque, A., Ranjith, P.G. and Shahinuzzaman, A., “Lateral Load Capacity of SinglePiles Socketed into Jointed Rocks — A Review”, 2nd Proceedings of First Southern HemisphereInternational Rock Mechanics Symposium, Vol. 1, 2008, pp 297-309.

6. Eberhardt, E. (2003), “Rock slope stability analysis – Utilisation of Advanced Numerical Tech-niques”, Geological Engineering/Earth and Ocean Sciences, UBC, Canada.

7. Francis, B. (2003), “Laterally loaded piles in jointed soft rock masses”, Masters of EngineeringScience Thesis of Monash University, Clayton, Australia.

8. Francis, B., Haberfield, C., and Kodikara, J. (2004) “Laterally Loaded Model Piles in Jointed SoftRock Masses”, Proceedings of the 29th Annual Conference on Deep Foundations, Vancouver,Canada.

9. Gabr, M.A., Cho, K.H., Clark, S.C., Keaney, B.D. and Borden, R.H. (2002), “P-y curves for lat-erally loaded drilled shafts embedded in weathered rock”, Draft Report No. FHWA/NC/2002/08,North Carolina University, Raleigh, NC.

10. Hoek, E. and Brown, E.T. (1988), “The Hoek–Brown criterion — a 1988 update”, Proceedings15th Canadian Rock Mechanics Symposium, University of Toronto.

11. Itasca (2007), “3DEC User’s Guide”, Itasca Consulting Group Inc., Minnesota, USA.12. Johnston, I.W. and Choi, S.K. (1986), “A synthetic soft rock for laboratory model studies”

Geotechnique 36 (2), pp. 251–263.13. Reese, L.C. (1997), “Analysis of laterally loaded piles in weak rock”, Journal of Geotechnical and

Geo-environmental Engineering, ASCE, Vol. 123, No. 11, pp. 1010–1017.14. Yang, K (2006), “Analysis of Laterally Loaded Drilled Shafts in Rock”, Doctor of Philosophy

Thesis of University of Akron, United States.15. Zhang, L., Ernst, H. and Einstein, H.H. (2000), “Non-linear analysis of laterally loaded rock-

socketed shafts”, Journal of Geotechnical and Geo-environmental Engineering, Vol. 126, No. 11,pp. 955–68.

420

Distinct Element Analysis on the Stability of a Stone Pagoda atMireuk Temple Site in Korea

H. KIM1 AND S. JEON2,∗1School of Civil, Urban & Geosystem Engineering, Seoul National University2Department of Energy Systems Engineering, Seoul National University

1. Introduction

The pagoda at Mireuk Temple Site is known as the oldest and largest one among existingstone pagodas in South Korea. It has highly historical and cultural value and shows how thetechnique of building a wooden pagoda was adapted to stones. It seems to have been nine-storied pagoda originally, but it was partially collapsed before the 17th century and roughlyreinforced with concrete in 1915 as Fig. 1 shows. At present, the pagoda is being dismantledfor repair and restoration because the possibility of additional collapse was anticipated bythe structural safety inspection in 1998.1

In order to restore such unstable structure safely, stability analysis on the pagoda duringand after the restoration is required. In this study, a numerical simulation of the scaled modeltest was carried out using 3DEC and the results were compared with experimental data toverify the adequacy of prediction using numerical model. We investigated the effect of stonefilling inside the pagoda on the overall structural stability. Finally, three restoration plans forthe pagoda were examined based on the analysis of stresses and displacements generated byself-weight load. In addition, the influence of material deterioration on the structural safetywas evaluated.

2. Restoration Plans for Pagoda

Before the stone pagoda was dismantled, only the east side portion remained relativelyundamaged. There were stacked masonry and concrete patching up the damaged part ofthe pagoda in the west side portion as seen in Fig. 1.

On the basis of that shape of pagoda, three restoration plans (RP1∼3, Figure 2) are nowunder consideration by cultural properties conservation team2 and each plan has the follow-ing features:

• RP1: All four sides of the first floor are completely restored to the original state(shape of the east side). Stones are stacked in the west and the south portion fromthe second to the sixth floor to restore the appearance of the pagoda just beforedismantling (the left and back side of model in Figure 2).• RP2: All four sides are restored to the original state from the first to the third floor.

The fourth to the sixth floor are built like the appearance before dismantling.• RP3: All six floors are completely restored to the original state.

The comparison with the scaled model test in the next section was carried out with RP2model and the effect of filling stones in Section 4 was investigated with RP3 model.




14.1

63 m

(a) Side view (From east) (b) Side view (From north)

Figure 1. Stone pagoda at Mireuk Temple Site before dismantling.

(a) RP1 (b) RP2 (c) RP3

Figure 2. Modelling of restoration plans.

3. Comparison with Scaled Model Test

Kwon2 made a stone pagoda model which was based on the restoration plan RP2 and scaleddown to 1/8th of the original size. Loads were measured by load cells located underneath thebottom plates of the model while an experimenter was stacking stones in traditional method.Granite that had been used in the building of the stone pagoda was replaced by the stone-likematerial made by baking the soil in the form of rectangular solid.

3.1. Mechanical properties of material for numerical analysis

We examined properties of the material using the same test methods for evaluating mechan-ical properties of rock. Due to the limited sample size, the surface properties such as shearstiffness, cohesion, and friction angle were obtained from experimental data for MireukMountain granite.3 The material properties used in the numerical analysis are listed inTable 1.

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Table 1. Material properties used in the numerical analysis.

Properties Block Surface

Density(kg/m3)

Young’smodulus

(×103MPa)

Poisson’sratio

Normalstiffness(MPa/m)

Shearstiffness(MPa/m)

Cohesion(MPa)

Frictionangle

(◦)

Stone-likematerial

2,040 23,900 0.19 4,960 5,910 0.03 28.3

3.2. Comparison between experimental and numerical results

Normal stresses were calculated from the observed load data at four filling stone portions(FS-1∼4) and three decoration stone portions (DS-1∼3) whenever one floor was built in thenumerical model. The results were compared with the experimental data as shown in Fig. 3.A ratio of absolute difference between the experimental and numerical value to numericalvalue (|σexp − σnum|/σnum) was used for evaluating the accuracy of the prediction. The ratioof filling stone portions was respectively 7%, 24%, 4%, and 17% on average. The ratio ofdecoration stone portions was 32%, 17%, and 8% on average. There was stress decreaseespecially in FS-4 portion while stones were stacked without decoration stones in the testmodel, and similar trend also emerged in the numerical result.

Blocks in the form of rectangular parallelepiped were stacked densely and the surfaceswere attached to each other in the numerical model. On the contrary, there are a lot ofopen spaces between blocks in test model. Therefore, the numerical model of interactionbetween blocks has some limit in predicting the behaviour of blocks, but differences betweenthe experimental and numerical value in this study can be evaluated to be relatively smallconsidering such assumption in numerical modelling and errors in experiment.

In conclusion, it can be stated that discrete element analysis using 3DEC is able to predictthe behaviour of the masonry structure and furnish data for the evaluation of overall stabilityin terms of stresses and displacements. However, there can be some discrepancy due to the

FS-1 FS-2

FS-3FS-4

DS-1

DS-2

DS-3

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7

0 10 20 30 40 50

Vertical stress (kPa)

Stor

y

Tested

Numerical

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(a) Measurement area (b) FS-1 (c) FS-2 (d) FS-3

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(e) FS-4 (f) DS-1 (g) DS-2 (h) DS-3

Figure 3. Comparison of numerical analysis with experimental measurements.

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difficulties in accurately identifying the representative properties of block surface. Differencesbetween two data in each graph (as seen in Fig. 3) would be smaller if the mechanical testresults of stone-like material were used instead of reference data in numerical analysis.

4. Effect of Filling Stones on Structure Stability

4.1. Mechanical properties of filling stones

Before the real scale modelling of the restoration plans was performed, we obtained coresamples from filling stones selected out of many dismantled stones and carried out labora-tory tests to find out the mechanical properties of the stones. Type of rock is granite, andthe weathered part in surficial portion of large stone block can be seen in Figure 4. Themechanical properties of stones are listed in Table 2.

Figure 4. Rock core samples obtained from filling stones.

Table 2. Mechanical properties of filling stones.

Density(kg/m3)

Compressivestrength(MPa)

Young’smodulus

(×103MPa)

Poisson’sratio

Cohesion(MPa)

Frictionangle(◦)

P-wavevelocity

(m/s)

S-wavevelocity

(m/s)

2,510∼2,580 39∼85 7.5∼19.8 0.28∼0.34 7∼12 57∼58 1,445∼2,340 718∼1,350

Suh et al.4, 5 carried out non-destructive close examinations for Three-Story Pagoda andDabo Pagoda in South Korea to assess structural safety and estimated the strength of stoneblocks from measured ultrasonic wave velocities. Those two stone pagodas were built about1 century after building of stone pagoda at Mireuk Temple Site and rock type of stone blocksis granite. So, the estimated properties can be used as comparison data for this study. Consid-ering the average estimated strength of Three-Story Pagoda is 46.3MPa (ranging from 13.4to 84.4MPa) and that of Dabo Pagoda is 39.6MPa (ranging from 9.3 to 131.4MPa), com-pressive strength of filling stones is in the range of the strength of stones used in two pagodas,and the minimum strength of core samples (39MPa) is similar to two average strengths.

4.2. Stability analysis of stone pagoda with and without filling stones

The numerical model with filling stones which was based on restoration plan RP3 was com-pared with the model without filling stones to examine the effect of stones filling inside of thepagoda. The test result of rock sample corresponding to the minimum compressive strength

424


was used. Its density is 2,510 kg/m3, Young’s modulus is 7.5× 103MPa, and Poisson’s ratiois 0.32. Because the predicted stresses were lower than the block strength and the possibilityof rock failure was low, isotropic elastic model was applied as block model. Coulomb slipmodel using the properties listed in Table 1 was applied as joint model.

The vertical stresses of pagoda with filling stones were evenly distributed over all floors.On the contrary, in the case of pagoda without filling stones, vertical stresses concentratedat the main stone walls of each floor in the form of arch as seen in Fig. 5(b). The maximumvertical stress of the former was 1.21MPa at the ceiling of cross aisle of the first floor and thatof the latter was 2.19MPa at similar position. In close investigation shown in Figure 6, themaximum vertical stresses at each step of pagoda with filling stones were much lower thanthat of pagoda without filling stones except for the fifth and the sixth floor. The maximumstress decrease was 74% at the top step of the first floor (1.29MPa decreased to 0.34MPa)and the averaged stress decrease was 38%. As a result, it was found that stones filling theinside of stone pagoda distributed the stresses generated by self-weight load evenly over entirepagoda and made the structure more stable than pagoda without filling stones.

(a) Pagoda with filling stones. (b) Pagoda without filling stones.

Figure 5. Vertical stress contours of stone pagoda.

0

500

1,000

1,500

2,000

2,500

1st 2nd 3rd 4th 5th 6th

Story

Max

. ver

tica

l str

ess

(kP

a)

Filling

Empty

Figure 6. Maximum vertical stresses at each step of stone pagoda.

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5. Stability Analysis of Restoration Plans

5.1. Stresses and displacements by self-weight load

Numerical analyses were carried out for three models based on restoration plans RP1∼3 asmentioned above. RP3 had about 150 more stone blocks than RP1, and overallself-weight load increased 12%. Looking at the analysis results, symmetrical stress distri-bution was observed as expected in RP3 model. On the other hand, asymmetrical stressdistribution was obviously developed in RP1 and RP2 model. In two models, the verticalstresses of northeast portion with decoration stones were larger than those of southwestportion without decoration stones (Fig. 7). Maximum vertical stresses of RP1∼3 were allfound at the ceiling of cross aisle of the first floor, and the stress magnitude was 0.9, 1.12,and 1.21MPa, respectively. The maximum vertical displacements occurred all at the samelocation where several blocks slipped slightly into empty space in the first floor, and thedisplacement magnitude was 15, 7, and 6mm, respectively. In comparison of the maximumvertical stresses at each floor as shown in Fig. 8, the stresses of RP3 were 18∼39% largerthan those of RP1 at the first, fifth, and sixth floor, and the differences of the other floorswere below 3%. As a result, small displacements and large stresses were developed symmet-rically in RP3 model, but relatively large displacements and small stresses were developedin RP1. However, even though there was asymmetry and concentration in stress distributiondepending on the restoration plan, the developed stresses were smaller than the strength ofstone blocks. So, all stone pagodas to be built based on restoration plan were found to bestructurally stable.

(a) RP1 (b) RP2 (c) RP3

Figure 7. Vertical stress contours at ceiling of the 1st floor and central cross section (SN direction).

5.2. Effect of material deterioration on stability

Natural weathering of stone blocks in the pagoda exposed to air is inevitable. For the inves-tigation of the effect of material deterioration on structural stability, we carried out 27numerical modelling in total, with reducing Young’s modulus of block and stiffness parame-ters (normal and shear) of joint model to 50% and 10%, respectively, of the present values ineach plan. And then analysis of variance (ANOVA) was performed. The purpose of ANOVA

426


0

200

400

600

800

1,000

1,200

1,400

1st 2nd 3rd 4th 5th 6th

Story

Max

. ver

tica

l str

ess

(kP

a)

RP1

RP2

RP3

Figure 8. Maximum vertical stresses at each floor.

Table 3. ANOVA (Maximum vertical stress).

Factor S ù V F

A 94,558.9 2 47,279.4 47.2B 25,069.3 2 12,534.6 12.5C 64,217.0 2 32,108.5 32.0

B×C 30,970.0 4 7,742.5 7.7E 16,036.0 16 1,002.3T 230,851.0 26

Table 4. ANOVA (Maximum vertical displacement).

Factor S ù V F

A 246.3 2 123.1 0.7B 1,194.4 2 597.2 3.4C 24,428.6 2 12,214.3 69.0

B×C 2,769.7 4 692.4 3.9E 2,833.2 16 177.1T 31,472.2 26

is to decide whether the differences between the results are simply due to random error orwhether there are systematic treatment effects that have caused results in one group to differfrom results in another.6 As seen in Table 3, F-ratios of restoration plan (factor A), Young’smodulus (B), and stiffness parameters (C) were all much above critical value F(2, 16, 0.05) =3.63 at the 5% significance level, and F-ratio of interaction B×C was large, too. Therefore,it was found that the type of restoration plan and material deterioration had much effecton the maximum vertical stresses and the interaction between Young’s modulus and surficialproperties of blocks also had significant effect. On the other hand, in Table 4, only reductionin surface stiffness (factor C) had effect on the maximum vertical displacement.

6. Conclusions

The following conclusions are deduced from the distinct element analysis on the stability ofthe stone pagoda at Mireuk Temple Site:

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• In comparison of numerical analysis results with the results of scaled model test,there were differences in load distribution ranging from 4 to 32% caused by thelimit of modeling interaction between blocks. But, it was found that the analysismethod could be effective in predicting the stresses or displacements of masonrystructure before actual restoration.• The stones filling the inside of pagoda distributed the stresses generated by self-

weight load evenly over entire pagoda and made the structure more stable thanpagoda without filling stones. Filling stones increased total load of structure by morethan 48% but decreased maximum vertical stress by 74%.• The location of stress concentration was investigated from the numerical analyses

of restoration plans RP1∼3. And the structural stability assessment was performedby their displacements and stresses. The developed stresses were smaller than thestrength of stone blocks, and all stone pagodas to be built based on restorationplan were found to be structurally stable. It was found, using ANOVA method, thatthe type of restoration plan and material deterioration had significant effect on thestresses of stone blocks. But, only the surface stiffness had effect on displacements.

In future, a study on local stress concentration caused by irregular shape of stones and adynamic analysis for earthquake will be carried out for in-depth research.

References

1. National Research Institute of Cultural Heritage, “Research on Repair and Restoration”, 2002.Web site: http://www.nricp.go.kr/eng/archi/repair.jsp.

2. Kwon, Y.H., Structure Stability of the Stone Pagoda of Mireuksa Temple Site, MS thesis, SeoulNational University, 2008.

3. National Research Institute of Cultural Heritage, Research Report on the Cause of Collapse ofStone Pagoda at Mireuk Temple Site, Korea, 2005, p. 178.

4. Suh, M., Song., I. and Choi, H., “The Structural Safety Diagnosis of Three-Story Pagoda in BulkukTemple using the Probability of Failure”, Journal of the Korean Geophysical Society, Vol. 4 No. 1,2001, pp. 57–69.

5. Suh, M., Song., I. and Choi, H., “The Structural Safety Diagnosis of Dabo Pagoda of Bulkuk Templeusing Analyses of Ultrasonic Wave Velocity”, Journal of the Korean Geophysical Society, Vol. 5 No.3, 2002, pp. 199–209.

6. Hendricks, W., “Quantitative Methods”. Web site: http://www.ilir.uiuc.edu/courses/Lir593/one_way_ anova _class_notes.htm.

428

Distinct Element Analysis of Staged Constructed UndergroundCavern in the Vicinity of a Fault

Nanyang Technological University, Singapore

1. Introduction

In this study, the influence of a major fault in the vicinity of a rock cavern is considered.Detailed numerical studies were carried out to investigate the effect of various fault locations,i.e. at left side of the rock cavern, at right side of the rock cavern and intersecting the rockcavern, on the stresses and displacements around the cavern. The influence of the cavernexcavation sequences on the cavern stability was also considered.

2. Geotechnical Properties

The two-dimensional Universal Distinct Element Code (UDEC) was adopted in this numer-ical study. The rock mass properties adopted for the numerical model were estimated usinga Q-value of 10, i.e. a typical lower bound Q-value for the Jurong Formation of Singapore(Zhao et al., 1999). Rock Mass Rating was estimated using the correlation shown below(Bieniawski, 1984):

RMR = 9 ln Q + 44

The rock mass deformation modulus Em, cohesion c, and friction angle φ were then estimatedusing the following equations (Bieniawski, 1984):

Em(GPa) = 10RMR−10

40

c(MPa) = 0.005(RMR − 1.0)

φ = 0.5RMR + 4.5

The rock mass deformation and shear strength parameters adopted in the study are sum-marized in Table 1. The rock mass was modelled using the Mohr-Coulomb model in UDEC.

Table 1. Properties of Jointed Rock Mass.

Parameter Value Unit

Unit Weight 26.5 kN/m3

Young’s modulus, Em 23340 MPaCohesion, c 0.32 MPaFriction angle., φ 37 DegreeKo 2.0 -

H.C. CHUA∗, A.T.C. GOH AND Z.Y. ZHAO

Analysis of Discontinuous Deformation: New Developments and Applications.

Edited by Guowei MA and Yingxin ZHOU. Published by Research Publishing Services.

Copyright c© 2009 by Society for Rock Mechanics & Engineering Geology (Singapore).

ISBN: 978-981-08-4455-4doi:10.3850/9789810844554-0064 429


4. Influence of Fault Locations

The influence of fault locations on the displacements and plastic point distributions around acavern are illustrated in Figures 2 and 3, respectively. It can be seen that the effect of the faultlocations becomes insignificant at horizontal fault distance of greater than 1.5 W away fromthe cavern centreline. Comparisons of cases involving fault distance of 0.5 W to each left andright side of the cavern centreline, as well as the case where the fault is intersecting the cavernindicate the least overall cavern displacements and number of plastic point for the latter. Inaddition, the presence of fault adjacent to the right cavern wall also shows relatively largeroverall cavern displacements, but smaller number of plastic points as compared to the casewhere the fault is presence adjacent to the left cavern wall. This suggests that constructing acavern through a fault may not always be considered as an unfavourable case. In general, thepresence of a fault to the left of a cavern can be considered as the most unfavourable case inthis study as it generally resulted in large overall cavern displacements as well as number ofplastic points around the cavern. Although the displacements of the cavern roof for the caseinvolving a fault intersecting the cavern is much greater than the displacement of the cavernroof for no-fault (NF) case, both displacements of the left and right walls seem to be similarin terms of trend and magnitude.

3. Numerical Model

Figure 1 illustrates the general characteristics of the numerical model adopted in this study toevaluate the effects of fault locations and cavern excavation sequences on the displacementsand plastic point distributions around a cavern. A cavern dimension of 20 m width and 30 mheight was used in the study, as illustrated in Fig. 1. To investigate the effects of fault locationsand cavern excavation sequences, the fault is modelled at left and right side of the cavern,as well as intersecting the cavern. The cavern has three major stages of excavation, i.e. topheading (−1), first benching (−2) and second benching (−3). The analyses were classifiedinto two broad cases, namely, sequential excavation of the top heading (i.e. left drift, middleheading and right drift), and full heading excavation followed by two stages of benching ofthe cavern.

L

Bench1

Bench2

M R

X=-0.5W X=0.5W

H=1.5W

=70o

Fault Left (FL) H=5.25W

Fault Middle (FM)

Fault Right (FR)

(Not to scale)

Figure 1. Illustrations of numerical model.


430

(a) Roof

(b) Left Wall (c) Right Wall

Figure 3. Variations of plastic point distributions around cavern with fault locations.

5. Influence of Excavation Sequences

Figures 4, 5 and 6 show the cavern displacements for cases involving fault and without fault.The total number of plastic points generated around the cavern after completing cavern exca-vation is presented in Table 2. As can be seen in Figure 4, top heading excavation sequences

Figure 2. Variations of displacements of cavern at (a) Roof; (b) Left Wall and; (c) Right Wall withfault locations.


431

(a) Fault Left (b) Fault Middle

(c) Fault Right (d) No Fault

Figure 4. Vertical displacement at Roof for various excavation sequences of cavern for fault (F) andno-fault (NF) case.

Table 2. Total number of plastic points around cavern.

Cases Fault Locations Excavation Sequences Plastic Points

Left L-M-R 2702R-M-L 2652

Full Heading 2964

Fault Middle M-L-R 2557M-R-L 2442

Full Heading 2436

Right L-M-R 2375R-M-L 2531

Full Heading 2466

No Fault − L-M-R 2352M-L-R 2277

Full Heading 2316


432



Figure 5. Horizontal displacement at Left Wall for various top heading excavation sequences of cavernfor fault (F) and no-fault (NF) cases.

of the cavern affected the vertical displacements of the cavern roof. However, opposite phe-nomena were observed for cavern wall displacements, except for the case where the fault ispresent at the right side of the cavern wall, as shown in Figures 5(c) and 6(c). It was gen-erally observed that the full top heading excavation usually resulted in the smallest cavernroof displacement, for all locations of fault. For cases involving sequential excavation of thetop heading in the vicinity of a fault, excavation sequences of R-M-L and M-R-L resulted inmuch smaller cavern roof displacements, as compared to the excavation sequences of L-M-Rand M-L-R.

Comparison between Figures 5(a) and 6(c) shows that the displacement of the left cavern

left and right side of the cavern walls, respectively. However, various top heading excavationsequences of a cavern seem to have negligible effect on the displacement of the cavern wall,except for the case where the fault is presence at the right side of the cavern wall. As expected,full top heading cavern excavation induced largest displacement of the right cavern wall as

wall is smaller than the displacement of the right cavern wall, for cases involving fault at


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Figure 6. Horizontal displacement at Right Wall for various top heading excavation sequences ofcavern for fault (F) and no-fault (NF) cases.

compared to excavation sequence R-M-L, whereas excavation sequence L-M-R induced thesmallest cavern wall displacement, as a result of minimal stress relief of the rock mass forexcavation sequence L-M-R.

In general, top heading excavation sequences R-M-L and L-M-R of a cavern for casesinvolving fault at the left and right side of a cavern, respectively, induced the smallest num-ber of plastic points. These observations demonstrate that the initial excavation of a driftfurther away from the fault could induce smaller stress relief of rock mass, in turn, min-imised the excessive stresses from being generated around the cavern. Although comparisonsbetween cases involving fault intersecting the cavern and no-fault case show negligible effecton the displacements of the cavern walls, the effect of fault on the displacement of the cavernroof and plastic points generated around the cavern is considered significant, as shown inFigures 4(b), 4(d) and Table 2.


434

For no-fault case, full top heading excavation resulted in the smallest displacement of thecavern roof whereas similar displacement magnitude of the cavern roof was observed forboth excavation sequences L-M-R and M-L-R, with negligible effect of excavation sequencesobserved for the displacement of the cavern walls. Among the effects of the three excavationsequences for no-fault case, the least number of plastic points was observed for excavationsequence M-L-R, followed by full heading excavation. These phenomena may be attributedto the more prominent arching effect that occurred around the cavern roof for excavationsequence M-L-R.

6. Conclusions

A series of numerical parametric studies has been conducted to investigate the effects ofan adjacent fault on stresses and displacements induced around an underground cavern.The geotechnical properties of sedimentary rocks of Jurong Formation of Singapore wereadopted.

For the effects of fault locations, the presence of fault facing the right wall of the cavernresulted in greater overall cavern displacements, as compared to the fault facing the leftwall of the cavern. However, the opposite was true for stresses induced around the cavern.It was also found that excavating a cavern through a fault induced almost similar trendand magnitude of overall cavern displacements as compared to no-fault case, where greaterplastic point distribution was observed for the case involving a fault intersecting the cavern.This suggests that constructing a cavern through a fault may not necessary be consideredas an unfavourable case. In addition, the effects of fault locations were observed to becomenegligible at fault distance of greater than 1.5W from cavern centreline.

For cases involving various top heading excavation sequences with fault adjacent to thecavern walls as well as intersecting the cavern, the effects of cavern excavation sequenceson displacement of the cavern roof is more significant than cavern walls, except for the casewhere the fault is at the right side of the cavern walls. While full top heading excavationsequence resulted in the smallest displacement of the cavern roof for fault and no-fault cases,excavation sequences L-M-R and R-M-L generally induced lesser stresses and wall displace-ments for cases involving a fault at each right side and left side of the cavern wall, respectively.

References

1. Bieniawski, Z.T. 1984. Rock mechanics design in mining and tunneling. Balkema, Rotterdam,272 pp.

2. Zhao, J., Lee, K.W., Choa, V., Liu, Q. and Cai, J.G. 1999. Underground cavern development in theJurong Formation of sedimentary rocks. Nanyang Technological University, Singapore, 100 pp.


435

Numerical Experiment on Thermo-mechanical Behavior ofJointed Rock Masses Under Cryogenic Conditions

S.K. CHUNG1, E.S. PARK1,∗, Y.B. JUNG1 AND T.K. KIM2

1Korea Institute of Geoscience and Mineral Resources, Korea2SK Engineering and Construction Co., Ltd., Korea

1. Introduction

One of the most important problems related to the underground storage of the cryogenicmaterial is to prevent the leakage of the liquid and the gas from the containment system tothe rock mass caused by the shrinkage of the rock mass around the caverns due to the tensilestresses (Monsen & Barton, 2001).

Failures of the underground storage system of the cryogenic material (e.g. LiquefiedNatural Gas) were mainly due to the thermal stresses generating cracks in the host rockand the thermal cracks contributed to induce the gas leakage and to an increase in the heatflux between LNG (boiled temperature: −162◦C) and the surrounding rock mass. If the stor-age is unlined and frozen down to −162◦C, the rock joints start to open followed by flowingof a part of the gas into the joints and continues cooling the rock wall. This will successivelyreopen the joints, and heavily increase the cooled area and the extent of the cooling front(Park et al., 2006). The way to prevent a hard rock mass from cracking under cryogenic con-ditions would be to locate the unlined storage cavern deep enough below the ground level sothat the geostatic stresses counterbalance the tensile stresses caused by cooling. On the otherhand, in-ground insulated concrete tanks have been developed as a reliable LNG storagetechnique (Amantini & Chanfreau, 2004). A new concept for storing LNG in a lined rockcavern has been developed and is based on the combination of a containment system againstthermal shock and a drainage system against freezing of the surrounding rock masses.

The thermo-mechanical behavior of the discontinuous rock masses would be different fromthat of the intact rock because joints in rock masses can affect the crack expansion undercryogenic conditions. Numerical simulations using the PFC 2D code were performed to figureout the thermo-mechanical behavior of the jointed rock mass around LNG storage cavern.An important goal of the numerical simulation was to investigate the effect of joints on theformation of new cracks during the cooling down phase.

2. Fracture Mechanics Related to Cryogenic Conditions

2.1. Generation of new cracks

From a previous study (Lee, 1993), it has been known that when the rock is cooled slowly,thermal cracks are generated because of the dissimilarity of the thermal expansion betweenthe rock components. Dahlstrom (1992) also mentioned that from the results of the AcousticEmission experiments on the granite by Shell, the generation of micro crack was started from−50◦C. When a thermal stress induced by cooling down exceeds the tensile strength of rock,micro cracks could be created.

However, there is an effect of compensating tensile stresses with the thermal stresses inducedby cooling down because the in-situ stresses (compression) are present in the rock masses. As




Figure 1. A modified criterion of restraint of crack generation.

a result, the crack generations due to the thermal shock would be suppressed. The criterionof the LNG storage suggested by Goodall et al. (1989) is given as follows.

In-situ stress+ tensile strength > thermal stress (1)

The tensile strength of the rock mass in the Equation (1) can be excluded because it is gen-erally assumed that the rock mass has no tensile strength for the numerical simulations.Consequently the cracks are not created when the induced thermal stress is smaller than theminimum compressive stress in the surrounding rock masses as shown in Fig. 1. Howeverthis modified criterion is considered as a quite conservative one. Further studies are neededto validate and improve it.

2.2. Propagation of pre-cracks

The expansion of pre-cracks in the discontinuous rock masses is much easier than the gener-ation of new cracks in the intact rock if the temperature in the rock mass falls down. It meansthat the propagation of the pre-cracks could be initiated under the low thermal contractionbecause the mechanical characteristics of the discontinuities are less competent than those ofthe intact rock.

Groundwater usually flows through the discontinuities of the rock masses and it wouldremain locally in the discontinuities if the groundwater in the rock masses around the cav-erns was badly drained during the construction of the caverns. As a result, the remaininggroundwater in the discontinuities could be frozen and the frost heaving pressures would begenerated during the LNG storage. And it affects the propagation of the pre-cracks withinthe rock masses.

The propagation mechanism of a pre-crack could be explained by the relationship betweenthe stress intensity factor (K) and the fracture toughness (Kc) at the crack tip as in Fig. 2.

3. Thermo-Mechanical (T-M) Analyses

3.1. Determination of basic properties

Generally a temperature drop under the cryogenic conditions shrinks the surrounding rockmasses, and consequently a tensile stress will occur. It means that a tensile strength of theintact rock is very important for the generation of crack during the cooling down stage.Therefore based on the tensile strength of the intact rock, the micro properties of the PFCmodel were determined by trial and error. About the same numerical samples, a total of 10

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Figure 2. A criterion of suppression of crack propagation.

(a) Numerical sample (b) Direct tensile simulation

Figure 3. A numerical sample and simulation for determining basic properties.

direct tensile simulations were performed (see Fig. 3). The sample has a tensile strength of7.6± 0.5 MPa for the numerical simulations.

3.2. Thermal stresses induced by a temperature drop

The T-M analyses were carried out to find the distribution of the induced thermal stressesonly without the initial stresses in a model. Fig. 4 shows the model configuration and theboundary conditions for the T-M analyses. The dimension of the model is 2m high × 2mwide, and the displacements of all the boundaries are constrained to induce the thermalstresses from shrinkage. Table 1 represents the results of the T-M analyses due to the temper-ature drop. When the temperature drops or the thermal expansion coefficient is increased,the magnitude of the induced thermal stresses becomes larger.

Based on the design concept of the underground LNG storage, a minimum temperatureof the surrounding rock mass does not have to drop below −50◦C after 30 years of LNGstorage operation. According to this criterion, if a maximum temperature drop is assumedto be −70◦C (initial temperature of rock mass: 20◦C), the thermal stress in the model iscalculated as a tensile stress equivalent to 6.42MPa with 2.62 × 10−6/◦C, 13.37MPa with5.78× 10−6/◦C respectively.

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Figure 4. Numerical model for T-M analysis.

Table 1. Induced thermal stresses according to thermal contraction coefficients.

Temperature drop (�T) Thermal contraction coefficient

2.62× 10−6/◦C 3.711× 10−6/◦C 5.78× 10−6/◦C

−40◦ 3.95MPa, 3.89MPa 5.33MPa, 5.24MPa 7.93MPa, 7.80MPa−50◦ 4.78MPa, 4.70MPa 6.49MPa, 6.39MPa 9.74MPa, 9.59MPa−60◦ 5.60MPa, 5.51MPa 7.66MPa, 7.54MPa 11.56MPa, 11.38MPa−70◦ 6.42MPa, 6.32MPa 8.82MPa, 8.69MPa 13.37MPa, 13.17MPa−80◦ 7.25MPa, 7.13MPa 9.99MPa, 9.83MPa 15.19MPa, 14.95MPa−90◦ 8.07MPa, 7.94MPa 11.15MPa, 10.98MPa 17.00MPa, 16.74MPa−100◦ 8.89MPa, 8.75MPa 12.32MPa, 12.13MPa 18.82MPa, 18.53MPa

Table 2. Calculated thermal stresses according to a temperature drop.

Temperature drop (�T) σxx(horizontal stress), σyy (vertical stress)

−40◦C 5.33MPa, 5.24MPa−50◦C 6.49MPa, 6.39MPa−60◦C 7.66MPa, 7.54MPa−70◦C 0.01MPa, 0.08MPa−90◦C 0.08MPa, 0.33MPa

3.3. Fracture analyses for an intact rock

Fracture analyses are performed to figure out the temperature of the crack initiation for anintact rock model. The model considered has a tensile strength of 7.6MPa and a thermalexpansion coefficient of 3.711 × 10−6/◦C under no initial stress. Table 2 represents the hor-izontal and the vertical stresses in the model caused by a temperature drop.

Figure 5 shows the fracture patterns caused by the temperature drop. As the temperaturedrop increases, the induced thermal stresses also become larger before the fractures are cre-ated. When the thermal stresses is larger than the tensile strength of the intact rock model(7.6MPa), the fracture initiated below −60◦C and the magnitudes of the induced stresses are

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Temp. drop: -70 Temp. drop: -80 Temp. drop: -90

Figure 5. Fracture patterns due to the temperature drop.

(a) 2D trace sections (b) One joint set model (c) Two joint sets model (d) Three joint sets model

Figure 6. Joint set models for thermo-mechanical analyses.

decreased steeply below −70◦C. And the fractures are distributed on the whole area of themodel at −90◦C.

3.4. Fracture analyses for jointed rock masses

In order to find out the effect of discontinuities on the crack occurrence and the propagationin the rock masses under the cryogenic conditions, the coupled T-M analyses were carriedout on a different number of joint sets and the mechanical properties of the joints.

The PFC model is built as the following steps. First, the intact rock is modeled as a sin-gle rock block of the model size. Then the fracture traces are integrated in the model. Thefracture model in PFC2D was generated from 2D trace sections extracted from the 3D DFNmodel developed for Aspo ZEDEX tunnel — see Fig. 6(a). A fracture is identified as a con-tact that exists between particles that fall on the opposite sides of the joint plane. The frac-tures are modeled by generating bands of the particles within the matrix. Band particles areassigned the micro-properties which are different to those possessed by the matrix particles.Figure 6(b)–(d) shows some PFC jointed rock mass models for the T-M analyses. The dimen-sion of the model is 30m high × 30m wide, and the displacements of all the boundaries areconstrained to induce the thermal stresses from shrinkage.

Figure 7 shows the results of T-M analyses on the different number of the joint sets withthe constant joint properties. As the number of the joint sets increase, the amount of thecrack occurrence has reduced. Cracks were formed and expanded at a lower temperature asthe joints exist in the PFC models.

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(a) One joint set model (b) Two joint sets model (c) Three joint sets model

Figure 7. Crack patterns occurred in the model with different number of joint sets (temp. drop =−60◦C).

(a) 1/10 of fresh rock (b) 1/10,000 of fresh rock (c) 1/10,000,000 of fresh rock

Figure 8. Crack patterns occurred in the model with different joint properties (One joint set model).

Figure 8 shows the results of the T-M analyses by one joint set model with the differentjoint properties when a temperature drop is −60◦C. As the joint properties decrease, theamount of crack occurrence has reduced.

As shown in Fig. 6 and 7, new cracks are created and propagated mostly along the jointsduring cooling down. It could be thought that because the mechanical properties of thejoints are much weaker than those of the surrounding rock, the joints opened wider due tothe contraction occurred by cooling down of the rock mass. Consequently an induced stressis concentrated at the tip of the crack and it causes a gradual expansion of the cracks.

4. Conclusions

The following conclusions are obtained from the T-M coupled analyses using the PFC2Dcode:

• The magnitude of the induced thermal stress becomes larger when the temperaturedrop as well as the thermal expansion coefficient is increased. But its magnitude isdecreased steeply below −70◦C from which the fracture initiates and then the cracksare created considerably on the model as the temperature drops further.• Based on the new design concept for the LNG storage, a minimum temperature of

the surrounding rock mass should be kept above −50◦C even after 30 years of theLNG storage operation. According to this criterion, if a maximum temperature drop

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is assumed to be −70◦C (initial temperature of rock mass: 20◦)C, the calculatedthermal stress was a tensile stress equivalent to 6.42 MPa with 2.62 × 10−6/◦C,13.37 MPa with 5.78× 10−6/◦C.• Cracks are formed and expanded at a lower temperature when a pre-crack exists in

the model. It would be thought that as the mechanical properties of the cracks aremuch poorer than those of the surrounding rock, a pre-crack becomes easily widerdue to the displacements occurred by the cooling-down of the rock mass, an inducedstress is concentrated at the tip of the pre-crack and it causes a gradual expansion ofthe cracks.• As the number of the joint sets increase or the joint properties decrease, the amount

of the crack occurrence has been reduced. So it could be thought that the presenceof the joints in the rock masses plays an important role in the creation and thepropagation of the cracks under the cryogenic conditions.

The results of the study are not entirely comparable with the observations from the realsites due to the complicated geological and the groundwater conditions. However, it couldbe possible to estimate qualitatively the mechanisms of fracture with the PFC2D models.

Acknowledgements

This study was funded by the Korea Institute of Construction & Transportation TechnologyEvaluation and Planning under the Ministry of Construction & Transportation in Korea(Grant No. 05-D10, Development of Water Control Technology in Undersea Structures).

References

1. Amantini, E. and Chanfreau, E., “Development and construction of a pilot lined cavern for LNGunderground storage”, 14th International conferences & Exhibition on Liquefied Natural Gas,Doha, Qatar, 2004, PO–33.

2. Chung, S.K., Park, E.S. et al., “Study on the design parameters for an underground LNG storagesystem in lined rock cavern and analysis of results from by a pilot test”, Research report by KIGAMsubmitted for SKEC, 2004, p. 144 (In Korean).

3. Dahlstrom, L.O., Swedenborg, S. and Evans, J., “Localization of underground hydrocarbon gasstorage caverns in respect to performance criteria”, Proc. World Tunnelling Conference, 2004, C29:1–8.

4. Glamheden, R. and Lindblom, U., “Thermal and mechanical behavior of refrigerated caverns inhard rock”, Tunnelling and Underground Space Technology, 2002, 17: 341–353.

5. Lee, H.W., A study on thermal cracking and temperature dependence of strength and deformationbehavior of rocks, Ph.D Thesis, Seoul National University, 1992, p. 196.

6. Monsen, K. and Barton, N., “A numerical study of cryogenic storage in underground excavationswith emphasis on the rock joint response”, Int. J. of Rock mechanics & Mining Sciences, 2001, 38:1035–1045.

7. Park, E.S., Chung, S.K., Kim, H.Y. and Lee, D.H., “Simulation of fracture mechanics for rockmasses under very low temperature conditions”, 4th Asian Rock Mechanics Symposium, Singapore,2006, p. 276.

443

UDEC Simulation of Block Stability Analysis around a LargeCavern

A. SOOKHAK1, A. BAGHBANAN1,∗, H. HASHEMALHOSSEINI1 AND M. BAGHERI2

1Mining Engineering Department, Isfahan University of Technology (IUT); Isfahan; IRAN2Division of Rock and Soil Mechanics, Royal Institute of Technology (KTH), Stockholm, Sweden

1. Introduction

Evaluation the volume of possible unstable blocks, analysis of block stability and estimationof minimum required support patterns for blocky rocks are important steps in undergroundexcavation design. One of the most serious problems in tunnel excavation is the accidentalfalling of rock blocks that are formed by the intersection of the tunnel surface and disconti-nuities in the rock mass. Prediction and prevention of falling blocks demands a removabilityanalysis on the rock blocks based on a precise characterization of discontinuities in the rockmasses around the tunnel. The volumes of possible unstable blocks are evaluated using Dis-crete Fracture

Network (DFN) method and the forces acting on blocks and designed support patterns arealso estimated using numerical methods such as Distinct Element Method (DEM) approach.1

In DFN modeling the fracture system geometry based on stochastic representations of frac-ture systems, using the probabilistic density functions of fracture parameters (e. g. orien-tation, size and location and aperture) formulated according to field mapping results. DFNpresents a more realistic representation of geology and fracture network geometry. The prob-lem in generated DFN models is calibration for making representative realizations of DFN.Using some statistical tests such William-Watson (W-W) test the most fitted DFN modelswith mapped fracture in the field are selected for block stability analysis. The W-W test is astatistical test of means for spherical data which is conducted on the composite data set todetermine the equivalency on the mean fracture orientations from two sets of observations.1

The DEM for modeling discontinues media is relatively new and focuses mostly on appli-cations in the fields of fractured or particulate geological media. The essence of the DEM isto represent the fractured medium as assemblages of blocks formed by connected fracturesin the problem domain, and solve the equations of motion of these blocks through contin-uous detection and treatment of contacts between the blocks. The blocks can be rigid orbe deformable with FDM or FEM discretizations. Large displacements caused by rigid bodymotion of individual blocks, including block rotation, fracture opening and complete detach-ments is straightforward in the DEM, but impossible in the FDM, FEM or BEM.2 The DEMapproach, either explicit or implicit, has become a powerful numerical modeling tool simplybecause of its flexibility in handling a relatively large number of fractures, for either purelymechanical problems or for coupled THM processes.2 The main difficulty for the DEM mod-eling is the uncertainty about the fracture system geometry, and the effect of this uncertaintyon stability of excavations.

The design of reinforce excavation usually requires a number of simplifying assumptions torender the problem tractable. A methodology is proposed which is built on the foundationsof block theory laid down by Ref. 3,4. This is consolidated into a rational approach andsupplemented with reinforcement design for unstable blocks. Methodologies for excavationdesign have been presented by Ref. 5. In this procedure, if an excavation is assessed to be




unstable it needs to be redesigned or a reinforcement scheme proposed. If a reinforcementscheme is proposed, it must also be assessed to indicate if modifications are required toimprove stability or to optimize design.

The behavior of jointed rock is characterized by the nature and disposition of discon-tinuities. The near by discontinuities to excavation define the surface block assembly andinfluence its stability. When a set of reinforcement is installed through the surface blocks, itintersects and reinforces the block faces. Block movement is defined by three translationaldisplacements towards the excavation. These displacements subject the reinforcement at thediscontinuities complex combinations of tension and shear with components of bending, tor-sion and compression. Experience has shown that reinforcement is most effective under lowstress conditions that accompany surface block instability. This leads us to believe that rein-forcement schemes for excavations in jointed rock can be designed using a number of simpleconcepts based on the examination of the assemblage of surface blocks. Examples of severalforms of reinforcement behavior are shown in Figure 1.5

In this study firstly the generated DFN models are verified by the W-W test and assignedto the DEM model as the geometric basis for the block stability analysis using UniversalDistinct Element Code (UDEC). The UDEC is a two-dimensional numerical program basedon the distinct element method for discontinuum modeling. UDEC simulates the responseof discontinuous media (such as a jointed rock mass) subjected to either static or dynamicloading. The discontinuous medium is represented as an assemblage of discrete blocks. Thediscontinuities are treated as boundary conditions between blocks; large displacements alongdiscontinuities and rotations of blocks are allowed. Individual blocks behave as either rigidor deformable material. The relative motion of the discontinuities is also governed by linearor non-linear force-displacement relations for movement in both the normal and shear direc-tions. UDEC is based on a “Lagrangian” calculation scheme that is well-suited to model thelarge movements and deformations of a blocky system.6

Two types of reinforcement model are provided in UDEC: local and global reinforcement.A local reinforcement model considers only the local effect of reinforcement where it passesthrough existing discontinuities. A global reinforcement model considers the presence of thereinforcement along its entire length throughout the rock mass. Surface support consists ofconcrete lining, steel sets, shotcrete, etc. that are placed on the surface of an excavation

Figure 1. Reinforcement actions at opening and shearing discontinuities, after.5

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Table 1. Input geometrical parameters of fracture sets in this study.

Orientation set Coefficient, k Fracture Intensity Length Distribution Parameter

(Dip/DipDirection) For Fisher Percentage (P20) Distribution Mean Std. Dev.Distribution (%) (m) (m)

1 (80/155) 7.63 22 0.062 Lognormal 4.8 1.72 (85/260) 9.81 11.3 0.032 Lognormal 4.5 1.43 (80/50) 9.81 11.3 0.032 Lognormal 4.5 1.44 (90/25) 7.63 22 0.062 Lognormal 4.8 1.7

5 (30/190) 7.63 22 0.062 Lognormal 4.8 1.76 (70/90) 9.81 11.3 0.032 Lognormal 4.5 1.4

and, in many cases, act to truly support, in whole or part, the weights of individual blocksisolated by discontinuities or zones of loosened rock.6 For a large excavation the combinationof shotcrete (for stability of small blocks) and bolt pattern for stability of all primary andsecondary blocks3 is applicable and used in this research work.

In this paper, estimating block volumes based on the DFN generation are first explained.Then the design methodology for estimating block stability and support pattern using DEMapproach are described. The results of block stability analysis and support design for a largeexcavation are presented in Section 4 and finally we draw some conclusion and discuss aboutresults in Section 5.

2. Estimation Method for Block Volumes Around Excavations

Dershowitz and Einstein introduced two major approaches for describing the assemblage ofgeometric joint characterization (disaggregate and aggregate). In the disaggregate approach,the joint characteristics such as fracture orientation and length are described separately bytheir distributions; however In the aggregate approach, the interdependence of fracture char-acteristics is captured through the joint system model.7,8 The DFN analysis is based on thesecond approach.

2.1. DFN generation

A large number of stochastic DFN realizations are generated based on the field results of acavern in southeast Sweden as reported in Ref. 8 as shown in Table 1. Using W-W statisticaltest the most fitted DFN models among the 1000 DFN realizations, with mapped fracture inthe field are selected for block stability analysis. The real fracture pattern in this site of studywas taken from one cross-cut tunnel perpendicular to the cavern axis. A FORTRAN codewas developed based on the W-W test to check the compatibility of DFN realizations withthe results of fracture mapping.

3. Block Stability Analysis

In this study, block stability around the cavity was analyzed by numerical DEM method.The geometric dimensions of a horseshoe cavern are 115 m length, 21 m width and 27 mheight, and it has been located horizontally in direction of N12E. The maximum measuredprincipal stress is horizontal direction and oriented E-W with magnitude of 6 MPa. Theintermediate stress is 4 MPa in N-S in horizontal plane, and minimum principal stress is in

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Table 2. Mechanical properties of intact rock and fractures.

Intact rock Elastic modulus (GPa) Poisson’s ratio Compressive strength (MPa)77.0 0.27 161

Fractures Basic friction Residual friction Normal stiffness Shear stiffnessangle angle (MPa/mm) (MPa/mm)

34 27 100 29

Table 3. Property of bolts and shotcrete.

Bolts

Tensile Bond Axial Shear Ultimate 12 “active” length (m)Capacity strength Stiffness Stiffness Shear

(MN) (MN/m) (N/m) (N/m) Capacity (N)0.31543 0.3467 2.483×109 0.285×109 0.029× 106 0.5

Shotcrete

Shear Unit Elastic Poisson’s Tensile Residual Compressivestrength Weight Modulus Ratio Yield Tensile Yield(MPa) (MN/m3) (Pa) Strength Yield Strength

(Pa) Strength (Pa) (Pa)2 0.024525 18×109 0.15 450× 106 450× 106 20× 106

vertical direction which varies by value of overburden. The input mechanical properties ofintact rock and fractures are presented in Table 2 and Mohr-Coulomb criterion has beenused for stability analysis by UDEC. It should be noted that these parameters serve onlyas data sources for generations of more realistic model for generic study not for a case ofsite application. The results presented and conclusions reached have therefore no link to theactual site condition at all.

3.1. DEM methodology for rock block stability

The DEM technique is based on the equations of motion of rigid or deformable bodies.Although the assumption of a rigid body is a simplification, in some circumstances such asthe cases involve large-scale movement on discontinuities this idealization is quit practical.8

In this study firstly the generated DFN models are verified by the W-W test and assigned tothe DEM model as the geometric basis for the block stability analysis using UDEC code. Wefirstly simulate the model without any excavation till equilibrium condition is satisfied. Thevolumes of possible failed blocks are calculated after cavern is excavated and different boltpatterns are examined until all blocks become stable. The common mechanical properties ofreinforcement and shotcrete are reported in Table 3.

4. Results

Volume of potential failure blocks and required support patterns for the most compatibleDFN realizations with field fracture mapping are numerically evaluated. Fig. 2 show the fre-quency (Fig. 2(a)) and cumulative (Fig. 2(b)) distribution of potential unstable block volumes.Although some large block volumes exist in some models, however the calculated mean valueof block volumes is 2.06 m3. As can be seen the unstable block volumes follow the lognor-mal distribution function (Fig. 2(a)). About 80 percent of the unstable block volumes are less

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a) b)

201612840

200

150

100

50

0

block volume

Freq

uenc

y

Loc -1.158Scale 2.080N 210

403020100

10095

80

60

40

20

0

block volume

PCD

F (%

)

Loc -1.158Scale 2.080N 2101.8 9.6

Figure 2. Distribution of key block volume from DFN-DEM analysis (a), and probabilistic cumulativedistribution graph of unstable blocks volume (b).

than 1.809 m3 and 95 percent are less than 9.6 m3. Which means that small block volumesare dominated (Fig. 2(b)).

Figure 3 shows an example of block stability analysis of DFN-DEM analysis. The redarrows show the displacement vectors and boundaries of blocks are represented in green line.Figure 4 shows a proposed bolt pattern around the wall and roof of cavern. The sequenceof support design procedure from unstable blocks, insufficient and minimum required boltpattern for stability of blocks are illustrated in figure 5. Both primary and secondary blocks(according to the Goodman and Shi definition) may cause the block instability hazard (Fig.5(a)) and therefore they both need some reinforcements. Beside the mechanical properties ofbolts and shotcrete, density and also length of installed bolts around the cavity are importantparameters in minimum required support pattern for stability of an excavated undergroundspace. The effects of two lather parameters (bolt density and lengths) on stability of cavernare considered in this research work. Therefore the total required bolt lengths are evaluatedfor different DFN models. Fig. 5(b) shows insufficient 3 m spacing and 1 m bolt length whichwas stabilized by a 6m length with 1m spacing pattern of bolts in Fig. 5(c).

Table 4 shows minimum required bolt patterns of 8 DEM models and Fig. 6 show thefrequency distribution (Fig. 6(a)) and also empirical cumulative distribution of required boltlengths (Fig. 6(b)) for all fifteen models.

Mean value of required bolt length is 228 m. Like the unstable block volumes distributionthe lognormal distribution function is well fitted with the required bolt length. More than 80percent of DEM models need only 309.3 m bolt which means that the bolt pattern with 1m

Table 4. Result of DFN-DEM analysis.

DFN number 1 2 3 4 5 6 7 8

blot length (m) 15 3 9 6 3 3 6 15bolt spacing (m) 5 7 7 1 5 3 3 1number of bolt 14 11 11 68 14 23 23 68total blot length (m) 210 33 99 408 42 69 138 1020

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Figure 3. Failed blocks around the cavern.

Figure 4. Position of bolts in wall and roof cavern.

spacing and 5m length is the mostly sufficient bolt pattern for cavern stability. However only5 percent of DEM models demand relatively heavy support pattern which is a pattern with1m spacing and 12 m length of bolts (Figure 6b).

5. Conclusions

Block stability analysis around a large excavation is analyzed with DFN-DEM approach.Different combination of geometric parameters of fracture sets are selected and unstableblock volumes and minimum required support pattern are estimated probabilistically usingDEM. For generated and calibrated DFN models, a series of numerical DEM modellingare performed to measure volume of potential unstable blocks and also minimum requiredsupport pattern.

The following results are deduced from our numerical experiment:

• DEM modelling is a very strong tool for stability analysis in large underground excava-tion in blocky rocks and make a more realistic evaluation for support pattern in engi-neering works.

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a) b)

c)

Figure 5. Unstable blocks without any support (a), insufficient of bolt pattern with bolt length of 3mand spacing of 1m (b), and stable model with bolt pattern of 6m length and 1m spacing (c).

a) b)

1500125010007505002500

20

15

10

5

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bolt length (m)

Freq

uenc

y

Loc 4.763Scale 1.154N 15

180016001400120010008006004002000

10095

80

60

40

20

0

bolt length (m)

PCD

F (%

)

Loc 4.763Scale 1.154N 15

309.3 781.2

Figure 6. Distribution of required bolt patterns in DFN-DEM analysis (a), and probabilistic cumula-tive distribution function of bolt length (b).

• Using W-W statistical test, most compatible DFN realizations with field mapping of frac-tures are selected.• Probabilistic design provides a flexible tool for engineers to design different support pat-

terns based on the importance of a project and confident level.

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• In our study, both unstable block volumes and total required bolt lengths follow lognor-mal distributions.• Small block volumes are dominated and 80 percent of the unstable block volumes are

less than 1.809 m3.• More than 80 percent of DEM models need only 309.3 m bolt which means that the bolt

pattern with 1 m spacing and 5 m length is the mostly sufficient bolt pattern for cavernstability at this confident level.

References

1. Baghbanan, A. and Sookhak, A. Hashemalhosseini, H. Bagheri, M., “Block stability analysisaround a large cavern using probabilistic approach” , EUROCK2009, Cavata, Croatia, (Acceptedfor publication).

2. Jing, L., “A review of techniques, advances and outstanding issues in numerical modeling for rockmechanics and rock engineering”, in J. Rock Mechanics and Mining Sciences, 2003, pp. 283–353.

3. Goodman, R.E. and Shi, G.H., “Block theory and its application to rock engineering”, 1985.4. Priest, S.D., “Discontinuity Analysis for Rock Engineering”, 1993.5. Windsor, C. and Thompson, A., “Reinforcement design for jointed rock masses, Presented at 33rd

US Symposium on Rock Mechanics”, 1992.6. Itasca Consulting Group Inc, UDEC User’s guide, ver 4.0, Minneapolis, Minnesota, 2004.7. Dershowitz, W.S. and Einstein, H.H., “Characterizing Rock Joint Geometry with Joint System

Models, Rock Mechanics and Rock Engineering”, 1998, pp 21-51.8. Bagheri, M. and Baghbanan, A. and Stille, H., “Some aspects on model uncertainty in the calcu-

lation of block stability using kinematics limit equilibrium”, in Proc American Rock mechanicsSymposium, SanFrancisco 2008.

452

The Application of Meshless Methods in Analysis of DiscontinuousDeformation

M. HAJIAZIZI∗

Assistant Professor, Semnan University, Iran

1. Introduction

The computational problems in engineering branch grow ever more in some fields like crackpropagation, fragmentation and large deformation in the simulation of manufacturing pro-cesses for solid and liquids, etc. One needs to model the large deformation and crack propaga-tion properly with arbitrary paths. The analysis of these problems with conventional compu-tational methods such as finite element and finite difference are not well proper. The analysisof large deformation problems by the methods based on meshes may require the remeshing ofthe domain in each step of the evolution. This strategy is proper for method based on meshesbut introduce numerous difficulties. The continuous remeshing of the domain in each step ofthe evolution leads to degradation of accuracy and complexity in the computer program andthe time-consuming mesh generation. Over the past three decades, many researchers havecome to realize that so-called meshless methods can be developed that eliminate the meshesand their difficulties. In the meshless methods there are only nodes.

Although must be taken to meshes in at least parts of the some of meshless methods, oftenbe treated without remeshing in shape function with minor costs in accuracy degradation.In this manner some of meshless methods basically require no meshes in background, suchas Finite Point Method (FPM).1 Therefore using methods based on meshes to solve largeclasses of problems or three dimensional problems are very awkward. About thirty yearsago until recently many researchers have developed types of meshless methods. T.P. Fries andH.G. Matthies2 published a special issue on meshless in July 2004 that is classification andoverview of meshfree methods. They introduced type of meshless methods such as SmoothParticle Hydrodynamics (SPH), Diffuse Element Method (DEM),3 Element Free Galerkin(EFG),4 Least Squares Meshfree Method (LSMM), Meshfree Local Petrov Galerkin (MLPG),Local Boundary Integral Equation (LBIE), Partition of Unity Methods (PUM), hp clouds,Natural Element Method (NEM),5 Meshless Finite Element Method (MFEM), Reproduc-ing Kernel Element Method (RKEM). In this paper EFG method is used with enforcing theessential boundary conditions is applied penalty method and the example is shown goodresults.

2. Element Free Galerkin Method (EFG)

Belytschko et al.6 modified the constructing shape function for Diffuse Element Method(DEM). They named it the Element Free Galerkin (EFG) method. The Moving Least Squares(MLS) approximation procedure related to construct shape function for EFG method. TheMLS approximation uh(x) is defined in the form of

h(x) =m∑

j=1

pj(x)aj(x) ≡ pT(x)a(x) (1)




where pj(x) are monomials of basis function in the space coordinates

xT = [x,y,z] (2)

PT(x) in 1D space is provided by

PT(x) = [1,x,x2, . . . ,xm] (3)

and in 2D space

PT(x,y) = [1,x,y,xy,x2,y2, . . . ,xm,ym] (4)

and in 3D space, we have

PT(x,y,z) = [1,x,y,z,xy,yz,zx,x2,y2,z2, . . . ,xm,ym,zm] (5)

where m is the number of terms of monomials (polynomial basis). a(x) is a vector of coeffi-cients and is obtained at any point x by minimizing J.

J =m∑

i=1

w(x− xI)[pT(xI)a(x)− uI]2 (6)

where J is a function of weighted residual and constructed using the approximated values ofthe field function. The stationary of J with a(x)

A(x)a(x) = B(x)u (7)

or

a(x)=A−1(x)B(x)u (8)

where

A(x)=n∑

I=1

wI(x)pT(xI)p(xI) (9)

B(x)=[w1(x)p(x1),w2(x)p(x2), . . . ,wn(x)p(xn)] (10)

u=[u1,u2, . . . ,un] (11)

Substituting the Equation (8) into (1) leads to

uh(x) =n∑

I=1

m∑j=1

pj(x)(A−1(x)B(x))jIuI (12)

or

uh(x) =n∑

I=1

�I(x)uI (13)

where the MLS shape function �I(x) is defined by

�I(x) =m∑

j=1

pj(x)(A−1(x)B(x))jI ≡ pT(x)A−1(x)B(x) (14)

The partial derivative of �I(x) can be obtained as follows

�I,i =m∑

j=1

pj,i(A−1B)jI + pjA−1(B,i − A,iA−1B)jI (15)

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The weight function wI(x) is positive and an important coefficient. In this paper it is

wI(x) ={

1− 6s2 + 8s3 − 3s4 for: dI <= r0 for: dI > r

(16)

where s = dI/r, dI = ‖x-xI‖, r = influence domain.The weight function is large for xI close to x and is small for xI far from x and is zero for

out of influence domain.

3. Displacement, Strain and Stress

The displacement of any point in the domain is obtained by

uh(x) = �(x)u (17)

where �(x) is shape function and it is defined by

�(x) = pT(x)A−1(x)B(x) (18)

ua is the displacement vector of nodes in the influence domain x.The displacement vector of nodes is obtained from equilibrium equation. The equilibrium

equation can be obtained from variational principle. The functional of the total potentialenergy of a material is given by

� = �b +�f +�p (19)

where �b is the elastic strain energy of block. �f is the potential energy of the body force feand �p is the potential energy of the concentrated force p. They are given by

�b =∑

e

te

∫∫�e

12εTDbεdxdy = 1

2

∑e

uTb (te

∫∫�e

BTb DbBbdxdy)ub (20)

�f = −∑

e

∫∫�e

uTfedxdy = −∑

e

uTb (∫∫

�eNTfedxdy) (21)

�p = −∑m

uTpm = −∑m

uTb (NTpm) (22)

We can obtain equilibrium equation from the stationary of functional � in (19). The equi-librium equation of solid materials is given by

KU = P (23)

where

K =∑

e

te

∫∫�e

BTb DbBbdxdy (24)

P =∑

e

∫∫�e

NTfedxdy +∑m

NTpm (25)

U = [u1,v1,u2,v2, . . . ,un,vn]T (26)

where te is the thickness of the material e, n is the total number of nodes in the problemdomain, U is displacement vector related to total number of nodes in the problem domain.

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The displacement of any point into problem domain is obtained by

u = N.ub (27)

where

N =(�1(x), 0, �2(x), 0, . . . , �n(x), 0

0, �1(x), 0, �2(x), 0, . . . , �n(x)

)(28)

and ub is the displacement vector of influence domain x. The Strain and Stress at any point xare given by

ε = Bbub (29)

σ = DbBbub (30)

where

Bb =⎛⎝�1,x(x), 0, �2,x(x), 0, . . . , �n,x(x), 0

0, �1,y(x), 0, �2,y(x), 0, . . . , �n,y(x)�1,y(x), �1,x(x), . . . , �n,y(x), �n,x(x)

⎞⎠ (31)

and for plain stress state

Db = (E/(1− υ2))

⎛⎝1 υ 0υ 1 00 0 (1− υ)/2

⎞⎠ (32)

and for plain strain state

Db = (E/(1− 2υ)(1+ υ))

⎛⎝1− υ υ 0

υ 1− υ 00 0 (1− 2υ)/2

⎞⎠ (33)

and

ε = [εx,εy,εxy]T (34)

σ = [σx,σy,σxy]T (35)

4. The Equilibrium Equation with Enforcing the Essential BoundaryCondition

The nodal value of the interpolation function uh(x) in element free Galerkin method is notequal to the nodal value of the function u(x). This is causes the shape function of EFG methodis not equal to kronecker delta. Namely

�I(xJ) �= δIJ (36)

Thus the essential boundary condition should be imposed. A simple and efficient way forimposing essential boundary condition is penalty method. The essential boundary condition

456


is

u = u on �u (37)

where u is the prescribed displacement on boundary u. The equation is obtained from weakform Galerkin with enforcing the essential boundary condition and using penalty method as7∫�

δ(Lu)Tc(Lu)d�−∫�

δuT. bd�−∫tδuT. tcd− δ

∫u

(.5)(u−Ou)T.α. (u−Ou)d = 0 (38)

Applying mathematical calculation on (38), the final equilibrium equation is

[K+ Kα]U = P+ Pα (39)

where Kα and Pα are obtained for the essential boundary condition using penalty method, as

Kα = α∫u

NTNd (40)

Pα = α∫u

NT ud (41)

where α is penalty factor. Ideally it is true to use infinite penalty factor. But if it is taken asinfinite or too large, the numerical problems will be encountered. Thus the penalty factor isa number that the constraints be properly enforced. Usually it is equal to 103E to 1013E andE is elasticity modulus.


Example. In this example an infinite plate with a circular hole is considered (Fig. 1). Theplate is subjected to a uniform tension in the x direction at infinity. The plane strain state is

1

2

3

4

5 678

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

x

y

Figure 1. The plate with a hole subjected to a tensile load in the x direction.

457


assumed. The uniform tension is 100 KN. 27 nodes are selected in domain. Radius of circleis equal to 1m with 5 nodes (No. 1, 5, 6, 7, 8) are selected on boundary of circle qudrature.

Distance between nodes in x and y directions 1m are assumed except for nodes on circlequadrature. 240 cells and 3 ∗ 3 Guass Qudrature point in each cell is selected in domain.The weight functions were Quadraticspline. The Poisson ratio and elasticity modulus are0.3 and 20 GPa respectively. The influence domain and penalty factor are 1.185 and 10 5Erespectively. The results from exact solution and present method are compared. To comparetwo methods the results for all nodes in x direction in Figs. 2 to 4 are presented. The resultsshow good concede between exact solution and EFG (MFree) method. The exact solution

0

0.000004

0.000008

0.000012

0.000016

0 1 2 3 4 5 6 7 8 9

Nodes number

Dis

plac

emen

t in

x di

rect

ion(

m)

Exact MFree

Figure 2. Displacements in x direction for nodes 1 to 9 in hole plate.

0.00E+00

5.00E-06

1.00E-05

1.50E-05

2.00E-05

2.50E-05

10 11 12 13 14 15 16 17 18

Nodes number

Dis

plac

emen

ts in

x d

irec

tion(

m)

Exact MFree


458


0.00E+00

5.00E-06

1.00E-05

1.50E-05

2.00E-05

2.50E-05

3.00E-05

19 20 21 22 23 24 25 26 27

Nodes number

Dis

plac

emen

t in

x di

rect

ion(

m)

Exact MFree


for displacement is8

ux = σ (1+ ν)

E

(1

1+ ν r cos θ + 21+ ν

a2

rcos θ + 0.5

a2

rcos 3θ − 0.5

a4

r3cos 3θ

)

uy = σ (1+ ν)

E

(−ν

1+ ν r sin θ − 1− ν1+ ν

a2

rsin θ + 0.5

a2

rsin 3θ − 0.5

a4

r3sin 3θ

)

6. Conclusions

The meshless methods are suitable and easy to use compared to methods based on meshes.The based on meshes method for large deformations problems such as analysis of discon-tinuous deformation lead in reduction of speed and degradation of accuracy and complexityof the calculations. The meshless methods can be used for the analysis of discontinuousdeformation. In this paper the Element Free Galerkin method is used and showed very accu-rate results. This method can be used in homogeneous and nonhomogeneous problems. Thepenalty method is used with enforcing the essential boundary condition that is an approachin the regard. The element free Galerkin method for discontinuous problems in the elasticstate and in the elstoplastic state is used. Comparing of exact solution and EFG (MFree)method in examples has shown efficiency and accuracy of EFG method.

References

1. Onate, E., Idelsohn, S., Zienkiewicz, O.C., Taylor, R.L. and Sacco, C., “Stabilized finite pointmethod for analysis of fluid mechanics problems”, Comput. Methods Appl. Mech. Eng., 139,1996, pp. 315–346.

2. Fries, T.-P. and Matthies, H.-G., Classification and Overview of Meshfree Methods, BraunschweigInstitut fur Wissenschaftliches Rechnen Technische Universitat Braunschweig, July, 2004.

459


3. Nayroles, B. Touzot, G. and Villon, P., “Generalizing the finite element method: diffuse approxi-mation and diffuse elements” Comput. Mech., Vol. 10, 1992, pp. 307–318.

4. Belytschko, T., Lu, Y.Y. and Gu, L., “Element-Free Galerkin Method.”, International Journal forNumerical Methods in Engineering, Vol. 37, 1994, pp. 229–256.

5. Sukumar, N., Moran, B. and Belytschko, T., “The natural element method in solid mechanics” Int.J. for Numerical Method in Engng., Vol. 43, 1998, pp. 839–887.

6. Belytschko, T., Lu Y.Y., Gu, L., “Element-Free Galerkin Method”, International Journal forNumerical Methods in Engineering, 229–256, 1994.

7. Liu, G.R., Mesh Free Methods: Moving beyond the Finite Element Method, CRC Press, Florida,2002.

8. Timoshenko, S.P., Goodier, J.N., Theory of Elasticity, McGraw-Hill, New York, 1970.9. Zhu, T., Atluri, N., “A modified collocation method and a penalty formulation for enforcing the

essential boundary condition in the element free Galerkin method”, Computational Mechanics,(21), 1998, pp. 211–222.

460

The Optimum Distance of Roof Umbrella Method forSoft Ground by Using PFC

YUSUKE DOI1,∗, TATSUHIKO OTANI2 AND MASATO SHINJI1

1Graduate School of Science and Engineering, Yamaguchi University, Yamaguchi, Japan2Nishimatsu Construction CO., LTD., Tokyo, Japan

1. Introduction

In recent years, the construction case which adopts the NATM method as a tunnellingtechnique has been increasing in the ground condition and the overburden are not suffi-cient. In these cases, the roof umbrella method such as the forepolling pipe method or thepipe roof method has been used as an auxiliary method to minimize the surface settlementof ground, in cases of the tunnel construction close to the existing structures on the con-straint of land use of urban area (see Photo 1). However, an ordinary design method of roofumbrella method established based on an experiential technique, so that it cannot considerthe effects of minimizing settlement of the ground. Moreover, all the ground soil betweeneach forepolling pipes along the tunnel cross section has intend to falling out by the roofumbrella method or the pipe roof method, as shown in a Photo 2, so that the reinforcementof the ground by forepolling pipe cannot be expected.

Photo 1. The tunnel construction with the pipe roof method.

In this study, the relationship between general material properties such as the Young’smodulus and Poisson’s ratio, the cohesion and the internal friction angle of the ground andthe micro-properties of PFC which was one application program of granular material analysis




Photo 2. The construction situation of forepolling pipes.

method has determined by the simple biaxial element test. And the numerical simulation byusing PFC in which steel pipe diameter and distance between pipes change was carried out therelationship between the cohesion and the height of shape. Finally we propose the rationaldesign of the roof umbrella method using this correlation.

1.1. Conventional design method of roof umbrella in Japan

Figure 1 shows the history of the diameter of forepolling pipe by using the pipe roof methodsin JAPAN. It is clear from this figure, the diameter of a steel pipe become was 200 mm or lessin 1960’s, recently it reaches the 1000mm. According to the construction records of the Japanpipe roof association,1 one of the reason of the large the diameter of a steel pipe is the removalpossibility by manpower in the soft ground including the boulder. Generally, the ratio ofconventional forepolling pipe interval L to the pipe diameter R is experientially designedless than L/R = 2, it depend the discontinuities and the physical properties of the ground.

The

pip

e di

amet

er (m

m)

Construction year

Figure 1. The history of the diameter of forepolling pipe by using the pipe roof methods in Japan.

462


)2/45sin(/ −=

oRL

L

R

2/45 −o

roofPipe

outfalls which groundThe

Figure 2. The experimental design chart of the determination of the pipe interval.

Figure 2 shoes the experimental design chart of the determination of the pipe interval.1 Thisratio L/R is a experimental ratio that the ground between each forepolling pipes along thetunnel cross section has intend to falling out.

2. Numerical Modelling by Distinct Element Method

We have to consider the ground behaviour of tunnel excavation in the low cohesive ground.It may not be suitable to carry out a numerical simulation with the continuum model. In thisstudy, distinct element method (DEM) is used, with the two-dimensional numerical discon-tinuum program called PFC2D to simulate the discontinuous behaviour of the part of thecrown in tunnel.

PFC2D simulates the mechanical behaviour of a material by representing it as an assem-blage of circular particles that can be bonded to one another. The contact-bond model and theparallel-bond model can be chosen as the bonding method of particles in PFC2D. As shownin Fig. 3, the contact-bond model can be added to the point between particles, the parallel-bond model can be added moment force to the contact-bond model can also be transmittedamong particles. In this study, the contact-bond model was adopted and the stress betweenparticles exceeded designated strength.2

BondContact − BondParallel −

field.in theup pastedhasIt point.at theup pastedhasIt

It has pasted at the point It has pasted at the area

The Contact-Bond model The Parallel-Bond model

Figure 3. The outline of bonded models.

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2.1. Model for the biaxial test

The most basic mechanical properties, for example, Young’s modus and Poisson’s ratio, arederived from laboratory tests. In the continuum model, such elastic properties can be inputteddirectly. In PFC2D, however, the mechanical behaviour of the assemblage is dominated bythe micro-properties of the particles and micro-properties of the particles and the bondsbetween them. These properties cannot be obtained laboratory tests. Thus, the relationshipbetween the micro-properties and the mechanical properties should be determined prior tothe simulation of the roof umbrella.

Figure 4 shows the model of biaxial compressive test. For the biaxial test, the top and thebottom walls, as loading platens, and the velocities of the lateral walls are controlled by aserve mechanics that maintains a specific confining pressure. Under the confining pressure aredetermined 0.05, 0.1, 0.4 Mpa, a set biaxial test is conducted with the associated stress-strainbehaviour. Macro-properties were obtained by using stress-strain behaviour.

Tables 1 and 2 summarized the relationship between micro and macro properties deter-mined by biaxial test. Moreover, Fig. 5 shows the relationship between macro properties

Figure 4. The model of biaxial compressive test.

Table 1. The relationship between micro and macro properties determined by biaxial test in the caseof change of the internal friction angle.

464


Table 2. The relationship between micro and macro properties determined by biaxial test in the caseof change of cohesion.

Figure 5. The relationship between the mean contact-bond normal strength and the cohesion.

(cohesion) value and a micro parameters (the mean contact-bond normal strength). It is clearfrom this figure this relation well expresses the physical properties of the material.

Photo 3. The modelled part.

465


LR

Figure 6. The roof umbrella model as a part of the crown Figure 7. The snapshot of the particlespart of the tunnel. just after the gravity calculation.

Figure 8. The detail particles distribution of Fig. 7. Figure 9. The arching shape.

2.2. Simulation model of roof umbrella

In the numerical simulation, as shown in a Photo 3, roof umbrella model as a part of thecrown of the tunnel was applied. The roof umbrella model consists of two parts, the pipepart and the ground part. The pipe part (the big particle) is simulated forepolling pipe, it isfixed. Its radius is defined as the diameter R = 812.6, 609.6 and 406.4 mm respectively3).And the interval between forepolling pipes L is defined the ratio (interval / diameter) as 1.8,2.5, and 3.1 respectively.

466


3. Result and Discussions

3.1. Definition of falling particles

The snapshot of the particles just after the gravity calculation is shown in Fig. 7. It is clearfrom these figure particles between forepolling pipes starting fall down by gravity. Figure 8shows the detail particles distribution of Fig. 7. From Fig. 8, a particle without contact withthe other can find in this figure. We defined these particles as falling particles. The contactforce calculation is performed to the particles of all models, and it marked as black particlesand expressed an arching shape of the ground by the roof umbrella as shown in Fig. 9. Byusing Fig. 9 we obtain the arching height H as the remaining continuous ground.

Figure 10. The relationship between the cohesion Figure 11. The relationship between theand the arching height. internal friction angle and the arcing height.

Figure 12. The radius and the ratio of the arching height to the interval of the forepolling pipe by thedifference of the radius of forepolling pipe.

467


3.2. Relationship between the macro properties and the arching height

Figure 10 shows the relationship between the cohesion and the arching height. Figure 11shows the relationship between the internal friction angle and the arcing height. It is clearfrom these figures if the forepolling pipe interval widens the arching height raise. And thereis the correlation of the inverse proportion from Fig. 10 between the cohesion and the arch-ing height. And the arching height becomes a little lower, when the internal friction angleincreases. However, the correlation is not clear. Figure 12 shows the radius and the ratio ofthe arching height to the interval of the forepolling pipe by the difference of the radius offorepolling pipe. It is clear from these figures that each figure indicates almost same shapeeven if the ratio the interval of the steel pipe to the radius of forepolling pipe is different. Byusing these figures, if the cohesion of ground is obtained, we can design the optimum intervalof the forepolling pipe.

4. Conclusions

A number of numerical simulation have been conducted in order to understand the optimuminterval of roof umbrella. The following results have been found;

• Relation between a macro parameter and the ratio H to LThe arching height H increases, as the interval of forepolling pipe L widens. Althoughthe conventional design scheme paying attention to angle of internal friction, the cor-relation between the cohesion and the ratio H to L is clear.• The proposal of the optimal design of roof umbrella method

By using the design chart of the interval of forepolling pipe to the cohesion, theoptimum interval of roof umbrella is proposed by using the material properties ofthe ground.

References

1. Tatsuhiko Otani, Masato Shinji and Tatsunori Chijiwa; “Proposal of Numerical Model and theDetermination Method of Design Parameters for Pipe Roofing Method”, Doboku Gakkai Ronbun-shuu F, Vol. 64, No. 4, pp. 450–462, (2008).

2. ITASCA PFC2D version 3.1 Manual, “Bonding Models”, Theory and Background, 2004.3. TH-Piperoof Kyokai “Piperoofing method Construction method explanation/ Addition data”, p.

41, 2002.

468

3DEC Investigation on Slope Stability at Norwich Part Mine

S.G. CHEN1,∗ AND B. SHEN2

1Southwest Jiaotong University, China2CSIRO Exploration and Mining, Australia

1. Introduction

Two highwall failures with a similar pattern in shape and volume occurred at chainageCH800 and CH700 of R10N Strip 17 at Norwich Park Mine, Australia, on the 13th and26th of October 2005. Based on the site investigation, both failures were typical rock wedgefailure. The wedges were cut by a vertical joint on the left and a curved fault on the rightas shown in Fig. 1. The rock wedges are about 15 m wide on the highwall face, 10 mdeep into the highwall and 30 m high. The rock wedge failures are believed to be affectedby: (1) unfavourable geological structures (the vertical joint set and the curved fault); (2)unfavourable block geometry; (3) shear strength along the joint/fault planes; and (4) watereffect on shear strength of joints/faults. Based on the understanding, a new design was pro-posed for subsequent Strips to improve the highwall stability.

Figure 1. One of the two similar rock wedge failure in R10N Strip17.

This study is to apply a commercial software 3DEC (Three-dimensional Distinct ElementCode), developed by Itasca, USA1, to simulate the experienced highwall instability in R10N




Strip 17 and assess the new design in R10N Strip 18. To determine the Factor of Safety ofthe new design in R10N Strip 18, the following steps have been taken in this study: (1) Theexperienced highwall failures in R10N Strip 17 with the old design are back analysed todetermine the joint/fault shear strength. The Factor of Safety of the failed highwall with theold design is assumed to be 1.0; (2) The joint/fault shear strength is then applied to the newdesign in R10N Strip 18. The critical joint/fault shear strength for the new design is obtained;(3) Comparing the critical joint/fault shear strength for the new design with that of the olddesign, the Factor of Safety is then derived. As the two rock wedge failures are similar, onlyone is analysed.

2. Computational Model and Rock Properties

Figure 2 shows a part of the computational model with the completed excavation of thehighwall where the rock wedge can be seen. A total of eight rock strata are introduced in themodel which is derived from the borehole logging data as shown in Fig. 3. The strata withgreen colour in the model represent three coal seams: Dysart Seam, Rider Seam and HarrowCreek Seam.

The rock wedge is located in the centre of the model which is simplified from the fieldsurvey data as shown in Fig. 4. The rock wedge is cut by five faces: the joint (left), the upperpart of the curved fault (right), the lower part of the curved fault (bottom), highwall (front)and the floor of Harrow Creek Seam (top).

The rock properties used in the model are listed in Table 1. The UCS (uniaxial compressionstrength) and density are taken from the borehole logging data as shown in Fig. 3. The UTS(uniaxial tensile strength) is assumed to be 1/50 of UCS. The Young’s modulus, Poisson’sratio and friction angle are based on the laboratory test results of coal and rocks from othermines in Bowen Basin2,3. The cohesion is derived from UCS and an assumed friction angle.Four types of joints are included in the model. The joint/fault properties are listed in Table 2.

The wedgeThe wedge

Figure 2. The computational model.

470


-30

-20

-10

0

10

20

30

40

50

60

70

Top soil

Upper rock

Horrow Creekseam

Middle rock

Rider seam

Lower rock

Dysart seam

Base rock

-30

-20

-10

0

10

20

30

40

50

60

70

Top soil

Upper rock

Horrow Creekseam

Middle rock

Rider seam

Lower rock

Dysart seam

Base rock

a

Sandp

Figure 3. Borehole logging data and simplified strata used in the model.

Surveyed data The wedge in the modelSurveyed data The wedge in the model

Figure 4. The failed rock wedge in S10N Strip 17: surveyed data (left) and the wedge in the model(right).

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Table 1. Rock properties.

Rock type Densitykg/m3

Young’smodulusGPa

Poissonratio

CohesionMPa

Frictionangle( ◦)

UTSMPa

UCSMPa

Base rock 2500 4.0 0.25 2.89 30 0.20 10.0Dysart seam 1300 1.5 0.30 0.70 40 0.06 3.0Lower rock 2500 4.0 0.25 2.17 30 0.15 7.5Rider seam 1300 1.5 0.30 0.70 40 0.06 3.0Middle rock 2500 4.0 0.25 1.44 30 0.10 5.0Harrow Creekseam

1300 1.5 0.30 0.70 40 0.06 3.0

Upper rock 2100 1.5 0.25 0.87 30 0.06 3.0Sandpass 1100 1.0 0.40 0.29 30 0.02 1.0

Table 2. Joint properties.

Joint type NormalstiffnessGPa

ShearstiffnessGPa

CohesionMPa

Frictionangle(◦)

TensilestrengthMPa

Artificialjoints

10 10 104 0 104

Materialinterfaces

10 10 0 30 0

Joints 10 10 0 or 0.25 ? 0Faults 10 10 0 or o.25 ? 0

3. Modelling Cases and Results

The experienced rock wedge failure in R10N Strip 17 with the old design is back-analysedto determine the joint shear strength. The shear strength is then applied to the new design tofind out the critical friction angle.

To investigate the effect of horizontal/vertical in situ stress ratio and the cohesion, fourcases as listed in Table 3 are modelled. In the first three cases, zero joint cohesion is assumedand the horizontal/vertical in situ stress ratio is varied to be 0.33, 1.0 and 2.5. The fourthcase has the same horizontal/vertical in situ stress ratio as Case 3 but with joint cohesionof 0.25 MPa. The effect of joint cohesion on the rock wedge stability can be examined bycomparing Case 4 with Case 3.

The joint/fault shear strength for the old design and the critical joint/fault shear strengthfor the new design are determined numerically when the rock wedge is moving from a stable

Table 3. Modelling cases.

Case 1 2 3 4

Horizontal/vertical in situ stress ratio 0.33 1.0 2.5 2.5Joint/fault cohesion, MPa 0.0 0.0 0.0 0.25

472


state to an unstable state. By specifying the cohesion, the joint/fault shear strength can bedetermined by the friction angle only.

The criterion on judging rock wedge failure in the model is based on the velocity, displace-ment and movement of the rock wedge. For example, Figs. 5 & 6 show the displacement,velocity and movement when the rock wedge is stable. It can be seen that the displacementand velocity are very small. However, as shown in Figs. 7 & 8 for a failure case, the dis-placement and velocity at the rock wedge are large, indicating the rock wedge has failed andbecomes unstable.

Table 4 lists the modelling results for all the cases. The joint/fault friction angles determinedfrom the back-analysis of the experienced rock wedge failure (with old design) are 32, 33, 33,

Figure 5. Displacement when the rock wedge is stable.

Velocity

Movement

Velocity

Movement

Figure 6. Velocity and movement when the rock wedge is stable.

473


Figure 7. Displacement when the rock wedge is failed.

Velocity

Movement

Velocity

Movement

Figure 8. Velocity and movement when the rock wedge is failed.

32 degrees, respectively for Cases 1 to 4. The Factor of Safety for the old design is assumedas 1.0.

The critical joint/fault friction angles for the new design are obtained to be 27, 30, 31 and29 degrees, respective, for Cases 1 to 4. The Factor of Safety for the new design is derivedusing the equation below.

FOS = ShearStrengthShearStress

= c+ σn tanφactual

c+ σn tanφcritical(1)

474


Table 4. Modelling results.

Case 1 2 3 4

Horizontal/vertical in situ stress ratio 0.33 1.0 2.5 2.5Joint/fault cohesion, MPa 0.0 0.0 0.0 0.25Old design, φactual 32 33 33 32Factor of Safety 1.0 1.0 1.0 1.0New design, φcritical 27 30 31 29Factor of Safety 1.226 1.125 1.081 1.091

where:

φactual – Joint/fault friction angle obtained from back-analysis of the failuresφcritical – Critical joint/fault friction angle of the new design.c – Joint/fault cohesionσn– Normal stress on the joint/fault planes (can be obtained from the model).

If the cohesion is zero, the above equation can be simplified as:

FOS = tanφactual

tanφcritical(2)

Using Equation (2) for Cases 1–3, and Equation (1) for Case 4, the obtained Factors ofSafety are 1.226, 1.125, 1.081 and 1.091, respectively for Cases 1 to 4.

4. Conclusions

The key findings from this study are listed below:

• The highwall failures in R10N Strip 17 are believed to have been caused by anunfavourable combination of the joints and curved faults that cut off an major rockwedge.• The rock wedge, when totally exposed after extraction of the Dysart Seam, slide due to

insufficient joint/fault shear strength, leading to the highwall failure.• The new design is predicted to marginally improve the stability of the highwall,• The predicted Factor of Safety for the new design is within a range from 1.08 to 1.23

depending on the horizontal/vertical in situ stress ratio.

References

1. Itasca Consulting Group, Inc (2003). 3DEC user’s guide.2. Boland J.N. and Harbers C. (1999). Mechanical testing of rock samples from the Gregory Mine.

CSIRO Exploration and Mining Report 603C, March.3. Medhurst T.P and Carvolth D.J. (1996). Strength and deformation properties of coal at Moura Pit

18BL. CSIRO Exploration and Mining Report 289C, July.

475

Evaluation of Deformations around a Tunnel by using FEM,FEBEM, UDEC, UDEC-BE and CFS

RAJBAL SINGH∗

Central Soil and Materials Research Station, Olof Palme Marg, Hauz Khas, New Delhi-110016

1. Introduction

The methods for the analysis of underground openings are analytical methods or closedform solutions, experimental methods and numerical methods. The numerical methods canbe divided into the following categories:

• Finite difference method (FDM).• Finite element method (FEM).• Finite element method with infinite elements (FEM-IE).• Boundary element method (BEM).• Coupled finite element and boundary element method (FEBEM).• Universal distinct element code (UDEC).• Universal distinct element code with boundary elements (UDEC-BE).

The numerical methods such as FEM and BEM are increasingly used for the analysis ofunderground openings. FEM can be used for the analysis of any shape of underground open-ing incorporating practically any type of material behaviour of geological media and anycomplex boundary condition. In this method, the fixation of external boundary to representinfinite domain as encountered in the case of underground openings introduces approxima-tion in the results. Also, a large domain is to be discretized and the preparation of data istedious.

The use of BEM requires only the boundary of excavation to be descretized. BEM can beused for openings of any geometrical shape, but with limited zones of different geologicalmedia. This method is more accurate and less tedious than FEM in analysing a particularcase of underground opening. In the case of opening where the material properties are likelyto vary significantly near the face of the excavation, it is difficult to adopt BEM. For suchcases, finite elements can be used near the face of excavations to take care of the complexitiesin material properties and boundary elements can be used away from the opening to takecare of the infinite conditions of the media. This could be particularly useful in the caseof underground excavation with boundary at infinity. The coupled FEBEM is being usedextensively for finding out accurate solutions.

The coupling of FEM and BEM was first proposed by Zienkiewicz et al.1 and Kelly et al.2

The application of FEBEM to linear elastic analysis of underground openings was discussedby Brady and Wassying,3 Beer and Meek,4 Varadarajan and Singh,5 Varadarajan et al.6,7 andSharma et al.8 Beer9 applied FEBEM to visco-plastic analysis of mine pillar problem by usingDrucker-Prager yield criterion. Singh et al.10 and Vardarajan et al.11 applied the FEBEM toelasto-plastic analysis of underground opening. Singh12 applied the method for elastic andelasto-plastic analysis of circular opening in layered rock media.

In this paper, the applications of finite element method (FEM), coupled finite element andboundary element method (FEBEM) and universal distinct element code (UDEC) have been




presented for tunnel excavation problem. Further, the displacement and stresses predictedby coupled FEBEM and FEM analysis have been compared with closed form solution (CFS)in terms of accuracy of results. These results have further been compared with UDEC andUDEC with boundary elements (UDEC-BE). The application and feasibility of appropriateanalysis approach can be judged by keeping in view all the methods.

2. Computer Program for FEM and FEBEM

A computer program FEBEM was developed for two-dimensional analysis of undergroundopenings with plane strain condition. Simulation of one step excavation was incorporated inthe computer program. The same FEBEM computer program is used for FEM analysis also.The main computer program of FEBEM was developed by Singh (1985) in three stages forlinear elastic analysis. In the first two stages, the computer program of FEM and BEM weredeveloped separately and tested for the analysis of circular opening by comparing the resultswith available closed form solutions. In the third stage, coupling programs of FEM andBEM was developed the computer software of FEBEM. The FEBEM program was developedusing 8-noded isoparametric finite elements and 3-noded parabolic boundary elements. Theprogram has the provision of automatic generation of nodal coordinates, nodal connectivityof elements and loading on nodal points.

3. UDEC Numerical Modeling

The universal distinct element code (UDEC) is a two-dimensional numerical computer pro-gram based on the distinct element method for dicontinuum modeling, UDEC simulates theresponse of discontinuous media (such as a jointed rock mass) subjected to either static ordynamic loading. The discontinuous medium is represented as an assemblage of discreteblocks. The formulation and development of the distinct element method embodied in UDEChas progressed in last three decades beginning with the initial presentation by Cundall13.

The program can best be used when the geologic structure is fairly well defined from theobservation or geologic mapping. A wide variety of joint pattern can be generated in themodel. A screen plotting facility allows the user to instantly view the joint pattern. Adjust-ment can easily be made before the final pattern is selected for analysis.

4. Tunnel Analysis by FEM, FEBEM, UDEC and UDEC-BE

When modelling infinite bodies (e.g. tunnels and underground cavern) or very large bodies, itmay not be possible to cover the whole body with blocks due to constraints on memory andcomputer time. Artificial boundaries are placed sufficiently far away from the area of interestthat the behaviour in that area is not greatly affected. It is useful to know how far away toplace these boundaries and what error might be expected in the stresses and displacementcomputed for the areas of interest. A series of numerical experiments were performed on anumerical model containing a circular tunnel in an elastic material.

4.1. Tunnel analysis by FEM and FEBEM

A two-dimensional analysis with plane strain condition was carried out. The excavation oftunnel was simulated in single step. The discretizations of the circular opening for FEBEMand FEM are shown in Figs. 1 and 2, respectively. For the present comparison purposes,the excavation of the circular tunnel opening in a geological medium of infinite extent was

478


Figure 1. FEBEM discretization for circular tunnel.

Figure 2. FEM discretization for circular tunnel.

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considered. The radius of tunnel was taken as 1.0 meter. The geological medium was consid-ered to be homogeneous and the behaviour was assumed to be linear-elastic. The modulus ofdeformation and Poisson’s ratio were taken as 1 GPa and 0.25 respectively, for hydrostaticstress condition of .01 MPa.

A comparison of displacements from FEM and coupled FEBEM with closed form solutionare shown in Fig. 3. Based on this study, Singh12 concluded that boundary between FE andBE interface can be fixed at 4 times the radius of the tunnel while the boundary can be fixedat 8 times the radius of the tunnel in the case of FEM. The circular tunnel model as usedearlier by Singh12 and Varadarajan et al.7 was adopted for FEM and FEBEM. The results ofthese analyses were compared with available closed form solution (CFS) by Pender14. Thesame properties of rock mass were taken for the purpose of comparison.

Figure 3. Fixation for boundary for FEM and FEBEM and comparison with CFS.

Figure 4. UDEC discretisation for circular tunnel.

480


Figure 5. Fixation for boundary for FEM, FEBEM and UDEC and comparison with CFS.

4.2. Tunnel analysis by UDEC

The boundary distance in the case of UDEC was also fixed at the distances of 2, 4, 6, 8, 10and 12 times the radius of the tunnel. The discretisation of UDEC is shown in Fig. 4. Theratio of boundary distances and tunnel radius has been plotted against resulting displacementfor CFS, FEM, FEBEM and UDEC as shown in Fig. 5. The results of CFS, FEM and FEBEManalysis are referred from Singh12 and Varadarajan et al.8 The UDEC analysis was performedduring the research fellowship at NGI by Singh.15

It is seen from the results in Fig. 5 that the boundary fixation does not make much differ-ence in FEBEM and the boundary between finite element and boundary element regions canbe taken as 3 to 4 times the radius of the tunnel. However, the boundary must be fixed at adistance of 10 to 12 times the radius of the tunnel in the cases of FEM and UDEC. However,the boundary should not be less than 8 times the radius of the tunnel for having sufficientaccuracy in the results.

The study shown by Singh12 is interesting to note that the boundary can be fixed at adistance of 8R in FEM while it can be fixed safely at 4R in the case of FEBEM with a betteraccuracy in the resulting deformations and stresses. It can, therefore, be advantageous touse boundary element boundary in the case of UDEC since the code is having a provisionfor boundary element. The rock mass properties (Table 1) were taken the same as used bySingh.12 The values of normal and shear joint stiffness were chosen for this model by trialand error increasing or decreasing in the model. About thirty models were run for fixing thefinal values of normal and shear stiffness. Bulk modulus and shear modulus of rock masswere calculated from modulus of deformation and Poisson’s ratio.

Figure 6 shows the comparison of resulting displacement in UDEC by using a fixed bound-ary (UDEC) and boundary element (UDEC-BE) and using circular boundary (UDEC-CB) inplace of usual squire boundary being used in most of the problems. There is not much differ-ence in the magnitude of displacement in the case of using BE boundary as compared withCFS. The BE boundary can be used safely at a distance of 3 to 4 times the radius of thetunnel. However, boundary can be fixed at a distance of 8 times the tunnel radius in UDEC.

The comparison of accuracy between FEBEM and FEM is given in Table 2. To evaluatethe accuracy of the results, the analytical solution for deep circular tunnel given by Pender14

was used. The percentage errors are indicated with respect to analytical results as also givenin Table 2. In the case of FEBEM, the percentage error is very small and is in the order of0.09 per cent for displacement. In the case of FEM, the error is 4.47 percent, which is about50 times higher than that observed in the case FEBEM. Similarly, the magnitudes of error

481


Table 1. Rock mass properties.

S.No. Rock Properties Magnitude with Unit

1. Modulus of Deformation, Ed 44.80 GPa2. Bulk Modulus, K 23.30 GPa3. Shear Modulus, G 18.98 GPa4. Poisson’s Ratio, ν 0.185. Joint Normal Stiffness, JKn 100 MPa/mm6. Joint Shear Stiffness, JKs 100 MPa/mm7. Cohesion, c 1.60 MPa8. Friction Angle, φ 419. Density, γ 2940 kg/m3

10. Permissible Tensile Strength, τ 8.50 MPa11. Height of Overburden, h 300 m12. In-situ horizontal stress, σh 8.82 MPa13. In-situ vertical stress, σv 8.82 MPa14. Stress ratio, k 1.00

Figure 6. Fixation for boundary UDEC and UDEC-BE and comparison with CFS.

in stresses predicted by FEM analysis are higher than those predicted by FEBEM. Thus, inall the cases, FEBEM gives more accurate results than FEM and this is due to the fact thatthe interface boundary between finite element and boundary element regions is fixed at fourtimes the radius of the opening while it is fixed at eight times the radius of the opening in thecase of FEM.

Similar conditions are applied between UDEC and UDEC-BE, which gives more accurateresults than UDEC without boundary elements as given in Table 2. The error in UDEC ascompared to CFS is 3.92 per cent which is about 50 times higher than the error of 0.08 percent in UDEC-BE.

5. Conclusions

The following conclusions are drawn on the basis of this research work:

• An efficient alternative is to utilize the advantages of both FEM and BEM by couplingthem especially for the analysis of underground openings with significant modification ofvariation in rock mass properties near the tunnel excavation. Finite elements may be used

482


Table 2. Stresses and displacements around a tunnel from different analysis.

S. No. Points of Discus-sion

CFS FEBEM FEM UDEC UDEC-BE

1. Displacements(PercentageError)

−0.01250 −0.012511(0.09 %)

−0.011949(−4.47 %)

−0.012010(−3.92 %)

−0.012510(−0.08 %)

2. Major Stress(PercentageError)

1.697901.69750(−0.02 %)

1.64590(−3.06 %)

— —

3. Minor Stress(PercentageError)

0.30270 0.30270(0.20)

0.29350(−2.85 %)

— —

near the excavation surface for taking care of the complexities in the material behaviorand boundary elements may be used away from it to take into account of infinite domain.• From the analysis of circular tunnel excavation and comparison of the results as predicted

by FEBEM and FEM analysis, it has been found that the FEBEM is more accurate andeconomical than FEM. Therefore, it is concluded that the coupled FEBEM can efficientlybe utilised for the analysis of tunnel excavations.• FEM and UDEC can be used for the analysis of tunnel excavation with little compromise

on the accuracy. However, boundary elements can be used effectively with both FEM andUDEC as external boundary in the discretisation to increase the accuracy in the analysis.

Acknowledgements

The author is extremely thankful to Dr. Suzanne Lacasse, Director, Norwegian GeotechnicalInstitute (NGI), Oslo, Norway for awarding Research Fellowship to carry out this study atNGI. I am also thankful to Mr. Eystein Grimstad, Dr. Nick Barton and Dr. Rajinder Bhasinof NGI for their useful suggestions, technical assistance and domestic help during the stayin Norway. The efforts put in by then officials in the Ministry of Water Resources, Govt. ofIndia are also gratefully acknowledged for timely approval for the fellowship in Norway.

References

1. Zienkiewicz O.C., Kelly D.W and Dettess P., “The coupling of finite element method and boundarysolution procedures”, Int. J. Num. Meth. Engg., 11, 1977, pp. 355–375.

2. Kelly D.W., Mustoe C.G.W. and Zienkiewicz O.C. “Coupling boundary element methods withother numerical methods”, in Banerjee P.K. and Butterfield R. (eds), Development in BoundaryElement Methods, Vol. 1 Applied Science Publications, London, Chap. 10, 1979, pp. 251–285.

3. Brady B.H.G. and Wassyng A., “A coupled finite element method of stress analysis”, Int. J. RockMech. Min. Sci. & Geomech. Abstr., 18, 1981, pp. 475–485.

4. Beer G. Meek J.L., “The coupling of boundary and finite element methods for infinite problems inelastoplasticity”, Proc. of Third Int. Seminar Irvine, California, edited by C.A. Brebbia, 1981, pp.575–591.

5. Varadarajan A. and Singh R.B., “Analysis of tunnels by coupling FEM with BEM”, Proc. 4th Int.Conf. Of Num. Meth. in Geomech., Edmonton, Canada, Vol. 2, 1982, pp. 611–618.

6. Varadarajan A., Sharma K.G. and Singh R.B., “Analysis of circular tunnel by coupledFEBEM”,Proc. Indian Geotechnical Conference, Madras, India, VI, 1983, pp. 113–118.

483


7. Varadarajan A., Sharma K.G and Singh R.B., “Some aspects of coupled FEBEM analysis of under-ground openings”, Int. Journal Numerical and Analytical Methods in Geomechanics, 9, 1985, pp.557–571.

8. Sharma K.G., Varadarajan A. and Singh R.B., “Condensation of boundary element stiffness matrixin FEBEM analysis”, Commun. Appl. Num. Methods, 1, 1985, pp. 61–65.

9. Beer G., “Finite element, boundary element and coupled analysis of unbounded problems in elas-tostatics”,Int. J. Num. Meth, Engg., 19, 1983, pp. 567–580.

10. Singh R.B., Sharma K.G and Varadarajan A., “Elastoplastic analysis of circular opening byFEBEM”, 2nd Int. Conf. on Comp. Aided Analysis and Design in Civil Engg. Roorkee, India,1985, pp. 128–134.

11. Varadarajan A., Sharma K.G. and Singh R.B., “Elastoplastic analysis of an underground openingby FEM and coupled FEBEM”, Int. J. Num. Anal. Meth. in Geomech., 1987, pp. 475–487.

Singh Rajbal, “Coupled FEBEM analysis of underground openings”, Ph.D. Thesis, Dept. ofCivil Engineering, Indian Institute of Technology, New Delhi, India, 1985, pp. 285.

12. Cundall P.A., “A computer model for simulating progressive large scale movements in blocky rocksystems”, Proc. of the Symp. Int. Soc. Rock Mech., Nancy, France, Vol. 1, Paper No. II-8, 1971.

13. Pender M.T., “Elastic Solutions for a Deep Circular Tunnel”,Geotechnique XXX, 2, 1980, pp.216–222.

14. Singh, Rajbal, “Report on research fellowship at NGI”, Submitted to Norwegian GeotechnicalInstitute (NGI), Norway, 1999, pp. 55.

484

Numerical Modeling of Undrained Cyclic Behaviour of GranularMedia Using Discrete Element Method

B. FERDOWSI∗, A. SOROUSH AND R. SHAFIPOUR

Department of Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, Iran

1. Introduction

In a granular medium such as sand, forces are typically transformed through the contactsbetween particles. The discrete nature of the behaviour of the medium under loading makesits constitutive relation very complex; hence many laboratory testing may be necessary inorder to understand it in detail. An alternative way to study the behaviour of granular mediais to model soil as an assemblage of particles, using a variety of simple shapes such as circulardiscs, oval shaped rods, and spheres. Analytical, physical and numerical modelling have beenused for this purpose.

Considering the difficulties in both the analytical and physical (photo-elastic and othermethods) models, which are only limited to very simple cases of loading, the best way tomodel assemblies of discs or spherical particles is the numerical technics, mainly discreteelement method (DEM). The DEM is an explicit finite difference scheme, introduced byCundall,1 which can handle particles of different shapes and has been used by an increasingnumber of researchers in the last two decades.2−6 DEM can be used to determine many ofdesired information such as internal stresses, strain and micromechanical characteristics atany stage of loading path. The major advantage of DEM is the wealth of micromechanicaland statistical information that can be generated along with macroscopic responses.

In this study, the undrained cyclic behaviour of cohesionless soils has been explored indetails using the DEM. Undrained tests on fully saturated geomaterials are generally knownas constant volume tests as a consequence of low particle and fluid compressibility. In prin-ciple, it seems that a drained strain-controlled constant volume test is equivalent to anundrained test. This idea has led some researchers to simulate drained loading at a constantvolume, and assume that the results are representative of undrained loading. References 4 and7–12 have used constant volume modelling to study the undrained (monotonic and cyclic)behaviour of cohesionless granular media.

Herein, the constant volume testing has been used. A series of cyclic undrained tests wereperformed on two dimensional assemblies of circular discs with various initial states includ-ing two different void ratios of e = 0.208 and e = 0.178 and confining pressures in the rangeof p = 60 to p = 550 kPas, covering an almost complete range of loose and dense specimens.Results of the analysis of the macro- and micro-scale responses are explained in details. Thestress-strain response, pore pressure build-up and stress path are evaluated as macro-scaleresponse descriptors. Evolutions of deviatoric fabric tensor and coordination number withprogress of cycles are studies as micro-scale responses.

2. Discrete Element Method

DEM models granular materials as individual elements which can make and break con-tacts with their neighbours and are capable of analyzing interacting bodies undergoing large




absolute or relative motions. It’s important feature is that it incorporates the Coulomb’s fric-tional law at contacts between elements. Slippage occurs when the tangential force at contactexceeds a critical value. The equilibrium contact forces are obtained from a series of calcula-tions by solving Newton’s law of motion followed by force displacement law at each contact.When all forces for each contact in the assembly are updated, forces and moment sums aredetermined on each element, and the above process is repeated in cycles.

In this investigation, the DEM code developed by Ref. 13 have been used for modellingthe constant volume strain-controlled loading of the specimens. The motion of each particleis calculated using Newton’s second law as follows:

mpvp = fg +∑

c

fc + fd (1)

Ipωp =∑

c

rc × fc +Md (2)

in which, mp and Ip are the particle mass and moment of inertia, respectively; vp and ωpare linear and angular velocity vectors, respectively; dot indicates time derivative; fg is bodyforce, which is ignored in this study; fc is contact forces between two contacting particles; rcis branch vector, a vector connecting the center of particle to the point of contact and finallyfd and Md are damping force and moment which are introduced into the model to help theparticles reach the equilibrium state.

Herein, DEM employs a rigid body-soft contact approach, where particles are assumedto be rigid, and particle deformations occur just in contacts. The Hertzian and simplifiedMindlin-Deresiewicz relations are used for the normal and tangential contact forces, respec-tively. Furthermore, non-viscose local damping introduced by Ref. 14 is employed

f id = −α|f i

unbal|sign(νip) (3)

Mid = −α|Mi

unbal|sign(ωip) (4)

where, f iunbal and Mi

unbal are, respectively ith component of unbalanced force and momentacting on particle p that are sum of the body, contact and fluid forces and moments.

Constant volume condition is employed by setting the strain rate of walls as ε11 = −ε22 sothat the specimen’s volume is kept constant to account for water incompressibility. It shouldbe noted that in the current code, walls represent rigid boundaries.

3. Input Parameters of the Simulation

A two-dimensional assembly consisting of 7000 discs with diameters ranging from 0.15 to1.18 mm (four different sizes corresponding to log normal distribution) is used in the numer-ical simulations. The gradation used here is in the range defined for Regular Ottawa Sand,15

as shown in Fig. 1. The specimen dimensions are 28×28 mms. Initially the disc generation isaccomplished by a random number generator that places nonoverlapping spheres of desiredsizes corresponding to the desired grain size distribution at random locations.

Table 1 shows the input parameters used for the numerical simulations. Density scalingwas adopted to increase the critical time-step and reduce the computational cost of the DEMsimulations.16

4. Sample Preparation and Testing

It is well known that density and cyclic strain amplitude are the major factor affecting thecyclic behaviour of sands.

486


Dmax=1.18 mmDmin=0.15 mmD60= 0.4 mmD30= 0.32 mmD10= 0.2 mmCu= 2Cc= 1.28

Figure 1. Grain size distribution of the assembly.

Table 1. Input parameters for numerical simulations (*Ks

was approximated by23

Kn).

Property Symbol Value

Density (gr/mm3) ρ 2× 1010

Normal stiffness (gr/mm) Kn 7× 107

Shear stiffness (gr/mm) K∗s 4.67× 107

Critical time step (sec) �tcr 1.0Damping coefficient α 0.6Friction coefficient for shearing load μ 0.5

After generating the specimens by randomly distributing the particles in the box, a straincontrolled loading was applied to prepare specimens with desired isotropic confining pres-sures. It should be noted that desired confining pressures at each void ratio are accessible byassigning different friction coefficients to the particles during the stage of applying confiningpressure. Afterwards, sinusoidal cyclic strain-controlled loading are applied to the specimens.The program of the specimen preparations is summarized in Table 2.

Cyclic tests are carried out by applying a constant amplitude sinusoidal strain-controlledloading on the specimens. Mean normal stress (p) and shear stress (q) are defined asEqs. (5) and (6), respectively.

p = σ1 + σ2

2, q = σ1 − σ2

2(5) and (6)

Table 2. Simulation’s program.

Void ratio Friction coeff. Confining Press. (kPa.mm) Cyclic strain amplitude (%)

0.208 0.5 150 0.50.4 113 2.50.3 60

0.176 0.5 550 2.50.3 460

487


5. Results

5.1. Macro-scale response

Figures 2 (a–c) show the plot of deviatoric stress (q) vs mean stress (p) for specimens at voidratio e = 0.208, having initial confining pressures of p = 150, 113, 60 kPas, respectively. Itis seen that with loading and unloading cycles there are gradual decrease in mean stress (p)and deviatoric stress (q). The behaviour is similar to medium dense sand general behaviour.At initial cycles of loading, the specimens exhibit quasi-steady state, except the one with thelowest confining pressure (i.e. p = 60 kPa) which initially reaches the phase transformationline. Continuing the cycles of loading, all the specimens liquefied. Specimens with initialconfining pressure of 150, 113 and 60 kPas liquefied at 15, 17 and 19 cycles of loading,

a

cb

Figure 2. Deviatoric vs. mean stress variation for specimens (e = 0.208) at (a) p = 150 kPa,; (b)113kPa and (c) 60 kPa.

��

P=150 kPa

P=113 kPa

P=60 kPa

Figure 3. Variation of generated pore pressure vs. number of cycles for specimens (e = 0.208) at initialconfining pressures of p = 150 kPa, 113 kPa and 60 kPa.

488


showing that increasing the initial confining pressure, intensify the compressive behaviourwhich leads to a sooner liquefaction.

Figures 3 illustrates the pore pressure build up of the simulations, calculated as �u =σ0−σ2 which is the difference between the initial confining pressure and the horizontal stresscalculated during the cyclic shearing. The rate of the pore pressure generation in the first cyclein the first cycle is highest in all of the specimens and it decreases with the number of cycles.Whenever the pore pressure approaches the initial confining pressure, initial liquefactionoccurs.

The plot of deviatoric stress vs. deviatoric strain at 0.5% deviatoric strain amplitude canbe seen in Figs. 4 (a–c) for specimens with three confining pressures. The assemblies, havinghigher initial confining pressures, show higher peak deviator stresses. Thereafter, devaitoricstress reduces gradually in successive cycles and finally reaches zero deviator stress level,which shows complete liquefaction.

It is known that one of the factors affecting the deformation and failure of sands inundrained cyclic loading is the cyclic shear strain amplitude.17 Figs. 5 ((a) and (b)) showthe stress path and deviatoricstress vs. cyclic strain plots of the assembly generated withe = 0.208, at initial confining pressure of 150 kPa under cyclic loading amplitude of 2.5%.It can be seen that increasing the strain amplitude, will result in a considerable increase inthe peak deviatoric stress. The deviator stress (q) reaches the critical state line at first cycle.

\

a b c

Figure 4. Variation of deviatoric stress vs. cyclic strain for specimens (e = 0.208) at initial confiningpressures of (a) 150 kPa, (b)113 kPa and (c) 60 kPa.

a b

Figure 5. (a) stress path (b) deviatoric stress vs strain for speciman (e = 0.208) at initial confiningpressure of 150 kPa, 2.5% cyclic strain amplitude.

489


ba

Figure 6. Stress path plot for speciman (e = 0.175) at initial confining pressures of (a) 550 kPa and(b) 460 kPa, 2.5% cyclic strain amplitude.

ba

Figure 7. Deviatoric stress vs cyclic strain plot for speciman with (e = 0.175) at initial confiningpressures of (a) 550 kPa and (b) 460 kPa, 2.5% cyclic strain amplitude.

Furthermore, the shape of the deviatoric stress-strain changes compared to specimens under0.5% cyclic strain.

In order to simulate the behaviour of dense sand under cyclic undrained loading, anotherset of specimens were prepared with void ratio of e = 0.175 with initial confining pressuresof 550 and 460 kPas. The specimens were then subjected to undrained cyclic loading of 2.5%strain amplitude. Figs. 6 (a and b) show the stress path plots of the assemblies. Representingthe dense sand behaviour, the specimens never liquefied. Once the stress state reached thephase transformation line, the stress-strain curve (Figs. 7 (a and b)) move back and forthalong and below the steady-state line and shear strain developed gradually.

5.2. Micro-scale response

Figure 8(a–c) show the plots of average coordination number (number of contacts per parti-cle) vs. mean normal stress (p) for specimens with the three initial confining pressures. Theplots show that the average coordination number (CN) decreases with progression of thecycles such that it finally becomes one (C.N. = 1), which is an indication of liquefaction orzero effective confining pressure, because in this case the discs are not capable of transferringtheir loads to each other. It can be seen that in the sequence of reaching the phase transforma-tion to initial liquefaction, the specimens become unstable, showing a noticeable fluctuationsin the values of coordination number.

490


a b c

Figure 8. Variation of average coordination number vs. mean normal stress for specimens (e = 0.208)at initial confining pressures of (a) 150 kPa; (b)113 kPa and (c) 60 kPa.

a b c

Figure 9. Variation of deviatoric fabric tensor vs. number of cycles for specimens (e = 0.208) at initialconfining pressures of (a) 150 kPa; (b) 113 kPa and (c) 60 kPa.

In the continuation of this study, the variation of deviatoric fabric tensor with the numberof cycles for specimen of void ratio (e = 0.208) at three confining pressures of 150, 113and 60 kPas are derived. It is now well established that shear deformation of granular mediaproduces an induced structural anisotropy which is developed primarily as a result of contactseparation occurring in directions which are approximately orthogonal to the direction of themajor principal stress.16

Structural anisotropy is defined by the distribution of contact orientations and charac-terized by the structural anisotropy or fabric tensor (ij). For spherical particles, the fabrictensor is given by Ref. 18.

ij = 12Nc

Nc∑k−1

nki nk

j (5)

where Nc the number of contacts, ni is the component of the unit branch vector in the idirection, and the branch vector is the vector joining the centroids of the two contactingparticles. The principal values, 1, 2 and 3, and the principal directions of the fabrictensor can be calculated by considering the eigenvalues and eigenvectors of the fabric tensor.The deviator fabric (1 − 3) quantifies the anisotropy of the microstructure.16,19 For thethree simulations of specimen with void ratio of e = 0.208, the fabric tensor was calculatedby considering all the contacts in the specimen. Regarding the results presented in Figs. 9(a–c), it can be seen that the quantity of the deviatoric fabric tensor, initially increases for allof the specimens till the deviatoric stress reach the phase transformation line. So all of theplots have a peak deviatoric fabric tensor when the q-p plot pass the phase transformation

491


line. By progression of the number of cycles, deviatoric fabric tensor decreases to the onsetof liquefaction. The deviatoric fabric tensor fluctuates about a value of about 0.002–0.008for the specimens, which a higher value are recorded for the specimen with higher initialconfining pressure.

6. Conclusion

In this study, the undrained cyclic behaviour of cohesionless soils has been explored in detailsusing the DEM. Assemblies of two dimensional 7000 discs are prepared under the con.ningpressures in the range of 60 to 550 kPas and different void ratios covering both loose anddense states. The prepared specimens are then subjected to cyclic undrained loadings.

Results of the analysis of the macro- and micro-scale responses are explained in details.The stress-strain response, pore pressure build-up and stress path are evaluated as macro-scale response descriptors. Evolutions of deviatoric fabric tensor and coordination numberwith progress of cycles are studied as micro-scale response.

The DEM simulations have modeled the liquefaction and undrained behavior of loose anddense assemblies very close to the observed behavior of real sands in laboratory experiments,while the presented results give a qualitative insight into the saturated sand response undercyclic loadings from a micromechanical point of view.

References

1. Cundall, P.A. and Strack, O.D.L., “A discrete numerical model for granular assemblies”, Geotech-nique, 29, 1, 1979, pp. 47–65.

2. Thornton, C. and Randall, C.W., “Applications of theoretical contact mechanics to solid parti-cle system simulations”, In Mechanics of granular materials, (Eds. Satake and Jenkins), Elsevier,Amsterdam, Netherlands, 133–142, 1988.

3. Ting, J.M., Corkum, B.T., Kauffman, C.R. and Greco, C., “Discrete numerical model for soilmechanics”, Journal of Geotechnical Engineering, ASCE, 3, 1988, pp. 379–398.

4. Ng, T.T., “Numerical simulation of granular soils under monotonic and cyclic loading: a particu-late mechanics approach”, PhD thesis, Rensselaer Polytechnic Institute, Troy, N.Y., 1989.

5. Bathurst, R. and Rothenburg, L., “Investigation of Micromechanical Features of Idealized Granu-lar Assemblies using DEM”, Proc. of 1st U.S. Conf. on Discrete Element Methods, Golden, Colo.,1989.

6. Itasca, Particle Flow Code, PFC3D, Release 3.0, Itasca Consulting Group, Inc., Minneapolis, 2003.7. Ng, T.T. and Dobry, R., “Numerical simulations of monotonic and cyclic loading of granular

soils”, International Journal of Numerical and Analytical Methods in Geomechanics, 18, 2, 1994,pp. 388–403.

8. Kishino, Y., “Quasi-static simulation of liquefaction phenomena in granular materials”, in Secondinternational symposium for science on form, Tokyo, 1999, pp. 157–174.

9. El-Metskawy, M., “Discrete element simulation for seismically-induced soil liquefaction”, PhDthesis, State University of New York at Bufflo, Bufflo, NY, USA, 1998.

10. Sitharam, T.G. and Dinesh, S.V., “Numerical simulation of liquefaction behavior of granular mate-rials using discrete element method.”, In: Proceedings of the Indian Academic Science (EarthPlanet Science), 112, 3, 2003, pp. 479–484.

11. Sitharam, T.G., “Discrete element modeling of cyclic behavior of granular materials”, Geotechnicaland Geological Engineering, Springer, 21, 2003, pp. 297–329.

12. Sitharam, T.G., Vinod, J.S., “Critical state behaviour of granular materials from isotropic andrebounded paths: DEM simulations”, Granular Matter, Springer, 11, 1, 2008, pp. 33–42.

13. Shafipour, R. and Soroush, A., “Fluid coupled-DEM modelling of undrained behavior of granularmedia”, Computers and Geotechnics, Elsevier, 35, 5, 2008, pp. 673–685.

492


14. Cundall, P.A., “Distinct element methods of rock and soil structures”, in Analytical andcomputational methods in engineering rock mechanics (Ed. E.T. Brown), Allen & Unwin, 1987,pp. 129–163.

15. ASTM, Standard specification for standard sand, C 778-02, Annual Book of ASTM Standards,14.04, 2006.

16. Thornton, C., “Numerical simulations of deviatoric shear deformation of granular media”,Geotechnique, 50, 1, 2000, pp. 43–53.

17. Mitchell, J.K., Fundamentals of soil behaviour, 3rd ed., Wiley, USA, 2005.18. Satake, M., “Fabric tensor in granular materials”, in Deformation and failure of granular materials

(Eds. P.A. Vermeer and H.J. Luger), Rotterdam, Balkema, 1982, pp. 63–68.19. Cui, L. and O. Sullivan, C., “Exploring the macro- and micro-scale response characteristics

of an idealized granular material in the direct shear apparatus”, Geotechnique, 56, 7, 2006,pp. 455–468.

493

A Fundamental Study on the SPH Method Application for ImpactResponse of RC Structural Members

J. FUKAZAWA AND Y. SONODA∗

Dept of Civil Engineering, Kyushu University, Fukuoka, Japan

1. Introduction

Many analytical studies have been performed on the impact resistance of civil engineeringstructures subject to impact loads, and it is reported that the elastic-plastic impact responseof structural members such as a RC beam and a RC slab could be estimated by existing FEanalysis software.1 However, the phenomena with discontinuous displacement such as thepenetration of collision objects have not yet been established. In general, these phenomenaare essentially difficult to be handled by FE method.2 For the discontinuous displacementfield, the particle method3,4 such as smoothed particle hydrodynamics (SPH) is consideredto be more appropriate. Consequently, this study performed a fundamental review on theapplication of the SPH method, aiming to establish an analytical method for evaluating theelastic-plastic response of protective structural objects and destruction of them. The SPHmethod was proposed for analyzing compressive fluids by Lucy, Monagham, et al5−7 in thelate 1970s. The SPH method can continue the analysis even if individual particles movewidely from their initial position by resolving the motion equation while varying the inter-acting force by the weight corresponding to particle distance. Accordingly, the method canhandle destruction phenomena such as cracking, penetration, and destruction, which are dif-ficult to handle with the FE method, and was applied to the ultrafast fracture problem dueto space debris in the 1990s. However, differing from the fluid problem where only restag-ing of particle group rough flow is sufficient and the ultrafast fracture problem subjected tothe generation of structural destruction, it is very important to obtain the strain field accu-rately at the initial phase of plasticity with small deformation at the beginning to evaluatethe possibility of solid state material destruction.

In this study, a fundamental review was made taking the theoretical characteristics of theSPH method. First, the modelling issues of the SPH method were considered. In addition,when performing elastic-plastic impact analysis, the means of ensuring sufficient analysisaccuracy by using the SPH method in the region where elastic particles and plastic particlesare mixed, was reviewed. Finally, simulation analyses were performed for the weight dropimpact test of RC slab conducted and possible application of the SPH method was discussed.

2. Basic Equations of the SPH Method

In the SPH method, the continuum is represented by a set of arbitrarily distributed particles,therefore physical quantities such as the acceleration and strain of each particle at any timeare approximated by the smoothing kernel function:

f (x) ≈∫

f (x′)W(x− x′,h)dx′ (1)

where f is an arbitrary function, W(x − x′,h) is the smoothing kernel function, h is thesmoothing length defining the area of influence of the smoothing kernel function W, x is




a coordinate value of an approximated point, and x′ is an arbitrary coordinate value. Thedomain integral representations in the support domain can be converted to discretized formsof summation over all the particles in the support domain by the following equation:

f (x) ≈N∑

J=1

mJ

ρJf (xJ)W(x− xJ,h) (2)

Where J is particles in the support domain, m is the mass of a particle, ρ is density, and Nis the total number of particles in the support domain. The derivative of the equation isgiven by:

� · f (x)−N∑

J=1

mJ

ρJf (xJ) ·�W(x− xJ,h) (3)

Equation (3) states that the derivative of one physical quantity is approximated using thegradient of the smoothing kernel function. This idea is a basic characteristic of the SPHmethod, and it is the most important equation for the SPH method. The smoothing kernelfunction must have some conditions. For example, it must be normalized over its supportdomain, must satisfy the Dirac delta function condition, and must be fully smoothed. Somesmoothing kernel functions are submitted based on the support domain and considering thestability of solutions in addition to the above conditions. This study applied the followingspline-type smoothing kernel function:

W( r

h

)= 15

7πh2×

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

23−( r

h

)2 + 12

( rh

)30 ≤ r

h< 1

16

(2− r

h

)31 ≤ r

h< 2

0 2 ≤ rh

(4)

where h is the smoothing length defining the influence area, and r is the distance betweenparticle I and particle J.

In the case of SPH, the computing time becomes extremely long as the total number ofparticles increases. Thus, the computational efficiency of the SPH is improved by a multi-thread technique in this study.

3. Analysis Accuracy of Impact Response Using SPH Method

3.1. Treatment of boundary particles

In the SPH analysis, internal forces arising from the particle interactions are calculated bya weighted mean using a smoothing kernel function. This calculation method causes a lackof approximation at the boundary particles shown in Fig. 1. Thus, the following two simplecorrection methods are applied and their efficiency is discussed.

(a) Virtual particle method: In order to improve the lack of surrounding particles, virtualparticles are arranged outside the free surface as shown in Fig. 2(a).

(b) Correction of kernel function: In order to correct the weighted mean directly, the weightof boundary particles is increased as shown in Figure 2(b).

First, the response of a 2-D cantilever beam 1000 mm long, 100 mm high, and 100 mm thickas shown in Fig. 3 under a total constant load of 1.5 kN (refer to Fig. 4) was calculated using

496


Smoothing length

Interior particles are calculated accurately.

Analysis accuracy of boundary particles decrease

due to the lack of neighboring particle.

Figure 1. SPH kernel and particle approximations.

Boundary

particle

Interior

particle

boundary

Virtual particle

Boundary

particle

Interior

particle

boudary

1 32

In the case of kernel approximation of boundary particle,

the weight of interior particles (ex;No1 3) are modified.

(a) Virtual particle (b) Modifying the kernel approximation

Figure 2. Boundary treatment.

fixed

load 1.5kN

L=1000mm

H=100mm

Figure 3. 2-D cantilever beam model.

0.0

1.0

2.0

3.0

0 2 4 6 8 10

Load ( k

N )

Time ( ms )

Figure 4. Load-time relation.

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the above two correction methods and their accuracies were compared with the theoreticalvalue. In this model, the material was assumed to be an isotropic elastic with Young’s mod-ulus E = 2.1 × 104 N/mm2 and Poisson’s ratio . ν = 0.3. The time increment was set as�t = 1.0× 10−7 sec which satisfies Courant’s condition.

The stress distribution near the fixed end and the displacement response at the free endof each model at 20 ms are shown in Figs. 5 and 6. These figures indicate that the stressdistribution of the cantilever beam was improved and closer to the Euler-Bernoulli beamassumption. However, correction using the kernel function gives an error four times largerthan when using the virtual particle method. Therefore, the correction method using thekernel function should be given an accurate strain distribution in advance and is not suitablefor problems with an arbitrary stress field. In the case of virtual particle method, the responsecould be simulated with certain accuracy. However, their computational efficiency is notalways reasonable due to the increase of particle number.

-50.0

-40.0

-30.0

-20.0

-10.0

0.0

10.0

20.0

30.0

40.0

50.0

-15.0 -10.0 -5.0 0.0 5.0 10.0 15.0

Dis

tance fro

m c

ente

r ( m

m )

Stress ( N/mm2 )

No correction diference26.7%

Virtual particle diference13.3%

Kernel correction diference43.3%

Theoritical value

Figure 5. Stress distribution in the cross section.

Figure 6. Displacement — time relations of the beam.

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Table 1. FE analyses model size.

Case Number of partitions Remarks

Height Length

1_1 2 20 1 Element: 5.0×5.0cm1_2 4 40 1 Element: 2.5×2.5cm1_3 5 50 1 Element: 2.0×2.0cm1_4 10 100 1 Element: 1.0×1.0cm1_5 20 200 1 Element: 0.5×0.5cm

3.2. Analysis accuracy of SPH method

Next, we applied no correction method using as many particles as possible within an appro-priate computing time and investigated the relation between analytical accuracy and degreeof segmentation. On this occasion, the conventional analytical FEM method with a linearshape function was also calculated and their accuracy and computing efficiency were com-pared.

(1) Accuracy of FE analysis

In order to compare the accuracy of FE analysis, the response of a 2-D cantilever beam 100mm high and 1000 mm long was compared with the theoretical value. Table 1 shows thefive mesh sizes of model used in this study (refer to Fig. 7). The displacement error comparedwith the theoretical solution is shown in Fig. 8. It can be seen that the displacement responseis simulated well with more than 4 or 5 elements in the beam height direction.

(2) Accuracy of SPH analysis

To compare the accuracy of SPH analysis with no correction method, the response of a 2-D cantilever beam was calculated using 4 kinds of particle sizes. The resulting stress errorcompared with the theoretical solution is shown in Fig. 9 and the displacement error isshown in Fig. 10. These figures show that the analysis results converge as the particle size

(b) Case No.1_2

(a) Case No.1_1

(c) Case No.1_3

(d) Case No.1_4

(e) Case No.1_5

Figure 7. FE analysis models.

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4

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

1_1 1_2 1_3 1_4 1_5

Dis

pla

cem

ent

(mm

)

Case No.

Theoritical value 4.762mm

Figure 8. Displacement error by FE analysis.

-50.0

-40.0

-30.0

-20.0

-10.0

0.0

10.0

20.0

30.0

40.0

50.0

-150.0 -100.0 -50.0 0.0 50.0 100.0 150.0

Dis

tance fro

m c

ente

r ( m

m )

Stress ( N/mm2 )

2.5mm

5mm

10mm

20mm

Theoritical value

Figure 9. Stress distributions by SPH analysis.

Figure 10. Displacement responses by SPH analysis.

500


becomes smaller. However, it is found that the SPH analysis is less accurate than the FEanalysis. Regarding the displacement of the SPH result with no correction method, in orderto obtain the response within 10% error, it requires at least 40 particles in the beam heightdirection. Therefore, when we apply the SPH method to structural response problems, weshould use about 10 times smaller segmented particles compared to the FE mesh using alinear interpolation algorithm.

4. Elastic-Plastic Analysis Using the SPH Method

For the elastic-plastic problem, the concept of SPH based on weighted mean using a smooth-ing kernel function must be considered due to the mixture of elastic particles and plastic par-ticles. To confirm the effect of the smoothing length, weighted mean strain in the axial direc-tion is compared for two smoothing lengths, h and 1/2h. Furthermore, the impact responseusing FE analysis is compared. In this calculation, Object A is assumed to be made of steel,assuming von Mises yield criterion with 300 N/mm2 yield stress. After the yield condition,strain hardening with an initial stiffness of 1/100 is considered. For object B, it is assumed tobe a concrete material that has a bi-linear stress-strain curve with 30 N/mm2 yield strengthand strain hardening with an initial stiffness of 1/100 is considered. Regarding the impactconditions, an initial impact velocity of 20 m/s is applied to object A as shown in Fig. 11.Figure 12 shows the equivalent stress distribution at the maximum displacement response.In this figure, the yellow area indicates the domain whose equivalent stress exceeds the yield

Obj.B (Concrete)

Obj.A (Steel)

Initial velocity 20m/s

Fixed 50mm

100mm

300mm

100mm

Figure 11. Impact analysis model by FEM.

N/mm2

31.0

15.0

23.0

7.0

Ela

stic

Pla

stic

Elastic & Plastic Area

Elastic Area

Figure 12. Impact responses result by FEM.

501


Table 2. Weighed mean error by smoothing length.

Case Elastic or plastic Smoothing length � Strain X(μ) Difference (%)

FEM SPH

A Elastic h 20.765 19.407 6.5B1 Plastic h −45.100 −86.371 91.5B2 Plastic h/2 −45.100 −41.136 8.8

μ

Figure 13. Stress response with no return mapping scheme.

stress of object B. Table 2 show the precise stress distribution in the elastic-plastic domain.From this table, it is observed that the smoothing length affected the weighted mean strainin the SPH analysis. Thus, it is preferable to change the size of smoothing length consider-ing the elastic-plastic stress state. Figure 13 shows the equivalent stress U equivalent plasticstrain relation in a particle as an example. We could find that the stress response oscillates inspite of the sufficiently-small time increment (�t = 1.0 × 10−7 sec by Courant’s condition).In view of these circumstances, the return mapping scheme is used in the SPH analysis atevery time increment. Figure 14 shows the result using the return mapping scheme and weconfirmed that the stress response of the plastic particle becomes stable, which can be tracedto the assumed stress-strain curve.

Finally, we simulated impact response of RC slab using the SPH method with the returnmapping scheme.

Figure 15 shows an overview of the RC slab analysis model and reinforcing bars arrange-ment. Figure 6 shows applied stress-strain curves. In this simulation, the effect of mechanicalcharacteristic of concrete after the tensile strength is considered by comparison with tensioncut-off model and no cut-off model. In the case of (c), tensile stress cannot be transferredover the 3N/mm2 tensile strength.

Figure 17 shows the effect of cut-off model on the displacement response of the RC slab.This figure clearly shows the tensile stress cut-off effect on the displacement response. There-fore, it is found that the elastic-plastic impact analysis of a RC structural member requiresspecial treatment for the tensile failure in order to obtain accurate analytical results.

502


μ

Figure 14. Stress response with return.

(a) Overview of RC Slab model (b) Reinforcing bar

Figure 15. RC slab model.

(a) Reinforcing bar (b) Concrete (No cut-off) (c) Concrete (Cut-off)

Figure 16. Stress-strain relations.

5. Conclusions

The following conclusions are deduced from this study

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Figure 17. Displacement response of RC slab.

(1) According to the fundamental analysis of kernel approximation, it is difficult to find acorrection method of free surface that has an assumed accuracy and reasonable efficiencyunder the arbitrary stress fields.

(2) When we apply the SPH method to structural response problems, we should use about10 times smaller segmented particles compared to the FE analysis.

(3) For the elastic-plastic problem, the SPH method requires appropriate return mappingscheme to obtain accurate results.

(4) In order to simulate the impact response of a RC structural member, special treatmentfor the tensile failure of concrete should be considered.

References

1. Impact problem committee of JSCE, Fundamentals and applications of impact test and analysis,Structural Engineering Series 15 of JSCE, Maruzen, 2004 (in Japanese).

2. Hughes, T.J.R. The Finite Element Method, Linear Static and Dynamic Finite Element Analysis,Prentice Hall Inc., 1987.

3. Cundall, P.A. Distinct element models of rock and soil structure, Analytical and ComputationalMethods in Engineering Rock Mechanics, 1987, pp. 129–163.

4. Koshizuka, S. et al. Numerical analysis of breaking waves using the moving particle semi-implicitmethod, International Journal for Numerical Methods in Fluids, 1998, Vol. 26, pp. 751–769.

5. Lucy, L.B. Numerical approach to testing the fission hypothesis, Astronomical Journal, Vol. 82,pp. 1013–1024

6. Monaghan, J.J. Particle methods for hydrodynamics, Computer Physics Report, 1985, Vol.3,pp. 71–124.

7. Monaghan, J.J. An introduction to SPH, Computer Physics Communications, 1988, Vol. 48,pp. 89–96.

8. Liu, G.R., Liu, M.B. Smoothed Particle Hydrodynamics, World Scientific Pub Co Inc., 2003.

504

2-D FEM Analysis of the Rock Fragmentation by Two Drill Bits

S.Y. WANG1,∗, Z.Z. LIANG2, M.L. HUANG3 AND C.A. TANG2

1Centre for Geotechnical and Materials Modelling,Department of Civil, Surveying and Environmental Engineering, The University of Newcastle, Australia2School of Civil & Hydraulic Engineering, Dalian University of Technology, Dalian, P.R. China3School of Civil Engineering, Beijing Jiaotong University, Beijing, P.R. China

1. Introduction

Rock fragmentation by a drill bit is the preferred technique for effective mechanical methodsand rock fragmentation. Usually, the kinetic energy is transferred to the drill bit by means ofa stress wave. The stress wave travels through the bit until it reaches the end in contact withthe medium to produce failure and fractures.1 Many experimental investigations on rockfragmentation have been carried out in the past decades.2 However, the process of fracturesinitiation, propagations and coalescence under mechanical indentation is rarely observeddirectly in the standard laboratory tests.

Numerical methods based on finite elements method (FEM) have been developed to simu-late the impact problem in rock drilling. Liu3 developed and used R-T2D (Rock-Tool inter-action) to simulate successfully the rock fragmentation process induced by single and doubleindenters. The heterogeneity of rock is considered and the progressive process of rock frag-mentation in indentation is reproduced. However, his work on rock fragmentation due toindenters is limited to quasi-static state. In the present study, the numerical code RFPA2D,4

was used to simulate the evolution of dynamic fractures initiation, propagation in rock dueto static, dynamic and coupled static and dynamic loading, respectively.

2. Brief Description of RFPA2D

Briefly, the code RFPA2D5 is a two-dimensional finite element code that can simulate thefracture and failure process of quasi-brittle materials such as rock. To model the failure ofrock material (or rock mass), the rock medium is assumed to be composed of many meso-scopic rectangle elements with the same size. Their material properties are different from oneto another and are specified according to a Weibull distribution. These elements are acted asthe four-nodded iso-parametric elements for finite element analysis. Elastic damage mechan-ics is used to describe the constitutive law of the meso-scale elements, and the maximumtensile strain criterion and the Mohr-Coulomb criterion are utilized as damage thresholds.

The effects of strain rate on the strength of rock have been conducted widely especiallythrough experiments. Based on a variety of experimental results of granite, Zhao6 proposedthat the Mohr-Coulomb is also applicable to dynamic loading conditions if only the increaseof cohesion with the strain rate is taken account. The relation between dynamic uniaxialcompressive strength and loading rate can be described with a semi-log formula as follows.

σcd = A log (σcd/σc)+ σc (1)

where σcd is the dynamic uniaxial compressive strength (MPa), σcd is the dynamic loadingrate (MPa/s), and σc is the quasi-brittle loading rate (approximately 5 × 10−2 MPa/s), σc is




the uniaxial compressive strength at the quasi-static loading rate. Parameter A is consideredas a material parameter. For different rock material, this parameter should be different, andshould be determined based on variety of laboratory test. The compressive/tensile strengthratio is assumed to be unchanged for dynamic loading conditions.

3. Model Setup

In the following numerical simulations, the indentation problem is simplified to a plain straincondition. A plane passing through the central axis of the indenter is considered, as shownin Fig. 1(a)–(c). Fig. 1(a) is for the case of static problem; Fig. 1(b) is for the case of dynamic(impact) problem, and Fig. 1(c) is for the case of coupled static and dynamic problem. Forthe static case, a displacement increment (0.005 mm/step) is applied on the indenters. For thedynamic (impact) case, an impact loading with peak stress of 90 MPa are applied as shown inFig. 2. For the coupled static and dynamic case, the axial pre-compression stress of 180N wasapplied firstly, and then the same impact loading shown in Fig. 2 was applied. The indenteris simulated as a homogeneous material whose elastic modulus is a few times higher thanthat of rock in order to safeguard against permanent deformation of the indenter. The rock

Static loading

(a)

Dynamic loading

(b)

Coupled static and dynamic loading

(c)

Figure 1. Mechanical models of numerical simulation: (a) Static case, (b) Dynamic (impact) case; (c)Coupled static and dynamic case.

506


P (

MPa

)

90

10

t (μs)

20 30 40 0

60

Figure 2. Dynamic impact compressive stress waves applied at the top surfaces of two indenters forcases (b) and (c).

Table 1. Material properties of specimens.

Homogeneity index (m) 3Mean compressive strength (σ0) 300(MPa)Mean elastic modulus (E0) 30,000(MPa)Poisson ratio (μ) 0.25

Table 2. Material parameters for indenters.

Homogeneity index(m) 20Mean compressive strength (σ0) 3000(MPa)Mean elastic modulus (E0) 300,000(MPa)Poisson ratio (μ) 0.25

specimen is considered as heterogeneous material. The basic parameters for rock specimenare listed in Table 1. For the dynamic case, the time step is 0.2 μ s, and the wave velocityis 3500 m/s. The strength and elastic modulus of indenters in the current models are givenenough high value, in order that the indenters will not fail during the rock fragmentation.The properties of indenters are listed in Table 2.

4. Numeiral Simulated Results

4.1. Numerically simulated rock fragmentation process due to staticloading

Figure 3 shows the comparison of the numerical simulated rock fragmentation due to staticloading. When the indenter acts on the rock, a high stress zone, corresponding to the high-light zone in Fig. 4 (stage A) some small fractures come into being immediately beneath theindenter. As the stress intensity builds up with an increasing load, one or more of the flawsnucleates the so call initial radial crack around the two corners of the truncated indenter instages B and C in Fig. 3. It is noted that due to the heterogeneity of rock, the initial radialcrack propagating path from the two indenters are not totally the same. The initial radial

507


(A) (B)

(C) (D)

(E) (F)

Figure 3. Numerical simulated of fracture process of specimen subjected to static loading.

crack from left indenter grows faster than that of right indenter (stages C and D). After-wards, along with the increasing stress, all of the cracks propagate downwards (Stages E andF). In addition, the stages A–F are corresponding to the points A–F in the force-penetrationcurve in Fig. 4. The associated seismicity, which is usually called the acoustic emission (AE)phenomenon in rock mechanics, is obtained by recording the counts of the failure elements.From Fig. 4 the force-penetration curve has almost a linear shape in the initial loading stage(before Stage B). It is noted that there are two evident force-drops in points C and D in Fig. 4.It indicates that the more cracks occur and propagate in Stages C and D in Fig. 3, which leadto the bearing ability of rock decreasing and then the compressive forces in these two stagedrop suddenly.

A

B C

D

E F

p

Figure 4. Force-penetration curve during rock fragmentation subjected to static loading.

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4.2. Numerically simulated rock fragmentation process due to dynamic(impact) loading

Figure 5 shows the numerical simulated of fracture process of specimen due to dynamic(impact) loading. At the stage of impact loading (t ≤ 20μs), the stress fields induced by thedouble indenters are equal to those induced by the case of static loading (t ≤ 40μs). Withcompressive stress waves from two impact points propagating 40μs ≤ t ≤ 60μs, the twocompressive waves meet and interfere and some incipient chip occurs especially concentratingon the inter-zone of two indenters, as the interaction of stress field between the two indentersenhance the fractures initiations in such zone. Along with the compressive waves developing,these tensile cracks between the two indenters propagate fast (60μs ≤ t ≤ 100μs). Mean-while, there are also some tensile cracks on the other side of the two indenters emerging.Comparing the static case, owing to interaction of two compressive stress waves, the incipi-ent chip initiation and propagation in the inter-zone of two indenters enhance inevitably thefragmentation of rock.

4.3. Numerically simulated rock fragmentation process due to coupledstatic and dynamic loading

In this section, the axial pre-compression stress of 480N was applied firstly, and then theimpact loading with peak stress of 90 MPa was applied. Figure 6 shows the numerical simu-lated of fracture process of specimen due to coupled static and impact loading. For the stageof pre-compression, it is the same with the static case. The pre-compression stress of 480Nis corresponding to point P in Fig. 4. In this stage, there are no fractures occur under thetwo indenters. Afterwards, at the stage of impact loading (t ≤ 20μs), comparing with thepure dynamic case in Fig. 5, due to the pre-compression, the indenters have entered the rock

10μs 20μs

40μs 60μs

80μs 100μs

Figure 5. Numerical simulated of fracture process of specimen subjected to dynamic (impact) loading.

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Pre-compression stress of 480N 20μs

40μs 60μs

80μs 100μs

Figure 6. Numerical simulated of fracture process of specimen subjected to coupled static anddynamic loading.

before the impact loading applied. At a result, more fractures initiate and propagate fromthe two indenters. With the impact loading applied (20μs ≤ t ≤ 60μs), the secondary radialcracks driven by tensile stresses run downward along the stress trajectories of the maximumprincipal stresses. Afterwards, more and more some incipient chip occurs especially concen-trating on the inter-zone between two indenters (60μs ≤ t ≤ 100μs). Comparing with thepure dynamic case in Fig. 5, the fractures in the inter-zone between two indenters propagateeven faster than the fractures from the two indenters in Fig. 6. It indicates that the rock canbe more fragmentized when the rock is subjected to the coupled static and dynamic (impact)loadings. It is noted that the different pre-compression stress and different peak stress ofimpact loading can affect the efficiency of fragmentation in indentation. The influence of thecombination of different pre-compression stress and peak stress of impact loading will bediscussed in another paper.

5. Conclusions

In this study, RFPA2D code has been applied to simulate rock fragmentation due to static,dynamic and coupled static and dynamic loadings. Although the reality is often much morecomplex than the numerical models applied, the study provides interesting indications forimproving performance of rock fragmentation in indentation. Numerical simulated resultsshow that comparing the static case, owing to interaction of two compressive stress waves,the incipient chip initiation and propagation in the inter-zone of two indenters enhanceinevitably the fragmentation of rock. In addition, due to the pre-compression, the rock canbe more fragmentized when the rock is subjected to the coupled static and dynamic (impact)loadings.

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Acknowledgements

The work described in this paper was partially supported Australian Research Council grantDP0881238, and the grant of National Natural Science Foundation of China 50804006, towhich the authors are very grateful.

References

1. Chiang, L. Dynamic force-penetration curves in rock by matching theoretical to experimental waveresponse. J Exp Mech 2004, 2, pp. 167–175.

2. Kou, S.Q. Some basic problems in rock breakage by blasting and by indentation. PhD thesis, LuleåUniversity of Technology, 1995.

3. Liu, H.Y. Numerical modeling of the rock fragmentation process by mechanical tools. PhD Thesis:32D, Lulea University of Technology, Sweden, 2004.

4. Zhu, W.C., Tang, C.A. Numerical simulation of Brazilian disk rock failure under static and dynamicloading Journal of Rock Mechanics & Mining Sciences, 2006, 43, pp. 236–252.

5. Tang, C.A. Numerical simulation of progressive rock failure and associated seismicity. InternationalJournal of Rock Mechanics and Mining Sciences, 1997, 34, pp. 249–261.

6. Zhao, J. Application of Mohr-Coulomb and Hoek-Brown strength criteria to the dynamic strengthof brittle rock. Int. J. Rock Mech. Min. Sci., 2000, 37, pp. 1115–112.

511

Determination Method of Rock Mass Hydraulic ConductivityTensor Based on Back-Analysis of Fracture Transmissivity andFracture Network Model

LI XIAOZHAO1,∗, JI CHENGLIANG1, WANG JU2, ZHAO XIAOBAO1, WANG ZHITAO1,SHAO GUANHUI1 AND WANG YIZHUANG1

1NJU-ECE Institute for Underground Space and Geoenvironment,School of Earth Sciences and Engineering, Nanjing University, Nanjing 210093, Jiangsu, China2Beijing Research Institute of Uranium Geology, Beijing 100029, China

1. Introduction

The Representative Element Volume (REV) and the 3-D hydraulic conductivity tensor arevery important for the evaluation of hydraulic properties of the rock mass. Usually, thestochastic discrete fracture network that characterize rock mass structure is built to ana-lyze the seepage properties.1,2 Although many researches have been done to describe rockmass fracture system during recent years,1−11 there are many problems. The measurement offracture width is difficult and inaccurate, which may cause great error for the hydraulic con-ductivity calculated based on the discrete fracture fluid flow model. Yu Qingchun3,4 decidedthe equivalent fracture width of the discrete fracture flow model by simulating the infiltra-tion test. M. Wang2 obtained the flow rate per unit hydraulic gradient by simulating thepacker tests in the borehole assuming that the difference between the average values of thetransmissivity for different fracture sets is not significant. The transmissivity of each fractureis obtained by simulating the borehole packer test based on the discrete fracture networkusing the data from televiewer and packer test in the borehole in this paper. Based on thetransmissivity calibrated fracture fluid flow model, the permeability property of rock mass inBeishan area, Gansu Provence, China, a high-level radioactive waste geological repository, isanalyzed.

Firstly, based on the geometry statistical parameters of fractures mapped in field, thestochastic discrete fracture network was established and calibrated using Monte-Carlomethod. Then, the transmissivity of the discrete fracture fluid flow models with differentsizes was calibrated by simulating the packer test in the 3# borehole. The existence of theREV scale was proved by analyzing the variation of the directional hydraulic conductivity ofthree directions with block sizes. The 3-D hydraulic conductivity tensor of the fractured rockmass near the borehole was decided using regression method. Finally, the principal hydraulicconductivity and the directions were decided.

2. Three-Dimensional Stochastic Fracture Network and Its Validation

The rock mass dealt within the paper is composed of granite. It is located Gansu Province,West of China, where a high-level radioactive waste geological repository lies in. Near the 3#borehole, 11 scan windows provide more than 800 fractures. Table 1 shows the summary offracture set delineation results obtained for the granite rock mass based on the data mapped.Empirical Fisher distribution is suitable to fit the statistical distribution of orientation offracture sets.5 The Lognormal distribution was found to be the best distribution to representfracture trace length. Packer test data were available from various depths of 3# borehole.




Table 1. Summary of fracture set delineation results.

Fractureset

number

Orientation (Dip direction/Dip angle)(◦)

Trace Length(m)

2-D Density(/m2)

Mean Fisher Distribution Mean Value Variance DistributionOrientation Parameter Value

1 258.48/85.29 16.169633 Fisher 1.095 0.478 Lognormal 0.392 205.67/86.81 17.760510 Fisher 1.162 0.576 Lognormal 0.223 319.73/73.88 11.303227 Fisher 1.483 0.602 Lognormal 0.17

Table 2. Summary of 2-D and 3-D intensity of fracturesets after calibration.

Fracture set no. 2-D density (/m2) 3-D density (/m3)

1 0.36 0.672 0.20 0.383 0.18 0.29

The borehole of length 500 m provides 400 deterministic fracture data through acoustic tele-viewer technique. For simplification, only the packer test between 171.5 m and 178 m wassimulated to calibrate the transmissivity of fracture fluid flow models with different sizes.

Based on the geometry statistical parameters showed in Table 1, the 3-D stochastic discretefracture network model with edge length 50 m was built using Monte-Carlo method. Themodel is established on the hypothesis that a fracture is a circular disc in space and thelocation of the center of the disc distribute randomly. To built the fracture model that cancharacterize the real structure of rock mass in site, the above process must be repeated toverify the validity of the model.1,2 The 3-D density of fractures of each set which decides thecount of fractures is a very important parameter during modeling. Generally speaking, theparameter can not be measured in field. The inverse-model method3,4 was used to determinethe 3-D density of each fracture (Table 2) of the model by comparing the 2-D density of themodel with the observed 2-D density of each fracture from the outcrops.

3. Three-Dimensional Fracture Fluid Flow Model and Calibration ofFracture Transmissivity of the Model

3.1. Pipe flow network model

A lot of fractures in the stochastic fracture network model do not connect with any otherfractures/boundary (isolated fractures) or connect with the only one fracture/boundary (dead-end fractures) and consequently are not involved in the fluid flow in the network. These frac-tures should be removed from the discrete fracture network when the fluid flow calculationwas conducted to speed up the computing process. After the fractures above-mentioned wereremoved from the network, new fractures that should be removed will occur. Therefore, theprocess that removes fractures from the network should be repeated until all the fractures inthe model connect with at least two other fractures or at least one fracture and a boundary.3

The model then was called connected fracture network model.

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Subsequently, pipe flow network model should be developed based on the connected frac-ture network model. M. Wang2,10 and Yu Qingchun3,4 suggest that it is reasonable to useequivalent one-dimensional pipes to represent conceptually the fluid flow in a 2-D fracturein order to simplify and minimize the complexity of fluid flow in a fracture. The intersectionlines between fractures in the connect fracture network model were determined. The centerpoint of a fracture disc and the middle point of the intersection line between the fracture andthe other one are set to be nodes. The pipes are placed between the center of the fracture discand the midpoint of the intersected line of the intersected fractures. Between two intersectedfractures, two equivalent single flow pipes as shown in Fig. 1 represent flow from one fractureto another. It is assumed that the fluid in a fracture flows to another or vice versa through thepipe. The pipe between two intersected fractures should be considered as a trapezoid zone asshown in Fig. 2.

The equivalent width of the flow pipe can be derived from the Darcian law. The bc inFig. 2 is the width at one end of flow path crossing the diameter of the fracture disc and canbe obtained by the Equation (1), bL is the other end of the flow path equal to the length ofthe intersection line between two fractures.

bc =(

2RNL

)×(

bL

bTm

)(1)

where R is the radius of fracture disc, NL is the total number of intersections between thefracture and the other fractures or boundaries, bTm is the average of lengths of all the inter-sections between the fracture and the other fractures or boundaries. According to the Darcianlaw, the equivalent width of the flow pipe can be derived. bm can be written as a function ofbc and bL such as Equation (2).

bm =(bL − bc

)ln(

bLbc

) (2)

Figure 1. Features of nodes and pipes in the model.

cbLb mb

Figure 2. The equivalent process of a flow sub-domain on a fracture.

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3.2. Calibration of fracture transmissivity of the flow model

The fracture transmissivity (T) reflects the permeability of the fractures in the model and canbe obtained by the Equation (3).

T = gb3

12νw(3)

where b is the fracture equivalent width, νw is the kinematic viscosity coefficient of water,g is gravitational acceleration constant. It can be obtained from Equation (3) that the fracturewidth has great influence on the permeability of the fracture. Usually, fracture transmissivityis calculated by the randomly generated fracture width that fits a certain distribution. Thismethod requires good measurement of fracture width in field. Generally speaking, the frac-ture width is difficult to measure in field because of the weathering of the rock outcrop andthe fillings in the fracture. Therefore, the fracture transmissivity of the discrete fracture fluidflow model is calibrated by simulating the borehole packer test in this paper. Then, basedon the calibrated flow model, the permeability of the rock mass near the borehole can beanalyzed.

In the 3-D conceptual linear pipe discrete fracture fluid flow model, the average value ofthe transmissivity for each fracture of all fracture sets needs calibration. The packer test forthe depth range 171.5–178 m of the 3# borehole will be simulated to calibrate the equivalentfracture transmissivity of the flow model. The data of fractures that lie in the range wereobtained through acoustic televiewer. The in situ observation data for the fractures exposedon the borehole walls indicated that the average values of the aperture and the extent of thefilling in fractures for different fracture sets to be almost the same.2,10 It provided supporton the point that the difference between the average values of the transmissivity for differentfracture sets can be ignored.

Cubic samples of sizes 6.5, 6.8, 7.1, 7.4, 7.7, 8.0, 8.3, 8.6, 8.9, 9.2, 9.5, 10.0, 10.5, 11,12 m were selected from the center of the cube with edge length 50 m that was developed andvalidated in Section 2 to calibrate the fracture transmissivity of the these models. In each ofthese samples, the borehole was placed vertically at the center of the cube to have the centerof the packer interval to coincide with the center of the cube. The observed fractures on theborehole from the televiewer were placed in the stochastic fracture network model to theconsidered depth length with the disc center coinciding with the borehole in the horizontalplane. For the observed fractures, the known dip angle, dip direction and location of eachfracture on the borehole were used in setting the place. The diameters of these determin-istic fractures were set to be the average values of fracture set they belong to respectively.The centers of observed fractures that intersect the borehole were set to be nodes whichthe hydraulic heads have been known. It is assumed that the fluid of the packer test in theborehole flow from the centers of the deterministic fractures to the boundary through the“stochastic-deterministic model”. It was decided to use the injection water pressure value asthe input to numerical discrete fracture fluid flow model and predict the injection flow rateas the output. The difference of hydraulic heads between the nodes on the borehole and thaton the boundary was set to be the injection pressure. The injection flow rate of packer testwas available to compare with the predicted flow rate to calibrate the fracture transmissivityof the discrete fracture fluid flow model. Similarly, the Equation (4) is suggested to calculatethe flow path width bc for a fracture that intersects a borehole in an interval of packer testsuch as fracture B in Figs. 3 and 4.

bc =(

2πRNB

)×(

bL

bTm

)(4)

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AB

C

Borehole

D

Figure 3. Intersections between deterministic fractures and stochastic fractures.

Borehole

bL Bbc

bL D

bL C

Figure 4. Trapezoid sub-domain on a fracture that intersect the borehole.

where R is the radius of borehole intersected fracture such as fracture A in Fig. 3, NB is thetotal number of intersections between the fracture A and the other stochastic fractures andbTm is average of the lengths for all the intersections in the fracture A.

The resulting injected flow rate was then compared with the observed injected flow rateof packer test between 171.5 and 178.0 m. This procedure was repeated for each modelusing different selected transmissivity (T) until excellent agreement was obtained betweenthe predicted and observed injected flow rates.

4. Estimation of REV Size and Determination of Hydraulic ConductivityTensor in 3-D

4.1. Estimation of REV size

A series of stochastic fracture network models with different sizes which the fracture trans-missivity has been calibrated in Section 3.2 were used to estimate the REV size for the graniterock mass around the 3# borehole. Each cubic block has two vertical faces perpendicular tothe N-S (North-South) direction. The calibrated fracture transmissivity value for the depthrange 171.5–178.0 m was assigned for the fractures in each block. To calculate the direc-tional hydraulic conductivity in a chosen direction, a unit hydraulic gradient was applied inthat direction using the two perpendicular planes of the cube corresponding to the chosen

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6 7 8 9 10 11 12

0.00E+000

5.00E-008

1.00E-007

1.50E-007

2.00E-007

2.50E-007

3.00E-007

3.50E-007

4.00E-007

N-S direction E-W directionVertical direction

Dir

ecti

onal

Hyd

raul

ic C

ondu

ctiv

ity(

m/s

)

Block Size(m)

REV Size

Figure 5. Directional hydraulic conductivity in S-N, W-E and vertical direction vs. block sizes.

a b c

Figure 6. Variation of directional hydraulic conductivity in 3-D space ((a): view along the directionhaving trend 341◦ and downward plunge=21◦; (b): view along the direction having trend 209◦ anddownward plunge 19◦; (c): view along the direction having trend 7◦ and downward plunge 31◦).

direction.10 The other boundaries of the block were specified as no flux boundaries in per-forming the flow calculations under the steady state. The directional hydraulic conductivitiesobtained in the directions N-S, E-W and vertical for each block are shown in Fig. 5.

It is clear from Fig. 6 that the hydraulic conductivities in different directions do not changewith the block sizes for block sizes greater than 9.5 m. This implies that the REV size for thegranite rock mass of this area can be considered to be about 9.5 m.

4.2. Determination of hydraulic conductivity tensor in 3-D

To calculate the hydraulic conductivity tensor for the rock mass near the 3# borehole, acube of 9.5 m was chosen from the center of the cube of size 50 m having the correspondingfracture network. The calibrated fracture transmissivity corresponding to the 9.5 m blockwas assigned to the model. As mentioned earlier, a unit hydraulic gradient was applied inthat direction using the two perpendicular planes of the cube corresponding to the chosendirection to calculate the directional hydraulic conductivity. The other boundaries of theblock were specified as no flux boundaries in performing the flow calculations using the dis-crete element method. Trough this way, the directional hydraulic conductivity was calculated

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Table 3. Values and directions of principal hydraulic con-ductivity near the borehole.

Principal hydraulicconductivity (×10-7)

(m/s)

Trend (◦) Downwardplunge

(◦)

K1 2.08 110 80K2 1.71 349 5K3 0.47 258 8

in more than 280 directions covering the 3-D space. Fig. 6 shows the spatial variation ofhydraulic conductivities in 3-D.

The Equation (5) can be derived from the Darcian law and the tensor form of hydraulicconductivity.2,10

m2i1K11 +m2

i2K22 +m2i3K33 + 2mi1mi2K12

+ 2mi2mi3K23 + 2mi1mi3K13 = Kd(−→mi) (5)

where K11, K22, K33, K12, K23 and K13 are six independent components of the hydraulicconductivity tensor, Kd(−→mi) is the ith directional hydraulic conductivity along the hydraulicgradient unit vector −→mi = (mi1,mi2,mi3)

T.The estimation of K11, K22, K33, K12, K23 and K13 can be obtained by applying the method

of regression. In addition, the three principal values of the hydraulic conductivity tensor andthe corresponding three principal directions were determined (Showed in Table 3).

Note that the hydraulic conductivity of the rock mass is significantly anisotropic. Thehighest hydraulic conductivity is obtained along the sub-vertical direction. This compareswell with the fracture system of the rock mass that has three sub-vertical fracture sets. Thegeometrical mean value of the three principal hydraulic conductivities also compares wellwith the equivalent hydraulic conductivity obtained from the equivalent continuum model.

5. Conclusions

• It is indicated that there were three fracture sets in this area through the statistic data.Based on the statistical model, the stochastic fracture network model which characterizesthe rock mass structure was built and validated. The 3-D density that was obtained bythe “back-analyze” method was used to generate fractures of each set.• Assuming that the transmissivity of each fracture in the model is the same, the parameters

of each model with different size was calibrated by simulating the packer test of the depthrange 171.5–178.0 m. The error evoked by the measurement of fracture width can beavoided by this method.• The REV size of the rock mass around the 3# borehole was determined to be about

9.5 m. The values and directions of principal hydraulic conductivities were obtained bythe method of regression. It is find out that the geometrical mean value of the threeprincipal hydraulic conductivities compares quite well with the hydraulic conductivityobtained through the radial flow continuum porous media assumption for the rock massaround the borehole. So the results obtained by simulation can be used to evaluate thepermeability of the rock mass around the borehole.• The permeability of this area is significantly anisotropic as the three principal hydraulic

conductivities reflect. The direction of the highest hydraulic conductivity is sub-vertical.

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This compares well with the fact that all the three fracture sets of this area are sub-vertical. So it can be concluded that the fluid flow in this area is controlled by the threemain fracture sets.

Acknowledgements

The authors acknowledge the financial support of National Natural Science Foundation ofChina (NSFC) (40702046) and “the eleventh five year plan” project of the Commissionof Science, Technology and Industry for National Defence (Ke Gong Ji [2007] 825). Thesupport received for this project from the Beijing Research Institute of Uranium Geology isappreciated very much.

References

1. Dershowitz W.S., Einstein H.H. Characterizing Rock Joint Geometry with Joint System Models[J]. Rock Mechanics and Rock Engineering, 1988, 21: 21–51.

2. Wang M., Kulatilake P.H.S.W., Um J., Narvaiz J. Estimation of REV size and three-dimensionalhydraulic conductivity tensor for a fractured rock mass through a single well packer test anddiscrete fracture fluid flow modelling [J]. International Journal of Rock Mechanics and MiningSciences, 2002, 39: 887–904.

3. Yu Qingchun, Wu Xiong, Ohnishi Yuzo. Channel Model for Fluid Flow in Discrete Fracture Net-work and its Modification [J]. Chinese Journal of Rock Mechanics and Engineering, 2006, 25(7):1469–1474.

4. Yu Qing-chun, Liu Feng-shou, Yuzo Ohnishi. Three Dimensional Planar Model for Fluid Flow inDiscrete Fracture Network of Rock Masses [J]. Chinese Journal of Rock Mechanics and Engineer-ing, 2005, 24(4): 662–668.

5. Kulatilake P.H.S.W. Fitting fisher distributions to discontinuity orientation data [J]. Journal ofGeological Education, 1985, 33: 266–269.

6. Priest S.D., Hudson J. A. Estimation of discontinuity spacing and trace length using scanline sur-veys [J]. International Journal of Rock Mechanics and Mining Sciences. 1981, 18(3): 183–197.

7. Zhang L., Einstein H.H. Estimating the mean trace length of rock discontinuities [J]. RockMechanics and Rock Engineering. 1998, 31(4): 217–235.

8. Lianyang Zhang, Einstein H.H. Estimating the intensity of rock discontinuities [J]. InternationalJournal of Rock Mechanics and Mining Sciences. 2000, 37: 819–837.

9. Dershowitz W.S. Rock joint systems Ph.D. Cambridge, Massachusetts: Massachusetts Institute ofTechnology, 1984.

10. Wang M. Discrete fluid flow modeling, field applications in fractured rocks. Ph.D. dissertation,The University of Arizona, Tucson, AZ, 2000.

11. Panda B.B. Investigation of relations between the geometrical properties and hydraulic propertiesof jointed rock through numerical simulation. PhD dissertation, The University of Arizona,Tucson,AZ, 1995.

12. Song J.-J. Estimation of Areal Frequency and Mean Trace Length of Discontinuities Observed inNon-Planar Surface [J]. Rock Mechanics and Rock Engineering. 2006, 39(2): 131–146.

13. Ki-Bok Min, Lanru Jing, Ove Stephansson. Determining the equivalent permeability tensor forfractured rock masses using a stochastic REV approach: Method and application to the field datafrom Sellafield, UK [J]. Hydrogeology Journal, 2004, 12: 497–510.

14. Ki-Bok Min. Fractured Rock Masses as Equivalent Continua-A Numerical Study. PhD dissertation,KTH Land and Water Resources Engineering, Stockholm, 2004.

520

Numerical Simulation of Scale Effect of Jointed Rock Masses

Z.Z. LIANG1,∗, L.C. LI1, C.A. TANG1 AND S.Y. WANG2

1School of Civil and Hydraulic Engineering, Dalian University of Technology, Dalian, P.R. China2Centre for Geotechnical and Materials Modelling, Department of Civil,Surveying and Environmental Engineering, The University of Newcastle, Australia

1. Introduction

Jointed rock masses are often encountered during excavation in civil engineering and miningengineering. The design of high rock slopes, typical of open pits, and tunnel excavationoften requires the evaluation of the rock mass strength containing joints. Many failures ofunderground openings during excavation are reported closely relate to joints. Joints usuallyoccur in sets which are more or less parallel and regularly spaced. And also there are usuallyseveral sets in very different directions so that the rock mass is broken up into a blockystructure.1

The importance of scale effect for the design of rock masses as well as other heterogeneousmaterials is a well debated issue. The combination of the Weibull statistical theory of randomstrength and energetic theory is one of the promising approaches to explain the phenomenaof scale effect. Scale effect rises originally from microstructures of brittle materials. Manyexplanations have been put forward to predict the strength and fracture behaviour of brittlematerials, such as rock, concrete and ceramics et al. Based on experiments on fatigue frac-ture of metals and heuristic arguments, Weibull2 introduced his probability distribution intothe theory of fatigue failure of metals and ceramics and obtained the first power law for thestatistical scale effect. For about a half of a century, almost all experimentally observed scaleeffects in all materials are attributed to Weibull theory. If it is only attributed to the heteroge-neous nature of the material, Weibull theory seems provides a satisfactory answer3. However,serious discrepancies appear from experiments conducted on quasi-brittle materials4, whichlack plasticity and are characterized by gradual softening in a fracture process zone thatcannot be negligible compared to structure size. The deterministic energetic scale effect isobtained for not too large structure sizes, and the Weibull statistical scale effect is obtainedas the asymptotic limit for very large structures.3 Scale effect of jointed rocks can be studiesby analytical analysis, experimental study and empirical investigation. Jaeger proposed anempirical formula to predict the rock mass strength. Numerical study has been come to be auseful tool to better understand the failure process as well as scale effect of brittle materialsnowadays. A direct approach to studying the scale effect phenomenon in a heterogeneousmaterial, such as concrete, is based on the square lattice network5−8. Bazant et al.,5 andSchlangen and van Mier et al.7 modified the Herrmann approach in several ways in order toapply it to concrete.

In this paper, progressive failure process of the heterogeneous jointed rock specimen wassimulated by a two dimensional numerical micromechanics model with a view to study thescale effect in strength. The rock mass specimens containing pre-existing layered joints andrandomly distributed joints were represented to undertake uniaxial compressive loading tests,and the specimens had the similar shape but different sizes. The size effect on peak strength,deformation and fracture patterns was discussed.





2.1. Numerical code and joint model

A numerical code RFPA was applied to investigate the failure behaviours of jointed rocks.RFPA is a numerical code developed by C.A. Tang and is used to simulate many problems onbrittle materials, such as rocks, concretes and ceramics. The RFPA code has been developedby considering the deformation of an elastic material containing an initial random distri-bution of micro-features to simulate the progressive failure in a more visual way, includingsimulation of the failure process, failure induced seismic events and failure induced stressredistribution. Difference from common Finite Element Method, the RFPA code can sim-ulate non-linear deformation of a quasi-brittle behaviour with an ideal brittle constitutivelaw for the local material by introducing heterogeneity of rock properties into the model.By introducing a reduction of material parameters after element failure, the RFPA code cansimulate strain-softening and discontinuum mechanics problems in a continuum mechanicsmode. When the elements reach the failure assumed criteria, the elastic modulus will reduceto a certain value. The macro-failure of rock is the accumulation results of the failure of ele-ments. Not only the peak strength but also the fracture process of the rocks can be obtainedby using this code. For further particulars, please refer to literature.9,10

In RFPA code, joint elements are considered to be a kind of special material, and they havesmaller shear strength and tensile strength compared with rock elements. Generally, they areassumed to have no tensile strength and fracture when subjected to tensile stress. The normalstiffness and the tangential stiffness of the joint elements are the same when they are in elasticstage before failure. However, when the tensile stress is beyond the tensile stress, the jointswill fracture in tensile failure mode and the normal stiffness is reduced to a value close to zero.When the shear stress satisfied the shear strength criterion of the joint elements, the jointswill fracture in shear failure mode, and the shear stiffness will be reduced to a small residualvalue. Thus, whichever failure mode the joint elements have, they can have a relatively largedisplacement in either normal or shear direction correspondingly. When the joint elements inshear failure mode are compressed in next step, the normal stiffness will be rebuilt up whichcan prevent element imbedding.

2.2. Numerical modelling

In this study, three kinds of rock masses were considered. For the first kind of rock masses,layered joints with the dip angle 30◦, 45◦ and 60◦ were arranged, and the scales of thespecimens are 1m×1m, 2m × 2m, 3m × 3m, 5m × 5m, 7m × 7m, 10m × 10m, 15m × 15m,20m × 20m and 30m × 30m respectively. In order to investigate the anisotropic mechanicalbehaviour of jointed rocks, the specimens of 5m × 5m and 10m × 10m were prepared byvarying the joint dip angle from 0◦, 15◦, 75◦, to 90◦. There were totally 32 cases for this kindof rock mass containing one set of joints. For the second kind of rock masses, two groups ofjoints perpendicularly were distributed, and the scales of the specimens were the same, 1m× 1m, 2m × 2m, 3m × 3m, 5m × 5m, 7m × 7m, 10m × 10m, 15m × 15m, 20m × 20mand 30m × 30m. For the last kind of rock masses, rock joints were randomly distributedthroughout the rock mass specimens by using Monte-Carlo method. The even value of thedistance between two sets of joints was 2.5m, and the even length of the joints was 2m. Theeven dip angle for there two sets of joints was 120◦ and 30◦.

Only uniaxial compressive loading was applied on the top of the specimens when thebottom of them was fixed in vertical direction. Compressive loading was implemented byusing a constant rate of displacement increment. In the numerical model, both the rocks and

522


the joints were considered to be heterogeneous. The heterogeneity index for the rocks wasassumed to be 2.0, and the index for joints, which were much more heterogeneous thanrocks, was assumed to be 1.5. The parameters used in the simulation were listed in Table1. The jointed rock specimen containing one set of joints with the dip angle 45 was shownin Fig. 1(a), the jointed rock specimen containing two sets of joints with dip angle 45 wasshown in Fig. 1(b), and the rock specimen containing two sets of randomly distributed jointswas shown in Fig. 1(c).

Table 1. Parameters used in simulation.

Parameters UCS(MPa) E (GPa) C/T Poisson ratio φ(◦) RS

Rock 120 20 10 0.25 30 0.1Joints 20 0.5 20 0.25 30 0.1

(a) (b)

(c) 30m×30m

20m×20m

15m×15m

10m×10m

7m×7m

5m×5m

20m×20m

15m×15m

10m×10m

7m×7m

5m×5m 3m×3m

1m×1m

30m×30m

20m×20m

15m×15m

10m×10m

7m×7m

5m×5m 3m×3m

1m×1m

30m×30m

Figure 1. Sketch of jointed rock specimens with different scales ((a) One sets of joints with dip angle45, (b) Two sets of joints with dip angle 45◦, and (c) Randomly distributed joints).

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The joints had the heterogeneity index 1.5, which were more heterogeneous than the rocks.Due to the small elastic modulus, the joint elements had a large deformation relatively alongthe joints surface even though they could not slide far way. However, the failed joint elementswould get a high stiffness if they were compressed to reach the maximal strain coefficient.The high stiffness would prevent the imbedding problem. In Table 1, UCS was the uniaxialcompressive strength, C/T represents the ratio of UCS to uniaxial tensile strength, and RSrepresents the coefficient of residual strength, the ratio of the strength of the intact rock tofailure rock.

3. Numerical Simulation and Results

3.1. Scale effect

The peak strength of the specimens tested was obtained as shown in Fig. 2, Fig. 3 andFig. 4. As found by other investigators, the peak strength under uniaxial compressive loadingdecreased with the increasing of jointed specimen scale. When the scale grew larger andreaches to a critical value, the peak strength tended to be a constant. The critical scalecan be regarded as REV (Representative Elementary Volume). Numerical results of rockmass with two perpendicular joints also showed the same principle, as shown in Fig. 3 andFig. 4. When the scale of the specimens increased to 20m, the peak strength would neverdecrease any more. The mechanical parameters of the specimen with this critical scale aremuch important for the evaluation of the total rock engineering. The relation of the peakstrength and specimen scale could be well fitted by a negative exponential function. Thefunction could be described as y = a + be−cx, where y represents the peak strength and x isthe specimen scale. Parameter a, b and c were all constant. When the scale tended to bezero, the peak strength would be an unexpected large value, and if it tended to be largeenough, the peak strength would be a constant value, which represented the critical valueREV. Many authors have given this function to describe the size rule of brittle materials.1−3,5

Pea

k st

reng

th (

m)

Specimen scale (m)

Figure 2. Relation between the peak strength and the specimen scale of the specimens containing onset of joints with different dip angle.

524


Scale (m)

Pea

k st

reng

th (

MP

a)

Figure 3. Plot of peak strength for the jointed specimens containing two sets of joints at differentscale.

Scale (m)

Peak

str

engt

h (M

Pa)

Figure 4. Plot of peak strength for the rock specimens containing two sets of randomly distributedjoints at different scale.

3.2. Anisotropic feature of peak strength

When we kept the scale unchanged and vary the dip angle, we got the relation between peakstrength and the dip angle. Figure 6 showed the simulated results of the specimens with thescale of 5m×5m and 10m×10m. Both curves showed a V shape. The peak strength decreasedas dip angle increases to 60◦. However, when the angle increased further, the peak strengthincreased. The specimens with the joints of 60◦ gave the minimal peak strength, while thespecimens with the joints of 90◦ had the maximal peak strength.

As found by other scholars, the dip angle with respect to the loading direction has muchinfluence on the strength of layered rocks. As the dip angles increase, the peak strength ofrock samples undergoes a decrease-increase process, and the strength of the jointed rocksshows a shoulders type feature.1,11

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(a) single set of joints, 45° (b) two sets of joints, 45°

Scale (m)

Peak

Str

engt

h (M

Pa)

0.66392.3343 7.3190 xy e−= + 2 0.985R =

Scale (m)

Pea

k S

tren

gth

(MPa

)

0.36232.8581 4.5393 xy e−= +2 0.971R =

(a) single set of joints, 45◦ (b) two sets of joints, 45◦

Figure 5. Fitted function for the scale effect of rock masses containing joints of 45◦.

g j

(a) 5m×5m (b) 10m×10m.

Dip angle (°)

Peak

str

engt

h (M

Pa)

(a) 5m×5m (b) 10m×10m.

Figure 6. Plots of peak strength vs. the dip angle.

3.3. Fracture patterns

It is very interesting to find that there are two different fracture patterns for the jointed spec-imens with dip angle 45◦. When the scale was less than 10m, the fracture of the specimenswas dominated by shear slide along a single joint surface. The strength of the specimensdepended on the strength of the joints (Fig. 7(a)).

When the scale was larger than or equal to 10m, shear fractures along joints were observedat the beginning loading stage, while the vital fractures were found perpendicular to the sur-faces of the joints. The failure showed a tensile fracture mode. In these cases, the coalescenceof these tensile fractures led to the final failure of the jointed rock masses. In this fracturemode, the peak strength would not depend on the shear strength of the joints, as shown inFigs. 7(b) and 7(c).

When the dip angle changed to be 60◦, there existed another fracture mode between tensilefracture mode and shear fracture mode. As shown in Fig. 8 for the specimen with joints angle

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(a) (b) (c)

Figure 7. Different fracture modes for specimens with different scale ((a) Shear failure (5m×5m, 45◦),(b) Tensile failure (10m×10m, 45◦), and (c) Tensile failure (30m×30m, 45◦)).

(a) 7m×7m, 60° (b) 10m×10m, 60°

Figure 8. Fracture patterns of the rock mass specimens containing on set of joints with dip angle 60◦.

60◦, both shear fractures and tensile fractures are observed in the specimen of 7m×7m. Itwas difficult to determine which fracture dominates the coalescence of the specimen.

As shown in Fig. 9, even though the rock masses containing two set of joints had thesame size rule as those containing one single set of joints depicted by a negative exponentialfunction (Fig. 5(b)), the fracture mode of them were not similar. For the larger specimens,the failure was influenced by two groups of joints. For the specimen with small scale, muchdifferent from the cases for small jointed specimens containing one single set of joints, notobvious shear fractures but tensile fractures were found.

Figure 10 gives the fracture process of the typical rock specimens. The fracture processhad three main stages: crack initiation stage, crack propagation and coalescence stage andcollapse stage. At the starting loading stage, the stress was not high enough to break thejoints. Stress concentration was found at the tips of the joints and at the intersections ofthe joints. At the first fracture stage, failure defects were scattered along the joints wherestress was built up. When stress increased, shear stress led to failure of the joints, and cracks

527


10m×10m, 45° 20m×20m, 60° 30m×30m, 60°

Figure 9. Fracture patterns of the rock mass specimens containing two sets of joints with dip angle45◦.

propagated along the joints when subjected to tensile stress. As uniaxial compressive stressapplied continuously, cracks went ahead and broke the rock bridges, which would lead tothe coalescence of the cracks.

It should be noted that the fracture angle of the specimens with idealized joints dependedon the joint dip angle. In Fig. 10(a), the vital rupture was along the joints and they had thesame dip angle. For the specimens containing two sets of perpendicular joints, even thoughcracks propagated along the joints, the vital rupture was formed perpendicular to the jointsby the coalescence of the cracks, as shown in Fig. 10(b). While in specimens containingrandomly distributed joints, as shown in Fig. 10(c), different from idealized distributed joints,the fractures occurred throughout the entire specimen in different directions. It seemed thatthe propagation of the cracks resulting from one set of joints had no influence on that ofthe cracks from another set of other joints. At the final stage, cracks propagated further andlinked to each other.

4. Discussion

Scale effect or size effect for jointed rock masses maybe originates from the joint distribution,the strength of joints and rocks, and the boundary condition. Many researchers consideredheterogeneities in rocks and other materials was the key factor of scale effect. The distri-bution of joints, including joint density, length, width as well as depth in three-dimension,can be regarded as one of the heterogeneities contained in rocks. As shown by the numericalsimulation, the strength of jointed rocks is influenced by the strength of rocks and joints.

The strength of the jointed rock masses decreases with the increasing of the scale/size,which can be predicted by a negative exponential formula. In our simulation, a power func-tion y = a + bxc is also optional for the effect. The failure mode of the jointed rock massescan be classified into tensile fracture perpendicular to the joints or shear fracture parallelto the joints. In relatively smaller specimens, rock specimens containing fewer joints have ahigher strength, so the peak strength depends on the shear strength of the joints. However,in relatively larger specimens, rock specimens containing more joints, due to the interaction

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(a) 5m×5m, 45°

(b) 30m×30m, 45°, single sets of joints

(c) 30m×30m, two sets of randomly distributed joints

Figure 10. Fracture process of the specimens with different scales containing joints.

between shear fractures appearing at the beginning loading stage, the coalescence of theseshear fractures induced by tensile stress will predominate in the failure.

Stress condition and loading style will have influence on scale effect of jointed rocks. Itshould be noted that in this study only uniaxial compressive tests were undertaken to inves-tigate the scale effect. It can be predicted that the fitted formulas will be different more orless and the scale effect of jointed rocks under confining pressure will be not so obvious asthe results under uniaxial loading. Further investigation on jointed rocks will be presented inanother paper.

5. Conclusions

The problem of scale effect of jointed rock masses is an important issue in civil and miningengineering. The scale/size effect of the rock mass specimens containing ideally distributedjoints and randomly distributed joints subjected to uniaxial compression was studied numer-ically by using a numerical code. By varying the dip angle of the joints, the numerical rockmass specimens showed a strong anisotropic feature. Numerical results confirmed that scale

529


effect or size effect for jointed rock masses maybe originates from the joint distribution,the strength of joints and rocks. The strength of the jointed rock masses decreased with theincreasing of the scale/size, which can be predicted by a negative exponential formula. Thefailure mode of the jointed rock masses can be classified into tensile fracture perpendicularto the joints or shear fracture parallel to the joints. The results will give help to understandthe mechanical behaviors of jointed rock masses.

Acknowledgements

This research was supported by National Basic Research Program of China (973 Program,Grant No. 2007BAF09B01 and 2007CB209400) and the National Natural Science Founda-tion of China (Grant No. 50804006 and 40638040).

References

1. Jaeger, C. Rock Mechanics and Engineering (2nd Edition). Cambridge University Press, London,1979.

2. Weibull, W. “The Phenomenon of Rupture in Solids”, Proc Rswed Inst Engng Res (Ing AkadHandl Sweden), 153, 1939, pp. 1–55.

3. Bazant, Z.P. and Yavari, A. “Is the Cause of Size Effect on Structural Strength Fractal or Energetic–Statistical?”, Engineering Fracture Mechanics, 72, 2005, pp. 1–31.

4. Walsh, P.F. “Fracture of Plain Concrete”, Indian Concr J, 46, 11, 1972, pp. 469–470, 476.5. Bazant, Z.P., Tabbara, M.R., Kazemi, M.T. and Pijaudier, C.G. “Random Particle Model for Frac-

ture of Aggregate or Fibre Composites”, ASCE J Engng Mech, 116, 1990, pp. 1686–705.6. Herrmann, H.J., Hansen, A. and Roux, S. “Fracture of Disordered Elastic Lattices in Two Dimen-

sions”, Phys Rev B, 39, 1989, pp. 637–48.7. Schlangen, E. and Van, M.J. “Experimental and Numerical Analysis of Micromechanisms of Frac-

ture of Cement-Based Composites”, Cement Concrete Comp, 14, 1992, pp. 105–18.8. Walraven, J.C. “Aggregate Interlock: A Theoretical and Experimental Analysis”, PhD Thesis, Delft

University of Technology, The Netherlands, 1980.9. Tang, C.A. “Numerical simulation on progressive failure leading to collapse and associated seis-

micity” Int J Rock Mech Min Sci, 34,1997, pp. 249–262.10. Tang, C.A. and Kou, S.Q. “Fracture propagation and coalescence in brittle materials” Engineering

Fracture Mechanics, 61, 1998, pp. 311–324.11. Tien, Y.M. and Tsao, P.F. “Preparation and mechanical properties of artificial transversely isotropic

rock”, Int J Rock Mech Min Sci, 37, 2000, pp. 1001–1012.

530

Influence of Cobblestone Geometrical Property on EquivalentElastic Modulus of Cobblestone-soil Matrix

M.Z. GAO1,2,∗, H.S. MA1,3, AND J. ZHAO1

1Ecole Polytechnique Federalede Lausanne (EPFL), LMR, CH-1015 Lausanne, Switzerland2College of Water Resource and Hydropower, Sichuan University, Chengdu 610065, China3School of Civil and Environment Engineering, University of Science and Technology Beijing,Beijing 100083, China

1. Introduction

In the past decades, the technology of mechanised tunnel excavation has been developedrapidly. Tunnelling boring machine (TBM) excavation is regarded as one of the most popu-lar methods of tunnel construction1. Although TBM is more versatile and capable of boringthrough greatly varying geologic conditions than the conventional excavation method, thegeological situation will greatly influence the effect and overall success of TBM.2 Mixed-faceground has been a prominent problem due to its adverse influence on TBM tunnelling.3 Itcan lead to cutter wear, jam of roller cutterhead, settlement, further to TBM performanceand cost overrun.4 In Metro Line 1 of Chengdu, China, such problems were encounteredduring excavation of tunnels where cobblestone-soil ground was presented (Fig. 1). The geo-logical situation is a typical mixed-face ground. On macroscopic scale, it can be regardedas continua, statistically homogeneous. The cobblestone and soil are usually firmly bondedtogether at the interface.5 The cobblestone strength generally is many times as the soil’s. Inorder to understand the whole material deformation performance in advance, the research ofthe cobblestone-soil is becoming more and more important in TBM’s good performance.

This paper is intended to address the problem of the equivalent elastic modulus of thematerials. Although many theory prediction models have been proposed for composite mate-rial in the past years, most of them based on assumptions which might deviate from actualsituation in the site.6 The assumptions will limit the use of existing theoretical models indealing with complex materials. Therefore, with the development of computing technology,the numerical methods are becoming increasingly important and widely used in research.7

For analyzing equivalent elastic modulus of materials, the research does not take intoaccount material failure process and only highlights the deformation. Thus, the numericalmodelling process was carried out in the framework of a linearly elastic. The numericalmodelling software is ANSYS. The cobblestone is assumed as the perfect ellipse shape. Thegeometrical shape is represented by the length ratio of the major axis to shorter axis ofcobblestone, and fixed at 1.5 in the current study. These data of cobblestones come fromChengdu Metro site by the screen separation and digital image technique method. Two fac-tors affected the equivalent elastic modulus were analysed:

(a) The angle between the cobblestones major axis and loading direction. It varies from 0 to90◦ at interval of 15◦.

(b) The cobblestone percentage content. It ranges from 0% to 60% at interval of 5%.




Figure 1. The ground situation of Chengdu Metro Line 1.

2. Cobblestone Shape and Distribution

The size distribution for cobblestones was done at Chengdu Metro site. Samples were takenfrom four different sites along the metro line 1 from north to south. Three group tests werecarried out at each site. The main composition of cobblestones includes granite, diorite andquartzite. Sorting features are relatively good. Most of the cobblestone is approximatelyellipsoidal or globular. The geometrical parameters were gained based on statistical analysis.

In order to get more credible data, two-dimensional digital image technique method wasused.8 The method is capable of obtaining the data at TBM site where it could be difficult totake samples. Origin image is the high resolution colour image. It is very easy to transformthe image data to the gray level image by assigning the brightness to a different integer valuewhich is named as the gray level at each pixel. Mostly used 256 gray images whose gray levelshave the integer interval from 0 to 255 were adopted. Because the cobblestones usually havehigher gray levels and the remaining matrices usually have lower gray levels, as a result, theinterfaces usually have some great changes of the gray levels. The interface between matrixand cobblestone can be identified from their different gray levels. Two different digital imagetechniques are adopted to acquire the actual interface in digital image processing: regionsegmentation method and edge detection method.9 The interface line can be transformed tovector data which can be opened by existing software AutoCAD. By this way the same dataas the size distribution can be acquired.

As the result of the statistic analysis, the diameter of cobblestones mostly ranges from 10 to80mm. A few bigger boulders can be found randomly. The maximum diameter encounteredis 512mm. The distribution of diameter can be supposed normal distribution (Fig. 2). Theaverage and variance value are 38.23mm and 26.50 respectively. The average length ratio ofthe major axis to shorter axis of cobblestone is 1.54. In the mechanical analysis later shownin the paper, it was fixed at 1.50.

3. Modelling

The model width and height are 300mm and 600mm respectively according to laboratorysample. For studying the effect of cobblestone on whole model, the model size should be atleast five times as the maximum size of cobblestone.10 The cobblestones with diameter below

532


0%

5%

10%

15%

20%

25%

30%

7.5 15 25 35 45 55 65 75 85 95 105 115 125 135 145 155 165 175 185 195 205

The diameter of cobblestones(mm)

Figure 2. The distribution of cobblestone diameter.

Figure 3. The DWF file (60% cobblestone at 30◦). Figure 4. The finite element mesh in ANSYS.

5mm were regarded as fine matrix. Therefore the cobblestones created lie in between 5mmand 60mm. All cobblestones were generated using Monte Carlo method. The size distributionof cobblestone is normal distribution following size analysis at site. Cobblestones generatedbeyond range from 5mm to 60mm were eliminated. The cobblestones were located in modelzone randomly. Any cobblestone can not overlap with others. The process can be performedautomatically by a self-editing Visual Basic program. The result files include DWF vector file(Fig. 3) and ANSYS command flow file. The command flow file can run directly in ANSYSfor meshing and calculating (Fig. 4).

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Table 1. Properties of materials.

Matrix Cobblestone

Elastic Modulus(MPa) 8.0 20000Poisson’s Ratio 0.35 0.25

Figure 5. The equivalent elastic modulus and cobblestone percent.

The computational models were assumed to be the perfect elastic material and their proper-ties are listed in Table 1. The left and right boundaries are free. In vertical direction, displace-ment is fixed at the lower boundary and displacement load is applied at the upper boundary,respectively. The model was discretized with finite element mesh (Fig. 4). The element sizewas set to 5mm and can be adjusted automatically.

4. Numerical Simulation Results and Discussion

4.1. Numerical simulation results

Due to the topology arithmetic, it is very difficult and slow to generate more cobblestoneswhen the cobblestone content is above 60%. Consequently the paper just models the cob-blestone content from 0% to 60%. The result also applies to the same range of cobblestonecontent.

As shown in Fig. 5 and Fig. 6, the following result can be shown:

(a) The equivalent elastic modulus increases obviously with increasing of cobblestone per-centage content.

(b) The equivalent elastic modulus also increases with decreasing of the angle between thecobblestones major axis and loading direction.

4.2. Compare between numerical and theoretical approach

Analytic solution for the equivalent elastic modulus can be derived from theoretical approachresembling the cube imbedded formulation deduced by Paul in 1960.11 During the derivingprocess, the strain was assumed uniform in the whole model and the length of each side ofmodel is regarded as 1. Analytic equation for composite matrix equivalent elastic modulus

534


Figure 6. The equivalent elastic modulus and cobblestone position.

Figure 7. The equivalent elastic modulus and cobblestone position.

with elliptical cobblestones which major axis is perpendicular to the loading direction wasderived with the same principle, and is shown as Eq. 1.

1E= mb

E1

⎛⎜⎜⎝π +

m ln 1−√

1−m2

1+√

1−m2√1−m2

⎞⎟⎟⎠+ 1− 2b

E1( |m| < 1) (1)

where m = E12a(E2−E1)

, E, E1 and E2 are the equivalent elastic modulus, matrix elastic modulusand the material imbedded elastic modulus respectively. a and b are length of the major axisand the shorter axis in an ellipse. As shown in Fig. 7, the theory solution is reasonably close tonumerical modelling result when the cobblestones major axis is perpendicular to the loadingdirection.

4.3. Discussion

During tunnelling in Chengdu Metro Line 1, the cobblestone-soil ground extensively affectsthe TBM performance. For example, cutter wear, jam of roller cutterhead, settlement, andcost overrun have already been encountered. To a certain extent, those problems were induced

535


by the lack of knowledge about such special ground. TBM in the cobblestone-soil groundwith good performance is essential for applying of TBM method in future tunnelling. Inorder to further realize cobblestone-soil, the equivalent elastic modulus of the material wasdiscussed based on numerical modeling and analytic theory.

The numerical simulation and analytic solution show the equivalent elastic modulusincreases with increasing of cobblestone percentage content. The trendline agrees with secondorder curve very well. The effect of cobblestone content on the equivalent elastic moduluswill be improved with increasing of cobblestone content. The situation also can be foundin Fig. 6. It is quite obvious that the interval of curve is progressively larger as the cobble-stone content changes from 0% to 60%. It is easy to understand the phenomena which theequivalent elastic modulus of material will be enhanced when another material with higherelastic modulus mixed.6,12 For cobblestone-soil material, the whole property deformed wascontrolled by property of soil itself yet when the cobblestone content is relative small. Whenthe cobblestone content exceeds 30%, it will extremely affect the whole property deformed.In that case, the cobblestones could form a column zone forced in vertical loading directionfrom top to bottom with small space in each cobblestone. The top loads will be enduredmainly by forced zone. As a result, the interface strength between cobblestone and matrixwill play a significant role in deforming performance of the cobblestone-soil material.

The equivalent elastic modulus also increases with decreasing of the angle between thecobblestones major axis and loading direction. The trend becomes obvious gradually as cob-blestone content increasing. The main reason could be that it is easier to form the verticalcolumn zone forced when the cobblestones major axis parallels to the loading direction underthe same cobblestone content condition. The cobblestone position just takes a more promi-nent role only when the percentage of cobblestone reaches a certain value. The value is relatedto performance of each material.

The digital image method shows most of cobblestones are close to horizontal direction inChengdu Metro site. The reason could be long-term geological process of river. However, thecobblestones are always simultaneous occurrence of different position in actual engineeringsite. Only one type of cobblestone position in actual engineering is impossible. Therefore theequivalent elastic modulus of cobblestone-soil should be the result of taking into accountdifferent positions in one model. The research of this paper is basically an attempt to pre-liminary estimates the deforming capability of cobblestone-soil for the benefit of the furtherdevelopment of TBM and its performance.

5. Conclusions

The study presented in this paper provides the following conclusions:

(a) The equivalent elastic modulus increases with increasing of cobblestone percentage con-tent. The trendline agrees with second order curve very well (Fig. 5).

(b) The equivalent elastic modulus also increases with decreasing of the angle between thecobblestones major axis and loading direction (Fig. 6).

(c) The theory solution for the equivalent elastic modulus is reasonably close to numeri-cal modelling result when the cobblestones major axis is perpendicular to the loadingdirection (Fig. 7).

(d) The digital image method shows most of cobblestones are close to horizontal directionin Chengdu Metro site.

536


Acknowledgements

The junior author thanks the China Scholarship Council for financial support. The authorsare very grateful to all the contractors for their helps on data collecting at Chengdu Metrosites.

References

1. Zhao, J., Keynote: Tunnelling in rocks-present technology and future challenges. Proceedings ofthe 33rd ITA-AITES World Tunnel Congress. Prague 2007.

2. Zhao, J., Gong, Q.M. and Eisensten, Z., “Tunnelling through a frequently changing and mixedground: A case history in Singapore”, Tunnelling and Underground Space Technology. 22, 4,2007, pp. 388–400.

3. Ranjith, P.G., Zhao, J. and Seah, T.P., A case study of effects of ground conditions on Tunnelboring machines. Proceedings of the 28rd ITA-AITES World Tunnel Congress. Australia 2002.

4. Blindheim, O.T., Grøv, E. and Nilsen, B., “The effect of mixed face conditions (MFC) on hardrock TBM performance”, Proceedings of the 28rd ITA-AITES World Tunnel Congress.Australia2002.

5. Hashin, Z., “Analysis of Composite Materials - A Survey”, Journal of Applied Mechanics,Transactions ASME. 50, 3, 1983, pp. 481–505.

6. Wang, M. and Pan, N., “Predictions of effective physical properties of complex multiphase mate-rials”, Materials Science and Engineering R. 63, 1, 2008, pp. 1–30.

7. de Borst, R., “Challenges in computational materials science: Multiple scales, multi-physics andevolving discontinuities”, Computational Materials Science. 43, 1, 2008, pp. 1–15.

8. Chen, S., Yue, Z.Q. and Tham, L.G., “Digital image-based numerical modeling method for pre-diction of inhomogeneous rock failure”, International Journal of Rock Mechanics and MiningSciences. 41, 6, 2004, pp. 939–957.

9. Yue, Z.Q., Chen, S. and Tham, L.G., “Finite element modeling of geomaterials using digital imageprocessing”, Computers and Geotechnics. 30, 5, 2003, pp. 375–397.

10. Holtz, W. and Gibbs, H., “Triaxial shear tests on pervious gravelly soils”, Journal of the SoilMechanics and Foundations Engineering Division, ASCE. 82, 1, 1956, pp. 1–22.

11. Paul, B., “Prediction of elastic constants of multiphase materials”, Transactions. AIME. 218, 1960,pp. 36–41.

12. Hashin, Z. and Monteiro, P.J.M., “An inverse method to determine the elastic properties of theinterphase between the aggregate and the cement paste”, Cement and Concrete Research. 32, 8,2002, pp. 1291–1300.

537

Comparative Studies of Physical and Numerical Modeling onRegular Discontinuities

ABBAS MAJDI1,∗, HESSAM MOGHADDAM ALI2, AND KAYUMARS EMAD3

1Associate Professor, School of Mining Engineering, University of Tehran, Iran2Head of Infrastructures, Transportation Research Institute (TRI), Ministry of Road and Transportation, Iran3Senior Extraction Engineer, IMPASCO Company, Ministry of Industries and Mines, Iran

1. Introduction

All in-situ natural rocks may consist of the two parts; discontinuities and rock substances. Assuch, in most cases due to the proximity of the discontinuities the corresponding propertiesrepresent the inherent in-situ rock mass. Such discontinuities geometrically are recognizedas the smooth surfaces or wavy curve shapes, and/or saw teeth-like joints within the rockmass. From strength point of view, when they are subjected to a perpendicular tensile load-ing are characterized low resistance compared with the force that required breaking the samerock mass in compression. In reality, the normal and shear stresses are transferred to suchdiscontinuities due to execution of earthwork engineering activities. Hence, determinationof the most crucial discontinuity with the lowest shear strength is essential to properly eval-uate rock structural stability surrounding any open excavations and or underground struc-tures. This strength depends on discontinuities’ conditions and therefore is strongly relatedto parameters such as roughness and thickness of filling materials, joint water, weathering ofthe discontinuous surfaces and other associated factors.

In this research, several laboratorial made plaster samples with regular discontinuous sur-faces were prepared and tested for determination of the critical shear surfaces. The labora-torial results in connection with the current shear strength criteria were used to illustrate therepresentative shear strength parameters. These representative parameters were further usedas the input data for a numerically prepared model on the basis of 3DEC software. Finallythe results of numerical modelling were back analysed and then compared with that attainedin real samples tested in laboratory.

2. Physical Modelling of Regular Discontinuity

In physical modelling, in order to study the behaviour of discontinuities with taking the effectof roughness of the jagged surfaces of the joints into account, several laboratory samples withartificial joints with different geometries were investigated.1 To date, various experimentshave been performed by changing the geometric conditions of the discontinuities.2 Artificialsamples preparation for the shear test is very similar to that is used for preparation of theregular rock samples. In this test, samples were prepared in a way that the top and bottomof the discontinuous surfaces of the joints are parallel to the discontinuity governing axis.Subsequently the samples were tested under the constant normal loading (N or σn) then, theshear force (τ ) was applied parallel to the discontinuous layer. In this research, the relativehorizontal and vertical displacements (u,v) of the discontinuity have been measured versusthe shear force as are discussed hereafter.




2.1. Samples preparation

In this research, samples were prepared with plaster materials, fine sands and water with w/sratio 1:1, 1:1, and 3:5. Where w represents weight of water and s is the weight of combinedsolid materials in the samples. The artificial samples were produced with jagged regular geo-metric roughness. During the shear test performance the two parameters; normal load anddiscontinuity surface roughness were considered variables as they are frequently happen innatural in-situ conditions. Dimensions of the sample preparation box were taken: (30 cm ∗30 cm ∗ 15 cm; W ∗ L ∗ H). Where; W is the width, L represents the length, and H is theheight of the samples. The height of each sample has been divided into two equal sections,each section with a length of 7.5 cm.

For preparing the samples with regular jagged surface, 3 templates with three differentjoint asperity angles; 15, 30 and 40 degrees were prepared (Fig. 1). Four samples were pro-duced for each angle and then tested at different vertical loading conditions.

In order to determine the basic friction angle of the joint surface (ϕb), two samples wereprepared in smooth discontinuity format called the saw-cut samples. In order to maintainthe consistency of the results, in all experiments, the physical and strength properties of thesamples were kept identical. After completion of the direct shear test and drawing the socalled τp − σn diagrams, the basic friction angle equal to 35 degrees was estimated.

2.2. Geometric and mechanical characteristics of the samples

JRC is one of the most representative geometric parameters which can be used to deter-mine the strength characteristics such as normal stiffness of the discontinuity. In this study,Z2-Method4 is used to evaluate JRC of the samples. This method is based on a statistical-probability relationship between the JRC and Z2 where Z2 is the root mean square of thefirst derivative of the discontinuity surface profile with a correlation coefficient of R = 0.986.

JRC = 32.2+ 32.47 log Z2 (1)

Z2 =⎡⎣ 1

m(�x)2

m∑j=0

(yj+1 − yj)2

⎤⎦

1/2

(2)

Type 1 i= 15 degree L = 2 cm



L = 30 cm

W =30 cm

H =15 cm

N

View of sampling format

Figure 1. Geometrical characteristics of joint roughness profile in 3 templates format (L = ∑�X)

and i = asperity angle) [after 3].

540


Table 1. Geometric and strength characteristics of the Samples under CNL loading [After 3].

Sample Jogged σc σt D (Fractal JRC σn τexp Up (mm) Vp (mm)Number Degree (Kg/cm2) (Kg/cm2) Dimension) (Kg/cm2) (Kg/cm2)

1-1 15 44.5 12.0 1.032 13.62 8.8 8.44 5.5 0.771-2 15 45.3 12.1 1.023 13.62 6.2 6.62 6.5 1.142-1 30 45.0 11.8 1.154 24.45 8.8 15.11 5.0 1.292-2 30 44.5 12.1 1.154 24.45 6.2 11.11 6.0 1.773-1 40 45.1 12.1 1.305 29.72 8.8 17.84 4.8 2.1373-2 40 44.9 12.2 1.305 29.72 6.2 16.00 5.5 2.689

Where in Eq. (2), m represents total data points spaced at a constant small distance �Xalong the discontinuity surface profile. yj−1 and yj+1 illustrate the amplitude of two datapoints with respect to the centre line that is, yj, at each side of �X on joint surface, and JRCis the joint roughness coefficient.

As it can be seen from the Eq. (2), Z2-value inversely corresponds to �X. in order toincrease the accuracy of the calculated JRC coefficient; the Z2-value should be kept as smallas possible.4 The results of this study including strength parameters such as the uniaxialcompressive strength, tensile strength and friction angle that obtained from the direct sheartests of the samples are also presented in Table 1.

3. Initial Estimation of the Shear Strength

In order to compare the results of physical modeling with the existing shear strength crite-ria, two approaches proposed by Patton,5 and Barton and Choubey6 were used to estimatethe samples joints shear strength. Patton’s bilinear criterion has the capability of evaluatingthe shear strength of the discontinuity in both shearing and sliding up conditions. In thesemodels, the testing conditions have been divided into two parts, (a) state of the normal stresslevel is low; in this case, due to non-restricted dilation, the slide up occurs, (b) state of thenormal stress level is high; due to the dilation constraint, degradation of the joints asperitiesoccurs and hence shear failure may take place across the discontinuity surface.5

τp = σn tan (ϕb + i) when σn < σt (3)

τp = c+ σn tan (ϕb) when σn > σt. (4)

Where C is the cohesive strength; ϕb represents the basic friction angle of the joint surface,and i is the asperity angle to the direction of shear force application.

It is rare that natural discontinuities behave exactly according to Patton’s criterion; how-ever a combination of the two different aforementioned mechanisms as shown by Equations(3) and (4) may control the natural discontinuities.5 Barton and Choubey presented the fol-lowing non-linear relationship to determine the natural joint’s shear strength.5,6

τ = σn tan[φr + JRCn log

(JCSn

σ

)]. (5)

Meanwhile taking the effect of non-conformity of the joints for shear strength estimation,Zhao7 proposed a new term, JMC, that to be incorporated within the Equation (5).

τ = σ tan[φr + JMC× JRCn × log

(JCSn

σ

)](6)

541


Where τ represents the shear strength of the joint, JRCn is the joint roughness coefficient,JCSn is the joint wall compressive strength, and JMC is coefficient of joint compliance whichillustrates the degree of non-conformity of the jaggy surface of the natural joints, σ and φr isthe normal stress and the residual friction angle respectively. Generally, the JMC-value variesfrom 1; in fully matched joint, to zero in the fully separated joint.7 However, the lowestpossible value for JMC in natural joint contact is equal to 0.3. The shear strength resultsobtained on the basis of the equations proposed by Patton, and Barton and Choubey areshown in Table 3. These results were compared with that obtained in laboratory.

4. Failure Mechanism of Laboratory Samples

The peak shear strength τp is attained at an early shearing stage and the corresponding sheardisplacement is Up or �U. Then, due to a gradual decrease of shear stress, it reaches to aconstant value which represents the residual shear strength, τr of the sample. It is obviousthat the corresponding maximum vertical movement occurs at the failure point which repre-sents the dilation, Vp or �V. Upper block horizontal movement velocity is another effectiveparameter when dealing with shear testing, and hence, 2.258 mm/min is taken in all tests.Constant Normal Loading (CNL) was adopted in all laboratory tests for which the maxi-mum value equal to 8.8 Kg/cm2 was assigned. The laboratory measured displacements andshear strength are shown in Table 1.

5. Numerical Modelling of Regular Discontinuity

In this paper, the numerical models were constructed on the basis of 3DEC discontinuousmethod. Hence, all numerical requirements including geometric, simulation of the discontin-uous surface, displacement velocity, etc. along with the corresponding mechanical propertieswere employed according to the general guidelines of the software. With regard to the com-plex behaviour of the discontinuity within the physical model, the quantitative evaluation ofnormal stiffness, Kn and shear stiffness, Ks were made by means of Bandis et al’s equations,8

and then the selection of the representative behavioural model was made.

5.1. Selection of discontinuity behavioural model

After construction the appropriate numerical models, among the two behavioural modelsproposed in 3DEC manual preference was given to the selection of Continuously YieldingModel due to the closeness of its results with the actual condition.9 Parameters needed todescribe the characteristics of the continuously yielding model are: normal and shear stiffnessof discontinuity, joint surface roughness and inherent friction angle of the joints.

5.2. Determination of normal and shear stiffness coefficients

To evaluate the normal stiffness of discontinuity, the relations proposed by Bandis et al8 andgiven by Equations (7) and (8) shown below are used to estimate the quantitative value ofKni, tan. Subsequently, the results obtained from Equation (8) is plugged into Equation (9) tocompute the value of Kn, tan. Kn values estimated for the laboratory samples discontinuity arepresented in Table 2.

et ≈ JRC(

0.04σc

σd− 0.02

)(7)

542


Table 2. Normal and shear stiffness values of sample discontinuity based on Bandis et al relations.

SampleNumber

JRC σn (Kg/cm2) σd (Kg/cm2) � V max (mm) ei Kn(MPa/mm)

Ks,avg(MPa/mm)

1-1 13.62 8.8 4.45 0.77 0.2726 105.70 15.551-2 13.62 6.2 4.53 1.14 0.2726 107.30 15.652-1 24.45 8.8 4.5 1.29 0.489 125.63 28.082-2 24.45 6.2 4.45 1.77 0.489 124.63 27.073-1 29.72 8.8 4.51 2.037 0.5944 135.06 41.873-2 29.72 6.2 4.49 2.889 0.5944 134.66 41.57

Kni, tan ≈ 7.15+ 1.75JRC+ 0.02(σd

ei

)(8)

kn, tan = kni, tan

(1− (6n − 6ni)

(6n − 6ni)+ (Kni, tan�Vmax)

). (9)

ei: average initial discontinuity aperture. ei in mm under a normal stress of approximately1 KPa could be estimatedVmax: maximum normal displacement produced by increasing σn from specified seatingpressure (mm)σc: uniaxial compressive strength (MPa)σd: joint wall compressive strength referred by Barton as JCS (MPa)Kn, tan: initial tangent normal stiffness (MPa/mm)Kn, tan: normal tangent stiffness (MPa/mm)

Bandis et al8 in a survey of the peak shear stiffness, measured on some 450 discontinuitiesranging in length from 200mm to faults extending for more than 100 km found that for agiven normal stress, shear stiffness was inversely proportional to the discontinuity length.Bandis et al10 also found that the initial tangent shear stiffness, Ksi, tan, with increasing nor-mal stress is associated with the following power function

Ksi, tan ≈ Kj(σn)ni (10)

where Kj and ni: are empirical constants termed the stiffness number and the stiffness expo-nent respectively. Subsequently, the tangent shear stiffness can be computed by using thefollowing relation5

Ks, tan = Ksi, tan

(1− τ ×Rf

τf

)2

(11)

where Rf is failure ratio given by τf /τult. The τf can be estimated from Equation (5) and τultis ultimate shear stress at large shear displacement when the failure ratio, Rf , is found. Theresult of this relationship is presented in Table 2.

5.3. Discussion on the results of regular joint modelling with 3DEC software

As it was mentioned earlier in this paper, on the basis of continuously yielding methodin 3DEC software with employing the geometric and strength properties of the laborato-rial samples the desired numerical models were constructed. The results of Bandis et al8,10

543


relations both shear and normal stiffnesses were used as the input data for the models.Three models with three different discontinuous indentation angles ranging as 15, 30 and40 degrees were prepared. In all numerical models, normal stresses and displacement veloc-ities used in laboratory tests were also employed. The upper part of model geometry with aconstant speed of 0.037 mm/s, moves in x-direction while the lower part of the model waskept fixed similar to the laboratory conditions. However, the possibility of normal movementin vertical direction was taken into consideration as well. Fig. 2(a) illustrates shear stress ver-sus the corresponding shear displacement resulted from model series 1–1, with indentationangle of 15 degrees and the normal stress of 8.8 Kg/cm2. The maximum estimated shearstrength is equal to 8.9 Kg/cm2 when the Continuously Yielding Method was used. Whereasthe maximum shear strength measured in laboratory direct shear test of the same model wasequal to 8.44 Kg/cm2.

The results indicate that the physical and numerical modelling values match in sampleseries 1–1 very well. The other 6 numerical models were also constructed on the basis ofphysical specifications of the laboratory models. Figure 2(a) illustrates a non-linear behaviourup to peak stress and followed by a gradual decrease in shear stress, then continues almostwith a constant residual stress.

Figure 2(b) illustrates the change of normal displacement versus the corresponding sheardisplacement of the sample series 3–1. As it can be seen from this diagram the normal dis-placement is negative at early stage of loading which expresses the closing state of the dis-continuity. Subsequently by rising the normal stress along with increasing the correspondingshear stress on the upper block, the resulting normal displacement increases and attains atmaximum when it reaches the failure point, then it remains constant. Then after if shearloading continues the shear displacement will increase while normal displacement remainsconstant as shown in Fig. 2(b). In order to evaluate other output parameters of the models,displacements contour-lines resulting from failure behaviour under the Continuously Yield-ing Method in remaining samples were studied separately.

a)

Shea

r st

ress

(M

Pa)

Shear displacement (MPa)

Max = 8.9 Kg/cm2 Nor

mal

dis

plac

emen

t (m

)

Shear displacement (m)

b)

Figure 2. Variation of; (a) Shear stress vs. shear displacement, (b) Normal displacement vs. sheardisplacement in continuously yielding model.

544


Table 3. Results of numerical and physical modelling, Barton and Choubey, and Patton rela-tionships on regular discontinuity samples.

Sample number τexp (Kg/cm2) τModelling (Kg/cm2) τBarton (Kg/cm2) τPaton (Kg/cm2) JRC

1-1 8.44 8.99 8.74 10.59 13.621-2 6.62 6.21 6.56 7.42 13.622-1 15.11 12.5 11.41 19.06 24.452-2 11.11 8.31 9.11 13.34 24.453-1 17.84 14.4 13.07 33.17 29.723-2 16.00 12.0 10.83 32.22 29.72

6. Comparison of Numerical and Physical Modelling Results

Table 3 represents the final results of numerical and physical modelling along with thoseobtained based on Barton and Choubey,6 and Patton5 relationships on regular discontinu-ity samples for the sake of comparison. Figure 3(a) illustrates the change of shear strengthobtained from both numerical and physical modelling with respect to the prescribed jointasperity angles, that is, 15, 30 and 40 degrees along with the corresponding JRC. As it canbe seen from Fig. 3(a), with increasing the indentation angle, JRC, and ei representing jointgeometrical characteristics, the results obtained from the two models, numerical and experi-mental, diverges from each other considerably.

A comparative analysis was also made using the results obtained from Barton andChoubey,6 and Patton5 shear strength relations with those computed on the basis of physi-cal and numerical modelling. Figure 3(b) illustrates the aforementioned comparative results.Figure 3(b) clearly reflects the proximity of the results obtained from all 4 methods up to30 degrees of the indentation angles, whereas afterwards the Patton’s method diverges andshows overestimation of the discontinuity shear strength significantly. However, Barton andChoubey’s method yields a very accurate result which confirms the validity of the method.

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

18.00

20.00

1 2 3 4 5 6

t

Numerical Modeling Expxrimental

1-1 1-2 2-1 2-2 3-1 3-2

JRC

13.62 13.62 24.45 24.45 29.72 29.72

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

1 2 3 4 5 6

t

Exp Paton Barton Model

1-1 1-2 2-1 2-2 3-1 3-2

JRC

13.62 13.62 24.45 24.45 29.72 29.72

ba

Figure 3. Comparative analysis of: (a) Numerical and physical modelling, (b) Numerical and physicalmodelling, Barton and Choubey, and Patton shear strength criteria.

545


7. Conclusions

A number of laboratorial samples with regular discontinuous surfaces were prepared andtested for determination of the critical shear surfaces. The laboratorial results in connec-tion with the current shear strength criteria were used to illustrate the representative shearstrength parameters. These representative parameters were further used as the input data fora numerically prepared model on the basis of 3DEC software.

After construction the appropriate numerical models, among the two behavioural modelsproposed in 3DEC manual preference was given to the selection of Continuously YieldingModel due to the closeness of its results with the actual condition. The results of numericalmodeling were compared with those obtained by laboratory testing. The comparative con-sideration revealed a minor difference between the two different models results. The resultsobtained from laboratorial physical and numerical modelling were further compared withthose computed on the basis of Barton and Choubey, and Patton shear strength relations.The comparative analysis reflects the closeness of the results obtained from all 4 methods upto 30 degrees of the indentation angles, whereas afterwards the Patton’s method diverges andshows overestimation of the discontinuity shear strength significantly. However, Barton andChoubey’s method yields a very accurate result which confirms the validity of the method.Final conclusion confirms the validity of Bandis et al’s equation for the normal and shearstiffness’ determination of regular discontinuities.

References

1. Benmokrane, B. and Ballivy, G., “Laboratory study of shear behaviour of rock joints under con-stant normal stiffness conditions". Proc. 30th U.S. Symp. Rock. Mech. 1989, pp. 50.

2. Hudson, J.A. and Harrison, J.P., Engineering Rock Mechanics (An Introduction to the Principles),Pergamon, 1997.

3. Askari, M. and Ahmadi. M., Post-peak shear behaviour of artificial joints, M.Sc. Thesis, Facultyof Engineering, Dept. of Mining Engineering, University of Tarbiat Modarres, 2006.

4. Tse, R. and Cruden, D.M., “Estimating Joint Roughness Coefficients". Int. J. Rock Mech. & Min.Sci., 16, 1978, pp. 303-307.

5. Priest S.D., Discontinuity Analysis for Rock Engineering, Chapman and Hal, London, 1997, p.473.

6. Barton, N. and Choubey, V., The shear strength of rock joints in theory and practice. RockMechanics 1/2:1–54. Vienna: Springer. Also NGI Publ., 1978, p. 119.

7. Zhoa. J. “Joint Surface Matching and shear strength", Part B: JRC-JMC Shear Strength Criterion,Int. J. Rock Mech. & Min. Sci., Pergamon. Vol. 34, No., 1997, pp. 179-185.

8. Bandis, S.C., Lumsden, A.C. and Barton, N.R., “Experimental studies of scale effects on the shearbehavior of rock joints"., Int. J. Rock Mech. & Min. Sci, 18, 1981, pp. 1–21.

9. Itasca Codes, “Manual of 3DEC", Theory and Background, Section Four, Interface, 2004, pp.1–36.

10. Bandis, S.C., Lumsden, A.C. and Barton, N.R., “Fundamentals of rock joint deformation"., Int.J. Rock Mech. & Min. Sci, 20, No. 6, 1983, pp. 68–249.

11. Ladanyi, B. and Archambault, G., “Simulation of the shear behavior of a jointed rock mass", Proc.11th. U.S. Symp. On Rock Mechanics, 1970, pp. 105–125.

546

Probabilistic Assessment of a Railway Steel Bridge

B. CULEK1, V. DOLEZEL1,∗ AND P. P. PROCHAZKA2

1University of Pardubice, Czech Republic2Association of Czech Civil Engineers, Prague, Czech Republic

1. Introduction

The structural engineering profession needs new approaches if we want to provide the bestpossible service to society. We have to consider the transition from the deterministic “way ofthinking” to open-minded probabilistic concepts accepting the random character of individ-ual variables involved as well as their interaction. Tools such as simulation techniques andpowerful personal computers will contribute to reaching such goals.1,2

A general method for lifetime performance analysis of existing steel girder bridges is pre-sented in.3 Only the superstructure components are considered. The formulation is estab-lished by identifying four distinct categories: limit state equations, random variables, deter-ministic parameters and constant coefficients.

The design fatigue life of a bridge component is based on the stress spectrum the com-ponent experiences and the fatigue durability. Changes in traffic patterns, volume, and anydegradation of structural components can influence the fatigue life of the bridge. A fatiguelife evaluation reflecting the actual conditions has value to bridge owners. Procedures are out-lined in the Guide Specifications for Fatigue Evaluation of Existing Steel Bridges to estimatethe remaining fatigue life of bridges using the measured strain data under actual vehiculartraffic. The paper4 presents the methodology with an actual case study of Patroon IslandBridge. The results indicate that most of the identified critical details have an infinite remain-ing safe fatigue life and others have a substantial fatigue life.

The paper5 focuses on the fatigue damage caused in steel bridge girders by the dynamictire forces that occur during the crossing of heavy transport vehicles. This work quantifiesthe difference in fatigue life of a short-span and a medium-span bridge due to successivepassages of either a steel-sprung or an air-sprung vehicle. The bridges are modelled as beamsto obtain their modal properties, and air-sprung and nonlinear steel-sprung vehicle modelsare used. Bridge responses are predicted using a convolution method by combining bridgemodal properties with vehicle wheel forces.

The lifetime performance of deteriorating structures, defined by their time dependent con-dition index and reliability index, is analyzed in Ref. 6

Authors in Ref. 7 describe approach of fatigue life assessment of thermo-mechanical fatigueproblems.

Fatigue cracks are one of the most devastating problems for orthotropic steel bridge decks.Well known examples of fatigue cracks are those observed in the bascule bridge Van Brieneno-ord in Rotterdam in summer 1997. These were cracks in the decks plate at the crossing ofthe crossbeam and the longitudinal girder and are known as the most dangerous cracks fortraffic safety. The number and amplitude of stress cycles, which are closely related to amountof axles and their loads, govern the fatigue phenomena.8 In this paper results of probability




Figure 1. Longitudinal section.

calculation of the lifetime of the railway steel bridge are presented. The bridge composedfrom continuous beams, (Fig. 1).

These calculations are subsequently compared with calculations performed through linearhypothesis by Corten-Dolan. Assessment of the service life comes from the real records ofthe response of traffic loads, which are obtained by strain-gauge measurements, which wererealized for above mentioned bridge. For calculations of service life there were chosen placesof probability creation of fatigue rifts.

2. Basis of Lifetime Assessment

Total number of strain gauges on the construction was 30 (in position over whole construc-tion). Critical place for lifetime assessment was on connection between longitudinal andcross beams The railway bridge was monitored 24 hours. During this period were measuredin total 151 passing trains, (Fig. 2).

T 13

T 14

T 25

T 28

T 22

T 39

TRAIN 5304

Str

es

s c

urv

es

[M

Pa

]

5 10 15 20

Time [hrs]

-20

-15

-10

-5

0

5

10

15

20

24

Figure 2. Strain gauge data record.

548


2.1. Service life assessment by the help of Corten Dolan Hypothesis

According to the Corten-Dolan (CD) hypothesis stands total number of oscillations to afailure:

H(CD) =

p∑i=1

ni

p∑i=1

ni

Np∗(σai

σap

)k∗w =Np

p∑i=1

ni

hz∗(σai

σap

)k∗w , (1)

where

ni is a number of cycles of a given amplitude,Np is a number of cycles of the highest amplitude until failure,hz is a total number of oscillations of all the amplitudes,σai is a size of given amplitude,σap is a size of the maximal amplitude,w is an exponent of the fatigue curve,k is a reduction of the fatigue curve.

For the cycle counting of stochastic process of the stress there was a two-parametricRain Flow counting method (RF) used. Fatigue curve was obtained empirically and it hasa functional shape:

log N = log a−w∗ log�σC, (2)

for T39K:

a = 11.4

w = −3.0

σC = 52.0 MPa

Service life Z of railway bridge in the position of strain gauge T39K was determined byclassical calculation (using of one linear fatigue curve and CD hypothesis) at:

ZT39K = 1.695819E + 05 hours, (19.35856 years)Calculation was done by virtue of computer system Zivot2, which was developed by

authors of this article. Calculation corresponds to 50% failure probability. This also takesinto account the influence of the mean value. Demonstration of the calculation is shown inFig. 3.

2.2. Assessment of the Service life by means of probability approach

Probability assessment is based on simulation of a response of loads, experimental determi-nation of fatigue curve and hypothesis chosen for the assessment of service life.

2.2.1. Simulation of the response of the load

Simulation of the response of the load is necessary because of keeping a stochastic characterof the response of the load. Important parts of the simulation are:

• method of the three parametric Rain Flow (RF),• generation of the two parametric matrices of RF,• backward transformation.

549


Figure 3. Assessment of the lifetime given by classical hypothesis.

2.2.2. Method of three-parametric rain flow (RF)

The method is composed from two parts. The first part is identical with two-parametricmethod (authored by N. E. Dowling, 1972), its description and principle is described, e.g.,in.9 The second part, which progresses similarly to the first one, includes recording the timesof amplitudes into files. Number of files with amplitude times corresponds with the numberof elements of the matrix. If the size of RF matrix will be [n, m], then number of the files willbe n ∗m. Times of particular amplitudes in cells correspond to the origins of rain running-down, (Fig. 4).

1

3

5

7

2

4

6

0.00

0.02

0.04

0.06

0.11

0.13

0.07

0.09

Time [s] σa [MPa] σa [MPa]

Left branch of RF Right branch of RF

Figure 4. Left and right branch of RF with marked numbers of particular amplitudes and appropriatetimes.

550


2.2.3. Generation of two-parametric matrixes of rain flow

We obtain RF matrix of a measured record by processing measured record by RF method. Inorder to create (simulate) a continuation of measured record it is necessary to obtain a newRF matrix, which is different from matrix source record responding probabilistic study. Thedifference in stochastic and deterministic access is given by various frequencies of amplitudesat a given mean value. Progress of RF matrix generation is as follows:

(a) Division of the matrix on separate mean values.(b) Evaluate the distribution function for every mean value (concerning only those, which

have frequency of the amplitudes nonzero).(c) On basis of this knowledge the distribution function generate stochastic numbers and

assigns them to those stochastic appropriate amplitudes, (Fig. 7).(d) Gradual generation of amplitudes and their inserting into new matrices. Generation runs

as long as frequencies of the amplitudes at given mean values of the new and originalmatrices are different.

(e) Result composition of generated matrices.

2.2.4. Backward transformation

Basis of backward transformation produce generated matrices. We create in reversal a simu-lated record of the stress from these matrices by following procedure:

(f) generation of new amplitudes,(g) assignment of the times to newly generated amplitudes,(h) arrangement of amplitudes according to the different times,(i) modification of the same times and alternation of maxima and minima.

Figure 5. Illustration of original and simulated record.

551


Generation of new amplitudes and assignment of times to newly generated amplitudesprogresses simultaneously. The assignment time progresses as follows:

(j) generation of appropriate amplitude and its mean value,(k) determination for which the components of the matrix are the generated amplitudes,(l) opening the file with times, belonging to given component of the matrix,

(m) random selection of one time instant,(n) assignment of time to generated amplitude.

The next step is re-creation of time sequence. One of the most effective methods forarrangement data upwardly according to size (in our case according to time) is Quicksortmethod,.13

Backward transformation carried out in terms of calculation of the service life assessmentthe most time-consuming. To make the calculation faster, a dynamic variables principle isused, and the whole signal is loaded to the RAM of computer. In this phase the velocityespecially depends on performance of processor of a computer. Illustration of original andsimulated record is in Fig. 5.

2.2.5. Fatigue curve

The best way on how to assess the service life it seems to determine the fatigue curve exper-imentally. Experimental determination of the curve is mostly limited by certain numberof measurements,10 by virtue of a limited set of points, which a regression curve meets.In case of logarithmic coordinates this curve possesses a linear character. By meeting thesepoints by a linear regression curve instead of the fatigue curve yields the result in logarithmicscale illustrated in Fig. 6.

3. Conclusion

In the first step the changes of mechanical properties arise from the structural changes occur-ring in the whole volume of the material during the cyclic loading. Depending on the cyclicdeformation resistance of the material, strengthening (resistance increase) or softening (resis-tance decrease) can take place. The processes leading to the changes of mechanical properties

log

N

Linear regression curve

Y= -1.4122x + 8.0742 R = 0.975

2

1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2

5.0

5.2

5.4

5.6

5.8

6.0

6.2

log [MPa]σ

Figure 6. Example of linear regression curve.

552


Figure 7. Lifetime assessment histogram and distribution function.

mentioned above are of the saturation type and occur during a relatively short time withregard to the life time of the material. The resulting structures in the saturation state dependon the stress amplitude and, from the structural point of view, especially on the cross slipfacility. In the course of the second stage, the processes initiated by an intensive plastic defor-mation particularly near the free surface of the material and leading to the nucleation offatigue cracks occur. Structural changes and corresponding changes in the surface relief arebriefly described and the assessment of the considered mechanisms of fatigue crack nucle-ation is carried out.

The third stage of the fatigue process is characterized by the propagation of cracks. Theclassification of the ways of fatigue crack propagation according to the mechanisms of theirgrowth and according to the dependence of the growth rate of fatigue cracks on stress inten-sity factor range is carried out. The final part of the lecture is devoted to the mechanisms offatigue crack propagation both in the initial stage of their growth (i.e. during the first stageof propagation) and during the second stage of propagation which characterized especiallyby the formation of fatigue striations. Attention is also paid to the effect of the structure(particularly of the cross slip facility) on the way of fatigue crack propagation. Both accessesof assessment of the service life are close to real life of the construction detail. In real life ofthe bridge there were found cracks in expected positions, (Fig. 7). First type (classical) is veryfast preview on the lifetime. This access is easy to use, it is very fast, but without any chanceto include more loads into calculation or to determine percentage probability of crack. Sec-ond approach is more complex, it allows including of other load factors into calculation,variability of traffic loads during time, etc. Probability calculation has higher requirementson PC performance.

Our result is that “classic” form of assessment of the service life is recommended onlyin those cases, when we have only one type of accidental load (but not combination ofaccidental loads) and only in such cases, when form of the stressed part corresponds totabular classification according to current norms.

Acknowledgements

This paper was prepared under financial support by Grant Agency of the Czech Republic No103/08/0922.

553


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3. Akgul, F. & Frangopol, D.M., Lifetime Performance Analysis of Existing Steel Girder Bridge Super-structures, J. Struct. Engrg., Vol. 130, Issue 12, pp. 1875–1888, 2004.

4. Alampalli, S. & Lund, R., Estimating Fatigue Life of Bridge Components Using Measured Strains,J. Bridge Engrg., Vol. 11, Issue 6, pp. 725–736, 2006.

5. MacDougall, C., Green, M.F. & Shillinglaw, S. Fatigue Damage of Steel Bridges Due to DynamicVehicle Loads, Journal of Bridge Engineering, Vol. 11, No. 3, pp. 320–328, 2006.

6. Neves, D.M., Luis, A.C. & Frangopol, D.M., Lifetime multi-objective optimization of mainte-nance of existing steel structures, International symposium on steel bridges, Constructional Steel-work Association, “Steel bridges: proceedings”, Prague 2006.

7. Skála, O., Hejman, M. &, Papuga, J., Hodnocení únavové životnosti pøi komplikovanémpùsobení mechanického namáhání a teplot, Computational mechanics 2006, 22nd Conferencewith International Participation, Czech Republic, 2006.

8. De Jong, F.B.P. & Boersema, P.D.E, Lifetime calculation for orthotropic steel bridge decks, Min-istry of Transport, Civil Engineering Division, Section Steel & Mechanical Engineering, Nether-lands, 2004.

9. Matsuiski, M., Endo, T.: Fatigue of metals subjected to varying stress, Japan Soc. Mech. Engrg,1969.

10. Culek B. jr., Culek B.: Determination of non-traffic loads of the Czech Railways bridges, Reliabil-ity, Safety and Diagnostic of Transport Structures and Means 2005, 2nd International Conference,Pardubice, 2005.

11. Bericht ERRI C 178/RP2 Vereinheitlichung der Methoden der Lebensdauerbestimmung dynamischbeanspruchter Bauteile, Utrecht, 1992.

12. Frýba, L., Dynamics of railway bridges, Thomas Telford, London 1996.13. Culek, B., Evaluation of fatigue life of steel structures under complex stress, Ph.D. thesis, Pardu-

bice, 2003.14. Doležel, V., Procházka, P.: Back analysis of underground structures. In: 11th Intern. Conference

RB 07, CI Premier, 14-17 November 2007, pp. 362–369, North Cyprus, 2007.

554

An Analysis of Dynamic Tensile Fracture in Concrete Under HighStrain Rate

M. KURUMATANI1, S. IWATA2, K. TERADA1, S. OKAZAWA3,∗ AND K. KASHIYAMA2

1Department of Civil and Environmental Engineering, Tohoku University2Department of Civil Engineering, Chuo University3Department of Social and Environmental Engineering, Hiroshima University

1. Introduction

Since concrete specimens under high strain-rate exhibit quite different mechanical behav-ior from those under quasi-static loading (low strain-rate),1−3 it is commonly recoginizedthat the overall or apparent material behavior of concrete is to be distinguished for dynamicand quasi-static problems. Also, the dynamic responses observed in experiments depend onthe rate of loading; that is, the overall peak strength and overall elastic modulus measuredfrom the speciems subjected to high strain-rate are greater than those under low strain-rate.4,5 In addition, the failure modes or the fracture patternsvary depending on strain-rates.4

Fig. 1 shows the representative examples of crack patterns — cracks pass through the mor-tar and propagate along the interfaces under low strain-rate while flatted cracks penetratethrough the aggregates under high strain-rate. It is thus a common practice that the concreteis regarded as a rate-dependent material. Accordingly, the numerical studies, which havebeen made to characterize the overall material behavior of concrete, rely on visco-elastic orviscoplastic type constitutive models to represent the rate-dependency.6−8 The use of rate-dependent models seems to make consistent with the specimens’ overall responses observedin experiments, but makes us to close our eyes to the underlying meso-scale mechanisms thatreproduce rate-dependent fracture behavior of concrete.

(a) Failure mode at low strain-rate (b) Failure mode at high strain-rate

Figure 1. Difference of failure modes between low and high strain-rates.

On another front, a key to understanding the deformation and strength characteristicsof concrete subjected to dynamic loading is thought to be the clarification of the fractureprocess with cracking under tensile loading. Although numerous experimental studies havebeen made for the investigation of the tensile fracture processes of concrete, the tensile teststends to provide the scattering results and, as a result, few reliable data are available in theliterature. This is probably due to the fact that the overall tensile strength of concrete spec-imens is extremely sensitive to the local tensile strength and heterogeneities. Thus, reliable




crack simulations of mesoscale heterogeneous structure of the concrete, which is composedof mortar matrix, aggregated inclusions and the interfaces, seem to be promising to find outif the constitutive responses under dynamic and quasi-static loading are essentially the same.That is, it might be possible to reproduce the rate-dependent mechanical behavior of concretespecimens without introducing rate-dependent constitutive models.

In this context, care should be exercised in crack simulations of concrete, since it is a quasi-brittle material, unlike pure brittle material such as glass. One of the main characteristics isthe formation of a fracture process zone (FPZ) that involves softening behavior due to micro-crack interactions. To represent the softening behavior, the so-called cohesive crack model9

is commonly introduced at fictitious fractured surfaces and is dispensable for the reliablesimulations of the fracture process, or equivalently crack nucleation and propagations, inconcrete’s meso-scale heterogeneous structure.

In this paper, in order to examine the strain-rate dependency, we perform the numericalsimulations for the tensile dynamic fracture behavior of concrete’s meso-scale structure. Theconcrete is regarded as a three-phase composite composed of mortar matrix, aggregatedinclusions and the interfaces, and its heterogeneity is explicitly considered in our numericalmodels. In the suggested analysis method, the standard dynamic explicit code for the FEMis used to solve the equations of motion and, at the same time, the discrete crack model isemployed together with the cohesive crack model. A numerical example is carried out toexamine the strengthening effects and failure modes obtained withlow and high strain-rates.

2. Dynamic Fracture Analysis Using Cohesive Crack Model

In this section, we present a method for dynamic fracture analysis of quasi-brittle materialsand a modeling method for propagating cracks. After explaining the finite element analysisof discrete cracks together with the cohesive crack model, an approximation procedure ofquasi-brittle softening behavior based on the cohesive crack model and the details of thediscrete crack simulation are illustrated.

2.1. Cohesive crack model

Concrete is a quasi-brittle material, involving softening behavior during the process of frac-ture, which is characterized by the formation of fracture process zone (FPZ) as shown inFig. 2(a). The FPZ is formed between the unbroken elastic domain and the fully separatedstress-free fractured domain, and the stress is transmitted in the FPZ due to interlocking ofaggregates and micro-crack interactions. The softening behavior in the FPZ is realized bythe decrease of the stress transfer due to the increase of the crack opening displacement.The cohesive crack model is one of the major fracture mechanics model to represent suchsoftening behavior of concrete. The stress transfer in the FPZ is substituted for the cohe-sive traction force between two fictitious fractured surfaces facing with each other as shownin Fig. 2(a). The relation between the cohesive traction andthe crack opening displacement(traction–separation law) is namedasa tension softening curve, and the area under the curveis called the fracture energy, which is also defined as the fracture energy necessary to create aunit crack area.

The traction–separation law prescribed in this study, in which the tensile strength and thefracture energy are introduced as material parameters,10, 11 is given by

‖tcoh(w)‖ = ft exp(− ft

Gtw)

on �PZ (1)

556


Traction-free crackFracture Process ZoneElastic domain

Γ PZΓ EL

κ (x) := || g ||

u [a]

u [b]

t coh

κ

Gf

ft

t coh

t coh

g

= pcoh gt coh

n

t coh ( )κ

(a) Schematic of fracture process zone and the cohesive crack model (b) Spring-based cohesive crack model

Figure 2. Fracture mechanics model for quasi-brittle materials.

where ‖tcoh(w)‖ is the norm of cohesive traction vector and w is a history parameter ofthe gap displacement ‖u[1] − u[2]‖ ever experienced during the loading process. Here, ft isthe tensile strength of material, which is the critical stress to form the FPZ, and Gf is thefracture energy, which is the energy dissipation per unit crack area. Since the original cohesivecrack model has been proposed for the analysis of statically propagating cracks, the tensilestrength ft and the fracture energy Gf are assumed to be rate-independent. Thus, the rate-independent fracture mechanics model is employed for local material response, whereas theoverall structural response is obtainedby solving the equationof motion for the specimen.

2.2. Application of the cohesive crack model to dynamic fracture analysis

In the original cohesive crack model, the cohesive traction is treated as a kind of dynamicboundary condition of distributed external loading, and the nonlinear softening behavior isapproximated by iterative convergent calculation.11 This conventional procedure, however,leads to numerical instability because the dynamic fracture simulations involve complicatedcracking behavior. In this study, in order to realize eficient and stable computations, werewrite the cohesive traction as a reaction force of cohesive spring expressed in the followingequation:

tcoh = pcohg = pcoh(u[a] − u[b]) (2)

where prmcoh is the nonlinear spring stiffness between cracked surfaces, which is given by

pcoh = ‖tcoh‖‖g‖ (3)

which can be approximated by means of the relative displacement of the previous time stepand is updated only when the crack opening displacement increases. Therefore, the dynamicfracture simulation is expected to be performed with adequate accuracy, when we employthedynamic explicit solution method with small time increment.

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2.3. Weak form of the dynamic fracture analysis using cohesivecrack model

In this paper, the discrete crack model, in which discontinuous deformation is expressed asthe separation of neighboring element edges, is employed for the modeling of progressive fail-ure. To be more specific, the penalty method is applied to impose the interfacial constraintsat the FE edges with double nodes. Figure 3(a) illustrates the schematic of our methodol-ogy. The penalty spring p0 has been used until the fracture is generated. Once the fractureoccurs, the penalty spring p0 is replaced by the cohesive spring pcoh, which represents thequasi-brittle softening behavior in the FPZ.

The week form for dynamic FE analysis using the penalty method and cohesive crackmodel is given by∫

�

ρδuu d�+∫�

∇δu:σ d�+∫�P

p0 δg · g d� +∫�PZ

pcohδg · g d�

=∫�

δu · b d�+∫�

δu · t d� ∀δu (∀δg) (4)

where ρ is the material density, u is the acceleration vector, δu is the virtual displacementvec-tor, σ is the Cauchy stress tensor, b is the body forcevector, t is theexternal tractionvector,� isthe whole domain, �t is the Neumann boundary, �p is the boundary where the penalty springis inserted and �PZ is the FPZ; that is the boundary where the cohesive spring is introduced.

2.4. Modeling of discrete crack propagations

The characteristics of dynamic fracture in concrete is that quite a few cracks and interfacialdebondings are generated over a wide range in the specimen. Since it is impractical to considerall these discontinuities as they are, we employ the conventional discrete crack approach inthis paper, in which cracks propagate along only finite element edges as schematized in Fig. 3.Here again, the penalty spring is used before fracture and the cohesive spring is applied afterthe fracture begins.

To perform the progressive failure analysis, we need to introduce the fracture criteria forinternal and interfacial fracture. In this paper, since the penalty springs are inserted along FEinterfaces, the traction vector can approximately be calculated by the product of the penaltyparameter and the relative displacement vector as follows:

tP = p0 g (5)

p coh

p0

Finite element mesh Penalty springs are inserted before fracture Cohesive springs are introduced after fracture

Crack path

Figure 3. Discrete crack approach with the penalty and the cohesive springs.

558


We determine the generation and propagation of internal cracking and interfacial debondingby using the traction vector tP. Discontinuities are formed when the traction tP reaches thefracture criteria postulated as

tP · n > 0 and‖tP‖ ≥ ft (6)

where n is the outward unit normal vector at FE interface.


A numerical example is presented to examine the influence of strain-rates on the rate-dependent fracture behavior in concrete by applying the suggested analysis method todynamic fracture simulation of concrete’s meso-scale structure; see Ref. 12 for the referencedexperiment.

We simulate dynamic tensile fracture of concrete’s test specimen subjected to low and highstrain-rates as shown in Fig. 4. The concrete is regarded as a meso-scale three-phase compos-ite composed of mortar matrix, aggregated inclusions and the interfaces, and its heterogeneityis explicitly considered in our numerical models. The material parameters of each phase areprescribed by educated guess; in particular, the tensile strengths of aggregate, matrix andinterface, f a

t , f mt and f i

t , are ordered as f at > f m

t > f it . The displacement are constrained

along the bottom and right edges, and the deformation rates are applied at the top surface.Either low or high deformation rate is applied and is gradually increased from zero as shownin Fig. 5(a) to prevent the impact shock loading. Although the modeling of contact or fric-tion is not incorporated, there is no possibility that the release force due to fracture causesoverlapping of separated segments because of the introduction of cohesive springs.

First, we present the relationship between the percentage of fractured surfaces and the dis-placement at loading position in Fig. 5(b). This graph illustrates that the amount of generatedcrack sunder high strain-rate is much more than that under low strain-rate. This is identicalwith the experimental observations in Ref. 4 and implies the validity of the suggested method.Fig. 5(b) also shows the overall load–displacement curve. As can be seen from this figure, thepeak strength evaluated with high strain-rate is greater than that with low strain-rate. Thesame applies to the overall elastic response; that is, the overall elastic modulus estimated with

23 mm

40 m

m

Velocity

5x5mm

Density

(kg/m3)

Young's modulus

(MPa)

Poisson's ratio

Matrix Inclusions

20000 40000

2000 2500

0.100.10

Tensile strength

(MPa)

Fracture energy

(N/mm)

Matrix Inclusions Interfaces

3.5 6.0 3.0

0.03 0.0003 0.03

Figure 4. Meso-scale structure of concrete and material parameters.

559


Displacement at loading surface (mm)

0 0.01 0.02

0

1

2

3

4

0

10

20

Rea

ctio

n f

orc

e /

load

ing s

urf

ace

(M

Pa) P

ercentag

e of fractu

red su

rfaces (%)

Low rate

High rate

60 MPa

: Low rate

: High rate

6 M

Pa

Time (sec)

Def

orm

atio

n r

ate

(m

m/s

ec)

0 0.001

0

10

20

0.0030.002

Low rate

High rate

0.2 mm/sec

20 mm/sec

(a) Time history of applied loading-rates (b) Load-displacement curve with the number of local cracking

Figure 5. Time history of loading rate and load–displacement curve with the number of local cracking.

0.0

6.0

(MPa)

Displacement at top surface

: 0.0083 mm

Displacement at top surface

: 0.0159 mm

(a) Simulation results at low strain-rate (b) Simulation results at high strain-rate

Figure 6. Fractured configuration with principal stress in meso-scale structure.

high strain-rate is 1.1 times as high as that with low strain-rate, which is consistent with theresult reported in Ref. 5.

It is, therefore, confirmed that the macros copic rate-dependent mechanical responses areproperly simulated by solving the equation of motion together with the cohesive crack model.Particularly it should be emphasized that no rate-dependent parameters is introduced in our

560


numerical analysis. At the same time, it can be seen from the results that the softening incli-nation in the case of high strain-rate is steeper than that of low strain-rate. This is probablydue to the fracture of aggregates.

Figure 6 shows the principal stress distributions together with crack patterns in the casesof low and high strain-rate, respectively. When the deformations become large, the crackswith low strain-rate propagate within the mortar phase and along the interfaces, while thecracks with high strain-rate penetrate through the aggregates. In addition, the formed crackpath with high strain-rate is flatter than that with low strain-rate, because the crack has littletime to seek the path of least resistance at high strain-rate. These results are in agreementwith those reported in Ref. 4, too.


We have performed the numerical simulations for dynamic tensile fracture behavior of con-crete’s meso-scale structure, and examined the rate-dependent fracture behavior in concrete.The analysis method is based on dynamic explicit code for the FEM, which is incorporatedwith the discrete crack model together with the cohesive crack model. The numerical exampledemonstrates that the strengthening effects and the different failure modes with high strain-rate is properly simulated without anyrate-dependent material model. In order to simulatethe rate-dependent fracture behavior in concrete, we have solved only the equation of motiontogether with the cohesive crack model for quasi-brittle materials. Consequently, it is quitelikely that the origin of rate-dependency is not the material responseof concrete,but rather itsstructural response.

Acknowledgements

This research is supported by “The Ministry of Education, Culture, Sports, Science and Tech-nology, Grant-in-Aid for Scientific Research B-19360207”.

References

1. Mindess S., Banthia N., Yan C., “The fracture toughnessof concrete under impact loading”, Cem.Concr. Res., 17, 1987, pp. 231–241.

2. Ross C.A., Tedesco J.W., Kuennen S.T., “Effects of strain rate on concrete strength”, ACI MaterialJournal, 92, 1995, pp. 1–11.

3. Malvar L.J., Ross C.A., “Review of strain rate effects for concrete in tension”, ACI MaterialJournal, 95, 1998, pp. 735–739.

4. Yan D., Lin G,. “Dynamic properties of concrete in direct tension”, Cem. Concr. Res., 36, 2006,pp. 1371–1378.

5. Rossi P, Toutlemonde F,“Effect of loading rate on the tensile behaviour of concrete: description ofthe physical mechanisms”, Mater. Struct., 29, 1996, pp. 116–118.

6. Zheng D., Li Q., “An explanation for rate effect of concrete strength based on fracture toughnessincluding free water viscosity”, Engng. Fract. Mech., 71, 2004, pp. 2319–2327.

7. Georgin J.F., Reynouard J.M., “Modeling of structures subjected to impact: concrete behaviourunder high strain rate”, Cem. Concr. Compos., 25, 2003, pp. 131–143.

8. Pedersen R.R., Simone A., Sluys L.J., “An analysis of dynamic fracture in concrete with a contin-uum visco-elastic visco-plastic damege model”, Engng. Fract. Mech., 75, 2008, pp. 3782–3805.

9. Hillerborg A., Modéer M., Petersson P.-E., “Analysis of crack formation and crack growthin concrete by means of fracture mechanics and finite elements”, Cem. Concr. Res., 6, 1976,pp. 773–782.

561


10. Wells G.N., Sluys L.J., “Anew method for modelling cohesive cracks using finite elements”,Int. J.Numer. Meth. Engng., 50, 2001, pp. 2667–2682.

11. Kurumatani M., Terada K., “Finite cover method with multi-cover-layers for the analysis of evolv-ing discontinuities in heterogeneous media”, Int. J. Numer. Meth. Engng., 79, 2009, pp. 1–24.

12. Fujikake K., Uebayashi K., Ohno T., Emori K., “Study on dynamic tensile softening characteristicof concrete material under high strain-rates”, http://library.jsce.or.jp/jsce/open/00037/2001/669-0125.pdf.

562

A New Equivalent Medium Model for P-Wave PropagationThrough Rock Mass with Parallel Joints

G.W. MA∗, L.F. FAN AND J.C. LI

School of Civil and Environmental Engineering, Nanyang Technological University, Singapore 639798

1. Introduction

Joints commonly exist in rock mass, which often occur as nearly parallel groups or sets. Suchset (or sets) of joints often control the hydraulic and mechanical behaviour of a rock mass. Ifa wave propagates through a jointed rock mass, its amplitude will be greatly attenuated (andslowed) due to the presence of the set (or sets) of joints. Thus, a study of wave propagationthrough jointed rock mass is of great importance to mine stability, waste isolation, predictionof earthquake motions and assessing the damage of rock structures under dynamic loads.

A usual method for analyzing the wave propagation through jointed rock mass is theequivalent medium method (EMM), which treat problems from a viewpoint of entirety andare utilized usually for highly jointed medium, in which the intact rock block and the con-tained joints together are treated as an equivalent continuous medium. Usually, the equiva-lent medium methods relate averages of constitutive variables, e.g. average strain and aver-age stress. Thus, a representative elementary volume (REV) is assumed, and the effectivemoduli are seeded to predict the aggregate effects of joints in the REV,1 so as to make acontinuum analysis of the jointed media practicable. Using a static approach, Zhao et al.2

developed an equivalent medium model to explain the wave phenomena when the incidentwavelength is substantially greater than the fracture spacing. It is observed that the pres-ence of joints resulted in anisotropy in the material properties of a rock mass, therefore,Schoenberg3 derived constitution expressions of a transversely isotropic medium to repre-sent a rock mass containing a single set of plane parallel fractures by a dynamic approach.In their works, the constitution of the effective medium model contained five effective elasticconstants in terms of the elastic properties of the intact rock, the fracture spacing, and thefracture specific stiffness. Hence, the wave velocities would be obtained by the equivalentmedium without fractures. Thomsen4 derived the expression of the phase velocity as a func-tion of the angle of incidence, and a comparison of the normalized group velocity with thedisplacement discontinuity model was carried out. However, as mentioned by Cook,5 theeffective moduli models do account for the effects of joints on seismic velocities, but theycannot account for their effect on attenuation. The effective moduli methods limit to theeffective elastic moduli of the rock mass, which is effective only if the frequency-dependenceis ignored.

Pyrak-Nolte6 recommends the no longer purely elastic equivalent medium, e.g. viscoelasticequivalent medium, to consider the attenuation. Johnston et al.7 investigated the attenuationof the seismic waves propagation through the dry and saturated rocks. Both the mecha-nism studies and Laboratory measurements are carried out. Pyrak-Nolte6 found that the realjoints can be expected as the interface possess elastic coupling with viscous by the labora-tory experiments on seismic wave transmission across natural joints. The deformation wavesof arbitrary shape propagation through viscoelastic solids have been reviewed by Kolsky,8

both the experimental and theoretical aspects of the stress wave propagation has been inves-tigated. However, how to introduce the viscoelastic equivalent medium model to represent




the parallel jointed rock mass and how to identify the material parameters of the viscoelasticequivalent medium model from the jointed rock mass and how to describe the attenuationusing the viscoelastic equivalent medium model have not yet to be introduced.

The purpose of this paper is to propose an equivalent medium model for a parallel jointedrock mass. This model is a viscoelastic continuum medium, which is suitable for the dynamicinvestigation of rock mass under a dynamic load. And it can be used to consider the effec-tive moduli of the jointed rock mass, the effective velocities of the stress wave propaga-tion through the rock mass and the attenuation of the stress wave across the jointed rockmasses. In order to identify the parameters of the present model, the current solution forstress wave propagation through a rock mass with single joint is used. The transmitted waveforms obtained from the present model are compared to the results for the displacementdiscontinuity method for theoretically verification.

2. Viscoelastic EMM for Rock Mass with Parallel Joints

The investigation of Cook5 also shows that the dynamic approach based on elastic equiva-lent medium model cannot account for attenuation of the stress wave propagation throughjointed rock mass. Therefore, a viscoelastic EMM introduced to eliminate the limitations ofthe traditional elastic EMM.

Figure 1 shows the viscoelastic equivalent medium model. The model can be regarded asan auxiliary spring placed in series with the Voiget model, which will be used to display eitherthe attenuation or the frequency-dependence of the transmitted wave. The constitution of themodel can be obtained as below

(Ea + Ev)σ + ηv∂σ

∂t− ηvEa

∂ε

∂t− EvEaε = 0 (1)

where σ is stress, ε is strain, Ea and Ev are the elastic moduli of the springs; ηv is the viscosityratio.

The equation of the longitudinal motion for a one-dimensional problem can be expressedin the term of velocity as

ρ∂v∂t= ∂σ

∂x(2)

Meanwhile, from the harmonious equation,

∂ε

∂t= ∂v∂x

(3)

vE

v�

aE

Figure 1. Viscoelastic equivalent medium model.

564


So that differentiating Eq. (1) with respect to x and t, then substituting for σ from Eq. (2)and substituting for ε from Eq. (3), we obtained

ρηv∂3v

∂t3+ ρ(Ea + Ev)

∂2v

∂t2− ηvEa

∂3v

∂x2∂t− EvEa

∂2v

∂x2= 0 (4)

with an arbitrary incident wave, at the boundary x = 0 as fellow,

v (x,t) |x=0 = v0 (t) (5)

The application of the Fourier Transformation to the motion Eq. (4) and the boundaryEq. (5) yields, [

−ω2ρηv + iωρ(Ea + Ev)]

v (ω)−[ηvEa − i

ωEvEa

]v,xx (ω) = 0 (6)

and

v (x,ω) |x=0 = v0 (ω) (7)

respectively, where we use the velocity in the frequency domain v(ω) to describe FourierTransformed time domain velocity. And ω means the frequency. i denotes imaginary sign,as we have i2 = −1. The comma in the Eq. (6) denotes the differential to x. v0 (ω) is theboundary condition of v0 (t) after Fourier Transformation.

Solving Eq. (6) on the boundary condition Eq. (7) and consider the convergence of thesolutions. The results can be written as

v(x,ω) = v0(ω) exp(−√B/Ax

)(8)

where B = −ω2ρηv + iωρ (Ea + Ev) and A = ηvEa − iEvEa/ω.An inverse Fourier Transformation will be applied to obtain the velocity in the time

domain,

v(x,t) = 12π

∫ ∞−∞

v0 (ω) exp(−√B/Ax

)exp (iωt)dω (9)

In Eq. (9), v0(ω) is obtained from the transformation of the arbitrary function v0(t), so themethod can be applied to solve the motion problems with any shapes of the incident wave.Furthermore, no assumption is introduced during the procedure, so no additional errors willbe introduced.

3. DDM for Rock Mass with Parallel Joints

A displacement discontinuity method (DDM) is introduced here to theoretically verify thepresent EMM. DDMs have been applied to study the wave propagation through jointedrock mass by several investigators. The essential assumption of these methods is that stressesacross the interface of the joint are continuous, but the displacements across the interface arediscontinuous.

Pyrak-Nolte et al.6 obtain the transmission coefficient for a P-wave at normal incidence,with equal material properties on either side of the joint is

T(ω) = 2k/z−iω + 2k/z

(10)

where z is the wave impendence, define as z = ρC, ρ is density of the intact rock and C is theP-wave velocity in the intact rock.

565


Therefore, if an incident P-wave is applied at the left boundary of rock mass with the formof u = u0 exp (− iωt) (assume at x = 0), the transmitted wave after the rock mass (assume atx = S) with one joint can be derived and written as

u = 2k/z−iω + 2k/z

u0 exp (−iωt+ inS) (11)

When a wave incident to a rock mass with multiple joints, it is recognized that the multi-ple reflection has significant effect on the transmitted wave forms. However, it is difficult toexplicitly determine the superposition of the reflected waves. One of the simplified methodswas by ignoring the multiple reflections as an approximation. Laboratory experiments car-ried out verified that this approximation is valid, when the joint space is larger relative tothe incident wave-length, in that case, the firstly arriving wave is not contaminated by themultiple reflections.6

4. Parameter Studies of Viscoelastic EMM

Equation (1) indicates that the dynamic terms (the strain rate and stress rate terms) onlyrelated to the damping ratio ηv. That means the elastic parameters are independent on themagnitude of the loading ratio. Thus, for a quasi-static loading procedure, the stress can beobtained from Eq. (1) as

σ = EvEa

(Ea + Ev)ε = Ecε (12)

where Ec is written for the moduli for the two springs in series, so that 1/Ec = 1/Ea + 1/Ev.The comparison between a representative element of rock mass and a representative ele-

ment of the equivalent medium is made. Figure 2 shows a representative element in a rockmass with parallel joints. S is the initial length of the total representative element. The normalstrain is

ε = SS= Sj +Sr

S(13)

where S denotes the deformation of the total element, Sj is the closure of the joint andSr is the deformation of the rock, respectively.

Based on the in series properties and the stress expressions of the joint and the intact rock,and consider the suggestion by Li and Ma9 the constitution of the representative element of

a b

Joint

Equivalent length

S

Rock

��

Figure 2. Representative element of linear jointed rock mass.

566


rock mass can be obtained as

σ = knSEa

knS+ Eaε (14)

Where, kn is the normal stiffness of the joint, E is the Young’s moduli, the subscript j meansthe properties of the joint and r denotes the properties of the rock.

Eq. (12) is derivate from the discontinuous rock mass, which can be represent by thecontinuous equivalent medium. Therefore, it should have the same stress-strain relationship.After the comparison of Eqs. (12) and (14), we can obtained,

Ev = knS (15)

In order to achieve the damping ratio ηv, the characteristic method will be used to analyzethe Eqs. (1)–(3). The velocity of the P-wave attenuates following

v = v∗ exp(−ρCv

2ηx)= v∗ exp (−ςx) (16)

where, Cv is the wave velocity along the characteristic line, which can be expressed as C2v =

Ea/ρ, v∗ = v(0,0) is wave velocity at the initial time on the boundary x = 0 and ς is theattenuation factor, which is defined as

ς (ηv) = −1S

ln

⎡⎣ 2k/z√

ω2 + (2k/z)2⎤⎦ (17)

Therefore, the attenuation factor can be measured from the wave propagation experimentor can be simulated from the theoretical study as the works done by Pyrak-Nolte et al.6 orCook.5 Meanwhile, the density ρ and the Young’s moduli of the spring Ea and Ev can beobtained by the regular statics experiment. Thus, the damping ratio can be obtained fromEq. (17).


Based on the above discussion, it can seen that the incident wave v (0, t) can be any arbitraryfunctions. Assume a half-cycle sinusoidal wave is applied as the incident wave at the left sideof a rock mass. e.g.

vI(t,0) ={

0.02 sin (100πt)0

, when0 ≤ t ≤ 0.01

others(18)

where 0.02 is the amplitude of the incident wave. 100π is the angular frequency of theincident wave.

In the present numerical examples, it is assumed that rock density ρ is 2650Kg/m3, P-wavevelocity C is 5830m/s, the normal joint stiffness kn is 3.5GPa. The Young’s moduli of theintact rock Ea = 90GPa. Fig. 3(a) and 3(b) respectively show the relationships between Evand ω, ηv and ω, when S is λ, 2λ, and 5λ. It is observed from two figures that either Ev orηv depend on the incident wave frequency and the joint spacing S. For a given S, Ev and ηvdecrease with the increasing ω. And for a given ω, Ev and ηv increase with the increase of S.

If an incident P-wave at the boundary with the form of Eq. (18) propagates through a rockmass. The transmitted waves can be obtained by the present viscoelastic EMM and DDM,respectively. Figure 4 shows the transmitted waves when the incident wave across the jointedrock mass using the two different methods. The “. . . ” curves denote the incident wave with

567


0 50 100 150 200

0

2000

4000

6000

8000

Ev (

GP

a)

Frequency ω (× 2π , Hz)

S=5λ

S=2λ

S=1λ

0 50 100 150 200

0

20

40

60

80

100

120

ηv (

GP

a)

Frequency ω (× 2π , Hz)

S=5λ

S=2λ

S=1λ

(a) Ev ~ ω (b) ηv ~ ω

Figure 3. The relation between parameters Ev, ηv and frequency ω.

the form of Eq. (18), the “—◦—” curves denote the transmission waves obtained by theviscoelastic EMM and the“——” curves denote the transmission waves from DDM.

6. Discussions

6.1. Comparison of transmitted waveforms

The transmitted waveforms obtained by using viscoelastic EMM are compared with thoseby using DDM, with the same incident wave propagates through a rock mass containing aset of joints. Figure 4(a)–(d) show the results for different joints numbers, respectively. It isfound from the figures that the waveforms of transmitted waves by two different methods arebasically the same. The comparison proves the validity of the present EMM can effectivelydescribe the dynamic property of the rock mass with a set of joints subjected to a normalincident wave.

The compassion procedure also shows that the present viscoelastic EMM is a more effec-tive method to obtain the transmitted waveforms than the DDM, especially when the rockmass contains multiply joints. For an example, when the rock mass with n joints. In orderto obtain the transmitted waveforms by the DDM, each transmitted waveform across indi-vidual joint should be calculated, therefore, it will cost n steps until the final transmissionwaveforms be obtained. However, for the present viscoelastic EMM, which consider thejointed rock mass as an entire continuous medium. Therefore, it can obtain the transmittedwaveform as simplily as the rock mass contain only single joint.

6.2. Comparison of transmission coefficients

The transmission coefficient is defined as the ratio of the amplitude of the transmission waveto the amplitude of the incident wave. It can be seen from Fig. 4 that the transmitted waveshave an obvious attenuation on the amplitude of the incident velocity.

When an incident wave in the form given by Eq. (18) with f = 50Hz propagates across arock mass with a set of linear deformational joints. The Transmission coefficients obtainedby the DDM and by the present viscoelastic EMM for the rock mass with one to four jointsare obtained, respectively. A comparison of the transmission coefficients is carried out. Theresults are listed in Table 1.

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0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.000

0.005

0.010

0.015

0.020 Incident wave

Transmitted wave (by EMM)

Transmitted wave (by DDM)

Velo

city (

m/s

)

Time (s)

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.000

0.005

0.010

0.015

0.020 Incident wave



Velo

city (

m/s

)

Time (s)

(a) Single joint (b) Two joints

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.000

0.005

0.010

0.015

0.020 Incident wave



Velo

city (

m/s

)

Time (s)

0.00 0.01 0.02 0.03 0.06 0.07 0.08 0.09 0.10 0.11 0.12

0.000

0.005

0.010

0.015

0.020 Incident wave



Velo

city (

m/s

)

Time (s)

(c) Three joints (d) Four joints

Figure 4. Transmission waveforms by EMM and DDM with different joint number (S = λ).

Table 1. Comparison of transmission coefficients.

Joint numbers 1 2 3 4

EMM 0.8365 0.7160 0.6225 0.5545DDM 0.8430 0.7235 0.6300 0.5575Difference (%) 0.7770 1.0475 1.2048 0.5410

It can be seen from Table 1 that the transmission coefficients for a rock mass with singlejoint is about 0.84, and for a rock mass with multiply joints, it fellows the |T|N-method. Italso can be seen from Table 1 that transmission coefficients obtained from EMM are veryclose to the results from DDM.

6.3. Comparison of effective velocities

The effective velocity for stress wave is a function of the length of the medium to the timedifference for the two peak velocities of incident and transmitted waves. Assume the incident

569


wave in the form of Equation (18) with f = 50Hz and the joint spacing is 1λ. From thewaveforms in Fig. 4, the effective velocities can be obtained.

For the present viscoelastic EMM, the effective velocity Ce for single joint case is5336.4m/s, for two joints case is 5360.9m/s, for three joints case is 5348.6m/s, and forfour joints case is 5354.8m/s. The difference is caused by the errors introduced during thenumerical process. Therefore, an average effective velocity Ce is proposed for the presentviscoelastice EMM, which is Ce = 5350.2m/s. Agree well to the effective velocity value inreference6 (DDM) where the interaction between joints and multiple reflections are ignored.The difference between the effective velocity based on EMM and DDM is less than 1.3%.

7. Conclusions

A viscoelastic equivalent medium model is proposed in the present paper for parallel jointedrock mass. In the present EMM, the linear visco-elastic property of the medium is considered,and the time delay property and the attenuation of the transmitted waves are considered. Thefollowing results can be concluded,

• By comparing the wave propagations through a rock mass with one joint and thecorresponding equivalent medium respectively, the visco-elastic parameters in thenew model can be theoretically estimated.• Meanwhile, the frequency-dependent of joint are obviously shown in the present

viscoelastic equivalent medium model.• By comparing the effective velocity of P-wave through a jointed rock mass using the

DDM and the present viscoelastic EMM respectively, it can be seen that the presentviscoelastic EMM can be used to calculated the time delay by the joint, while thetraditional static approaching cannot.• By comparing the transmitted waveforms of P-wave through a jointed rock mass

using the DDM and the present viscoelastic EMM respecitively, it can be seen thatthe attenuation can be taken into account in the present viscoelastic EMM, whilethe effect is lost in the previous effective medium model (both elastic static approachand elastic dynamic approach).• It can be seen from the verification of the new model that the present viscoelastic

EMM can replace the displacement discontinuity method to study the wave propa-gation in a rock mass with parallel joints.

Above all, a viscoelastic EMM is proposed in this paper using a continuous medium todescribe a discontinuous medium, which simplifies the dynamic response of parallel jointedrock mass and makes the wave propagation solutions explicit.

References

1. Singh, B., “Continuum Characterization of Jointed Rock Masses”, International Journal of RockMechanics and Mining Sciences, 10, 1973, pp. 311–349.

2. Zhao, X.B., Zhao, J. and Cai, J.G., “P-wave Transmission Across Fractures with Nonlinear Defor-mational Behaviour”, International Journal for Numerical and Analytical Methods in Geomechan-ics, 30, 11, 2006, pp. 1097–1112.

3. Schoenberg, M., “Reflection of Elastic Waves from Periodically Stratified Media with InterfacialSlip”, Geophysical Prospecting, 31, 1983, pp. 265–292.

4. Thomsen, L., “Weak Elastic Anisotropy”, Geophysics, 51, 10, 1986, pp. 1954–1966.

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5. Cook, N.G.W., “Natural Joint in Rock: Mechanical, Hydraulic and Seismic Behaviour and Prop-erties under Normal Stress”, International Journal of Rock Mechanics and Mining Sciences, 29, 3,1992, pp. 198–223.

6. Pyrak-Nolte, L.J., Myer, L.R. and Cook, N.G.W., “Anisotropy in Seismic Velocities and Ampli-tudes from Multiple Parallel Fractures”, Journal of Geophysical Research, 95, B7, 1990,pp. 11345–11358.

7. Johnston, D.H., ToksOz, M.N. and Timur, A., “Attenuation of Seismic Waves in Dry and SaturatedRock, II: Mechanisms”, Geophysics, 44, 4, 1979, pp. 691–711.

8. Kolsky, H., “Stress Waves in Solids”, Journal of Sound and Vibration, 1, 1, 1964, pp. 88–110.9. Li, J.C., Ma, G.W. and Zhao, J., “An Equivalent Viscoelastic Model for Rock Mass with Parallel

Joints”, submitted.

571

Stability Analysis of Transformer Cavern and the CorrespondingBus Duct System at Siyah Bishe Pumped Storage Power Plant

ABBAS MAJDI1,∗, KAYUMARS EMAD2 AND HESSAM MOGHADDAM ALI3

1Associate Professor, School of Mining Engineering, University of Tehran, Iran2Head of Infrastructures, Transportation Research Institute (TRI), Ministry of Road and Transportation, Iran3Senior Extraction Engineer, IMPASCO Company, Ministry of Industries and Mines, Iran

1. Introduction

Siyah Bishe pumped storage power house is located near the Siyah Bishe village betweenTehran- Chalous road 120 km. away from Tehran. It has been designed for producing 1000Megawatt Hydro- power to balance the national electrical network needs during the peakelectricity consumption in the country. Detailed engineering geological site investigationsrevealed significant joints and micro-faults within the rock mass surrounding the transformercavern and the corresponding bus duct galleries. Since the surrounding rock mass due to stageexcavations may behave accordingly, hence, 3DEC1 software has been used to simulate thethree dimensional discontinuous nature of the rock masses for construction of the desirednumerical models.

Rock mass joints and strength characteristics obtained from the site investigations alongwith the laboratory test results have been used as input data for the numerical models.Then, the outputs were used for stress and displacement analysis at different stages of theexcavations. For long term stability analysis, the predicted displacements obtained from thenumerical analyses were compared with that calculated by Sakurai’s critical strain approach.Subsequently, regions surrounding the cavern where the predicted displacements were morethan the critical displacements have been classified as unstable zones.

Next, the results obtained from the numerical models and those predicted by Sakurai’smethod were back analysed with the in-situ displacements provided by the monitoring systemafter the support installations.

The analyses focus with first priority on the type of support requirement for the final exca-vation stage of the caverns. For this condition, wedge and Finite Element Analyses (FEA)were also performed, and the respective deformations were estimated adopting the contrac-tor’s preferred excavation sequence which is described further in this paper. The tabulateddeformation results can be compared to values actually measured on the site and will thusallow judging the adequacy of the parameter assumptions used for stability analyses.

2. Geometric and Mechanical Characteristics of the Numerical Model

Transformer cavern is excavated in sedimentary layered strata with a thickness ranging from0.2m to 3.5m. Physical and mechanical properties of the constituent rocks were obtainedlaboratorial and displayed in Table 1 in which D is a factor that depends upon the degree ofdisturbance to which the rock mass has been subjected by blast damage and stress relaxation.It varies from 0 for undisturbed in situ rock masses to 1 for very disturbed rock masses.2

With regard to the type of excavation, D-value” representing the rock mass characteristicafter excavation was taken equal to 0.7 for the numerical model construction purpose.




Table 1. Rock mass properties in Siyah-Bishe pumped storage project.3

D = 0 D = 0.7

Rock Type GSI σc rock(MPa)

mi φ(◦) C (MPa) σcm(MPa)

E (GPa) φ(◦) C (MPa) σcm(MPa)

E (GPa)

Quartzite-Sandstone

53 85 20 53 1.6 22 11 46 1.1 14 7.1

Red Shale 48 50 9 41 0.98 7.9 6.3 32 0.66 4.7 4.1

Table 2. In-situ test results (Flat Jack) in Siyah-Bishe pumped storage project.3

Rock type Poisson’s ratio – ν Young’s Modulus – E (GPa)

Sandstone (quartzite-sandstone) 0.2 15Red shale 0.25 7.5

The modulus of elasticity and Poisson’s ratio required as part of the numerical modellinginput data were taken from the In-situ stress measurements made by using Flat Jack methodand represented in Table 2.3 Three dimensional models have been constructed with in a 300m ∗ 300 m ∗ 300 m geo-structural frame shown in Fig. 1 The underground spaces includingthe transformer cavern and the corresponding bus ducts are located at the centre of themodel.

In order to suitably employ the joint characteristics including; dips and orientations ofjoints and bedding planes in the model 650 in-situ joint measurements have been madethrough mapping of the cavern roof.4 Then joint analysis was performed by using Dips and

300 m 300 m

300 m 27 m

161 m

16 m 51 m

25 m

132 m

47 m

Major discontinuities (bedding planes, joints and faults in 3D model -Siyah-

Bishe pumped storage project)

Figure 1. 3DEC model representation with major discontinuities along with the corresponding under-ground spaces.

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Figure 2. Graphical illustrations of joint analysis results for selecting the major discontinuities to betaken in the model — Siyah-Bishe pumped storage project.

Table 3. Geometric and mechanical properties of discontinuity in Siyah-Bishe pumped stor-age project.

Discontinuitynumber

Discontinuity properties

Dip Dip Friction Ks (GN/m) Kn (GN/m) Joint spacing (m)direction (Degree) angles(Degree) (Degree)

Bedding planes 147 60 25 7.5 20 variableJoint-J1 295 70 30 7 20 0.5Joint-J2 175 60 30 7 20 1.2Joint-J3 92 76 35 7.5 20 1.0Joint-J4 284 81 27 7 20 1.5Joint-J5 324 53 30 7 20 0.75

Wedge Software to determine the governing discontinuities and the corresponding orienta-tions to be taken into the model as part of the input parameters (Fig. 2). Table 3 representsthe geometrical characteristics and mechanical properties of the governing discontinuities ofthe rock mass. For a design of the required rock supports the most relevant joint orientationswere given more weight than the others. Major discontinuities considered in the numericalmodels are illustrated in Fig. 1.

3. Displacement Analysis of the Walls and Roofs of the Excavation

Change of displacements measured at 132 points in some regions of the models’ roof andwalls are referred henceforth as observational or check and or history points. Some of thesepoints were characterized on the basis of unstable and critical points resulted from the pri-mary numerical models prepared for this project. Some complementary history points werealso determined from the bus duct cross cuts along with the extensometers measuring points.The effect of stage excavations of transformer on both stress and displacement changes ofhistory points were checked. Then the model has been reconstructed to represent post con-struction behaviour as compared with the monitoring results.

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N

BD1

BD2

BD4

BD3

e

f

g

h

History Points of Cavern Roof

History Points of Bus Duct Roof

US

DS

a

b

c

d

Figure 3. Locations of observational points installed on the roof of the bus ducts and Transformercavern — Siyah-Bishe pumped storage project.

Vertical displacem

ent (cm)

6 m

Roof (0)

10 m 15 m 17 m

6

10

15

17

Step of software run

0

Displacements after drilling the roof of Transformer cavernInitial displacements

Figure 4. Effect of stage excavation on cavern deformational behaviour (check point c in Figure 3)-Siyah-Bishe pumped storage project.

In this paper the check points locations are primarily located on the transformer cavernroof and on the roof of the T-junction of transformer cavern and the corresponding bus ductsas well (Fig. 3). Multiple point extensometers were installed at the observational points up to

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Vertical displacem

ent (cm)

Displacements after drilling the Boss DuctInitial displacements

10 m

6 m

2 m

Roof (0)

Step of software run

6

2

10

0

Figure 5. Effect of stage excavation on bus ducts deformational behaviour (check point f in Fig. 3) –Siyah-Bishe pumped storage project.

20 m and 10 m into the transformer cavern and the T-junction roofs respectively. One mea-suring point was assigned for each meter of the extensometer length. Hence, for consistentcomparative analysis of the results, these very points have also been taken into account inthe numerical models. Bus duct numbers with brevity 1BD, 2BD, 3BD and 4BD have beendisplayed in Fig. 3. Variation of vertical displacements reflecting the effect of stage excava-tions obtained from model analysis at 0, 6, 10, 15, and 17 meters depths into the transformercavern roof at check point c are illustrated in Fig. 4. Cavern roof at 0 m-and at 17 m depthswith 36mm and 6mm respectively exhibits the maximum and the minimum vertical displace-ments correspondingly. The extent of the roof vertical displacement clearly indicates that theeffective length of the required rock bolts must be more than 17 meters. Similarly, the effectof stage excavations have been analysed for check point f at T-junction of transformer cavernand the corresponding bus ducts. The respective vertical displacements in T-junction of busduct 2 at 0, 2, 6, and 10 meters depths into the roof are displayed in Fig. 5. In this casethe T-junction roof at 0 m- and at 10 m depths with 11.8mm and 5mm respectively exhibitsthe maximum and minimum vertical displacements correspondingly. Whereas the maximumvertical displacement at the depth of 10m prior to bus duct excavation was 3mm.

4. Modelling of Excavation and Support Members

Each section of the transformer cavern with a length of 10m has been excavated in ten crosssectional stages. Hence, the representative longitudinal model has been divided into 16 sec-tions. As such the sequence of excavation for each model was performed in 160 stages. Exca-vation is simulated according to the currently executed or envisaged excavation sequence.

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Table 4. Shotcrete mechanical properties in Siyah-Bishe pumped storageproject.

MechanicalPropertiesof Shotcrete

Density(kg/m3)

Young’sModulus(GPa)

v UCS(MPa)

Kn(GPa)

Ks(GPa)

C(MPa)

φ

2500 21 0.2 30 3.5 2.1 0.5 35

The support is simulated so that adequate safety is reached while not overloading the sup-port members.

Shotcrete, Rock Bolts and Tendons for support members were simulated in model. TheYoung’s Modulus in Shotcrete is set to 21,000 MPa and ν to 0.2 (Table 4). The liners areinstalled after excavation of each stage at the final surface excavated during that stage. Therock bolts are modelled as fully bonded bolts and each anchor behaves as a single element.

The rock support was designed to satisfy the support requirements determined by theabove analyses. In Table 5 rock support design with each member is shown. It has to exertsufficient support pressure so that wedges are held with sufficient safety. On the other sidethe support members must not be overstressed in the course of subsequent excavation stagesand related deformations.

5. Stability Analysis of Transformer Cavern and Bus Ducts

Discontinuous numerical modelling has been used to perform deformational analysis of theroof, floor and the walls of the transformer to assure the total stability of the structure. Itis obvious that in designing any underground structure a design criterion must be fulfilled.Hence, for long term stability analysis, estimating the critical deformation after the supportinstallation is crucial. In this paper critical strain method proposed by Sakurai5 has beenused to compute the critical strain of the prospected sections of the transformer and thecorresponding bus ducts. At 4 locations on the longitudinal profile of the Cavern 4-pointsmultipleextensometers were installed so that the distance between every two adjacent pointsis taken as 2 meters. Hence, the total length of each extensometer was taken 8m into the

Table 5. Transformer cavern and bus ducts rock support items in Siyah-Bishe pumped storage project.

Rocksupportitems

Rock bolts Tendons Shotcrete

Grid (m2) Capacity Length Grid (m2) Capacity Length Thickness(kN) (m) (kN) (m) (cm)

Transformercavern roof

3 140 6 6 890 15 20

U/S wall 3 140 6 and 8 10 300 12 15D/S wall 9 140 5 12 890 12 15Shafts ofbus barsystem

4 140 8 – – – 15

Bus ductgalleries

4 140 5 – – – 10

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Figure 6. Transformer cavern monitoring plan in longitudinal section — Siyah-Bishe pumped storageproject.

roof. The installation location of the extensometers were chosen some how to be the mostcritical with regard to shear planes or the critical discontinuities. The results obtained fromthe extensometers were back analysed as a base of comparison for the numerical analysis.14 locations on the peripheral area of the cavern were chosen for monitoring purposes.4 at roof and 4 at each corresponding side at the spring line both on the upstream anddownstream sides (Fig. 6).

With regard to the critical strain concept the following three different risk levels are given5:

log εc = −0.25 log E− 0.85 I

log εc = −0.25 log E− 1.22 II εc = uc

Rlog εc = −0.25 log E− 1.59 III.

Where, εc represents the critical strain, E is the modulus of elasticity, uc stands for crit-ical displacement, and finally R illustrates the radius or width of underground excavation.If εc num represents the critical strain obtained from the numerical modelling analysis andεc meas stands for the critical strain computed based on in-situ stress measurements then the

0

2

4

6

8

1.0

1.2

0 200 400 600 800 1000 1200 1400 1600

0

1

2

3

4

5

6

7

8

9

10

Recording sequence

Critica

l Disp

lacem

ents (m

m)

Mo

del a

nd

Instru

men

ts Disp

lacem

ents (m

m) Recording sequence

WALL

U/SD/S

a

0

2

4

6

8

10

12

0 200 400 600 800 1000 1200 1400 16000

2

4

6

8

10

12

Recording sequence

WALL

U/SD/S

Critica

l Disp

lacem

ents (m

m)

b

Sakurai’s critical displacements (mm)

Predicted model displacements (mm)

Measured displacements (mm) from the first ring extensometer

Figure 7. Comparison of the measured displacements with; Sakurai’s critical displacements and pre-dicted model displacements at first ring of instrument; (a) 140 m from south end wall in Fig. 6, (b) 78m from south end wall in Fig. 6 - Siyah-Bishe project.

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following limits can be considered:

εc num or εc meas = εc ⇒ Equilibrium state (1)

εc num or εc meas > εc ⇒ Critical state (2)

εc num or εc meas < εc ⇒ Safe state. (3)

Depending up on type of the projects one can use any of the above given risk levels. Thenregardless which risk level is employed one of the three states given in Eqs. (1) to (3) isresulted. Care must be taken into account that the desired modulus of elasticity must beobtained from in-situ measurements. Fig. 7 represents the comparative analysis of the resultsobtained from the numerical modelling performed in this research with those obtained fromin-situ measurements and that calculated from the Sakurai’s method.

6. Conclusions

Numerical simulation has been performed to evaluate the transformer cavern and the corre-sponding bus duct systems behaviour. Rock mass joints and strength characteristics obtainedfrom the site investigations along with the laboratory test results have been used as inputdata for the numerical models. Then, the outputs were used for stress and displacementanalysis at different stages of the excavations. For long term stability analysis, the predicteddisplacements obtained from the numerical analyses were compared with that calculated bySakurai’s critical strain approach. Subsequently, regions surrounding the cavern where thepredicted displacements were more than the critical displacements have been classified asunstable zones. The results clearly reflected the role of support systems in minimizing theundesired displacement, in particular, for long term stability of the cavern.

Hence, the analyses focus with first priority on the type of support requirement for thefinal excavation stage of the caverns.

References

1. Itasca Consulting Group, Hnc., “3DEC, 3 dimensional distinct element code”, Version 2.00, UserManual.

2. Hoek, E., Carranza-Torres, C., and Corkum, B., “Hoek — Brown failure criterion — 2002 edition,www.rocscience.com.

3. Soils engineering services (SES), Tehran, Iran, Reports of rock mechanics laboratory tests, February,2004.

4. Tablyeh construction engineering company, “Reports of joint mapping and geotechnical investiga-tions in TC, PHC and bus ducts- Phase III”, Program No. 3, Vol. I, Report January 2005.

5. Sakurai, S., “Lessons learned from field measurements in tunnelling”. Tunnelling and UndergroundSpace Technology, Vol. 12, No. 1999, 4, pp. 453-460

580

Process Zone Development Associated with Cracking Processes inCarrara Marble

L.N.Y. WONG1,∗ AND H.H. EINSTEIN2

1Nanyang Technological University2Massachusetts Institute of Technology

1. Introduction

In this study, laboratory compressive loading tests have been conducted on prismatic Carraramarble specimens, which contained a pair of pre-existing artificial flaws. Cracks initiatingfrom the pre-existing flaws are found to be preceded by development of individual linear tocurvilinear white patches. Observation with the scanning electron microscope (SEM) showsthat the macroscopic white patches in marble, which develop prior to the initiation of macro-scopic cracking, consist of microcracking zones (process zones).

2. Background

Crack-tip plasticity is an established concept used to account for the inelastic material defor-mation. The size and shape of the associated process zone are related to the material param-eters and the loading conditions.1−5 One of the key assumptions is that the local materialyielding occurs adjacent to a crack tip. Yielding ahead of the crack tip continues as the crackpropagates and lengthens.

As shown in previous experimental studies by the same authors,6,7 which consisted ofcompressive loading of prismatic Carrara marble containing pre-existing artificial flaws, indi-vidual white patches developed from the pre-existing flaws in response to loading prior tothe initiation of any macroscopic observable cracks (Fig. 1). Once developed, most of thewhite patches (free of any observable macroscopic cracks) then propagated (increased inlength), widened and intensified in colour as the applied loading further increased. It is inter-esting to note that instead of initiating and propagating from the pre-existing flaw as a singlemacroscopic continuous crack, multiple individual tensile cracks usually developed as shorten-echelon crack segments. These en-echelon cracks lengthened, and eventually linked up toform a continuous crack as loading further increased. It has to be emphasized that the aboveobservations were based on video recordings at a macroscopic scale.

A similar white patch development was also observed in other experimental studies onmarble.8−10 The white patch development was suspected to be due to the presence of inducedmicrocracks8 or “deviation and failure of crystalline grains”.10 However, no experimentalattempts (e.g. microscopic imaging) were made by these authors to confirm their hypotheseson the nature of the white patches.

To investigate the nature of these white patches, the present study consists of loadingidentical Carrara marble specimens with pre-existing artificial flaws to different stress levels,at which only white patches but no observable cracks have developed. The microstructuralcharacteristics of the corresponding white patches are then examined by using the scanningelectron microscope (SEM) imaging technique. The SEM imaging technique has also beensuccessfully used by others to observe the details of cracks in rocks (e.g. 11–14).




i 1 k l i bl i f h i b fl i b f l di b

Figure 1. Crack coalescence in Carrara marble. (a) View of the region between flaw tips before load-ing. (b) A white patch developed between flaw tips in response to loading. (c) A short tensile crackinitiated within the white patch. (d) Multiple short tensile cracks developed. (e) A continuous crackformed due to the lengthening and coalescence of the short tensile cracks. The distance between thetips of the pre-existing flaw is 13 mm.

Figure 2. Artificial flaw pair geometry prepared for the white patch study in marble specimens. Eachflaw is 13 mm long and 1.3 mm wide. Compressive loading direction is vertical. The specimen is com-pressively loaded vertically.

3. Experimental Procedures

Three identical Carrara marble specimens (152 mm × 76 mm × 32 mm) containing a pairof pre-existing flaws (13 mm long, 1.3 mm wide) as shown in Fig. 2 are prepared. The flawsare created using an OMAX abrasive jet.

These marble specimens are loaded to respectively 50%, 70% and 90% of the specimenfailure stressa, which lead to the development of white patches of different extent (Fig. 3).The loaded marble specimens are then trimmed down and polished to obtain flat and smoothsurfaces. Carbon coating is applied to the specimen before being examined in the scanningelectron microprobe (SEM). The microprobe model used for this study is a JEOL JXA-733SEM.

aThe average failure stresses of Carrara marble specimens containing the same flaw pair geometrydetermined from previous tests16 was 52.0 MPa. This stress value is used for computing the percentageof failure stress to be applied to the marble specimens.

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Figure 3. Sketches of development of white patches (grey lines) in three different marble specimens,which have been subjected to uniaxial (vertical) compression loading of up to (a) 50%, (b) 70% and (c)90% of the specimen failure load.15 The thick black lines represent the pre-existing flaws. The inducedwhite patches indicated by arrows are later examined by the SEM as shown in Fig. 5.

Figure 4. (a) The grain boundary of the elliptical grain A is identifiable where there is an adequatecolor contrast between adjacent grains; (b) Grain boundary cracking (inter-granular cracking), whichappears black in color, occurs around grain B; (c) Grain C is dissected into three parts by multiplealmost parallel intra-granular cracks (indicated by arrows), which terminate at the boundary of grainC. Intra-granular cracks refer to those cracks completely embedded within mineral grains, but not alonggrain boundaries. (d) An intra-granular crack (main crack) cuts through grain D and grain E. (e) Theintra-granular cracks in grain F are controlled by two dominant set of cleavages (indicated by arrows).

4. Results

4.1. White patch development — macroscopic observation

In response to the applied uniaxial loading, individual white patches of conventional wingappearance, which are identifiable even with unaided eyes, emanate from the flaw tips. Thelength and width of the white patches increase with the applied stress level, and this is accom-panied by an intensification of the white colour. All of these white patches are free of anymacroscopic cracks as observed with a 10× handlens. It was observed in a previous study7

that further loading of specimens with identical configuration would eventually lead to thedevelopment of tensile wing cracks along these white patches.

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Figure 5. Assemblages of SEM images for individual white patches observed in the marble speci-mens. The bottom right black regions are the pre-existing flaws. Refer to Fig. 3 for the regions underexamination.15

4.2. White patch development – microscopic observation

Figure 5 shows the SEM images taken along the white patches as indicated in Fig. 3. Thecrack density distribution is classified into one of the four classes – background (B)b, low (L),medium (M) and high (H) as shown in Fig. 6 (overlay of crack density distributions ontocrack traces). The classification is carried out by visually comparing the actual crack distri-bution against the four reference grids shown in the top left portion of Fig. 6. Throughoutthe discussion below, a microcracking zone is defined as a region with microcrack densitiesof class L or higher. The characteristics of the microcracking zones observed at differentloading levels are discussed below. Also refer to Fig. 4 and the corresponding caption for theidentifications and descriptions of the typical microcracking features.

50% loading (Fig. 6(a)) – The individual white patch close to the flaw consists of an under-lying zone having multiple short microcracks (inter-granular and intra-granular), but notwith any observable dominant cracks. The development of dominant cracks occurs at subse-quent higher loading levels. As shown in Fig. 6(a), the crack density inside the microcrackingzone belongs to class M in the core and class L in its vicinity. Note that crack density classH has not yet developed at this level of loading. Outside the microcracking zone, the crackdensity drops to class B. Figure 6a also shows that the amount of microcracking decreaseswith distance away from the flaw face (black region in the bottom right). At a sufficiently

bNote that the microcracks constituting the background crack density (B) were due to two sources —inherent microcracks and microcracks due to the cutting action of water abrasive jet.

584


Figure 6. Overlays of crack density distribution over the sketches of crack traces shown in Fig. 5. Thekeys of the different classes of crack density (qualitative) — background (B), low (L), medium (M) andhigh (H) are shown in the top left region of the figure.15

remote distance (not shown in the figure), individual grains completely free of microcrackssuch as those illustrated in Fig. 4(a) exist.

70% loading (Fig. 6(b)) — The individual white patch close to the flaw consists of anunderlying zone, having a central elongated core of class H, and regions of classes L andM on its flank. The central region consists of a dominant undulating generally continuouscrack, which usually follows grain boundaries. This central dominant crack is also flankedby multiple much shorter intra-granular cracks (classes L and M). The density of these intra-granular cracks, which usually trend more or less parallel to the dominant crack, is thehighest next to the central dominant crack (class H). It drops off rapidly with distance fromthis central crack (note that a large portion of the central region of class H is flanked byregions of relatively low crack density (class L & class B). The crack density also drops in adirection orthogonal to the flaw face.

90% loading (Fig. 6(c)) – Similar to the specimens subject to 70% loading, an individualwhite patch develops close to the flaw tips and it can be correlated with a microcrackingzone. The microcracking zone again consists of a central elongated densely cracked zone(class H) and multiple much shorter orthogonal intra-granular cracks of classes M and L atits flank. The density of the microcracks is also the highest in the central core and decreaseswith distance from it. However, the extent of regions of class M and class L, which flankthe central class H region (Fig. 6(c)) is larger for the 90% case (Fig. 6(c)) than that for 70%case (Fig. 6(b)). In addition, the region of class H inside the white patch due to 90% loadingis composed of multiple long microcracks (indicated by arrows in Fig. 6(c)), which trendalong with the general orientation of the white patch, instead of only one single dominant

585


Figure 7. Schematic representation of microcrack density along and away from an induced whitepatch.

crack as was the case for 70% loading (Fig. 6(b)). Based on the above observations, theproperties of the microcracking zone underlying a while patch are schematically representedin Fig. 7. The three small sketches show that the microcrack density (ρ), which can be definedas number of microcracks per unit area, generally decreases away from the white patch. Ata distance sufficiently far away from white patch, the microcrack density (ρo) drops to thatof the background level (class B). In addition, the microcrack density (ρ1 > ρ2 > ρ3 > ρo)and the width of the microcracking zone (d1 > d2 > d3) decrease in a direction orthogonalto the flaw face.

To summarize, white patches develop and evolve in response to the applied loading alongthe trajectories of future tensile wing cracks. At a 50% load level, an elongated zone scatteredwith microcracks (classes M and L) forms adjacent to the flaw face around the tip regions. Asthe applied loading progressively increases, a microcracking zone consisting of a dominantcontinuous crack becomes identifiable. This dominant crack with a core region of class His flanked by multiple microcracks with crack density decreasing away from the dominant

Figure 8. Schematic illustration of the tensile wing crack development in marble associated with theevolution of microcracking zones (modified from 15).

586


Figure 9. Idealized representation of crack propagation and the associated process zone. Note thatthe actual size and shape of the process zone depends on the material parameters and the loadingconditions.

feature and from the flaw face (classes M and L). As the applied loading increases further,the microcracking zone lengthens and widens (larger extent of regions of classes M and L).The density of microcracks next to the central core region also becomes higher. The sketchesin Fig. 8 summarize the evolution of the microcracking zones underlying the white patch.There is a strong indication that these regions of high crack density (H) are the locationswhere macroscopic tensile cracks develop (see Background, Section 2), hence leading to theinitiation of unconnected tensile cracks (en-echelon cracks) within the white patch in marble(sketch e in Fig. 8) before the occurrence of a continuous observable macroscopic crack(sketch f in Fig. 8).

5. Discussion

In conventional treatment of fracture mechanics, local material yielding is assumed to occuradjacent to a crack tip. Yielding ahead of the crack tip continues as the crack propagates andlengthens (Fig. 9). The present study in Carrara marble shows that the initiation of macro-scopic cracks in marble is associated with the enlargement and coalescence of microcrackingzones and also with an increase of crack density in areas in close proximity to, but not neces-sarily adjacent to, the crack tips. In other words, yielding is not restricted to finite zone aheadof the crack tip region (Fig. 8). Similar observations were made in a recent study on granite.17

This phenomenon in rocks is likely to be attributed to the inherent fabric and heterogeneousnature of rocks. The presence of local defects, including crystal cleavages, open/closed grainboundaries, inherent microcracks due to past geological history, can alter the local stress fieldand thus the development of a process zone, significantly.

6. Conclusions

The scanning electron microprobe (SEM) imaging technique is used to study the microscopicdevelopment of white patches in marble. Consequently, a relation between the macroscopicand microscopic mechanisms in marble in the formation of tensile cracks is established inthe present study. The microscopic imaging study in marble showed that the macroscopicwhite patches consist of extensive microcracking zones. The initiation of macroscopic cracksin marble is associated with an enlargement and coalescence of the microcracking zones andalso with an increase of crack density.

References

1. Irwin, G.R., “Plastic Zone near a Crack and Fracture Toughness”, Sagamore Research ConferenceProceedings, Vol. 4, 1961, Syracuse University Research Institute, Syracuse NY, pp. 63–78.

587


2. Dugdale, D.S., “Yielding in Steel Sheets Containing Slits”, Journal of the Mechanics and Physicsof Solids, Vol. 8, 1960, pp. 100–104.

3. Barenblatt, G.I., “The Mathematical Theory of Equilibrium Cracks in Brittle Fracture”, Advancesin Applied Mechanics, Vol. VII, Academic Press, NY, 1962, pp. 55–129.

4. Dodds, R.H., Jr., Anderson, T.L. and Kirk, M.T., “A Framework to Correlate a/W Effectson Elastic-Plastic Fracture Toughness (Jc)”, International Journal of Fracture, Vol. 48, 1991,pp. 1–22.

5. Nakamura, T. and Parks, D.M., “Conditions of J-Dominance in Three-Dimensional ThinCracked Plates”, Analytical, Numerical, and Experimental Aspects of Three-Dimensional Frac-ture Processes, ASME AMD-91, American Society of Mechanical Engineers, New York, 1988,pp. 227–238.

6. Wong, L.N.Y. and Einstein, H.H., “Systematic Evaluation of Cracking Behavior in SpecimensContaining Single Flaws under Uniaxial Compression”, International Journal of Rock Mechanicsand Mining Sciences, 46, 2, 2009, pp. 239–249.

7. Wong, L.N.Y. and Einstein, H.H., “Crack Coalescence in Molded Gypsum and Carrara Marble:Part 1 — Macroscopic Observations and Interpretation”, Rock Mechanics and Rock Engineering,42, 3, 2009, pp 475–511.

8. Chen, G., Kemeny, J.M. and Harpalani, S., “Fracture Propagation and Coalescence in MarblePlates with Pre-cut Notches under Compression”, In L.R. Myer, N.G.W. Cook, R.E. Goodman& C.F. Tsang (Eds). Symposium on Fractured and Jointed Rock Mass, Lake Tahoe, CA, 1995,pp. 435–439.

9. Martinez, A.R. Fracture Coalescence in Natural Rock, SM Thesis, Massachusetts Institute of Tech-nology, 1999, p. 341.

10. Li, Y.P., Chen, L.Z. and Wang, Y.H., “Experimental Research on Pre-cracked Marble under Com-pression”, International Journal of Solids and Structures, 42, 2005, pp. 2505–2516.

11. Sprunt, E.S. and Brace, W.F. “Direct Observation of Microcavities in Crystalline Rocks”, Interna-tional Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts, 11, 1974,pp. 139–150.

12. Tapponnier, P. and Brace, W.F., “Development of Stress-Induced Microcracks in Westerly Granite”,International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts, 13,1976, pp. 103–112.

13. Kranz, R.L., “Crack Growth and Development During Creep of Barre Granite”, Interna-tional Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts, 16, 1979,pp. 23–35.

14. Sagong, M. and Bobet, A., “Micro-fractographic Characterization of Tensile and Shear Cracks”,In P.J. Culligan, H.H. Einstein & A.J. Whittle (Eds). Soil and Rock America 2003, Cambridge,MA, 2003, pp. 937–944.

15. Wong L.N.Y. and Einstein, H.H., “Crack Coalescence in Molded Gypsum and Carrara Marble:Part 2 – Microscopic Observations and Interpretation”, Rock Mechanics and Rock Engineering,42, 3, 2009, pp. 513–545.

16. Wong, N.Y., Crack Coalescence in Molded Gypsum and Carrara Marble, PhD Thesis, Mas-sachusetts Institute of Technology, Cambridge, MA, 2008, pp. 876.

17. Miller, J.T., Crack Coalescence in Granite, SM Thesis, Massachusetts Institute of Technology,2008, p. 474.

588

Simulation of Stress Singularity Around the Crack Tips for LEFMProblems Using a New Numerical Method

G.R. LIU1,2 AND N. NOURBAKHSH NIA1,∗1Center for Advanced Computations in Engineering Science (ACES), Department of Mechanical Engineering,National University of Singapore, 9 Engineering Drive 1, Singapore 1175762Singapore-MIT Alliance (SMA), E4-04-10, 4 Engineering Drive 3, Singapore, 117576

1. Introduction

The strain smoothing technique was applied in the finite element method settings and thesmoothed finite element method (SFEM) was developed using cell-based smoothing domainscreated by further dividing elements.1 A node-based smoothed finite element method (NS-FEM)2 has also been proposed using smoothing domains constructed based on nodes inFEM settings. The NS-FEM is found instable temporally due to its “overly soft” featurerooted at the use of a small number of smoothing domain in relation to the nodes3–5 andcannot be used to solve dynamic problems. To eliminate the temporal instability, Liu et al.6

proposed the edge-based smoothed finite element (ES-FEM) which uses smoothing domainsconstructed based on edges of the elements. The significance of the ES-FEM is that it is oftenfound much more accurate than the linear FEM using the same mesh and even more accuratethan the FEM using quadrilateral elements with the same set of nodes. Most importantly, theimplementation procedure in ES-FEM offers a very convenient way to create the displace-ment field using the simple point interpolation method (PIM). This is because we need onlythe shape functions values on the edges of the smoothing domains in the ES-FEM formula-tion in computing the stiffness matrix. Therefore, we can create a desired displacement fieldin a very flexible manner using the PIM for various applications, such as simulating a propersingular stress field to be discussed in this work. This paper develops a singular ES-FEM forsimulating the stress singularity for linear fracture problems, considering the first (opening)mode. In the present ES-FEM we use a triangular mesh that can be generated automaticallyfor problems with complicated geometry. The numerical results have shown that the strainenergy, displacement and J-integral obtained using the present singular ES-FEM method ismore accurate than the standard linear FEM and even the quadratic FEM with standard6-node crack tip elements.

2. The Idea of Singular ES-FEM

2.1. Reproducing stress singularity at the crack tip

Currently, the most widely used standard method for simulating the stress singularity at thecrack tip is to use the so-called (quadratic) 6-node crack-tip element using mapping based onelements with the mid-edge nodes being shifted by a quarter edge-length towards the crack-tip. In the ES-FEM method, however, no mapping is needed and the stress singularity can becreated by a simple point interpolation method with extra basis functions of proper fractionalorder polynomials. In the present ES-FEM, we use 3-node triangle elements for areas withoutsingularity, and one layer of specially designed singular 5-node triangular elements containingthe crack-tip to produce the stress singularity behaviour at the crack tip. In these elements as




(a) an additional node is added on each edge of the triangular elements connected to the crack tip

(b) coordinate for an edge connected to the crack-tip

Figure 1. Node arrangement near the crack tip. Dash lines show the boundary of a strain smoothingdomain for an edge directly connected to the crack tip node.

shown in Fig. 1 we have added in a node on each edge of the triangular elements connectedto the node at the crack tip. The location of the added node is at the one quarter length ofthe edge, as shown in Fig. 1.

Based on this setting, the displacement field, for example the component u, at any point ofinterest on an edge which is directly connected to crack tip node (node 1 in Fig. 1.) can becreated using.

u = c0 + c1r+ c2√

r (1)

where r is the radial coordinate originated at the crack-tip (node 1), and ci (i = 0, 1, 2) arethe constants yet to be determined. After using Equation (1) at node 1, 2 and 3 we can solvethe simultaneous system of three equations for ci. By substituting them back to Equation (1),we shall obtain:

u =⎡⎢⎣1+ 2

rl− 3

√rl︸︷︷︸

φ1

−4rl+ 4

√rl︸︷︷︸

φ2

2rl−√

rl︸︷︷︸

φ3

⎤⎥⎦⎧⎨⎩

u1u2u3

⎫⎬⎭ (2)

where l is the length of the element edge, and cii (i = 1, 2, 3) are the shape functions forthese three nodes on the edge. It is clear that the shape functions are (complete) linear in rand “enriched” with

√r that is capable to produce a strain (hence stress) singularity field of

an order of 1/2 near the crack-tip, because the strain is evaluated from the derivatives of theassumed displacements. To perform the point interpolation using the 5-node triangle elementat the crack-tip, it is assumed that in the radial direction the displacements vary in the samefashion as given in Equation (1). In the tangential direction, however, it is assumed to varylinearly. Figure 2 shows two 5-node elements parts of which form one edge-based smoothingdomain. (1-C1-2-C2-1).

For any point on line 1− γ − β the displacement calculated as Eq. (13);

u = u1 φ1 + uγ φ2 + uβφ3 (3)

where

uγ =(

1− lγ−4

l4−5

)u4 + lγ−4

l4−5u5 & uβ =

(1− lβ−2

l2−3

)u2 + lβ−2

l2−3u3 (4)

590


Figure 2. Two 5-node elements at the crack tip node 1, and interpolation within element 1-2-3-4-5.

in which li−j is the distance between points i and j. Because of the fact thatlγ−4l4−5= lβ−2

l2−3= α:

u = φ1︸︷︷︸N1

u1 + (1− α)φ3︸︷︷︸N2

u2 + αφ3︸︷︷︸N3

u3 + (1− α)φ2︸︷︷︸N4

u4 + αφ2︸︷︷︸N5

u5 (5)

The general form of shape functions for the interpolation at any point within the 5-nodecrack-tip element can be written as:

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

N1 = ϕ1

N2 = (1− α)ϕ3

N3 = αϕ3

N4 = (1− α)ϕ2

N5 = αϕ2

(6)

Because in our singular ES-FEM, we do not need derivatives of shape functions, Equation (6)is all we need in computing the stiffness matrix and creating the numerical model.

2.2. Stiffness matrix evaluation

In order to calculate the stiffness matrix in the present ES-FEM, strain smoothing domainsare constructed associated with the edges of each element. Each 3-node triangular elementsare divided into three equal sub-triangular areas corresponding to three edges of the triangleand two such sub-triangular areas sharing with the same edge form a smoothing domain.For a 5-node crack-tip element, however, we divide the original smoothing domain to somesub-smoothing domains to better capture the singular strain field as shown in Fig. 3.

Based on the ES-FEM procedure, the stiffness matrix of the whole model is the summationof the sub-matrixes of the stiffness matrix associated with all the strain smoothing domains.

KIJ =Ns∑

k=1

KIJ(k) where KIJ(k) =∫As

k

BTI DBJdA (7)

where KIJ is the assembled stiffness matrix and KIJ(k) is the stiffness matrix of the smoothing

domain of the edge k and Ask is the kth strain smoothing area associated with edge k. B

TI can

591


a) One smoothing domain 1-C1-2-C2-1 b) Two sub-smoothing domains: 1-B1-4-B2-1 and 4-B1-C1-2-C2-B2-4 (S-SD=2)

c) Three smoothing cells: 1-D1-E-D2-1, E-D1-B1-4-B2-D2-E, 4-B1-C1-2-C2-B2-4 (S-SD=3)

Figure 3. Further division of the smoothing domain associated with edge 1-4-2 into smoothing cells.

be calculated using

BI(xk) =⎡⎢⎣bIx(xk) 0

0 bIy(xk)bIy(xk) bIx(xk)

⎤⎥⎦ : (bIx(xk) = 1

Ask

∫�(k)

NI(x)nkh(x)d� (h = x, y)) (8)

in which NI is the shape functions of the element, �(k) is the integration domain and nkh is

the outward normal vector matrix on the boundary �(k) and has the form

nk(x) =⎡⎣nx 0

0 nyny nx

⎤⎦ (9)

It should be noticed that for the boundary segments associated with the standard 3-nodetriangular element, one Gaussian point at the midpoint of one boundary segment is enough,while for a 5-node crack tip element more than one Gaussian point is needed.

2.3. J-integral evaluation

For a two-dimensional, planar, elastic solid including a sharp crack which is typically shownin Fig. 4 J-integral can be defined by Ref. 7

J = −∫

AJ

((σij∂ui

∂x1−wδ1j

)∂q∂xj

)dA (10)

When σij is the stress, ui is the displacement vector referred to a Cartesian coordinatesystem located at the crack tip, and w is the strain energy density. In addition, AJ is the areaenclosed by the segments �J1, �−, �J2 and �+ as shown in Fig. 4; in which �− and �+ are

592


Figure 4. A closed paths around the crack tip.

respectively parts of bottom crack face and top crack face. Furthermore, q is a sufficientlysmooth weighting function on AJ which takes a value of unity on �J1 and zero on �J2.

2.4. Numerical examples

Three geometries under the first fracture mode have been examined to investigate the prop-erty of singular ES-FEM. These geometries are shown in Figs. 5–7. The Young modulus forall structure is E = 2 × 107 and Poisson’s ratio ν = 0.3. All the problems have been ana-lyzed using different methods including FEM-T3, FEM-T6, standard ES-FEM and singularES-FEM using one and more sub-smoothing domains (S − SD = 1,2,3,4), and the results interms of strain energy and displacement were studied. For the first example in addition to thestrain energy and displacement, J-integral was also studied to examine the power of singularES-FEM to evaluating the J-integral.

2.5. Results and discussion

The results of strain energy and displacement for the first example have been plotted in Figs. 8and 9, respectively. The compact tension specimen has also been solved using FEM-T3, FEM-T6 and standard ES-FEM and the results in terms of the strain energy and displacement havebeen plotted in Figs. 10 and 11. Similarly the double cantilever beam under tension loadwas studied and the results in terms of strain energy are shown in Fig. 12. As it mentionedpreviously, the first example has been chosen to investigate the Singular ES-FEM’s power toJ-integral evaluation. The results have been depicted in Fig. 13.

Figure 5. Example 1 (Plate with a edge crack under a tension load).

593


Figure 6. Example 2 (Compact tension specimen). Figure 7. Example 3 (Double Cantilever beam).

0.00811

0.00813

0.00815

0.00817

0.00819

0.00821

0.00823

200 700 1200 1700 2200

DOF

Str

ain

en

erg

y

FEM-T3

ESFEM-T3

FEM-T6

Singular ES-FEM(S-SD=1)




Reference Solution

Figure 8. Strain energy for the rectangular plate with an edge crack computed using different methods.

From all the results of strain energy and displacement, it can be seen clearly that the resultsof the present singular ES-FEM with S − SD = 1 for each crack tip edge are much moreaccurate and convergence much faster than the FEM-T3 and standard ES-FEM. Besides, byincreasing the number of smoothing cells to 2, 3 or 4 strain smoothing cells (only for thecrack tip elements) the results can be further improved. when S − SD = 2 the results aremore accurate and converge much faster not only than FEM-T3 and ES-FEM, but also eventhan the FEM-T6 with standard crack tip elements. It can be also observed that there is nosignificant change in the results when S− SD ≥ 2 for the each crack tip edges. Similarly, theresults of J-integral calculation shows that singular ES-FEM yields to much more accurateresults compared to the FEM-T3 ad standard ES-FEM-T3 for all cases and by increasingthe number of sub-cells from 1 to 2, further improvement in the results will be observed.In this case the results are very closer to the analytical value in comparison with FEM-T3,ES-FEM-T3 and even Singular ES-FEM using one sub-smoothing domain. Moreover, whenperforming the integration along the lines surrounding the strain smoothing area in equation(8) using 7 Gaussian points gives almost the same results as using 5 Gaussian points. As aconclusion, 5 Gaussian points can be recommended for our Singular ES-FEM.

594


0.00811

0.00813

0.00815

0.00817

0.00819

0.00821

200 700 1200 1700 2200

DOF

Dis

pla

cem

en

t

FEM-T3ESFEM-T3FEM-T6Singular ES-FEM(S-SD=1)Singular ES-FEM(S-SD=2)Singular ES-FEM(S-SD=3)Singular ES-FEM(S-SD=4)Reference Solution

Figure 9. Displacements for the rectangular plate with an edge crack computed using different meth-ods.

0.006

0.0065

0.007

0.0075

0.008

0.0085

0.009

0.0095

0.01

0.0105

0.011

800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800

DOF

Str

ain

en

erg

y

FEM-T3ES-FEM-T3FEM-T6Singular ES-FEM(S-SD=1)Singular ES-FEM(S-SD=2)Singular ES-FEM(S-SD=3)Singular ES-FEM(S-SD=4)Reference Solution

Figure 10. Strain energy for the compact tension specimen computed using different methods.

8.00E-05

8.50E-05

9.00E-05

9.50E-05

1.00E-04

1.05E-04

1.10E-04

1.15E-04

1.20E-04

1000 1200 1400 1600 1800 2000 2200 2400 2600 2800

DOF

Dis

pla

cem

en

t

FEM-T3ES-FEM-T3FEM-T6Singular ES-FEM(S-SD=1)Singular ES-FEM(S-SD=2)Singular ES-FEM(S-SD=3)Singular ES-FEM(S-SD=4)Reference Solution

Figure 11. Displacement for the compact tension specimen computed using different methods.

595


3.40E-04

3.50E-04

3.60E-04

3.70E-04

3.80E-04

3.90E-04

4.00E-04

4.10E-04

700 900 1100 1300 1500 1700 1900

DOF

Str

ain

en

erg

y

FEM-T3ES-FEM-T3FEM-T6Singular ES-FEM(S-SD=1)Singular ES-FEM(S-SD=2)Singular ES-FEM(S-SD=3)Singular ES-FEM(S-SD=4)FEM-T6-VERY FINE MESH

Figure 12. Strain energy for the cantilever beam computed using different methods.

0.7

0.75

0.8

0.85

0.9

0.95

1

100 600 1100 1600 2100DOF

Norm

ali

zed

J

FEM-T3ES-FEM-T3Singular ES-FEM (S-SD=1,GP=5)Singular ES-FEM (S-SD=1,GP=7)Singular ES-FEM (S-SD=2,GP=5)Singular ES-FEM (S-SD=2,GP=7)Analytical Solution

Figure 13. Normalized J-integral calculated by different methods.

3. Conclusions

The following conclusions are deduced from the experimental results:

• The singular ES-FEM have much more accurate results in term of the strain energy,displacements in comparison with the standard ES-FEM-T3, FEM-T3 and evenFEM-T6 with the standard 6-node crack tip elements.• Increasing the number of sub-smoothing domains from one to two can yield to some

improvement between the results. However, for S-SD>2 the results do not consider-ably change.• The singular ES-FEM works well with the J-integral. Using one strain smoothing

domain and 2 sub- smoothing domains presents excellent results in comparison withFEM-T3 and ES-FEM-T3.• We now, for the first time, have a basically linear displacement method, singular

ES-FEM, that works very well for simulating desired singular stress field.

596


Acknowledgements

“This work is partially supported by A*Star, Singapore (SERC Grant No: 052 101 0048)”. Itis also partially supported by the Open Research Fund Program of the State Key Laboratoryof Advanced Technology of Design and Manufacturing for Vehicle Body, Hunan University,P.R. China under the grant number 40915001.”

References

1. Liu, G.R., Dai, K.Y. and Nguyen, T.T., “A smoothed finite element method for mechanics prob-lems”, Computational Mechanic, 39, 2007, pp. 859–877.

2. Liu, G.R., Nguyen, T.T., Nguyen, X.H. and Lam, K.Y., “A node-based smoothed finite elementmethod for upper bound solution to solid problems (NS-FEM)”, Computers and Structures, 87,2009, pp. 14–26.

3. Puso, M.A. and Solberg, J., “A stabilized nodally integrated tetrahedral”, International Journal forNumerical Methods in Engineering, 67, 2006, pp. 841–867.

4. Puso, M.A., Chen, J.S., Zywicz, E., Elmer and W., “Meshfree and finite element nodal integrationmethods”, International Journal for Numerical Methods in Engineering, 74, 2008, pp. 416–446.

5. Nagashima, T., “Node-by-node meshless approach and its applications to structural analyses”,International Journal for Numerical Methods in Engineering, 46, 1999, pp. 341–385.

6. Liu, G. R., Nguyen-Thoi, T. and Lam, K. Y., “An edge-based smoothed finite element method (ES-FEM) for static free and forced vibration analysis”, Journal of Sound and Vibration, 320, 2009, pp.1100–1130.

7. Li, F. Z., Shih, C. F. and Needleman, A., “A comparison of methods for calculating energy releaserates”, Engineering Fracture Mechanics, 21, 2, 1985, pp. 405–421.

597

Modeling of Three-Dimensional Hydrofracture in PermeableRocks Subjected to Differential Far-Field Stresses

L.C. LI∗, C.A. TANG, G. LI AND Z.Z. LIANG

School of Civil and Hydraulic Engineering, Dalian University of Technology, Dalian 116024, China

1. Introduction

Hydraulic fracturing is one of the most important stimulation techniques of the energy indus-try. Other applications of hydraulic fracturing include the disposal of waste drill cuttingsunderground, heat production from geothermal reservoirs, goafing and fault reactivation inmining, and the measurement of in situ stresses.1

Since the pioneering work by Khristianovic and Zhelto, there have been numerous contri-butions on the analysis of hydraulic fracturing. The early efforts naturally focused on analyti-cal solutions for fractures of simple geometry, either straight in plane strain or penny-shaped.However, as pointed by Savitski,2 all these solutions are approximate as they contain strongassumptions about either the opening or the pressure field.

In recent years, the limitations of analytical models have shifted the focus of researchtowards the development of numerical algorithms to model the three-dimensional propaga-tion of hydraulic fractures. In the 3D modelling, fracture geometry models can be classifiedas pseudo-three-dimensional (P3D) and three-dimensional (3D). P3D models were developedin the 1980s, and extended the work of Simonson et al. to multiple layers. P3D models area crude, yet effective, attempt to capture the physical behavior of a planar 3D hydraulicfracture at minimal computational cost. There have also been attempts to model fully 3Dhydraulic fractures with limited success.1,3 A number of open questions still need to be prop-erly addressed in the modeling of hydraulic fractures. These include: (i) how to efficientlymodel 3D or “out of plane” effects. The success of fracture stimulation is largely dependenton the shape and the propagation behavior of the created hydraulic fracture. A recent trendis therefore to develop coupled non-planar fracture models and their use for interesting para-metric studies to understand the complex fracture growth; (ii) related to (i), how better todevelop an efficient and physically realistic code for 3D modelling. Although the computa-tional burden on 3D model systems is excessive, 3D models are essential in both complexsituations and the validation of the pseudo-three-dimensional models (P3D); (iii) the influ-ence of heterogeneity in rock on the fracture pattern or hydraulic fracture path cannot betaken into account in most of the existing flow-coupled models. It is well known that rock isa heterogeneous geological material containing many natural weaknesses. When rock is sub-jected to hydraulic loading, these pre-existing defects can induce crack or fracture growth,which can in turn change the structure of the rock and alter the fluid flow properties ofthe rock.4

As a contribution towards the recent trend that more and more coupled non-planar frac-ture models is develop and used for hydraulic fracturing studies, a 3D model based on Tang’swork4 is proposed in this paper. The governing equations and the solution strategy aredescribed. Example simulations are presented.




2. Outline of the Improved Flow-Stress-Damage (FSD) Model

In this study, a numerical code, RFPA3D based FEM, is employed to conduct the numeri-cal test. The heterogeneity in rocks can be taken into account by assuming the mechanicalparameters randomly distributed according to the Weibul’s function through Monte-Carlomethod.4 The fluid pressure in rock mass and its changes in time and space are one of thebasic factors affecting rock stability. The fundamental assumption behind the model pre-sented here is that the rock is fully saturated and the flow of the fluid (water) is governedby the Biot’s consolidation theory. As isotropic conditions are considered for the hydraulicbehavior at the elemental scale, according to the Darcy’s law of seepage flow in porous media,the following equation of the isothermal seepage flow in rock mass can be obtained.

k · ∇2p = S∂p∂t− α∂εv

∂t(1)

where k = permeability, p = pore pressure, S = Biot coefficient, a = Biot’s coefficient andεv = volumetric strain.

The equations of equilibrium and the strain-displacement relations can be expressed as

σ+ij,jfi = 0 (2)

ε=ij12

(μi,j + μj,i) (3)

where fi = component of body force and ui = component of displacement in the i-direction.The governing equations for mathematical model of an isotropic linear poroelastic mediumdeformation considering the fluid pore pressure can be expressed as

(λ+G)μj,ji +Gμi,jj + fi + (αp),i = 0 (4)

where λ = Lame’s constant, G = shear modulus, δij = Kronecker delta.The elastic damage constitutive law of element under uniaxial compressive stress and ten-

sile stress is illustrated in Fig. 1. When the stress of the element satisfies the strength criterion(such as the Columb crierion), the element begins to fail. In elastic damage mechanics, theelastic modulus of element may degrade gradually as damage progresses, and the elasticmodulus of damaged material is defined as

E = (1−D)E0 (5)

where D = damage variable, E = elastic modulus of the damaged materials and E0 = elasticmodulus of the undamaged materials.

According to different failure modes, damage variable D can be described as following.When the tensile stress in an element reaches its tensile strength σt, that is σ3 > σt, thedamage variable can be described as

D =

⎧⎪⎨⎪⎩

0 (ε < εt0)1−σrtεE0

(εt0 ≤ ε ≤ εut)

1 (ε > εut)

(6)

The parameters in above equation are defined in Fig. 1(a). The variation of damage variableis obtained when the element is subjected to uniaxial tensile stress. In RFPA3D code, rockspecimens are subjected to 3-D stress loading, according to Mazars investigation, we extend

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it from one dimensional damage model to a 3-D model. In Mazars study, effective strain canbe defined as

ε =√〈ε1〉2 + 〈ε2〉2 + 〈ε3〉2 (7)

where ε1, ε2, and ε3 are principle strains and 〈x〉 is a function and it can be defined as

〈x〉 ={

x x ≥ 00 x < 0

(8)

In order to describe the element damage under compressive or shear stress condition, wechoose the Mohr-Coulomb criterion as the second damage criterion

σ1 − σ31+ sinφ1− sinφ

≥ σc (9)

where φ = friction angle; σc = uniaxial compressive strength. The damage variable underuniaxial compression is described as

D ={

0 ε1 < εc0

1− 1−σrcE0ε1

ε1 ≥ εc0(10)

where σrc = compressive residual strength and εc0 is can be described in Fig. 1(b).In the mathematical model, the stress is directly associated with the changes of permeability

of rock. During elastic deformations, rock permeability decreases when the rock compacts,and increases when the rock extends. Most of the theories regarding stress induced variationof permeability are only valid in pre-failure region. The permeability variation for an intactrock element (when D = 0) in elastic state can be described as

ke = k0 exp[−β

(σii/3− p

H

)](11)

where the k0 = initial permeability of rock element, β = coupling coefficient, H = Biot’scoefficient and σii/3 = average total stress.

However, it is noted that when a rock element is loaded to a state where macro frac-tures begin to form, the permeability will undoubtedly increase dramatically. This is oneof the important concerns in the model. In the post-peak stage, dramatic change in rockpermeability can be expected as a result of generation of numerous micro fractures. Thereremains the difficulty of determining the permeability of the fractured material. In order toapply appropriate post-peak hydraulic characteristics, the use of a strain-based formulation

0tεutε

tσ

rtσ

ε 0

cσ

rcσ

0cε

σ

1ε

(a)tensile mode (b)compressive mode

Figure 1. Elastic damage constitutive law for element.

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Intact rock element b

b

b

Figure 2. Schematic illustration of permeability of a rock element under due to brittle failure.

for permeability variation is more suitable.5 We assume that a damaged rock element maybe represented hydraulically as a unit of rock containing three orthogonal fractures. This isshown conceptually in Fig. 2. Assuming that the three fractures are planar and have parallelsides, the aperture of the fractures is approximately given by

b ≈ V3l2≈ εvV

3 3√V2= εv

3√V3

(12)

where V is the change of volume of the element due to dilatation, and l is the side lengthof the element before dilatation. The so-called cubic law gives the flow rate between smoothparallel plates as

q = b3ρlg12μl

Jf = b3ρlg12μl

Hl

(13)

where H is the fluid (water) head loss across the two ends of the conduit, μl is the viscositycoefficient of the fluid, b is the aperture, l is the length of the plates in the direction of theflow, and g is the acceleration due to gravity. In Eq. 13, the hydraulic conductivity is givenby the term b2ρlg/12μl. Therefore the hydraulic conductivity for a damaged rock element(when D > 0) can be expressed as

kd = b2ρlg12μl

=3√

V2ρlg108μl

ε2v (14)

In RFPA3D, we rely on FEM parallel computing to perform the seepage and stress analysisof the model. Linear elastic calculating is implemented with a library package MPI (MessageProcess Interface) and Fortran 77 language. The stress is then examined and those elementsthat are strained beyond the pre-defined strength threshold level are assumed to be dam-aged irreversibly. For a damaged element, the element stiffness and strength will be reducedwhereas its permeability will be changed accordingly. The model with new parameters willbe re-analyzed. The number of failed elements and associated energy released, which can betreated as indicators of the acoustic emission (AE) activities accompanying the failure, arealso numerically simulated.4

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3. Numerical Simulation and Discussion

3.1. Discussion of test results

Numerical simulations of a number of cases have been performed to investigate the behav-ior of the propagated fracture geometry from the wellbore under different stress regimes.A hydraulic fracturing section is located at the center of a wellbore of 40 mm diameter asshown in Fig. 3. The wellbore is assumed to be located in a block of 1000mm×1000mm×1200 mm dimensions, which has been discretized into a 100×100×120 (1,200,000) ele-ments. Hydraulic pressure is applied along the boundary of the interior hole of hydrofrac-turing section. The rate of pressurization is kept constant throughout the numerical tests at0.2 MPa/step. Four different cases are simulated to illustrate the influence of far-field stresson the hydraulic fracturing behavior. The applied far-field stresses are shown in Table 1, withthe borehole axis aligned with one of the far-field stress directions. The properties of the rockand fluid are given in Table 2.

50mm

Hydrofracturingsection

Figure 3. Schematic of the numerical model.

Table 1. Far-field stress for the four cases.

σx(MPa) σy(MPa) σz(MPa)

Case 1 1 1 4Case 2 4 4 1Case 3 1 3 5Case 4 1 1 1

Table 2. Mechanical parameters for numerical simulation.

Elasticmodulus(MPa)

Compressivestrength(MPa)

Tensionstrength(MPa)

Frictionangle (deg)

Poisson’sratio

Coefficient ofpermeability

(cm/s)

8000 100 10 30 0.25 1e-10

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3.2. Discussion of test results

Figure 4 shows the evolution of pore pressure during the hydraulic fracturing process ofCase 1 (at section Z = 625 mm). Generally speaking, there is no preferential location alongthe wellbore wall for the fracture to initiate since the geometry of the sample is symmetri-cal, and the magnitude of far-field stresses σx and σy is equal. Therefore, the location andorientation of the fracture initiation is unpredictable. Typically three stages of the hydraulicfracturing process are noted. They are:

1. Elastic deformation leading to fracture initiation. In this stage, stresses accumulate(step = 1–37), as the internal pressure increases. A fracture was initiated along the pre-ferred direction first, i.e. along σz. The height (diameter) of the initiated fracture mouth atthe wellbore wall was 50 mm, which was then propagated stepwise. Pressures developedat the fracture mouth during fracture propagation.

2. Stable fracture propagation. In the second stage, cracks propagate stably (step = 38–57,is defined as the state of cracks propagated under continually increasing hydraulic pres-sure). Due to the further increase in the borehole pressure, the diameter of the borehole

Step =38 Step =53 Step =58 fracture location

Figure 4. Hydraulic fracturing mode of case 1.

Figure 5. Pore pressure contour. Figure 6. Minimum principal stress contour.

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p p g y

(a) hydraulic fracturing mode of case 2

(b) hydraulic fracturing mode of case 3

(c) hydraulic fracturing mode of case 4

Step =45 Step =57 Step =62

Step =30 Step =40 Step =44

Step =38 Step =53 Step =59 fracture location

fracture location

fracture location

Figure 7. Hydraulic fracturing process and the final failure pattern of case 2, 3 and 4.

continues to increase, and hence to widen the fracture and drive the fracture to propa-gate. A planar fracture parallel to σ z came into being gradually.

3. Fracture growth leading to breakdown. In the third stage, cracks propagate unstably(step = 58, is defined as the state of cracks propagated without increasing hydraulicpressure). Beyond step 58, the pressure reaches its peak level (breakdown pressure Pb =11.6 MPa), which indicates the unstable propagation without increasing pressure. Fig-ures 5 and 6 are pore pressure and corresponding minimum principal stress contourunder breakdown pressure. As the fracture initiated and propagated, the highest tensilestresses immediately concentrated at the immediate vicinity of the fracture edge.

Figure 7 is the numerical results for the case 2, 3 and 4. The fracture pattern closelyresembles the experimentally observed hydraulic fracture path obtained in hydraulic frac-ture tests.6 From the figures, one can conclude that the propagation of hydraulic fractures iscontrolled by the far-field stress orientation and pre-existing field of defects. The hydraulicfracture deterministically selects a path of least resistance through the material with statistical

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p p g y

Figure 8. AE location at different visual angle during the hydraulic fracturing process in case 4.

Figure 9. AE counts with load step.

features, and the random location of the individual inhomogeneities results in an irregularhydraulic fracture trajectory. In reality, a perfect transverse fracture (in other words a perfectplanar fracture perpendicular to any stress direction) is not possible in highly heterogeneousreservoir rock. As soon as the fracture goes slightly out of plane, the shear component startsdeveloping to reorient the fracture further towards the preferred direction for fracture prop-agation with minimum resistance. For example, though the cracks in case 1, 2 and 3 initiateand propagate in a plane, they branch out after growing for a short distance. One can alsonote that isolated fractures also open within the rock mass. Such fractures should repre-sent the existence of weak elements. Figures 7(c) indicate multiple major traces without anypreferred orientations are formed. There are significant branching and isolated fracturing.Comparison of the present results with experiments6 shows that the present model can pre-dict the initiation and development of fractures fairly accurately. These results indicate thatthe crack pattern depends on the homogeneity when the far-field stress ratio is close to one.

AE location in case 4 is shown in Fig. 8. One can note that although a few events are stilloccurring throughout the volume of the sample, most events are clustered near the nucle-ation zone in the central part of the sample. The AE counts during fracture propagation areplotted in Fig. 9. The results clearly show that the fracture initiated along the non-preferreddirection (Case 4) requires relatively high pressure for propagation. This indicate that thefractures propagate unstably once initiated, which agree well with the point, concluded by

606


Detournay and Carbonell,7 that at slow pressurization rates and uniform far-field stress con-dition fracture initiation always results in unstable propagation.

4. Conclusions

It is a very general perception that the fracture initiated in the non-preferred direction andplane turns and twists during propagation and tends to be aligned with the preferred direc-tion and plane. However, if a perfectly planar fracture is perfectly oriented along the non-preferred direction, theoretically the fracture should propagate in plane though may requirehigher pressure than that for the fracture in the preferred direction. In the field, however, therock formation is extremely heterogeneous which is more likely to induce out of plane frac-ture growth. It is very important to optimize the well trajectory, perforation direction andfracture configurations for a given stress condition in the field to avoid the treatment failuresrelated to complex fracture growth.

A number of cases studies have been successfully conducted by the numerical tool, RFPA3D.The phenomenological approach has rightly facilitated the coupled fluid flow and deforma-tion analysis capability that is crucial to accurately model the fluid driven propagating behav-ior of hydraulic fractures. Although many of the conclusions that are given here may even becommon sense, the reproduction of these phenomena in a numerical simulation is significantfor several reasons. Firstly, to the best knowledge of the authors, no convenient experimentalmethod has been available for obtaining the stress or strain field during the hydraulic frac-turing process until now. Numerical simulation provides supplementary information on thefailure-induced stress redistribution. Secondly, although the AE technique has been used tomonitor micro-fracturing in rocks for many years, very few theoretical methods have beenavailable to simulate the locations of AE event sources. The numerical simulation of AEprovides a further means by which to check results that are obtained with AE techniques.Finally, and most importantly, the successful reproduction of experimentally observed failurephenomena with a numerical method will help us to make further progress in the field ofhydraulic fracturing.

Acknowledgements

The study presented in this paper was jointly supported by grants from the China NationalNatural Science Foundation (Grant Nos. 50820125405, 40638040 and 50804006) and theNational Basic Research Program of China (Grant No. 2007CB209404).

References

1. Adachi, J., Siebrits, E., Peirce, A. and Desroches, J., “Computer simulation of hydraulic fractures”,Int. J. Rock Mech. & Min. Sci., 44, 2007, pp 739–757.

2. Savitski, A.A. and Detournay, E., “Propagation of a penny-shaped fluid-driven fracture in animpermeable rock: asymptotic solutions”, Int. J. Solids Struct., 39, 2002, pp 6311–6337.

3. Mofazzal Hossain, Md. and Rahman, M.K., “Numerical simulation of complex fracture growthduring tight reservoir stimulation by hydraulic fracturing”, J. Pet. Sci. Eng., 60, 2008, pp 86–104.

4. Tang, C.A., Tham, L.G., Lee, P.K.K., Yang, T.H. and Li, L.C., “Coupled analysis of flow, stressand damage (FSD) in rock failure”, Int J Rock Mech Min Sci, 39, 4, 2002, pp 477–89.

5. Yuan, S.C. and Harrison, J.P., “Development of a hydro-mechanical local degradation approachand its application to modelling fluid flow during progressive fracturing of heterogeneous rocks”,Int J Rock Mech Min Sci, 42, 2005, pp 961–984.

607


6. Doe, T.W. and Boyce, G., “Orientation of hydraulic fractures in salt under hydrostatic and non-hydrostatic stress”, Int. J. Rock Mech. Sci. Geomech. Abstr, 26, 6, 1989, pp 605–611.

7. Detournay, E. and Carbonell, R., “Fracture mechanics analysis of breakdown process in minifracor leak-off tests”, Proceeding of Eurock 94, Rotterdam: Balkema, 1994, pp 399–407.

608

Crack Propagation Analysis Using Wavelet Galerkin Method

S. TANAKA1,∗, S. OKAZAWA1 AND H. OKADA2

1Graduate School of Engineering, Hiroshima University2Faculty of Science and Technology, Tokyo University of Science

1. Introduction

In this paper, crack propagation analyses using wavelet Galerkin method are presented.Wavelet Galerkin method is one of the methodology to solve partial differential equations.Scaling function and wavelet function are used as the basis function in Galerkin formula-tion. The wavelet functions have the so-called multiresoution properties. High spatial reso-lution wavelet functions can be superposed where high stress concentration region such ashole edges or crack tips. In this study, B-spline scaling function/wavelet function are usedas the wavelet Galerkin basis functions to solve two dimensional crack propagation anal-yses. B-spline scaling function/wavelet function are piecewise polynomial function and arecompact support basis functions. The B-spline wavelet basis functions are tractable to solvesolid/structural analyses in the Galerkin formulation because the integration and differenti-ation operations are relatively easy.7 On the other hand, there are some difficulties to treatdiscontinuous displacement of crack faces because displacement continuity is assumed in thewavelet Galerkin method. Then, new enrich functions based on the concept of X-FEM1,3 areintroduced as the wavelet Galerkin basis functions to represent displacements discontinuityof crack faces and near crack tip asymptotic solution. Stress intensity factors (SIFs) calcula-tion for the two dimensional mixed-mode crack problems are carried out by the interactionintegral method8. The interaction integral method can split KI and KII components of SIFsfrom energy release rate obtained by J-integral.

To proceed crack propagation analyses, crack grows angle θ ’ and crack growth rate�a areimportant parameters. In this study, crack angle θ ’ is determined by the maximum circumfer-ential criterion2 and crack growth rate �a is assumed constant. In this paper, mathematicalformulations of the wavelet Galerkin method and strategy for the crack propagation anal-ysis is presented. Two dimensional crack propagation analysis is presented as a numericalexample.

2. Crack Propagation Analysis Using B-Spline Wavelet Galerkin Method

2.1. B-spline wavelet Galerkin method

In this chapter, mathematical formulation and discretization to the B-spline wavelet Galerkinmethod for two dimensional crack problems are presented. Illustrations of B-spline waveletGalerkin discretization are shown in Fig. 1. B-spline wavelet Galerkin method can consider asa one of the fixed grid finite element analyses. In the B-spline wavelet Galerkin method, solidor structure are discretized by equally spaced structured cells. The interpolation functions(B-spline scaling function/wavelet) are periodically located along the coordinate axes. Sub-cell approach is adopted to accurately represent boundary of body or hole edges. The highresolution wavelet function are superposed where high stress concentration region such ashole edge or crack tips. Gauss integration is carried out to the cell or sub-cells to integrate




(a) (b)

Figure 1. B-spline wavelet Galerkin discretization to the two dimensional crack problem [(a) 2Dboundary value problem to be solved, (b) B-spline wavelet Galerkin discretization].

stiffness matrices. Linear B-spline scaling function/wavelet used in this study are shown inFig. 2.

(a) (b)

Figure 2. Basis functions of B-spline wavelet Galerkin method [(a) Linear B-spline scaling function,(b) Linear B-spline wavelet function].

(a) (b)

Figure 3. B-spline wavelet Galerkin discretization of near crack tip region [(a) Illustration of nearcrack tip region, (b) B-spline wavelet Galerkin discretization].

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B-spline wavelet Galerkin discretization of near crack tip region is shown in Figs. 3(a) and3(b). In the Bspline wavelet Galerkin analyses, displacements can represent by superposingscaling function and different length scale wavelets. Level m+1 displacement uwx

m+1(x) for twodimensional crack problems are shown as,

uwxm+1(x) =

∑k,1

�m,k,1(x)um,k,1 +3∑

i=1

∑k,1

� im,k,1(x)vi

m,k,1 +∑

k,1∈Js

H(x)�m,k,1(x)bm,k,1

+∑

k,1∈Cs

�m,k,1(x)4∑

n=1

γn(x)cnm,k,1 +

3∑i=1,

∑1∈Cw

� im,k,1(x)

4∑n=1

γn(x)dnm,k,1 (1)

The first term �m,k,1(x) is level m linear B-spline scaling function and um,k,1 is their coef-ficient (k, l is position of the scaling function), second terms �1

m,k,1(x), �2m,k,1(x), �3

m,k,1(x)

are level m linear B-spline wavelet function and v1m,k,1, v2

m,k,1, v3m,k,1 are their coefficients,

respectively. The third term of equation (1) is enrich function to represent discontinuity dis-placements of crack faces. Here, H(x) is Heaviside step function as,

H(x) ={

1 ∈ �+−1 ∈ �− (2)

where �+ and �− are upper and lower region of crack faces shown in Fig. 3(a). bm,k,1are their coefficient and k, l are location of the enrich function. The discontinuous enrichfunction is located on the triangle nodes in Fig. 3(b). The forth term and fifth term are enrichfunctions to represent near crack tip asymptotic solution for the scaling function �m,k,1(x)and wavelet functions �1

m,k,1(x), �2m,k,1(x), �3

m,k,1(x), respectively. The function γi(x) (i =1,2,3,4) represents near crack tip asymptotic solution of elastostatic crack tips as,

γ1(x) = √r cosθ

2, γ2(x) = √r sin

θ

2, γ3(x) = √r sin

θ

2sin θ , γ4(x) = √r cos

θ

2sin θ , (3)

where (r,θ ) is polar coordinate with the origin at the crack tip shown in Fig. 3(a). Enrichfunction of the forth and fifth term are located in the radius re corresponds to square nodesCs and cross nodes Cw in Fig. 3(b).

The linear B-spline wavelet function shown in Fig. 2(b) did not have the so-called Kro-necker delta property, penalty formulation are adopted to prescribe the displacement bound-ary condition. The principle of virtual work for the penalty formulation are shown as,∫

�

ε(δuwx):D:ε(uwx)d�−∫t

δuwx · tdt + α∫u

δuwx · (uwx − u)du = 0 (4)

where, uwx, δuwx are the displacement and its variation and ε(uwx), ε(δuwx) are the straincomponents and its variation. D is elastic tensor, α is penalty parameter. The displacementsuwx

m+1(x) of eq. (1) into eq. (4), we obtain the simultaneous linear equation as,

(K+ Kα)Uwx = f + fα (5)

where Uwx is unknown vector, K is global stiffness matrices and f is right hand side vector.Kα and fα is stiffness matrices and right hand side vector of penalty formulation.

Enrich functions of eq. (1) have the discontinuous function or trigonometric function.Spatial integration rule is needed. In this study, sub-cell integration scheme is adopted shownin Fig. 3(b). 2×2 integration are adopted to each sub-cells.

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2.2. Calculation of stress intensity factors

In this chapter, calculation of stress intensity factors for the two dimensional mixed-modecrack problems is presented. Illustration of interaction integral method is shown in Fig. 4. Theinteraction integral method8 can evaluate the SIFs KI and KII by splitting the energy releaserate obtained by J-integral.6 We consider two independent equilibrium state of an elasticsolid. One is actual state 1 J(1)(u(1)

i ,ε(1)ij ,σ (1)

ij ) and the other is auxiliary state J(2)(u(2)i ,σ (2)

ij ,σ (2)ij ).

The auxiliary state is determined by the asymptotic solution of elastostatic crack problems.The superimposed state J(1+2)(u(1+2)

i ,ε(1+2)ij ,σ (1+2)

ij ) are assumed by the sum of two states as,

J(1+2)(u(1+2)i ,ε(1+2)

ij ,σ (1+2)ij ) = J(1)(u(1)

i ,ε(1)ij ,σ (1)

ij )+ J(2)(u(2)i ,ε(2)

ij ,σ (2)ij )+ I(I+2) (6)

where third term on the right-hand side I(1+2) is interaction integral of state 1 and state 2.The interaction integral I(1+2) can be written as,

I(1+2) =∫

A

[σ

(1)ij

∂u(2)i

∂x1+ σ (2)

ij

∂u(1)i

∂x1−W(1+2)δ1j

]∂q(x)∂xj

dA (7)

where the local coordinate x1 are taken to be parallel to the crack faces and nj is normal in

Fig. 4 (b). In eq. (7), W(1+2)(σ

(1)ij ε

(2)ij = σ (2)

ij ε(1)ij

)is interaction strain energy. Function q(x) is

continuous function and has the property q(x) = 1 on 1 and q(x) = 0 on 0. In B-splinewavelet Galerkin discretization, the function q(x) is defined in the circle of radius R, thecentre located on the crack tip as shown in Fig. 4(c).

The interaction integral in eq. (7) can be written to be,

I(1+2) = 2E′(K(1)

I K(2)I + K(1)

II K(2)II

)(8)

where K(1)I , K(1)

II and K(2)I , K(2)

II are stress intensity factors for state 1 and state 2. E’ is Young’smodulus,

E′ =⎧⎨⎩

E, for plane strainE

1− vfor plane stress

(9)

(a) (b) (c)

Figure 4. Illustration of interaction integral method [(a) superposition of two states, (b) domain inte-gral region, (c) definition of q(x) function].

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where ν is Poisson’s ratio. If the state 2 (auxiliary state) is chosen pure mode I (K(2)I =

1, K(2)II = 0), we obtain the mode I SIF of the state I (actual state) as,

K(1)I =

E′

2I(1+2). (10)

Otherwise, if the auxiliary state is chosen pure mode II (K(2)I = 0, K(2)

II = 0), the mode II SIFfor state 1 are determined as,

K(1)II =

E′

2I(1+2). (11)

2.3. Procedure of crack propagation analysis

In this chapter, procedure of crack propagation analysis using wavelet Galerkin method ispresented. Flowchart of crack propagation analysis is shown in Figure 5 (a). The proceduresare follows, (a) analysis of initial cracks are carried out, (b) calculation of SIFs KI and KII ofthe cracks are employed, (c) determine crack length �a and crack propagation angle θ ’ fromthe SIFs (d) crack definition and enrich function relocation are employed in accordance withcrack geometry, and (e) analysis of the cracks are carried out again. Procedures from (b) to(e) are carried out repeatedly for finite steps. In this study, the crack length �a is assumedconstant, and maximum circumferential criterion is used to obtain crack propagation angleθ ’. The angle θ ’ are obtained by eq. (12) from SIFs KI, KII,

θ ′ = 2 tan−1 14

(KI

KII±√

KI

KII+ 8

). (12)


Crack propagation analysis for edge crack in rectangular plate with a hole is carried out.Illustration of the rectangular plate is shown in Fig. 6(a). Dimension of the plate is 2W =100 (mm), 2H= 150 (mm). Diameter of the hole is D= 20 (mm). The initial edge cack length

(a) (b)

Figure 5. Crack propagation procedure for the wavelet Galerkin method [(a) Flowchart of crackpropagation analysis, (b) Determination of crack propagation angle θ ’ and crack propagation velocity�a].

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(a) (b) (c)

Figure 6. Crack propagation analysis for edge crack of rectangular plate [(a) Rectangular plate to besolved, (b) Crack path and location of enrich function, (c) Comparison of crack paths].

is a = 20 (mm). Distance between initial crack tip and hole edge is L = 15 (mm). Plane stresscondition is assumed in this analysis and Young’s modulus E = 3300 (MPa), Poisson’s ratioν = 0.33 are used as the material data. Uniform tension load 1.0 (MPa) is enforced on theleft side of the plate.

The rectangular plate is divided 243×162 equally spaced structured cells. 2×2 gaussquadrate rule is adopted to each cells to integrate the stiffness matrices. To accurately rep-resent hole edge, the cells near the edge are divided 32×32 sub-cells and no integration iscarried out to the centre of sub-cells located in the hole.

Two kinds of crack growth rate (�a = 1.5 (mm), �a = 3.0 (mm)) are used. Crack pathand location of enrich functions are shown in Fig. 6(b). In this analysis, enrich function torepresent near crack tip asymptotic solution of the scaling function and wavelet function ineq. (1) are located internal area re = 1.35 (mm) from crack tip. As the crack growth, the cracktip approaches to the hole edge. Crack paths for (�a = 1.5 (mm), �a = 3.0 (mm)) shownin Fig. 6(c) are compared with numerical result of.9 These analyses are good agreement withthe conventional numerical result.

4. Conclusion

In this study, B-spline wavelet Galerkin analysis for the two dimensional elastostatic crackpropagation analysis is adopted. To represent crack geometry, enrich functions are intro-duced in the B-spline wavelet Galerkin displacement function. Interaction integral method isused to calculate stress intensity factors for two dimensional mixed-mode crack problems.Maximum circumferential criterion is adopted to predict the crack angle. Crack propaga-tion analysis for rectangular plate with a hole is presented. The numerical results are goodagreement with the conventional numerical results.

References

1. Belytschko T., Black T., Elastic crack growth in finite elements with minimal remeshing Interna-tional Journal for Numerical Methods in Engineering, 45, 1999, pp. 601–620.

2. Erdogan F., Sih G. C., On the Crack extension in plates under plane loading and transverse shear,Transactions of ASME, Journal of Basing Engineering, 85, 1963, pp. 519–527.

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3. Moes N., Dolbow J., Belytshko T., A finite element method for crack growth without remeshing,International Journal For Numerical Methods in Engineering, 46, 1999, pp. 131–150.

4. Moran B., Shih C.F., Crack tip and associated domain integrals from momentum and energy bal-ance, Engineering Fracture Mechanics, 27, 6, 1987, pp. 615–642.

5. Murakami Y. ed., Stress intensity factors handbook, Pergamon Press, 1987.6. Rice J. R., A path independent integral and the approximate analysis of strain concentration by

notches and cracks, Journal of Applied Mechanics, 35, 1968, pp. 379–386.7. Tanaka, S., Okada, H., On the analysis of stress concentration problems using wavelet Galerkin

method (3rd report, Adaptive Analysis), Transactions of the Japan Society of Mechanical Engi-neers, Series A, 73–725, 2007, pp. 42-49.

8. Yau J. F., Wang S. S., Corten H. T., A mixed-mode crack analysis of isotropic solids using conser-vation laws of elasticity, Journal of Applied Mechanics, 47, 1980, pp. 335–341.

9. Norikura T., Murakami Y., Application of the body force method to the analysis of stress intensityfactors and the prediction of crack propagation path under two-dimensional mixed boundaryconditions, Transactions of the Japan Society of Mechanical Engineers, A, 49–443, 1983, pp. 818-828.

615

Simulation of Multiphase Fluid Motion in Pore-scale Fractures

M.B. LIU1,∗ AND J.Z. CHANG2

1Institute of Mechanics, Chinese Academy of Science, Beijing 100190, China2School of Mechatronic Engineering, North University of China, Taiyuan, Shanxi 030051, China

1. Introduction

Small scale environmental and geophysical flows are very important, but are usually difficultto simulate because of the associated multiple fluid phases and multiple physics, as well as theexistence of complex geometries and arbitrarily moving interfaces. For example, fluid motionin the vadose zone is very critical for groundwater recharge, fluid motion and contaminanttransport. Flow through fractures and fractured porous media can lead to exceptionally rapidmovement of liquids and associated contaminants.1,2 The physics of fluid flows in unsatu-rated fractures and porous media is still poorly understood due to the complexity of multiplephase flow dynamics. Experimental studies of fluid flow in fractures and fractured porousmedia are limited, and in computer simulations it is usually difficult to take into account thefracture surface properties and microscopic roughness. A broadly applicable model must beable to simulate a variety of phenomena including film flow with free surfaces, stable rivulets,snapping rivulets, fluid fragmentation and coalescence (including coalescence/fragmentationcascades), droplet migration and the formation of isolated single-phase islands trapped dueto aperture variability.

Realistic models for multiphase fluid flows in fracture and fractured porous media must beable to handle moving interfaces, large density ratios (e.g., ≈1000:1 for water and air), andlarge viscosity ratios (e.g., ≈100:1 for water and air). These requirements combined withthe complex geometries of natural fractures present severe challenges to mechanistic mod-els. Grid based numerical methods such as finite difference methods, finite volume methodsand Eulerian finite element methods require special algorithms to treat and track the inter-face between different phases. However, continuum grid based numerical models usually donot take account of the detailed void and obstacle geometries, fluid-fluid interface dynam-ics within pores and complex fluid-fluid-solid contact line dynamics. They rely on consti-tutive equations that describe the coarse-grained behaviour and can, at least in principle,be derived from the results of pore scale simulations or experiments. Therefore, small-scalesimulations with mechanistic models are needed to develop a better understanding of thetemporal and spatial dynamics of multiphase flow through pore-scale structures such as frac-tures and fractured porous media. Pore-scale flows have been studied extensively using gridbased methods including finite difference method,3 finite volume method,4 and finite elementmethod,5 However, due to the difficulties associated with geometrically complex boundaries,fluid-fluid-solid contact line dynamics, and fluid-fluid interface dynamics, it is difficult toapply conventional grid based multiphase simulation methods coupled with interface track-ing algorithms to pore-scale multiphase flow modelling.

Dissipative particle dynamics is a meso-scale particle method. Though it may be less com-putationally efficient than the grid-based methods, it is advantageous in simulating pore-scalemultiphase flow modelling in fractures. DPD is a Lagrangian method, and conserves massexactly. In DPD method, there is no explicit interface tracking — the motion of the fluid isrepresented by the motion of the particles, and fluid surfaces or fluid-fluid interfaces move




with the particles. In this paper we will demonstrate the application of the DPD in simulatingmultiphase fluid flow in fractures with a number of numerical examples.

2. Basic Concept of Dissipative Particle Dynamics

Dissipative particle dynamics6,7 is a relatively new mesoscale technique that can be used tosimulate the behaviour of complex fluids. In DPD simulations, a complex system can be sim-ulated using a set of interacting particles. A particles represent clusters of molecules that inter-act via conservative (non-dissipative), dissipative and fluctuating forces. Because the effectiveinteractions between clusters of molecules are much softer than the interactions between indi-vidual molecules, much longer time steps can be taken relative to molecular dynamics (MD)simulations. A longer time steps combined with a larger particle size makes DPD much morepractical to simulate hydrodynamics than MD. DPD is particularly promising for the sim-ulation of complex liquids, such as polymer suspensions, liquids with interfaces, colloidsand gels. Because of the symmetry of the interactions between the particles, DPD rigorouslyconserves the total momentum of the system, and because the particle-particle interactionsdepend only on relative positions and velocities, the resulting model fluids are Galilean invari-ant. Mass is conserved because the same mass is associated with each of the particles, andthe number of particles does not change.

It is convenient to assume that all of the particles have equal masses, and use the mass ofthe particles as the unit of mass. Newton’s second law governs the motion of each particle.The time evolution for a certain particle, i, is given by the following equation of motion

dri

dt= vi,

dvi

dt= fi = f int

i + f exti , (1)

where ri and vi are the position and velocity vectors of particle i, f exti is the external force

including the effects of gravity, and f inti is the inter-particle force acting on particle i. The

particle-particle interaction is usually assumed to be pairwise additive and consist of threeparts: a conservative (non dissipative) force, FC

ij ; a dissipative force, FDij ; and a random force,

FRij , i.e. f int

i =∑j �=i

Fij = ∑j �=i

FCij + FD

ij + FRij . Fij is the inter-particle interaction force exerted on

particle i by particle j, which is equal to Fji in magnitude and opposite in direction. Thissymmetry of the interactions ensures that momentum is rigorously conserved. The pairwiseparticle interactions have a finite cutoff distance, rc, which is usually taken as the unit oflength in DPD models.

The dissipative force FDij represents the effects of viscosity, and is given by FD

ij = −γwD(rij)(rijgvij)rij, where γ is a coefficient, rij = ri − rj, r = rij = |rij|, rij = rij/rij, vij = vi − vj

and wD(rij) is the dissipation weight function. The random force FRij represents the effects

of thermal fluctuations, and is given by FRij = σwR(rij)ξijrij, where σ is a coefficient, wR(rij)

is the fluctuation weight function, and ξij is a random variable. The fluctuation-dissipation

relationship8 requires wD(r) = [wR(r)]2 and γ = σ2

2kBT , where kB is the Boltzmann constantand T is the temperature. One straightforward choice for the dissipative and random weightfunctions is wD(r) = [wR(r)]2 = (1− r)2, r < 1.

The conservative force, FCij , is a “soft” interaction acting along the line of particle centres,

and has the form FCij = aijwC(r)rij, where aij is the magnitude of the repulsive interaction

strength between particles i and j. For particles from different media, the strength coefficientcan be different. wC(rij) is the weight function for the conservative force. In previous DPD

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implementations, a conservative force weighting function in a simple form of wC(r) = 1− rhas been used. Because the fluid generated by DPD simulations with this purely repulsiveconservative force is a gas, it cannot be used to simulate the flow of liquids with free sur-faces, the behaviour of bubbly liquids, droplet dynamics and other important multiphasefluid flow processes. Including a long-range attractive component in wC(r) is necessary forsuch applications.

3. Modification of the Interaction Potentianl Function

We constructed a new particle-particle interaction potential U(r) by combining the commonlyused SPH cubic spline smoothing functions W(r,rc) with different interaction strengths A andB, and different cutoff distances rc1 and rc2, multiplied by an interaction strength coefficient a

U(r) = a(AW(r,rc1)− BW(r,rc2)) (2)

The DPD conservative particle-particle interaction forces are thus given by FCij = −dU(r)

dr rij.The constructed interaction potential function U(r) consists of short-range repulsive andlarge range attractive interactions (when A > B, and rc1 < rc2) and allows the behaviorof gases, liquids, solids and multiphase systems to be simulated. A certain set of parame-ters A, B, rc1 and rc2 in equation (2) determines the shape of the particle-particle interactionpotential which describes the property of the corresponding fluid. The magnitude of the con-servative force weight function and the location of the transition point from repulsion toattraction should be easily adjustable to allow the behavior of different fluids to be simu-lated. Figure 1 shows the conventional DPD potential function, U(r) = 0.5 − (r− 0.5r2),the cubic spline potential functions, U(r) = W(r,1.0), and two particle-particle interac-tion potential functions resulting from the cubic spline, U(r) = 2W(r,0.8) −W(r,1.0) andU(r) = 2W(r,0.8)− 0.9W(r,1.0).

In DPD simulations, the effects of solid walls are usually be simulated by using fixed par-ticles to represent the solid matrix near the solid-fluid interface. In our implementation, the

0 0.2 0.4 0.6 0.8 1

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

r

U(r

)

0

U(r) = 0.5-(r-0.5r2)

U(r) = 2W(r, 0.8) - W(r, 1.0)

U(r) = W(r, 1.0)

U(r) = 2W(r, 0.8) - 0.9W(r, 1.0)

Figure 1. The conventional DPD potential function, U(r) = 0.5 −(r− 0.5r2

), the cubic spline

potential functions, U(r) = W(r,1.0), and two potential functions resulting from the cubic spline,U(r) = 2W(r,0.8)−W(r,1.0) and U(r) = 2W(r,0.8)− 0.9W(r,1.0).

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entire computational domain is covered by a grid and grid cells are labeled “0” for fluidfilled regions and “1” for solid filled regions. DPD particles are injected into the compu-tational domain randomly until a pre-defined particle number density is reached, and thesystem is then run to equilibrium using a DPD simulation. The particles within the solid cells(marked as “1”) are then ‘frozen’ to represent the solid matrix. In order to reduce computa-tional effort only the frozen particles that are within 1 DPD unit (or rc) from the solid-fluidinterface are chosen as boundary DPD particles. The random distribution of the frozen wallparticles describes the solid surface roughness in a natural way. A reflective boundary wasused in addition to the interactions between fluid and wall particles. The implementation ofno-slip boundary conditions with randomly distributed frozen wall particles was found tobe very flexible, especially for problems with complex geometries such as porous media andfracture geometries.9

4. Multiphase Flow in a Fractured Junction With Fractal Surfaces

The geometry of the fracture junction is shown in Fig. 2, in which the fracture walls wererepresented by self-affine fractal surfaces characterized by a Hurst exponent.10 The size ofthe computational domain is 128 × 3 × 128 in the x, y and z direction. The fracture wallswere represented by 7116 frozen wall particles. The coefficients used in the DPD model wereσ = 3.0 and kBT = 1.0(γ = 4.5). The interaction strength between the fluid particles wasaf = 18.75, and aw, the strength of the interactions between the fluid and wall particles, canbe changed to mimic different wetting behaviors. A modified velocity-Verlet time integrationalgorithm was used for time integration.7 The parameters for the SPH potential and weightfunctions were A = 2.0, rc1 = 0.8, B = 1.0 and rc2 = 1.0. The particle-particle interactionpotentials were given by U(r) = af (2W1(r,0.8)−W2(r,1.0)) for fluid-fluid particle interac-tions and U(r) = aw (2W1(r,0.8)−W2(r,1.0)) for fluid-wall particle interactions.

Figure 2 shows the particle distribution of an injection flow into the fracture junction fromthe top fracture aperture, with an injection rate of 200 particles per 100 steps, an interactionratio aw

/af of 5, and gravitational forces, gx = −0.02, and gz = −0.02 along negative x,

and z direction. A few particles evaporated from the bulk fluid and the flow was a two-phase flow with co-existing liquid-gas phases. The particles near the fracture walls movedinto the aperture faster than those far from the walls, and the fluid exhibited a strong wettingbehaviour with a small contact angle. In contrast to grid-based methods in which the contactangle is imposed on the fluid, the contact angle in DPD is estimated from the position of thewall and liquid particles. The strong wettability of the fluid leaded to a continuous film flow

(a) (b) (c) (d)

Figure 2. Sequential images of an injection into the fracture junction at (a) 1000, (b) 20000, (c) 30000and (d) 40000 steps obtained using DPD method.

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(a) (b) (c) (d)

Figure 3. Sequential images of flow of a liquid droplet through the fracture junction at (a) 300, (b)1100, (c) 2500, and (d) 4300 steps obtained using DPD method.

formed along the fracture wall surface (Figs 2(b), (c), and (d)). A gas bubble was entrappedin the bulk liquid due to the wetting effects and fracture aperture variation (Fig. 2(c)).

Figure 3 shows the sequential images of flow of a liquid droplet through the fracturejunction with an interaction ratio aw

/af of 3, and gravitational forces, gx = −0.05, and gz =

−0.05, along negative x, and z direction. The gravitational force was large enough to offsetthe capillary and viscous forces. Therefore, the liquid drop broke at the fracture junction,and entered the left horizontal and lower vertical apertures. Some liquid was disconnectedfrom the bulk fluid, and formed thin films along the fracture walls. The contact angles of thewetting fluid varied at the upstream and downstream of the bulk liquid, and demonstratedas advancing and receding contact angles. The advancing and receding contact angles variedwith position and time. Further investigation revealed that the contact angle was closelyrelated to af , aw and g, whereas af , aw and g, characterized the dynamic balance of viscous,capillary, and gravitational forces. This velocity dependent contact angle behavior was alsoobserved in real systems.11

5. Multiphase Flow Through a Porous Media Overlying a Fracture WithFractal Surfaces

The geometry of this case is shown in Fig. 4, which is a heterogeneous granular porousoverlying a fracture with self-affine fractal surfaces. The size of the computational domain is128 × 2 × 128 in x, y and z direction. The fracture walls were represented by 9734 frozenwall particles. In x, y and z direction, periodic boundary was applied, where on fracturesurfaces, no-slip boundary was imposed. The coefficients used in the DPD model were σ =

(a) (b) (c) (d)

Figure 4. Sequential images of an injection into a porous media overlying a fracture at (a) 1000, (b)20000, (c) 40000, and (d) 55000 steps obtained using DPD method.

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3.0 and kBT = 1.0(γ = 4.5). The interaction strength between the fluid particles was af =18.75, and the interaction ratio is aw

/af = 0.5. The parameters for the SPH potential and

weight functions were A = 2.0, rc1 = 0.8, B = 1.0 and rc2 = 1.0. Figure 4 shows theinjection of liquid into the porous media with an injection rate of 100 particles per 100steps, and a downward vertical gravitational force, g, of 0.05. Since the interaction betweenfluid particles, af , is larger than the interaction between fluid and wall particles, aw, theliquid exhibits a non-wetting behaviour with a contact angle larger than π

/2. The contact

angle varied with position and time due to the dynamic balance between viscous, capillary,and gravitational forces. The injected fluid particles moved downward, through the porousmedia, as the density of the injected particles and the concomitant pressure increased, andthen reached the fracture aperture. The gravitational force was large enough for some liquidto break from the bulk liquid and form some small liquid drops.

6. Multiphase Flow in a Fracture Network

A numerical study using a volume of fluid (VOF) method was presented by Huang and his co-workers to investigate the unsaturated multiphase flow through a fracture network, togetherwith a flow experiment based on the same fracture network fabricated using polymethyl-methacrylate.12 Here a DPD simulation was also conducted for the same fracture networkgeometry. The size of the computational domain is 100 × 3× 103 in DPD unit in x, y and zdirection. The fracture walls were represented by 13844 frozen wall particles. In x, y and zdirection, periodic boundary was applied, where on fracture surface, no-slip boundary wasimposed. The coefficients used in the DPD model were σ = 3.0 and kBT = 1.0(γ = 4.5). Theinteraction strength between the fluid particles was af = 18.75, and the interaction ratio isaw/

af = 5. The parameters for the SPH potential and weight functions were A = 2.0, rc1 =0.8, B = 1.0 and rc2 = 1.0. In the simulation, the gravitational force was taken 0.3 in DPDunit. In the flow experiment, the apparatus was tilted 2.5◦ in the plane of the photograph.This corresponds to a diagonal gravitational force with a component along both the leftwardhorizontal and downward vertical directions. The injection was conducted at two DPD cellsalong the left fracture wall of the top entrance. This corresponds to the injection of waterinto the top entrance of the channel network using a syringe pump positioned next to the lefttop fracture aperture.

(a) (b) (c)

Figure 5. Comparisons of the injection flow into the channel network at an intermediate stage fromDPD model (a), VOF model (b), and (3) experimental observation (5c).

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Figure 5 shows the comparisons of the injection flow into the channel network at anintermediate stage obtained from DPD model (Fig. 5(a)), VOF model (Fig. 5(b)) and (3)experimental observation (Fig. 5(c)). It is clear that the DPD simulation result agrees qual-itatively with the experimental observation and VOF result provided in.12 There are somediscrepancies if precisely examining the figures from the DPD simulation, VOF simulationand experiment. This is understandable since even for very simple fracture geometries, themultiphase flow can exhibits very complex spatial and temporal behaviours. Small pertur-bations in the simulation and experiment can give rise to quite different flow modes. TheVOF simulation assumed constant fracture aperture with smooth fracture surface, and theadvancing and receding contact angles were prescribed to be constant. The fracture walls inthe experiment are more or less rough rather than smooth in micro-scale, which can affectthe wetting behaviour and flow modes of the fluid. The DPD simulation used randomly dis-tributed frozen wall particles, whose positions determine the roughness of the fracture wallsurfaces, and affect the advancing and receding contact angles.

7. Conclusions

This paper presented the simulations of multiphase flow in complex pore-scale fracturegeometries using a dissipative particle dynamics method. This dissipative particle dynam-ics method employed conservative particle-particle interactions that combine short-rangerepulsive and long-range attractive interactions to simulate gases, liquids, solids and mul-tiphase systems, depending on the average particle density, the temperature and the detailsof the particle-particle interactions. The interaction strength between the fluid particles, andbetween the fluid and wall particles are closely related to the wetting behaviour and thecontact angles.

The simulations revealed that multiphase flow in pore-scale fracture geometries is compli-cated due to the interplay of viscous, capillary and gravitational forces, fracture geometry,and the inflow conditions. Different flow modes can coexist in a complex fracture geometrysystem, including continuous or discontinuous film flow, entrapment of one phase in anotherone, stationary or moving droplets. Small perturbations of the flow regime or fracture surfacecan give rise to quite different flow modes. The advancing and receding contact angles canvary spatially and temporally, depending on the dynamic balance of viscous, capillary andgravitational forces.

References

1. Scanlon, B.R., Tyler S.W. and Wierenga P.J., “Hydrologic issues in arid, unsaturated systems andimplications for contaminant transport”, Reviews of Geophysics, 35, 4, 1997, pp. 461.

2. Nativ, R., Adar E., Dahan O. and Geyh M., “Water recharge and solute transport through thevadose zone of fractured chalk under desert conditions”, Water Resources Research, 31, 2, 1995,pp. 253.

3. Anderson, J.D., Computational fluid dynamics: The basics with applications, McGraw Hill, NewYork, 2002.

4. Chung, T.J., Computational fluid dynamics, Cambridge University Press, 2002.5. Zienkiewicz, O. C. and Taylor R.L., The finite element method, Butterworth-Heinemann, 2000.6. Hoogerbrugge, P. J. and Koelman J., “Simulating microscopic hydrodynamic phenomena with

dissipative particle dynamics”, Europhysics Letters, 19, 1992, pp. 155.7. Groot, R.D., “Dissipative particle dynamics: Bridging the gap between atomistic and mesoscopic

simulation”, Journal Of Chemical Physics, 107, 11, 1997, pp. 4423.8. Espanol, P. and Warren P., “Statistical mechanics of dissipative particle dynamics”, Europhysics

Letters, 30, 4, 1995, pp. 191.

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9. Liu, M.B., Meakin P. and Huang H., “Dissipative particle dynamics simulation of pore-scale flow”,Water Resources Research, 43, 2007.

10. Meakin, P., Fractals, scaling and growth far from equilibrium, Cambridge university press, 1998.11. Cox, R. G., “Inertial and viscous effects on dynamic contact angles”, Journal of Fluid Mechanics,

357, 1998, pp. 249.12. Huang, H., Meakin P., Liu M. B. and McCreery G. E., “Modeling of multiphase fluid motion in

fracture intersections and fracture networks”, Geophysical Research Letters, 32, 2005.

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An Analysis of Model Tests on Rock Cavern Damage Induced byUnderground Explosion

ZHANG XINGUI1,∗, MA GUOWEI2, WU WEI2, YAN LIE1, LI MANGYUAN1 AND CHENG QINGSHENG1

1School of Civil & Architectural Engineering, Guangxi University, Nanning, Guangxi, 530004, China2School of Civil & Environmental Engineering, Nanyang Technological University, 639798, Singapore

1. Introduction

The rock cover thickness required for construction of underground ammunition facilities canbe designed based on currently existing design manuals, mainly US DoD 6055.STD1, NATOAASTP-1 Part III 2, and UK JSP 482.3 It is found that rock cover requirement is scatteredfrom 0.8 Q1/3 to 1.2 Q1/3 (Q: Charge Weight in kg) according to different manuals, wherethe effects of loading density and rock mass strength have not been incorporated. The damagepattern and intensity of rock cover are not addressed in the design manuals. Due to the factthat full scale underground explosion tests are extremely expensive, it is almost impossibleto evaluate rock cover damage based on full scale tests.

The traditional analysis simplifies rock cover failure using a quasi-static approach, whichignored dynamic failure feature of the rock mass.4−7 The design manuals for rock coverare empirical and lack significantly theoretical basis. A rock cover damage model based ondynamic analysis is of special interests for underground ammunition storage design.

The objective of the proposed blast test project is to investigate rock cover damage inducedby underground explosion to support the development of rock cover design criteria for under-ground ammunition storage. The effects of rock mass strength, loading density, and coverdepth on the dynamic failure of the rock cover will be studied through model tests of under-ground explosions, and to find the critical overburden of rock chambers under the conditionsof scheduled underground cave room and ammunition quantity through explosion tests onrock chambers. The model tests include:

(a) Two different rock types, i.e. soft rock and hard rock;(b) Rock chamber with different depths;(c) Detonation with different loading densities.

It is expected that the test results can provide some insight understanding on rock coverdamage induced by underground explosion.

2. Experimental Setup and Scheme

2.1. Design of geometric parameters

The test model and the prototype shall be of geometric similarity, which is a basic require-ment for homogeneous analogue and shall only involve similarity of independent geometricalquantities, such as length, height and distance which are parameters directly influencing thetest results. For non-independent items, such as area, volume, sectional module, their simi-larities are not necessary unless specifically required. Those geometrical quantities that areirrelevant with the main parameters during the model tests are also not required to satisfy the




similarity conditions. Therefore, a test model was established with a scaling rate of 1:200.The chamber in the prototype model was designed as a square shape. However, round cham-ber was selected in the test for the convenience of drilling. A typical test model and theindication of dimensions are shown in Fig. 1, and the dimension of test models are listed inTable 1.

Figure 1. A typical test model and the indication of dimensions.

Table 1. Dimension of test models.

Parameters Model 1 (cm) Model 2 (cm) Model 3 (cm) Model 4 (cm) Model 5 (cm)

Width 58 68 78 88 100Height 44.5 52 59.5 67 76Thickness 50 50 50 50 50Diameter 4 4 4 4 4Thickness ofoverburden

13.5 16 18.5 21 24

W1 27 32 37 42 48W2 27 32 37 42 48h 27 32 37 42 48

Table 2. The physical and mechanical properties of the graniteand marble blocks.

Physical and mechanical properties Granite Marble

Nature weight density 2499 kg/m3 2596 kg/m3

Dry density 2495 kg/m3 2594 kg/m3

Water absorptivity 0.21% 0.15%Uniaxial compressive strength 167 kN 119 kNPoisson’s ratio 0.22 0.23Modulus of Elasticity 39.92 GPa 71.50 GPa

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2.2. Physical and mechanical properties of the rock specimens

Thirty granite and marble blocks were blasted and manufactured from Gongcheng County,Guilin City, Guangxi Zhuang Autonomous Region, P.R.China. The physical and mechanicalproperties of the granite and marble blocks are given in Table 2.

3. Indoor Test Devices

Sensor

The blast test was near-field blasting vibration velocity test for rock particle. It was requiredthat the measurement frequency range of sensors should be sufficient. Thus high frequencyaccelerometers were then used which were able to measure the shock waves at the blocksurface.

Data collection device

The test employed UBOX-1 portable data collection device produced by China SichuanTuopu Digital Devices Co., Ltd, particularly designed for site exploration, vibration, impulse,noise tests and a mini device used for signal recording and analysis.

Topview2000.BM

The software, Topview2000.BM, is used for demolition vibration analysis to provide signalrecording test, data analysis, data management, document management, report productionand so on.

WSD-2 digital sonicator

WSD-2 digital sonicator was used for acoustic wave or ultrasonic non-destructive test instructures and non-ferrous materials such as concrete, rock mass, borehole specimens, ceram-ics and graphite, etc. And it can be used for intensity test, deficiency test of interior structures,detection of splits, and parameters tests of elasticity of materials, etc.

4. Test Procedures

4.1. Peak particle velocity test

Arrangement of test measuring points

Five points for installation of sensors were installed symmetrically against both horizontaland vertical central axial lines on the top surface of the rock blocks. The distance betweenany two points in horizontal or vertical direction is set as h. The layout of five points is shownin Fig, 2.

Powder charge

TNT explosive, made into 12 cm long cylindrical form, was put at the centre of chargehole with detonation tube being controlled by electromagnetic pulse blaster. And the chargehole was sealed with wood stopper. The quantity of explosive and blasting energy weredetermined by required quantity of the test and calculated through scale conversion based onparameters from standards.

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Figure 2. Layout of measuring points.

Control of boundary effect

In order to simulate an underground explosion, the charge hole was closed and fastened bysteel rail, wood plate, rebar and bolts, etc.

Installation of sensors

The five accelerometers were pasted onto the rock surface at the measuring points withhydrous gypsum slurry and connected to a data recorder

Detonation and data recording

TNT explosive was ignited by the electromagnetic pulse blaster. Then test data was down-loaded from the data recorder to a computer and the results were analyzed by Viewtop 2000software. Each rock block was blasted for three times with different charges.

4.2. Integrality test

Arrangement of test measuring points

Integrity test was carried out for each rock block by an acoustic detector prior to the blasttest. The measuring points were selected at the upper portion of two side surfaces of the rockblock to measure the wave acoustic velocity, as shown in Fig. 3.

Site test

The procedure of site test is as follows: erection of acoustic wave detection instrument, acous-tic sensor coated with Vaseline and acoustic sensor clung to measuring point.

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Figure 3. Arrangement of measuring points for acoustic wave examines.

5. Test Results Analysis

5.1. Analysis of peak particle velocity at the top surface

Effect of cover thickness

From the test data, relationship between the cover thickness of granite and marble specimensand the peak particle velocity with different loading density is plotted in Fig. 4, which showsthe peak particle velocity at the top surface versus the cover thickness for the granite andmarble specimens, respectively. With different loading density and rock properties, the peakparticle velocity at the top surface decreases within the increase of the cover thickness.

Effect of loading density

The effect of the loading density on the peak particle velocity attenuation is also shown inFig. 4 for the granite and marble specimens respectively. It is clear that, considering the samecover thickness, the peak particle velocities under 25 kg/m3 loading density are obviously

05

101520253035404550556065

0.135 0.16 0.185 0.21 0.235

Peak p

art

icle

velo

cit

y

(m/s

)

Cover thickness of specimens (m)

loading density of 25kg/m3 marbleloading density of 25 kg/m3 graniteloading density of 10 kg/m3 granite loading density of 10 kg/m3 marble

Figure 4. Contrast of integrative factors.

629


larger than those under 10 kg/m3. It indicates that the higher loading density, the largervibration intensity.

Effect of rock types

The rock type also affected slightly the measured peak particle velocities at the top surface.Fig. 4 compared the peak particle velocities for the two different rocks. It is observed thatthe peak particle velocity in the granite specimen was slightly lower than that in the marblespecimen. It is worth mentioning that the granite has much higher compressive strength. Thereason that the peak particle velocity in granite specimen was lower is probably due to theinherent cracks in the marble specimens which largely dissipated the blast energy.

Contrast analysis of integrative factors

Design of underground ammunition facilities require many relative factors such as coverthickness, loading density, rock types etc. From Fig. 4, two factors, loading density and rocktypes, is in direct proportion to peak particle velocity reflecting degree of rock breakagein the test. When loading density increases and rock medium became harder, peak particlevelocity is higher, vice versa. But cover thickness is in reverse trend. When cover thicknessincreases, peak particle velocity decreases. Under the same explosive condition, the higherloading density, the greater explosive energy releases; the harder rock, the better mediumthrough which stress wave can transmits; the thinner covering layer, the less distance of wavetransmission, thus peak particle velocity is greater, vice versa.

5.2. Integrality test results analysis

Figure 5 and 6 indicate that the acoustic wave velocity after the two blast tests is smaller thanthe original one regardless that the loading density is 10 kg/m3 or 25 kg/m3. In the explosiontest with loading density of 10 kg/m3, the wave velocity reduction rate η of measuring point1 at the direct top of the hole is about 10%, while that of point 2 a little bit far away fromthe hole is smaller than 10%. In the explosion test with the loading density 25 kg/m3, thewave velocity reduction rate η at the measuring points 1 and 2 was larger than 10%.

Figure 5. Acoustic wave velocity data contrastive analysis before and after explosion test with loadingdensity 10kg/m3.

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Figure 6. Acoustic wave velocity data contrastive analysis before and after explosion test with loadingdensity 25 kg/m3.

5.3. Analysis of break-up tests

The energy release of scaled explosion test became larger when the charge weight or loadingdensity was increased. Break-up tests was conducted to examine the capacity of the rockblocks to resist close-explosion. The accelerometers were removed from the rock surfacesince it was doubted that the damage of rock might cause the accelerometers to fail. Thecharge weight was increased gradually until it was able to break the rock blocks.

The break-up tests results reveal that except the charge weight of granite type I and marbletype I is less than 31.25g, which is calculated under loading density of 50kg/m3, explosivequantity of other pieces are more than 50kg/m3 and gradually increases along with incrementof cover thickness of caverns. On the whole, the necessary explosive quantity for marble isless than that for granite.

6. Conclusions

The relation between Q and H is included in current design criteria for cover thickness ofunderground powder magazine is

H = β ·Q 13 (1)

Where, β refers to relation coefficient of Q and H, which is defined within 0.8-1.2.The initial test plan selected β within 0.6-1.0 and the cover thickness was thus designed

by scaling down the charge condition of 100 ton TNT equivalent with loading density of 10kg/m3, 25 kg/3 and 50 kg/3 respectively. From the present break-up tests, the coefficient β inthe existing design guide is slightly larger or conservative.

Based on the acoustic test results, it can be concluded that the empirical damage criteriarecommended by Bauer and Calder (1978)8−9 was very conservative. The acoustic test resultsin the present study showed that the rock blocks did not experience severe damage with themeasured peak particle velocity as high as 63.5 cm/s. However, it should be mentioned hereis that the measured peak particle velocity is from the free rock surface which may be largerthan the free field quantity due to surface reflection.

From the current test results, the rock blocks were maintained the integrity and no majordamage was observed for all the cases when the loading density was 10 kg/m3 and 25 kg/m3.It implied that the coefficient β in the range of 0.6–1.0 is sufficient and the cover thicknessdesigned based on Eq. (1) is very safe to the loading density level below 25 kg/m3.

631


When the loading density increased to about 50 kg/m3 or above, the coefficient β reducedcorrespondingly and its range is between 0.43 and 0.65 which depends on the loading density.This means that the cover thickness requirement as specified in the empirical formula whichgives β in the range of 0.6–1.0 is rather conservative.

References

1. Ammunition and Explosive Standards DOD 6055.9-STD. 1999, US Department of Science: Wash-ington D.C.

2. Nato, Manual on NATO Safety Principles for the storage of Ammunition and Explosives. 2000.p. AC/258, AASTP1-PartIII.

3. ESTC, UK MOD Explosive Safety Regulations — JSP482. July, 2003.4. Raina, A.K., A.K. Chakraborty, M. Ramulu and J.L. Jethwa, Rock mass damage from under-

ground blasting, a literature review, and lab- And full scale tests to estimate crack depth by ultra-sonic method. Fragblast, 2000. 4(2): pp.103-125.

5. Maxwell S.C. & Young R P Seismic imaging of blast damage. Int. J.Rock Mech. Min. Sci &Geomech. Abstr., 1993, 30(7):1435-1440.

6. John S. Rinehart. Stress Transients in Solid. Hyper Dynamics. San Fe, New Mexico, 1975.7. Kutter H.K.and Fairhurst C.F. On the Fracture Process in Blasting, Int. J. Mech. Min. Sci. 1971,

Vol. 8:181-202.8. Singh, S.P. & Lamond R. D. Investigation of Blast Damage and Underground Stability. In: Pro-

ceeding of 12th Conference on Ground Control in Mining. 1993: 366-372.9. Bauer, A. and Calder, P.N. Open Pit and Blasting Seminar. Mining Engineering Department,

Queens University, Kingston, Ontario, 1978.

632

Microscopic Numerical Modelling of the Dynamic Strength ofBrittle Rock

G.F. ZHAO AND J. ZHAO∗

Ecole Polytechnique Federale de Lausanne (EPFL), Rock Mechanics Laboratory,EPFL-ENAC–LMR, Station 18, CH-1015 Lausanne, Switzerland

1. Introduction

Rock, representing both rock materials and rock mass, is the key research object of rockmechanics. Unlike any other man-made materials, rock usually has experienced a geolog-ical history involving appreciable mechanical, thermal and chemical actions over millionsof years. Complex structures can be detected in rock materials as well as in rock mass (asshown in Fig. 1), which make the mechanical properties of rock more complex than thoseof other man-made materials. This structural complexity of rock plays an important rolein rock mechanics, especially the strength and fracture pattern of rock materials. Dynamiceffect of rock material, influence of strain rates on strengths and deformational modules etc.,is one of the most important research issues in rock dynamics. It is a key element in thesolution of many engineering problems involving dynamic loading conditions. In order todescribe the dynamic strength of brittle rock, based on experimental results Zhao1 developedthe dynamic version of Mohr-Coulomb and Hoek-Brown criteria by adding a loading-rate-dependent term. However, the mechanism governing the rate-dependent behaviour of rockmaterials is still not clear now. Different kinds of models, such as heat activation theory,2

spring-dashpot models3 and sliding crack model4 are try to explain the dynamic effect. Inthis paper we try to explain the dynamic effect based on the microscopic mechanical responseof rock material.

In recent years, researchers have realized that it is important to consider the microstruc-ture of a material when studying its macroscopic mechanical properties. For example, it isdifficult to assign a unique value of fracture toughness to rock materials without cognizanceof their microstructural characteristics.5 Microscopic experimental observations will offerpromising explanation to dynamic effects of rock material failure. The three-dimensionaltexture of granite was observed in microscope through using an ultra-bright synchrotronradiation (SR)–CT system and time-dependent fracturing behaviour was studied.6 The scan-ning electron microscope (SEM) is another used device, e.g., micro-cracking and propagationof concrete at different temperatures was studied through SEM.7 Recent developments inlaboratory-based micro X-ray diffraction have extended X-ray examination of geo-materialsto the microscopic level (50–500 um).8 This will help researchers perform further studyingon micromechanics of rock materials. However, these experimental methods have been lim-ited by the instruments. For example, CT and SEM are only applicable at low loading rateswhich could not well study the dynamic responses of rock materials. This would become abarrier of performing study on the dynamic effect of rock materials.

With the rapid advancements in computing technology, numerical methods provideextremely powerful tools for rock mechanics. There exist a large number of numerical meth-ods which have been applied to study the microscopic mechanical behaviours of rock materi-als. The most well-known model is Bonded-particle model (BPM) which can describe damage




Figure 1. Microstructure of sandstone (left) and structure of rock mass (right).

mechanisms and time-dependent behaviours by adding a damage-rate law.9,10 The contin-uum based methods also have been used, e.g., combining continuum damage model withelement delectation technique and the Weibull distribution model FEM was successfully usedin micromechanics study of rock materials.11 Similar works were also done by Prisco andMazars to analyze the crush-crack of concrete12 and Du et al.13 to study the influence ofstrain rates on the dynamic tensile strengths of concrete. The SPH method was also used tosimulate the dynamic behaviours of rock material at microscope.14 However, both the con-tinuum based methods and discontinuum based methods share their own limitations. As thecontinuum based methods are good at the pre-failure stage while bad at the post-failure stage,the discontinuum based methods have reserve performance. Recently, the authors proposeda microstructure based model of elasticity and developed the corresponding numerical modelbased on the theory. The proposed numerical model could reconstruct the elasticity solutionbefore failure and smoothly change into discontinuum response after failure happen. Thismakes it a promising choice to study the microscopic mechanical response of rock materials.For this reason, the model will be used to study the dynamic strength of brittle rock in thispaper. The modelling results show that the model could produce the dynamic Hoek-Browncriterion through a simple micro failure law. From the modelling results it is also observedthat the microstructure of rock materials plays an important role for the dynamic strength ofbrittle rock.

2. The Distinct Lattice Spring Method

2.1. The real multi-dimensional internal bond model (RMIB)

The RMIB model15 is an extension of VMIB16 in which materials are discretized into massparticles linked through distributed bonds. The microstructure of the model is shown inFig. 2 in which spherical particles are distributed randomly in space. The particles are notrestricted to the same size. Whenever two particles are detected in contact, they are linkedtogether through bonds between their center points. The multidimensional internal bond ofVMIB is adopted, that include one normal bond and one shear bonds. The shear bond is avector spring in 3D case and a normal spring in 2D case. The model is a useful description forfracture modelling of materials such as rock and concrete. Due to the explicit considerations

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y

z

x

Contact point i

L spring

R spring

(b) Multi-dimensional internal bond (a) Continuum element

Internal bond

Figure 2. Microstructure of the real multi-dimensional internal bond model (RMIB).

of the microstructure of the material, the proposed micromechanical model has the poten-tial to give more realistic modelling of material failure behaviours than a phenomenologicalmodel does.

Based on Cauchy-born rules and the hyperelastic theory, the relationship between themicromechanical parameters and the macro material constants, i.e. the Young’s modulusand the Poisson ratio can be obtained as follows:

kn = 3E

α3D(1− 2ν)(1)

ks = 3(1− 4ν)Eα3D(1+ ν)(1− 2ν)

where k is the normal stiffness of the spring, k shear stiffness, E Young’s modulus, v Poissonratio and α3D is a microstructure geometry coefficient which can be obtained from:

α3D =∑

l2iV

(2)

where li is the original length of the ith bond, V is the volume of the geometry model. Thedetails of this model can be found in.15

2.2. The Distinct Lattice Spring Model (DLSM)

The Distinct Lattice Spring Model17 is a numerical method based on the RMIB model ratherthan elasticity equations. In DLSM, material is discretized into mass particles with differentsize. Whenever the gap of particles are following a given threshold value, they are linkedtogether through bonds between their center points. The threshold value will influence thelattice structure of the model. Different threshold value would produce different lattice struc-tures. The particles and springs make a whole system which represents the material. For thissystem, its motion equations can be expressed as

[K]u+ [C]u+ [M]u = F(t) (3)

where u represent the vector of particle displacement, [M] the diagonal mass matrix,[C] the damping matrix, F(t) the vector of external forces on particles. In DLSM, the motion

635


equations of particle system are solved through the explicit center finite differences scheme.The interaction between particles is represented by one normal spring and one shear spring.The shear spring is a multi body spring which is different from the conventional lattice springmethods. The multi-body shear spring is introduced to make the model can handle problemswhich Possion’s ratio is beyond 0.25. The behaviour of normal spring is in a conventionalway. For example there existing one bond between particle i and particle j. The unit normaln(nx,ny,nz) points form particle i to particle j. The relative displacement is calculated as:

uij = uj − ui. (4)

Then vector of normal displacement and interaction force between two particles can be givenas

unij = (uij • n)n and Fn

ij = knunij (5)

where kn is the stiffness of the normal spring. The multi-body shear spring between twoparticles is introduced through a spring with a multi-body shear displacement vector. Theshear displacement between two particles is evaluated by a local strain state. Assume thestrain of each particle is evaluated as [ε]i and the strain state of bond is given as the averagevalue of two particles.

[ε]bond =[ε]i + [ε]j

2(6)

where [ε] =⎡⎣ εxx εxy εxzεyx εyy εyzεzx εzy εzz

⎤⎦. The shear displacement vector of the bond is given as

usij = [ε]bondnT − (([ε]bondnT) · n)n. (7)

Then the shear interaction between two particles is given as

Fsij = ksus

ij (8)

where ks is the stiffness of the shear spring. Equations (7) and (8) are available for unbrokenbonds.

When the normal or multi-body shear displacement of the bond is exceeding the prescribedvalue, the bond will be broken. After failure happen the bond will change into a contact bondwhere only normal springs with zero strength is applied. At current stage, only a simplefracturing law is adopted and more comprehensive study on the fracture is needed. We cansee that there are only two spring parameters and two failure parameters in the model. Thismakes the model can be used easily. It is much suitable for microscopic modelling as theless inputted parameters the better and easier to focus on the microstructure influence onmechanical response.

3. Microscopic Study the Dynamic Strength of Brittle Rock

The computational model used in this paper is shown in Fig. 3(a) and a constant vertical andopposite displacements is imposed on the particles that make up the top and bottom of the

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(a) The numerical model (b) Micro-discontinuities (c) tensile failure (d) Compressive failure

Figure 3. The computational model and failure pattern of uniaxial tensile and compressive test.

specimen to create a piston-like effect. The applied constant rate of strain is

ε1 = 2h0

∂

∂t(uz(t)) = 2vz(t)

h0(9)

where h0 is the sample height and uz(t) is the imposed displacement. The dimension of thesample is 10×10×20 mm and the size of particle is around 1 mm. There are about two thou-sands of particles in the model. It is targeting at studying the macroscopic dynamic responseof Bukit Timah granite based on a microscopic numerical model. The inputted Young’s mod-ulus and Possion’s ratio are 73.9 Gpa and 0.15. The limited value of bond’s tensile and shearstretching are 0.0002 mm and 0.0024 mm. In order to reproduce the interlocking effect ofrock materials, two types of microstructures are used in the paper. The first one is ran-domly enriching the strength of bonds of a given percent. The second one is to enrich thebonds which do not cut by the randomly distributed discs (as shown in Fig. 3(b)). From thesimulation results it shows both these two method can produce the required inter-locking

0 0.002 0.004 0.006 0.008 0.01 0.0120

2

4

6

8

10

12

14

16x 10

7

strain

stre

ss (

Pa)

confining stress = 4Mpaconfining stress = 8Mpaconfining stress = 12Mpa

0 0.002 0.004 0.006 0.008 0.01 0.0120

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

8

strain

stre

ss (

Mpa

)

confining stress = 4Mpa

confining stress = 8Mpaconfining stress =12Mpa

(a) The random bonded model (b) The micro-discs model

Figure 4. The strain-stress curves of the different microstructure model.

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effect. The tensile failure and compressive failure of the computational model are shown inFig. 3 (c) and (d).

Uniaxial tensile, uniaxial compressive and tri-axial tests are performed under these threedifferent strain rates as 10−1, 10 and 101 Fig. 4 shows the strain-stress curves of the differentmicrostructure model under different confining stress. Increasing the confining pressure thecorresponding strength will increase which is in agreement with experimental observations.The results show that the microstructure have influence the strength of the model. The uni-axial compressive tests under different strain rate are performed. The results are given inFig. 5 which shows that the dynamic strength will increase with the increasing of strain rate.It should be mention that there is no rate law used in the model, the dynamic effect is aresult rather than a constitutive effect. Figure 6 shows the comparison of the failure envelope

0 0.5 1 1.5 2 2.5 3 3.5 4

x 10-3

0

2

4

6

8

10

12

14x 10

7

strain

stre

ss (

Pa)

strain rate = 10-1

strain rate = 100

strain rate = 101

0 1 2 3 4 5 6

x 10-3

0

2

4

6

8

10

12

14

16

18x 10

7

strain

stre

ss (

Pa)

strain rate = 10-1

strain rate = 100

strain rate = 101


Figure 5. The strain — stress curves of uniaxial compressive tests under different strain rates.

-20 -15 -10 -5 0 5 10 150

20

40

60

80

100

120

140

160

180

S3 (Mpa)

S1

(Mpa

)

Hoek-Brown with strain rate 10-1

DLSM with strain rate 10-1

Hoek-Brown with strain rate 100

DLSM with strain rate 100



-20 -15 -10 -5 0 5 10 150

50

100

150

200

250

S3 (Mpa)

S1

(Mpa

)

Hoek-Brown with strain rate 10-1

DLSM with strain rate 10-1






Figure 6. The Hoek-Brown failure criterion and the envelope obtained from numerical simulation.

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obtained from numerical simulations with that of Hoek-Brown failure criterion. The resultsreview that the numerical modelling could reproduce the Hoek-Brown failure criterion whichderived from experimental data based on a simple microscopic failure law.

4. Conclusions

The dynamic strength of brittle rock is studied through a microscopic numerical modelDLSM, some conclusions are derived from the numerical modelling:

• The existing of microstructure and inter-locking have great influence on the strength ofrock materials.• Different microstructures will produce different strength.• The dynamic effect of brittle rock material could be reproduced through a microscopic

model which does not include a rate-dependence law.• The Hoek-Brown criterion could be reproduced through the numerical model with a

simple micro-failure criterion.

Acknowledgements

Financial support from the China Scholarship Council to the first author is gratefully acknowl-edged. The research is also partially supported by the Swiss National Science Foundation(200021-116536).

References

1. Zhao, J., “Applicability of Mohr-Coulomb and Hoek-Brown strength criteria to the dynamicstrength of brittle rock”, International Journal of Rock Mechanics and Mining Sciences, 37(7),2000, pp. 1115–1121.

2. Kumar, A., “Effect of Stress Rate and Temperature on Strength of Basalt and Granite”, Geophysics,33(3), 1968, pp. 501–510.

3. Chong, K.P., Hoyt, P.M., Smith, J.W. and Paulsen, B.Y., “Effects of Strain Rate on Oil-Shale Frac-turing”, International Journal of Rock Mechanics and Mining Sciences, 17(1), 1980, pp. 35–43.

4. Li, H.B., Zhao, J. and Li, T.J., “Micromechanical modelling of the mechanical properties of agranite under dynamic uniaxial compressive loads”, International Journal of Rock Mechanics andMining Sciences, 37(6), 2000, pp. 923–935.

5. Nasseri, M.H.B. and Mohanty, B., “Fracture toughness anisotropy in granitic rocks”, Interna-tional Journal of Rock Mechanics and Mining Sciences, 45(2), 2008, pp. 167–193.

6. Ichikawa, Y., Kawamura, K., Uesugi, K., Seo, Y.S. and Fujii, N., “Micro- and macrobehaviorof granitic rock: observations and viscoelastic homogenization analysis”, Computer Methods inApplied Mechanics and Engineering, 191(1–2), 2001, pp. 47–72.

7. Wang, X.S., Wu, B.S., Wang, Q.Y., “Online SEM investigation of microcrack characteristics ofconcretes at various temperatures”, Cement and Concrete Research, 35(7), 2005, pp. 1385–1390.

8. Flemming, R.L., “Micro X-ray diffraction mu XRD: a versatile technique for characterization ofearth and planetary materials”, Canadian Journal of Earth Sciences, 44(9), 2007, pp. 1333–1346.

9. Cho, N., Martin, C.D. and Sego, D.C., “A clumped particle model for rock”, International Journalof Rock Mechanics and Mining Sciences, 44(7), 2007, pp. 997–1010.

10. Potyondy, DO., “Simulating stress corrosion with a bonded-particle model for rock”, InternationalJournal of Rock Mechanics and Mining Sciences, 44(5), 2007, pp. 677–691.

11. Tang, C.A. and Kaiser, P.K., “Numerical Simulation of Cumulative Damage and Seismic EnergyRelease During Brittle Rock Failure-Part I:Fundamentals.” Int. J. Rock Mech. & Min. Sci, 35(2),1998, pp. 113–121.

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12. Prisco, M. and Mazars, J., “Crush-crack a non-local damage model for concrete”, Mechanics ofCohesive-Frictional Materials, 1(4), 1996, pp. 321–347.

13. Du, J., Kobayashi, A.S. and Hawkins, N.M., “FEM dynamic fracture analysis of concrete beams”,Journal of Engineering Mechanics, 115(10), 1989, pp. 2136–2149.

14. Ma, G.W., Dong, A.A. and Li, J., “Modeling strain rate effect for heterogeneous brittle materials”,Transactions of Tianjin University, 12 (SUPPL), 2006, pp. 79–82.

15. Zhao, G.F., Fang, J., Zhao, J., “A new microstructure-based constitutive model for failure model-ing of elastic continuum”, European Journal of Mechanics, A/Solids, 2009, (submitted).

16. Zhang, Z.N. and Ge, X.R., “Micromechanical consideration of tensile crack behavior basedon virtual internal bond in contrast to cohesive stress”, Theor. Appl. Fract. Mech., 43(3), 2005,pp. 342–59.

17. Zhao, G.-F., Fang, J.N. and Zhao, J., “A 3-D distinct lattice spring model for elasticity anddynamic failure”, Communication in Numerical Methods in Engineering, 2009, (Submitted).

640

Fault Studies and Coal-gas-outburst Forecast in Coal Mines

H.Q. CUI∗, X.L. JIA, Z.P. XUE, AND F.L. YANG

Henan Polytechnic University, Jiaozuo, China 454003

1. Introduction

Coal-gas-outburst accidents in coal mines have been the most serious geological hazards inChina. The situation has become worse than ever with the mining of deeper coal seams athigher production rates in a more complex underground mining environment, in order tomeet the huge national demand for coal-fuelled power plants.1 A program named “four inone” has been carried out to prevent coal and gas outburst in all coal mines with outburstproneness in China.2 The program includes four steps: (1) Outburst proneness forecast ofcoal seam, (2) Application of preventing methods, (3) Available examination of applied pre-venting methods, and (4) Application of safe measures. The most basic step in the program isto forecast outburst proneness, which is directly related with what and how many preventingmethods and measures should have to be applied in a mining area. Therefore, precision offorecasting outburst proneness of a special mining area has been required as high as possibleto save unnecessary preventing engineering under safe consideration. According to coal min-ing practices, it has been found that fault studies are very useful to more accurately forecastdangerous mining sites where coal-gas-outburst accidents may occur if preventing measureswould not be taken suitably. Based on fault studies, safer mining design and efficient preven-tion of coal-gas-outburst accidents could pursue in coal mining operation.

2. Relationship Between Faults and Coal-gas-outburst Accidents

2.1. It is around faults where coal and gas outburst usually takes place

Based on reported cases in Henan, more than seventy percent of coal-gas-outburst accidentstook place around different-scale faults. Fig. 1 is two examples in Henan that coal and gasoutburst accidents concentrate in influencing zones of the faults known as Xingdian NormalFault and Niuzhuang Reverse Fault. Eight and eleven outburst accidents took place respec-tively around the two faults.

2.2. Reactivated faults result in coal and gas outburst

It is common features that faults have activated more than once since they were developed ata special geological period. The reactivated faults are easier to result in coal and gas outburstbecause there is more developed tectonically disturbed coal or soft coal in their influencingzones 4,5. The soft coal usually provides necessary physical condition of outburst accidents.Furthermore, human mining action may change the stress and strain states around a faultingzone and promote it reactivation. The reactive faulting zone reduced by human mining mayresult in coal and gas outburst as well. It has been proved that the stress concentration andreactivation of fault surface resulted from the roadway heading to it should be the key causeof the outburst accident. Fig. 2 is the modeling result of an outburst accident happened in




N50 m 40 m

N

H3

H2

H1

ba

Xindian normal fault

Niuzhuang

reversefault

Niuzhuang

syncline

Rail

dip

Ra

ilra

ise

fault coal line normal fault reverse fault coal and gas outburst positionsyncline

Belt conveying roadway

Figure 1. Concentrated coal and gas outburst accidents around faults (After Guo, D. and Han, D.,1998).

Figure 2. Distribution of maximum principal stress around a fault end while a coal roadway headingto it.

Henan through finite-difference method, showing that the stress concentration and the highdeviatoric stress values at end of the reverse fault above the coal roadway provide the keycondition of coal-gas outburst with high compressive gas in coal seam of hanging-wall.

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2.3. Different fault types and fault systems have different criticality forcoal and gas outburst

Different fault types have different function on controlling gas content in coal seam. In gen-erally, the faults with tensional features hold less gas content than those with shear or com-pressive features. When the faults are currently under control of compressive or shear stressfields, more serious outburst accidents may take place around them because of their sealingfaulting zones and more blocked gas content.6 Because normal fault systems, such as half-graben and horst-and-graben, are helpful for gas escape in coal seam, few of them result incoal-gas-outburst accidents; the reverse fault systems, otherwise, are more prone to promotecoal and gas outburst. Furthermore, there may be different gas content zones in differentparts of a fault system, and different dangerous degree for outburst. Figure 3 is an exampleof reverse fault systems, where the part có is of outburst proneness, and the part cñandcòareusually quite safe because there is no deep coal seam which supplies them with enough gasresource. There are different zones of gas content around faults, known as gas content losszone, reduced zone and increased zone. The gas content increased zones adjacent to the faultare more dangerous areas of outburst than other zones.

1

2

3

2 2

3

4

5

� � �

sealing fault plane gas flow

1-gas loss zone 2-gas content reduced zone 3-gas content increased zone4-gas content normal zone 5-surface

Figure 3. Different gas content zones in a reverse fault system.7

2.4. Around vanishing end of fault is the most dangerous sitewhere outburst accident may occur

Vanishing ends of faults usually concentrate higher stress, and around the ends there are moreaccompanied or induced joints and fractures than at other parts of the faults.8 Therefore, itis possible to centralize more free gas with high pressure around vanishing ends of fault incoal seam. It is why many outburst accidents took place around the vanishing ends of faultsin Henan (Fig. 1). Figure 4 is a maximum principal stress contour map that shows stressconcentration around vanishing ends of a fault under general triaxial stress state. The stressconcentration around vanishing ends of fault is quite clear, and high stress difference betweenhanging-wall and footwall is obvious as well (Fig. 5). All these features provide advantagefor coal and gas outburst at vanishing ends of faults during coal mining.

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Figure 4. Stress concentration around vanishing ends of fault (Fault surface dip: 45◦. Modeling soft-

ware: FLAC3D).

Figure 5. Relationship between maximum principal stress difference and distance away from the faultplane.

3. Safer Mining Design and Better Preventing MeasuresBased on Fault Studies

Based on fault studies, safer mining design and better measures for preventing coal–gas-outburst accidents could be made around faults. For example, excavation layout and working

644


order could be arranged more helpful for stress relief around vanishing ends of faults, andeven stress relief blast could be applied when a coal roadway is heading to the vanishing endof fault. Preventing engineering and measure for coal-gas outburst could be reduced aroundthe faults and fault systems with tensional features.

4. Conclusions

The following conclusions are deduced from above examples and analysis:

• Coal-gas-outburst accidents have direct relationship with faults. Fault studies canhelp to more accurately forecast the dangerous sites where outburst proneness maybe high in a mining area.• Serious coal-gas-outburst incidents usually take place around reactivated faults. The

reactivation of a fault can be induced in geological history or while currently coalmining.• Different mechanical types of faults and fault systems have different effects on coal

and gas outburst. Contractional faults and contractional fault systems are more help-ful for holding gas content and for promoting coal and gas outburst than extensionalfaults and extensional fault system.• The most dangerous sites, where coal-gas-outburst accidents may occur, are located

near to the vanishing ends of faults because there is higher stress concentration there.

References

1. Liu, M., Mitri, H., and Wei, J., Recent Trends of Coal and Gas Outburst Accidents in China, 27thInternational Conference on Ground Control in Mining, 2007. Web site: http://218.196.244.90/.

2. Coal Industry Department, Detailed Rules for Preventing Coal and Gas Outburst, China CoalIndustry Publishing House, Beijing, China, 1995.

3. Guo, D. and Han, D., Research on the Types of Geological Tectonic Controlling Coal Gas Out-bursts, Journal of China Coal Society, 4, 1998.

4. Wang, N., Study on Geological Structure Projecting Mechanism, Journal of Jiaozuo Institute ofTechnology, 3, 2005.

5. Zhang, Y., Zhang, Z., and Cao, Y., Deformed-coal Structure and Control to Coal-gas Outburst,Journal of China Coal Society, 3, 2007.

6. Tang, Y. and Cao, Y, Study on the Outburst Prone of Different Fault Sides, Journal of Coal Science& Engineering, 6, 2002.

7. Yu, Q., Prevention and Treatment of CMM, China University of Mining and Technology Press, XuZhou, Jiangsu, China, 1992.

8. Su, S., Affection of Faults on Stress Field and its Significance on Engineering, Chinese Journal ofRock Mechanics and Engineering, 2, 2002.

645

Suggestion of Equations to Determine the Elastic Constants of aTransversely Isotropic Rock Specimen

CHULWHAN PARK1, CHAN PARK1, ∗, E.S. PARK1, Y.B. JUNG1 AND J. W. KIM2

1Korea Institute of Geoscience and Mineral Resources, Daejeon, Korea2Cheongju University, Cheongju, Korea

1. Introduction

Anisotropy which can easily be found in stratified sedimentary rocks and foliated meta-morphic rocks is one of characteristics to realize the rock mass or the intact rock. Someof volcanic rocks show it despite of apparent isotropy. These rocks which have undergoneseveral formation processes may present more than one direction of planar anisotropy, andthese directions are not necessarily parallel or perpendicular to each other. It has alreadybeen counted that rock anisotropy is important in civil, mining and petroleum engineeringbefore Amadei (1996) pointed out in Schlumberger Lecture Award paper. Improvements ofthe computer and the analysis program enable to simulate a rock mass to be an anisotropicbody more virtually in the design of underground structures.

Orthotropy which implies the different properties in the three mutual perpendicular direc-tions may represent the rock mass such as coal with bedding plane and cleat. And rockmasses usually have several types of planar anisotropy in a rock mass. But it has been knownthat the identification of nine independent elastic constants of an orthotropic body is veryhard and three or more specimens are need for laboratory test. For this reason, orthotropicrock may be referred imaginary and anisotropic rock are generally regarded as a transverselyisotropic body which can reduce the number of independent elastic constants to five (Wittke,1990).

Many researchers have studied the varieties of strengths and deformabilities of theanisotropic rock. It is still difficult to characterize five constants by compression tests inlaboratory, even though the numerical design with a model simulation is not complicatedany more. If many specimens from a rock mass are given, its five constants can be deter-mined without any assumption. But this might still not be easy because it will be practicallydifficult to get many field specimens which should be uniform each other. In engineering,the limited amount of or even only one specimen is occasionally given. With a single spec-imen of the transversely isotropic rock, the number of independent strains to be obtainedin measurement is only four and one equation should be assumed. Eq. (1) has been usedafter Saint-Venant’s approximation in 1863 and many researchers have approved its valid-ity through many kinds of tests. FLAC program (Itasca, 2005) is still using this equation initself.

1G2= 1

E1+ 1+ 2ν2

E2(as defined by

1G∗ in this paper) (1)




But some studies reported that this empirical relation was not revealed to be acceptableand the modified equations were also suggested. Worotnicki (1993) reviewed the reportedresults of over 200 tests and analyzed that quartzfeldspathic and basic/litic rocks, low tomoderated degrees of anisotropy, G2 conforms closely to Saint-Venant’s approximation. Healso found actual G2 can be twice to three times lower than this for rocks in other classes andhe suggested the possibility of a new equation as Eq. (2), independent of Eq. (1). Talesnickand Ringel (1999) investigated by torsional shear tests that G2 can be much higher than thatfrom Eq. (1) and G2 can be defined as Eq. (3) with a correction factor.

G2 = G1E2/E1 (2)

G2 = G ∗ (2E1 − E2)/E1 (3)

Very recently Gonzaga et al. (2008) proposed the methodology to determine elastic con-stants for a single specimen with nine strains measurement. The predicted strains have alsoerrors to experimental data for St-Marc limestone by using Eqs. (1) and (3). And the non-linear behavior caused by porosity and microcracks was taken consideration into this method-ology. Duevel and Haimson (1997) also studied non-linearity and anisotropy for pink LacDu Bonnet granite.

Throughout testing experiences in our laboratory, Eq. (1) generally yields the acceptableranges of constants for rocks with steep angle of dip. But the unreasonable ones have occa-sionally been obtained in most cases of the flat specimens, which may have cause for the needof another approximation replacing Eq. (1). This study aims to suggest new equations andmeasuring directions of four independent strains in order to determine the five constants ofa single specimen for a transversely isotropic rock. The equations are originated by a differ-ent approach of Eq. (1) and do not consider the nonlinearity. Validation of each equation isdiscussed in simple numerical models and actual rocks under uniaxial compression tests aswell.

2. Theory Review

The stress - strain relations for a linear elastic material are defined by generalized Hooke’slow.(Flugge, 1972) The elastic moduli represent a tensor of the fourth order and the numberof components is 81. By using theory of elasticity, independent moduli are 9 for an orthotropyand only 2 for an isotropy.

The rock material may be realized an anisotropy rather than an isotropy on the presence ofbedding, stratification, foliation, schistosity or jointing. Orthotropy which implies the differ-ent properties in the three mutual perpendicular directions may represent the rock mass suchas coal with bedding plane and cleat. But it is very hard to identify the 9 independent elasticconstants of an orthotropic body and three or more specimens are need in laboratory test.For this reason, orthotropic rock may be referred imaginary and anisotropic rock is generallyregarded as a transversely isotropic body which can reduce to 5 of independent elastic con-stants. (Wittke, 1990) Eq. (4) is the constitutive equations for a transversely isotropic body.There are 5 constants of the second order, even expressed in the first order, and G1 ( = G13)is dependent and two equation are duplicated. Thus 5 constants (E1, E2, ν1 = ν13 = ν31,ν2 = ν21 and G2 = G21) and 4 equations are independent each other. ν12 is different fromν21, but is a dependent constant on symmetry of the compliance matrix as in Eq. (5) which

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is already applied in expansion of Eq. (4).⎡⎢⎢⎢⎢⎢⎢⎣

ε1ε2ε3�23�31�12

⎤⎥⎥⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎢⎢⎣

1/E1 −v2/E2 −v1/E1 0 0 0−v2/E2 1/E2 −v2/E2 0 0 0−v1/E1 −v2/E2 1/E1 0 0 0

0 0 0 1/G2 0 00 0 0 0 1/G1 00 0 0 0 0 1/G2

⎤⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎣

σ1σ2σ3τ23τ31τ12

⎤⎥⎥⎥⎥⎥⎥⎦

(4)

νij/Ei = νji/Ej (5)

In this reason, two or more specimens with different shapes should be tested if manyspecimens from a rock mass are given. This experiment with multi-tests and multi-shapedspecimens may be a hard work and it will also be practically difficult to get many field spec-imens which should be uniform each other. In engineering, the limited amount of specimenor even only one specimen is occasionally given. In the single specimen test, one suppositionsuch as Eq. (1) might be essential to identify the 5 independent elastic constants.

3. Directions of Strain Measurements

As in Eq. (4), number of the independent strains obtained is 4 in maximum for a singlespecimen in compression test. Fig. 1 explains the directions of strain measurements and theirnotations as well. Strain rosette is placed on the infinitesimal plane A in the direction of x,y, and 45 degree (actually −45 degree in mathematics) when compression load acts on ydirection and one gauge is on B in z direction.

Strains in other direction are all dependent on those by using the transformation in theoblique coordinate. There of course may be other sets of 4 gauges, but they absolutelymeasure smaller amounts of strains, which can increase the errors in measurements. It wasreported that the maximum principal strain on A acts almost same as in y directions, withindependence of the amounts of elastic constants. (Park, 2001) In order to reduce the error instrain measurements and to upgrade the accuracy in experiments, it would be recommendedto set another 4 strains in same way on the other sides of planes A− B.

The angle, φ is the angle of dip of the transversely isotropic plane (e.g. bedding plane) andcan be defined as the anisotropic angle. The axis of symmetry of rotation has the angle ofπ + φ.

4. First Suggestion

As described above, 4 strains from the adequate measurement require one more equation.Throughout our experiences and many other researches, Eq. (1) have not yield acceptablevalues of constants not only for the rock engineering range that ν varies 0.11 ∼ 0.46 but alsofor the thermodynamic constraints of −1∼1 (Amadei, 1996). In this moment the fifth equa-tion would now like to be suggested as Eq. (6) by using Ea, the apparent Young’s moduluswhich can be derived directly from strain measurements as in Fig. 1.

This equation is independent of Eq. (4) and also looks independent of Eq. (1) becauseits origination and terms are totally different each other. This may not consider the elasticconstraints (if yes, it may be dependent equation and useless any more) except the threecritical cases that φ = 0, φ = π/2 and E1 = E2.

1Ea= 1

E1sin2 φ + 1

E2cos2 φ (6)

where, Ea = σy/εy

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(a) strain rosette on plane A

(b) lateral strain on plane B

Figure 1. Axis of a transversely isotopic specimen and strain gauges.

However, the resultants by applying Eq. (6) are just same as that by applying Eq. (1) insolution of 5 constants with 4 strain measurements. After the identity of two approximationsis become aware from the analysis result, it has been proved in mathematics by using thedirection cosines.

As both are identified as identical each other, the historical equation can be characterizedby analyzing Eq. (6), which is that the apparent Young’s modulus is a function of E1, E2 andφ. And it is a monotonous increasing function by the differential (dEa/dφ, Fig. 2), if E1 isbigger than E2.

Nasseri et al. (2003) concluded the relation of Ea − φ under 2 categories that the oneshows U or W shapes with peak(s) and the other shows monotonous by collecting data fromreported test results. They carried out unconfined and confined compression tests on fourtypes of Himalayan schists of which two show U shapes and other two show monotonousas in Fig. 4.

It will be concluded that the suggested equation in this paper is compatible on accountof identity of the historical equation and those may be useful to determine 5 independentelastic constants in single specimen test of the rocks with the monotonous relation. And it isalso found that the other equation should be needed for the rocks with the non-monotonousrelation.

App

aren

t You

ng’s

Mod

ulus

Anisotropic Angle0 10 20 30 40 50 60 70 80 90

E2

E1 > E2

E1 < E2

Figure 2. Apparent Young’s modulus on angle.

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Figure 3. Relations of Ea − φ in literatures (top; diatomite under uniaxial compression after Allirotand Boehler, 1979, bottom; artificial rock block I after Tien and Tsao, 2000).

Figure 4. Relations of Ea − φ of four types of Himalayan schists (after Nasseri et al., 2003).

5. Secondary Suggestion and Validation

5.1. Suggestion of Equations

Based on the effectiveness of Eq. (6) and the need of new one, two more equations aresuggested here as Eqs. (7) andby using the apparent Poisson’s ratio which can directly be

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derived from strain measurements as in Eq. (9). These equations may not consider the elasticconstraints except the three critical cases like in Eq. (6). Each has an advantage to apply tothe rocks whether the Ea−φ relation is monotonous or not. They may have a role to replacethe previous equations at least in rocks of which the Ea − φ relation is not monotonous.

νa = 0.5(ν12 + ν21) sin2 φ + ν2 cos2 φ (7)

νa = 0.5(ν12 + ν21)(

2φπ

)+ ν2

(π − 2φπ

)(8)

where,νa = −0.5 (εx + εz)/εy (9)

5.2. Model of a transverse isotropy

A model of the transversely isotropic body is constructed in order to verify the validity of thesuggested equation. Throughout experiences and literatures of many experiments, the rangeof elastic modulus of rock is several tens GPa and E1 is generally bigger than E2 (Wittke,1990). Based on the assumptions, the elastic moduli in two directions of the transverselyisotropic model are assumed 50 GPa and 40 GPa respectively. Poisson’s ratio of the isotropicplane is assumed to be 0.25, and ν21 expressed as ν2 is 0.22 at first and a variable. Those twomay have a value between 0.1 and 0.4 respectively.

The shear modulus of the anisotropic plane is also assumed a variable. If it is as the sameas G∗ (new notation of G2 only when calculated by Eq. (1), 17.86 GPa in this model),strains in Fig. 1 are definitely controlled by Eq. (1) or Eq. (6) and the other fifth equationsare not needed nor existed any longer. Rock may have the value around this but not same,furthermore it has already been found that many types of rocks show the non-monotonousrelation of Ea − φ. In this consideration, let G2 of a model be 16.96 GPa which is just 5%smaller than G∗. Their new notations in the first order and values in the model are listed inTable 1.

Table 1. Independent elastic constants.

independent the first order valueconstant notation in the model

E11 E1 50 GPaE22 E2 40 GPaν31 ν1 0.25ν21 ν2 0.22 or varG12 G2 16.96 GPa

5.3. Validation of the suggested equations

If the value of G2 of the model is bigger or smaller than G∗, some error will occur for Eq. (1).And it will be evaluated whether the error is acceptable or not. The validation flowchart isshown in Fig. 5. The effectiveness of the suggested equation can be evaluated by the errorbetween the input in the first step and the output from the last step of the every elasticconstant.

Six strains from step C in the flowchart can be generated following by Eq. (4), but the inde-pendent strains obtained here are only four. By using Mohr’s strain circle, four independent

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Figure 5. Flowchart of validation of equation.

strains in X− Y direction can be derived in step D. They are 4 strains including Z-directionstrain, which are just same as the strain measurements obtained from actual test as in Fig. 1.Step E to transform them in 1 − 2 − 3 coordinates system can make it easy to calculate theconstants by the four constitutive equations and one of suggested equations.

5.4. Validation in models

For example in a model with φ = 20, strain in y direction is measured as much as 24.72μ-strain when 1 MPa is applied in y direction, and others are listed in the first column ofTable 2. Converted strains and apparent constants are in the second column and the thirdcolumn respectively.

Final output with applying Eq. (1) are in the first column in Table 3 and the relative error(RE) for every constant, which can be defined as Eq. (9) are also mentioned. The sum and theaverage of RE are 67.4% and 16.8% respectively without counting the value of G2 becauseit is always identical to the input at any model. Three other resultants which are almost sameas the input may be acceptable, but ν1 shows a large difference with 51.5% of RE even G2is given 5% smaller than the expected one in Eq. (1).

RE = 100%× (1− output/input) (10)

With applying Eq. (7), the whole results are presented in second column. Not only the sumor the average, but also the every RE is much smaller than those with Eq. (1), which explainsthat Eq. (7) may be more compatible and become a tool to solve the constants for some kindsof rocks.

It is revealed that the results in third column with applying Eq. (8) present smallest error.The sum and the average of RE are analyzed as much as 17.7% and 4.4% respectivelyand the largest one is 13.6% in ν1 which is much smaller than those with applying othertwo equations. This analysis explains that Eq. (8) is the most compatible equation and alsobecomes an alternative in this model. It is also induced that E2 and ν2 tend to be analyzedwith small errors and E1 and ν1 are calculated with large differences to the input values inmodels of flat angle. In the other models with steep angle, constants tend to be analyzed viceversus and Eq. (7) is the most compatible one. It is true that this tendency is practically foundin actual tests.

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Figure 6. Guide map to select the most compatible equation.

6. Map to Select the Compatible Equation

It is verified that the secondary suggestions are useful and has higher compatibilities in modelseven if they are constructed with less consideration of theory of mechanics. They may bemore applicable than Saint-Venant’s because of the possibility to apply in models whethertheir Ea − φ relations show monotonous or not. It has been found that applying Eq. (7)results in the closer constants for a large angle model and Eq. (8) for a small angle modelwhere G2 is given just the smaller value than G∗. In models of E1/E2 = 1.2 and ν1 = 0.25,the most compatible equation among three ones can be selected by comparing RE with thevarious inputs of ν2 and φ.

Figure 6 is the guide map to select the adequate equation for this model on the variationsof ν2 and φ. This map indicates that Eq. (1) is the most applicable equation for model withmid angle (marked SV), Eq. (7) for larger angle and Eq. (8) for smaller angle. It is foundthat both equations in boundaries of transition show the almost same results in the model.It is also disclosed in the large angle model that resultants and REs are analyzed in thealmost same values for 3 equations while Eq. (7) is the best. Saint-Venant’s approximation

Table 2. Strains in a model with φ = 20.

Strains in Step C Strains in Step D Apparent(m–s/MPa) (m–s/MPa) Value

εy = 24.72 ε1 = −2.52 Eapp = 40.45 GPaε(45) = 9.90 ε2 = 21.43 νapp−x = 0.2348εx = −5.80 ε12 = 18.95 νapp−z = 0.2201εz = −5.44 ε3 = −5.44 νapp = 0.2275

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Table 3. Results and errors in a model with φ = 20.

constantwith Eq. (1) with Eq. (7) with Eq. (8)

amount error amount error amount error

E1 44.8 GPa 10.31% 47.5 GPa 4.99% 48.6 GPa 2.72%E2 39.9 GPa 0.16% 40.0 GPa 0.07% 40.0 GPa 0.04%ν1 0.121 51.53% 0.188 24.96% 0.216 13.62%ν2 0.232 5.37% 0.225 2.46% 0.222 1.32%

RE Sum 67.4% 32.4% 17.7%RE Average 16.8% 8.1% 4.4%

is preferable even though Eq. (7) shows the smaller error on account of its renownless inthe case of the large angle model. There exist the blanks of no compatible equation for themodels with extremely acute angle which shows the RE average with more than 15% andone of 4 individual RE with more than 25%. This means that it is very difficult to analyzethe true values of constants in the small angle rock specimen.

7. Application to Rock Tests

It is definitely occurred that the measured strain data occasionally results in the unacceptablemagnitudes of constants, which may be induced by applying Eq. (1), especially ν1 and E1for a specimen with flat angle through the testing experiences. Samples from two differenttunnel projects may be good examples to explain this fallacy which has been revealed in themodel analysis. Every specimen of tested granite and mudstone has one dominant foliation asgeneral in class-B rock (Gonzaga et al., 2008). And testing data on St-Marc limestone fromliterature is also to be presented.

7.1. Seoul gneiss

Two specimens of high strength gneiss from Seoul subway construction site were tested withdouble sets of strain measurements as in Fig. 1. Both specimens have same anisotropic angleof 17 and can be assumed to have same characteristics as much as the similarity of their

Figure 7. Stress-strain curves for two specimens of Seoul gneiss.

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Table 4. Strain measurements in gneiss specimens (unit; μ-strain/MPa).

specimen planes εy ε45 εx εz

Gneiss – Aon A and B 19.059 9.510 −5.485 −7.492on A’ and B’ 21.566 9.791 −5.491 −5.694

average 20.312 9.651 −5.488 −6.593

Gneiss – Bon A and B 19.755 8.220 −5.694 −6.582on A’ and B’ 22.302 8.595 −6.835 −6.804

average 21.028 8.408 −6.264 −6.693Imaginary Specimen – C average of two specimen 20.670 9.029 −5.876 −6.643

Table 5. Resultants of deformability analysis of gneiss.

specimen equation E1 E2 ν1 ν2 G∗

Gneiss − AEq. (1) 176.44 46.12 3.2493 0.2531 26.09Eq. (7) 55.41 45.90 0.3345 0.3050 18.82Eq. (8) 43.13 45.80 0.0387 0.3263 16.87

Gneiss − BEq. (1) 69.84 46.18 0.7876 0.2893 20.62Eq. (7) 54.17 46.10 0.3865 0.3066 18.71Eq. (8) 46.97 46.05 0.2022 0.3185 17.59

Imaginary Specimen CEq. (1) 100.07 46.15 1.4857 0.2712 23.03Eq. (7) 54.78 46.00 0.3607 0.3058 18.76Eq. (8) 45.35 45.93 0.1266 0.3216 17.29

strengths, 155 MPa and 160 MPa respectively. Stress-strain curves are illustrated in Fig. 7and their tangent values in the unit of μ-strain/MPa in the linear zone are described in Table 4for both and their averages. Strains are measured as much as 19.059, 9.510, −5.485, −7.492μ-strain/MPa on plane A and B, and 21.566, 9.791, −5.491, −5.694 μ-strain/MPa on planeA’ and B’ in order for the specimen A. Their averages are listed in the first row in Table 4.Imaginary specimen (C) of 17 angle can be constructed with the strains as much as averagesof two tested specimens.

The independent elastic constants as the analysis resultants with applying each suggestedequation are listed in Table 5. It is clearly disclosed that ν1 and E1 are absolutely analyzedunacceptable with Eq. (1). Higher values and lower values of ν1 are analyzed by applyingEq. (7) and Eq. (8) respectively. It will be the next task to choose the best one of the twowhich may result in the compatible values of constants.

It is not correct that Eq. (8) is the best when being chosen from the guide map in Fig. 6because G2 for tested specimen is not smaller than G∗ derived by Eq. (1). For specimen A,G2 is as much as 26.09 GPa from actual test, and G∗ is evaluated 18.82 or 16.87 in applyingEq. (7) or Eq. (8). If model of which G2 is larger than G∗ instead of model mentioned beforeis investigated, Eq. (7) can be the best on another guide map. It will be another evidence toshow that Eq. (7) is the best that the similarity of ν1 values for 2 specimens with Eq. (7) isbetter than that with Eq. (8).

The average values of constants, 54.79 GPa, 46.00 GPa, 0.3605 and 0.3058 in order withapplying Eq. (7) are the correct results in this sample and they are almost same values asthose of an imaginary specimen C. There may be large differences when Eq. (1) or Eq. (8) isapplied. It is noted that the resultants with 3 equations should be re-evaluated whether G2 issmaller than G∗ and whether another guide map is needed.

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7.2. Busan mudstone

Two specimens of the greenish gray colored silty mudstone from Busan area were testedwith angles of 26 and 22, in the high strengths of 196 MPa and 188 MPa respectively.Experiment was conducted with double sets of strain measurements as well and the datawere reported in Korean journals (Park and Park, 2002, and Park, 2001) where the directionof strain measurements was focused. Average tangent values in the linear zone are describedin Table 6. Imaginary specimen can also be constructed with averages of strains and angleof 24.

The analysis resultants are listed in Table 7. It is found that all of constants are analyzedalmost same values except ν1 and may have the acceptable amounts with every equation. Itwill also be the next task to choose the best one of three equations. Generally the equationwhich yields the medium values is the best one. Eq. (1) is the most compatible one in this case.

Another way to choose is the application of the guide map by comparing G2 from test(independent of the fifth equation) to G∗ derived from 4 constants with every equation. G∗with Eq. (7) is 14.66 GPa just 4% larger than G2, which means that Eq. (8) is the compatibleone on the guide map as Fig. 6. Thus the constants with Eq. (7) cannot be the best compatibleequation. And G∗ with Eq. (8) is analyzed 13.91 GPa just 2% smaller, which means thatEq. (7) will be compatible on the other guide map where G2 is larger than G∗, and Eq. (8)cannot be chosen either.

Based on the two methods to choose the best equation, Eq. (1) is the best and one of Eq. (7)and Eq. (8) cannot be the best in this sample.

In this sample, 8 independent equations can be constructed because of the difference ofangles of two specimens, which enables to solve 5 independent constants by root meansquare method to minimize the error of strain measurements. This kind of the optimumsolution can produce the true constants. The last row in Table 7 mentions its resultants, the

Table 6. Strain measurements in mudstone specimens.

specimen angle εy ε45 εx εz

Mudstone – A 26 27.020 7.553 −6.724 −7.032Mudstone – B 22 21.230 4.801 −5.992 −6.349

Imaginary Specimen – C 24 24.125 6.177 −6.358 −6.690

Table 7. Resultants of deformability analysis of mudstone.

specimen equation E1 E2 ν1 ν2 G∗

Mudstone – AEq. (1) 26.55 40.84 0.2211 0.2746 13.23Eq. (7) 27.52 40.96 0.2654 0.2626 13.59Eq. (8) 26.03 40.76 0.1970 0.2814 13.03

Mudstone – BEq. (1) 28.43 52.76 0.2426 0.3162 15.13Eq. (7) 30.42 52.93 0.3300 0.2972 15.87Eq. (8) 27.88 52.71 0.2189 0.3217 14.91

Imaginary Specimen – CEq. (1) 27.56 46.05 0.2306 0.2928 14.14Eq. (7) 28.95 46.20 0.2926 0.2778 14.66Eq. (8) 26.95 45.98 0.2034 0.2998 13.91

Optimum Solution 20.6 52.2 0.36 0.24 14.14

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amounts of constants are far from those with the suggested approximations. These resultsmay be evaluated the erred ones because E2 is far from Ea. It may be considered that thesuggested equation yield the better results than the optimum solution in this case. It may alsobe concluded that the optimum analysis for multi-tests on similar shaped specimens cause anerroneous result because strains and strain measurements always involve the errors.

7.3. St-Marc limestone from literature

There is one literature to use the directions of strain measurement as in Fig. 1 to examineSt-Marc limestone with φ = 44.5 where 9 strains were measured (Gonzaga et al., 2008). 4independent ones were included and analyzed as much as 16.162, 6.821, −4.161, −4.774μ-strain/MPa in order, of which 45 degree directional strain measured in the other directionof Fig. 1 was here transformed. Ea, νa and G2 are interpreted as 61.87 GPa, 0.2764 and24.64 GPa respectively.

Table 8. Resultants of deformability analysis of St-Marc lime-stone.

equation E1 E2 ν1 ν2 RE sum G∗

Eq. (1) 69.0 56.3 0.373 0.234 40.1% 24.64Eq. (7) 71.5 57.8 0.422 0.213 25.6% 25.87Eq. (8) 71.7 57.9 0.427 0.211 25.8% 25.98

true value 70.1 61.6 0.40 0.19 − −

Table 8 presents the resultants by applying three equations and the true values of theconstants measured in the literature. The amounts of all constants may be analyzed in theacceptable ranges. Each of ν2 showing the higher value because its true value is known ashigh as 0.40, is also acceptable. All three approximations may be adequate on this analysis.The RE sum comparing to the true value is calculated as much as 40.1%, 25.6% and 25.8%for every suggested equation, which means Eq. (7) can be the best. If true values are notknown, Eq. (8) may be most compatible because the guide map indicates in this case.

8. Conclusions

Test of two or more uniform specimens can identify the exact values of the five independentelastic constants of a transversely isotropic rock. It is found from the analysis of actual testthat the optimum analysis for multi-tests on similar shaped specimens may cause an erro-neous result because strain measurements always accompany the errors. In engineering, theymight be often measured with a single specimen and Saint Venant’s approximation is essentialon the lack of one relation in the constitutive equations.

The directions of 4 strains suggested in Fig. 1 may minimize the errors in measurementand 2 sets of strain gauge can increase the accuracy. The first suggested equation has beenapproved identical to Saint-Venant’s approximation. Both equations present that the appar-ent Young’s modulus is dependent on and monotonously increasing to the anisotropic angle,which may conclude that they are not applicable to rocks with the non-monotonous relation.They may also result in the unacceptable constants in the case of a small angle specimen.

The second and the third equations defined by the apparent Poisson’s ratio are alsoapproved useful through the analysis of numerical models and actual tests of rocks. Each

658


equation has an advantage to apply to the rocks whether the relation between the apparentYoung’s modulus and the anisotropic angle is monotonous or not. Each equation obviouslyhas a role as an alternative to replace the historical equation which may yield the unreason-able results especially in the case that anisotropic angle of a specimen is small. The adequateequation can be selected by the amounts of ν2 and φ as shown in the guide map. If G2 islarger than G∗ after analysis, the adequate equation can be found from the other guide map.Even though one of those two suggestions can show the smaller amount of error as in modelanalysis, Saint-Venant’s approximation is preferable in the case that the anisotropic angle ofa specimen is large and the resultants has amounts in the engineering ranges.

Acknowledgements

This study was funded by the Korea Institute of Construction & Transportation TechnologyEvaluation and Planning under the Ministry of Construction & Transportation in Korea(Grant No. 05-D10, Development of Water Control Technology in Undersea Structures).

References

1. Allirot, D. and Boehler J.P., “Evolution of mechanical properties of a stratified rock under confin-ing pressure”, I 4th ISRM Congress, Montreal, 1979, pp. 15–22.

2. Amadei, B., “Importance of anisotropy when estimating and measuring in situ stresses in rock”,International Journal of Rock Mechanics and Mining Science, 33, 3, 1996, pp. 293–325.

3. Duevel, B. and Haimson, B., “Mechanical characterization of pink Lac Du Bonnet granite: Evi-dence of nonlinearity and anisotropy”, International Journal of Rock Mechanics and Mining Sci-ence, 34, 3, 1997, pp. 117e1–e18.

4. Flugge W., Tensor analysis and continuum mechanics, Springer-Verlag, Berlin, 1972.5. Gonzaga, G.G., Leite, R. and Corthesy R., “Determination of anisotropic deformability parame-

ters from a single standard rock specimen”, International Journal of Rock Mechanics and MiningScience, 45, 2008, pp. 1420–1438.

6. Goodman R.E., Introduction to rock mechanics, John Wiley & Sons, 1980.7. Park, Chulwhan, “Analysis of elastic constants of an anisotropic rock”, Tunnel and Underground

Space as KSRM Journal, 11, 1, 2001, pp. 59–63.8. Park, Chulwhan and Park, Chan, “Suggestion of testing method to determine elastic constants of

an anisotropic rock”, Proc. Seminar on Standard Method of Rock Testing in Civil Engineering byKSRM Rock Testing Commission, 2002, 101 pp.

9. Talesnick, M.L. and Ringel, M., “Completing the hollow cylinder methodology for testing oftransversely isotropic rocks; torsion testing”, International Journal of Rock Mechanics and MiningScience, 36, 6, 1999, pp. 627–639.

10. Tien, Y.M. and Tsao, P.F., “Preparation and mechanical properties of artificial transverselyisotropic rock”, International Journal of Rock Mechanics and Mining Science, 37, 4, 2000, pp.1001–1012.

11. Wittke W., Rock mechanics – Theory and applications with case histories, Springer-Verlag, Berlin,1990.

12. Worotnicki, G., “CSIRO triaxial stress measurement cell”, Chap 3–13 in Comprehensive RockEngineering (edited by Hudson J. A.), Pergamon Press, 1993, pp. 329–394.

659

Numerical Analysis of Deep Excavation Affected by TectonicDiscontinuity

L. MICA1,∗, V. RACANSKY1,2,∗ AND J. GREPL1

1Brno University of Technology, Faculty of Civil Engineering, Institute of Geotechnics2Keller – speciální zakládání, spol. s r.o., Czech Republic

1. Introduction

Retaining structures are in most cases excavated in soils (coarse grained or fine-grained soils)whereas rock environments are less common. This can change in the near future. As thespace is scarce especially in cities, the use of underground is more and more common. Theconsequence of this may be fact that during excavation of the foundation pit, rock environ-ment formed mainly by solid rocks will be reached. This environment was subject to manygeological process which have altered the rock properties. The most significant are tectonicprocesses. They usually cause many discontinuities (cracks, clefts). This was the backgroundfor the paper analysing retaining structures which are built in the rock environment affectedby tectonic disturbances.

The main content of the paper is the numerical analysis of the retaining structure in therock environment disturbed by joints. An influence of some factors on displacements andbending moments of retaining structure (diaphragm wall) is studied. Particularly an incli-nation, thickness and location of the joint (fault) from the retaining wall is analysed. A FEanalysis of the shale rock affected by tectonic disturbances is presented. Joints were modelledby clusters, instead of using interface elements, due to significant fault thickness of up to 3 m.

2. Material Parameters of the Tectonic Fault

Parameters for the fault zone are based on the laboratory tested samples 1 collected duringthe excavation of Mrázovka tunnel in Prague. The material is made by mixture of flat spallsof the shale and fine-grained silty matrix. The size of the shale is less than 10 cm. Studiedtectonic disturbance was found in Ordovician series of the shale rock. Triaxial testing (diam-eter of the sample was 38 mm) and large-diameter set (1× 1× 0.75 m) were conducted. Forthe standard sample size triaxial test, it was necessary to exclude the grains whose size wasmore than 4 mm. For large-diameter tests, no such exclusion was necessary. The laboratoryresults have showed following: the material without grains over 4 mm has a peak frictionangle of 32.9◦ and the material with grains over 4 mm has a peak friction angle of 34.9◦.This illustrates the fact that fine-grained matrix of the fault has a dominant influence on theshear strength. The oedometric module has been evaluated from the large-diameter test fromthe consolidation phase. This module has been recalculated for Young’s modulus “E”. Inputdata are summarized in the Chapter 3.2.

3. Numerical Analysis

3.1. Description of model

In the first step, just before carrying out the numerical analysis, the retaining structure wasdesigned neglecting the presence of the fault. On the basis of this step four times strutted

∗Corresponding author. E-mail: [email protected], [email protected]



diaphragm wall has been designed. Its thickness was 600 mm, length was 27 m for a depth ofthe excavation of foundation pit which is 25 m. Different positions of the tectonic disturbancewere defined for this geometrical setup. The tectonic disturbance has been chosen with angles30◦, 45◦, 60◦ and 75◦ at three positions. The thickness of tectonic disturbance has beenselected 3 m. Nine different position of the fault zone has been defined. Dimensions of theproblem are given in Fig. 1(a).

FE code PLAXIS V94 was used for calculations. A 2D plane strain analysis using 15-nodedtriangular elements for modelling of soil behaviour was performed. Drained soil behaviourwas considered in the analysis for the construction period of the excavation. Structure, retain-ing wall, was modelled as plate element. Struts were modelled by bar element. A width ofthe model is 130 m and the depth is 85 m. Generated mesh contained approx. 3500–4500elements (Fig. 1(b)).

The construction was divided in 11 construction stages:

1. Generation of initial stresses,2. Construction of diaphragm wall3. Excavation at the depth – 4.0 m4. Construction of 1st level of strut at depth 3 m5. Excavation at the depth – 8.5 m6. Construction of 2nd level of strut at depth 8.0 m7. Excavation at the depth – 14.0 m8. Construction of 3rd level of strut at depth 13.5 m9. Excavation at the depth – 19.5.0 m

10. Construction of 4th level of strut at depth 19.0 m11. Final excavation to depth 25 m

Mechanical behaviour of the rock environment has been approximated using Mohr-Coulomb and Hardening soil models with material characteristics given at Chapter 3.2. Alsothe material parameters for the material of the fault and of structural elements are given inChapter 3.2.

3.2. Input data

The analysis of the influence of tectonic disturbance has been carried out for subsoil formedby shale. They are characteristic for the area of Prague (Czech Republic). Behaviour of shalehas been extensively studied2 during the excavation of tunnel “Mrázovka” (1999-2004).Tests have been carried out on the material of the rocks as well as on tectonic disturbance.On basis of this information the geological profile has been compiled. The profile was usedfor numerical analysis with following data field:

0.0 – 4 .0 m Deluvial Sediments4.0 – 14.0 m Partly weathered Shale14.0 – m Unweathered Shale

Material characteristics based on [3] which were used in the numerical analysis are sum-marized in Table 1.

Input data for material which forms tectonic fault are based on1 and are given at Table 2.Similarly as for previous materials this material has been described by Mohr-Coulomb modeland Hardening soil model.$

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Fault zonePos. II

Pos. I

Pos. III

a

-4.0

-8.5

-14.0

-19.5

-25.0

-17.0

-26.0

-32.0

Diaphragm wallStrut

Diluvial Sediments

Partly weathered Shale

Unweathered Shale

(a) Geometry used for the analysis (cross-section)

(b) Finite element mesh – Pos. III_4

Figure 1. Geometry.

The diaphragm wall was defined by axial stiffness (EA) and bending stiffness (EI). Thethickness of diaphragm wall is 0.60 m and Young modulus of the concrete used in the analysiswas 30 GPa. The axial stiffness of the prop is EA = 1.6e− 8 kN per 3 m.

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Table 1. Model parameters – soils and rocks.

Parameter Unit MC model

Deluvial Sediments Partly weathered Shale Unweathered Shale

γsat/γunsat kN.m−3 21/21 23.5/23.5 25.2/25.2Eref kN.m−2 10 000 80 000 400 000c′ kN.m−2 22 20 40ϕ′ ◦ 5 26 32ψ ◦ 0 0 0ν′ - 0.3 0.35 0.265

Table 2. Model parameters – tectonic fault - joint.

Parameter Unit Material model

MC model HS model

γsat/γunsat kN.m−3 21/21 21/21Eref kN.m−2 44 600 -

Eref50 kN.m−2 - 31 000

Erefoed kN.m−2 - 31 000

Erefur kN.m−2 - 93 000

c′ kN.m−2 8/0 8/0ϕ′ ◦ 32.9 32.9ν′or νur - 0.3 0.2ψ ◦ 16/0 16/0m - - 0.6Rf - - 0.9

4. Results

Series of numerical analysis have been performed in order to examine the influence of jointson bending moment in the retaining structure. Diaphragm wall with four levels of struts wasmodelled in the shale rock mass of Liben region. Joint of 3m of thickness was modelled withfour different angles (30◦, 45◦, 60◦ and 75◦) and three different depth levels with respect tothe structure. The geometry of the variants, which has been analysed is given in Fig. 2. Fordescription of mechanical behaviour of the joint two constitutional models have been chosen:MC model and HS model. In addition to the values given in Table 2 the analysis with theparameters „c“ and „ψ“, which both equal 0, has been also done. The rock mass outside thearea of the failure has been modelled by MC model.

Model without any tectonic fault was regarded as a reference one. Bending moments cal-culated in models with the tectonic fault were always compared to the reference solution. Ithas been assumed, that the reinforcement of the diaphragm wall wall has been done the sameway in both surfaces (i.e. for maximal bending moment of the wall).

Numerical analysis shows, that in case of given geological conditions there is not a sig-nificant difference between modelling of the tectonic fault by MC and HS model. For bothcases it is possible to observe the same trend in the change of the bending moments. A differ-ence has been in the bending moment values. The MC model has given slightly higher values

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Model

Pos. I.

30° -I_1

45° -I_2

60° -I_3

75° -I_4

Pos. II.

30° -II_1

45° -II_2

60° -II_3

75° -II_4

Pos. III.

30° -III_1

45° -III_2

60° -III_3

75° -III_4

Figure 2. Scheme of the models.

(Fig. 3). Considering these facts and the fact that the HS model gives better description ofthe soil behaviour compared to the MC model (standard MC model uses constant Young’smodulus of elasticity which is independent on the stress level and also on strains). Resultsfor HS model only will be presented. Results from the 11th stage which give the most criticalresults are shown.

Faults located in position II have the most significant influence on the wall behaviour. Anincrease (in percents) of bending moments for selected fault inclinations is: 30◦ - 45%; 45◦- 64%; 60◦ - 57%; 75◦ - 53%). For faults located in position I, the change of the maximalbending moment has not exceeded 10 %. However the change of bending moments was upto 100% for angle of 45◦ and above. This was caused by the bending moment redistributionat the point where the fault crosses the wall. The bending moment on the intrados has risenwhereas the moments on the extrados has decreased. For the fault position III the maximalbending moment has slightly changed compared to the reference case (Fig. 4(b)). However,an anomaly can be observed in model III_1, where the bending moment has significantly

-27

-24

-21

-18

-15

-12

-9

-6

-3

0-600 -400 -200 0 200 400 600 800

De

pth

of th

e w

all (m

)

Bending moment kN/m

-27

-24

-21

-18

-15

-12

-9

-6

-3

0-600 -400 -200 0 200 400 600 800

Bending moment kN/m

0

II_1

II_2

II_3

II_4

HS model MC model

Figure 3. Comparison of bending moment for MC and HS Model – Pos. II.

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-27

-24

-21

-18

-15

-12

-9

-6

-3

0-300 -200 -100 0 100 200 300 400 500

Dep

th o

f the

wa

ll (m

)

Bending moment kN/m

0

I_1

I_2

I_3

I_4

-27

-24

-21

-18

-15

-12

-9

-6

-3

0

-400 -200 0 200 400 600Bending moment kN/m

0

III_1

III_2

III_3

III_4

4a) Redistribution of moment at Pos. I. 4b) Bending moment for Pos. III.

Figure 4. Differences between bending moment for position I and III compared with model withoutthe fault.

increased at the wall toe. This is most probably caused low quality mesh generated for thismodel in the toe zone.

Regarding wall displacements, following was observed: models with faults in position Ihad the highest value of horizontal displacement at the point where the joint crosses thewall. Models with faults in position II had the highest value at wall toe. Models with faultsin position III had nearly identical horizontal displacements as the reference case.

Further, the same analysis was performed with tectonic fault strength parameters c = 0 kPaand ψ = 0◦. Same conclusions regarding bending moments and displacements are valid,however the values of bending moments are higher.

5. Conclusion

An FE-analysis of a deep excavation influenced by a tectonic fault is reported. Primary aim ofthe analysis was to evaluate the influence of the different positions and thickness of tectonicfault on the behaviour of the retaining wall. Motivation for this study was the fact, that aconventional analysis used for the design cannot take inclined layers into account. Followingconclusions can be drawn:

(1) For given geological conditions it has been shown that the worst position of the faultis the position II, with maximal rise in bending moment for joint angle of 45◦. Bendingmoments for faults in positions I and III have not changed significantly.

(2) Neglecting the presence of the tectonic fault when using conventional design methods(limit equilibrium method, beam on elastic (elasto-plastic) subgrade) is not on the safe sidefor design of the retaining wall. Example may be the case with the fault in position II wherethe maximal bending moment has increased by 64%. This increase can already have seriousconsequences.

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The analysis shows that the fault can significantly influence the retaining wall design. Ithas to be stressed out that these conclusions are partial. Author’s aim is to continue with thistopic and reach more general conclusions, which can be helpful for technical practice.

Acknowledgements

This contribution was financially supported by the project of the Czech Science Foundation(GA CZ) No. GA103/09/1262 and by the research project of The Ministry of Education,Youth and Sports (MŠMT CZ) No. MSM0021630519. Authors appreciate this support.

References

1. Mašín, D., Vliv výplnì tektonických poruch na deformace tunelu, Master thesis, Faculty of Science,Charles University in Prague, Prague, Czech Republic, 2001, (in czech).

2. Hudek, Exploratory galéry of the Mrazovka tunnel in Prague, Tunel, Vol. 6, No. 3, 1997, CzechRepublic, pp. 13–16.

3. Doležalová, M., Zemanova, V., Danko, J., The Mrazovka expoloratory adit – modelling of rockmass mechanical behaviour according to field measurements, Tunel, Vol. 8, No. 1, Czech Republic,pp 8–15.

4. Plaxis 2D – Version 9.0, Edited by R.B.J. Brinkgreve & W. Broere, D. Waterman, 2008.

667

The Finite Element Analysis for Concrete Filled Steel TubularColumns under Blast Load

J.H. ZHAO∗, X.Y. WEI AND S.F. MA

School of Civil Engineering, Chang’an University, Xi’an 710061, China

1. Introduction

The responses of blast load are always taken into consideration for the significant buildingand protective construction. Presently, concrete filled steel tubular (CFST) is widely used inconstruction because it has the beneficial qualities of both concrete and steel. In order tostudy the mechanical behavior of the CFST column under blast load, the dynamic responsesof a square CFST column under surface explosion were simulated by the nonlinear finiteelement program ANSYS/LS-DYNA. The JHC model was used for concrete material andthe MAT_PLASTIC_KINEMATIC model which accounted for the strain rate used for steel.The failure behavior of the CFST column at scale distance equal to 1.0 was analyzed. Theresults indicate that the inner concrete was seriously damaged, however, the deformation ofconcrete was restricted by the steel tube. It shows that CFST column has excellent ductilityand blast resistance. The time-history curve of displacement of key nodes at different scaledistance are compared, which indicates that the deformation of column obviously decreaseswith the increase of scale distance.

2. Numerical Simulation

2.1. Numerical model

As shown in Fig. 1, the responses of CFST column under surface blast occurred at variousstand-off distances are investigated. The clear height of the CFST column is H = 3 m. Assum-ing the column has square cross section and the width, the depth and the thickness of thesteel tube is 500 mm, 500 mm and 10 mm, respectively. The top and bottom of the column isconsidered as fully fixed. A 3-D numerical model of concrete filled steel tubular column wasset up. Solid elements are used to model both the concrete and the steel. There are a total of40460 elements in the numerical model. Convergence test is conducted and it was found thatfurther refinements in mesh density did not significantly improve global response.

2.2. Material model

The Johnson-Holmquist (J-H) material model is used for concrete. This model can be used forconcrete subjected to large strains, high strain rates, and high pressures. The equivalent stressis expressed as a function of pressure, strain rate and damage. A more detailed descriptioncan be found in LS-DYNA theoretical manual.8 The parameters of concrete used in this studyare shown in Table 1.

The MAT_PLASTIC_KINEMATIC material model is used for steel. Isotropic, kinematic,or a combination of isotropic and kinematic hardening may be obtained by varying a parame-ter β between 0 and 1. For β = 0 and β = 1, respectively, kinematic and isotropic hardening




Figure 1. Concrete filled steel tubular column under blast.

Table 1. Concrete material parameters (g-mm-ms).

Parameter MID RO G A B C N

Value 1 2.25E-3 1.38E4 0.75 1.65 0.007 0.76Parameter FC T EPS0 EFMIN SFMAX PC UCValue 40 3.92 0.001 0.01 7 13.33 7.3E−4Parameter PL UL D1 D2 K1 K2 K3Value 800 0.1 0.038 1 1.74E4 −3.88E4 2.98E4

Table 2. Steel tube material parameters (g-mm-ms).

Parameter MID RO E PR SIGY ETAN BETA SRC SRP FS VP

Value 2 7.85E-3 2.1E5 0.3 345 1180 0 40.4 5 0.3 0

are obtained. Strain rates effect is accounted for using the Cowper-Symonds model whichscales the yield stress by a strain rate dependent factor. The parameters of steel used in thisstudy are shown in Table 2.

2.3. Blast loading model

The explosive process is not included in this study. The blast pressures are generated usingprocedures outlined in TM5-1300 and the loading functions corresponding to these blastpressures are then applied to the numerical model. TM5-1300 is widely used by blast engi-neers for preliminary design purpose. It adopts the cube-root scaled distance for consideringvarious stand-off distances and charge weight. The scaled distance is defined as

Z = R/W1/3 (1)

in which R is the distance from the source and W is the weight of explosives.Figure 2 shows a free-field typical pressure-time history. At any point away from the burst,

the pressure disturbance has the shape shown in Fig. 2. The shock front arrives at a givenlocation at time tA and after the rise to the peak value, Ps0 the incident pressure decays tothe ambient value P0 in time to which is the positive phase duration. This is followed bya negative phase with a duration t−0 . The negative pressure has a maximum value of P−s0.

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Figure 2. Free-field pressure-time variation.

Usually the negative phase is less important in the design than is the positive phase. Hence,only the positive phase of blast pressure is considered in the numerical simulation.

The empirical pressure-time history in Ref. 2 is used herein:

P(t) = Ps0(1− t/t0) exp (− bt/t0) (2)

in which b is the parameter of the shock wave.The shock waves propagate with supersonic velocity and finally it hit the building. They

reflect from the building with amplified overpressures and it can be determined from TM5-1300. Assuming the stand-off distance is 5 m, three blast scenarios are considered, i.e., thescaled distance Z = 0.7, 1.0 and 1.3. The blast pressure is uniformly loaded on the columnsurface.

3. Numerical Results

Numerical simulations are carried out to estimate the blast response and damage of the CFSTcolumn subjected to explosive blast loading based on the transient dynamic finite elementprogram LS-DYNA.

3.1. Results of scaled distance = 1

Figure 3–5 shows the deflection in X direction and maximum principal stress of concrete oftime t = 2 ms, 5 ms, 9 ms, respectively. It is observed that the maximum deflection occurs atthe middle of the column. It is expected because the column has symmetrical supports andit is under uniform load. The deflection increases with time and reach its maximum valueof 117 mm when t = 9 ms. From the stress contour of the column, it can be found that thetensile damage occur first at the top and bottom of the concrete. The maximum principalstress reaches the tensile strength of concrete. When time increases to 9 ms, the concrete atthe middle of the column is also damaged and erosion occurs. However, the ratio betweenthe deflection and the height of column is 3.9%. Hence, it can be concluded that the steeltube effectively restricted the lateral deflection of the column and thus can improve the blastresistances.

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(a) Column deflection in x direction (b) Maximum principal stress of concrete

Figure 3. Deflection and stress of t = 2 ms.

(a) column displacement in x direction (b) maximum principal stress of concrete


3.2. Comparison of Displacement

Figures 6(a) and (b) shows the deflection in x direction of the column for scaled distance z =0.7m/kg1/3 and z = 1.3m/kg1/3, respectively. It can be seen that the maximum deflectionsof the column decrease significantly with increase of the scaled distances.

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(a) column deflection in x direction (b) maximum principal stress of concrete


(a) scaled distance z=0.7m/kg1/3 (b) scaled distance z=1.3m/kg1/3

Figure 6. Deflection in x direction.

4. Conclusion

The following conclusions are deduced from the numerical results:The Johnson-Holmquist (J-H) material model can be applied to simulate reasonably both

the compressive crush zone and tensile damage.When scaled distance is 1.0 m/kg1/3, the ratio between the deflection and the height of col-

umn is 3.9%. It can be concluded that the steel tube effectively restricted the lateral deflectionof the column and thus can improve the blast resistances.

The maximum deflections of the column decrease significantly with increase of the scaleddistances.

673


Acknowledgements

The supports of the Fund for the Doctoral Program of Higher Education of China(20040710001) and Shaan Xi Province Natural Scicence Foundation (SJ08E204) are grate-fully acknowledged.

References

1. Georgios Giakoumelis, Dennis Lam. Axial capacity of circular concrete-filled-tube columns. Journalof constructional Steel Research, 60, 2004, pp. 1049–1068.

2. Ben Young, Ehab Ellobody. Experimental investigation of concrete- filled cold-formed high strengthstainless steel tube columns. Journal of Constructional Steel Research, 62, 2006, pp. 484–492.

3. Zhang, F.G. and Li, E.Z. A computational model for concrete subjected to large strains, high strainrates, and high pressures. Explosion and Shock Waves. 2002, 22(3), pp. 198–202.

4. LSTC. LS-DYNA keywords manual, Version 970, Livermore Software Technology Corporation,Livermore, CA, 2003.

5. Wei, X.Y. Dynamic response of concrete and masonry structure under explosive and impact loads.Reports of post PhD, 2007.

6. TM5-1300. Structures to resist the effects of accidental explosions. US Army, USA, 1990.7. LSTC. LS-DYNA theoretical manual, Livermore Software Technology Corporation, Livermore, CA,

1998.

674

Numerical Simulation of Performance of Concrete-Filled FRPTubes under Impact Loading

C. WU1,∗, T. OZBAKKLOGLU1, G. MA2 AND Z.Y. HUANG3

1School of Civil, Environmental and Mining Engineering, The University of Adelaide, SA, Australia2College of Engineering, Nanyang Technological University, Singapore3College of Civil Engineering, Hunan University, Changsha, Hunan, 410082, China

1. Introduction

Fiber reinforced polymer (FRP) composites have found increasingly wide applications in civilengineering due to their high strength-to-weight ratio and high corrosion resistance.1 Oneimportant application of FRP-composites is as a confining material for concrete, in both theretrofit of existing reinforced concrete (RC) columns and in concrete-filled FRP tubes in newconstruction. As a result of FRP confinement, both the compressive strength and the ultimatestrain of concrete under static loads can be greatly enhanced. However, little research hasbeen carried out to investigate FRP confined concrete under impact loads.

Numerical modelling techniques have been used to simulate the performance of FRP con-fined concrete under static loads. An explicit finite element model with a concrete materialmodel based on the K&C concrete material model released in 1994 has been used to analyzetest data by Suter and Pinzelli2 on concrete cylinders wrapped by aramid. Malvar et al.3 hassuccessfully used numerical modeling to reproduce the strength enhancement observed in thetest specimens for various level of confinement under static loads. It is believed that thesedeveloped models can also be used to model the behaviours of FRP confined concrete underimpact loads.

In this paper the Karagozian & Case (K&C) Concrete damage Model — Release III inLSDYNA4 was validated by static test data on the performance of concrete specimens withtwo different sizes confined with single-layer and double-layer CFRP sheets. The concretedamage model was further verified by the recorded data from Split Hopkinson Pressure BarTest (SHPB) on CFRP confined concrete specimens under impact loads. The validated numer-ical model was then used to conduct parametric studies on how CFRP confinement affectsdynamic strength of concrete under impact loads. Using the simulated data DIF formulae forCFRP confined concrete specimens under different strain rates are derived.

2. Material Model for Concrete

Concrete is a common construction material and a lot of available material models havebeen developed to model the performance of concrete under static loads. In this paper theKaragozian & Case (K&C) Concrete damage Model — Release III with the equation of state“Tabulated Compaction” in the LS-DYNA program was used to simulate the performance ofthe confined concrete. The K&C Concrete Model is a three-invariant model, uses three shearfailure surfaces, includes damage and strain-rate effects, and has origins based on the Pseudo-TENSOR Model (Material Type 16). The most important parameters for Concrete damageModel are compressive damage scaling parameter (B1), tensile damage scaling exponent (B2)and damage scaling coefficient for triaxial tensile (B3) which are determined in the processof comparison and correction with the results of static test. Based on the recorded data for




Figure 1. Plain concrete specimens.

Figure 2. A comparison of stress-strain curves.

plain concrete specimens with a diameter of 100 mm with height of 195 mm as shown inFig. 1, it was found that the input parameters B1, B2 and B3 are equal to 1.35, −0.65and 1.15, respectively. Figure 2 shows a comparison of simulated and tested compressivestress-strain curves. It can be seen the numerical results match well with the tested data,demonstrating that the concrete damage model with the input parameters B1, B2 and B3 canwell simulate the performance of concrete material.

3. Simulation of Static Tests on FRP Confined Concrete Specimens

With the above concrete damage model, concrete specimens confined with CFRP sheets asshown in Fig. 3 were simulated. The CFRP tubes were manufactured from the carbon fibersheets with epoxy resin. The Young’s modulus of CFRP is 240 GPa and tensile strength is3800 MPa. The thickness of CFRP sheet used in this study is 0.117 mm. CFRP sheet wasmodelled using an elastic-brittle material model in the LS-DYNA program. Figure 4 shows

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Figure 3. CFRP confined concrete specimens.

Figure 4. Numerical model of CFRP confined concrete.

the numerical model of the FRP confined specimen. Figure 5 shows the stress-strain curvesof static test and numerical simulation for single-layer (SL) confined concrete specimen. It isnoted that the elastic parts of simulated and tested results are almost coincident. The resultof simulated ultimate compressive strength is 79.1 MPa, slightly higher than that of the statictest 72.5 Mpa. Generally, the two stress-strain curves match very well. Figure 6 shows a

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Figure 5. Stress-strain curves for SL specimen.

Figure 6. Stress-strain curves for DL specimen.

comparison of the results for double-layer (DL) CFRP confined concrete specimens. It alsodemonstrates that the numerical result agrees well with the test data.

4. Simulation of Impact Tests on FRP Confined Concrete Specimens

The above validated numerical method is used to simulate SHPB tests on FRP confined con-crete specimens with diameter 75 mm and height 75 mm. A typical SHPB experimental setup and specimen failure are shown in Fig.s 7 and 8, respectively. Figure 9 shows the numeri-cal model of SHPB test in LS-DYNA program including striker bar, input bar, specimen andoutput bar. In the numerical simulation the stress and strain of each element are more orless different due to possible non-uniform stress/strain inside the specimen. Therefore, theaverage result of the all elements of the specimen is used for the stress and strain of the con-crete specimen. Figure 10 shows a comparison of the axial strain time histories of input barbetween the simulated and recorded data. As shown, the simulated result agrees well withthe measured data. Figures 11 and 12 show the simulated and recorded stress and strain timehistories of the CFRP confined concrete specimens and the simulated results matching wellwith the measured data is observed again.

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Figure 7. SHPB Test set up.

Figure 8. Specimen failure.

Figure 9. Numerical model of SHPB test set up.

5. Dynamic Increase Factor

Using the above model, parametric studies are conducted to investigate the stress strainrelationships of FRP confined concrete specimens under static and impact loads. Then, thedynamic increase factor (DIF) which is defined as the ratio of dynamic carrying capacity overstatic carrying capacity of specimens can be estimated. If Pd is the dynamic load carrying

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Figure 10. SHPB Test simulation.

Figure 11. Specimen stress-time curves.

Figure 12. Specimen stress-time histories.

capacities and Ps is the static loading carrying capacity, the DIF for capacity is written as,

DIF = Pd

Ps. (1)

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Figure 13. Dimensions of striker bars (mm).

Figure 14. Incident waves from tapered striker bars.

To achieve DIF in different strain rates for FRP confined concrete, different length of strikerbars should be used to enable a range of incident stress waves. The lengths of striker bars usedin the parametric studies are 150 mm, 300 mm, 600 mm and 900 mm, respectively. Figure 13shows the shapes of the four striker bars which will play an important part in the dynamicincrease factor research. With the four striker bars incident stress waves of short and longtime durations have been successfully simulated. Figure 14 shows incident waves produced byimpact loads from the four striker bars onto the incident bar at the same speed. As shown, theshorter incident wave has a shape similar to a half-sine waveform with total time duration ofabout 110 ms, while the other three incident waves have the total duration of about 175 ms,285 ms and 395 ms, respectively, demonstrating that the duration is directly proportionalwith the length of striker bar. The maximum strains of four groups are almost the sameexcept for the longest striker bar having a slight increase. Thus with variation of strike barlengths DIF of FRP confined concrete specimens with varying strain rates can be achieved.

A series of numerical analyses has been conducted on concrete specimens confined withSL and DL CFRP tubes under impact loads. Figures 15 and 16 show a comparison of resultsfor SL and DL CFRP confined concrete under static and dynamic simulation. It is found thatDIF is 1.12 at strain rate of 85 and 1.21 at strain rate of 112 for SL CFRP confined concreteand 1.16 at strain rate of 140 and 1.27 at strain rate of 176 for DL CFRP confined concrete.Using the simulated results the relationships between DIF and average strain rate for SL and

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Figure 15. SL CFRP confined concrete simulation.

Figure 16. DL CFRP confined concrete simulation.

DL CFRP confined concrete specimen are plotted in Fig. 17. As shown at the same strainrate DIF for SL CFRP confined concrete is slightly smaller than that for DL CFRP confinedconcrete. The main reason leading to this situation is because CFRP tube will fail earlierbefore concrete crashing under a high strain rate for SL CFRP confined specimens while onthe contrary, for DL CFRP confined specimen concrete crashing becomes earlier before CFRPfailure due to the high confinement ability of DL CFRP sheets. Figure 18 shows DIF versusaverage strain rate for SL and DL CFRP confined concrete in a semi logarithmic scale. It isobserved that the strain rate effect can be expressed by bilinear approximations respectively.

Using the simulated data DIF for SL CFRP confined concrete can be derived as

DIF = 1.01+ 0.001 log (ε) 0� ε � 30 s−1 (2)

DIF = 0.975+ 0.001 log (ε) 30� ε � 200 s−1. (3)

For DL CFRP confined concrete:

DIF = 1.01+ 0.002 log (ε) 0� ε � 30 s−1 (4)

DIF = 0.986+ 0.001 log (ε) 30� ε � 200 s−1. (5)

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Figure 17. SL CFRP confined concrete specimens.

(a) SL CFRP confined concrete (b) DL CFRP confined concrete

Figure 18. Semi logarithmic plot of DIF versus average strain rate for SL and DL CFRP confinedconcrete.

Figure 19. Comparison between plain concrete and CFRP confined concrete.

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A comparison of DIF versus strain rate for plain concrete and CFRP confined concrete isshown in Fig. 19. The data for plain concrete was tested by Zheng et al.5 As shown, DIF forplain concrete is much higher than CFRP confined concrete.

6. Conclusions

The following conclusions are deduced from the experimental results:

• A numerical model for CFRP confined concrete specimens under static loads and impactloads is validated.• Dynamic increase factors for CFRP confined concrete specimens under different strain

rates are derived from the numerical simulation.• DIF for SL DIF CFRP confined concrete is smaller than that DL CFRP confined concrete,

but they are smaller compared with plain concrete.

Acknowledgements

The numerical analysis was conducted as part of an advanced master research project ‘FRPconfined concrete under impact load’ by Mr Lei Guo.

References

1. Mirmiran, A. and Shahawy, M., “Behavior of Concrete Columns Confined by Fiber Composites”,Journal of Structural Engineering, May, 1997, pp. 583–590.

2. Suter, R. and Pinzelli, R., “Confinement of concrete columns with FRP sheets”, Proc. 5th Int.Conf. on Fibre Reinforced Plastics for Reinforced Concrete Structures, University of Cambridge,Cambridge, UK 2001, pp. 793–802.

3. Malvar, L.J., Morrill K.B. and Crawford J.E., “Numerical Modeling of Concrete Confined byFiber-Reinforced Composites” Journal of Composites for Construction, July/August 2004, pp.315–322.

4. Malvar, L.J., Crawford, J.E. and Morrill, K.B., “K&C Concrete Material Model Release III— Automated Generation of Material Model Input.” K&C Technical Report TR-99-24-B1, 18August 2000.

5. Zheng, S., Haussler-Combo, U., and Fibl, J., “New Approach to Strain Rate Sensitivity of Concretein Compression” J. Eng. Mech., 125(12), 1999, pp. 1403–2410.

684

Estimating Hydraulic Permeability of Fractured Crystalline RocksUsing Geometrical Parameters

R. VESIPA∗, Z. ZHAO AND L. JING

Royal Institute of Technology, Stockholm, Sweden

1. Introduction

Flow analysis in rock masses is very important for many engineering and environmentalproblems, such as safety assessment of high-level radioactive waste repositories,1 geothermalenergy and oil/gas reservoirs, hydrocarbon storage in rock caves and hydropower projects.2,3

In crystalline rock masses, the rock matrix has almost negligible permeability, and thehydraulic proprieties of the whole rock mass can be assumed to entirely depend on its fracturesystem.4 For such geological systems, the rock mass permeability analysis usually deals withtwo problems. The first one is to estimate whether the fracture connectivity is adequate topermit flow through the regions concerned, i.e. to judge if the fracture system is percolating.The other problem is to estimate the flow rate for the percolating fracture networks for backcalculation of the permeability.

In order to estimate the probability of a connected fracture cluster to be hydraulicallyconductive, percolation theory have been extensively used.5−11 Basically, some parametersdescribing network connectivity, such as fracture density per unit area or intersection densityper unit area, were defined. Then, models with different values of this parameter are builtand the hydraulic proprieties are estimated. Finally, a relation between the chosen parametersand the likelihood of having a percolating system is established.

In order to estimate the flow rate, two approaches can be used: a deterministic approachor a stochastic approach.

Since the significant spatial variability of the actual geometry of in-situ fracture networks,deterministic approaches for fluid flow simulations in the fracture systems are usually of lim-ited applicability. Therefore, stochastic approaches are often required, based on distributionsof geometrical parameters of the fracture networks, such as length, aperture, orientation andlocation, which are used for generating realizations of fracture network models for percola-tion and fluid flow analyses.

For estimating permeability of percolating fracture networks, approaches of different com-plexity can be used. The first approaches were to estimate the permeability from the geometryof the fracture network, without building a numerical model of the network and consideringreal connectivity, but just considering the fracture network parameter value distributions.1

A second one is the approach of Discrete Fracture Network (DFN) for Monte Carlo simu-lations of fluid flow with a large number of DFN models, using the parameter distributionsfrom the field mappings.

The objective of this paper is to develop a direct technique for estimating the equivalenthydraulic permeability of fractured crystalline rocks without using numerical methods forfluid flow simulations, but using only the fracture network models.

The problem was assumed to be in 2D, the rock matrix was considered impermeable; thusthe fluid moves through the connected fractures only. The effects of fracture roughness andstress were not considered for simplicity at this stage. The cubic law (Eq. (1)) was adopted to




define the flow rate q of a fluid with dynamic viscosity μ along a smooth rock fracture witha constant aperture a, length l and pressure difference between the two fracture ends �p.

q = a3

12μ· �p

l. (1)

2. Equivalent Permeability Estimation

Equivalent permeability is often represented as a permeability tensor,12 as defined in Eq. (2).The flow in the i-th direction qi is related to the pressure gradient in the j-th direction �pj bythe permeability tensor term kij. [

qyqx

]=[

kyy kyxkxy kxx

] [�py�px

](2)

This paper purposed a geometrical method to estimate all the terms of the permeabilitytensor as shown in Eq. (2), without using direct numerical flow simulations.

For a fracture network model with N linear channels (fractures) connecting two oppositeboundaries of a DFN model (Fig. 1), the equivalent permeability of the network as a porousmedium in a specific direction y is given by Eq. (3d), which is obtained by letting the flowrate in the y-direction qy of the fractured network shown in Fig. 1(a) (and represented by Eq.(3a)) equal to the flow rate of the equivalent continuum medium as shown in Fig.1(b) (andrepresented by Eq. (3b)), through the mathematical manipulation shown in Eq. (3c).

qy =N∑

i=1

a3i

12μ

(p1 − p2

li

)(3a)

qy = wKyy,Eq

(p1 − p2

L

)(3b)

qy =N∑

i=1

a3i

12μ

(p1 − p2

li

)=

N∑i=1

a3i

12μLL

(p1 − p2

li

)=(

p1 − p2

L

) N∑i=1

a3i

12μLli

(3c)

Kyy,Eq = qyL

w (p1 − p2)= 1

w

N∑i=1

a3i

12μLli= 1

wNk (3d)

(a) (b)

Figure 1. Equivalence between a fracture network (a) and a porous continuum (b).

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In Eq. (3d), k is the average permeability of one channel, and is defined as Eq. (4),

k = 1N

N∑i=1

a3i

12μLli

(4)

where N is the number of equivalent channels.The next step is to model the actual network realization of randomly distributed fractures

as shown in Fig. 2, as an “equivalent channel network”.At this stage of the research, we assumed that a randomly distributed and percolating frac-

ture system can be simplified as a sum of two regular networks of parallel and persistentfractures in the two principal directions of its permeability tensor. In Fig. 2 a randomly dis-tributed DFN is simplified as the sum of two perpendicular sets of regular fractures (channels)of large length and uniform apertures.

Equation (3d) is used to evaluate the equivalent permeability in the vertical direction,Kyy,Eq, but the fracture network is made up of n fractures. The number of equivalent channelsand the mean permeability are given by Eq. (5) and (6).

N =

n∑i=1

ly,i

L(5)

k =n∑

i=1

a3i

12μ

n∑i=1

ly,i

n∑i=1

li

(6)

where ly,i is the length of the projection of the fracture in the vertical direction.The proposed approach is a more advanced step forward than that as reported in1 where

a model was developed to use the vertical projection of the fracture length,1 but the average

permeability was calculated not considering the correction factor in permeability,n∑

i=1ly,i/

n∑i=1

li,

due to the fracture inclination.

Figure 2. Equivalence between randomly distributed fracture system and regularized channel (frac-ture) systems.

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3. Testing and Results

In order to validate the method purposed above, the estimated values of permeability obtainedwith realistic DFN models were compared to results using code UDEC with flow simulations.

3.1. Fracture network construction

At first randomly generated network models (Fig. 3) were built. The geological data and themodel construction technique used have been extensively published in previous work.12 Inthe following paragraphs, only a brief description is given.

A square DFN model is characterized by a model side length L and a randomly gener-ated fracture network, based on distribution functions of fracture length, fracture orienta-tion, fracture aperture and centre. The number of fracture is defined by the fracture densityparameter.

Fracture length is assumed to follow a power law distribution, while fracture orientationis assumed to follow the Fisher distribution. Fracture centre position follows a Poisson distri-bution, while fracture aperture in this first analysis stage has been set as constant. The MonteCarlo method is used to generate the previously listed parameters.

3.2. Approaches and results of equivalent permeability estimation

The first step of analysis, for both the developed geometrical method and UDEC flow mod-elling is the regularization of the network as reported in.12 Regularization process removesdead ends and isolated fractures, since these features do not contribute to fluid flow in thefracture systems. During the study it has been noted that including dead ends on permeabilitycalculation for the proposed geometrical method can lead to an error estimation up to 70%.

The next step is to apply Eqs. (5, 7, 8) to estimate the permeability tensor components bythe developed geometrical method and by UDEC code for fluid flow simulations. In order to

Figure 3. DFN model and hydraulic boundary conditions in the UDEC modelling of fluid flow withpressure gradient in y-direction. The flow directions are shown as well.

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Table 1. Values of hydraulic permeability estimations withvarying block numbers.

Blocks Kyy,UDEC Kyy,Est ε �

[1]3[m/Pa·s/m]

3[m/Pa·s/m] [%]

3[m/Pa·s/m]

1–1000 4.21E-12 2.79E-12 −51 1.42E-121001–2000 1.14E-11 8.78E-12 −30 2.6E-122001–3000 1.63E-11 1.44E-11 −13 1.81E-123001−5000 2.13E-11 2.05E-11 −4 8.49E-135001−7000 2.53E-11 2.56E-11 1 2.74E-13

7000 + 3.07E-11 3.14E-11 2 7.08E-13

measure the fluid flow through the DFN models using UDEC code, two linearly independentboundary conditions are applied. Each model is also rotated a given number of times withfixed rotation angles (e.g. 30◦, as reported in12). For simplicity, only the results for compo-nent Kyy,Eq are compared here due to page limit. The hydraulic boundary conditions in theUDEC models are shown in Fig. 3.

It was found that the density of the fractures, which determines the block numbers inthe DFN models, plays a significant role on the accuracy of permeability estimation by theproposed geometrical method (since the UDEC modelling was considered reliable). Thereforethe results are compared as functions of the block number of the DFN models, by increasingthe fracture density. The results obtained by using the two methods are compared in Fig. 4

Figure 4. Comparisons between the proposed geometrical method and UDEC modelling with varyingblock numbers.

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Figure 5. Change of estimation error of the proposed geometrical method as function of block num-ber.

and Table 1, both for estimated value Kyy,Est and for measured value (Kyy,UDEC ). An errorε, calculated with Eq. (7), and the absolute difference, calculated by Eq. (8), are listed as welland plotted in Fig. 5.

ε = Kyy,UDEC − Kyy,Est

Kyy,UDEC· 100 (7)

� = Kyy,UDEC − Kyy,Est (8)

4. Discussions and Concluding Remarks

The results show that the number of blocks is a parameter determining whether the rockmass permeability can be modelled with the proposed geometrical method with reasonableaccuracy. For the DFN models used in this study, if the number of blocks is smaller than1000 the relative error is too big to permit an accurate estimation by the proposed geometri-cal method. Different lower limits may vary on site-specific conditions of the fracture systemgeometry. Therefore, the proposed geometric method can be used when fracture density isadequately high, well connected and percolating, as an alternative method at early stage ofsite investigations. The DEM fluid flow modelling techniques, on the other hand, have nosuch limitations. However, they suffer from the fact that much more extensive numericalmodelling efforts must be spent to determine the equivalent hydraulic permeability of frac-tures rocks through fluid flow simulations and it is time consuming and costly, especially forlarge scale projects. A proper combination of the geometrical method and numerical flowsimulations using DFN or DEM approaches may be a better approach.

The geometrical method as reported in this paper is at its early stage of development andis simple, since it involves only network generations from geological parameter distribution,and a pre-processing of fracture system regularization as reported in literatures. However,it provides a fast approach for initial estimation of in-situ permeability of subsurface rockmasses using preliminary fracture mapping data, useful for preliminary site characterizationand design of subsurface rock engineering projects..

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Acknowledgments

We would like to acknowledge the financial supports from Swedish Nuclear Fuel and WasteManagement Co (SKB) through the DECOVALEX-2011 project.

References

1. Oda, M. “An equivalent continuum model for coupled stress and fluid flow analysis in jointedrock masses” Water resources research 22(13), 1845–1856, 1986

2. Lin, H & Lee, C. “An approach to assessing the hydraulic conductivity disturbance in fracturedrocks around the Syueshan tunnel, Taiwan” Tunneling and Underground Space Technology 24,222–230, 2009

3. Zhang, X; Powrie, W; Harkness R &Wang S. “Estimation odf permeability for the rock massaround the shiplocks of the Three Gorges Project, China” Int. j. of Rock Mechanics and miningSciences 36, 381–397, 1999

4. Lee, C; Deng, B. & Chang, j. “A continuum approach for estimating permeability in naturallyfractured rocks” Engineering Geology 39, 71–85, 1995

5. Sisavath, S; Mourzenko, V; Genthon, P; Thovert, J & Adler, P. “Geometry, percolation and trans-port proprieties of fracture networks derived from line data” Geophys.J. Int. 157, 917–934, 2004

6. Mo, H; Bai, M; Lin, D & Roegiers J. “Study of flow and transport in fracture network usingpercolation theory” Applied Mathematical Modeling 22, 277–291, 1998

7. Balberg, I; Berkowitz, B. & Drachsler, G. “Application of a percolation model to flow in fracturedhard rocks” Journal of Geophysical Research 96, 10015–10021, 1991

8. Nakaya, S. & Nakamura, K. “Percolation conditions in fractured hard rocks: A numericalapproach using the three dimensional binary fracture network model” Journal of GeophysicalResearch 112, B12203, 2007

9. Berkowitz, B. “Analysis of fracture network connectivity using percolation theory” Mathematicalgeology 27, 467–483, 1995

10. Berkowitz, B. “Percolation approach to the problem of hydraulic conductivity in porous media”Transport in porous media 9, 275–286, 1992

11. Ahn, J; Furuhama, Y; Li, Y & Suzuki, A. “Analysis of radionuclide transport through fracturenetworks by percolation theory” Journal of nuclear science and technology 28(5), 433–446, 1991

12. K.B. Min, L. Jing & O. Stephansson. “Fracture system characterization and evaluation of theequivalent permeability tensor of fractured rock masses using a stochastic REV approach” Int. J.Hydrogeology, 12(5), 497–510, 2004.

691

Mutual Effect of Tectonic Dislocations and Tunnel Linings DuringTunnelling

K. WEIGLOVÁ∗ AND J. BOŠTÍKA

Brno University of Technology, Faculty of Civil Engineering, Institute of [email protected]

1. Introduction

In the implementation of underground structures and in the course of their lifetime externalforces arise that have the effort to change the mutual position of the structure and the sur-roundings. The difference in behaviour of the elements and whole system is determined bythe genesis and a complex process of the development of rock mass, which during its exis-tence was subjected to a number of external and internal influences.1–2 The most importantof them are the tectonic processes that caused the origin of discontinuities (flaws, cracks).

In the first part of this paper a prognosis of behaviour of underground structure and its sur-roundings is presented in the systems with extreme and exceptional conditions, which werestudied using physical models. The research was focused on the third limit state of cracks,the visible impairment of the structures considered, under the conditions of the principalorientations of surfaces of the dislocation that are the decisive factor basically affecting theinteraction of the underground structure and its surroundings.

In the second part of the paper 2D numerical analysis of the circular tunnel (tunnels) inthe rock mass impaired by the tectonic dislocation (fault zone) is described. On the modelproblem the effect of selected factors on the stability of the tunnel (tunnels) is studied. Itinvolves the occurrence of fault zone in the rock mass, its acclivity, thickness and distancefrom the tunnel (tunnels). The analysis was carried out by the finite element method (FEM).

2. Physical Models

For practical solution of this complex problem we started from the parametric studies ofmodels:

A — Geo-Brno-III-2 M 1 : 100 (three models a, b, c were built – see Figs. 1, 2, 3)B — Geo-Brno-III-2 M 1 : 10 (see Fig. 4)

The phenomena of physical dislocations decide about the heterogeneity, anisotropy andbehaviour of the rock mass. Besides the type, shape and density of the surfaces of dislocationsanother important parameter, deciding about the mutual arrangement of the blocks in themass, is the space orientation of surfaces and their unbrokenness.

To make the models fulfil the condition of the perfect similarity it was necessary to choosethe model scale. In its choice we also had to take into consideration the dimensions of themodelling stands.




2.1. Model GEO — BRNO III-2, 2008 M 1:100

For the approximate space observation the scale 1 : 100 was chosen as the most suitable.Three models of the underground structure were made, denoted as a, b, c:

a Blockdiagram: without the surfaces of dislocation (Fig. 1)b Blockdiagram: the surfaces of dislocation with the axis of stope of the underground struc-

ture are non-parallel (Fig. 2)c Blockdiagram: the surfaces of dislocation with the axis of stope are almost horizontal,

slightly inclined (Fig. 3)

Figure 1. Model Geo-2008 III a Blockdiagram a without dislocation surfaces — starting state.

Figure 2. Model Geo-2008 III b Blockdiagram b of dislocation surface with the axis of the stope ofthe underground non-parallel structure.

Figure 3. Model Geo-2008 III c Blockdiagram c of the surface of horizontal dislocation — aftervariable advance.

694


The most interesting result of the experiment appeared during the tunnelling in model c. Inthe case of safe face and variable purchases of tunnelling with almost horizontal dislocationsurfaces the limit state of failure of the underground structure and surrounding rock isobserved.

2.2. Model GEO — BRNO III-2, 2008 M 1:10

Based on results of three conception models having been built on a scale 1:100 (a, b, c)preparation of an essentially more complex model started and spatial problems were solvedfor tunnelling of the underground structure with a variable purchase with almost horizontalorientation of the dislocation surfaces. The dimensions of the model were 200×200×50 cm3.

For physical modelling a real sector was chosen with low overburden layers. In the rockbelow the tunnel at the depth of 200 cm two circular utility tunnels are placed with the profileof 200 cm (cable utility tunnel KK, and circular sewer KS). The underground structure wasmodelled with the diameter of 400 cm and circular lining (Fig. 4). Figs. 5 to 9 depict adevelopment of construction of the model underground structure.

Figure 4. Model Geo-2008 III M 1:10 State after finishing the building of the model.

Figure 5. Model Geo-2008 III M 1:10 State after finishing the building of the model.

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i 6

Figure 6. Model Geo-2008 III M1:10 Technological shield tunnelling in the model.

Figure 7. Model Geo-2008 III M1:10 Formation of cracks during tunnelling.

After mounting and calibration of 7 pressure cushions the creation of model has beencarried out according to the time schedule. Altogether 34 layers of equivalent material wascreated with the total weight of material for building of 1 665 kg.

Monitoring the stress state and reshaping during the building and the subsequenttunnelling was implemented by means of pressure cushions, tensometric measurement,electromechanical resonance (string) gauges, mechanical recorders, electromagneticthermometers (core and contact ones), electric gauges, and geodetic and photogrammetricmeasurements.

After a ten-week observation of the model with the space orientation of dislocation sur-faces the progress of the technology of advancing was begun and carried out with variableengagement.

The results of the technological procedure of advancing of the underground structureshowed the following: considerably reduced material cohesion, contact stresses substantiallyincrease, certain regions of interaction of the underground structure with the surroundingsare disappear, flaws and cracks appear, the surface and direction of dislocations are com-pressed (e.g. by preventing transversal deformations inside the mass) and these pressurescaused buckling and interaction of tectonic prestress (Fig. 10).

696


Figure 8. Model Geo-2008 III M 1:10 Finishing the tunnelling in the model.

Figure 9. Model Geo-2008 III M 1:10 Formation of cracks after finishing tunnelling.

3. Numerical Analysis of the Effect of the Fault Zone

By means of numerical modelling (finite element method) the effect of the damaged zonebetween the two parallel tunnels of circular profile,3 was studied. In the same manner asin the experimental model, the tunnels are situated in small depth below the surface of theterrain, which, within the study carried out, is constant, being approximately 1.7 multiple ofthe diameter of tunnel D. A more detailed overview about the geometric arrangement of theproblem and parameters, whose effect was studied, is shown in Fig. 11.

Mechanical behaviour of the rock environment was approximated by means of the Mohr-Coulomb material model with material characteristics also given in Fig. 11. Calculationswere carried out in the plane strain state using Plaxis program,4 in the following variants:

– simulation of the parallel tunnelling of two tunnels,– simulation of successive tunnelling (left tunnel→ right tunnel) and– simulation of successive tunnelling (right tunnel→ left tunnel).

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Figure 10. Model Geo-2008 III M 1:10 Formation of cracks after finishing tunnelling.

Figure 11. Analysed task.

Unbalanced forces were applied to the respective (active) parts of the model either to fullextent or up to the failure. The reinforcement of the tunnels by lining was not considered inthe computation.

698


o

Figure 12. An example of discretisation of region solved (x = 1.5, t = 3 m, α = 45◦).

In the cases when loading was applied (as a consequence of tunnelling) to full extent, thesafety factor (SF) was calculated. SF was stated by the method of reduction of the strengthparameters.

In Fig. 12, for illustration’s sake, the discretisation of a part of the rock mass is given, whichhad been considered in the mathematical models. Shapes and dimensions of all models wereidentical, namely they were rectangles 82 by 37.82 m.

Individual solved cases were then compared. Further the evaluation the safety factor ismentioned.

3.1. Parallel excavation of two tunnels

A summary overview of the stability of the tunnels in this case is given in Table 1, in whichthe calculated safety factors for the monitored parameters are stated. Free cells denote caseswhen the load due to the tunnel excavation was not applied in full extent.

Table 1. Safety factor — parallel tunnelling of tunnels.

tunnels separation 1.5D tunnels separation 2D t/α 0 15 30 45 60 75 90 0 15 30 45 60 75 90 1 1.05 1.05 1.07 1.01 1.00 1.06 1.11 1.14 1.14 2 1.03 1.04 1.00 1.03 1.10 1.13 1.12 3 1.01 1.02 1.00 1.08 1.11 1.12 4 1.06 1.09 1.11 tunnels separation 2.5D t/α 0 15 30 45 60 75 90 1 1.09 1.02 1.08 1.13 1.13 1.15 1.15 2 1.02 1.01 1.11 1.13 1.15 1.15 3 1.10 1.12 1.15 1.15 4 1.07 1.11 1.15 1.15

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The change of the safety factor depends on the inclination of the damaged zone up to about14 %, on the thickness of the failure and distance between the tunnels. From the table it isevident that with increasing distance of the tunnels also the extent of 100 % load increaseswithin the cases considered. With the increase of the thickness of tectonic discontinuitiesthere is a drop in the safety factor. In a larger distance of the tunnels (2.5D) and a steeperinclination of the damage zone (75, 90◦) the value of SF practically does not change with thethickness of the damage zone (t).

For the case of the rock mass not weakened by the damage zone SF is, as expected, mostfavourable and for the distance of the tunnels 1.5D, 2D and 2.5D SF was calculated as 1.07,.1.15 and 1.15, respectively. The last mentioned value corresponds with the value of a steeplyrunning damage zone at the largest considered distance of the tunnels.

3.2. Successive excavation of tunnels

Individual dependencies of SF vs. t, SF vs. α and SF vs. distance of tunnels for the caseof successive excavation are again pictured in the tabular form, only for the distance 1.5D.Table 2 corresponds to the case of excavation of first the left tunnel and then the right tunnel.In Table 3 the stated value of SF then corresponds to the opposite order of tunnel excavation.

From the results it can be seen that the region of stable tunnels in the final stage (i.e. afterdriving both tunnels) delimited by the variable parameters of the analysis is in the simulationof the successive excavation of tunnels more or less in agreement with the region determinedat the simulation of the contemporaneous excavation of the tunnels, even though in thevalues of SF it is possible to observe certain differences.

From the view of an only one tunnel excavation the region of the stable tunnel is moreextensive in the case of the left tunnel (i.e. the tunnel situated over the failure zone), so thatit is possible to admit a milder inclination of the failure zone and/or its higher thickness. Thedifference mentioned is more significant at a shorter distance of the tunnel from the faultzone.

Table 2. Safety factor – gradual tunnelling of tunnels (left→ right), x = 1.5D.

left tunnel both tunnels t/α 0 15 30 45 60 75 90 0 15 30 45 60 75 90 1 1.08 1.09 1.10 1.15 1.15 1.14 1.13 1.05 1.05 2 1.01 1.04 1.04 1.12 1.13 1.13 1.11 1.03 1.04 3 1.02 1.09 1.11 1.11 1.08 1.07 1.02 4 1.01 1.05 1.09 1.08 1.07 1.00

Table 3. Safety factor – gradual tunnelling of tunnels (right→ left), x = 1.5D.

right tunnel both tunnels t/α 0 15 30 45 60 75 90 0 15 30 45 60 75 90 1 1.09 1.04 1.03 1.10 1.09 1.10 1.13 1.02 1.05 1.05 2 1.01 1.06 1.08 1.09 1.12 1.00 1.03 1.04 3 1.02 1.05 1.07 1.09 1.03 4 1.02 1.05 1.07 1.00

700


4. Conclusions

4.1. Physical modelling

During the excavation of underground structure (at a two-component shield coating) thecourse of the values necessary for the coefficient KZP (ratio of the purchase to the tunneldiameter) was monitored.

As an optimum safety limit of KZP for our model on the basis of the experiment it wasstated the value

KZP = 0.32− 0.37 (1)

Another significant result of the experiment is the value of the coefficient KSt — i.e. theratio of the equivalent material volume to the surface settlement surface (equals to the ratioof the volume of equivalent material and the value of the volume of the tunnel area)

KSt = 0.25− 0.31 (2)

In conclusion of the experiments carried out it is possible to conclude that in the case ofunderground structure under extreme conditions it is for purposeful and economic master-ing of the possible extraordinary contingency suitable to study not only all documents ofengineering geological investigation, results of measurements in situ, but also to confront themathematical modelling with the physical modelling.

4.2. Numerical analysis

From the results obtained by numerical analysis it can be judged that the occurrence ofdiscontinuities, represented here by the tectonic discontinuity zone of a certain thickness, canlead to a significant influencing of the response of the mass to the advance of undergroundworks. The rate of influencing is then determined by not only the inclination and thicknessof the fault zone, but also by its localisation with respect to the tunnels.

As a part of the extents considered of the individual parameters of analysis it is possible toadmit milder inclination and/or higher thickness of the discontinuity zone. The instability oftunnels at a low inclination of this zone it is evidently done due to the fact that the damagezone either crosses the stope or runs in its immediate neighbourhood, which appears aboveall with its higher thickness.

In the excavation of a single tunnel, from the point of view of tunnel stability, the occur-rence of a tectonic discontinuity zone above the tunnel (the right tunnel) appears as lessfavourable.

Acknowledgements

This contribution was financially supported by the project of the Czech Science Foundation(GA ÈR) No. 103/07/P323 and by the research project of The Ministry of Education, Youthand Sports (MŠMT ÈR) No. MSM0021630519. Authors acknowledge this support.

References

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Author Index

A Ahn, T. Y. 127 Ali, Hessam Moghaddam 539, 573 An, X. M. 47, 237, 297

B Baghbanan, A. 445 Bagheri, M. 445 Bai, J. G. 135 Bakun‐Mazor, D. 193 Bao, H. R. 99, 161 Bicanic, N. 283, 291 Boštík, J. 693

C Cai, Y. C. 405 Chang, J. Z. 617 Chen, G. Q. 227 Chen, G 343 Chen, P. W. 73, 83, 91, 211 Chen, S. G. 367, 469 Chen, Y. L. 297 Cheng Q. S. 625 Chong, W. L. 413 Chua, H. C. 429 Chung, S. K. 437 Cui, H. Q. 641 Culek, B. 547

D Dai, K. D. 211 Ding, X. L. 325 Doi, Y. 461 Dolezel, V. 547 Dong, Z. H. 325

E Einstein, H. H. 581 Emad, Kayumars 539, 573

F Fan, L. F. 563 Ferdowsi, B. 485 Fukazawa, J. 495 Fu, G. Y. 351

G Gao, M. Z. 531 Glaser, S. D. 193 Goh, A. T. C. 429 Grepl, J. 661

H Hagedorn, H. 373 Hagiwara, I. 265 Hajiazizi, M. 453 Hamasaki, E. 153, 395 Haque, A. 413 Hashemalhosseini, H. 445 Hatzor, Y. H. 13, 193 He, L. 47, 305, 351 Hong, B. N. 315 Hori, S. 185 Hou, Y. L. 227 Huai H. J. 211 Huang, M. L. 505 Huang, T. 177 Huang, Z. Y. 675

I Irie, K. 153

N Nakai, T. 273 Nakamura, K. 39 Nia, N. Nourbakhsh 589 Ning, Y. J. 73, 83, 91 Nishimura, T. 387 Nishiyama, S. 39, 153, 255, 265

O Ohnishi, Y. 39, 153, 255, 265, 273 Okada, H. 609 Okazawa, S. 555, 609 Otani, T. 461 Ozbakkloglu, T. 675

P Park, Chan 647 Park, Chulwhan 647 Park, E. S. 437, 647 Pearce, C. J. 283, 291 Peng, X. C. 177 Prochazka, P. P. 359, 547

Q Qiu, K. H. 145

R Racansky, V. 661 Ranjith, P. G. 413 Ryu, S. H. 127

S Sasaki, K. 265 Sasaki, T. 255, 265 Shafipour, R. 485 Shahinuzzaman, A. 413 Shao, G. H. 513 Shen, B. 469 Shi, G. H. 1, 135 Shi, G. B. 135 Shimaoka, K. 153 Shimauchi, T. 39 Shinji, M. 461 Singh, Rajbal 477 Song, J. J. 127 Sonoda, Y. 495 Sookhak, A. 445 Soroush, A. 485 Stadelmann, R. 373 Su, H. D. 247 Sun, S. R. 315 Sun, B. P. 135

T Tajiri, Y. 395 Takeuchi, N. 395 Tanaka, S. 609 Tang, C. A. 505, 521, 599 Terada, K. 555

V Vesipa, R. 685

W Wang, J. 513 Wang, M. 135 Wang, S. Y. 505, 521 Wang, Y. 135 Wang, Y. Z. 513 Wang, Z. T. 513 Wei, L. J. 217 Wei, X. Y. 669, 169

Iwata, S. 555

J Jeon, S. 421 Jia, X. L. 641 Jiang, W. 109 Ji, C. L. 513 Jiao, L. Q. 351 Jing, L. 685 Jung, Y. B. 437, 647

K Kaneko, F. 185 Kashiyama, K. 555 Kim, H. 421 Kim, J. W. 647 Kim, T. K. 437 Kobayashi, T. 273 Kourepinis, D. 283, 291 Koyama, T. 153, 255, 265, 273 Kulatilake, P. H. S. W. 59 Kurumatani, M. 555

L Lee, C. I. 127 Li, G. 599 Li, J. C. 563 Li, L. C. 521, 599 Li, L. X. 297 Li, X. J. 27 Li, X. Z. 513 Li, Y. 381 Liang, Z. Z. 505, 521, 599 Lin, S. Z. 145 Lin, Y. L. 217 Liu, G. R. 37, 589 Liu, M. B. 617 Liu, Y. Q. 91 Lu, B. 27, 325

M Ma, G. C. 185 Ma, G. W. 47, 73, 83, 237, 297, 305, 351, 381, 563, 625, 675 Ma, G. S. 27 Ma, H. S. 531 Ma, S. F. 669 Majdi, Abbas 539, 573 Mangyuan, Li 625 Maruki, Y. 273 Mica, L. 661 Miki, S. 255, 273

Weiglová, K. 359, 693 Wong, L. N. Y. 581 Wu, W. 381, 625 Wu, A. Q. 27 Wu, C. 675

X Xia, C. C. 201 Xie, X. L. 247 Xu, C. B. 201 Xue, J. 119 Xue, Z. P. 641

Y Yagi, K. 273 Yan, L. 625 Yang, F. L. 641 Yang, J. 73, 83, 91 Yang, Q. G. 27 Yang, W. J. 315 Yoshinaka, R. 265

Z Zhang, C. H. 227 Zhang, G. X. 177 Zhang, H. 367 Zhang, H. H. 297 Zhang, X. G. 625 Zhang, Y. 169 Zhao, G. F. 633 Zhao, J. 531, 633 Zhao, J. H. 669 Zhao, X. B. 513 Zhao, X. 201 Zhao, Y. B. 367 Zhao, Z. 685 Zhao, Z. Y. 99, 161, 169, 429 Zheng, H. 109 Zhu, H. H. 405 Zhu, L. 315 Zhuang, X. 405