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IhWSTIGATION OF NONLLYEAR ELASTICITY AL- ANALYSIS OF SILO FILLING
by
Jikang Zhou
Deparmient of Civil and Environmental Engineering Faculty of Engineering Science
Submitted in partial fuIfiIlment of the requirements for the degree of
Master of Engineering Science
Faculty of Graduate Studies The University of Western Ontario
London, Ontario March, 1999
O Jikang Zhou 1999
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Many existing nonlinear elastic models and their application in practice have been
reviewed and categorked. Surprisingly, very fev energy-conserving nonlinear elasticity
models for soils have found their way into multipurpose elastoplastic k i t e element (FE)
codes. This is due to either the very complex expressions for these models making it
difficult to derive their tangential constitutive relations, or, the number of parameters for
the model whose measurement requires advanced testing.
A nonlinear elasticity model for ganular material onghally developed by Boyce
(1980) has been studied thoroughiy. The rnodel is proved to be energy-consemative. Only
three parameters are needed and they are easily obtained by conducting conventional
tests. Direct definition of Poisson's ratio is avoided in the model, which is an advantage
since it is not constant in nature. The tangential form of the stress-strain relations for the
model is derived- This constitutive matrix is found to be symmetric. The model has been
implemented into a research FE code named AFENA. Numerical problems encountered
in the implementation are discussed and treated. Simple cornputer tests have been
perfurmed using both MENA and smalI test programs. The results are in agreement.
Parameter evaluation for the model is studied. Approaches to obtaining
parameters for problems with different stress paths are suggested. The focus is on the silo
filling problem. Using experimental data for Ioose sands fiom the literature, and
comparing with published parameters, the magnitudes of the parameters are developed
for loose sands. Parameters for other commody used elastic models are also studied-
FE analyses are perfomed on the silo filling problem using these elastic models
for the filling matenaI. Both planar silos (plane strain) and cylindrical axially symrnetric
silos are studied. Ta11 silos as well as squat silos are considered- The silo wall is treated as
either a rough rigid boundary or a flexible matenai. Solid-structure interaction is
modelled using weak eiements for the filling material close to the wall. The
'construction' technique is used to model the filling process. The horizontal and shear
stress distributions adjacent to the vertical wall are calculated. The complete stress States
in the silo system are given. Comparkons of the results show that the Boyce model
behaves differently from the other elastic models. Poisson's ratio may play a key role
during the analysis.
This thesis is a theoretical study with respect to the implementation of nonlinear
elastic models and application of these models in silo filling. The numencal predictions
~ 1 s t h- ex~di.t.d o!zeyi !Q m.i.r.d preseixep. mis g t ~ ~ d y coiild p iv ide a
valuable reference for analyzing experimental data in the future and consequently
enhance silo design.
Many individuals have assisted the author during his Master of Engineering Science
candidature. The author would like to tbank ail of them for their valuable contribution.
The author would especially like to express his sincere gratitude and appreciation to his
advisor, Dr. Ian D. Moore, for his guidance, encouragement and support throughout the
duration of this study and preparation of this dissertation.
The author wishes to thank Dr. Ashraf El Damatty for his valuable advice.
This research was conducted as part of a collaborative research project on silo structures
beiwçeri Cie C'r;vcrsiiy u i Wtsism Ciiiàiiù iiïd Ediiiûüür& Ui i i~s i~ i iÿ . The iï ivû~~tiï i t i i i
of Dr. Jin Y. Ooi and Professor J. Michael Rotter in that investigation and the research
support provided by NSERC are geatly acknowledged.
In addition, The author would like to express his gratitude to his fellow graduate students
at the RISC Lab, Allen Li, Paul Dittrich, Graeme Skinner, Mohamed Abdel-Fattah,
Tatsuo Iryo, Bahaa Taleb, and Michael Law for their fiendship, encouragement and
support. Thanks are due to Henry Sangam, Wenxing Zhou and Craig Lake for their kind
assistance. They al1 helped to make the author's candidature more enjoyable. Thanks to
Ms. Elizabeth Milliken for her help as the GRC administrative assistant.
The author wishes to acknowledge the tremendous support provided during this course of
study by his parents, Fenling and Mudan, and his brother, Weikang. -
Finally, the author wishes to thank his fiancée, Cuiqiong, for her limitless patience and
constant support.
TABLE OF CONTENTS Page
CERTEICATE OF EXAhrIINATTON
ABSTRACT
ACKNOWLEDGEMENTS
TABLE OF CONTENTS
LIST OF TABLES
LIST OF FIGURES
CHAPTER 1 INTRODUCTION
1.1 General
1.2 Objectives of the Research
1.3 Outline of the Dissertation
CEIAPTER 2 ELASTIC BEHAVIOUR OF SOLS
2.1 Introduction
2.2 Basic Concepts of Isotropie Elasticity
2.3 Elasto-plastic Behaviour of Soils
2.4 Poisson's Ratio
2.5 Isotropy of Elastic Behaviour
2.6 BuIk Modulus and Shear Modulus
2.7 Unloading and Reloading
2.8 Energy Conservation
2 Classification of Elastic Models
2.10 PoroeIastic Model
2.1 1 Janbu Model
2.12 Lade and Nelson Model
2.13 Boyce Model
2.14 Elastic Models for Bulk Solid Stored in Silos
CEFAPTER 3 LMPLEMENTATION OF THE BOYCE MODEL
3.1 introduction
3.2 Derivation of the Tangential Form of the Stress-Strain Relation for
the Boyce Model
3.3 Symmetry of the Constitutive Matrix
3.4 Separating Elastic StrainandPlastic Strain
3.5 Treatment of Some Numerical Problerns
3.5- 1 Introduction
3.5.2 y,, = O or y,, 2 O (q=O)
3.5.3 p=o
3 -5.4 Choosing adequate incremental stress and strain
3.5.5 Other sets of the Boyce parameters
2 h Tp~t &&!pm~ -3h-g the Rcyce Mn&! d .-
3 -6.1 Introduction of MENA
3.6.2 Test Mode1
3.6.3 Isotropie compression and triaxial compression cases
3.6.4 One dimensional compression cases
3.6.5 Effect of numerical treatrnent
C W T E R 4 PARAMETER EVALUATION
4.1 introduction
4.2 Parameter Evaluation for the Boyce Model
4.2.1 Approaches to obtaining parameten for the Boyce mode1
4.2.2 Selected sand data fkom the literature
4.2.2.1 Material stored in silos
4.2.2.2 Loose Sacramento River Sand
4.2.2.3 Loose Santa Monica Beach Sand
4.2.3.4 Loose Ottawa Sand
4.2.2.5 Summary
vii
4.2.3 Published parameters
4-2-4 Discussion
4.3 Parameter Evaluation for the Lade and Nelson Model
4.3- 1 Loose Santa Monica Beach Sand
4.3.2 Loose Sacramento River Sand
4.3.3 Loose Ottawa Sand
4.4 Parameter Evaluation for the Janbu Model
4.5 Parameter Evaluation for the Poroelastic Model
4.6 Parameter Evaluation for the Linear Elastic Model
1.7 Summary and Discussion
C W T E R 5 APPLICATION IN SILO FILLLNG
5.1 Introduction
5 -2 Problem Definition
5.2.1 General
5.2.2 Material properties
5.2.3 Silo geometry
5.3 Finite Elernent Simulations
5.3.1 Finite element meshes
5.3.2 Simulation of the silo filling process
5.3.3 Solution procedures
5.3 -3.1 Stress correction
5 -3 -3 -2lncremental Ioad steps
5.4 Results from Finite Element Analyses
5 -4.1 General
5 -4.2 Approaches to reducing oscillations
5.4.3 Predicted stress distributions adjacent to the wall
in the ta11 silo
5.4.3.1 Axially syrnmetric analysis with ngid wall
5.4.3.2 Plane strain analysis with ngid wall
5.4.3 -3 Axially symrnetric analysis with flexible wall
5.4.4 Predicted stress distributions adjacent to the wall
in the squat siio
5.4.4.1 Axially symrnetric analysis with ngid wall
5.4.4.2 Plane strain analysis with rigîd wafl
5.4.4.3 Axially syrnmetric analysis with flexible wall
5 -4.5 Predicted wall deflections
5.4.6 Stress contours
5.4.7 Plastic zones
5.4.8 Impact of Poisson's ratio of bulk solid
5.4-9 Effect of weak elements
54-10 Effect of stress correction
Summary and Conclusions
CHAPTER 6 SUMMARY AND CONCLUSIONS
6.1 Five Elastic Models for GranuIar Material
6.2 Implementation of the Boyce Mode1
6.3 Parameter Evaluation fijr the Five Models
6.4 Application in Silo Filling
6.5 Recommendations for Future Work
APPENIDIX A CONSERVATION OF ENERGY FOR THE
BOYCE MODEL
APPENDIX B PROOF OF SYMMETRY OF THE CONSTITUTIVE
MATRDC: FOR TKE BOYCE MODEL
REFERENCES
VITA
LIST OF TABLES
Table 4- 1 Experimental data for Loose Sacramento River Sand. (after Lade, 1977)
Table 4-2 Experimental data for Loose Santa Monica Beach Sand. (after Lade and Nelson, 1987)
Table 4-3 Experimental data for Loose Ottawa Sand. (after Dakoulas and Sun, 1992)
Table 4-4 Parameters of the loose sands for different models
Table 5-1 Elastic parameters of the bulk solid to be used in finite element calculations.
Table 5-2 Plastic and other parameters of the bulk solid to be used in fmite elernent calculations.
Table 5-3 Parameters of the steel silo wall.
Page 79
LIST OF FIGURES
Figure 2-1 Closed stress cycle in effective stress space. Page
32
Figure 2-2 Poisson's ratios determined directly from resuIts of triaxial tests on Loose Santa Monica Beach Sand (afier Lade and Nelson, 1987). 33
Figure 3 4 Flow chart of the procedure for calculating the increments of strain 5 1 and stress at each integration point.
Figure 3-2 Comparison between the secant solution and the incremental solution using different p,, in computer simulations of elastic behaviour during isotropic
compression tests. Boyce model, K, = 404(kPa)"~', G, = 673(k~a)O-~', n = 0.62.
A & = 3 x l ~ - ~ . 52
Figure 3-3 Comparison between the secant solution and the incremental solution using different p,, in computer simulations of elastic behaviour during isotropic
compression tests. Boyce model, K, = 404(kPa)~-~', G, = 673(kfa)*.",n = 0.62.
A & = 3 x 53
Figure 3-4 Cornparison between the secant solution and the incremental solution using different incremental strains in computer simulations of elastic behaviour during isotropic compression tests. Boyce rnodel, KI = 4 0 4 ( k ~ a ) ~ - ~ ' ,
G, = 673(k~a)~-", n = 0.62. p,, = O-OOSkPa. 54
Figure 3-5 Comparison bebveen the secant solution and the incremental solution using different incremental stresses in computer simulations of elastic behaviour during isotropic compression tests. Boyce model, KI = 4 0 4 ( k ~ a ) ~ - ~ ' ,
G, = 6 7 3 ( k ~ a ) ~ . ~ ' , n = 0.62. p,, = O.OO5kPa .
Figure 3-6 Comparison between the secant solution and the incrernental solution using different p,, in computer simulations of elastic behaviour dunng isotropic
compression tests. Boyce model, K, = 40O(kfa)O", G, = 185(k~a)O.", n = 0.4.
A & = 3 x IO-'. 56
Figure 3-7 Cornparison between the secant solution and the incremental solution using different p,, in computer simulations of elastic behaviour during isotropic
compression tests. Boyce model, K, = 400(k~a)"', G, = l85(kfi)O", n = 0.4.
A E = 3 x
Figure 3-8 Comparison between the secant solution and the incremental solution using different incremental strains in computer simulations of elastic behaviour
during isotropic compression tests. Boyce model, K, = 400(kPa)~-~,
G, = 185(k~a)O.', n = 0.4. ph,, = O.OO5kPa . 58
Figure 3-9 Comparison betwcen the secant solution and the incremental solution using differcnt incremental stresses in computer simulations of elastic behaviour during isotropic compression tests. Boyce model, Ki = 400(kfi)O",
G, = 185(k~a)O-', n = 0.4. pmin = O.OO5kPa.
Figure 3-10 Comparison of stress-strain predictions for isotropic compression followed by triaxial compression; solutions fiorn MENA and from direct stress integration. Axially symrneûic problem, elastic behaviour.
Figure 3- 1 l Comparison of stress-strain predictions for uniaxial compression; solutions from MENA and from direct stress integration. PLane strain problem, elastic behaviour.
Figure 4-1 Triaxial compression tests with constant stress ratios, and with loading and unloading (after El Sohby and Andrawes, 1973).
Figure 4-2 ; sv~ûpic ~ û i i i p ~ ~ s i û i i ûf Lûûse Sâca.ïeilto K Ï C ~ SâiiY p ï k ~ û ï ï loading, unloading and reloading branches (after Lade, 1977).
Figure 4-3 Unloading portion of isotropic compression tests of Loose Sacramento River Sand and its curve fitting for the Boyce model. (Coefficient of correlation r2=0.9969)
Figure 4-4 Stress-strain and volume change behaviour of Loose Santa Monica Beach Sand in triaxial compression tests with several unloading-reloading cycles (after Lade and Nelson, 1987).
Figure 4-5 Poisson's ratios calculated from computer simulations of triaxial compression using the Boyce model with different K, . For Santa Monica Beach
Sand, G, = 3 0 8 ( k ~ a ) ~ - " ~ , n = 0.45.
Figure 4-6 Isotropie compression tests on Loose Ottawa Sand (afier Dakoulas and Sun, 1992).
Figure 4-7 Unloading portions of isotropic compression of Loose Ottawa Sand and its curving fimng for the Boyce model. (Coefficient of correlation r2=0.9823)
Figure 4-8 Effect of applied constant stress ratio on strain ratio for a loose sand (after El Sohby and Andrawes, Fig.(S)a, 1973).
Figure 4-9 Variation of Young's modulus in triaxial and cubical triaxial tests for Loose Santa Monica Beach Sand (afier Lade and Nelson, 1987).
xii
Figure 4-10 Variation of coefficient of correlation, modulus number and exponent with Poisson's ratio in triaxial tests on Loose Santa Monica Beach Sand (after Lade and Nelson, 1987).
Figure 4-1 1 Cornparison between the expenmental data and the predicted results for Loose Santa Monica Beach Sand using the Janbu model.
Figure 4-12 Pararneter evaluation for the poroelastic model using regression analysis for Loose Sacramento River Sand (Coefficient of correlation r2=0.9864)
Figure 4- 13 Parameter evaluation for the poroelastic model using regression analysis for Loose Ottawa Sand (unloading). (Coefficient of correlation r2=0.9904, 0.9643,0.9750,0.9860 for Ist, 2nd, 3d and 4' unloading respectively)
Figure 4-14 Parameter evaluation for the poroelastic model using regression analysis for Loose Ottawa Sand (overall unloading). (Coefficient of correlation r2=0.9475)
Figure 5-1 Geometrical dimensions of the ta11 silo.
Figure 5-2 Filling process for bulk solid in the ta11 silo.
Figure 5-3 Finite element mesh for the ta11 silo with rough rigid wall: (a) b u k Solid; (b) boundary conditions.
Figure 5-4 Finite element mesh for the ta11 silo with flexible wall.
Figure 5-5 Finite element mesh for the squat silo.
Figure 5-6 Cornparison of stress-strain predictions for isotropic compression test; solutions f?om AFENA with substeps and without substeps. Axially symmetric problem, elastic behaviour.
Figure 5-7 Smoothing approach used for reducing oscillations.
Figure 5-8 Horizontal stress distribution adjacent to the rigid wall (r=2.23m) in the tail silo. Axially symrnetric analysis.
Figure 5-9 Shear stress distribution adjacent to the rigid wall (r=2.23m) in the ta11 silo. Axially syrnmetric analysis.
Figure 5-10 Horizontal stress distribution adjacent to the rigid wall (x=2.23m) in the ta11 silo. Plane strain analysis.
Figure 5-1 1 Shear stress distribution adjacent to the rigid wall (x=2.23m) in the ta11 silo. Plane strain analysis.
Figure 5-12 Horizontal stress distribution adjacent to the flexible wall (r=2.23m) in the ta11 silo. Axially symmetric analysis, Thickness=lcm. 133
Figure 5- 13 Shear stress distribution adjacent to the flexible wall (r=2.23m) in the ta11 silo. Axially syrnrnetric analysis. Thickness= 1 cm. 134
Figure 5-14 Horizontal stress distribution adjacent to the wall in the ta11 silo. Rigid wall and flexible wall. Axially symrnetnc analysis. Lade and Nelson model. 135
Figure 5-15 Shear stress distribution adjacent to the wall in the ta11 silo. Rigid wall and flexible wall. Axially symmetric analysis. Lade and Nelson model. 136
Figure 5- L 6 Horizontal stress distribution adjacent to the ngid w a l l ( ~ 2 . 4 lm) in the squat silo. Axially symmetric analysis. 137
Figure 5- 17 Shear stress distribution adjacent to the ngid wall ( F U 1 m) in the squat silo. Axially symmetric analysis. 138
Figure 5- 18 Horizontal stress distribution adjacent to the rigid wall (FU 1 m) in the squat silo- Plane strain analysis. 139
Figure 5-19 Shear stress distribution adjacent to the ngid wall (r=2.41m) in the squat silo. Plane strain analysis. 140
Figure 5-20 Horizontal stress distribution adjacent to the flexible wall(~2.41m) in the squat silo. Axially symmetric analysis. Thiclaiess= 1 cm. 141
Figure 5-2 1 S hear stress distribution adjacent to the flexible wall (r=2.4 1 m) in the squat silo. Axially syrnmetric analysis. Thichess= k m . 142
Figure 5-22 Wall deflection of the ta11 silo. Axially symmetric analysis. Thickness= 1 cm. 143
Figure 5-23 Wall deflection of the ta11 silo. Axially symmenic analysis. Thickness=OScm. 144
Figure 5-24 Wall deflection of the squat silo. Axially symmetric analysis. Thickness= 1 cm. 145
Figure 5-25 Contom of horizontal stress cm predicted for the ngid ta11 silo: (a) Lade and Nelson model; (b) Boyce model. 146
Figure 5-26 Contours of shear stress oc predicted for the rigid taIl silo: (a) Lade and Nelson model; (b) Boyce model.
xiv
Figure 5-27 Contours of vertical stress O= predicted for the rigid ta11 silo: (a) Lade and Nelson model; (b) Boyce model.
Figure 5-28 Contours of horizontal stress cm predicted for the rigid squat silo: (a) Lade and Nelson model; (b) Boyce model.
Figure 5-29 Contours of shear stress predicted for the rigid squat silo: (a) Lade and Nelson model; (b) Boyce model.
Figure 5-30 Contours of vertical stress predicted for the rigid squat silo: (a) Lade and Nelson model; (b) Boyce model.
Figure 5-3 1 Plastic zones within the bulk solid for the rigid ta11 silo: (a) Lade and Nelson model; (b) Boyce model.
Figure 5-32 Plastic zones within the bulk solid for the rigid squat silo: (a) Lade and Nelson model; (b) Boyce model.
Figure 5-33 Impact of Poisson's ratio on the horizontal stress adjacent to the ngid wn,!!. Linex e!astic mode!.
Figure 5-34 Impact of Poisson's ratio on the shear stress adjacent to the rigid wall. Linear elastic model.
Figure 5-35 Horizontal stress distribution adjacent to the ngid wall in the ta11 silo. Janbu model. With stress correction and without stress correction.
Figure A-1 Stress path from A to B shown in 1, - JJ, diagram
CHAPTER 1
INTRODUCTION
1.1 General
Silos are a class of structures used to store many materials. Larger silos store gain or
cernent, while small silos are used for anything fiom hay silage on the fann to powders in
the processing industry. Most of these materials are bulk solids. Complex relations exist
between bulk solids material properties, the silo filling and discharge processes, the
imperfections of silo walls and other factors leading to behaviour which is different from
many other structural forms. The rate of failures of silos has been reported to be between
100 and 1000 times higher than that of buildings (Weare, 1989), causing heavy economic
Iosses in many countries around the world.
Silo problems are multidisciplinary. One problem of importance to silo design is the
prediction of wall pressures in silos subject to bulk solids loading. .An analytical solution
established by Janssen (1895) a hundred years ago is still being used in many countnes to
calculate the wall pressures in silos f i e r initial filling. This mode1 as well as other
ana1ytical models (e-g. Homes, 1972; Walten, 1973; Abdel-Sayed et al., 1985) are simple
and can be employed for the analysis of simple geometrical shapes where the bulk
matena1 is in the at rest condition. The finite element analysis has been proved to be a
usefùl and powerfûl tool to investigate silo structures with any geornetly under different
working conditions. Many attempts have &eady been made in this area (e-g. Bishara and
Mahmoud, 1976; Mahmoud and Abdel-Sayed, 1981; EibI et al. 1982; Ooi and Rotter,
1990; Ooi and She, 1997)-
The core of numerical methods used to solve problems in solid rnechanics is the choice of
constitutive laws for rnaterials. Many constitutive models have been proposed in the past
forty years. Much of the work on constitutive modelling of particulate solids has been
performed by the geotechnical engineering research cornrnunity. One group of
geotechnicd models is based on an assumption that the total deformation of soil can be
decomposed additively into an eiastic part and a plastic part. This elasto-plastic behaviour
of soils has been developed using nurnerous experimental observations and characterizes
some important features of soil properties. The simplest elastic mode1 for granular
material is a linear elastic mode1 sirnilar to the one developed by Hooke for metal
p!&icIry. H~wewr , it h- hppn f~ in r ! th.at ,ian.u!îr hi-!!k so!i&, !ik sand, exhihit
stifhess that is significantly stress dependent. Nonlinear elastic constitutive models have
therefore been proposed by a nuniber of workers (e-g. Janbu, 1963; Schofield and Wroth,
1968; Duncan and Chang, 1970; Boyce, 1980; Duncan et al., 1980; Lade and Nelson,
1987) to capture the increases in modulus that occur as stress increases.
In the elastic regime, there is no energy dissipation or generation when the sample is
cycled over a closed stress or strain path. Energy conservation is therefore a critical
concem when judging the validity of an elastic model. Sorne popular nonlinear elastic
models like the Janbu model and the poroelastic mode1 violate the principle of
conservation of energy (Zytynski et al-, 1978). For granular material, another important
issue is Poisson's ratio. It has been observed that Poisson's ratio is not constant for
granular matenal like sand during loading, partkularly at low stress levels (e.g. Duncan
and Chang, 1970; Lade and Nelson, 1987). A constant Poisson's ratio is utilized in most
of the elastic rnodels, such as those proposed by Janbu, and Lade and Nelson. If these
models are used, predictions of particulate solid response at Low stress levels and the
solid-structure interaction Like that which occurs in a silo during filling may not be very
satis factory.
Boyce (1980) proposed a nonlinear, energy-consenrative elastic rnodel. f i s model
avoids using Poisson's ratio. In fact, the implicit Poisson's ratio is not constant. However,
this model is defined in a secant form, and f5rther development is needed to re-express
the rnodel in a tangential form, like that needed for implementation in rnost nonlinear
finite element codes. Qtrthermore: the model has rarely been used and there is little
guidance in the literature regarding parameter evaluation. The use of this mode1 to
simulate simple Ioading problems like that seen in the triaxial compression test has yet to
be exarnined, and work is needed to establish the impact of variable Poisson's ratio. The
Boyce rnodel is likely to provide predictions of solid response in silos which are different
f?om those resulting fiom the use of other cornmonly used elastic models like Janbu, and
Lade and Nelson.
1.2 Objective of the Research
The pnmary objectives of this dissertation are:
1) to review the elastic behaviour of soils and the modelling of that behaviour with focus
on granular material;
2) to derive the tangential f o m of the Boyce model and implement it into a f d t e
element code MENA;
3) to obtain parameters for the Boyce model and other models for several typical loose
sands;
4) to examine the performance of these models in the modelling of silo filling.
1.3 Outline of the Dissertation
Chapter 2 reviews various properties of soils, including Poisson's ratio, isotropy of elastic
behaviour, bulk moduius and shear modulus, unloading and reloading behaviour and
energy conservation. Five elastic constitutive models are discussed in detail, namely
..- . i d u , Lade ana Nelson, poroeiasric, Eoyce, anà hear eiasiic modris. f i a p b r 3 prrsenis
the details of a numerical implementation procedure for the Boyce model. This involves
deriving the tangential form of the model, checking the symrnetry of the constitutive
rnatrix, treating some numerical problems when this model is used in finite element
analysis, and performing a senes of computer simulation tests to illustrate the agreement
betsveen results from direct stress integraion using secant form of the Boyce mode1 and
those fiom a finite element code where this mode1 is implemented. In chapter 4,
approaches to obtaining the parameters of the Boyce model are k s t discussed with focus
on the study of parameters suitable for analysis of the silo filling problem. Data seIected
from the literature for three loose sands, Sacramento River Sand, Santa Monica Beach
Sand and Ottawa Sand are reviewed and used to evaluate the parameter values for the
Boyce model, and these are discussed in relation to published parameters. Parameters of
the four other elastic models are also evaiuated. Chapter 5 predicts stress distributions
adjacent to the silo wall in a filling process modelled using finite element analysis. Issues
like silo geometry, wail flexibility, plane strain cases and axially symmetric cases are
considered. Predictions using the Boyce mode1 and the four other models are compared.
Details of the numencal simulation of silo f i l hg and the resulting solid-silo interaction
are discussed. Poisson's ratio may play a key role in the predictions. Finally, chapter 6
contains a sumrnary and conclusions drawn £kom the work presented in this dissertation
together with recomrnendations for bture work.
ELASTIC BEHAVIOUR OF SOILS
2.1 Introduction
In this chapter, various aspects of the elastic behaviour of soils are reviewed. The basic
concept of isotropic elasticity is introduced fïrst. The importance of the elastic regime in
elasto-plastic stress-strain relations is then explained. Next, elastic properties OC soils
such as Poisson's ratio, isotropy during unloading, bulk modulus and shear modulus,
unloading and reloading behaviour and energy conservation are descnbed. Classification
of the elasticity models is provided. Individual nonlinear models such as the poroelastic
model, the Janbu rnodel, the Lade and Nelson rnodel and the Boyce model will be
discussed in detail. Of particular interest are constitutive models for ganular solids, such
as sand. At the end of the chapter, the silo filling problem will be examined and
conclusions drawn about use of the elastic models in that context-
2.2 Basic Concepts of Lsotropic Elasticity
If the deformation of an engineering matenal due to the extemal forces disappears with
removal of the forces, the matenal response is termed elasric. If the elastic properties are
the same in al1 directions in the material, it exhibits isotropy. Homogeneity means the
matter of an elastic body is homogeneous and continuously distributed over its volume so
that the smallest element cut fiom the body possesses the same specific physical
properties as the body (Timoshenko and Goodier, 1970).
Imagine an elastic soi1 elemental rectangular parallelepiped with the sides parallel to the
coordinate axes and subjected to the action of normal stress q uniformly distributed
over two opposite sides. The unit contraction of the element is given by
where E is the rnodrdus of elasticity. This contraction of the element in the x direction is
accompanied by laterd strain components (elongation)
where v is called Poisson S ratio. If E and v are taken as constants for the material, this
material is linear elastic. Otherwise, it is nonlinear elastic.
The deformation charactenstics of a soi1 can not only be expressed through modulus of
elasticity and Poisson's ratio, but also through brtlk modtrlus K and shear modulus G,
which are defined as
K = E
3(l- îv) '
htroduce definitions for mean stress p, deviatoric stress q, volumetric strain E, and
deviatoric strain E, , using
where o; ,o and o3 are principal stresses; q, , - , a,, , q2, a, and are the six
cornponents of the stress tensor; q. 5 and 5 are principal strains; q , , E , and
+, are the six cornponents of the strain tensor, bulk modulus and shear modulus can also
be defined as follows
It is often convenient to use K and G instead of other elastic parameters (e-g. E, v )
because they may be evaluatcd independentiy and they may be directly related to the state
of stress.
In contrast to elasticity, plasticiv can be defined as the ability of a material to change
shape continuously under the idluence of an applied stress, and to retain the new shape
on removal of the stress.
2.3 Elasto-plastic Behaviour of Soils
Traditionally, geotechnical engheers soIve problems of soils and soil-structure
interaction using basic theories of soil mechanics and empincal formulae. For example,
Boussinesq's tquation based on the theory of elasticity is used to obtain stress
distribution and settlements of foundations. The limit equilibrium method is employed to
analyze stability of slopes. Since the 196OYs, with the advent of cornputers and finite
elernent techniques, advanced analyses of geotechnical en=%eering pro b lems have been
available. These analyses include the prediction of defornations and stress distributions
in soils, structures and at their interfaces. An essential step in these predictions involves
using appropriate srress-sirain reiaiions or consiiiuiive modeis For descn'oing soii
behaviour.
Many constitutive modeIs have been proposed in the past forty years. One group of
models is based on an assumption that the total deformation of soil can be decomposed
additively into an elastic part and a plastic part. This elasto-plastic behaviour of soils has
been developed using nurnerous expenmental observations and characterizes some
important features of soil response.
In general, plastic deformation dominates the total deformation of soils under loading,
which means that the elastic recoverable strains are relatively small in the presence of
plastic strains. However, the elastic regime still plays an important role in the prediction
of displacements. Firstly, alrnost al1 soil deposits are overconsolidated to a certain degree
in nature. The behaviour of overconsolidated soils is largely elastic, so when load is
exerted, the initial behaviour of soils is elastic. Secondly, when unloading occurs due to
operations such as excavation, the behaviour of soils is largely elastic, and is generally
modelled as purely elastic. Moreover, in the theoretical derivation of the final elasto-
plastic constitutive relation, the elastic properties of the soi1 must be defined in order to
invert the stress-strain relations-
Hornogeneous isotropic linear elasticity has often been used in the past to predict the
working load response of a variety of three-dimensional problems encountered in
geotechnical engineering. Under certain conditions, such as at stress levels significantly
~P!QW &!IL-, S p e S r rl;gtT;_hcecfi i_m_me&tc sen!pm-nt he c&~!-ted hy thic
elastic procedure @avis and Poulos, 1968; Gibson, 1974; Selvadurai, 1979; Poulos and
Davis, 198 1). However, many expenmental studies indicate that soil density or void ratio
and the state of stress and stress history acting on the soil al1 affect the elastic properties.
Nonlinear elastic models have therefore been developed to characterize soil behaviour
(Janbu, 1963; Duncan and Chang, 1970; Wroth, 1972; Vermeer, 1978).
Given the complexities of soil behaviour, no mode1 can accurately predict the behaviour
of every kind of soil. Simplified idealized models have been developed to solve problems
with certain ranges of conditions. Chen and Mizuno (1990) proposed three conditions for
the evaluation of those models. The theoretical evaluation aims to ensure the models are
consistent with the theoretical requirernent of the basic principles of continuum
mechanics. The objective of the experimental evaluation is to judge the performance of
the mode1 to reproduce experimental data fiom a variety of available tests, and to
examine the ease of determining material parameters fiom standard tests. The numerical
and computational evaluation considers the feasibility of deriving constitutive relations
and implementing these relations into computer calculations, particular into h i t e -
element computer codes.
2.4 Poisson's Ratio
Poisson's ratio v is very important in linear stress-strain relations, and it is an explicit
component in some non-linear elastic models. Generally, it is taken as a constant for a
soil with a certain range of density such as loose sand and dense sand, although most
. . cûn;:iZi:;vc modc! d c ~ d o p c ï s ûû:c k i t ,Pois;onl; rutk i; nû: û cons:ûïi: z: ûu!!. IIGK~Yc~,
the variations in Poisson's ratio are generally considered to be of little or modest
significance for a given type of soil, and its variation is assumed to have little effect on
predictions for most practical problems (Lade and Nelson, 1987).
Expenmental results show that Poisson's ratio varies with shear strain (Yokota and
Konno, 1980). There is a tendency for Poisson's ratio to increase slightly with increasing
magnitudes of shear strains. This is due to the dilation of sand as it is subjected to shear
stress. The influence of confining pressure in triaxial tests was found to be negligible
(Yokota and Komo, 1980; El Hosri, 1984).
Analysis of the tests performed on Ham River Sand (Dararnola, 1980) shows v is 0.20 at
void ratio e=0.57 and 0.3 1 at e=0.75. This means void ratio o r density of the sand has an
effect on v; v increases with increasing e.
From equations (2-3) and (2-4), - I < v 5 0.5 is required for positive b u k rnodulus and
shear modulus.
Duncan and Chang (1970) found that v varies f?om 0.26 to 0.41 fkom the results of tests
on loose sand. Values of v greater than 0.5 have been observed under repeated loading
(Allen and Thompson, 1974). Figure 2-2 shows Poisson's ratio versus stress from the
triaxial compression tests on Loose Santa Monica Beach Sand performed by Lade and
Nelson (1987). The values of Poisson's ratio change from 0.2 to 0.32 in the low stress
range.
Therefore, a single value of Poisson's ratio c m o t characterize the behaviour of the sand
accurately. The importance of variation of Poisson's ratio in the silo problem will be
discussed later in section 2.14.
2.5 Isotropy of Elastic Behaviour
The elastic behaviour of homogeneous soils during unloading is isotropic, even when
strains during loading may display anisotropic behaviour. Rowe (1971) showed results of
expenments on Mersey River Sand, Silver Sand and Fine Quartz Sand. AU of them
exhibit isotropic behaviour during unloading despite the initially anisotropic fabric which
exists in those sands. Rowe also concluded that Poisson's ratio can be considered to be
isotropie durîng unloading.
Sirnilar conclusions can be drawn for clay (Knzek, 1977, Wong and Mitchell, 1975).
2.6 Bulk Modulus and Shear ModuIus
Experimental evidence shows that bulk modulus K and shear modulus G for a soi1 depend
on both the state of stress and the stress history. Both the mean stress and the deviatoric
stress may influence these elastic moduli. There is at present no evidence of any
influence of the third stress invariant (Loret, 1985; Lade and Nelson, 1987).
Loret (1983) anaIyzed the experimental data presented by Bishop et al. (1965) and
Tatsuoka (1973). The results show that K can be expressed as a h c t i o n of the mean
stress p,
where K, and n are matenai constants, and p, is atrnospheric pressure expressed in the
same units as p.
Pappin and Brown (1980) found that K increases at higher q l p values. A sirnilar
phenornenon can be found in triaxial compression tests with constant effective confining
pressure (Lade and Nelson, 1987). In these tests, the bulk modulus was seen to Vary with
the deviatoric stress.
Hardin and Black (1969) suggested an empincal expression for the shear modulus for
clays and sands deterrnined from wave propagation velocities and fkom small amplitude
cyclic simple shear tests
G = 3230-OCR~
where o* is the mean stress in k~lrn ' ; OCR is the overconsolidation ratio; e is the void
ratio and the exponent k is a material constant. Seed et al. (1984) expressed a general
form for G
angularity of grains and other factors; p is a material constant that is usually taken to be
Brown and Hyde (1975) also reported that granular matenal exhibits significant changes
in volume when subjected to shear stress, and found evidence indicating that G increases
with the normal stress. Rowe (1971) and Vermeer (1978) proposed another expression
for a non-constant shear modulus based on their own test data
where Go and ,û are expenmental constants. Rowe (1971) measured values for ,û
1 behveen 7 and f .
From a theoretical point of view, the elastic behaviour of a mass ofgranular material can
be considered as resulting fkom particle deformation. Modelling the soi1 as a regular array
of spheres in contact, classical elastic theory (Timshenko and Goodier, 1970) can be used
to quanti@ the relative displacement Y between two adjacent spherical elastic particles in
contact in terrns of the normal force P as
Y= I;P"', (2- 15)
where I; is a matenai constant. Thus, the stifiess of the contact is given by
The power of the force P can have other values depending on the geometry of the contact
hehveen each pair o f adjacent particles. for example, a point a line or an area; clearly this
geometry will be cornplex in real ganular materials.
If the shear modulus in an energy conservative mode1 takes the sarne form as equation (2-
14), the bulk modulus shouId be coupled, Le., it should be related to both the mean stress
and the deviatoric stress. This tvill be discussed in detail Iater in section 2.13.
2.7 Unloading and Reloading
It is generally agreed that elastic behaviour cm be observed in the unloading and
reloading paths of a loading process on soils. It also occurs in initial loading but may be
obscured by the presence of plastic strains. Parameters for most elasticity models can be
ob tained fkom fitting to the unloading-reloading curves.
~Many researchers have pointed out that due to the complications of soi1 behaviour,
unloading and reloading are not purely elastic (Arnold and Mitchell, 1973; Loret, 1985;
Lade and Nelson, 1987; Dakoulas and Sun, 1992). Pure elasticity only exisr in an
infitesimal unloading, irnplying parameters should be detemiuied using measurement of
the slope of the stress-stmin curve just at load reversal (Loret, 1985).
Lade and Nelson (1987) argued that large amounts of plastic strains occur during
unloading since the predicted dope of the volume change cuve is better defined for
reloading than for unloading in their model. However, Dakoulas and Sun (1 992) found
that there is a good agreement between predictions for unloading using the same model in
reloading curves deviate from the unloading curves after reapplication of about 50% of
the previous maximum stress. They attributed this phenornenon to plastic deformation
taking place during reloading.
Other researchers hold the same view as Dakoulas and Sun (1992), that unloading is
more purely elastic than reloading. El Sohby (1973) performed triaxial compression tests
with constant principal stress ratios and concluded that the strains during unloading are
generally independent of the number of unloading cycles which means the deformation
during unloading is elastic, and can be taken to represent the elastic component during
reloading.
Some researchers have concluded that plastic deformation occurs during reloading
depending on the stress paths. in their triaxial compression tests with constant mean
stresses conducted on Ottawa Sand, Dakoulas and Sun (1992) observed that the plastic
deformation was not as significant as that in the isotropic compression test during the
reloading phase. In those triaxirl compression tests, only the shear component is
unioaded and the hydrostatic component remains unchanged, while a11 stresses becorne
small during the isotropic unloading. It is not surprising that greater arnounts of plastic
deformation occur during reloading for load paths involving changes in both stress
components.
If ~zb"zç--!c&m n r r i i r c frnm 2 s@ess !cye!, t h FtrrE-g~a~~ c ~ ~ - v e ciw~~ii~p a a ------ ----
hysteresis loop. Bell (198 1) suggested that this resulted fiom crushing and repositioning
of grains. Energy will dissipate during such an unloading-reloading cycle. Holubec
(1968) snidied the elastic properties of Ottawa Sand in triaxial compression and found
that large hysteresis loops usually occur when samples are unioaded to zero shearing
stress corn stress conditions that are larger than approximately 80% of the maximum
stress he used. If a soi1 element is completely unloaded f?om this level, large hysteresis
loops are produced. Holubec suggested that only cycles f?om stress levels lower than
approximately 80% of the maximum stress should be used for the purpose of studying the
elastic deformation.
In practice, different models use different portions of the unloading-reloading branches.
The poroelastic mode1 assumes that the unloading and reloading portions are identical.
The rnodiiïed Janbu model (Duncan and Chang 1970) adopts a similar approach. For the
Lade and Nelson mode1 and the Boyce model, the unioading portion is considered during
evaluation of the parameters.
2.8 Energy Conservation
Due to deformation, energy is stored or released in the material body. A relationship
between the stress and the stored energy exists, and this relation is of fundamental
importance in developing constitutive laws for the material. In the elastic regime, there is
no energy dissipation or jeneration for a closed stress or strain path Like that shown in
Figure 2-1. Energy conservation is therefore a critical concem when judging the validity
o r an eiasiiç modai. iiC iiieiè & S ~ S a j i ü r N èi1~fgy ùî p~ic i ïr id iUii~iiùii kiiii 'i*'hi~:i i h ~
strain could be obtained, and the important features of the material are well captured, it is
good to use this elasticity mode1 with an appropriate plasticity rnodel to predict
deformation and other behaviour of soils in geotechnical applications.
For linear elasticity, the stored energy fùnction is a cubic polynomial. Borja and
Tamagnini (1948) proposed a class of stored energy fûnctions that can be employed to
derive the elastic constitutive law for their modified Cam-Clay model. But for most
nonlinear elasticity models, it is not easy to obtain an explicit expression of such a
Function. An alternative approach is to examine energy conservation through a closed-
loop stress path.
Boyce (1 980) obtained an equation that cari accomplish this task. The strain energy W in
an incrernental form can be expressed as
dW = pdq + q d ~ , .
For a closed-loop stress path, the following condition should be satisfied
Using Stokes' formula gives
This relation is based on the assumption that the formulae of E, and E, are independent
of the third stress invariant.
Lade and Nelson (1987) expressed the strain energy in another form
where Il is the first stress invariant and J, is the second deviatoric stress invariant
defined as
I I = O. + G" + o, (2-2 1)
where O-- , c,, and c.. are normal stresses, and
where r,, r, and r, are shear stresses, s,, s,, s,, s,, s,, s,are components of
deviatoric stress tensor.
They showed that if an elastic mode1 is energy conservative, the expressions for the bulk
rnodulus K and shear modulus G should satisQ the following equation
It may be expected that equations (2-19) and (2-23) are the same since they both are
based on energy conservation. This is now demonstrated. The mean stress p and
deviatoric stress q c m be expressed as
q = , /34 -
Substituting equations (2-24) and (2-25) into (2-9) and (2-IO),
There fore,
The stress invariants 1, and J, are independent, which mems
Thus, equations (2-28) and (2-29) can be written as
To satisfji equation (2-19), the nght sides of (2-3 1) and (2-32) should be equal.
Rearrangernent then leads to equation (2-23).
Zytynski et al. (1978) used a simple process to prove that the models of Janbu (1963),
Roscoe and Burland (1968), Duncan and Chmg (1970) and Simpson (1973) violate the
principle of conservation of energy. In his approach, these rnodels can be characterized as
d ~ , = K(p)dp y (2-3 3)
4 = cK(p)dq, (2-34)
of p; c is a constant related to Poisson's ratio. For the stress path
in q-p space s h o w in Figure 2-1, the path ACB can be expressed as
4 = L(P) 9
where Kfp) i
and the path BDA
cti
For the path ACB,
and for the path BDA,
(E,)BD..~= - [cK(P)~;_(P)~P - (2-40)
For the whole cycle, A E, = (E,),, + (E,), = 0, but A E,. = (q-),, + (gr), * 0 -
Permanent shear deformation appears for such a closed-loop stress path. Consequently
energy must be dissipated.
2.9 Classification of Elasticity Models
Elastic models c m be generally classified as either linear or non-linear. Besides simple
linear elasticity, a piecewise linear relationship can be used (e.g. Chen and Mizuno,
1990), which rneans for each Iinear interval, different material constants are employed-
Gibson (1967) found that Young's modulus varies linearly with depth in many natural
soi1 deposits, and a constant modulus gradient was therefore suggested.
By studying the reversibility of the strain and of the energy input for a general closed
path, nonlinear elastic material models can be classified as Cauchy elastic, hyperelastic
and hypoelastic models.
In Cauchy elastic models, the current state of stress is uniquely determined from the
current state of strain (or vice versa), Le., E~ = %(O). The strain recoverability is
guaranteed, but change of energy may occur for certain loading-unloading cycles. This
type of mode1 includes those proposed by Girijavallabhan and Reese (1968), Hardin and
Drnevich (1 972) and Nova and Wood (1979).
Hyperelastic models overcome the shortcoming of the Cauchy elastic models. A stored
energy or potential function is defined so that the stress tensor qJ can be expressed as
a' a;, = - . Strain is recoverable and enerm is conserved in
&]
Loret (1985), a necessary and sufficient condition for
hyperelastic is
these models. According to
an elastic material to be
where I, is the third stress invariant. Models proposed by Evans and Pister (1966), Ko
and Masson (1976), Vermeer (1978), Boyce (1980), Loret and Luong (1982), Morz and
Noms (1982), Houlsby (19S5), Lade and Nelson (1987), Molenkarnp (1988) and Bo j a et
al. (1 997) are hyperelastic.
The hypoelastic models represent the incrernental stress tensor as a fûnction of the
current stress or strain and strain rate, i.e., c$ = F, (O,, 6, ) or 6 = f;;, (.qm, ) . S train is
generally not recoverable. Energy may also be generated or dissipated over a closed
stress path. Among these models, there is a class composed of modified linear elastic
models. The elastic constants are replaced by variable tangentid moduli which are taken
to be functions of the stress and strain invariants. Among these modek, the Duncan and
Chang (1970) model, which was developed fiom the Janbu modeL (1963), has proven
very popular since it is simple and easy to fit to the experimental data. However, as
mentioned in section 2.8, this model does not conserve energy through a stress cycle.
Therefore, it may lead to errors in some applications.
2.10 Poroelastic Model
This model is generally used to provide the elastic part of predictions obtained using the
Cam-Clay model (Roscoe and Burland, 1968; Atkinson 1981). The elastic moduli K and
G are dependent on effective confining stress as well as void ratio. They c m be written
3K(1- 3 ~ ) G = /3 A T \ tL'->l
2(1+ v) ,
where e is void ratio, iris the dope of a swelling line in e - l n p figure for an isotropie
compression test and v is constant Poisson's ratio. From equations (2-42) and (2-3),
Young's modulus c m be expressed as
Although this mode1 is widely used, it is not an energy conservative model (Zytynski et
a[. 1978). Bo j a and Tamagnini (1998) proposed a coupled hyperelastic model for their
modified Cam-Clay rnodel based on the same idea.
2.11 Janbu Model
After studying a number of test results, Janbu (1963) proposed the following expression
of Young's modulus
where gi is minimum principal effective stress; Ki is the modulus number and n is the
rnodulus exponent. The parameter Ki is measured fiom the initial slope of the stress-
strain curve for triaxial compression. But this slope is often influenced by plastic
deformation. Duncan and Chang (1970) suggested use of the slope of an unloadinj
reloading cycle K, fiom a triaxial compression test
The value of K, is normally one to three times higher than Ki. Instead of using
minimum principal stress & some researchers use mean stress p in equation (2-46)
(Holst et al, 1997).
Poisson's ratio is constant in this model. The model is also not energy-conservative.
Duncan et al. (1980) improved this model by using values of b u k modulus which Vary
with confining stress rather than constant values of Poisson's ratio. The Young's modulus
relationships remain the same as equation (2-46). The bulk modulus is expressed by
where K, is the b u k modulus number and rn is the bulk modulus exponent. The
relationship beiween stress and strain is assumed to be governed by the generalized
Hooke's Law of elastic deformations and is expressed in tenns of Yoirng's modulus and
b u k modulus.
One of the advantages of this improved mode1 is it avoids using constant values of
Poisson's ratio. However, it is still not energy-conservative.
2.12 Lade and Nelson Mode1
Lade and Nelson (1 987) developed an isotropie model for nonlinear elastic behaviour of
granular rnaterials. The model uses constant Poisson's ratio and variable Young's
rnodulus expressed as follows
where M, A are material constants. The parameter R is related to Poisson's ratio v,
Equation (2-48) is actually similar to the equation proposed by Loret (1985)
where K, is the secant bulk modulus; KI and G: are initial bulk modulus and shear
modulus and n is a material constant. If K, , KI and G: are replaced by equations (2-3)
and (2-4), and q is replaced by J,, equation (2-48) and (2-50) are almost the same.
The Lade and Nelson model is energy conservative. Yu and Dakoulas (1992) generalized
this model for cross-anisotropic soils. The three Poisson's ratios needed in the
generalized model are still assumed constant.
2.13 Boyce Mode1
Boyce (1980) developed a non-luiear elastic rnodel for granular matenais. The bulk
modulus K and shear rnodulus G are expressed as
7
4- K = QI-" I (1- PF) , (2-5 1 ) P-
G = ~ , p ' - " , (2-52)
where p i s defined as
,O= (1- n)K, /6G,, (2-53)
K/, G/ and n are material constants; p and q are mean stress and deviatoric stress.
This model is an isotropie time-independent model. Though it was originally developed
for predicting the behaviour of granular material under repeated loading, it can also deal
with other types of loading.
In the model, the buk moduIus and shear modulus are proportional to the normal stress
raised to a power less than one. This is consistent with the experimental observations as
well as the theoretical consideration described in section 2.6. The bulk modulus is
expressed as a h c t i o n of deviatonc stress. This coupling feature not only makes the
model energy conservative, but also reflects the influence of shear stresses on volume
changes that has been observed in experiments.
The energy conservation of this model can be proved using the approach developed by
Lade and Nelson (1987), and this is shown in Appendix A. The model also satisfies
equation (2- 19).
As discussed in section 2.4, a single value of Poisson's ratio is not able to characterize the
nonlinear elastic behaviour of the ganular material accurately. The Boyce model avoids
using Poisson's ratio explicitly, which is obviously an advantage.
Boyce (1980) suggested the relationships for bulk modulus K and shear modulus G as
funchons of mean stress p and shear stress q; however, no fürther development has been
---.-A niit ;ri A a mirhl;cl-iaA 1;tat-hiva Th- ; r r r ~ ~ r n ~ n t = l c r r r i c t i t i i t ; ~ ~ ~elat innc have n _ ~ t C I C U A A ~ U vuc A A A UA% ~ L C V L A J A L ~ C I ALCIACLCUII- A ..Y UIYL -ICI---- - v - - - - - - --- -
been established for inclusion in finite element codes capable of solving geotechnical
engineering problems. ïherefore, the potential of this rnodel is unknown.
In this study, a tangential (incremental) constitutive relation has been derived based on
equations (2-5 1) and (2-52). Subsequently, the new constitutive relations have been
implemented into a finite element code. Details of the mathematical development and
irnplementation will be presented in Chapter 3. Chapter 4 will provide the evaluation of
the parameters of this model for specific applications from conventional tests. It will be
shown that Poisson's ratio evaluated using the Boyce rnodel displays characteristics like
those observed in tests on loose sand (Chapter 4).
Chen and Mizuno (1990) classified the Boyce formulation as a Cauchy elastic model.
Actually, it is hyperelastic since it satisfies equation (2-41).
2.14 Elastic Models for Bulk Solid Stored in Silos
The physical behaviour of the bulk solid stored in a silo is the most important issue that
decides the pressure distribution at the walls of a silo during filling, in the at rest
condition and during discharge. No constitutive models have been developed particularly
for silo applications. However, appropriate models for materials stored in silos should be
chosen with respect to their ability to describe silo-relevant stress-strain histories. For
models which cm well describe these histories it is assurned that they also yield
reasonable results in an application of finite element analysis to silos.
The stress-strain history for bulk solid placed into silos with stiff walls is similar to
conventional one dimensional soil consolidation where horizontal strains are close to
zero. Even for silos with flexible walls, the deflection of the wall is very small compared
to the silo geometry. Therefore, neglecting the shear stress developed in the material due
to the wall constraint, the stress and strain state in the matenal c m be considered as one
dimensional strain, which implies that Poisson's ratio is likely to have a key impact on
the horizontal stress, suice in such situations, a ratio between horizontal stress and
vertical stress in a soil element may be expressed as
The wall pressure stems fiorri the horizontal stress in the material adjacent to the wall and
consequently it is affected by the values of Poisson's ratio the matenal will have during
filling.
Normally, the height of a silo is between several meters to several tens of meters.
particle of b u k solid is placed into a silo, it is subjected to the confining pressure around
it and the gravity load on it Eom other particles. Stresses are zero at the initial stage to a
value less than one or two hundred kPa, depending on its eventual depth. For such levels
of stress, Poisson's ratio of loose granular material varies significantly during loading, as
shown in Figure 2-2. From equation (2-54), if Poisson ratio v changes from 0.15 to 0.3, K
will change Eom 0.18 to 0.43 correspondingly. Such variation is expected to have an
Most elastic models including poroelastic, Janbu, Lade and Nelson, and linear elastic
model employ a constant Poisson's ratio. The predictions using these models are
therefore expected to be different fiorn those using the Boyce model, which may feature
the variable Poisson's ratio at low stress level. Chapter 5 will present cornparisons
between predictions in the silo filling problem using finite element technique with
different elastic models.
More challenging work in silo research features predictions of wall pressure during
discharge, which is basically an unloading process. Unloading is largely an elastic
behaviour, as described in section 2.7. During discharge, the stress condition in the upper
part of the silo is similar to the condition for filling, while passing the transition zone, the
verticaI stresses are decreasing and the horizontal stress are increasing. Dynarnic
phenomena are observed, which may originate from energy release in the material. An
appropriate elastic model for the stored matenal is obviouçly vital in this case. Besides
the issue of variable Poisson's ratio, energy conservation of elastic mode1 is also very
important. Energy conservative models such as the Boyce model are likely to provide
more accurate predictions than those models which do not conserve energy.
Figure 2-1. Closed stress cycle in effective stress space
Figure 2-2. Poisson's ratios determined directly fiom results of triaxial tests on loose Santa Monica Beach Sand (after Lade and Nelson, 1987).
CHAPTER 3
IMPLEMENTATION OF THE BOYCE MODEL
3.1 Introduction
This chapter presents the details of a numerical irnplementation procedure for the Boyce
model. Firstly, a constitutive relation with tangential form is derived from equations (2-
51) and (2-52) through octahedral-based stress-strain relations (Murray, 1979). E s non-
linear incremental stress-stlain relation is implemented into a h i t e element computer
code for solution of geotechnical problems. Then, symmetry of the constitutive ma& is
discussed. Next, the numerical treatment of certain issues is addressed. Finally, a series of
CP1 cornputer simuiation tests are periormed. tne çompuisoris oi Giz ï é ~ u i i s < j b i ~ ï i ~ d bÿ
using the fuite element code involving this tangential form and the exact solutions using
secant expressions are carried out. The results are in agreement.
3.2 Derivation of the Tangential Form of the Stress-Strain Relation for
the Boyce Mode1
uitroducing the atmospheric pressure p , the bulk modulus K and shear modulus G in the
norrnalized formulation of the Boyce mode1 are as follows
Mean stress p and deviatonc stress q can be wntten as
where q , q and 0, are p ~ c i p a l stresses; O,, , - , a, , a12 , a, and cj , are the six
components of the stress tensor.
To extend its practical use in geotechnical engineering, the mode1 should be implemented
into a fulite element program. A tangential form of the constitutive relation must
therefore be derived instead of the secant form defined above. That is described in the
remainder of this section.
The volurnebic strain E, and deviatoric strain gX are d e h e d as
where E,, 5 and q are principal strains; q , and 9, are the six
components of the strain tensor. On the other hand, the elastic volumeaic and deviatoric
strains are also related to mean and deviatoric stresses
The denvations in sections 3 -2 and 3 -3 concem elastic strains. Elasto-plastic constitutive
response is discussed Iater in section 3.4, where the strains associated with elastic and
plastic response are separated.
The incrernents of strain invariants c m be expressed relative to increments of stress
invariants using equations (3-1) (3-2) (3-7) (3-8) (Boyce, 1980)
1
where
The octahedral shear stain y,, and octahedral normal strain a, are given in the form
From equations (3-6) and (3-14), the following are obtained
Solving simultaneous equations
L* = fi&x, 1
E, = EL.
(3-9) and (3- 1 O),
p=aE,+bÉ,,
q = c&+dé-.,
where
The stress increment tensor d q cm be decomposed into the deviatonc and hydrostatic
parts ds, and dcr,&,
where d;, is the Kronecher delta, widi value 1 if i = j and O if i * j . There is a relation
behveen deviatoric stress and deviatoric stain
s, = 2G.e,,
w here
Differentiating equation (3-2 1 ),
w here
From equations (3- 16) and (3- 17),
4 Differentiating the relation y:cL = -e,e,, the following are obtained
3
4 en dy, = y-- . de,. ' Yocr
Substituting (3-26) into (3-25),
b 4 e, d a , = a& + ------dem. fi 3
Note
.kÙ = -
Rewrîting (3 -22), differentiating it and substituting (3 -28),
Substituting (3-28) and (3-29) into (3-27),
3b 1 1 = a4dEu + --- (eu - 7 , 6 ~ )&M.
3& y,, 3
Note that es, = el, t e12 + e,, = 0,
4b da, = a & d ~ ~ + eu&
3 A y ,
Substituting (3-33) and (3-30) into (3-30),
dq, = 2(m- dowt . e, i Ge de,) + do,, 6, = ( h e , + %)dom. + 2Gde,
Substituting (3-29) into (3-3 1 ),
4b I da,= [(2rnev + b; . ) (~6~ + e,) c 2G(6,6,, - - d & ) J d ~ ~ . (3 -3 2)
3firoct 3
Equation (3 -32 ) provides the required incremental stress-strain relation. It can be written
in matrix fonn as
where
7
---
---
- ci- s Cs N +
rC1 @z @.Y-
2 m
1 - ' di- - 1 - f CU s 1 ;s = t - t -
3.3 Symmetry of the Constitutive Matrix
The constitutive matrix C given in (3-36) looks non-symmetric. For exarnple,
2 C ( 1 3 = 2mBe, ,e, - + Zame,, i Bex + a - TG,
3
2 C(2J = ZrnBe, ,eE + 2ame2 + Be,, + a - TG,
3
C(1,4) = ZmBe, ,el, + Bel2, C(4,l) = 7mBe,,e,, + 7amel2.
where
and so C(1,2) is not equal to C(2,l) and C(1,4) is not equal to C(4,l) unless Zam=B.
Parameters a and h arise fiom the solution of simultaneous equations (3-1 7) and (3- 1 8).
m is the denvative of G with respect to p (Equation (3-24)), and B is Linked to octahedral
strain y,, (Equation (3-37)). It is not easy to see directly that Zam is equal to B.
However, the constitutive matrix (3-36) is indeed symmetric. This is proved in Appendix
When implementing the constitutive matrix (3-36) into cornputer codes, care should be
taken even it is theoretically symmetric. Following substitution of equations (3-l), (3-1 l),
(3-12) and (3-13), equation (B-8) (see Appendix B) can be written as
Due to the numerical treatrnent of initial loading discussed later in section 3.5, there
would be discrepancy between 2am and B at the beaoinning of the load path. This is
because only rninirnump is given when 2am is computed according to the left-hand side
of equation (3-38), while both minimum p and y, are used to calculate B, which is the
right-hand side of equation (3-38).
This problem can be satisfactorily resolved by using (3-38) to derive an alternative
expression
which is used instead of equation (3-14) to compute y,, . This means that the symmetry
of the constitutive matrix is maintained numerically.
3.4 Separating Elastic Strain and Plastic Strain
Elastic-perfectly-plastic models decompose the total incremental strain into an elashc
component and a plastic component, i.e., É= ke + E, . The Boyce mode1 is a mess and
strain dependent model, as shown in equation (3-32). When the matenal yields, the
elastic constitutive matrix should be calculated using the elastic values of y, and eu
instead of using total strain. Failure to exclude plastic strains fiom the elastic constitutive
matrix results in a solution that is divergent. Therefore, the cumulative elastic strain E,
must be stored in the finite element irnplementation and updated after each increment.
The flow chart shown in Fipure 3-1 presents the procedure for calculating the increments
of strain and stress at each integration point and then updating the current States of stress
and elastic strain.
Note that emgineering shear strain and tensorial shear strain are different. h finite element
prograrns, engineering shear strain is comrnonly used, however, when using the Boyce
model to calculate constitutive relation, tensorial shear strain is employed.
3.5 Treatment of Some Numerical Problems
3.5.1 Introduction
When implementinj the model into the finite element computer program, certain
relation, difficulties are encountered when p=O andor y,=O. Another important issue is
the choice of the size of strain and stress increment needed to obtain an adequate solution.
In this section, the numencal treatment of these problems will be descnbed. The same set
of pararneters,
which is the set of the Boyce pararneters for Loose Ottawa Sand discussed in Chapter 4,
is adopted in the following investigation (sections 3.5.2, 3.5.3 and 3.5.4). In section 3.5.5
trials for other sets of parameters will then be briefly discussed.
3.5.2 = O o r y, = O ( ~ 0 )
Equation (3-14) reveals that the condition ywt = O is satisfied when q2 = E.,~ = 5, = O
and ql = E.,~ = q3. However, since equation (3-39) is used to compute y,, , the problem
shifis to circumstances when p=O andor q=O. Notice in the constitutive relation (3-32),
terms containing 7, in the denominators also contain parameter b in the numerators.
Parameter b is a function ofp and q according to equations (3-19), (3-1 1), (3-12) and (3-
13). Deviatoric stress q is finally divided out in these terms. Thus a very srnall q value is
substituted for zero, Say IO-", to prevent overfiow.
3.5.3 p=O
Zero mean stress p=O occurs, for example, at the beginning of any load path where the
initial stresses in the element are zero. Because the parameters a and b in equation (3-32)
feature p in the denorninator, this problem must be fixed nurnerically. The approach is to
use a minimum p (p, , ) instead of zero. h y p values less than the minimum value are
set to p,, .
Different p,,'s have been tried in a cornputer simulation for the problem of hydrostatic
compression with parameters in (3-40). Either incremental stresses or incremental s k n s
are specified in the simulation test. The first step in the evaluation of (3-36) involves
calculations of incremental strains from incremental stresses (or vice versa) using the
curent constitutive matrix. The total strains and total stresses are then updated for use
when obtaining the constitutive relation for the next load step. Using equations (3-3) and
(3-4), mean stress and deviatoric stress can be computed, which are used to obtain the
bulk modulus and shear modulus according to equations (3-1) and (3-2). The volumetric
strain and deviatoric strain can be calculated either 6-om equations (3-5) and (3-6), which
is called the incremental solution, or fkom equations (3-7) and (3-8), which is called the
secant solution- The latter should obviously provide an exact answer for the problem.
Figures 3-2 and 3-3 demonstrate that for a given incremental strain, it is not always the
mallest p,, that produces the most accurate solution. On the contrary, with a
comparatively large incremental strain, the smaller pmn produces a worse solution, as
shown in Figure 3-2. From the test, a p,, value between 0.01-0.001 was selected for
cases where incremental strain less than 5 p ~ is used. Though these values are for the
case of isotropic compression, they c m be reasonably used in other cases when p=O is
encountered. Cases shown in section 3.6 will exhibit that good solutions can be obtained
when these numerical treatments are employed.
3.5.4 Choosing Adequate Incremental Stress and Strain
As mentioned above, choosing a small incremental strain is always a good idea. This can
aIso be seen fkom Figure 3-4, where a cornparison has been made behveen the secant
solution and the incremental ones using different incremental sirains but with the same
p,,, , 0.005Wa Conclusions can be drawn that the magnitude of AE should be less than
5 p . To limit computational effort, AE is generally selected to be between O-Sp~and 5 p ~
Incremental stresses O.lkPa and OSkPa with the same p,, of O.OO5kPa are used in a
cornputer test that sirnulates isotmpic compression. Figure 3-5 illustrates that the solution
with A a=O.lkPa is very close to the secant one. Therefore, incremental stress less than
O. I kPa should be chosen in calculations for these Boyce parameters.
3.5.5 Other Sets of the Boyce Parameters
Another set of the Boyce parameters, n = 0.4, K, = 400(kP,)*, and G, = 185(kPa)", is
examined. Fipure 3-6 shows that for the same relatively large incremental strain of
A r = 3 x 10-j, use of a lxger minimum rnean stress p,, , O.5kPa, is better than use of a
smaller one, O. 1 kPa. For this set of parameters, if incremental strain O.lp&--Ip& is used,
p,, = O.OO5k. is again a good choice, as illustrated in Figures 3-7 and 3-8. Figure 3-9
shows that incremental stress should be taken as around 0-OlkPa, while pmn = O.OO5kPa
is used.
Other trials also demonstrate that different sets of the Boyce parameters need different
treatment, Le. different minimum values of some variables, to obtain good solutions.
3.6 Test Problems Using the Boyce Mode1
3.6.1 Introduction of AFENA
The Boyce mode1 has been implemented into a research program AFENA. The program
AFENA is an algorithm for the solution of boundary and initial value problems using
h i t e element techniques. It was originally coded by Carter and Balaam in 1980s. Since
then, many functions and models have been implemented so that it may be used to obtain
approximate solutions to various problems in soiid mechanics, particularly problems in
earth structures, soil-structure interaction prob lems, as well as other more general field
problems.
3.6.2 Test Model
To check the accuracy of the implementation, a series of' computer simulation tests have
been perfonned. Three types of analyses were used.
1. Exact solutions. These solutions are acquired using the constitutive relations defined
by equations (3- 1) and (3-2).
2- Direct integration of tangential constitutive matrix (3-36). In this approach,
. - iï,crcmcfi~o~ su.& a-e S F > S C i ~ S ~ fiEi. TLle cczjiikü~e iiidiiix (3-36) is
calculated using the curent total strains and stresses. Incremental stresses can be
computed thereafter according to equation (3-33). The total strains and total stresses
are then updated for use when obtaining the constitutive relation for the next step.
Alternatively, incremental stresses can be specified first. Following the same
procedure to calculate the constitutive matrix (3-36) as above and then invert it, the
incremental strains can be obtained. With the appropriate treatment of numericai
issues descried in section 3.5 (such as use of p,, and incremental load steps), these
solutions are in good agreement with the exact solutions.
3. Tangential Finite elernent analysis. This group of solutions is from AFENA involving
the implemented Boyce model. A unit block with two six-node elements is used.
Different boundaries and forces are added to simulate different problems. An example
is s h o w in Figure 3-10. The same load steps as for the direct stress integration
approach are applied. M e r carrying out the calculation following finite element
procedure, the stress and strain states in al1 Gauss points are obtained. In this unit
block, the stress and strain states for each Gauss point are the sarne. Therefore, only
one particular point is selected and its stresses and strains are used for cornparison.
If the results of these three types of analyses for the same problem are the sarne, it means
the h i t e element code involving the implemented Boyce mode1 may also provide the
exact problem solutions.
Parameter values in (3-40) are employed in all tests. In finite element analysis, the
cohesion of the material is set to a very hi& value so that there is no plasticity involved
in the tests.
3.6.3 Isotropie Compression and Triaxial Compression Cases
The fint test problem features isotropie compression followed by aiaxial compression. It
is an axially symmetnc problem. Figure 3-10 shows the two predictions of vertical stress
O, versus vertical strain E, . One solution is fiom MENA, represented with a solid fine,
and the other is firom direct stress integation shoum with triangles. In MENA, the
problem is realized by putting srnooth rigid boundaries on the two adjacent sides of the
block. The other two sides are firstly subjected to equal pressures that are gradually
increased from zero to 12OkPa. Only one of the sides is then pressured to 240kPa-
Vertical stress q and strain r, are shown fiom one particular Gauss point in each
incremental step. In the direct stress integration, the same block can be represented as a
point, three equal normal stresses are applied to this point fiom zero to l2OkPa and then
only one of them is increased to 230kPa. Verticaf stress and strain are also computed, The
cornparison in Figure 3-10 illustrates that the results are in ageement for this case.
3.6.4 One Dimensional Compression Cases
The second test problem features one dimensional compression behaviour of a block
subjected to a gradualiy increased pressure fiom zero to L87kPa on the top block
boundary as s h o w in Figure 3-1 1. The other three sides are constrained with smooth
rigid boundaries. It is a plane strain probIem. In the direct stress integration, normal stress
q is exerted to a point graduaily &orn zero to 187kPa; rneanwhile, normal strain E, is
always kept zero. The two predictions are in agreement again.
3.6.5 Effect of Numerical Treatment
In these two tests, a minimum deviatoric stress q of kPa and a minimum mean stress
p of 0.005 kPa are used. The incremental stress is set to 0.03kPa. The agreement in the
cwmparisons in Figure 3-10 and Figure 3-1 1 illustrates that the treatrnents of the
numerical probiems aforementioned in section 3.5 are appropriate.
Compute the coordinate of the Gauss point
I
Cornpute incrernental straïns fiom shape functions and nodal displacements
I
Get history variables of stress and elastic strain
I
Compute Boyce constitutive mat& DE I
Check whether yielding occurs. If so, compute 1 elasto-plastic constitutive matrix Dp 1
1 Compute elastic part of the incrernental 1
1 Compute cumulative stress, strain and its elastic part to date 1
Figure 3-1. Flow chart of the procedure for calculating the increments of strain and stress at each integration point.
500 1
secant solution incremental solution
400
300 - -
p,,,=0.001 kPa
200 -
IO0 -
O 0.0 O. 1 0.2 0.3 0.4 0.5 0.6 0.7
Volumetric Strain, E, (%)
Figure 3-2. Cornparison between the secant solution and the irrcremental solution using dif5erent p,_ in computer simulations of elastic behaviour during isotropie
compression tests. Boyce Model, ~,=404(lcPa)~-~~, ~ ~ = 6 7 3 ( k E ' a ) ~ . ~ ~ , ~ 0 . 6 2 .
A F ~ x 1 O".
- 0.0 O. 1 0.2 0.3 0.4 0.5 0-6 0.7
Volumetric Strain, E, (%)
Figure 3-3. Compûrison between the secant solution and the incremental solution using different p,, in cornputer simulations of elastic behaviour during isotropie
compression tests. Boyce Model, ~ , = 4 0 4 ( k ~ a ) * . ~ ~ , ~,=673(kPa)~-~', ~ 0 . 6 2 .
~ ~ 3 x 1 0 " .
- 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Volumetric Strain, E,, (Oh)
Figure 3-4. Cornparison between the secant solution and the incremental solution using different incremental strains in cornputer simulations of elastic behaviour during isotropie compression tests. Boyce Model, ~ , = 4 0 4 ( k ~ a ) ~ . ~ ~ ,
~,=673(k~a)~-",n=0.62. pmin=O.OOSkPa.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Volumetric Strain, zv (%)
Figure 3-5. Cornparison between the secant solution and the incremental solution using different incrernental stresses in computer simulations of elastic behaviour during isotropie compression tests. Boyce Model, ~ , = 4 0 4 ( k ~ a ) ~ - ~ ~ ,
~ , = 6 7 3 ( k ~ a ) ~ . ~ ~ , ~ 0 . 6 2 . p,,=O.OOSkPa.
0.0 0.1 0.2 0.3 0.4 0.5
Volumetric Strain. E,, (%)
Figure 3-6. Cornparison between the secant solution and the incremental solution using diEerent p,, in cornputer simulations of elastic behaviour during isotropie
compression tests. Boyce Model, ~,=400(kEb)~.~, ~ ~ = 1 8 5 ( k ~ a ) ~ - ~ , ~ 0 . 4 .
A F ~ X 1 O".
Figure 3-7. Cornparison between the secant solution and the incremental solution using different pm,,, in cornputer simulations of elastic behaviour during isotropic
compression tests. Boyce Model, ~,=400(kPa)~.', G,=1 85(kE'a)0.", ~ 0 . 4 .
AE=~x~o" .
500
400 h
as 1 Y
Q - 300
b 1 I I
/ secant solution incremental solution
-
- cn Co !!! ü3
C 200 - - . - s t
1' : O O
100 - -
0.0 O. 1 0.2 0.3 0.4 0.5
Volurnetric Strain, E, (%)
A secant solution incrernental solution
-
O - 0.0 0.1 0.2 0.3 0.4 0.5
Volumetric Strain, E,, (%)
Figure 3-8. Cornparison between the secant solution and the incremental solution using different incremental strains in computer simulations of elastic behaviour during isotropie compression tests. Boyce Model, ~~=400(kPa )~ -~ ,
G,=185(l~Pa)~-~, n=0.4.pm,=0.005kPa.
500
secant solution incremental solution
400 -
300 -
200 -
100 -
O 0.0 0.1 0.2 0.3 0.4 0.5
Volumetric Strain, E, (%)
Figure 3-9. Cornparison between the secant solution and the incremental solution ushg different incremental stresses in cornputer simulations of elastic behaviour during isotropic compression tests. Boyce Model, ~,=400(kPa)~",
GI=l 85(k~a)~.', ~ 0 . 4 . pm,n=0.005kPa
a2 (O-,
Results from MENA D Results from test program
cl (O+ 240kPa)
Figure 3- 10. Cornparison of stress-strain predictions for isotropic compression followed by triaxial compression; solutions from AFENA and from direct stress Litegration. Axially symmetnc pro blem, elastic behaviour.
Results from AFENA D Results from test program
Figure 3- 1 1. Cornparison of stress-strain predictions for uniaxial compression; solutions Eom AFENA and fiom direct stress integration. Plane strain problem, elastic behaviour.
CHAPTER 4
PARAMETER EVALUATION
4.1 Introduction
In this chapter, approaches to obtaining the parameters for the Boyce rnodel will be fïrstly
discussed. The focus will be the snidy of the parameters for the silo filling problem. The
parameters for the Boyce mode1 can not be obtained directly but c m be back calcdated
f7om Lab data. Sand data in the literature have been reviewed and selected data will be
used to evaluate the parameter values. Published parameters for the model are then
compared with the results f?om the experimental data. A set of parameters for use in the
dlG i,::ing aye:,is Yc I;rcsc;;-cd &iscrssiûi,.
Parameters for the four other elastic models will be then studied. The selection of
Poisson's ratio v will be discussed in some detail when the pararneters for the Lade
model are examined, since in each of the other three elastic models, v is assumed to be
the same constant. Young's modulus E varies with stress in both the Janbu model and the
poroelastic model, but is constant in the linear elastic model. Pararneters used in the
expression of E in the three non-linear elastic models will then be evaluated.
4.2 Parameter Evaluation for the Boyce Model
4.2.1 Approaches to Obtaining Parameters for the Boyce Model
The combination of two types of conventional tests for the same granular material can be
used to determine the three parameters of this rnodel, n, KI and G,. Pararneters KI and n
can be obtained fkom an isotropic compression test involving several unloading and
reloading cycles. Data fiom triaxial compression tests, also with unloading and reloading,
can be used to acquire G!.
If the mode1 is a good characterization of the granular rnaterial response, the same
material should have the same parameter values, whatever the stress or strain paths, and
whatever the stress levels which are used. However, in reality, the extent to which one set
of parameters can be applied is unknown. To rninimize inaccuracy, tests feaniring stress
paths similar to those that develop in the practical problems are always suggested.
Fer & 4 9 fi!!kAg ~ ~ c b ! ~ ~ , th- S E ~ ~ ~ P S \Ari!! ge~pyd!>r he !biM tz A^'MIcPE~ c~ngjdeing
a silo with a height of 20 meten and a specific gravity of the stored material around
20kN/m3. Neglecting sidewall friction, the stresses at the base of the silo will reach
400kPa. To obtain Gi, several tests with constant stress ratio (q la, = consr.) are
recommended for the silo filling problem (Figure 4-1). This is because the stress state of
granular material in most parts of a silo is sirnilar to that of natural soi1 deposits, which
means an aImost constant vertical and horizontal stress ratio exists. In the unloading
portions of these tests, the strain ratio E, should be constant, since granular material
exhibits elastic behaviour during unloading.
The following derivation presents how to obtain a set of Boyce model parametes using
the data from the isotropic compression tests and triaxial compression tests with constant
stress ratios.
For isotropie compression tests, the deviatoric stress q equals zero. Therefore,
By using data for volumetric strain E, venus confiiing stress p during unioading, and a
proper regession algorithm. KI and n can be obtained-
For a triaxial compression test with constant stress ratio, equations (3-l), (3-2), (2-53),
(3-7) and (3-8) yield
In axially symrnetric problems,
Substitution of equation (4-3) in (4-2) yields
By using data for the stress ratio 5 / o, versus the strain ratio E, / E, during unloading
portions, KI/ Gl and n can be detennined. Thus Gl can be found fiom KI and K ~ G I
acquired from the two experirnents. The n value fiom this test can be compared with the
one from the isotropic test so that the consistency of the mode1 can be checked.
4.2.2 Selected Sand Data from the Literature
4.2-2.1 Materials Stored in Silos
Silos are used to store many materials. Except fodder and liquids, most of these materials
are bulk solids, for example, grain such as wheat, rice and beans. The mining industry
also uses silos to store ore and rninerals in various stages of processing.
Sand is a typical buk solid matenal. Here data for loose sands are used to analyze the
response of the silo filling process, since the geotechnical community has reported many
constitutive studies for this granular matenal. During the filling process these types of
materiais are initiaiiy in a ioose state, so the focus of' acremion wiii bt: iesit data ibr iwse
sand.
4.2-2.2 Loose Sacramento River Sand
Lade (1977) provided data for Loose Sacramento River Sand in isotropic compression
with primary loading, unloading and reloading branches (Figure 42). It is a fine uniform
saturated sand with an initial void ratio of 0.87 and relative density of 38%. The test was
conducted until the confining pressure reached a very high value, 14MPa. Two
unloading-reloading cycles are presented in the figure. One starts fiom 4MPa and the
other 14MPa. The former is chosen for parameter estimation. Several points taken
directly from the experiment are selected. Because data points at low stress are not clear
in Lade's publication (1977), points up to stresses of lOOOkPa are chosen to force the
regression to be more accurate. The values of these points needed for regression are listed
in Table 4- 1. Figure 4-3 shows the relationship benveen volurnetric strain and mean stress
and a regression line. Parameter values of &=223(Wa)" and ~ 0 . 4 8 are obtained.
Atmosphenc pressure pa = 10 13 kPa has been employed.
4.2.2.3 Loose Santa Monica Beach Sand
Figure 4-4 shows the stress-strain and volume change behaviour of a drained triaxial
compression test on Loose Santa Monica Beach Sand with several unloading-reloading
cycles (Lade and Nelson, 1987). This sand is composed of subangular to subrounded
grains consisting mainly of quartz and feldspar. It has the following characteristics: Mean
diameter, 0.27mm; specific gravity of p i n s , 2.66; maximum void ratio. 0.87; minimum
-.--J -A: v u i u LPLIÛ, 3.58. Gïie ûf i k t G S f ~ ü î ü ~ d 8 !ÛGX GE;CC~;~~EZ %ith .=id X:~S 2f'-s!
corresponding to a relative density of 20%. This test was performed under a constant
confining pressure of 235kPa. Experimental points are selected from the third unloading
1 part of this figure. From equations (3-2) and (3-8), gs = - pn-'4 -
jG, P, TabIe 4-2 shows the
data required for analysis using the least-squares approximation method. Parameter
values of Gi=308(kPa)" and n=0.45 are obtained.
A cornputer program using the Boyce mode1 similar to those described in section 3.6.2 is
written to simulate the process of the triaxial compression test performed by Lade and
Nelson ( 1987). Hydrostatic force is added incrementally to 1 17.7kPa, and deviatoric
stress is exerted until the first stress invariant reaches 2400kPa. Using the parameter
values evaluated earlier, GI=308(kPa)" and ~ 0 . 4 5 , different Ki values are examined and
Poisson's ratio is calculated according to equation
4 v= -- 4 ' (4-5)
where 4 and 6 are incremental principal strains for samples under triaxial compression.
Figure 4-5 shows that all curves depict fast increase of Poisson's ratio immediately afier
the deviatoric loading, followed by decreases in Poisson's ratio with larger deviatoric
stress. The curve with K[=liO(kPa)" is considered closest to the experimental data shown
in Figure 3-2.
The significance of this simulation is that the Boyce model may be capable of capturing
the true nature of the behaviour of granular material, particularly the varying Poisson's
ratios during loading. The model simulates the experimental data shown in Figure 2-2
following selection of an appropriate set of Boyce model parameters.
4.2.2.4 Loose Ottawa Sand
Dakoulas and Sun (2992) conducted a number of tests on fine Ottawa silica sand. This
sand has a mean diameter of Dj, = 0.1 1 mm and specific gravity of 2.65. The maximum
void ratio is 0.95 and the minimum 0.57. The specimens of loose sand have relative
density of 30% and were fblly sanirated. The tests were performed under drained
conditions. An isotropie compression test for this loose sand is shown in Figure 4-6.
Experimental data of the first four unloading portions are selected and shown in Table 4-
3 and a regression line is obtained (see Figure 4-7). The stress range chosen here is from
zero to 400kPa. From the test, Ki=404(kPa)" and n=0.62. Dakoulas and Sun (1992) also
conducted some tnaxial compression tests with unloading and reloading cycles, but
unfortunately they omitted the portions of unloading and reloading cycles to improve the
ciarity of their figures.
4.2.2.5 Summary
No set of test results was found in the literahire to obtain a complete set of the Boyce
mode1 parameters for the sarne material, though a complete set of parameters for Loose
Santa Monica River Sand cm be obtained indirectly. However, the parameters calculated
using these different data sets have magnitudes that are reasonably consistent. These
parameters are therefore used in the pararneaic studies that follow.
4 7 9 m,,Ll:,L,A m , , , , , ~ . . ~ ~ *.&-a L U V L I ~ I L C U 1 aa ULLICCCL J
Loret (1985) pointed out that the ratio of the Boyce pararneters K, I jG, equals 0.14 for
one specific loose sand. His result was based on the data of El Sohby and Andrawes
(1973, Fig. (5)a). The fi,pre is redrawn in Figure 4-8. Apparently he selected hvo points
frorn this figure and then used equation (4-4). Solving the sirnultaneous equations, the
ratio K, /3G, = 0.14 was obtained, and a negative value of the parameter n (n=-3.5) was
established. The latter is obviously very different From the results in section 4.2.2.
Carefully checking the content of the original paper (El Sohby and Andrawes, 1973), the
data in the figure which Loret quoted is not consistent with another figure (El Sohby and
Andrawes, Fig. (3)a, 1973), which is redrawn in Figure 4-1. From Figure 4- 1, when the
stress ratio ( O, / uj ) equals 1.5, the strain ratio ( E, / 4 ) in the unloading part should be
0.66; when a, / q equals 2.0, E, / 4 in the unloading part is 0.50. But in Figure 4-8, the
corresponding strain ratios are 0.78 and 0.62. Thus K, / 3G, = 0.14 is not the actual value
for that matenai. If the true values fiom Figure 4-1 are used to recalculate the problem,
the value of the parameter n is 0.71, and K, / 3G, equals 0.22 for this matenal.
It should be noted that the n value is very sensitive to strain ratios obtained fiom the tests.
More then two triaxial compression tests should be conducted to obtain better values.
Careful measurement is definitely needed.
A French paper by Ragneau, Aribert and Sanad (1994) presents a set of parameters for
sand. They suggest K1=4OO(kPa)", Gl=185(kPa)n and n-0.4. Each individual parameter is
in the range of those values presented in section 4.2.2.
Sanad, Ragneau and Aribert (1997) adopted the following set of parameters of the Boyce
mode1 for Sacramento River sand, n=O.43, KI=753 3 5(kPa)" and G1=56S0 1 (ma)". These
are nor nonnalized values. If normalised, KI= 1034 1 @Pa)", G1=7756(kPa)". These are
much greater than the results presented in section 4.2.2.
It is not clear why this research g o u p reported very different values in two different
studies, but it might be two different denvations of the constitutive relations of the Boyce
model.
3.2.4 Discussion
From section 4.2.2 and section 4.2.3, and fkom an examination of the available literatures,
preliminary conclusions c m be drawn regarding the magnitudes of Boyce model
parametes for loose sands: KI ranges fiom lSO(kPa)n to 450(kPa)", G, frorn lSO(Wa)" to
7OO(kPa)" and n From 0.35 to 0.70. The ratio K, /3Gl equals 0.18 for Loose Santa
Monica Beach Sand if GI=308(kPa)" and K,=LiO(kPa)" are used according to section
4.2.2.3. This is comparable with the value of 0.22 for loose sand based on testing by El
Sohby and Andrawes (1973). If an average ratio 0.20 is used, Gr can be estimated to be
372(kPa)" and 673(kPa)" for Loose Sacramento River Sand and Loose Ottawa Sand
respectively.
4.3 Parameter Evaluation for the Lade and Nelson Mode1
As described in Chapter 2, Young's modulus in the Lade mode1 is expressed as
invariant of the stress tensor and the second invariant of the deviatoric stress tensor
defined as (A- 1) and (a-2) in Appendix A. The parameter R is related to v as follows
Poisson's ratio v is important in equation (4-6) and is relatively hard to determine. It is
treated as a constant in this mode[.
4.3.1 Loose Santa Monica Beach Sand
Using experimental results like those shown in Figure 4-4, the deviatoric strain E, and
the deviatonc stress q can be calculated, Table 4-2. Using equation (3-8), substituted with
the retation between shear modulus G and Young's moduluç
r
Young's modulus E can be expressed as
A value of Poisson's ratio is fïrstly assumed, and then the values of E are computed
according to equation (4-9). By plotting (E / P , ) versus [(I l l p,)' + R ( 4 / p:)] in a log-
log diagram, the parameters M and A in equation (4-6) c m be determined. Figure 4-9
shows that the intercept of the linear regression line with [ ( I , / p,)' + R ( 4 / p f ) ] = 1 is
the value of Ad, and the slope of the line is A. The regression analysis can also provide the
value of the coefficient of detemination r' for this specific Poisson's ratio, which is
shown in Figure 4- 10.
By varying Poisson's ratio in the range from O to 0.5, corresponding values of r'. M and
A can be obtained. Variation of r' ,LM and /2 with Poisson's ratio in triaxial tests on Loose
Santa Monica Beach Sand is shown in Figure 4-10, from which, a maximum r' = 0.7924
is obtained for v = 026. For this Poisson's ratio, M = 600 and A = 027, as shown in
Figure 4-9. This work was conducted by Lade and Nelson ( 1987).
A direct rnethod can also be used to detemine Poisson's ratio. By measuring the
incremental volumetric strain and axial strain after stress reversal in triaxial tests,
Poisson's ratio can be calculated fkom the equation
The average value and the range of Poisson's ratio for Loose Santa Monica Beach Sand
detemined £iom this method are also shown in Figure 4-10. The average value of v is the
same as the one corresponding to the maximum coefficient of correlation. Interestingly,
within the range v,, f S . where vag is the average value and s is the standard deviation
of Poisson's ratio, r' , M and A are relatively insensitive to Poisson's ratio.
3.3.2 Loose Sacramento River Sand
Lade and Nelson (1987) provided the parameters for Loose Sacramento River Sand using
a procedure similar to the one outlined in section 4.3.1. Poisson's ratio was estimated to
be 0.2. Parameters A4 and A are 500 and 0.28 respectively.
4.3.3 Loose Ottawa Sand
Dakoulas and Sun (1992) does not report rneasured values of Poisson's ratio for fine
Ottawa Sand. Poisson's ratio is assurned to be equal to 0.2 using information from the
literature. One of the sources referenced is the work of Hardin (1978). He performed
resonant column tests on Ottawa Sand and found v to lie in the range fiom 0.1 1 to 0.23.
He concluded 'any value within this range is accurate enough for most purposes.' Lade
and Kim (1988) also provide information on Poisson's ratio. They found a value of
Poisson's ratio close to 0.2 for the elastic parts of unloading-reloading stress paths for
smds.
To evaluate the parameters M and A, Dakoulas and Sun maintained that it is sufficient to
use the results of isotropic compression tests. They compared the unloading data from a
variety of tests and reported that M = 590 and A = 0293 for Loose Ottawa Sand.
4.4 Parameter Evaluation for the Janbu Mode1
In the h b u formula, Young's rnodulus is expressed as
where Ku, is modulus nurnber, n is modulus exponent, , p is mean stress and pu is
atmospheric pressure. Janbu ( 1963) found values ofn fiom 0.35 to 0.50 for sand and silty
sand with porosities of 35%-50%. The modulus number varies from 50 to 500.
Substituting the relation between bulk modulus and Young's modulus
E K =
3(l- 2 V) '
and also equation (4-1 1) or (4-12), equation (3-7) can be
and then
For Loose Sacramento Ri1 rer Sand, Lade (1977) i ised the experirnental results and
equation (4-15) to determine the values of the parameters. He obtained the modulus
number K, =960, and the exponent n=0.57.
Introducing equation (4- LZ), equation (4-9) can be written as
This equation can be employed to evaluate the parameten of the Janbu mode1 for Loose
Santa Monica Sand. Poisson's ratio v = 0.26 from section 4.3.1 is used. Applying the
least-squares method, the value of modulus number K, is found to be 785, and n is
found equal to 0.58. A cornparison between the expenmental results in Table 4-2 and the
predicted ones using equation (4- 17) with these parameters is shown in Figure 4- 1 1.
Isotropic compression tests were performed on Ottawa Sand by Dakoulas and Sun
(1992). Using the experirnental data shown in Figure (4-6), and performing a regression
analysis similar to that illustrated in Figure 4-7 gives K, = 723 and n=0.38.
4.5 Parameter Evaluation for the Poroelastic Model
In the poroelastic model, the value of Young's modulus depends on the current values of
specific volume v and mean stress p
E = ~ P U ( ~ - ~ V ) / K , (4- 1 8)
where Poisson's ratio v is assurned to be a constant; and K is the dope of a swelling line
on a u- Inp figure obtained fiom an isotropic compression test. Atkinson and Bransby
(1973, pp. 133) have suggested that the values of p should not exceed 7OOkPa for sand,
because only below this value is the v - lnp relation observed to be linear.
For Loose Sacramento Sand, the initial specific volume v,= 1.87. Using equation
v = v,(l- E , ) , (4- 1 9)
the current specific volume can be found. Figure 4-12 shows the fint unloading cycle (as
shown in Figure 4.2) for the material plotted as v venus In p. The siope of the linear fit to
the unloading curve K is 0.00827 for this material.
The same relations are shown in Figure 4-13 for the fmt four unloading portions of
Figure 4-6 for Loose Ottawa Sand. The values of Kare found to lie in the range 0.00328
to 0.00367. If al1 the data are drawn in one figure, the linear regression gives K = 0.00336
(Figure 4- 14). It is also apparent fiom Figure 4- 14 that an exponential fitting of the data
is likely to be better than the linear one.
No isotropic compression data is available for Loose Santa Monica Beach Sand.
Therefore, the values of the parameters for the poroetastic model c m o t be calculated.
4.6 Parameter Evaluation for the Linear Elastic tMode1
The linear elastic model has been widely used in engineering practice. Assuming uniform
matenal properties, only two independent constants, Young's modulus and Poisson's
ratio, are needed. However, the behaviour of sand is really non-linear, as has been shown
in Figures 4-2, 4-4, and 4-6. Therefore, using equation (4-9) or (4-I4), a range of
Young's modulus values c m be obtained for each sand, as shown in Table 4-1. Average
values of Young's modulus (E,,) are adopted by taking the mean of a11 available
calculated values. It is suggested that a piecewise constant elastic Iinear model be
considered in practice (Chen, 19901, Le., for a certain stress intervaI, one constant
Young's modulus is used.
4.7 Sumrnary and Discussion
The parameters of the three loose sands for each model are listed in Table 4-4. It can be
seen that the values are generally reasonable and consistent. For the sake of brevity in the
following discussion, the three sands are denoted as Sand 1, II and III respectively.
An examination of equations (4- 1) and (4- 16) reveals that the sum of the exponents in the
Boyce model and the Janbu mode1 should be one for the same material. From Table 4-4,
the sums of the values of n in both models are 1.05, 1.03 and 1 .O respectively, which are
very close to one.
The values of exponents n in the Boyce mode1 are similar for Sand I and II, so are the
values of A in the L2de model, while Sand III appears to be different.
The parameters Ki and GI of the Boyce model for Sand 1 and II are comparable. But they
are much Iess than the ones for Sand III. However, values of parameter M in the Lade
model are similar for Sand II and III, and 20% less for Sand 1.
An examination of parameter K, in the Imbu mode1 indicates that the value for Sand I
is higher than those for Sand CI and III. This is consistent with the average values of
Young's modulus back calculated for the linear elastic model.
In general, the exponents in those models with exponential forrns are indicative of the
level of stress dependence. The values of K, in the Janbu model and K in the
poroeiasric mociei are vcry senriiive io Fuisauii's raiiù, ë~ jhùwri iii tqui i iù i i~ (4-16) ai?:
(4- 18).
Sand I is stiffer than the other two sands in terms of Young's modulus based on the
values in Table 4-4.
Sand III has the best quaIity of the experimental data among these three sands from the
references. The stress Ievel for Sand I is very high.
Table 4-1. Experimental Data for Loose Sacramento River Sand, afier Lade (1977)
Table 4-2. Experimental Data for Loose Santa Monica Beach Sand, &er Lade and Nelson (1 987)
Table 4-3. Experimental Data for Loose Ottawa Sand, after Dakoulas and Sun (1992)
1 3rd unloading 1 1 4th unloading 1
Table 4.4 Parameters of the loose sands for differer t rnodels
I Ottawa Sand
Sand III
n=O.62,&=404(kPa)",
GI=673(kPa)"
v=0.2sz.i=j 90.~=0.32'"
I
Poroelastic
Linear
Elastic
- -
Note: Al1 the parameters in Table 4-4 are back calculated fiom the lab data in the
Literature except those with superscripts which were given in the literature.
(1) after Lade and Nelson (1 987);
(2) after Dakoulas and Sun (1992);
(3) after Lade (1 977).
Sacramento River Sand ' Santa Monïca Beach Sand
v=0.2, ~=0.00827
v=O.Z,E=60- 1 36MPa
E, =94MPa
Sand 1
Boyce
Lade
Imbu
v=O.26,E=25-115MPa
E, =66MPa
Sand 11
n=0.48,Kf=223(Wa)",
Gl=372(kPa)"
v = o . z ~ = jo0,~=0.28'"
v=0.2, K, =960,n=O.57'~'
n=0.45 ,KF 170(kPa)",
G1=308(kPa)"
v=0.26~=600,A=0.27'"
~ 0 . 2 6 , K, =78S.n=U.S8
i
Figure 4- 1. Triaxial compression tests with constant stress ratios, and with loading and unloading. R=cr,/q (after El Sohby and Andrawes, 1973).
VOLUMETRIC STRAIN. eV (%)
Figure 4-2. Isotropie compression of Loose Sacramento River Sand with primary loading, unloading and reloading branches (afier Lade, 1977).
Figure 4-3. Unloading portion of isotropic compression tests of Loose Sacramento River Sand and its curve fining for the Boyce model. (CoeEcent of correlation 1=0.9969)
1200
1000
800
600
400
200
- 1 I l I 1 & I
- Boyce model experirnental data
-
- -
-
i / -
- -
0- I I
0.0 0.2 0.4 0.6 0.8 1 .O 1.2 1.4 1 -6
Volurnetric Strain, (%)
Figure 4-4. S~ess-strain and volume change behaviour of loose Santa Monica Beach Sand in triaxial compression tests with several unloading-reloading cycles ( d e r Lade and Nelson, 1987).
O 5 10 15 20 25
Dimensioniess pressure, I,/p,
Figure 4-5. Poisson's ratios calculated from computer simulations of triaxial compression using the Boyce mode1 with different KI. For Santa Monica
Beach Sand, ~ , = 3 0 8 ( k ~ a ) ~ - " , n=0.45.
Figure 4-6. lsottropic compression tests on Loose Ottawa Sand (after Dakodas and Sun 1992).
Experimental Data 2 400 - Boyce Mode1 c b* " c1-2 300 - W w t- rn 0 200 - - a O
-
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Volumetric Strain, €,(%)
Figure 4-7. Unloading portions of isotropie compression of Loose Ottawa Sand and its cuve fitting for the Boyce model. (Coefficient of correlation $=0.9823)
Points selected by Loret (1985)
( theoret *al)
Figure 4-8. Effect of applied constant stress ratio on strain ratio-loose sand (after El Sohby and Andrawes, Fig.(S)a, 1973).
d
TESTS
Figure 4-9. Variation of Young's modulus in triaxial and cubical triaKial tests for Loose Santa Monica Beach Sand (after Lade and Nelson, 1987).
PO WON'S RATIO DETf RMINED DIRECT LV FROM TESTS
Figure 4- 10. Variation of coefficient of correlation, moduius number and exponent with Poisson's ratio in triaxial tests on Loose Santa Monica Beach Sand (after Lade and Nelson, 1987).
Figure 4-1 1. Cornparison between the experimental data and results predicted for Loose Santa Monica Beach Sand using the Janbu model.
500
400
- - results from the Janbu mode1 experimental data
- a /
Figure 4- 12 Parameter evaiuation for the poroelastic mode1 ushg regression andysis for Loose Sacramento River Sand. (Coefficient of correlation ?=0.9864)
First unloading-reloading cycle
Third unloading-reloading cycle
1.836 1 1
Second unloading-reloading cycle
1-838 a
Fourth unloading-reloading cycle
Figure 4- 13. Parameter evaluation for the poroelastic mode1 using regressiong analysis for loose Ottawa Sand (unloading). (Coefficient of correlation r2=0.9904,0.9643, 0.9750, 0.9860 for lst, 2nd, 3rd, and 4th unioaduig respectively)
Figure 4- 14. Parameter evaiuation for the poroelastic mode1 using regression analysis for loose Ottawa sand (overail unloading). (Coefficient of correlation 2=0.9475)
CHAPTER 5
APPLICATION IN SILO FILLING
5.1 Introduction
Five elastic models, the Iinear elastic, Janbu, Lade and Nelson, poroelastic, and Boyce,
have been reviewed and parameters of these models for loose sand have been evaluated
in the preceding chapters. The linear elastic model and the Janbu model have previously
been incorporated in a general-purpose finite element code AFENA (Carter and Balaam,
1990). Ln this study, the Lade and Nelson model, the poroelastic model and the Boyce
EX!&! h2ve heen i~p!empnteA i~ l s code. This progrlm is cayh!e cf undertaking
analyses of both plane strain and axially symmetric boundary value problems.
In this chapter, their performance when used to mode1 silo filling is investigated. Finite
element analyses are carried out to predict the stress distributions as well as other results
like wall deflection, stress contours and plastic zones in silos with different geometry and
wall flexibility. The definition for the silo filling problem is firstly given, including
material properties and silo geometry. Finite element simulations for this problem are
then presented with focus on the simulation of the silo filling process and the incremental
load steps used in the analysis. Results fkom finite element analyses are reported and
differences among predictions using these elastic models are examined. Effects of
Poisson's ratio, stress correction, and weak elements are discussed in detail.
5.2 Problem definition
5.2.1 General
This project exarnining the modelling of silo filling using the finite element method
commenced with participation in a collaborative study conducted with the Silos Group at
the University of Edinburgh, Scotland. The definition of the problem and matenal
parameten were given in the problem description developed by Rotter er al. (1995) as
part of an international effort to examine the performance of existing procedures for bulk
solids analysis. A plane strain finite element analysis of the concentic filling of a
parallel-sided silo was planned. The plane strain model represents a long rectanglar
planform silo. Since cylindrical silos are more cornmon in reality, axially symmetrical
analysis will also be conducted in the present snidy. Tne geomeuy of h e siio is iypiciii vr
a ta11 silo. Squat silos will also be studied since they are becoming more popular.
5.2.2 Material propeties
In the problern description developed by Rotter et al. (1995), Loose Sacramento River
Sand was chosen for the filling material. The elastic behaviour of this material is modeled
using the Janbu formula (Lade, 1977; Schmidt and Wu, 1989). As discussed in chapter 4,
the experimental data for Loose Sacramento River Sand provided by Lade (1977) is not
very suitable to deduce the parameters for the other models in this study. Data for Loose
Ottawa Sand have been selected as the most suitabie for parameter evaluation.
The silo filling analysis will be performed with a number of different elastic models, but
the same plastic rnodel. Plasticity will be modelled using an associated flow rule and a
Mohr-Coulomb failure criterion, with both angle of interna1 fiction gj and angle of
dilation y, 35'. It is important to represent the friction angle between the b u k solid and
the wall as different fiom the angle of intemal fiction of the material within the silo. A
nurnber of diKerent procedures c m be used to mode1 the silo-solid interaction:
1, Joint elements, e.g. Goodman et al. (1 969);
2. Explicit modek, e.g. El Sawy and Moore (1996);
3. Continuum rnodsls, e.g. Desai et al. (1984).
In this study, weak continuum elernents are ernployed. A thin layer of weak elernents
with thickness 0. lm and both friction angle #w and dilation angle y, 18' is incorporated
down the wall of the silo. The value 18' is the angle of wall friction. This layer
experiences shear failure during filling. The waii constraint is the prirnary source of shear
and the critical failure plane is vertical (parallel to the wall). Details about this treatment
will be discussed in section 5-49.
A value of cohesion of O. 1kPa for both normal material and weak material is included to
limit (but not eliminate) certain numerical problems. These will be discussed later in
section 5.4.2.
The unit weight of the solid is taken as 13.72kN/rn3. For cases where a flexible wall is
examined, the material for the wall is steel, which has modulus of elasticity 2 x 108kPa
and Poisson's ratio 0.3. Al1 the material parameters used for calculations are listed in
Tables 5- 1,s-2 and 5-3.
5-23 Silo geometry
The geometrical dimensions of a full-scale ta11 silo with a width of Sm and a height of
25m are shown in Figure 5-1. Material is typically filled to a point where the highest
solids/wall contact is 20m above the base (&er filling, the vertical deformation of the
materiai surface will only be of magnitude 0.001m as calculated later). A conical shape
inclined with the angle of repose 3j0 of the material is formed afier filling, as shown in
Figure 5-2. The highest solids/wall contact is decreased to 10m for the squat silo, while
the width remains the same, 5m.
Both tigid and flexible walled silos are examined. Ooi and Rotter (1990) introduced a
nanmptrr r- ---- --- 1 = - s - - F- R I E lu r t~ ~h-racterire the relative wall flexibility, where E and Ei
are moduli of bulk solid and wall material respectively; R is the radius of the silo and r is
the thickness of the wall. Ooi and Rotter ( 1990) suggested that squat flexible silos usually
fa11 in a range (0.0 l c 7 -=z 0.2) where stored solid-silo interaction may be expected to play
a role in determining the magnitude of the wall pressures. Using the values Es = 89 MPa,
E, = 2 x 10'MPa and R=2.5m from Tables 5-1, 5-2 and 5-3, a thickness of lcm for the
wall of the squat silo leads to pO.11, which is within the range suggested. For the ta11
silo, calculations will be performed using two different wall thicknesses, k m and OScm,
to hrther explore the effect of silo wall thichess.
5.3 Finite Element Simulations
5.3.1 Finite Etement iMeshes
The finite element mesh used for the ta11 silo is shown in Figures 5-3 and 5-4. Due to
symmetries of both the silo geometry and the filling process, only half of the structure
and filled material are analyzed. A smooth boundary condition is used along the
centreline of the structure. The base of the silo is modelled as a rough ngid boundary. For
the silo wall, if it is treated as rigid, a rough ngid boundary is employed to replace the
wall, as shown in Figure 5-3b; othenvise, the wall is modelled using the eight-noded
rectangular elements, Figure 5 4 . Six-noded triangular elements are used to mode1 the
bulk solid. These elements are inclined in layers at the angle of repose 35', as illustrated
in Figure j 4 a . Figure 5-5 shows fie mesh for Uie squat siio.
The vertical stress inside the thin steel waIl will be linearly distributed through the wall
thickness (or very nearly so). Therefore, one sûip of eight-noded rectangular elements
will be adequate for simulating the wall behaviour since that element provides a linear
stress distribution (like that needed to represent 'thin shell' behaviour). The aspect ratio
of the element is large. An earlier trial showed that the same results were obtained if two
snips of eight-noded elements were used to mode1 the wall. Moreover, because the stress
distribution in the waIl is not the main concern in this study, the accuracy of intemal wall
stresses is not important.
53.2 Simulation of the Silo Filling Process
The filling process is modelled with progressive element activation in fi@ steps for the
tall silo and fifty-five steps for the squat silo using a layer by layer 'construction'
technique. AI1 eIements of the frlled buik material are excavated first, and filled from the
base to the top in layers as s h o w in Figure 5-3a. The bulk matenal is subjected to p v i t y
load.
In AFENA, 'excavation' means elements are removed from the present mesh or
'deactivated'. The excavated elements are not incorporated into the global stiffness
matrix. Conversely, 'fiIl' means elements are added to the present mesh or 'activated'.
Stress z d SZ& SEOS cf e l rh Gwrr point i e r t o d in history variables and updated
after each incremental load step. In each new load step, only those activated elernents
with their current stress and strain States will be taken into account. These techniques can
be used to simulate the construction or excavation of many different earth-structures.
5.3.3 Solution Procedures
5.3.3.1 Stress Correction
The elasto-plastic finite element analysis is undertaken using an incremental approach. A
problem can arise in this type of analysis in which the stress state predicted at the end of
an elasto-plastic loading incrernent does not lie on the yield surface. The stresses must be
corrected back to the yield surface because such discrepancies are cumulative and afier
many load steps, the final results c m deviate significantly from the failure cntenon.
Several techniques for correcting the yield stresses have been proposed in the past (e.g.
Nayak and Zienkiewicz, 1972; Christian et al-, 1977; Sloan and Randolph, 1982).
Associated changes in the elastic strains are not considered when projecting back the
stresses in these methods. Potts and Gens (1985) developed a method in which the elastic
strains accompanying any stress correction are also corrected. This is important when the
Boyce mode1 is used, since that constitutive relation is a fünction of elastic strains. If &
is the stress correction, an associated change in the elastic strains is given by
2- = [ D ~ ] - ' +, (5- 1 )
where [D,] is the elastic constitutive rnatrix. Assuming no change in the total strains
during the correction process implies that the elastic strain change must be balanced by
an equal and opposite change in the plastic strains. Therefore,
The plastic strain increments are proportional to the gradient of the plastic potential,
G(& h) = O . where h is a hardening or softening parameter. That is
&, = a,(Z//), (5-3)
where a, is a scalar quantity. Cornbining equations (5-2) and (5-3) gives
The value of a, can be expressed as
where F ( q h ) = O is the hnction of the yield surface; a, and ho are initial stress state
and hardeningkofiening parameter before the increment is added. (h, is zero in al1
caIcuiations presented in the present study.)
The issue of stress correction has a significant influence on the prediction of stress
dismbution afier the completion of silo filling. Details will be discussed later in section
5.4.1 o.
5.3.3.2 Incremental Load Steps
For analyses using the models other than Boyce, for each filling layer, a hundred
incrernenrai steps are empioyed to obtain gooa resuits, which means tne graviry ioaa is
divided into a hundred sub-Ioads and added gradually. Analyses with 20, 50 and 150 sub-
incremental load steps for each filling layer were investigated dunng early trials and the
results indicated that analyses with 100 sub-loads can produce good results economically.
For analyses using the Boyce rnodel, 150 incremental steps are used. For the ta11 silo,
each layer is about 0.4m hi@ with weight 5.6 k~lrn'. Divided by 150, each incremental
load step is 0.04 kN/m2. For the squat silo, each incremental load step becomes
0.02kN/rn2. These values satise the demands of the numerical integration scheme
developed for the Boyce mode1 with Loose Ottawa Sand parameters as discussed in
section 3.5.
In a typical finite element analysis, based on the formulation and solution of stiffhess
equations, an increment of displacement associated with the applied load increment is
obtained fkom solving the stiffhess equations. The displacement solution is used to
evaluate the incremental strains. The increment in stress is then deterrnined fiom the
incremental strains using constitutive relations
A n = D - A S . (5-6)
For nonlinear models other than Boyce, D is a function of current stress D = il@), and
using D - D(oo) to calculate the incremental stress proves satisfactory, where cr, is the
stress tensor at the end of the previous loading step. However, it has been found that
tirrther computationai enon is needed k r anaiyses using rine ëoyce mociei. This is
because the Boyce mode1 is not only stress dependent, but also strain dependent, Le.
D = D(a, E) . The use of D = D(a,,, E,) , where E, is the strain tensor at the end of the
previous loading step, requires the use of much smaller load increments. Otherwise,
stress and strain States cm develop where the D matrix is singular. This prevents
inversion of the elastic constitutive matrix in the finite element solution procedure shown
in Figure 3- 1.
Instead of reducing load step sizes, an improved integration scheme based on the work of
Sloan (1987) is used to solve this problem. The increment in strain obtained from
incremental displacement is subdivided into a number of smaller components, and (5-6)
is undertaken in a senes of substeps. Each sub-increment in stress is computed from the
small sub-increment in strain, and total stresses and elastic strains are then updated and
used to calculate the next sub-step until the total number of sub-steps are finished. The
advantage of this treatment is that the stifiess equations are solved just once for each of
150 incremental load steps in one filling layer, and the additional effort needed to
evaluate (5-6) is lirnited to that step in the solution algorithm. The alternative is to use
much small load steps for each layer of activated elements, but this is much more tirne-
consuming.
The effectiveness of this treatment is demonstrated by re-solving the load case defined in
section 3.6. Figure 5.6 shows the comparison of the two solutions.
E A n-.- .-1t .- 4%~- v:-:te W I n - * - t A . . * t - v e ; e 3,- L \ C ~ U I L ~ II VU I UULC UAFU~;UL n a n u a j
5.4.1 General
Seven categories of analyses have been perforrned with combinations of different silo
geometry and material rnodels. They are
axially syrnrnetnc analysis of the ta11 silo with rigid wall;
plane strain analysis of the tall silo with rigid wall;
axially symrnetric analysis of the ta11 silo with flexible wall (thickness=Icm);
axially symrnetric analysis of the ta11 silo with flexible wall (thickness=0.5crn);
axially symrnetric analysis of the squat silo with ngid wall;
plane strain analysis of the squat silo with ngid wall;
axially symmetric analysis of the squat silo with flexible wall (thickness= 1 cm).
From the results of these analyses, stress distributions inside the bulk solid have been
obtained. For the plane strain cases, horizontal stress O-, vertical stress ct,, and shear - -
stress a, are presented; while for the axially symmetry cases, radial stress o,, vertical
stress a- - and shear stress oc are given. The normal pressure on the wall p,, stems fiom
the horizontal stress or radial stress in the adjacent bulk material, while the frictional waI1
traction p, e s e s from the shear stress. For analyses with flexible wall, the deflection of
the wall c m be assessed by investigating the displacement of the finite elernent nodes on
the wall. Plastic zones developed in the stored solid, where shear stresses reach the failure
surface, and stress contours are also shown for some typical cases. The Poisson's ratio of
the bulk solid is expected to have a key impact on those predictions, and this will be
examined in more detail.
5.4.2 Approaches to reducing oscillations
The stress distributions along the silo wall are taken by evaluating the stresses along a
vertical section very close to the wall. A contouring algorithm (Sloan, 1991) is used to
compute the stresses on that section, which constructs a trianpulation involving the
integration points where stresses are available, and employing linear interpolation within
those triangles. Using this technique, stress distributions at a distance 2.23m From the
centreline of the silo are obtained for the tall silo, while for the squat silo, this distance is
2.41m. These sections are very close to the silo wall and the stress distributions along
these sections are close to those along the wall (This can be seen in the stress contours
discussed later in section 5.4.6.). A11 stress distributions along these sections obtained
using this technique feature a degree of numerical oscillation. These oscillations
complicate the process of companng the performance of the different elastic models. In
order to facilitate those comparisons, some smoothing approaches have been used.
One approach is to introduce a small amount of cohesion into the solid. A value of O. 1 kPa
is selected as a compromise between a value high enough to almost eliminate the problem
(say IkPa) and the correct value of OkPa for this cohesionless solid.
Having included a cohesion of 0.lkPa for the bulk solid, the resuits still exhibit
oscillation. A typical curve with oscillations is s h o w in Figure 5-7. Each 'saw tooth'
corresponds to a layer of filled material. A further effort is employed to decrease these
9+!!4t,i~m~- Eey- Ccp-!!y spvpcaJ pci&tc p2m-h hZLf cf &e Czcxï-r t_fi&'= E@C~PC ~ r e
smoothed by using the average value for each 'saw tooth' as a representative value.
These representative values are then linked together so that a smooth stress distribution is
acquired. Figure 5-7 shows a c w e directly fkom the triangulation procedure and a
srnoothed curve using this technique.
5.4.3 Predicted Stress Distributions Adjacent to the Wall in the Ta11 Silo
5.4.3.1 krially Symmetric Analysis with Rigid Wall
Figure 5-8 gives the predicted horizontal stress distributions adjacent to the rigid wall in
the taIl silo venus height. The horizontal stress increases graduaily fkom almost zero at
the top of the silo to a peak value at about 7m above the silo base, which is 35% of the
total height. These stresses then decrease to 20 to 26kPa near the base. Predictions using
the linear elastic, lanbu, Lade and Nelson, and poroelastic models are very close.
However, the stress distribution using the Boyce model is noticeably smaller than the
others except the area adjacent to the silo base. The difference in the peak values is about
15%.
The predicted shear stress distributions adjacent to the wall in the same silo are shown in
Figure 5-9. Again' the prediction using the Boyce model is much smaller than those using
the other rnodels. The ratio of shear stress to horizontal stress for each position from
Figures 5-8 and 5-9 is around 0.3, whatever modeis are used, which is consistent with the
friction angle 18" along the wail. For this value of fnction angle, the ratio of shear stress
to normal stress should be tan(lgO) = 032. Since the section taken at r=2.23m is close to
the weak iayer aiong the waii (r=2..iÛm-2.jÛm) but nor within riiar iayer, the stress raiio
predicted is reasonable.
While the predictions obtained using models other than Boyce are very sirnila., there is a
slight deviation between the prediction using the linear elastic mode1 and those using the
other three models. For positions higher than 7.5m above the silo base, both horizontal
stress and shear stress predicted using the linear elastic model are slightly higher than
those using the other three models, while for positions lower than that height, the trend is
reversed.
At the position about O.6m above the silo base, a relatively small but sudden decrease of
stress can be observed. Because the bulk solid at that corner has constraints from both the
wall and the base, the stress conditions are more complex than those in other positions.
The stress discontinuities observed there may be a numerical issue, and are not expected
to affect the results in other places.
5.43.2 Plane Strain Analysis with Rigid Wall
For plane strain cases, the difference between the predictions using the Boyce model and
those using each of the others is less discemible than that seen for the axially symmetrïc
analyses (particularly above height 8m), Figures 5- 10 and 5- 1 1. Below that height,
predictions using the Boyce mode1 are smaller than the others, the difference being less
than 10%. The solution obtained using the linear elastic mode1 deviates noticeably fiom
those resulting from the other four models above 8m. The difference can also reach as
5.4.3.3 Axially Symrnetric Analysis with Flexible Waii
The horizontal and shear stress distributions for the ta11 axially syrnrnetric silo with
flexible steel wall (thickness=lcm) are shown in Figures 5- 12 and 5-13 respectively. The
shapes of these stress distributions are very similar to those with purely ngid wall, as
illustrated in Figures 5-8 and 5-9. The magnitudes of these stresses are slightly smaller
then those with ngid wall, reflecting the stress release induced by the outward deflection
of the wall. However, the wall with this thickness is still very stiff. Reducing the
thickness of the wall to 0.5cm further decreases the horizontal and shear stresses as the
wall deflects fùrther. This is seen in Figures 5-14 and 5-15, where cornparisons are made
using the Lade and Nelson model. The difference arnong these predictions with different
wall stiffness is small at both the top and bottom of the silo, but is relatively large in the
middle.
At the height less than 1.5m above the base, discontinuities of stress distributions appear
once again. Cornpared with those in the silo with rigid wall s h o w in Figures 5-12 and 5-
13, the discontinuities have broadened. The wall flexibility appean to intemi@ the
complexity of the stress state at the wall-base corner.
5.4.4 Predicted Stress Distributions Adjacent to the Wali in the Squat Silo
5.4.4.1 Axially Symmetric Analysis with Fügid Wall
T-e heig!!! ~f the squat si!^ Is !Om, ha!f of the heieht o f the ta11 silo, w h i l e the radius or
width remains the same. The horizontal and shear stress distributions adjacent to the rigid
wall in this squat silo are s h o w in Figures 5- 16 and 5-17. Once again, the performance
of the Boyce mode1 is quite different from the other elastic models. Generally the
horizontal and shear stresses predicted using the Boyce model are rnuch smaller than
those using other models except those positions below 0.8m above the base. The
maximum discrepancy can reach 25% at the height 3.5m above the base, which is 35% of
the total height. The shapes of the stress distribution c w e s are sirnilar for al1 analyses of
the tall silo, Figures 5-8 and 5-9. However, in the squat silo, the peak values are reached
at the height 0.8m above the base for the Boyce model, while for other models, it is 3.2m.
The ratio of shear stress to horizontal stress for each position on the section taken is 0.32
for al1 rnodels, exactly equal to the value of tangent of the wall fiction angle. The section
taken is 2.41111 distant from the centreline and is within the weak material zone.
The linear elastic model again behaves somewhat differently f?om the other three models.
The discrepancy is Iarger than that seen for the ta11 silo. The points where the curve of the
linear elastic model and those of other three models cross are not as noticeable as those of
the ta11 silo. There also exists some deviation in the stress distributions in the height fiom
0.8m to 4m above the base arnong the Janbu, Lade and Nelson, and poroelastic models.
Discontinuities in stress distributions around the wall-base corner are not so significant as
ti.̂ se seen k the !=il! si!̂ .
5.4.4.2 Plane Strain Analysis with Rigid Wall
Results from plane strain analysis for the squat silo are basically similar to those for the
ta11 silo, Figures 5- 18 and 5-19. However, the difference between the predictions using
the Boyce model and other four models in the bottom half of the silo height is broadened,
as is the difference in predictions using the linear elastic model and other four models for
the top half of the silo.
5.4.4.3 Axially Symmetric Analysis with Flexible Wall
The stress distributions adjacent to the flexible wall (thickness=lcm) developed in the
squat silo shown in Figures 5-20 and 5-21 are very slightly smaller than those in the squat
silo with ngid wall shown in Figures 5- 16 and 5- 17. Again, the wall with this thickness is
very stiE.
5.4.5 Predicted Wall Deflections
The silo wall deforms when subjected to b c solid load. Predicted horizontal wall
deflection for the ta11 silo with wall thickness Icm is given in Figure 5-22. The prediction
with the Boyce mode1 is noticeably smaller than the others. But ail predictions peak at a
height of O.8m above the base.
Reducing the thickness to OScm, the horizontal wall deflection for the ta11 wall shown in
Fimire 5-77 a l m n c t dnrihiec the d e f l e ~ t j ~ n nf the WB!! with thicherp !cm. l n the classical ' -- --A*--- -- ----- ---
theory, the expansion d of a ring under intemal pressure can be expressed as
where pr is the uniform interna1 pressure; R is the radius of the ring; E is the Young's
modulus of the ring material; and t is the thickness. Given the same pr, R and E, when
thickness r doubles, deflection d should halve. In fact, horizontal stresses, shear stresses
only have a few percent difference for the two different thicknesses (one is k m , the other
is OScm), as shown in Figures 5-14 and 5-15. Circumferential stresses also don? change
much. Therefore, the values of pr almost remain the sarne. The prediction shown in
Figure 5-23 is consistent with that theory.
For the squat silo with lcm thick wall, the shapes of the horizontal deflection cuves, as
illustrated in Figure 5-24, are also similar to those of horizontal stress distributions shown
in Figure 5-20.
The discontinuities in the horizontal deflections around the base of the silo wall are
apparent. But this is consistent with the discontinuities in the stress distributions around
that area.
5.4.6 Stress Contours
Contours of horizontal, shear and normal stresses for the rigid ta11 silo are shown in
c;-t*-c a L b U L C 3 d C - 3 q Cd, 5-76 UV -nJ -.- 5-27 ~ s ~ P c S ~ J ~ L ~ I - E i n c ~ the three m ~ ( i e ! ~ 0-ve c ~ n t ~ i ~ r -
similar to those predicted using the Lade and Nelson model, cornparisons are only made
between the predictions using the Lade and Nelson rnodel and those using the Boyce
model. It is seen that in the same position, horizontal stress and shear stress predicted
using the Boyce model are smaller than those using the Lade and Nelson model. To keep
the equilibrium of the vertical forces, vertical stresses at those positions predicted using
the Boyce model are larger than those using the Lade and Nelson model, as illustrated in
Figure 5-27. The peak horizontal and shear stresses occur below the middle of the silo
height. Figure 5-25 shows that in the same height, fkom the centreline to the wall,
horizontal stress decreases more rapidly for the Boyce model than for the Lade and
Nelson model. The effect of the wall constraint can be obsewed in Figure 5-26, where
shear stresses developed in the area adjacent to the wall gradually decrease to zero at the
centre of the silo. Formation of the arching mechanism c m be seen from the vertical
stress contours in Figure 5-27. Vertical stress decreases very quickly along the section at
the same height f?om the centreline to the side.
Figures 5-25, 5-29 and 5-30 show the contours of horizontal, shear and vertical stresses
for the rigid squat silo. The same observations c m be made as those developed in the ta11
rigid silo.
5.4.7 Plastic Zones
For analyses using the Lade and Nelson model, pIastic zones developed aIong the whole
of the weak stnp at the side of the silo, as well as at two locations in the remaining
rn~te-A: &P hzse c!nçe tc the centerline, the t ~ p si~rfrtce ~f the silo solid. These are
shown in Figure 5-31(a) for the ta11 silo and Figure 5-32(a) for the squat silo. An
examination of the pIastic zones developed after each layer is filled shows that the
surface layer always exceeds yield; however, folIowing the subsequent filling steps, most
of the stored solid return to the elastic regime due to the overburden, while some remains
plastic.
Much more extensive zones of plastic material are developed during analysis using the
Boyce model, as demonstrated in Figure 5-3 1(b) for the ta11 silo and Figure 5-32(b) for
the squat silo. This issue wilI be discussed fùrther in the following section.
5.1.8 Impact of Poisson's Ratio of bulk solid
The preceding sections show that the predictions using the Boyce model are quite
different from those using other elastic models. The key role that rnay play here is
Poisson's ratio of the b u k solid. Constant Poisson's ratio is used in the linear elastic,
Janbu, Lade and Nelson, and poroelastic models. However, variable Poisson's ratios of
stored material are expenenced during the filling process, as discussed in section 2.14.
Normally these values are smaller than the average Poisson's ratio value for al1 stresses,
since the stTess level for bulk solid stored in a silo is usually small. As a result, the
horizontal and shear stresses are smaller for the Boyce model than those with other
models, equation 2-54.
This may also be employed to explain the fact that more extensive plastic zones
developed using the Boyce model than the others. Low Poisson's ratio leads to low value
of lateral coeficiefit of earth pressure K, again according to equation 2-54. Therefore, the
ratio of major principal stress a, to rninor principal stress q becomes larger, resulting in
greater likelihood of shear failure.
In the following pararneûic study, the effect of varying Poisson's ratio is explored. In
Figures 5-33 and 5-31, horizontal and shear stress distributions are presented for varying
values of Poisson's ratio of the stored solid in analyses using the linear elastic model.
Srnaller values of Poisson's ratio do result in smaller horizontal stress and shear stress
distributions. In reality, Poisson's ratio of every particle of the stored solid is changing
during the filling process; therefore, the consequent wall pressure developed is a
cumulative outcorne, so is the fictional wall traction.
5.4.9 Effect of Weak Elements
Carefully checking the stress state at each integration point within those weak elements, it
is f o n d that the stress state reaches plastic failure immediately after each filling step. In
the whole filling process, the stress state in the weak elements is always in the plastic
regirne. This phenornenon results from not only the wall constraint, but also the filling
mode, i-e. the elements are filled in inclined layers. The weak elements are in the bottom
of the inclined layers, so once a layer is filled, these elements are subjected to self weight,
FE!! cco~tt-b-t~ s_n_d ific!-pd fcrwe fin= adjacent e!ements, which make them fail.
Because the stress states on these elements are travelling on the failure surface
throughout the filling process, the plastic strain is predominant in the total strain. ï h e
total strain wiIl be large consequently, which is expected since the particles adjacent to
the wall should have much more relative rnovement than those inside the silo.
Joint elernents are ofien used by researchers (e-g. Goodman et al., 1968; Ooi and Rotter,
1990) to rnodel the wall fiction characteristic between soils and structures. The joint has
independently specified shear and normal elastic stifkess and yield occurs according to
some failure criterion. Coupled shear and normal displacement occur during plastic
defornation. While joint elements are ofien used to rnodel soil-structure interaction, it is
relatively difficult to obtain their elastic stiffhess parameters. Furtherrnore, Joint elements
can develop severe numerical problems (typically when stifhess ma& terms for the
solid and structure differ by more than several orders of magnitude) and can lead to
compatibility problems when used with higher order solid elernents. Explicit modelling
like that developed by El Sawy and Moore (1996) is necessary when large deformation
and rotation of the solid-structure interface impact the behaviour, but greatly increase
computational effort. The use of weak elements is a relatively straightfonvard approach,
which maintains compatibility between solid and structure.
5.4.10 Effect of Stress Correction
As discussed in section 5.3.3.1, a stress correction technique is employed in al1 the
calculations. The stresses are corrected back to the yield surface at the end of eacb elasto-
correction. Figure 5-35 shows the horizontal stress predictions using the Janbu mode1
with stress correction and without stress correction. The difference in the peak values is
15%. This difference is significant. Furthermore, it is found that the ratio of shear stress
to horizontal stress does not obey the failure critenon without stress correction. Such
errors are even more severe for solutions obtained using the Boyce mode1 without stress
correction.
5.5 Summary and Conclusions
A senes of elasto-plastic finite element analyses have been performed on the silo filling
problem using the elastic models discussed in the preceding chapters. The progressive
element activation technique sirnulates the process of silo filling. Weak elements appear
to be effective in simulating the solid-wail interaction in association with the inclined
layer filling procedure. Using this technique, the stress states in the weak elements down
the wall are always on the plastic surface, resulting in the ratio of shear stress to
horizontal stress consistent with the tangent of the angle of wall friction- Correct
model!ing of wall fiction is facilitated by use of the stress correction technique; this
corrects stresses back to the yield surface afier each incrernental step.
The performances of these models in this problem faIl into three categories. The Boyce
model behaves quite differently from the other four models. The predictions of horizontal
stress and shear stress distributions adjacent to the wall as well as other positions using
the Boyce model are smaller than those using the other four models; this is balanced by
:Yc 5 c t thlf v - f i i ~ ~ ! S ~ P S S ~ S ;kco-~ 5 the c~rresp~n<l_ing positions. Much more
extensive plastic zones develop in the bulk solid during analysis using the Boyce model.
Stress contours reveal that stresses decrease much more rapidly for the Boyce solutions
than for the others from the centerline to the side at the sarne height. En the four other
models except Boyce, the linear elastic model behaves slightly differently to the other
three. No notable difference is observed among predictions using the Janbu, Lade and
Nelson, and poroelastic models, particularly for the ta11 silo.
The flexibility of the silo was found to have some effect on the pressure distribution.
Stress release due to the outwarci deflection of the wall Ieads to reduced values of
horizontal and shear stresses adjacent to the wall. The shapes of the stress distribution
curves for the Boyce mode1 are also slightly changed.
The differences between the stress distributions produced by the Boyce model and the
others become larger in the squat silo than in the ta11 silo. In the plane strain analysis, the
differences tend to be smaller compared to the axially syrnrnetric analysis; and the linear
elastic model produces pressure distributions that are noticeably different from the other
three.
The Poisson's ratio of the stored solid may have a key impact on the pressure
distributions. The stress levels for the bulk solid in silos are norrnally less than îOOkPa,
Figure 5-27. At such low stress levels, expenmentai results indicate that Poisson's ratios
for sand are both low and variable (e-g. Lade and Nelson, 1987). It was seen in Chapter 4
"e 5 q f c e EQ&! ir the one of fies- &e which may ht capahle o f capturing this
feature; the other four models assume a constant Poisson's ratio. The deflection of the
silo wall is small compared to the silo geometry; therefore the ratio of horizontal stress to
vertical stress is pnmarily controlled by a factor involving Poisson's ratio (Equation (2-
54)). The small value of Poisson's ratio also produces more extensive plastic zones as
failure surfaces are more easily reached when the horizontal stress to vertical stress rzho
Is smaller. This rnay be important during silo discharge. It reflects a significant difference
in the 'initial7 (prior to discharge) stress state and yield status.
It will not be clear how well these models predict silo problems until cornparisons are
made with full-scale test measurements. Only at that t h e can the true value of the Boyce
mode1 be evaluated.
Table 5-1. Elastic parameters of b u k solid for calculation
Modei
Boyce
Table 5-2. Plastic and other parameters of bulk solid for calculation
Parameters
n =0.62, K =404(kPa)", G , =673(kPa)"
Lade & Nelson
Janbu
-
( Dilation angle of b u k materid (vi 1
~ 0 . 2 , M=590, /2 =0.32
~ 0 . 2 , K ,=723, n =0.38
1 Friction angle of weak material 4 , / 18"
1 Dilarion angle of weak materiai yIv I
Atmospherk pressure ( P a )
Angle of repose 4,
cohesion (kPa)
Table 5-3. Parameters of steel silo wail
3 5 O
O. 1
Young's modulus v a ) Poisson's ratio
2x10~ 0-3 d
Figure 5-1. Geometrical dimensions of the tall silo.
Figure 5-2. Filling process for bulk solid in the tail silo.
one layer, inclined at the angle of repose
weak elements
smooth ngid boundary
rough rigid boundary
Figure 5-3. Finite element mesh for the ta11 silo with rough rigid wall:(a) bulk solid, (b) boundary conditions.
smooth rigid boundary
rough rigid boundary
I buik solid: six-noded continuum elernents
. ... -
waI1: eight-noded " continuum elernents
weak elements: six-noded continuum
Figure 5-4. Finite element mesh for the ta11 silo with flexible wall.
ngid boudary for ngid or a cohmn of eight-no&
4 - L c ~ i y i r C!C"":S fir &xi& ;vs!!
Figure 5-6. Cornparison of stress-strain predictions for isotropie compression; solutions from MENA with substepping and without substepping. Axiaiiy sy mrnetrïc problem, elastic behaviour.
Curve directly from AFENA - Smoothed curve
Figure 5-7. Smoothing approach used for reducing oscillation.
t -.-.-..* Linear Janbu - Lade & Nelson - Poroelastic
7- Boyce
il^***^
Figure 5-8. Horizontal stress distribution adjacent to the rigid \val1 (r=2.23m) in the tall silo. AxialIy symmetric analysis.
. - - - . - - - Liner Janbu - Lade & Nelson - Poroelastic
-- Boyce
Figure 5-9. Shear stress distribution adjacent to the rigid wall(~2.23rn) in the tail silo. Axially symmetric analysis.
- Lade & Nelson - Poroelastic -- Boyce / -
/ - -
Figure 5-10. Horizontal stress distribution adjacent to the ngid wall (x=2.23m) in the ta11 silo. PIane strain anaiysis.
Figure 5-1 1. Shear stress distribution adjacent to the rigid wall (x=2.23m) in the tall silo. Plane strain analysis.
-.-.-p.. Linear
Janbu - Lade & Nelson - Poroelastic -- Boyce
FigureS-12. Horizontal stress distribution adjacent to the flexible wall ( ~ 2 . 2 3 m ) in the ta11 silo. Axially symmetric anaiysis. Thidmess= 1 cm.
. . * - - * . - Linear Janbu - Lade & Nelson - Poroelastic
-- Boyce
Figure 5- 13. Shear stress distribution adjacent to the flexible wall (r-2.23rn) in the ta11 silo. Axially symmetric analysis. Thickness=lcm.
Rigid wall -- Flexible wall (t=l cm) .--...-- Flexible walI (t=OScm)
Figure 5-14. Horizontal stress distribution adjacent to the wail in the ta11 silo. Rigid wall and flexible wall. Axidy symmeûic analysis. Lade and Nelson model.
Rigid wall -- Flexible wall ( ~ l c m ) .-.--..- Flexible wal l (~0.5crn)
Figure 5-15. Shear sûess distribution adjacent to the wall in the ta11 silo. Rigid wail and flexible wall. Axially symmetric analysis. Lade and Nelson model.
. - - - - . * . Linear Janbu - Lade & Nelson - Poroelastic
-- Boyce
Figure 5- 16. Horizontal stress distribution adjacent to the ngid w a l l ( ~ 2 . 4 1 m) in the squat silo. Axially çymmetric analysis.
- + Linear
Figure 5- 17. Shear stress distribution adjacent to the rigid w a l l ( ~ 2 . 4 1 m) in the squat silo. Axially symmetnc analysis.
Figure 5- 1 8. Horizontal stress distribution adjacent to the rigid wall(x=2.4 1 m) in the squat silo. Plane strain analysis.
- - - . - . - Linear Janbu - Lade & Nelson - Poroelastic
-- Boyce
Figure 5-19. Shear stress distribution adjacent to the rigid wdl (x=2.41m) in the squat silo. Plane strain analysis.
. - - - . - * - Linear Janbu - Lade & Nelson - Poroelastic
-- Boyce
Figure 5-20. Horizontal stress distribution adjacent to the flexible wal1(1-2.4 1 m) in the squat silo. Axially symmetric analysis. Thickness=lcm.
Figure 5-2 I. Shear stress distribution adjacent to the flexible wail (FU lm) in the squat silo. Axially symmetric anaiysis. Thickness=lcm.
0.00000 0.00005 0.000 1 O 0.000 15 0.00020 0.00025
Horizontal deflection of the silo wall (m)
Figure 5-22. Wall deflection of the ta11 silo. Axially symmetric analysis. Thickness =l cm.
- .--.-.-- Linear
Janbu - Lade & Nelson - Poroelastic -- Boyce
0.0000 0.000 1 0.0002 0.0003 0.0004 0.0005
Horizontal deflection of the silo wall (m)
Figure 5-23. Wall deflection of the tall silo. Axially symrnetric analysis. Thickness=O.Scm.
Linear Janbu - Lade & Nelson - Poroelastic
-- Boyce
0.00000 0.00002 0.00004 0,00000 0.00008 0.00010 0.00012
Horizontal deflection of the wall (m)
Figure 5-24. Wall deflection of the squat wdi. Axially s y m m e ~ c andysis. Thickness=l cm.
Figure 5-23. Contours of horizontal seess orn predicted for the rigid ta11 silo: (a) Lade and Nelson model; (b) Boyce model.
Figure 5-26. Contours of shear stress a, predicted for the rigid tall silo: (a) Lade and Nelson model; (b) Boyce model.
Figure 5-27. Contours of vertical Suess a, predicted for the ngid tall silo: (a) Lade and Nelson model; (b) Boyce model.
Figure 5-28. Contours of horizontal stress predicted for the rigid squat silo: (a) Lade and Nelson model; (b) Boyce model.
Figure 5-30. Contoun of vertical stress o, predicted for the ngid squat silo: (a) Lade model; (b) Boyce model.
Figure 5-3 1. Plastic zones within b u k solid for the ngid ta11 silo: (a) Lade and Nelson model; (b) Boyce model.
Figure 5-33. Impact of Poisson's ratio on the horizontal stress adjacent to the rigid wall. Linear elastic model.
Figure 5-34. Impact of Poisson's ratio on the shear stress adjacent to the rigid wall. Linear elastic model.
- Stress correction No stress correction
O 10 20 30 40 50
Figure 5-35. Horizontal stress distribution adjacent to the rigia wall in the tail silo. Janbu model. With stress correction and without stress correction.
CHAPTER 6
SUMMARY AND CONCLUSIONS
The objective of this study has been to investigate nonlinear elastic models for granular
material like sand and their performance in analysis of the silo filling problem. A
literature review was fmtly presented in Chapter 2 examining contributions to the
nonlinear eIastic behaviour of soils with focus on granular material, and also elastic
models proposed by various workers to charactenze soi1 properîïes. Two major issues
discussed were Poisson's ratio and the need for energy conservation. Five elastic modeIs,
linear elastic, Janbu, poroelastic, Lade and Nelson, and Boyce were reviewed in detail.
A summary of the work performed and the conclusions of this study will be made
examining each of the following four aspects in tum: 1) characteristics of the five elastic
models; 2) implementation of the Boyce model; 3) parameter evaluation of these five
models; 4) performance of these models in the modelling of silo filling. Finally,
suggestions will be made regarding future research.
6.1 Five Elastic Models for Granular Material
The simplest elastic model for granular matenal is linear elasticity. Two independent
constants, Young's rnodulus and Poisson's ratio, are needed. This model has limited
applicability for bulk solids, as it cannot describe many feahxes of these materials. This
model is adopted in silo analysis (e.g. Dickinson and Jofned, 1987; Ooi and Rotter, 1990)
for its simplicity and the time saved during numerical computations.
The Janbu mode1 descnbes Young's modulus of granular solids as dependent on minor
principal stress or first stress invariant raised to a power less than one. This model has
been widely used (e.g. Rotter et al., 1995), because it is also simple, and it captures a key
feature of particulate soi1 behaviour. However, it utilizes a constant Poisson's ratio
(gmnular matenal exhibits varying Poisson's ratio during loading, particularly at low
stress levels) and it violates the principle of conservation of energy during loading-
unloading cycles.
Young's rnodulus in the Lade and Nelson model is expressed as a power fùnction
involving both the first and the second stress invariants. Although this model is energy
-nne-mr-t;~~o qnrl -ln rrrn~r;de crr\r\d r ~ c t v l t ~ bA 2 9 ~ ~ q p ! i C & ~ P , ~ [ ~ L ~ ~ C ! Q I S L T ~ Sm, b V L 1 3 * 1 V U C A V CI C U L U b U I A F I U V L U C 3 - V u LI- . - -
1992), the use of a constant Poisson's ratio may not produce good results in other
applications.
The poroelastic model is generally used to provide the elastic part of predictions obtained
using the Cam-Clay model (Roscoe and Burland, 1968; Atkinson, 1981). The elastic
moduli K and G are dependent on both the fint stress invariant and void ratio. This model
has also been widely used. Like the Janbu model, it is not energy conservative and uses a
constant Poisson's ratio.
The hyperelastic model proposed by Boyce appears to capture a number of important
features of granular materials. Firstly, the bulk and shear moduli of this model are
proportional to the normal stress raised to a power less than one, which is consistent with
both experimental observations and theoretical considerations. Secondly, deviatoric stress
is coupled in the expression of bulk moduius, not only making this model energy
conservahve, but aiso making it consistent with the experimental observations. Thirdly
and perhaps most importantly for the silo filling problem, Poisson's ratio is not explicitly
d e k e d in this model. The calculated Poisson's ratio obtaïned fkorn bulk and shear
response in a cornputer simulation of triaxial compression varies in a manner consistent
with the test measurements reported by Lade and Nelson (1987). This is a preliminary
finding. Further testing should be conducted regarding this feature.
6.2 Implementation of the Boyce Mode1
A tmgentiai form of rne sness-snain reiacions For h s Soyço tiiudeizl i i ~ be~ri ùeiivd
using an approach through octahedral-based stress-strain relations (Murray, 1979). This
tangential stress-strain matrix c m be directly incorporated into finite elernent codes.
Although it appeared at first inspection to be non-symmetric, a closer examination
revealed that the matrix is symmetric, as required for a hyperelastic model. Since the
rnatrix form of this constitutive reIation is both stress and strain dependent, elastic strains
and plastic strains must be separated during elasto-plastic modeliing. In an incrernental
finite element analysis, the cumulative elastic strain must be stored and updated after
each increment. If a stress correction technique is used to maintain stresses in elasto-
plastic solid at the yield surface, associated elastic strains must also be corrected.
Three numerical problems can develop when thïs mode1 is used. Two variables,
octahedral strain y,, (or deviatoric stress q) and mean stress p, appear in the denominator
of various terms in the constitutive relations. Diff~culties are encountered when these hvo
variables are zero. Minimum values of q and p are used respectively (For details of
selection of q,, and p,, see Chapter 3). It is also important to choose an adequate strain
or stress increment for the nonlinear analysis. Incremental strain or stress must balance
the need for accuracy and economy. These choices are dependent on the material
parameters and the value selected for p,,.
The mode1 has been implemented into the finite element code named AFENA. A series
of numencal tests including isotropic compression, triaxial compression and one
dimensional compression have been performed. Cornparisons were made between results
f r î ~ smzl! rk?ilsrti~n tests using direct inte,mi_-'~n and those from AFENA. The two
predictions are in good agreement.
6.3 Parameter Evaluations for the Five Models
Constitutive relations have to be verified and parameters have to be selected with respect
to the engineering application being examined. During silo filling, the stress-strain
history of the bulk solid is similar to I-D consolidation (zero lateral saain), since the
deflection of the silo wall is small compared to the geometry of a silo. The stress level is
generally less than 400kPa. Several isotropic compression tests and triaxial compression
tests with constant stress ratios are suitable for obtaining parameters for the Boyce mode1
for examination of silo filling. Three loose sands fiom the literature, Sacramento River
Sand, Santa Monica Beach Sand and Ottawa sand were selected for parameter evaluation.
No set of test results was found in the literature which can be used to obtain a cornplete
set of the Boyce model parameters for the same material. However, the magnitude of
these parameters can still be obtained. Following consideration of the pubiished data, a
complete set of parameters for the Boyce mode1 was estimated for each of the three sand
materials.
Parameter evaluation for the other four models was also performed for each of these three
loose sands. Special attention was paid to evaluating Poisson's ratio for the Lade and
Nelson model, because these values of Poisson's ratio were used for a11 the four mode1s
except Boyce.
Q W Y P C C ; ~ ~ on-lygep e=$~@ fer In! D-QQ!~ ?Q &tain the hest fitting &.Ca* -diI."&& YU-.
parameters to those published experimental data. Unloading portions of these
experiments were chosen, since unloading is more purely elastic than reloading in the
opinion of many researchers.
6.4 Application in Silo Filiing
The performance of each of these five elastic models in seven categones of finite element
analyses of silo filling has been exarnined. Part of the problem definition was originally
developed by Rotter et al. (1995). In the present study, both plane strain analysis (planar
silos) and axially symmetnc analysis (cylindncal axially symmeûic silos) were
conducted. A ta11 silo and a squat silo were snidied. Both ngid wall and flexible wall
were considered; the former was treated as a rough ngid boundary, while the Iater was
modelled as a saip of eight-noded rectangular elements. Six-noded elements were used to
model the bulk solids, which were divided into two parts: the normal part for the majority
of the material and a 'weak' part with limited thickness along the wall. The intemal
fnction angle and dilation angle of the weak elements were taken as the same as the wall
fnction angle. As to the rnatenal properties, data for Loose Ottawa Sand was selected and
the parameters of the five elastic models for this sand were employed. If the wall
flexibility was taken into account, the wall rnatenal was steel. A Mohr-Coulomb failure
criterion with an associated flow mle was used during modelling of plastic response in
these elasto-plastic analyses.
The filling process was modelled with progressive element activation from the bottom to
tep ~1u,.-f~::e cf s i b l ~ k g 2 !:>or hy !zy?r ' ~ c ~ ~ ~ ~ r ~ i c c ' o c b i q u e &ch lwer is a
stip of elements inclined at the angle of repose. This approach simulates the process of
silo filling and consequently the results are different from others without this feature (e-g.
Holst et a[., !999). A stress correction technique, which also corrects associated elastic
strain changes, was used during the calculation. The use of this technique guaranteed that
the stress states of points, which have yielded, remain on the yield surface at the end of
each elasto-plastic loading increment. Thus, the ratio of shear stress and horizontal stress
at any point within the weak elements, which are always in the plastic state during the
whole filling process, is the tangent of the wall fiction angle. Appropriate incremental
load steps were employed in al1 analyses. For analyses using the Boyce model, an
improved integration scheme was adopted, where the incremental strains calculated fiom
the incremental displacements are divided into a number of substeps to better integrate
the constitutive relations to obtain incrernental stresses.
It can be seen fiom these analyses that: the predictions obtained using the £ive elastic
models fa11 into three categories. The Boyce model behaves quite differently from the
other four models; while among these four models, results produced by the linear elastic
model are slightly different from the other three. The three models, Janbu, Lade and
Nelson, and poroelastic, generate very similar results, particularly for the ta11 silo.
The predictions of horizontal stress and shear stress distributions adjacent to the wali as
well as the other positions obtained using the Boyce model are smaller than those
resulting fiom the other four models. To maintain force equilibrium, vertical stresses
ii.trese in the cere~px!&g p o s i t i o ~ ~ Moch more extensive plastic zones develop in
the bulk solid during analysis using the Boyce model. Stresses decrease much more
rapidly for the Boyce solution than for the othen &om the centerline to the wall at any
given height.
Silos with more ffexibIe walls tend to be subjected to decreased horizontal and shear
stresses, because stress release occurs as a result of the outward deflection of the wall. In
the squat silo, the difference between the stress distributions produced by the Boyce
model and the others becomes larger than in the tail silo. PIane strain analysis produces
smaller differences among these models than axially symmetric analysis. The shapes of
the stress distribution curves are mostly the same, with some difference among different
rnodels, silo geometries and analysis types.
Poisson's ratio may play a key role on both the pressure distributions and plastic zones
developed in the silo. The stress histories experienced in silos mean that the ratio of
horizontal stress to vertical stress is primarïly controlled by a factor involving Poisson's
ratio. Only the Boyce mode1 in these five models is capable of capturing the feature that
Poisson's ratios for the b u k solid are both low and variable at the low stress levels that
exist in silos.
No experimental data for silo f i l h g pressures were examined in this study. The results
must evaluated relative to future available full-scale test results. in particular, the validity
of the Boyce model for the silo filling problem must be examined using field
mes?s-me*. uu-x:, the c~x!-siczs ri,r~.m h m the ccmpmrisms hehveen lhese
models could provide a better understanding of the properties of the granular bulk solid
and of their interaction with the silo wall; moreover, they could be used as a valuable
reference for analyzing experimental data and consequently enhance silo design.
6.5 Recommendations for Future Work
The numerical modelling of granular bulk solids in silos is a challenging task. However,
this problem is of great industrial significance. This study presents the predictions of
symmetrical filling pressures. Although few silos fail during symmetncal filling, it is a
very important starting point for al1 O ther numerical simulations. Knowing the stress state
in a silo d e r filling is vital for prediction of the discharge process, which were the
conimonly causes of structural failures. The 'constmction' and 'excavation' techniques
mentioned in Chapter 5 c m be used to simulate other processes such as eccentric filling,
symrnetric discharge and eccentrïc discharge. nie effect of wall imperfections can also
be examined using finite element anatysis. Eight-noded elernents are employed in this
study to simulate silo walls; shell elements could be used in fiiture studies to examine
wall during silo filling and discharge. Cornparisons can be made between numerical
predictions and pubfished filling pressure measurements.
Constitutive modelling of ~ranu1a.r bullc solids coufd be m e r improved using plastic
models which capture smin hardening such as Lade's model (1977) instead of the Mohr-
Coulomb failure criterion. More expenmental work should be performed to evaluate
parameters for the Boyce model, as well as to check the performance of this model in
predicting Poisson's ratio.
Appendix A
CONSERVATION OF ENERGY FOR THE BOYCE MODEL
The Boyce model is an energy-conserving non-linear elasticity model, which means
energy is neither generated nor dissipated in closed-loop stress or stain paths. This can be
proved as follows.
Stress invariants 1, and J, are defined as
where a,, o, and - are normal stresses, and
where r _ , r, and r, are shear stresses, s, , s,, s= , Sc, %=, s, are components of
deviatoric stress tensor.
From (A- 1) and (A-2), incremental values are given by
dI* = dqr i do, + da,,
1 = ?[(ox - do. - dqV) + ( T ~ - %)(dqv - d- ) + (q - ~ , ) ( d - - do,)]
The definition of deviatoric stress in Cartesian form is
Substituting (A-5) into (A+,
dl, = sxdax + sVdov - - + 5dq. + 2(r,dr, + r&, + r,dr,).
For strain invariants 1' and J;, incremental values can also be derived
l =Ex+ E , + - ,
dl' = dgx c dgv + d s ,
where E,, gV, 5, E*, E,= and E, are components of strain tensor , and ex, ey , ez, e.*,
e , and e, are components of deviatoric strain tensor. The definition of deviatoric stress
in Cartesian form is
Stress tensor a, can be expressed in tems of the volumenic and deviatoric strains (see
equations (3-20), (3-2 1 ), (3-22) and (3-23) in chapter 3),
q, = ad3 + 2 ~ e , ,
where K and G are bulk modulus and shear modulus.
Surnrning the first three equations of (A-12)
1, = xi;,
Also, sican be expressed as
(A- 12)
(A- 13)
(A- 1 4)
Substituting (A- 14) into (A-2) and (A-91,
J, = ~ G ' J ; . (A- 15)
The elastic work per unit volume along stress path ACB in Figure A-l is calculated as
(A- 1 6)
Substituting (A- 10) and (A- 12) into (A- 17),
Substituting (A-8) and (A- 10) into (A- 18),
dW = ~ l ; d l ; + 2 ~ d l i .
Substituting (A- 13) and (A- 15) into (A- i 9),
Therefore, (A- 16) c m be written as
Similarly, for stress path ADB,
ADB
(A- L 8)
(A- 19)
According to the principle of conservation of energy, the work must be independent of
the path, W d c B = W4,. For the cycle ACBDA,
The integral in (A-23) can be written as
N and Q= - , which means P and Q can be derived from a potential hnction.
@
(A-23) is satisfied.
The incremental energy d W can be written as
where C and f i are the independent variables. Since dl2 = 2mfi, (A-25)
becomes
and
Since 2w a'w
d L d G = daaI , the following constraints on the expression for K and G are
obtained
As long as K and G satisQ equahon (A-29), the work is path independent.
For the Boyce model,
4 where ,û= (1- n)-. 6G,
From (A-32) and (A-33),
Therefore,
Substituting (A-30), (A-32) and (A-34) into the left side of equation (A-29),
Substituting (A-3 1 ), (A-33) and (A-38) into the right side of equation (A-29),
Since (A-39) is equal to (A-40), equation (A-29) is satisfied. Thus the work is stress path
independent. Similarly, it c m be proved the work is also strain path independent. In
summary, the mode1 is energy conservative.
Figure A-1 . Stress paths fiom A to B shown in r, - diagram
PROOF OF SYMMETRY OF THE CONSTITUTIVE
MATRIX FOR THE BOYCE MODEL
Starting h m the basic equations (3-8) and (3-2),
Rearranging (B- 1 ),
Multiplying both sides of (B-2) with (1-n) and then rnultiplying both the numerator and
denominator of the rïght-hand side by KI,
Note that ,û= (1 - n) K, / 6G,, therefore (B-3) can be written as:
Multiplying the lefi-hand side of (B-4) with Pa and the right-hand side wirh P (7)""
Substituting (3- 12), (3- 13) and (3- 16) into (B-6),
Multiplying both sides (B-7) with 2& produces
3(- - 3G,
D,' )
3 Gr dG Note that a = 1
- (1 - n)G,p,"pVn (Equations (3-2) (Equation (3-19)), rn = - - -- K, D,' d~ 3 Gr
1 K A
-- 4b
K, D,' - - 3 Gr
and (3-24)) and B = (Equations (3- L9) and (3-37)). It is now 3& 3J2y,
simple to show that 2arn = B .
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