Empirical relations for earthquake response of€¦ · sincere thanks and appreciation to Prof. Dr. Bishal Nath Upreti, Tribhuvan University in Nepal for the great support and encouragement

Empirical relations for earthquake response of

slopes

Subodh Dhakal Delft, The Netherlands March, 2004

Empirical relations for earthquake response of slopes

By

Subodh Dhakal Thesis submitted to the International Institute for Geo-information Science and Earth Observation in partial fulfilment of the requirements for the degree of Master of Science in Geo-information Science and Earth Observation, Geological Engineering specialization. Degree Assessment Board: Dr. H.R.G.K. Hack Chairperson and supervisor, Head Section Engineering Geology, ITC Delft M. Huisman, M.Sc. Supervisor, Section Engineering Geology, ITC Delft A. Scarpas, M.Sc. External Examiner, Section of Structural Mechanics, TU Delft P.M. Maurenbrecher, M.Sc. External Examiner, Section of Applied Earth Sciences, TU Delft S. Slob, M.Sc. Section Engineering Geology, ITC Delft Drs. J.B de Smeth Acting EREG – Programme Director, ITC Examination executed on: March 05, 2004 ITC-TU Delft Delft, The Netherlands

INTERNATIONAL INSTITUTE FOR GEO-INFORMATION SCIENCE AND EARTH OBSERVATION

ENSCHEDE, THE NETHERLANDS

I certify that although I may have conferred with others in preparing for this assignment, and drawn upon a range of sources cited in this work, the content of this thesis report is my original work. Signed: Subodh Dhakal Date: 09/03/2004

Disclaimer This document describes work undertaken as part of a programme of study at the International Institute for Geo-information Science and Earth Observation. All views and opinions expressed therein remain the sole responsibility of the author, and do not necessarily represent those of the institute.

ACKNOWLEDGEMENTS I am extremely grateful to ITC and NFP, as organizations, for providing me opportunity to carry out this research. I would like to extend my special and sincere gratitude to Dr. Robert Hack for giving fruitful suggestions on selecting the research topics as well as for invaluable supervision, support, and encouragement throughout the research period. I am highly grateful to Ir. Marco Huisman for his close and continuous guidance throughout the research period in all the affairs related with the research. I would like to acknowledge all the ITC staff members, here in Delft as well as in Enschede, for their direct or indirect support throughout the course. My special gratitude to Dr. Paul van Dijk, Program Director of EREG, for helping me to get opportunity to continue my M.Sc. research. Discussion with him during first six months in Enschede regarding my interest and my work back in my home univer-sity was supportive and unforgettable. Discussion with Ir. Siefko Slob during data interpretation phase was much helpful and my sincere appreciation to him. The willingness of Ir. Wolter Zigterman throughout the course to advise in many aspects is appreciable. Discussion with PhD Zhu Sicai at dif-ferent phases of work was encouraging. I am grateful to all the TU Delft staff members for their support to make the research successful. The discussion with Dr. Dominique Ngan-Tillard, M.Sc. P.M. Maurenbrecher and M.Sc.Laurent Gareau about their research projects was encouraging and appreciable. My special thanks and appreciation to Ir. W. Verwaal for his support at need during the study period. I appreciate the close relationship with all the ITC and TU Delft colleagues, it was unbelievable. I am also grateful to my employer, Tribhuvan University, for nominating me to pursue this course. My sincere thanks and appreciation to Prof. Dr. Bishal Nath Upreti, Tribhuvan University in Nepal for the great support and encouragement to take part in this course. I would also like to thank my seniors as well as colleagues in the Department of Geology, Tri-chandra campus, Tribuvan University. I am grateful to my family especially my father and mother for their support and encouragements over the years despite their incredible hard time. I want to owe a great debt to my wife Kalyani for sacrific-ing many things and giving moral support throughout the study period.

ABSTRACT

The finite difference program FLAC is used to find the relationships between the slope geometry, earthquake input signals (mainly frequency of the wave) and the material properties of the slope on the amplification of vibration on the surface. A sinusoidal wave with acceleration of 1 m/s2, input fre-quency of 5 Hz and amplitude of 1 m is applied for the duration of 0.25 sec in the first three models used for the calculations. For the fourth model, however influence of varying input frequencies is also investigated applying the input frequencies of 3 Hz, 5 Hz, 10 Hz and 15 Hz. Other material properties are varied whenever necessary to examine the response of that particular material properties. The re-sponse of three models are first observed before the simulation of the last model, the response of which is to be used for finding out the relationship between the input parameters and the amplification. The first three models revealed that the edge of the slope crest is generally more vulnerable for ampli-fication than farther away from it as in most of the cases the magnitude of amplification at the edge of the slope crest is more than that farther away. Regarding the influence of the slope angle, friction an-gle, and the material cohesion, the first three models show that if the slope angle is higher than the friction angle, the slope starts to fail when the material cohesion is not enough to resist the slope. In addition, the response of the slope after it fails is very different to that before it fails. Simulation with the fourth model show that higher slopes are amplified most by the lower input fre-quency whereas the reverse is true for the smaller slopes. The overall magnitude of the amplification is maximum with input signals of higher frequency and lower slope heights. The horizontal amplification as much as 17 are obtained for the normal limestone slope with 20 m height when an input signal of 15 Hz frequency is applied. The amplification peaks repeat for the same input frequency but for different slope heights. It is considered reflection of the harmonic effect. The exact effect of bulk modulus how-ever could not be found out, as the relationship could not be explained properly. A clear resonance effect is seen when the amplification is plotted against the shear modulus as well as against the ratio of the slope height to wavelength of the input signals. The slope height to wavelength ratio of 0.07 to 0.23 is seen to be most vulnerable for seismic amplification. The horizontal amplifications in the order of 6.5 are obtained for that range of slope height to wavelength ratio. The generation of standing waves at certain harmonic frequencies could be the reason behind such a high amplification. An em-pirical function is recommended to show the relation of slope height, shear wave velocity and the input frequency on amplification. This type of relationship is believed to have great importance for seismic microzonation study as the results might be used along with the digital elevation model for separating different zones of hazard levels. However, one should be aware that the relationship is for the particu-lar combination of material properties and slope geometry and the generalization for all type of mate-rials might give misleading results. Keywords: FLAC, earthquake, dynamic modelling, amplification, horizontal acceleration, slope height, material properties, input frequency

TABLE OF CONTENTS

CONTENTS PAGES 1. Introduction ................................................................................................................................. 1

1.1. Research Motivation...............................................................................................................1 1.2. Research Hypothesis ..............................................................................................................2 1.3. Research Objectives ...............................................................................................................2 1.4. Methodology ..........................................................................................................................4

1.4.1 Literature study...............................................................................................................4 1.4.2 Learning Softwares.........................................................................................................4 1.4.3 Generation of arbitrary slope geometry and assigning material properties ....................4 1.4.4 Introducing design earthquake signals and writing FLAC script ...................................5 1.4.5 Modelling for Dynamic Loading....................................................................................5 1.4.6 Reading data obtained from calculation from computer screen .....................................5 1.4.7 Data analysis and interpretation .....................................................................................6 1.4.8 Result formulation..........................................................................................................6

1.5. Scope of the study ..................................................................................................................6 1.6. Limitations of the present study .............................................................................................6 1.7. Structure of the thesis .............................................................................................................7

2. Theoretical background on ground response analysis............................................................. 8

2.1. Introduction ............................................................................................................................8 2.2. Types of seismic waves ..........................................................................................................8 2.3. Seismic waves and materials interaction .............................................................................10 2.4. Factors influencing ground response....................................................................................11

2.4.1 Source effect.................................................................................................................12 2.4.2 Path effect.....................................................................................................................12 2.4.3 Site effects ....................................................................................................................13

2.4.3.1 Topography and geometry of the ground .................................................................13 2.4.3.2 Material properties ...................................................................................................15 2.4.3.3 Other effects .............................................................................................................16

2.5. Methods for estimating site effects.......................................................................................16 2.5.1 Experimental Methods ................................................................................................16 2.5.2 Numerical methods ......................................................................................................17

2.5.2.1 Simple methods .......................................................................................................17 2.5.2.2 Advanced methods ..................................................................................................18

3. Dynamic loading and numerical modelling ............................................................................ 19

3.1. Dynamic Loading due to earthquake....................................................................................19 3.2. Numerical Modelling Techniques ........................................................................................19

3.2.1 Fast Langrangian Analysis of Continua (FLAC) .........................................................22 3.2.1.1. Constitutive Models ............................................................................................22 3.2.1.2. Dynamic analysis in FLAC .................................................................................23

3.2.1.3 General Methodology for solving dynamic problems .............................................30 4. Numerical simulation for dynamic loading using FLAC....................................................... 33

4.1. Introduction ..........................................................................................................................33 4.2. Slope geometry.....................................................................................................................33 4.3. Discretization and generation of mesh .................................................................................34 4.4. Model Formulation...............................................................................................................34

4.4.1 Grid generation.............................................................................................................34 4.4.2 Constitutive models ......................................................................................................35 4.4.3 Material properties .......................................................................................................35 4.4.4 Initial equilibrium.........................................................................................................35 4.4.5 Boundary conditions ....................................................................................................36 4.4.6 Dynamic Input..............................................................................................................36 4.4.7 Dynamic Damping .......................................................................................................36 4.4.8 Output...........................................................................................................................37

5. Ground response data as observed from FLAC Calculation ................................................ 38

5.1. Introduction ..........................................................................................................................38 5.2. Description of slope geometries ...........................................................................................38

5.2.1 Slope model 1 and 3 .....................................................................................................38 5.2.2 Slope model 2...............................................................................................................39

5.3. Variation of ground response with the friction angle...........................................................40 5.3.1 Slope model 1...............................................................................................................40 5.3.2 Slope model 2...............................................................................................................40 5.3.3 Slope model 3...............................................................................................................41

5.4. Variation of ground response with the material cohesion ....................................................42 5.4.1 Slope model 1...............................................................................................................42 5.4.2 Slope model 2...............................................................................................................43 5.4.3 Slope model 3...............................................................................................................44

5.5. Variation of ground response with the slope angle ..............................................................45 5.5.1 Slope model 1...............................................................................................................45 5.5.2 Slope model 2...............................................................................................................46 5.5.3 Slope model 3...............................................................................................................46

5.6. Variation of ground response with the distance from the edge of the slope crest ................47 5.6.1 Slope model 1...............................................................................................................47 5.6.2 Slope model 2...............................................................................................................48 5.6.3 Slope model 3...............................................................................................................48

5.7. Important conclusions of the calculation results obtained from first three slope models.....49 6. Relation of amplification with earthquake input signals, material properties and slope geometry............................................................................................................................................. 51

6.1. Introduction ..........................................................................................................................51 6.2. Bulk modulus versus amplification ......................................................................................52

6.2.1 General shape of function.............................................................................................55 6.2.2 Influence of slope height and input frequency .............................................................58

6.2.3 General relationship between slope height, input frequency and amplification...........65 6.2.4 The rate of change of amplification with bulk modulus...............................................66

6.3 Shear modulus versus amplification.....................................................................................68 6.3.1 Ratio of slope height to wavelength versus amplification............................................68 6.3.2 Shear modulus versus amplification.............................................................................69 6.3.3 Relation of shear modulus on amplification for different wavelengths .......................70

6.4 Modelling the function to show the variation of amplification with material properties, slope height and earthquake input signals .................................................................................................72

7. Conclusion and recommendation............................................................................................. 79 References .......................................................................................................................................... 81 Appendices

Appendix A-1: Script example used for the calculation (script for calculation no. 684, Appendix C-9) ......................................................................................................................................................85 Appendix A-2: Sample example of the screen capture of calculation results in FLAC (1) .............88 Appendix A-3: Sample example of the screen capture of calculation results in FLAC (2) .............89 Appendix B-1: The input and output parameters for slope model 1 ................................................91 Appendix B-2: The input and output parameters for slope model 2 ................................................93 Appendix B-3: The input and output parameters for slope model 3 ................................................95 Appendix C-1: Calculation matrix used for simulation (changes in bulk modulus, frequency 3 Hz)97 Appendix C-2: Calculation matrix used for simulation (changes in bulk modulus, frequency 5 Hz)...........................................................................................................................................................99 Appendix C-3: Calculation matrix used for simulation (changes in bulk modulus, frequency 10 Hz).........................................................................................................................................................101 Appendix C-4: Calculation matrix used for simulation (changes in bulk modulus, frequency 15 Hz).........................................................................................................................................................103 Appendix C-5: Calculation matrix used for simulation (changes in bulk modulus, slope angle 35.5° and frequency 3 Hz). ......................................................................................................................105 Appendix C-6: Calculation matrix used for simulation (changes in bulk modulus, slope angle 35.5° and frequency 5 Hz). ......................................................................................................................107 Appendix C-7: Calculation matrix used for simulation (changes in bulk modulus, slope angle 35.5° and frequency 10 Hz). ....................................................................................................................109 Appendix C-8: Calculation matrix used for simulation (changes in bulk modulus, slope angle 35.5° and frequency 15 Hz). ....................................................................................................................111 Appendix C-9: Calculation matrix used for simulation (changes in shear modulus).....................113 Appendix C-10: Calculation matrix used for simulation (changes in wavelength)........................115 Appendix D-1: Bulk modulus versus maximum amplification for different slope heights, input frequencies and slope angles (full data can be seen on calculation numbers 1 to 673 in appendices C-1 to C-8). ....................................................................................................................................117 Appendix D-2: Bulk modulus versus maximum amplification for different slope heights, input frequencies and slope angles (slope angle 35.5º) ...........................................................................119

Appendix D-3: Comparison of maximum amplification for the slope angle of 25.5º and 35.5º from the plot of slope height versus maximum amplification for different input frequencies................121

LIST OF FIGURES

FIGURES PAGES Figure 1.1. Flow chart of the methodology adopted for the study. .......................................................3 Figure 2.1. Types of seismic waves (source: Rey et al, 2001). .............................................................9 Figure 2.2. Interaction of waves in the interface between two types of materials (source: Rey et al,

2001) ............................................................................................................................................11 Figure 3.1. Types of numerical modelling methods (source: extracted from Jing, 2003)...................20 Figure 3.2. Types of dynamic loading boundary conditions available in FLAC (source: ITASCA,

2000). ...........................................................................................................................................26 Figure 3.3. Variation of normalized critical damping ratio with angular frequency (source: .............29 ITASCA, 2000). ...................................................................................................................................29 Figure 3.4. General solution procedure in FLAC (source: ITASCA, 2000) .......................................31 Figure 4.1. (a) Overlayed quadrilateral elements used in FLAC; (b) typical triangular element ........34 with velocity vectors; (c) nodal force vector (source: ITASCA, 2000). ..............................................34 Figure 5.1. General sketch of the geometry of slope model 1 and 3. ..................................................39 Figure 5.2. General sketch of geometry of the slope model 2.............................................................39 Figure 6.1. Geometry of the slope model used for the numerical calculation.....................................51 Figure 6.2. Bulk modulus versus amplification for frequency of 3 Hz (with slope angle 25.5 degree).

......................................................................................................................................................53 Figure 6.3. Bulk modulus versus amplification for frequency of 15 Hz (with slope angle 25.5 degree).



......................................................................................................................................................54 Figure 6.6. Schematic diagram showing general shape of the function for bulk modulus versus

amplification.................................................................................................................................55 Figure 6.7. Bulk modulus versus maximum and minimum amplifications for the input frequency of 3

Hz, slope angle 25.5 degree..........................................................................................................56 Figure 6.8. Bulk modulus versus maximum and minimum amplifications for the input frequencies of 5

Hz, slope angle 25.5 degree..........................................................................................................57 Figure 6.9. Bulk modulus versus maximum and minimum amplifications for the input frequencies of

10 Hz, slope angle 25.5 degree.....................................................................................................57 Figure 6.10. Bulk modulus versus maximum and minimum amplifications for the input frequency of

15 Hz, slope angle 25.5 degree.....................................................................................................58 Figure 6.11. Variation of amplification with slope height for different input frequencies with slope

angle of 25.5º, input frequency 3 and 5 Hz. .................................................................................59 Figure 6.12. Variation of amplification with slope height for different input frequencies with slope

angle of 25.5º, input frequency 10 and 15 Hz. .............................................................................59 Figure 6.13. Variation of amplification with slope height for different input frequencies with slope

angle of 35.5º, input frequency 3 and 5 Hz. .................................................................................61 Figure 6.14. Variation of amplification with slope height for different input frequencies with slope

angle of 35.5º, input frequency 10 and 15 Hz. .............................................................................61 Figure 6.15. Variation of maximum amplification with slope height for different slope angles. .......62 Figure 6.16. Variation of maximum amplification with the input frequencies for slope angle of 25.5º,

slope height of 10 m, 20 m and 25 m. ..........................................................................................63 Figure 6.17. Variation of maximum amplification with the input frequencies for slope angle of 25.5º,

slope height of 30 m, 35 m and 40 m. ..........................................................................................63 Figure 6.18. Variation of maximum amplification with the input frequencies for slope angle of 35.5º,

slope height 10 m, 20 m and 25 m. ..............................................................................................64

Figure 6.19. Variation of maximum amplification with the input frequencies for slope angle of 35.5º, slope height 30 m, 35 m and 40 m. ..............................................................................................64

Figure 6.20. General sketch for the relationship between the maximum amplification and input frequency for different slope heights............................................................................................65

Figure 6.21. General sketch to show the relationship between the maximum amplification and slope heights for different input frequencies. ........................................................................................66

Figure 6.22. Variation of the slope of the shape function with the input frequency for the slope angle of 25.5º. ........................................................................................................................................67

Figure 6.23. Variation of the slope of the shape function with the input frequency for the slope angle of 35.5º. ........................................................................................................................................67

Figure 6.24. Ratio of slope height to wavelength versus amplification for the slope angle of 25.5º and input frequency of 5 Hz................................................................................................................69

Figure 6.25. Formation of standing waves due to interference of waves. N in the figure denotes node and AN denotes anti node (source: Henderson, 1998). ................................................................69

Figure 6.26. Shear modulus versus amplification, slope angle of 25.5º and frequency 5 Hz. ............70 Figure 6.27. Shear modulus versus amplification for different input wavelengths (hence varying

frequency, slope height of 20 m)..................................................................................................72 Figure 6.28. Observed versus modelled amplifications for different slope heights (frequency 3 Hz) 76 Figure 6.29. Observed versus modelled amplifications for different slope heights (frequency 5 Hz) 76 Figure 6.30. Observed versus modelled amplifications for different slope heights (frequency 10 Hz)77 Figure 6.31. Observed versus modelled amplifications for different slope heights (frequency 15 Hz)77

LIST OF TABLES TABLES PAGES Table 2.1. Relation between surface geology and local amplification (source: Bard et al, 2000).......15 Table 3.1. Different aspects of Numerical Modelling Methods (after Coggan et al, 1998) ...............20 Table 3.2. Basic differences between explicit and implicit solution methods (ITASCA, 2000).........24 Table 5.1. Material properties used for the simulation. .......................................................................39 Table 5.2. Variation of displacement, velocity, and acceleration histories with friction angle (slope

angle 25.5º, model 1)....................................................................................................................40 Table 5.3. Variation of displacement, velocity, and acceleration histories with friction angle (slope

angle 30.5º, model 2)....................................................................................................................41 Table 5.4. Variation of displacement, velocity, and acceleration histories with friction angle (slope

angle 25.5º, model 3)....................................................................................................................41 Table 5.5. Variation of displacement, velocity, and acceleration histories with material cohesion (slope

angle 45º, model 1).......................................................................................................................42 Table 5.6. Variation of displacement, velocity, and acceleration histories with material cohesion (slope


angle 42.3º, model 2)....................................................................................................................43 Table 5.8. Variation of displacement, velocity, and acceleration histories with material cohesion (slope


angle 25.5º, model 3)....................................................................................................................44 Table 5.10. Variation of displacement, velocity, and acceleration histories with material cohesion

(slope angle 90º, model 3). ...........................................................................................................44 Table 5.11. Variation of displacement, velocity, and acceleration histories with slope angle (model 1).

......................................................................................................................................................45 Table 5.12. Variation of displacement, velocity, and acceleration histories with slope angle (model 2).

......................................................................................................................................................46 Table 5.13. Variation of displacement, velocity, and acceleration histories with slope angle (model 3).

......................................................................................................................................................47 Table 5.14. Variation of velocity and acceleration histories with distance from the edge of slope crest

(model 1). .....................................................................................................................................48 Table 5.15. Variation of velocity and acceleration histories with distance from the edge of slope crest

(model 2). .....................................................................................................................................48 Table 5.16. Variation of velocity and acceleration histories with distance from the edge of slope crest

(model 3). .....................................................................................................................................49 Table 6.1. Input parameters used for the simulation to get the relation of bulk modulus on

amplification.................................................................................................................................52 Table 6.2. Maximum amplifications for the combination of slope height and input frequency..........60 Table 6.3. Input parameters used for the numerical calculation to show variation of amplification with

shear modulus...............................................................................................................................68 Table 6.4. Detail of the input parameters as well as observed and modelled values of amplification and

regression co-efficients.................................................................................................................75 Table 6.5. Regression parameters for different input frequencies. ......................................................78

LIST OF SYMBOLS AND ABBREVIATIONS USED IN THE TEXT

The following lists of symbols and abbreviations are used in the text. However, it is tried to define in the text itself where they are introduced. ip= Incident angle of P-wave is = Incident angle of S-wave rp = Refracted angle of P-wave rs = Refracted angle of S-wave ic = Angle of critical incidence P-wave = Primary wave S-wave = Shear wave Vp = P-wave velocity Vs = S-wave velocity C = Wave speed w = Angular frequency k = Stiffness matrix of the structure m = Mass matrix h = Slope height f = Input frequency f0 = Fundamental elastic frequency (eigenfrequency) of slope model λ = Wavelength ρ= Mass density Amax. = Maximum amplification Hz = Hertz MPa = Mega Pascal GPa = Giga Pascal Pa = Pascal G = Shear modulus K = Bulk modulus

EMPIRICAL RELATIONS FOR EARTHQUAKE RESPONSE OF SLOPES

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1. Introduction

1.1. Research Motivation

Earthquakes are one of the most devastating natural hazards in the world that causes great loss of life and property every year. The countries in earthquake prone areas, including Japan, India, China, Phil-ippines, Turkey, Iran, and USA, are suffering a lot due to frequent medium to large-scale earthquakes. The 8.9 Richter scale earthquakes that occurred in Japan in 2nd march 1933 is the highest measured earthquake in the last century (Hu et al, 1996). The seriousness of earthquakes can be imagined by one of the examples of Hyogoken Nambu Earth-quake in 1994. Hyogoken Nambu Earthquake was occurred around the city of Kobe and surrounding region in the Osaka Bay, which resulted in more than 5000 people dead, 25000 injured, and the total estimated losses of 200 billion dollars (Bielak et al., 1998). This is just one of the examples of the loss due to earthquake. Many of such examples can be found in the literatures, which compelled lot of peo-ple from the earthquake prone countries to loose their life and property. The effect of earthquakes however is dependent upon many factors including the geotechnical properties of the site at the prox-imity of the event as well as the topography and geometry of the site. It is generally reported that the largest amplifications of the motion at the crests of the mountains (relative to the base) is as large, or larger than the amplification normally caused by the presence of near surface unconsolidated layers. The widely accepted belief that the structures built on hard rock are less susceptible to damaging ground motions than those on unconsolidated materials could therefore be a misleading generalization. It clearly indicates the importance of topography with respect to seismic amplification. The slope ge-ometry and the geotechnical properties of material similarly influence the amplification of the site. Earthquakes can play a triggering role for the slope stability problems as well, and can reactivate the old landslides bringing about sequence of disasters. Observations of ground motion during earthquakes have indicated that the source, path and site effects can contribute significantly to the severity of ground motion within a given basin (Aki 1988, 1993 in Bielak et al, 1998). Therefore, the demand of time is to find out the relation of slope geometry and material properties on its amplification during the earthquake with utmost accuracy prior to the disastrous events, and widened the space enough for the counter major works. Plenty of research has been carried out in the past throughout the world in the field of ground response analysis, seismic microzonation as well as the effect of topography on ground motion. Some of the latest research in the field of ground response relation with variable materials and topography are: Ha-venith et al., 2003, Loria (2003), Paolucci (2002), Bard and Thomas; 2000, Hack et al. (2000), Leenders (2000), Castro (1999), Athanasopoulos et al. (1998), Aki (1988, 1993), Sanchez-Sesma (1990), Geli et al. (1988), Bouchon (1973), Boore (1972), among others. It is generally seen that the ridges of mountains are amplified more than the base, but the effect of surface topography and the ma-


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terial properties are not fully understood that is why no empirical relationships for these effects exist until recently (Sanchez-Sesma, 1990). Under these circumstances, present research is aimed to find out some empirical relation of the earthquake input signals, slope geometry and material properties on the seismic amplification. For this purpose, 2D finite difference numerical modelling program, FLAC (Fast Lagrangian Analysis of Continua) software is used.

1.2. Research Hypothesis

Following are the general hypotheses made before actual research starts:

The slope geometry and material properties have a great influence on the amplification of a slope during dynamic loading due to earthquake. As the slope steepness increases, the amplification of the slope during dynamic loading in-

creases and vice versa. The slope height will also have great influence on amplification. The edge of the slope crest will be amplified most than the other parts farther away from the

edge. The higher the friction angle and the cohesion, the lower the amplification of the ground.

Higher the elastic modulus of the material, lesser the amplification. All of these effects however are dependent upon the frequency and wavelength of input sig-

nals. It should be possible to show such relationships empirically.

1.3. Research Objectives

The ultimate major objective of this research is to recommend the empirical relations of the input sig-nals, slope geometry and material properties to the amplification of the slope due to dynamic loading so that the results can also be integrated in GIS environment for seismic microzonation analysis. To fulfil this objective, the proposed research will be directed to:

Carry out numerical calculation for dynamic loading in some arbitrary slope geometries and the material properties to analyse their general relationship with the slope amplification in terms of displacement, velocity, and acceleration histories. The slope geometry mainly denotes for the slope angle and the slope height in this case. Carry out the dynamic analysis in the same slope geometries to find out the velocity, accelera-

tion, and displacement histories at different selected parts of the slope to find out the variation of those histories with the distance from the edge in the crest of the slope. Carry out the dynamic analysis in the same slope geometries to find out the relationship of the

material properties mainly shear modulus and bulk modulus as well as that of the frequency and wavelength of the earthquake signals with the maximum amplification.


3

Figure 1.1. Flow chart of the methodology adopted for the study.

Learning FLAC software

Literature study related with dynamic modelling, FLAC program, earthquake

characteristic and ground response analy-sis

Generation of arbitrary slope geometry and assigning material properties

Reading calculation result data from computer screen

Introducing design earth-quake signals and writing

FLAC script

Numerical calculation for dy-namic loading

In FLAC

Making database in Microsoft Excel

Data analysis and interpretation

Result formulation

Thesis writing


4

Recommend the empirical relations of the slope geometry; earthquake input signals as well as that of the material properties with the slope amplification during dynamic loading.

1.4. Methodology

The methodology adopted for this research consists of different stages as is shown in flow chart (Fig-ure 1.1) and each stage is described in the following sections. The methodology for dynamic loading (numerical calculation in FLAC) is given in Figure 3.4 in Section 3.2.1.3.

1.4.1 Literature study

The research starts from the literature study. This is done with the help of available books, journals, previous theses, reports and internet sites related with the earthquake engineering, seismology, ground response analysis and the numerical modelling. The literature study is aimed to synthesize the sum-mary of the researches done so far in the field of seismic amplification and the ground response analy-sis using the analytical and numerical methods as well as to evaluate their relationship with the present research. Another important part of this stage of study is the in-depth theoretical understanding about the numerical modelling methods with focus on the finite difference method (FDM).

1.4.2 Learning Softwares

The present research is the outcome of the 2D finite difference numerical modelling software FLAC. So in-depth understanding of that software is must to obtain the required results. Moreover, some mis-takes in the input of the parameter may give the totally misleading results. Therefore, after the litera-ture study the research is focused on learning the aforementioned software in terms of the theoretical and the practical aspects. The FLAC manual published by ITASCA (ITASCA, 2000) is followed for this purpose.

1.4.3 Generation of arbitrary slope geometry and assigning material properties

Since the research is not based on the real field data, arbitrary slope geometries are made for the mod-elling purpose. The slope geometry and the material properties are selected more close to the reality as far as possible (however to investigate the effect of variation of those parameters, sometimes non-realistic values as well may have been introduced). Total of four slope models are studied for the pre-sent research. The ground response in terms of amplification for different slope angles and the material properties are assessed. The first three models are used just to observe the variation of displacement, velocity, and acceleration histories with the variation of input parameters whereas the fourth model is used to find out the relation of those input parameters with the amplification of the ground. For the sake of simplicity and due to limited time, simple slope geometries are used for this study.


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1.4.4 Introducing design earthquake signals and writing FLAC script

Like the slope geometry and the material properties, the earthquake signals are also generated arbitrar-ily. Certain design earthquake with particular acceleration, frequency, amplitude, and wavelength are introduced. Then FLAC script is written as needed by the program which contains command for grid generation, slope geometry, material properties, design earthquake parameters, the measurement pa-rameter (displacement, velocity, acceleration histories) and the co-ordinates of the place of interest (slope crest in this study) where the output is required to find out.

1.4.5 Modelling for Dynamic Loading

As already explained in Section 1.3, the major objective of this research is to find out the empirical relation of the slope geometry and material properties on the amplification of slope. This can be done by the application of different numerical calculation techniques like continuum, discontinuum, or hy-brid techniques. FLAC is the main software tool used for the numerical calculation in this study. It is a continuum fi-nite difference program performing Langrangian analysis. This means that every derivative in the set of governing equations is replaced directly by an algebraic expression written in terms of the field variables (e.g. stress or displacement) at discrete point in space (ITASCA, 2000). For dynamic analy-sis, it uses explicit finite difference scheme to solve the full equation of motion using lumped grid point masses derived from the real density surrounding zone (rather than fictitious masses used for the static solution). To find out the response of the slope during the dynamic loading by earthquake and ultimately to quantify the effect, dynamic loading is applied to the arbitrarily generated slope geometry. The dis-placement, velocity, and acceleration histories at the different parts of the slope geometry in the same slopes are assessed. For the sake of simplicity, the material in the slope is considered homogeneous and continuous. The calculation is performed to find out the relation of shear modulus, bulk modulus and slope height. The calculation is also performed for various frequency and wavelength of the input signals to investigate their effect on overall amplification. Moreover, to investigate the variation of the amplification with the distance from the edge in the crest of the slope, responses at different distances from the edge of the slope are measured. The general relationship of the material cohesion, friction angle and the slope angle with the amplification is observed, which helped to understand their impor-tance on amplification. During calculation, only one input parameter is changed for a single calcula-tion to quantify the sole effect of that particular parameter alone.

1.4.6 Reading data obtained from calculation from computer screen

The required output parameters (as indicated in the FLAC script) as resulted from the FLAC calcula-tion are read from the computer screen and noted to make the data set in Microsoft Excel. The dataset later on is used for the data analysis and the interpretation.


6

1.4.7 Data analysis and interpretation

The data obtained from the numerical calculations are stored in a dataset and are analysed using Mi-crosoft excel. First, the general relationship between the input parameters and the output as observed from the calculation data are analysed and tried to find out their overall importance on the amplifica-tion. The observed relationship is later on tried to relate with the theoretical concepts. The variations are observed also by making the plot of the input parameter versus output. The plot for the input ver-sus output for different parameters are interpreted to answer the questions about why is that type of relation obtained and how. Those plots are later on tried to generalize in terms of schematic drawing as well so far as possible.

1.4.8 Result formulation

After the analysis and interpretation of data, the relationship between the input parameters and the am-plification is formulated based on the data analysis result and its theoretical interpretation. The empiri-cal relation is optimised to show the relation between the input and output parameters. The best-fit functions with realistic theoretical explanations are recommended as an empirical relation.

1.5. Scope of the study

The rock mass is generally discontinuous, anisotropic and is continuously loaded by dynamic move-ments of the earth such as earthquakes and tectonic movements. The nature of rock mass is quite com-plicated in itself and therefore it is not always possible to arrive at an analytical solution for the prob-lem such as stress strain analysis (Desai et al, 1972). The alternative is the use of numerical calculation methods. The scope of numerical modelling techniques is therefore quite high in the field of Rock me-chanics and Earthquake engineering. Lot of research has been done in the past using the numerical calculation methods to find out the ground response analysis as well as the effect of irregular topography and material properties to the amplification of slope as indicated in Section 1.1. The researches however are related with some par-ticular case study and are mostly focused on the topography effect for the comparison of amplification at the crest and the base of the slope, or to the top of the mountain and base of the valley. The re-searches, related with the relation between the material properties and the amplification, however are hardly found. Hence, present study is probably one of the very few attempts to find the empirical rela-tionship between the material properties and the seismic amplification in the slope.

1.6. Limitations of the present study

Very simple slope geometries are used for the simulation and the discontinuities in the rock mass are not considered, which may have a great influence on the amplification of the slope during dynamic loading. The material type is considered homogeneous and the anisotropy in the slope is not consid-ered at all, for the sake of simplicity and the time constraints. Moreover, present study is purely a nu-


7

merical simulation results and only prescribed parameters for the frequency and acceleration of the waves are introduced as input.

1.7. Structure of the thesis

This thesis consists of seven chapters. As already seen, the first Chapter is related with the introduction about the research, objective, methodology and the scope of the research. In the second Chapter, some of the theoretical aspects of the ground response analysis and the previous researches in the field of ground response analysis are described. The third Chapter explains some of the important aspects of the dynamic modelling including the available methods for dynamic modelling. The fourth Chapter explains the aspects of the numerical modelling method used in this research. This chapter describes in detail about the methodology of using FLAC program, which is the software used for this research. Chapter five gives the explanations of the data obtained from the FLAC calculation for the input pa-rameters versus different outputs including the maximum velocity, displacement as well as the velocity and acceleration histories in the horizontal and vertical directions. This reflects the preliminary idea about the role of material cohesion, friction angle, and the slope angle on the overall amplification of the slope. The detail interpretation of this relationship however is not performed in this chapter. Sixth Chapter is aimed for the description and interpretation of the relationship between different input pa-rameters and the output as well as the formulation of result by carrying out data analysis. An empirical function of slope height, shear wave velocity and input frequency on amplification is also recom-mended in this Chapter. Chapter seven gives the overall conclusion of the study and recommendations for the future research. References and appendices are given at last.


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2. Theoretical background on ground response analysis

2.1. Introduction

Ground response analyses are most important to predict the ground surface motions during dynamic loading, to evaluate liquefaction hazards and to evaluate the earthquake induced forces which can cause the instability of earth and earth retaining structures (Kramer, 1996). Ground response can be generally expressed in terms of acceleration, velocity, or displacement parameters. A number of tech-niques have been developed so far for ground response analysis in terms of one, two, and three dimen-sional ground response. Such analysis can be carried out also by using numerical methods. Finite dif-ference method treat seismic waves propagation in a bounded domain in which the region is discre-tized into a number of grid points that have individual material parameters. The wave motion at each grid point is evaluated by solving equations of motions of discretized form (Takenaka et al, 1998). The basis of finite difference technique is the replacement of partial derivatives in space and time appear-ing in the equations by finite difference approximations. The response of structure rapidly increases when it enters in resonance, leading to serious damage. The response of those structures depends on their eigenfrequencies as well as the characteristics of seismic event. Moreover, the soil above the bedrock plays a very important role in determining the ground sur-face motion as they generally enhance the peak ground accelerations. It is generally seen that weaker materials are subjected to high peak ground accelerations than the stronger. It is suggested in the litera-ture that the irregular morphology is also responsible for the enhancement of ground response (e.g. Paolucci, 2002, Athanasopoulos et al. (1998), Aki (1988, 1993), Sanchez-Sesma (1990), Geli et al. (1988), Bouchon (1973), Boore (1972); amongst others).

2.2. Types of seismic waves

Different types of seismic waves can be generated by an impulse such as an earthquake or an explo-sion. There are two category of seismic waves generated in this way:

Body waves, and Surface waves

The body waves move through the interior of the earth. The stiffness and density of the material through which waves travel, determine the velocity of body waves. Surface waves are generally con-


9

sidered as the result of the interaction between the body waves and the surface & superficial layers on the earth. P and S-waves are the major types of body waves whereas Love and Rayleigh waves are the major types of surface waves. Different types of surface and body waves are illustrated diagrammatically in Figure 2.1. Figure 2.1. Types of seismic waves (source: Rey et al, 2001). P-waves are also called primary, compressive, or longitudinal waves and are analogous to sound waves. The compression and rarefaction are the common phenomenon to occur into the materials


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through which P-waves pass. The displacement of the individual particles is parallel to the direction of the travel of the P-wave. They can pass through the solid as well as liquid medium. S-waves are also called secondary, shear, or transverse waves and they cause shearing deformation as they travel through the material. The motion of particle in this case is considered perpendicular to the travel direction of wave. They cannot pass through the liquid medium. Based on the direction of parti-cle movement, S-waves can be divided into two components: SV (vertical plane movement) and SH (horizontal plane movement). Surface waves are most destructive as usually they have the strongest vibrations. The amplitude of surface waves decrease exponentially with depth (Kramer, 1996). Interactions of SH-waves with the soft superficial layers give rise to love waves and do not have the vertical component of particle mo-tion whereas Rayleigh waves are produced due to the interaction of P-and SV-waves with the surface and concern with vertical & horizontal particle motion. The depth to which Rayleigh waves induce significant motion is inversely proportional to the frequency of the wave. The velocity of Rayleigh waves depends upon the frequency content: low frequency Rayleigh waves travel faster than the high frequency one.

2.3. Seismic waves and materials interaction 1

A wave which encounters the sudden change in material properties along its way will respond by re-flecting and refracting them at the boundaries (see Figure 2.2) and reach different parts of the earth through different path (Kramer, 1996). At the boundary between two media, with contrasted acoustic impedance, the product of density and seismic wave velocity, P and S seismic waves are reflected and refracted into two others P and S waves. The reflection or refraction depends on the angle of incidence and the impedance contrast. The wave propagation across interfaces is based on the theory of Huygens' principle: all points on a wavefront can be regarded as point sources for the production of new spherical waves. The new front is the envelop of the secondary wavelets. From this principle, Snell and Descartes derived the refrac-tion's law, initially proposed in optic but also applicable to elastic wave: The ratio of the sinus of the incident angle to the sinus of the refracted angle is equal to the ratio of the velocity of both media. Ap-plying this law to both reflected and refracted P and S-waves gives (Equations 2.1 and 2.2):

,1 ,2

( ) ( )p p

p p

Sin i Sin rV V

= ………………………………………………………………(Eq. 2.1)

,1 ,2

( ) ( )s s

s s

Sin i Sin rV V

= ........................………………………………………… (Eq. 2.2)

1 The text in this section is taken from Rey et al, 2001 unless source is indicated.


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Figure 2.2. Interaction of waves in the interface between two types of materials (source: Rey et al,

2001) The angle of incident wave governs the angle of energy of the outgoing wave in the next layer and the angle of reflection & energy back into the layer of origin. The orientation of an inclined body wave can strongly influence how energy is reflected and transmitted across an interface that acts as the boundaries between the geological materials (Kramer, 1996). According to Snell’s law, waves propa-gating upward through horizontal layers of successively lower velocities will be refracted closer and closer to a vertical path. For an angle of critical incidence (ic), Sin[r] becomes equal to 1 and the re-fracted wave propagates along the interface at the velocity V2. This wave, which is called the critical refraction, sends back critical reflections at angle ic. For an angle of incidence greater than critical in-cidence (ic), the incident wave is no longer refracted and it is completely reflected. Static and dynamic deformations are two types of deformations that can be found during an earth-quake. Static deformation refers to the permanent displacement of the ground due to the waves whereas the dynamic deformation denotes for the radiation of the waves from the seismic event as it ruptures. Usually, earthquake has maximum effect near the fault with maximum distortion. Regarding the attenuation, geometrically surface waves attenuate much more slowly than the body waves. The attenuation of seismic wave also depends on the plasticity of the material (Kramer, 1996).

2.4. Factors influencing ground response

Generally, ground response is controlled by the three dimensional geometry of the subsurface and sur-face material as well as by the interaction between source, path and site effects. The site conditions have strong influence on the important characteristics of the event including amplitude, frequency con-tent, and duration. According to literatures, ground response is dependent on the following three fac-tors:

Source effect Path effect


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Site effect

Those three effects are described in the following sections.

2.4.1 Source effect

The spatial and temporal behaviour of the slip on the fault or faults that rupture in an earthquake is central to predicting the ground motion (Brown, 2001). The source effect is dependent on the input motion characteristics (earthquake source) as well as the frequency and wavelength of the source wave. In general, the ground response is maximum near to the epicentre and closer to the hypocenter than the farther distance and is directly proportional to the earthquake intensity itself. Similarly, the earthquake magnitude also influences the ground motion. Generally, the higher the magnitude, longer the total duration. However, for strong motion duration this is not so at sites near the fault for very strong earthquakes because those parts far away from the site close to the fault will add almost nothing to the maximum peak acceleration from the nearby part of the fault due to attenuation (Hu et al, 1996). Resonance is the most important factors for controlling the magnitude of amplifications of the ground and structure. It is defined, as the condition in which the period of vibration of an earthquake induced ground shaking is equal to the natural period of vibration of the structure. When resonance occurs, the shaking response of the structure is enhanced and the amplitude of the vibration of the building or any other structure rapidly increases. Tall buildings and other large structures respond most to ground shaking that has a high period of vibration and small structures respond most to low period shaking (Day, 2002). The resonance effect can also be explained in terms of frequency. When the natural fre-quency of the structure coincides with the frequency of input signals, resonance occurs. However, resonance is the combine effect of source, path, and site conditions. The type of the incident wave influences the ground motion as well. Amplification is higher for the incident s-waves than for p-waves and larger for SV waves than for SH waves (Geli et al., 1988).

2.4.2 Path effect

Path effects modify the seismic wave field as it propagates through the complex crust of the earth and have a strong, often dominant influence on strong ground motion. Though seismic waves are referred to as elastic waves, anelastic effects due to energy losses (like inter-particle friction), which give rise to the attenuation of seismic waves, cannot be neglected. The effect of attenuation on ground motion is profound because the same soft materials near the earth’s surface that lead to strong amplification of ground motion can also lead to rapid attenuation (Aki and Richards, 1980). Damping is the important parameter to be considered for the path effects. If the damping ratio is high, the ground response at farther distance from the epicentre will be influenced more as the energy content of the wave will be reduced greatly and the ground amplification will be reduced. Damping affects the response at high frequency more than at lower frequencies. The attenuation of the wave amplitude with distance are caused by (Kramer, 1996):


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Material or viscous damping: Absorption of the energy by the materials through which it trav-els. Geometry of the wave propagation (radiation damping): Spreading of wave energy over a

greater volume of material as it travels away from the source. Damping occurs even at very low strain levels, and thus damping ratio never becomes zero. According to Kramer (1996), the damping ratio increases with the reduction of the ground plasticity index, void ratio, geologic age, and confining pressure.

2.4.3 Site effects

Site effects generally refer to the effects on ground motion when seismic waves interact with the com-plex geological environment in the shallowest 100 or so meters of the earth’s crust. The low seismic velocities and impedances in shallow sediments can lead to extremely large and locally varying ampli-tudes during ground motion. Moreover, in this domain wave propagation during ground motion is of-ten non linear and can lead to strong amplitude dependent attenuation effects (The National Academies press, 2003). It has been often reported that the buildings located on hilltops suffer much more inten-sive damage than those at the base of the hills. Lot of literatures can be found on such phenomenon, which is well documented for the particular devastating earthquake at particular area. It is generally seen that the ridges of mountains are amplified more than the base, but the surface topography effects are not fully understood that is why no empirical relationships for these effects exist until now (San-chez-Sesma, 1990). The important factors determining the site effects as observed by various research-ers in the past that are relevant with present study are explained in this section. 2.4.3.1 Topography and geometry of the ground Boore (1972), using a finite difference technique, examined calculated SH waves incident on three different topographical models using the spectral ratios of the amplitudes at various points on the to-pographic features to the amplitude expected if the topographic features were not there. He found that the amplitudes were amplified on the crest and are either slightly amplified or attenuated at the flank depending on the geometry of the model and the position of the flank. He also found that the amplifi-cation or deamplification is dependent on the relationship between the dimensions of the topographic feature and the wavelength of the incident wave i.e. the effects are frequency dependent. Davis et al. (1973) carried out the field instrumentation program in the crest and base of three different mountains to observe the topography effects on the amplification of the seismic waves. He found that the crests of mountain high are amplified significantly higher than the base but in the mean time he recommend that the amount of amplification and the periods at which it occurs are probably the func-tion of the relationship between the wavelengths of the incoming signal and the dimension of the mountain. However, he did not estimate the effect of the height, slope and general shape of the moun-tain. He found some correlation between the amplification and the half width of the mountain. He found that the shear waves cause resonance of the mountain if the wavelengths are comparable with the dimensions of the mountain (i.e. half width).


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Havenith et al., (2003) studied the influence of topographical and site-specific amplification effects in the Ananevo rockslide in north-eastern Tien Shan Mountain and pointed out that the site effect could be the cause of large-scale gravitational movements during strong earthquake. They also mentioned that the strongest amplifications occur at the top of the slope and along the crest. Loria (2003) carried out the numerical assessment for the earthquakes of Colombia, 1999 and El Sal-vador, 2001 using the finite difference software FLAC, and tried to find out the effect of topography on the amplification of the ground. According to her, it gives an idea that there is a faster change in amplification along steeper slopes, but more research should be done in this field. She also concluded that the FLAC responses are comparable with the ones previously assessed by Hack et al. (2000) with the 3-D FEM DIANA for the horizontal components. Paolucci (2002) addressed the amplification of the seismic waves by surface irregularities using ana-lytical and numerical methods. He estimated the fundamental frequency of the simple topographic pro-files using Rayleigh’s method and concluded that the resonance usually occurs for the wavelengths slightly larger than the base of the slope. According to him, the topography effects for slope angles less than 15° can be neglected and that when the base of the topographical feature is elongated, a wide amplification band is obtained whereas resonance dominates the frequency response for isolated cliffs. Hack et al. (2000) established a damage inventory, as well as a ground response analysis with a seis-mic microzonation of Armenia and Pereira, and a general analysis of the topographic site effects. Leenders (2000) evaluated the soil response and amplification factors for the topography of the Brasi-lia Nueva area in Armenia. He found that maximum horizontal amplification occurs on the top of the hill in general and in particular along the edges. Castro (1999) estimated a concentration of damage towards the slope edges, in a range of around 40 m from the edges for the earthquake of 1999 in Colombia. According to Athanasopoulos et al. (1998), the concentration of damage due to 1995 earthquake around the town of Egion, Greece was in the cen-tral part of the city, which is elevated region adjacent to the crest of high (>80 m) and steep escarp-ment. There is however minor or no damage along the low elevation region of the town, which lies at the foot of the escarpment. Geli et al., (1988) carried out the experimental and theoretical study to find the effect of topography on earthquake ground motion. They studied the effect of topography on earthquake ground motion and concluded that there exists a significant amplification at the hilltop with respect to the base for the fre-quencies corresponding to wavelengths about equal to the mountain width. According to them, the hillsides undergo complex amplification-attenuation patterns especially in the upper parts of the hill. They also mentioned that the complex subsurface layering might not be responsible for large crest/base amplification. In the mean time, they also warned that topography effects cannot be isolated from other effects as ground layering, and therefore the amplification on top of the geomorphologi-cally complex sites cannot be predicted by a priori estimations based solely on topography. It is also found that the amplification is lower for incident P-waves than for incident S waves and subsequently larger on horizontal components than on the vertical components. The approximate frequency for which there is maximum amplification corresponds to the wavelengths comparable with the mountain widths. It is also commonly seen that the observed amplifications are often much larger than the theo-retical predictions with two-dimensional homogeneous models (Geli et al, 1988). This discrepancy


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may be due to the three dimensionality of the topographic features, or probably due to the irregular subsurface layering and/or neighbouring topography. Bouchon (1973) studied the effect of steepness of the slope and that of topography on the amplifica-tion and found that amplification occurs towards the top of the ridge and attenuation along the ridge flanks. The values of both of them increase regularly with the steepness of the slope ridge. He also found that a zone of amplification takes place near the top in the case of ridge whereas in the case of depression, a zone of attenuation occurs near the bottom. 2.4.3.2 Material properties Material property has strong control over the amplification or attenuation of the seismic waves. Linear cyclic threshold shear strain is greater for highly plastic soils than low plasticity grounds (Kramer, 1996). Softer material sites amplify low frequency motions more than a stiffer site and the reverse oc-curs for high frequency event. According to Kramer (1996), the surface motions can be similar at dif-ferent sites, but amplitude and frequency content of the motions can vary from place to place. Amplifi-cation is high for the loose materials than the stiffer. The presence of softer soil layers above the bed-rock helps for higher amplification of the ground. Several attempts have been made at providing more precise quantitative relationships between surface geology and local amplifications (see Table 2.1) but their comparison is not straightforward as the method of measurement of the amplification is different (Bard et al, 2000). Borcherdt and Gibbs (1976) used average horizontal spectral amplification with respect to granite in the frequency range of 0.5 to 2.5 Hz whereas Shima (1978) used ratio of peak ground motion in the 0.1 to 10 Hz frequency range with respect to loam ground. In the other hand, Midorikawa (1987) used mean ground amplification in the 0.4 to 5 Hz frequency range with respect to pre tertiary rocks. The relations between the geological unit and amplification factor given in Table 2.1 were obtained from sites where detailed intensity data was readily available.

Table 2.1. Relation between surface geology and local amplification (source: Bard et al, 2000). Geological unit Relative amplification factor Borchert and Gibbs (1976): relative to granite Bay mud 11.2Alluvium 3.9Sanat Clara Formation 2.7Great Valley sequence 2.3Franciscan Formation 1.6Granite 1Shima (1978): relative to loam Peat 1.6Humus soil 1.4Clay 1.3Loam 1Sand 0.9Miorikawa (1987): relative to pre-Tertiary rocks Holocene 3Pleistocene 2.1Quaternary volcanic rocks 1.6Miocene 1.5Pre-Tertiary 1


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2.4.3.3 Other effects In addition to the factors described above, local spatial variability (incoherence) play important role for the response of some structures. The factors for such effect are (Kramer, 1996):

Ray-path effects Wave-passage effect Extended source effect

Ray path effects are caused by the scattering of waves due to heterogeneity along the travel path. Co-herency decreases with increasing distance between measuring points and with increasing frequency. In the wave passage effect, different waves reach different surface points at different times and pro-duce the time shifts between the motions at those points. Due to extended source effect, differences in the relative geometry of the source and sites produce different time shifts and motions at those sites

2.5. Methods for estimating site effects

There exist various techniques for the evaluation of site effects. The effectiveness of each technique is purpose specific and depends on the importance of engineering project as well as the size of the pro-ject. In general, there are three approaches of site effect estimation (Bard et al, 2000):

Experimental methods Numerical methods, and Empirical methods

2.5.1 Experimental Methods *

Based on different kinds of data available, two approaches of experimental methods are available: Macroseismic observations and Microtremor data. Macroseismic observations techniques was first used for the city of Tokyo to analyse the 1854 Tokyo earthquake as early as in 1913. The principle of this method is: If the site of interest has already undergone destructive earthquake and that detailed macroseismic observations are available, a detailed analysis of the data in the light of topographical and geotechnical maps will help to find out the qualitative hazardous zones. Microtremors are terms used to denote the vibrations of the ground caused by natural or ambient dis-turbances such as wind, sea waves etc. and are recorded by using high sensitivity seismometers. It has been found that the spectral features of microtremors exhibit some correlation with the geological con-dition of the site, which is the basic principle of using this technique. H/V ratio (Nogoshi-Nakamura’s technique) is the mostly used technique among the microtremor method. It is based on the principle of H/V ratio i.e. the ratio between the fourier spectra of the horizontal and vertical components of mi-crotremors of the same stations. Nakamura proposed that the spectral ratio is a reliable index of the


17

site’s response to S waves, which provides reliable information not only about the resonant frequen-cies but also about the corresponding amplification. It is found that these ratios are much more stable than the raw noise spectra. Nakamura version of the Microtremor method is one of the most inexpen-sive and convenient techniques to reliably estimate the fundamental frequencies of soft deposits. Some examples however show that Nakamura’s technique fails for higher harmonics and that the peak am-plitude is somewhat different from the amplification measured by the spectral ratios.

2.5.2 Numerical methods *

If the geotechnical characteristics of the area are known, site effects can be estimated through numeri-cal analysis. But the most important aspect of using numerical techniques is that it requires in depth understanding both of the analytical models and of the numerical schemes being used, otherwise the results obtained from this method will be less reliable than even the simpler approximations. 2.5.2.1 Simple methods * These methods are available only for the estimation of the amplification on soft soils but the under-standing for the surface topographic effects is not satisfactory enough from this technique. * Source: Bard et al, 2000 unless source is indicated. Hand calculations The amplification of the soft soil is related to the resonance effects, which show up in the frequency domain in the form of peaks in the transfer functions. Since the strongest effect generally occur at the fundamental frequency, the simplest numerical methods which do not require the use of any computer but only some hand calculations, aim at estimating the fundamental period of the soil and the corre-sponding amplification. However, this type of simple estimation of the amplification is possible only for the sites that can be approximated as a one layer over bedrock structure. The estimation of the fun-damental period is relatively easy for which only the S-wave velocity and the thickness of the surface layer are needed whereas to estimate the amplification, damping and the additional knowledge of the bed rock velocity is required. Simple computer analysis There exist numerous of simple analytical methods, which can be used for the computation of the seis-mic response of a given site with the help of a small personal computer. The most widely method makes the use of multiple reflection theory of S waves in horizontally layered deposits referred as 1D soil amplification. A soil column is excited by an incoming plane S wave, generally considered to be vertically incident, and corresponding to a surface bed rock motion representative of what is likely to occur in the area. The parameters required for such analysis are: the shear wave velocity, density, damping factor and thickness of each layer, which can be obtained from direct in situ measurements or


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from drillings and subsequent laboratory measurements or with known approximate relationships with other known properties. Some generic, average constitutive models have been proposed for different kinds of materials but the behaviour of the soils at the given site may strongly depart from these aver-ages. 2.5.2.2 Advanced methods * The structure or geometry in advanced methods can be 1D, 2D or 3D and the mechanical behaviour of the soil can be very complicated (like visco-elastic, non linear soil behaviour etc.). Advanced methods can be generally classified into four groups:

Analytical methods, which may be used only for a very limited number of simple geometries. Ray methods, which are high frequency techniques and are difficult to use when wavelengths

are comparable to the size of the heterogeneities. Boundary element techniques, which are the most efficient when the site of interest consists of

limited number of homogeneous geological units. Domain based techniques (including finite difference and finite element methods), which al-

low accounting for very complex structures and rheologies (mechanical behaviour) but are comparatively heavy from computational point of view.

* Source: Bard et al, 2000 unless source is indicated. The advantages of advanced numerical methods can be seen in their flexibility and versatility. They allow to carry out the phenomenological and parametric studies as well as can be used to assess the uncertainty in a site’s seismic response to the imperfectly known site conditions and its mechanical parameters. The disadvantage of numerical methods is that generally the cost of using numerical methods may be high perhaps well in excess of the cost using the instrumental means because they need the detailed geotechnical or geophysical investigations for the site to provide constitutive proper-ties needed as input parameters.


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3. Dynamic loading and numerical modelling

3.1. Dynamic Loading due to earthquake

The earthquake releases and transmits the seismic acceleration waves through the ground. Such a tran-sient dynamic loading help to increase the shear stress in the slope and also help to increase the water pressure in the pores and fractures, which ultimately decreases the frictional force that resist the slope mass. The slope response after the dynamic loading by an earthquake depends upon many factors in-cluding the magnitude of seismic acceleration, earthquake duration, strength characteristics of the ma-terials and the dimension of the slope (Abramson et al, 1996).

3.2. Numerical Modelling Techniques

Stereographic and kinematic analyses as well as Limit equilibrium methods are the conventional meth-ods of rock slope analysis. All of those methods have their own limitations. It is reported in the literatures that the stereographic and kinematic methods are mostly suitable for the preliminary design or design of non-critical slopes and considers the critical orientations neglecting other important joint properties. Limit equilibrium method does not consider the insitu stress and the factor of safety calcu-lations give no indication of instability mechanisms. In modern world, advances in computing power and the availability of commercial numerical modelling softwares is making the simulation of potential rock slope failure mechanisms standard component of a rock slope investigation (Stead et al, 2001). The rock mass is generally discontinuous, anisotropic, inhomogeneous as well as not elastic. More-over, they are under stress and continuously loaded by dynamic movements of the earth such as earth-quakes and tectonic movements. It is not always possible to arrive at an analytical solution for the problem such as stress strain analysis (Desai et al, 1972). This type of problems involving complex behaviour of materials and the dynamic loading due to earthquake can be modelled using different numerical calculation softwares available. Many numerical modelling methods have been developed so far, out of which some of them relevant to present research are described in this section. Numerical methods that are being used for rock mechanics problems can be broadly divided into three (Stead et al, 2001; Jing, 2003 among others):

Continuum methods


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Discontinuum methods, and Hybrid continuum/discontinuum methods

In the continuum methods, the displacement compatibility must be enforced between internal elements which is not required in discontinuum methods, rather is replaced by the contact conditions between blocks with specially developed constitutive models for point contacts or fractures (Jing, 2003). The choice of continuum, discrete or hybrid methods depends on many problem specific factors like the problem scale and fracture system geometry. Continuum methods Discontinuum methods Hybrid methods Figure 3.1. Types of numerical modelling methods (compiled from the article of Jing, 2003). Some of the input parameters as well as advantages and disadvantages of the three methods are given in Table 3.1. Each of the three methods has their own subdivisions. The subdivision of the numerical calculation methods is given in Figure 3.1, which is based on the text from Jing, 2003. Those numerical calculation methods indicated in the figure are being used also for the ground re-sponse and dynamic modelling. Continuum modelling includes finite element and finite difference methods, whereas discontinuum modelling includes distinct element and discrete element methods. Both 2D and 3D continuum codes are available. Two-dimensional code assumes plain strain condi-tions, which may not be valid for the inhomogeneous rock slopes with varying structure and lithology (Stead et al., 2001).

Table 3.1. Different aspects of Numerical Modelling Methods (after Coggan et al, 1998)

Numerical Calculation Methods

Finite Difference Method (FDM)

Finite Element Method (FEM)

Boundary Element Method (BEM)

Discrete Element Method (DEM)

Discrete Fracture Net-work (DFN) Method

Hybrid FEM/BEM Hybrid DEM/DEM Hybrid FEM/DEM Other hybrid Models


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Analysis Method

Critical input parameters

Advantages Limitations

Continuum modelling (eg.Finite Ele-ment, Finite Difference)

Representative slope geometry; constitu-tive criteria (e.g. elastic, elasto-plastic, creep etc.); groundwater charac-teristics; shear strength of surfaces; insitu stress state.

Allows for material defor-mation and failure. Can model complex behaviour and mechanisms. Capability of 3D modelling. Can model effects of groundwater and pore pressures. Able to as-sess effects of parameter variations on instability. Recent advances in comput-ing hardware allow complex models to be solved on PC’s with reasonable run times. Can incorporate dynamic analysis.

Users must be well trained, experienced and observe good modelling practice. Need to be aware of mod-els/software limitations (e.g. boundary effects, mesh as-pect ratios, symmetry, hardware memory restric-tions). Availability of input data generally poor. Re-quired input parameters not routinely measured. Inability to model effects of highly jointed rock. Can be diffi-cult to perform sensitivity analysis due to run time constraints.

Discontinuum Modelling (e.g. distinct element, discrete ele-ment)

Representative slope and discontinuity geometry; intact constitutive criteria; discontinuity stiff-ness and shear strength; groundwa-ter characteristics; insitu stress state.

Allows for block deforma-tion and movement of blocks relative to each other. Can model complex behaviour and mechanisms (combined material and discontinuity behaviour coupled with hydro-mechanical and dynamic analysis). Able to assess effects of parameter varia-tions on instability.

Experienced user required to observe good modelling practice. General limitations similar to those listed above. Need to be aware of scale effects. Need to simulate representative discontinuity geometry (spacing, persis-tence etc.). Limited data on joint properties available (e.g. jkn, jks).

Hybrid/Coupled Modelling

Combination of in-put parameters listed above for stand alone models.

Coupled finite ele-ment/distinct element mod-els able to simulate intact fracture propagation and fragmentation of jointed and bedded media.

Complex problems require high memory capacity. Comparatively little practi-cal experience in use. Re-quires ongoing calibration and constraints.

FLAC is a 2-D finite difference code being widely used in recent years, which incorporates dynamic modelling as well. FLAC3D is a 3-D continuum code that can be used for the 3-D analyses of rock slopes. Discontinuum method considers the rock slope as a discontinuous rock mass by considering it as an assemblage of rigid or deformable blocks. The discrete element method is the most popular among the discontinuum methods. UDEC is one of the widely used distinct element code that provides the displacements induced within the rock slope using force displacement law (stead et al, 2001). The influence of external factors like earthquake can also be simulated in UDEC. Distinct element methods are effective for the analysis of rock slopes with multiple joint sets where the mechanism of failure is controlled by those discontinuities. Hybrid approaches are also increasingly applied in rock slope


22

analysis; but have been used widely in underground rock engineering including coupled boundary-/ finite element and coupled boundary-/distinct element solutions. Coupled particle flow and finite dif-ference analysis using FLAC3D and PFC3D is the recent advances in hybrid methods.

3.2.1 Fast Langrangian Analysis of Continua (FLAC)1

FLAC (Fast Langrangian Analysis of Continua) was originally developed by Peter Cundall in 1986, and commercially released by ITASCA C.G., Inc. Capacity of performing large two-dimensional cal-culations without excessive memory requirements and the facility for dynamic modelling makes it ver-satile in rock mechanics and Earthquake Engineering. Speed of calculation however is a linear func-tion of the number of elements used for calculation. FLAC is a finite difference program performing Langrangian analysis. This means that every deriva-tive in the set of governing equations is replaced directly by an algebraic expression written in terms of the field variables (e.g. stress or displacement) at discrete point in space. On the other hand, finite ele-ment method (FEM) has a central requirement that the field quantities vary throughout each element in prescribed fashion using specific functions controlled by parameters. The formulation consists in ad-justing these parameters to minimize error terms or energy terms. However, both of the methods pro-duce a set of algebraic equations to solve and the resulting equations are identical for both the meth-ods. Finite element often combines the element matrices into a large global stiffness matrix whereas not done so in finite difference since it is relatively efficient to regenerate the finite difference equations at each step. For the explicit method, small time step is required which means that large numbers of steps must be taken. Overall, explicit methods are best for ill behaved systems example non-linear, large strain, physical instability but not efficient for modelling linear, small strain problems. FEM codes usually represent steady plastic flow by a series of static equilibrium solutions. The best FEM codes will give a limit load (for a perfectly plastic materials) that remains constant with increasing applied displacement. Either stress or displacement may be applied at the boundary of the solid body in FLAC. Displacements are specified in terms of prescribed velocities at given grid points. 3.2.1.1. Constitutive Models 1 Any of the following ten constitutive material models can be used in FLAC:

“Null” model, which represents removed or excavated material, stresses are zero, and no body forces act on these zones. Isotropic elastic model, which is valid for homogeneous, isotropic, continuous materials with

linear stress-strain behaviour with no hysteresis on unloading.

1 This section is taken from FLAC manual (ITASCA, 2000) unless source is indicated.

Transversely isotropic elastic model, which is used to simulate layered elastic media where there are distinctly different elastic moduli in directions normal and parallel to the layers. Plasticity models: Seven plasticity models are available in FLAC:


23

• Drucker-Prager, • Mohr-Coulomb (conventional to represent shear failure of rocks and soils), • Ubiquitous-joint, • Strain-hardening/softening, • Bilinear strain-hardening/softening ubiquitous-joint, • Double-yield, • Modified Cam-clay.

In addition, one can create a specific constitutive model, or modify the behaviour of the existing ones by using the FISH function which is a programming language set in FLAC that enables the user to de-fine new variables and functions. These functions may be used to extend FLAC’s usefulness or add user-defined features. FLAC provides two executable codes:

Single-precision version: The calculations are primarily based upon single-precision variables and are most efficient for most analysis. Double-precision version: It gives more accurate solutions for situations when the accumu-

lated value of a variable after many thousands of calculation steps is much larger than the in-cremental change in the variable, single precision limitations will prevent further changes in the variable. This code is recommended also for models with grids containing many zones with coordinates that are large compared to typical zone dimensions.

3.2.1.2. Dynamic analysis in FLAC 1 FLAC use explicit finite difference scheme to solve the full equation of motion using lumped grid point masses derived from the real density surrounding zone (rather than fictitious masses used for the static solution). Dynamic modelling can be executed in FLAC using either of two methods: Equivalent Linear Method and Fully Non Linear Method. The “equivalent-linear” method is common in earthquake engineering for modelling wave transmis-sion in layered sites and dynamic soil-structure interaction. In equivalent Linear Method (Seed & Idriss, 1969 in ITASCA, 2000), a linear analysis is performed with same initial values assumed for damping ratio and shear modulus. The maximum cyclic shear strain is recorded for each element and used to determine new values for damping and modulus by reference to laboratory derived curves that relate damping ratio and secant modulus to amplitude of cyclic shear strain. The new values of damp-ing ratio and shear modulus are then used in a new numerical analysis of the model. The whole proc-ess is repeated several times until there is no further change in properties. At this point, it is said that ‘strain compatible’ values of damping and modulus have been found and the simulation using these 1 This section is taken from FLAC manual (ITASCA, 2000) unless source is indicated. values is representative of the response of the real site. In contrast, only one run is done with a fully non-linear method (apart from parameter study), since non-linearity in the stress strain law us followed directly by each element as the solution marches on in time.


24

The equivalent linear method takes liberties with physics but is user friendly and accepts laboratory results from cyclic tests directly. In the fully non-linear method, damping and apparent modulus on strain level is automatically modelled. It correctly represents the physics but demands more user in-volvement and needs a comprehensive stress strain model in order to reproduce some of the subtler phenomenon. Characteristics of the Equivalent Linear Method:

Uses linear properties for each element that remain constant throughout the history of the shaking. The interference and mixing phenomenon that occur between different frequencies compo-

nents are missing. Does not provide information on irreversible displacements and the permanent changes like in

liquefaction. Elasticity theory used by linear method relates the strain tensor (not increments) to the stress

tensor. Hence, plastic yielding is modelled inappropriately. The material constitutive model is built into the method, which consists of a stress strain curve

in the shape of an ellipse. Characteristics of the Fully Non-Linear Method:

The method follows any prescribed non-linear constitutive relation. If a hysteretic-type model

is used and no extra damping is specified, then the damping and tangent modulus are appro-priate to the level of excitation at each point in time and space, since these parameters are em-bodied in the constitutive model. If Rayleigh or local damping is used, the associated damping coefficients remain constant throughout shaking. Interference and mixing of different frequency components occur naturally. Irreversible displacements and other permanent changes are modelled automatically. A proper plasticity formulation is used in all the built-in models, whereby plastic strain incre-

ments are related to stresses. The effects of using different constitutive models can be easily studied.

The dynamic analysis in FLAC is based on the explicit finite difference scheme and 2D plain strain or plane stress dynamic analysis is possible. In finite difference scheme, the governing equations are sub-stituted by finite differences in space, written in terms of field variables at discrete points i.e. the NODES - these are the key geometrical feature. In explicit scheme, non-linear solutions are produced in the same time as for linear problems (but longer solution times for implicit solutions). In Implicit scheme, solution at every node is dependent on solutions at four neighbour nodes - the solution at each node is not known until the entire solution is known. The basic difference between the explicit and the implicit methods are given in Table 3.2.

Table 3.2 Basic differences between explicit and implicit solution methods (ITASCA, 2000). Explicit Implicit Time step must be smaller than a critical value for stability

Time step can be arbitrarily large, with un-conditionally stable schemes


25

Small amount of computational effort per time step

Large amount of computational effort per time step

No significant numerical damping introduced for dynamic solution

Numerical damping dependent on time step present with unconditionally stable schemes

Provided that the time step criterion is always satisfied, non-linear laws are always followed in a valid physical way

Always necessary to demonstrate that the abovementioned procedure is: a) stable; and b) follows the physically correct path (for path sensitive problems)

Matrices are never formed. Memory require-ments are always at a minimum. No band-width limitations

Stiffness matrices must be stored. Ways must be found to overcome associated problems such as bandwidth. Memory requirements tend to be large

Since matrices are never formed. Large dis-placements and strains are accommodated without additional computing effort

Additional computing efforts needed to fol-low large displacements and strains

There are three aspects to be considered when using FLAC model for dynamic analysis:

Dynamic Loading and Boundary conditions. Mechanical damping. Wave transmission through the model.

Dynamic Loading and Boundary Conditions FLAC models a region of material subjected to external and / or internal dynamic loading by applying a dynamic input boundary condition at either the model boundary or at internal grid points. Wave reflections at model boundaries are minimized by specifying quiet (viscous), free field or three-dimensional radiation-damping boundary conditions (see Figure 3.2). In FLAC, the dynamic input can be applied in one of the following ways:

an acceleration history; a velocity history: a stress (or pressure) history; or a force history.


26

Figure 3.2. Types of dynamic loading boundary conditions available in FLAC (source: ITASCA,

2000).

(a) Flexible base

(b) Rigid base

free

fiel

d

free

fiel

d

quie

t bou

ndar

y

quie

t bou

ndar

y

quiet boundary

structure

3D dam-ping

external dynamic input (stress or force only)

Internal dynamic input

free

fiel

d

free

fiel

d

quie

t bou

ndar

y

quie

t bou

ndar

y

quiet boundary

structure

3D dam-ping

external dynamic input (velocity or acceleration)


free

fiel

d

free

fiel

d

quie

t bou

ndar

y

quie

t bou

ndar

y

quiet boundary

structure

3D dam-ping

external dynamic input (stress or force only)



27

The boundary conditions available in FLAC are described in the following section. a) Quiet Boundaries In the static analysis, fixed or elastic boundaries can be realistically placed at some distance from the region of interest. However, in dynamic analysis, such boundary conditions cause the reflection of outward propagating waves back into the model and do not allow the necessary energy radiation. Use of larger models may be one of the solution but it leads to the computational burden (since large mem-ory and the time required for large calculations) though due to damping material will absorb most of the energy in the waves reflected from distant boundaries. Hence, the best option is the use of the quiet (or absorbing) boundaries. The viscous boundary developed by Lysmer and Kuhlemeyer (Lysmer and Kuhlemeyer, 1969 in ITASCA, 2000) is used in FLAC. It is based on the use of independent dash points in the normal and shear directions at the model boundaries. It is almost completely effective at absorbing body waves approaching the boundary at an angle of incidence >30°, for lower angles of incidence or for surface waves, there are still energy absorption but is not perfect. The advantage of this system is that it operates in time domain. Another procedure to obtain the efficient absorbing boundaries to use in time domain was proposed by Cundall et al. (Cundall et al. 1978 in ITASCA, 2000) and based on the superposition of solutions with stress and velocity boundaries in such a way that reflections are cancelled. It requires adding the re-sults of two parallel overlapping grids in a narrow region adjacent to the boundary but is difficult to implement for a block system with complex geometry and thus is not used in FLAC. Quiet boundary conditions can be assigned in the X- and Y- directions of the deformable blocks. This can also be ap-plied for the rigid block model but first deformable blocks have to be created at the boundary. b) Free Field Boundaries In the case of surface structures like dams, the boundary conditions at the sides of the model must ac-count for the free field motion, which would exist in the absence of the structure. Free field is created and insitu conditions prior to the dynamic analysis are assigned. The equilibrium calculation for the free field should be performed before the free field is attached to the main model to bring the free field stresses to equilibrium. The free field boundary conditions require that lateral boundaries of the main model must be vertical and straight. The free model is connected to the main model and the free field grid can only be connected to deformable blocks. Mechanical Damping Damping is due, in part, to the energy loss because of internal friction in the intact material and slip-page along interfaces within the system. Natural dynamic systems contain some degree of damping of the vibration energy within the system; otherwise, the system would oscillate indefinitely when sub-jected to driving force. FLAC uses dynamic algorithm for two classes of mechanical problems: Quasi-static and Dynamic, out of which the former require more damping. The damping in numerical simulation should attempt to reproduce the energy losses in the natural sys-tem when subjected to a dynamic loading. In rock and soil, the natural damping is mainly hysteretic


28

(i.e. independent of frequency). Due to the following problem, reproducing hysteretic damping is dif-ficult:

When several waveforms are superimposed, many simple hysteretic functions do not damp all components equally. Hysteretic functions lead to path dependence, which makes result difficult to interpret

However, if a constitutive model contains an adequate representation of the hysteresis that occurs in real material, then no additional damping is necessary. Damping in FLAC required both for static and dynamic problems but static problems require more damping than the dynamic ones. Mass propor-tional (or viscous) and stiffness proportional both types of damping are available in FLAC. The former one is applied to centroids of rigid blocks or grid points of deformable blocks, a force that is propor-tional to the mass (or velocity) but in the opposite direction whereas the latter is applied to the contacts or stresses in zones a force that is proportional to the incremental force or stress in the same direction. The mass proportional damping tends to act on lower frequency modes generally associated with the movements of several blocks or grid points, in contrary the stiffness proportional component damps higher frequency inter-block vibrations. The critical damping ratio, λ, at any natural (angular) fre-quency of the system, ω, can be find out by using the relation (Eq. 3.1): λ = ½[(α/ω)+ βω] Eq. (3.1) Adaptive damping scheme in FLAC, whereby the mass-damping constant, α, can be automatically adjusted to the changing conditions of the problem, monitors the rate of change of energy in the model during solution. For dynamic analysis, the damping should attempt to reproduce the frequency inde-pendent damping of natural materials, which for example for geological materials, is generally 2% to 5% of the critical damping. a) Rayleigh Damping It is generally used to provide damping that is approximately frequency independent over a restricted range of frequencies. Rayleigh damping embodies two viscous elements (where the absorbed energy is dependent on frequency) but the frequency dependent effects are arranged to cancel out at the frequen-cies of interest. The equation of Rayleigh damping is expressed in a matrix form (Eq. 3.2): C = αm + βk Eq. (3.2) where, α is mass proportional damping constant, and β is the stiffness proportional damping constant. Stiffness proportional damping causes a reduction in the critical time step for the explicit solution scheme. In FLAC, the internal time step calculation takes account of stiffness proportional damping but it is still possible for instability to occur if very large block deformations occur. In this case, it is necessary to reduce the time step manually. For a multiple degree-of-freedom system, the critical damping ratio (ξi), at any angular frequency of the system (wi) can be given by equations 3.3 and 3.4 (Itasca, 2000):


29

α + β wi2 = 2wi ξi Eq. (3.3) ξi = ½ (α/wi + β wi) Eq. (3.4) where, α = the mass-proportional damping constant, and β = the stiffness-proportional damping constant. Figure 3.3 shows the variation of the normalized critical damping ratio with angular frequency. Mass-proportional damping is dominant at lower angular-frequency ranges, while stiffness-proportional damping dominates at higher angular frequencies.

Figure 3.3. Variation of normalized critical damping ratio with angular frequency (source: ITASCA, 2000). b) Local Damping It is originally developed as a means to equilibrate static simulations but some of its features make it attractive for dynamic simulations also. It operates by adding or subtracting mass from grid point or structural node at certain times during a cycle of oscillation hence there is overall conservation of mass (because amount added is equal to amount subtracted). Mass is added when the velocity changes sign and subtracted when it passes a maximum or minimum point. The critical damping D is given (Kolsky, 1963 in ITASCA, 2000), by the equation: αL= π*D Eq. (3.5) where, αL is local damping coefficient and hence there is no need to specify frequency. It appears to give good results for simple cases; it is frequency independent and need not to estimate the natural fre-quency of the system being modelled but it is untested for complicated situations involving many su-perimposed waveforms. Wave transmission through model


30

In a dynamic analysis, numerical distortion of the propagating wave can occur as a function of the modelling conditions. The frequency content of the input wave and the wave speed characteristics of the system will affect the numerical accuracy of wave transmission. According to Kuhlemeyer & Lys-mer (1973), the spatial element size, ∆L, must be smaller than approximately 1/10th to 1/8th of the wavelength associated with the highest frequency component of the input wave i.e. ∆L ≤λ/10 Eq. (3.6) where λ is the wavelength associated with the highest frequency component that contains appreciable energy and can be calculated from the wave speed by the formula, f = C/λ Eq. (3.7) for elastic continuum system. For the discontinuum system with a single set of planar joints oriented normal to the compression waves and in which solid material is rigid (or much stiffer than the joints),

* np

s kC =ρ

Eq. (3.8)

where Cp is the p-wave speed, S is joint spacing; Kn is joint normal stiffness and ρ is mass density. The necessity of very fine spatial mesh and corresponding small time step for Kuhlemeyer and Lysmer requirement may lead to the high time and memory for analyses of dynamic input with a high peak velocity and short rise-time. If the history is filtered and the high frequency component is removed, a coarser mesh may be used without significantly affecting the results. The filtering procedure can be accomplished with a low-pass filter routine such as the Fast Fourier Transform technique. 3.2.1.3 General Methodology for solving dynamic problems 1 Dynamic analysis in FLAC is viewed as a loading condition on the model and as a distinct stage. Al-ways, a static equilibrium precedes dynamic analysis. The general methodology of dynamic analysis in FLAC is given schematically in Figure 3.4. Generally, there exist four stages in dynamic analysis:

Zone size adjustment: It is to be ensured that the model conditions satisfy the requirements for accurate wave transmission. This test is to be performed before the static solution is performed because blocks cannot be recognized after the calculation starts.

1 Source: ITASCA, 2000

Specification of appropriate mechanical damping, representative of the problem materials and input frequency range. Applying dynamic loading and boundary conditions. Facilities to be set up to monitor the dynamic response of the model.


31

Figure 3.4. General solution procedure in FLAC (source: ITASCA, 2000) FLAC has three optional features:

Thermal option Creep material model Dynamic analysis

Start

MODEL SETUP 1. Generate grid, deform to desired shape 2. Define constitutive behaviour and material properties 3. Specify boundary and initial conditions

Step to equilibrium step

Examine the model response

PERFORM ALTERATIONS for example

- Excavate material - Change boundary conditions

Step to solution

Examine the model response

Parameter Study needed

Model makes sense

Acceptable result

End

Results unsatisfactory

More tests needed

Yes

No


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After one is determined to perform dynamic analysis, the model set-up precedes the analysis. The model set-up is a crucial step in the numerical modelling as the final output of the result is dependent solely on the input given during this stage. First, the geometry of the model is to be defined by which the area of the model is delineated and is followed by the constitutive model. As explained in Section 3.2.1.1, there are ten constitutive models available in FLAC. The choice of the constitutive model de-pends on the type of material being used for the modelling and the purpose of the research. The boundary conditions and the material properties are to be defined after choosing the appropriate constitutive models. The boundary conditions are described in Section 3.2.1.2 under the heading of dynamic loading and boundary conditions. The material properties include shear modulus, bulk modulus, mass density, internal friction angle, cohesion, and tensile strength of the material being used for the modelling. If the elastic constitutive model is chosen, density, shear modulus and bulk modulus are sufficient to be defined. After defining the model set-up, the gravity is introduced and solved for certain time steps. The model is then brought into rest and the second phase of calculation starts, which is the dynamic analysis. The earthquake waves recorded from some particular earthquake or some other design earthquake in the form of veloc-ity, acceleration, stress or force history can then be introduced into the model and the suitable damping is applied to compensate for the loss of energy within the model. The velocity, acceleration, displace-ment, stress or forces histories within different parts of the model can then be seen and analysed by defining the co-ordinates of the interested parts of the model. The result can then be analysed and in-terpreted theoretically to reach the conclusion.


33

4. Numerical simulation for dynamic loading using FLAC

4.1. Introduction

Numerical calculation for dynamic loading for the slope with simple geometry is carried out to inves-tigate the influence of earthquake input signals, slope geometry and material properties on the amplifi-cation during dynamic loading. The histories of the displacement, velocity, and acceleration are stud-ied in some of the selected zone of the slope (but one of the slope models is used to see the variation of acceleration only so that precise relationship between the input parameters and amplification could be obtained). The concentration of the measurement of amplification however is towards the crest of the slope as the literatures are saying those parts are amplified more than the other parts (see Section 2.4.3). The numerical calculations in FLAC have two basic objectives:

Investigate the general influence of the slope angle, internal friction angle, and material cohe-sion on the amplification during dynamic loading as well as to investigate the influence of the distance from the slope edge to the amplification. Finding out the relation of material properties (mainly shear modulus and bulk modulus),

slope height and input frequency on the overall amplification during dynamic loading.

Among the parameters of concern, only one variable is kept changing for one time simulation, and rest of the parameters kept constant to see the sole effect of that particular changed parameter. The general solution procedure in FLAC discussed in Section 3.2.1.3 is followed in this study. Looking initial re-sults, numerical calculations for more slope models and for varying input parameters are carried out according to the necessity to reach the conclusion. The detail description about how the calculations are performed in FLAC including general features of the slope geometry, material properties and the earthquake input signals that are used for the calculation is presented in this chapter.

4.2. Slope geometry

As already stated in Chapter 1, the present study is based on the numerical simulation of slopes to find out the relation of slope geometry and material properties with the amplification of the slope during dynamic loading. Therefore, for the modelling purpose some of the simple arbitrary slope profiles are generated. The dimensions of the model kept small to moderate to reduce the calculation time and the memory needed for the computer. The details of the slope geometry for three of the four slope models


34

are described in Section 5.2 and that of the fourth model are described in Section 6.1. The general ef-fect of the slope angle on the overall ground motion is observed in models 1,2 and 3 whereas the effect of slope height, wavelength and input frequency on overall amplification is studied in detail for the fourth model described in Chapter 6.

4.3. Discretization and generation of mesh

Like in other FEM programs, in FLAC, the structure is built out of finite amount of elements by dis-cretization resulting in a mesh. The mesh is composed by quadrilateral zones (or elements), internally subdivided in two sets of triangular elements of constant-strain as shown in Figure 4.1. The four trian-gular sub-elements are termed a, b, c and d. The deviatoric stress components of each triangle are maintained independently and the force vector exerted on each node is taken to be the mean of the two force vectors exerted by the two overlayed quadrilaterals (ITASCA, 2000). The zones or elements are organized in a row-column fashion. To refer to a particular zone, a pair of numbers representing the column and row should be used. Figure 4.1. (a) Overlayed quadrilateral elements used in FLAC; (b) typical triangular element with velocity vectors; (c) nodal force vector (source: ITASCA, 2000).

4.4. Model Formulation

4.4.1 Grid generation

Before the generation of the grid, “configure dynamic” command is to be given to let the program know that it has to perform dynamic analysis out of three types of analyses FLAC can do as mentioned in Section 3.2.1.3. To generate the grids, letter “g” in front of the required dimensions in co-ordinates (first the width and then the height separated each other by comma) is sufficient (for example: if the slope width is 100 m and height is 60 m, the command for grid generation would be: g 100,60). The grid will decide at how many points the calculations are to be performed.

(a) (b) (c)

a b

d c

b

∆s

a

(b) u f

(a) u f

(1) n i

(2) n i

(2) S

(1) S

Fi


35

4.4.2 Constitutive models

A constitutive model is a mathematically formulated expression to show a relationship between stress and strain and is used to express the prediction of the mechanical behaviour of engineering materials on the application of dynamic load. The detail of the constitutive models available in FLAC is de-scribed in Section 3.2.1.1. When selecting a constitutive model for a particular analysis, two consid-erations are important:

The known characteristics of the material being modelled. The intended application of model analysis.

As the Mohr Coulomb plasticity model is used for this research, this model is described in detail in this section. It represents a material that yields when subjected to shear loading. This model is based on the plain strain conditions and the shear criterion is characterized by the friction angle and cohesion represented. Mohr coulomb plasticity model should be used when the stress levels are such that failure of intact material is expected. Mohr-Coulomb model is most applicable for geotechnical studies and Mohr-Coulomb parameters for cohesion and friction angle are usually available more often than other properties for geo-engineering materials. This model assume linear elastic and perfectly plastic brittle weakening behaviour that means the stress strain response assumes that the material behaves in a linear elastic manner before yielding, and perfectly plastic behaviour after yielding. Most of the geological materials show such type of stress strain relationship and hence is more close to reality. Due to all of these reasons, Mohr-Coulomb plasticity model is used in this study.

4.4.3 Material properties

As already explained in the Section 2.4, the material properties will have great influence on the ground response during dynamic loading. Bulk modulus, shear modulus, density, friction angle, cohesion, and tensile strength are needed as material properties in Mohr-Coulomb constitutive model. Present re-search is only numerical simulation of the slope and do not have exact fieldwork area and hence there is no particular material type. The material properties though are chosen more close to reality. The ma-terial properties for some particular limestone given in UDEC manual are considered in the present study. However, during the course of calculation, material properties are to be changed, hence some-times the material properties used for the simulation might be unrealistic. For example: very low bulk modulus, and very high shear modulus or so on. The detail of the material properties used for different models are described in the upcoming sections whenever necessary.

4.4.4 Initial equilibrium


36

The gravity acceleration of 9.8 m/s2 is applied to the model with ‘set grav’ command and run until an equilibrium state is obtained but the lateral structures present if any and the tectonic stresses are not considered to keep the models simple to compensate for the limited time. The equilibrium state of the geometry can be assured with displacement and velocity histories made at some selected zones of the slope model generally upper part of the slope ridges. The horizontal and vertical displacement histories against the unbalanced force become constant when reaching the equilibrium. The velocity histories tend to zero after reaching the initial equilibrium state. The initial equilibrium however can be directly obtained by giving ‘SOLVE’ command. Once static equilibrium is reached, the displacements and velocities are redefined as zero to remove kinetic energy that could affect the dynamic calculation stage afterward.

4.4.5 Boundary conditions

Different types of boundary conditions available in FLAC are described in Section 3.2.1.2. Boundary conditions should be differentiated for static and dynamic stage calculation. Static calculation is done prior to the dynamic analysis with ‘dyn off’ command. For the static analysis, first and the last column in the x-direction and first row in the y-direction are fixed with ‘fix’ command. This helps to prevent the model to move as a body and generate lateral stresses during the ‘consolidation’. For the dynamic calculation however, free-field boundaries are applied to the two vertical sides of the model to avoid distortions in the waves transmissions and to make the model close to reality since the slopes are not isolated with the other topographical features in the lateral sides. The response of earth-quake motion of a structure founded on a deformable soil can be significantly different from the re-sponse of the same structure on a rigid foundation (rock), mainly through the increase in natural peri-ods, a change in the amount of system damping due to wave radiation and damping in the soil and modification of the effective seismic excitation. The base is kept rigid with the introduction of quiet boundaries at the first row in the x- and y- direction.

4.4.6 Dynamic Input

Dynamic input can be applied in terms of velocity, acceleration, or stress history. Before applying dy-namic input, the histories are reset to avoid the effect of static calculation. At the beginning of the dy-namic stage, the dynamic mode is re-activated and the dynamic time reset to zero, to avoid any calcu-lation errors that could be induced from the previous stage (static equilibrium). For this research, a sinusoidal wave function with acceleration of 1 m/s2 and duration of 0.25 sec. is defined using the FISH function available in FLAC. The frequency of the input signals however were kept changing whenever necessary to see the effect of varying frequencies on the amplification. Later on, the defined input function is applied through the bottom of the model (first row).

4.4.7 Dynamic Damping


37

The detail about the mechanical damping is described in Section 3.2.1.2. Before applying the dynamic damping, dynamic analysis is prepared using ‘dyn on’ command after introducing dynamic input and applying necessary boundary conditions. For the dynamic analysis, Rayleigh damping is applied to compensate the energy dissipation through the medium. According to literature (FLAC manual, Paolucci, 2002; Lokmer et al., 2002; etc), the damping for the geological materials lies between 2 % and 5%, and hence 5% Rayleigh damping is used for this numerical dynamic analysis.

4.4.8 Output After introducing dynamic input and the Rayleigh damping, calculation is ordered to the program. The displacement, velocity or acceleration histories at the co-ordinates of interest can be ordered with ‘hist’ command followed by the required parameter (e.g. displacement, velocity etc.) and then followed by the co-ordinates of interest e.g. edge of the slope crest with co-ordinate 40,30 (see Appendix A-1). For this research, the displacement, velocity and acceleration histories at the crest of the slope at the regular distance of 3-5 m from the edge of the slope is studied to look the variation because those are the places of interest where one may want to find out the effect of particular earthquake to the structure he wants to make. Also, because those places are most vulnerable for damage due to earthquake mo-tion according to literatures. The history curve are plotted with ‘plot hold hist’ command and the histo-ries are compared with each other by plotting them together with the dynamic time using ‘vs’ com-mand in FLAC. A sample script used in this study is given in Appendix A-1 and some of the screen captures of the FLAC results are shown in Appendices A-2 and A-3. The results obtained in this way are interpreted and analysed to find out the required relationship of slope geometry and material properties to the displacement and velocity histories as well as amplifica-tion of the slope. The detail of the observed data for the displacement, velocity and acceleration histo-ries is presented in Chapter 5 and the detail interpretation and analysis of the amplification due to the variation of shear modulus, bulk modulus, slope height, as well as the frequency and wavelength of input signals is given in Chapter 6.


38

5. Ground response data as observed from FLAC Calculation

5.1. Introduction

Before finding the relationship between the input parameters and the amplification of slope, prelimi-nary idea about the influence of some of the parameters is necessary. Before starting the actual model, the extent of the influence of the slope angle, friction angle, and the material cohesion as well as that of the distance from the edge of the slope is to be understood. For example: there could be failure of the slope if it is too steep to resist it by the friction angle and cohesion of the material in the slope. The result can be misleading if we compare amplification before failure and after failure. Therefore, it is a good idea to understand the program how it reacts about such phenomenon before jumping to the ac-tual modelling aiming to find out the relationship. This chapter deals about the data obtained from the numerical simulation of 3 slope models for the displacement, velocity and acceleration histories meas-ured at different distances from the slope along in the crest (without qualitative interpretation). The input accelerations for all of the three slope models described in this section are 1 m/s2. In fact, these are the preliminary attempt to finding out the relationship between different input parameters and the output. In other words, these are the observation models to understand the aspects of FLAC and hence these data are much valuable as they show guidelines for the last model (described in Chapter 6) used to formulate the relationship between different input parameters and the amplification. Moreover, it gives idea about the relative importance of horizontal and vertical motion of the slope.

5.2. Description of slope geometries

As described in Section 5.1, three different slope models are used to know the initial and general char-acteristics of the relationship between different input parameters and amplification. Out of three mod-els, model 1 and 3 have similar geometry but only the material properties are different, whereas model 2 has same material properties but different geometry (see Table 5.1). Each of the geometries of the models is described in the following sections.

5.2.1 Slope model 1 and 3

Slope model 1 has a maximum width of 60 m and maximum height of 20 m. The base of the slope is 20 m thick (see Figure 5.1 for detail). The slope angle is kept changing to see the variation of the dis-placement and velocity histories with the change in slope angle.


39

Figure 5.1. General sketch of the geometry of slope model 1 and 3. The geometry for slope model 3 is also same as that of the model 1 but the material properties are dif-ferent as indicated in Table 5.1. Friction angle and material cohesion however are kept changing in all three models to see their effect on the output displacement, velocity and acceleration.

Table 5.1. Material properties used for the simulation. Slope model

Density (Kg/m3)

Bulk modulus (GPa)

Shear modulus (GPa)

Friction angle (º)

Cohesion (MPa)

Tensile strength (MPa)

1 2000 22.6 11.1 42.0 * 6.7 * 1.58 2 2000 22.6 11.1 42.0 * 6.7 * 1.58 3 2000 26.8 7 27.8 * 27 * 1.17 * These parameters are changed whenever necessary to investigate their effect on amplification.

5.2.2 Slope model 2

Slope model 2 has a maximum width of 30 m and slope height of 10 m. The base of the slope is 10 m thick (see Figure 5.2 for detail). The slope angle, material cohesion, and the friction angle in the slope are changed to investigate the effect of those parameters on the overall displacement, velocity, and acceleration. Figure 5.2. General sketch of geometry of the slope model 2.

40 m

20 m

15 m

60 m

Slope angle: changing

20 m

10 m

11 m

30 m

Slope angle: changing


40

5.3. Variation of ground response with the friction angle

Apart from the parameters to be changed (as indicated in sections before), all other variables are kept constant and only the friction angle is changed to investigate the effect of the friction angle on the re-sponse of the slope during dynamic loading. The preliminary results show that for a lower friction an-gle up to around 20-25°, there is some effect on amplification. For greater friction angles, there are not much significant changes in the amplification. It is also remarkable that the change in ground response is higher for greater slope angle. When the slope angle exceeds friction angle, material cohesion is de-termining whether the slope fails or not. In this preliminary stage of study, the response was calculated for a range of slope angles up to 90° to investigate the response of the slope though slope may start failing by so high slope angle. The detail of the observation is described in the following sections.

5.3.1 Slope model 1

Generally, for slope model 1, the change in amplification due to change in friction angle is either very low or sometimes very hard to notice. The general pattern of variation of amplification if any is, higher due to lower friction angle and lower due to higher friction angle. For example: for slope angle of 25.5º, the horizontal acceleration due to friction angle of 5º is 2.5 m/s2, whereas that due to friction angle of 30º is 2.3 m/s2. The detail of the calculation results for the change in friction angle with a slope angle of 25.5º is given in Table 5.2. Details of the simulation and the results for the displace-ment, velocity and acceleration histories are given in Appendix B-1.

Table 5.2. Variation of displacement, velocity, and acceleration histories with friction angle (slope angle 25.5º, model 1).

Friction angle (º)

Input ac-celeration (m/s2)

Vertical accelera-tion (m/s2)

Horizontal accelera-tion (m/s2)

Horizontal displace-ment (m)

Vertical displace-ment (m)

Horizontal velocity (m/s)

Vertical velocity (m/s)

Maximum velocity (m/s)

5 1.00 0.85 2.50 1.50E-02 1.00E-04 6.20E-02 5.00E-03 3.18E-0210 1.00 0.85 2.50 1.50E-02 1.00E-04 6.20E-02 5.00E-03 3.18E-0215 1.00 0.85 2.40 1.50E-02 1.00E-04 6.20E-02 5.00E-03 3.18E-0220 1.00 0.85 2.40 1.50E-02 1.00E-04 6.20E-02 5.00E-03 3.18E-0225 1.00 0.85 2.40 1.50E-02 1.00E-04 6.20E-02 5.00E-03 3.18E-0230 1.00 0.82 2.30 1.45E-02 1.00E-04 6.00E-02 4.50E-03 3.18E-0235 1.00 0.82 2.30 1.45E-02 1.00E-04 6.00E-02 4.50E-03 3.18E-0240 1.00 0.82 2.30 1.45E-02 1.00E-04 6.00E-02 4.50E-03 3.18E-0260 1.00 0.82 2.30 1.45E-02 1.00E-04 6.00E-02 4.50E-03 3.18E-02

5.3.2 Slope model 2

The general variation pattern for the slope model 2 is similar as that for the slope model 1 though with a different magnitude. The calculation for slope model 2 with a slope angle of 30.5º reveals that the displacement in horizontal direction is 1.5 cm for a friction angle of 5º compared to 1.45 cm for a fric-tion angle of 30º. The displacement, velocity, and acceleration in vertical direction are very low but


41

the change due to variation in friction angle is noticeable. The velocity in horizontal direction is 6.4 cm/s for a friction angle of 5º whereas 6.2 cm/s for a friction angle of 30º. The horizontal acceleration however remains constant on 1.5 m/s2 even on changing friction angle. The detail of the numerical calculation results due to the variation of friction angle for the slope angle of 30.5º is given in Table 5.3. The detail of the variation of ground response with the friction angle for slope model 2 is pre-sented in Appendix B-2.


Friction angle (º)








Maximumvelocity (m/s)

5 1.00 0.85 1.50 1.50E-02 1.00E-04 6.40E-02 1.00E-03 3.77E-0210 1.00 0.85 1.50 1.50E-02 1.00E-04 6.40E-02 1.00E-03 3.77E-0215 1.00 0.82 1.50 1.50E-02 1.00E-04 6.40E-02 1.00E-03 3.77E-0220 1.00 0.81 1.50 1.50E-02 1.00E-04 6.40E-02 1.00E-03 3.77E-0225 1.00 0.81 1.50 1.50E-02 1.00E-04 6.40E-02 1.00E-03 3.77E-0230 1.00 0.80 1.50 1.45E-02 8.00E-05 6.20E-02 8.00E-04 3.77E-0235 1.00 0.81 1.50 1.45E-02 8.00E-05 6.20E-02 8.00E-04 3.77E-0240 1.00 0.80 1.50 1.45E-02 8.00E-05 6.20E-02 8.00E-04 3.77E-02

5.3.3 Slope model 3

The calculation for a slope with 25.5º angle reveals that in addition to vertical components, the change in displacement and velocity in the horizontal direction with the change in friction angle is also not very significant. The horizontal acceleration however is fluctuating with minimum of 2 m/s2 for a fric-tion angle of 35º and maximum of 2.2 m/s2 for that of 10º. For other slope angles also, the general pat-tern of variation of ground response is similar giving higher value due to lower friction angle and lower values due to higher friction angle. Detail of the calculation results for slope angle of 25.5º is given in Table 5.4 and the detail for all the slope angles used for simulation is given in Appendix B-3.


Fric-tion angle (º)

Input accel-eration (m/s2)

Vertical ac-celeration (m/s2)

Horizon-tal accelera-tion (m/s2)



Horizon-tal veloc-ity (m/s)



10 1.00 0.85 2.20 1.42E-02 1.00E-04 7.80E-02 6.00E-03 3.37E-0215 1.00 0.82 2.20 1.42E-02 1.00E-04 7.50E-02 6.00E-03 3.37E-0220 1.00 0.81 2.10 1.42E-02 1.00E-04 7.50E-02 6.00E-03 3.37E-0225 1.00 0.81 2.10 1.40E-02 9.00E-05 7.30E-02 5.00E-03 3.37E-0235 1.00 0.80 2.00 1.40E-02 9.00E-05 7.30E-02 5.00E-03 3.37E-02


42

5.4. Variation of ground response with the material cohesion

To investigate the effect of the material cohesion on the acceleration as well as the velocity and dis-placement histories, numerical calculation is performed by changing the material cohesion and keeping all other material properties constant. The calculation performed for three slope models reveals that, in general, the acceleration as well as the displacement and velocity vectors increased with the decrease in material cohesion. The calculation also shows that the change is greater for higher slope angles than for lower slope angles. The detail of the numerical calculation results for the cohesion variation is de-scribed in this section.

5.4.1 Slope model 1

For slope model 1, with a slope angle of 25.5º, there is not much significant difference in the maxi-mum velocity vector even in between the material cohesion of 67 Pa and 6.7 MPa whereas for slope angle of 45º, there is significant change (see Table 5.5). However, there is significant change in the vertical and horizontal accelerations even for the slope angle of 25.5º. The increase in acceleration his-tory is more or less regular with the decrease in material cohesion for most of the calculations per-formed for different slope angles (see Appendix B-1 for detail). The magnitude of velocity and dis-placement and their variation due to change in material cohesion is generally very low for slope model 1.

Table 5.5. Variation of displacement, velocity, and acceleration histories with material cohesion

(slope angle 45º, model 1). Material cohesion (Pa)









10 1.00 6.00 4.00 1.50E-02 8.00E-04 6.30E-02 1.10E-02 1.02E+0067 1.00 6.60 4.20 1.50E-02 8.00E-04 6.30E-02 1.00E-02 9.37E-01

670 1.00 9.00 3.50 1.50E-02 6.00E-04 6.20E-02 1.00E-02 8.42E-016700 1.00 4.80 5.80 1.50E-02 5.00E-04 6.30E-02 1.10E-02 3.19E-02

67000 1.00 0.90 2.00 1.50E-02 1.00E-04 6.30E-02 5.00E-03 3.18E-026700000 1.00 0.90 2.10 1.56E-02 1.00E-05 6.40E-02 5.00E-03 3.18E-02 It is observed that the slope starts failing when very low value of material cohesion is applied and the slope angle is very high. For example: With material cohesion of 67 Pa in the slope with a 75.9º slope angle, the velocity and accelerations are exceptionally high. There is significant change in the velocity, displacement, and acceleration histories with the change in material cohesion for the slope angle of 75.9º. A similar pattern is observed in the slope angle of 90º. Before the failure of slope, the highest velocity was 2.90 m/s for material cohesion of 6.7*103 Pa, however it is 3.18 cm/s for material cohe-sion of 6.7*104 Pa. Decrease in cohesion by 10 times, was not showing very big difference for lower slope angles but for 90º slope angle, this difference is pretty high (might be due to the failure of the slope). The horizontal acceleration varies from 2.2 m/s2 to 3 m/s2 for the same range of cohesion varia-tion. It is remarkable to note that after the failure of slope, the velocity and acceleration in the vertical


43

direction are exceptionally higher than that in the horizontal direction. For detail, see Table 5.6 and Appendix B-1.

Table 5.6. Variation of displacement, velocity, and acceleration histories with material cohesion

(slope angle 90º, model 1). Material cohesion (Pa)









6700 1.00 9.00 3.00 2.20E-01 6.00E-01 9.00E-01 2.39E+00 2.90E+0067000 1.00 1.50 2.20 1.50E-02 1.00E-04 6.60E-02 9.00E-03 3.18E-02

670000 1.00 1.50 2.20 1.50E-02 1.00E-04 6.50E-02 8.00E-03 3.18E-026700000 1.00 2.00 2.20 1.57E-02 1.00E-04 6.80E-02 1.00E-02 3.18E-02

5.4.2 Slope model 2

The calculation for slope model 2 reveals that the change in velocity and displacement vectors as well as the acceleration is more sensitive after some particular critical value of cohesion. As can be seen, the slope starts failing for a material cohesion of 1*104 Pa giving maximum velocity of 1.03 m/s but there is not much effect for cohesion of 6.7*104 Pa (see Table 5.8 or calculation numbers 102 and 103 in Appendix B-2) giving the maximum velocity of 3.98 cm/s only. It is also observed that the steeper slopes are more affected by the change in cohesion compared to the gentler slopes. This is evident from the fact that there is no significant change in the velocity and displacement vectors due to the change in material cohesion for a slope angles of 55º and less but significant variation for a slope an-gles of 68.2º and more. For detail, see Table 5.7.

Table 5.7. Variation of displacement, velocity, and acceleration histories with material cohesion (slope angle 42.3º, model 2).

Material cohesion (Pa)









67 1.00 3.50 2.50 1.42E-02 8.00E-05 6.40E-02 2.00E-03 3.84E-02670 1.00 2.60 1.60 1.42E-02 8.00E-05 6.40E-02 2.00E-03 3.79E-02

6700 1.00 0.90 1.60 1.42E-02 8.00E-05 6.40E-02 2.00E-03 3.78E-0267000 1.00 0.82 1.40 1.40E-02 6.00E-05 6.20E-02 1.00E-03 3.77E-02

670000 1.00 0.82 1.40 1.40E-02 6.00E-05 6.20E-02 1.00E-03 3.77E-026700000 1.00 0.90 1.50 1.45E-02 1.00E-04 6.30E-02 2.00E-03 3.78E-02

Table 5.8. Variation of displacement, velocity, and acceleration histories with material cohesion (slope angle 90º, model 2).


44










670 1.00 6.00 3.00 3.00E-01 6.00E-01 1.00E+00 2.50E+00 3.13E+006700 1.00 3.60 2.00 2.00E-01 3.00E-01 8.00E-01 1.40E+00 1.65E+00

10000 1.00 4.20 1.60 1.30E-01 2.20E-01 5.00E-01 8.50E-01 1.03E+0067000 1.00 1.00 1.50 1.45E-02 1.00E-05 6.40E-02 2.00E-03 3.98E-02

670000 1.00 1.00 1.50 1.45E-02 1.00E-05 6.40E-02 2.00E-03 3.98E-02

5.4.3 Slope model 3

The general relation of the material cohesion with the velocity and displacement histories for model 3 is similar as that for slope models 1 and 2. For the greater slope angles, the change is higher in magni-tude as well as more sensitive. For example for a slope angle of 25.5º, the maximum velocity vectors for material cohesions of 270 Pa and 2.7*105 Pa are 3.61 cm/s and 3.37 cm/s respectively, whereas for the same values of cohesion, the maximum velocity vectors are 4.12 m/s to 3.69 cm/s with a slope an-gle of 90º. For even lower cohesion, the slope starts failing with extraordinary values of velocity, dis-placement, and accelerations for a slope angle of 90º. The acceleration in vertical direction is maxi-mum after slope fails, which is 9.0 m/s2 for a slope angle of 90º with a material cohesion of 270 Pa whereas the horizontal acceleration for the same value of material cohesion is 2.5 m/s2. For detail, see Tables 5.9 and 5.10.

Table 5.9. Variation of displacement, velocity, and acceleration histories with material cohesion (slope angle 25.5º, model 3).










27 1.00 5.00 3.00 1.50E-02 8.00E-04 8.00E-02 1.00E-02 8.77E-02270 1.00 4.00 3.00 1.50E-02 8.00E-04 8.00E-02 1.00E-02 3.61E-02

2700 1.00 4.00 3.20 1.50E-02 8.00E-04 7.80E-02 8.00E-03 3.59E-0227000 1.00 3.50 4.00 1.50E-02 4.00E-04 7.80E-02 7.00E-03 3.55E-02

270000 1.00 0.82 2.20 1.42E-02 1.00E-04 7.80E-02 6.00E-03 3.37E-022700000 1.00 0.80 2.10 1.42E-02 1.00E-04 8.00E-02 5.00E-03 3.37E-02

Table 5.10. Variation of displacement, velocity, and acceleration histories with material cohesion (slope angle 90º, model 3).


45










270 1.00 9.00 2.50 1.30E-01 8.20E-01 6.00E-01 3.30E+00 4.12E+002700 1.00 8.00 2.50 2.00E-01 8.00E-01 7.00E-01 3.30E+00 3.83E+00

27000 1.00 4.20 2.50 1.60E+00 3.90E-01 6.50E-01 1.54E+00 1.84E+00270000 1.00 1.30 2.30 1.50E-02 3.00E-04 8.10E-02 9.00E-03 3.69E-02

27000000 1.00 1.20 2.30 1.45E-02 5.00E-04 8.60E-02 9.00E-03 3.69E-02

5.5. Variation of ground response with the slope angle

The numerical calculations performed for three slope models in general reveals that the velocity, dis-placement, and acceleration vectors have direct relationship with the slope angle. It is however de-pendent on the other material properties of the slope like the bulk modulus, shear modulus, material cohesion, friction angle, and tensile strength. It is evident from the fact that the calculation for the slope model 1 with higher values of shear modus and friction angle does not show significant change in the velocity and displacement histories with a change in slope angle. At the same time, it is also ob-served that there should be another controlling factor, height of the slope, in conjunction with the slope angle for such change because the same material properties used for the model 2 gives significant changes in the maximum velocities. Similarly, the lower shear modulus and friction angle used for slope model 3 (for which geometry is same as that of model 1) also gives significant change.

5.5.1 Slope model 1

As already explained, there is no significant change in the displacement and velocity histories with slope angles for slope model 1. The maximum velocity remains almost constant with a magnitude of 3.183 cm/s for a slope angle up to 90°. The acceleration histories however are changed significantly. The horizontal acceleration varied from 1.8 m/s2 for a slope angle of 25.5° to 2.2 m/s2 for a slope angle of 90°. The displacement and velocity histories remain almost constant for the change in slope angle. For detail, see Table 5.11.

Table 5.11. Variation of displacement, velocity, and acceleration histories with slope angle (model

1).


46

Slope angle (º)









25.50 1.00 0.80 1.80 1.55E-02 1.00E-05 6.20E-02 5.00E-03 3.18E-0229.70 1.00 0.82 1.80 1.55E-02 1.00E-05 6.30E-02 5.00E-03 3.18E-0238.60 1.00 1.00 2.00 1.56E-02 1.00E-05 6.40E-02 5.00E-03 3.18E-0245.00 1.00 0.90 2.10 1.56E-02 1.00E-05 6.40E-02 5.00E-03 3.18E-0253.00 1.00 1.20 2.00 1.56E-02 1.00E-05 6.40E-02 6.00E-03 3.18E-0263.00 1.00 1.20 2.10 1.57E-02 1.00E-05 6.40E-02 3.00E-03 3.19E-0275.90 1.00 1.20 2.20 1.57E-02 1.00E-05 6.50E-02 6.00E-03 3.18E-0281.00 1.00 1.50 2.20 1.57E-02 1.00E-04 6.60E-02 8.00E-03 3.18E-0290.00 1.00 2.00 2.20 1.57E-02 1.00E-04 6.80E-02 1.00E-02 3.18E-02

5.5.2 Slope model 2

There is significant change in maximum velocity for slope model 2 by the change in slope angle (for example: 3.774 cm/s for slope angle of 30.5º compared to 3.983 cm/s for a slope angle of 90°. The vertical displacement and velocities for lower slope angles are insignificant but are reasonable for higher slope angles. There is significant amplification of horizontal acceleration but the variation with the slope angle is not linear. For example: horizontal acceleration is 1.3 m/s2 for a slope angle of 42.3º whereas 1.5 m/s2 for a slope angle of 78.7º (see Table 5.12 for detail).

Table 5.12. Variation of displacement, velocity, and acceleration histories with slope angle (model 2).

Slope angle (º)









30.50 1.00 0.80 1.30 1.45E-02 1.00E-05 6.20E-02 2.00E-03 3.77E-0235.50 1.00 0.80 1.30 1.45E-02 1.00E-04 6.30E-02 1.00E-03 3.77E-0242.30 1.00 0.90 1.30 1.45E-02 1.00E-04 6.30E-02 2.00E-03 3.78E-0255.00 1.00 0.85 1.40 1.45E-02 1.00E-04 6.40E-02 2.00E-03 3.83E-0268.20 1.00 0.80 1.40 1.50E-02 1.00E-04 6.40E-02 1.00E-03 3.89E-0278.70 1.00 1.20 1.50 1.50E-02 1.00E-04 6.40E-02 2.00E-03 3.93E-0284.30 1.00 1.70 1.50 1.50E-02 1.00E-04 6.40E-02 3.00E-03 3.96E-0290.00 1.00 1.00 1.50 1.50E-02 1.00E-04 6.50E-02 3.00E-03 3.98E-02

5.5.3 Slope model 3

Similar to slope model 2, there is also significant change in maximum velocity for model 3 with the change in slope angle. For example: maximum velocity of 3.37 cm/s for a slope angle of 25.5° degree, whereas 3.69 cm/s for a slope angle of 90°. The displacement, velocity, and acceleration in vertical direction are not much significant. The amplification of horizontal acceleration is significant. The hori-


47

horizontal acceleration for a slope angle of 25.5º and that for a slope angle of 90º are 2.1 m/s2 and 2.5 m/s2 respectively. For the intermediate slope angles, the displacement, velocity and acceleration histo-ries remain in between those two extremities but some of the histories show irregular pattern of change as well. For example: the horizontal acceleration remains unchanged in 2.5 m/s2 for the slope angle of 38.6º through 90º (see Table 5.13). This might be because the friction angle is less than 38.6º and for all slope angles greater than friction angle, the amplification remains unchanged.

Table 5.13. Variation of displacement, velocity, and acceleration histories with slope angle (model 3).

Slope angle (º)









25.50 1.00 0.80 2.10 1.42E-02 1.00E-04 8.00E-02 5.00E-03 3.37E-0238.60 1.00 0.80 2.50 1.42E-02 1.00E-04 8.00E-02 6.00E-03 3.39E-0253.00 1.00 1.20 2.50 1.42E-02 1.00E-04 8.00E-02 6.00E-03 3.49E-0275.90 1.00 1.20 2.50 1.45E-02 1.00E-04 8.60E-02 7.00E-03 3.62E-0281.00 1.00 1.20 2.50 1.45E-02 5.00E-04 8.50E-02 7.00E-03 3.65E-0290.00 1.00 1.20 2.50 1.45E-02 5.00E-04 8.60E-02 9.00E-03 3.69E-02

5.6. Variation of ground response with the distance from the edge of the slope crest

Lot of numerical calculations were performed to investigate the effect of the distance from the edge of the slope crest on the amplification during dynamic loading. The calculations show that in general the amplification is decreased with the distance from the edge of the slope. The change pattern however is sometimes not consistent for the higher slope angles and lower values of cohesion and friction angle. The variation of the amplification with distance from edge of the crest of the slope is described in this section.

5.6.1 Slope model 1

There is slight variation in the displacement and velocity vectors with the distance from the crest of the slope. For greater slope angles (greater than friction angle), variation in the response is sometimes ir-regular. However, in most of the cases, the amplification is higher at the slope crest edge than farther away. As for example, for a slope angle of 25.5º, horizontal acceleration at the edge of the crest is 1.8 m/s2, compared to 1.5 m/s2 at 5 m away from the edge. However, the difference in horizontal accelera-tion for slope angle of 75.9º is irregular: horizontal acceleration is 2.1 m/s2 at the edge and maximum of 2.2 m/s2 at 15 m away from the edge of the slope crest (see Table 5.14 for more detail). This could be the reason that for slope angles greater than the friction angle of the material in the slope, the re-sponse is sometimes irregular.


48

Table 5.14. Variation of velocity and acceleration histories with distance from the edge of slope crest (model 1).

Slope angle (º)

Input accelera-tion (m/s2)

Distance from the slope edge (m)

Vertical velocity (cm/s)

Horizon-tal veloc-ity (cm/s)


Horizontal-acceleration (m/s2)

25.50 1.00 0 0.50 6.20 0.80 1.8025.50 1.00 5 0.50 6.20 0.75 1.5025.50 1.00 10 0.50 6.20 0.70 1.5025.50 1.00 15 0.50 6.20 0.65 1.5075.90 1.00 0 0.60 6.50 1.20 2.1075.90 1.00 5 0.40 6.40 1.00 2.0075.90 1.00 10 0.30 6.40 0.90 2.0075.90 1.00 15 0.30 6.40 0.55 2.2081.00 1.00 0 0.80 6.60 1.50 2.2081.00 1.00 5 0.50 6.50 1.00 2.0081.00 1.00 10 0.30 6.50 0.70 2.0081.00 1.00 15 0.20 6.50 0.50 2.00

5.6.2 Slope model 2

Similar to slope model 1, there is slight variation in the velocity and displacement histories with the distance from the crest edge for slope model 2 with lower slope angles. Regarding the acceleration, there is regular decrease in vertical and horizontal acceleration with increase in distance from the edge of the crest. The horizontal acceleration at the edge of the slope crest for a slope angle of 25.5º is 2.8 m/s2 and that at 11 m away from edge is 1.6 m/s2. For the slope angle of 78.7º, the horizontal accelera-tion is 1.2 m/s2 at the edge of the crest whereas 0.58 m/s2 at 11 m away from the edge. The horizontal accelerations for a 84.3º slope angle however give irregular pattern: 1.4 m/s2 at the edge of the slope as well as 11 m away from the slope edge with lower values in between (see Table 5.15).

5.6.3 Slope model 3

The variation pattern of the displacement and velocity histories with the distance from the crest edge is similar for slope model 3 as that of the model 1 and 2. The variation in the horizontal acceleration with the distance from the edge of the slope crest is only slight. For example: for a slope angle of 25.5º, at the edge of the crest, horizontal acceleration is 2.1 m/s2 whereas that at 10 m away from the edge is 2.0 m/s2. The decrease in horizontal acceleration with the distance from the slope edge, for a slope angle of 81º, however, is not regular. For detail, see Table 5.16.



49

Slope angle (º)

Input accel-eration (m/s2)

Distance from the slope edge (m)


Horizontal velocity (cm/s)

Vertical-acceleration (m/s2)

Horizontal acceleration (m/s2)

30.50 1.00 0 0.20 6.20 0.80 2.8030.50 1.00 3 0.20 6.20 0.70 2.5030.50 1.00 6 0.20 6.20 0.60 2.5030.50 1.00 9 0.20 6.20 0.650 2.5030.50 1.00 11 0.20 6.20 0.60 1.6078.70 1.00 0 0.20 6.40 1.20 1.2078.70 1.00 3 0.20 6.40 0.80 0.8078.70 1.00 6 0.10 6.40 0.75 0.7578.70 1.00 9 0.10 6.40 0.58 0.5878.70 1.00 11 0.10 6.40 0.58 0.5884.30 1.00 0 0.30 6.40 1.70 1.4084.30 1.00 3 0.30 6.40 0.90 1.3084.30 1.00 6 0.20 6.40 0.70 1.1084.30 1.00 9 0.10 6.40 0.55 1.1084.30 1.00 11 0.10 6.40 0.50 1.40


Slope an-gle (º)

Input acceleration (m/s2)

Distance from the edge of slope crest (m)


Horizontal velocity (cm/s)

Vertical acceleration (m/s2)

Horizontal acceleration (m/s2)

25.50 1.00 0 0.50 8.00 0.80 2.1025.50 1.00 5 0.50 8.00 0.79 2.1025.50 1.00 10 0.50 8.00 0.75 2.0025.50 1.00 15 0.50 8.00 0.70 2.0075.90 1.00 0 0.50 8.20 1.20 2.5075.90 1.00 5 0.30 8.20 1.10 2.3075.90 1.00 10 0.20 8.20 0.70 2.3075.90 1.00 15 0.20 8.20 0.65 2.3081.00 1.00 0 0.70 8.50 1.20 2.5081.00 1.00 5 0.40 8.50 1.20 2.3081.00 1.00 10 0.30 8.50 0.80 2.3081.00 1.00 15 0.30 8.50 0.70 2.40

5.7. Important conclusions of the calculation results obtained from first three slope models

The preliminary calculation results obtained from the three slope models, described in Section 5.2 through Section 5.6, are of great importance for the fourth model, which is to be used to find out the relationship between different input parameters and the output. These calculations help to understand the FLAC program and its response towards the material properties and the slope geometry in terms of


50

displacement, velocity, and the acceleration histories when dynamic load is applied. Some of the im-portant conclusions drawn from these calculation results are:

When the slope angle is very high compared to the friction angle, slope starts failing (after ex-ceeding the critical value of material cohesion). The ground response is generally higher due to lower friction angle and lower due to higher

friction angle. However, the magnitude of variation with the change in friction angle is not very high. The variation is not much significant for higher friction angles. The effect is sig-nificant if the slope angle is higher but the pattern of change is irregular due to the failure of slope. The ground response obtained for slope angles more than the friction angle might not give accurate and realistic result. Generally, the displacement, velocity, and accelerations are maximum at the edge of the slope

crest than far away from it. Nevertheless, variation in horizontal acceleration is not regular if the slope angle is higher than friction angle. The slope starts failing if the material cohesion is too low compared to the slope angle. In ad-

dition, the change in ground response is greater for the higher slope angles than for lower. The changes in ground response are large around critical values of material cohesion. The slope height is one of the important factors to be considered to investigate the variation of

ground response with material properties and the slope geometries, as is observed that for the same material used, slope model 1 and slope model 2 (which have different slope heights) show different results when simulated for similar ranges of slope angles. The relationship of ground response with the slope geometry and material properties is not lin-

ear.


51

6. Relation of amplification with earth-quake input signals, material proper-ties and slope geometry

6.1. Introduction

Ground motion caused by earthquakes is generally characterized in terms of ground surface displace-ment, velocity, and acceleration. Geotechnical engineers use acceleration rather than other two because acceleration is directly related to the dynamic forces that earthquake induce on the soil or rock mass (Day, 2002). For geotechnical analyses, the measure of the cyclic ground motion is represented by the maximum horizontal acceleration at the ground surface, Amax (also called peak horizontal ground ac-celeration). However, for most earthquakes, horizontal acceleration is greater than the vertical and thus peak horizontal ground acceleration is turned to be the peak ground acceleration (PGA). The horizon-tal acceleration is used in this study to find out its relationship with various input parameters. The amplification in this study refers to the ratio of the maximum output acceleration in the crest along the horizontal direction of the slope, to the input acceleration. The slope geometry is shown in Figure 6.1. The parts of the slope geometry that can be varied are indicated in the figure by the term “chang-ing”. The data obtained from the simulation in FLAC for the amplification of the slope is analysed to reach some conclusions about the relationship between different input parameters and amplification. Slope angle, slope height, bulk modulus, shear modulus and the frequency (or sometimes wavelength) of the input signals are the parameters that are continuously changed by varying only one parameter for one calculation to determine the effect of that particular parameter. The relationship between those parameters and the amplification of the slope from the data analysis and its interpretation is described in this chapter. Figure 6.1. Geometry of the slope model used for the numerical calculation.

Changing

20 m

14 m

100 m

25.5º or changing (if indicated)


52

6.2. Bulk modulus versus amplification

The calculations performed by varying the bulk modulus is indicated in the matrix presented in Ap-pendices C1- C8, which also consists of the detail of the parameters used for calculation as well as the values of amplifications observed (calculation numbers 1 to 673). Slope angle and some of the mate-rial properties used for this purpose are given in Table 6.1. The calculation is performed for slope heights of 10 m, 20 m, 25 m, 30 m, 35 m, and 40 m. For all those sets of slope height and bulk modulus combination, calculation is repeated for the input frequencies of 3 Hz, 5 Hz, 10 Hz, and 15 Hz. To find out the effect of slope angle, the calculation is first performed for the slope angle of 25.5º and then repeated for the slope angle of 35.5º. In this way, the relation of amplification with the bulk modulus for the set of slope height and input frequencies are obtained and are compared for slope an-gles of 25.5º and 35.5º.

Table 6.1. Input parameters used for the simulation to get the relation of bulk modulus on amplifica-tion.

Slope angle

Density (Kg/m3)

Shear modulus (GPa)

Bulk modulus

Friction angle (º)

Cohesion (MPa)


Input fre-quency

Slope height

25.5º or 35.5º

2000 10 Variable 42.0 6.7 1.58 Variable Variable

The values of amplifications as obtained from FLAC calculation are read from the computer screen with the naked eye, and the accuracy of the reading is in the order of about ± 0.3 (the precision of about decimal 3). That is why, the values only slightly different from each other could not be differen-tiated, and the curves are straight in those cases. The plots of the bulk modulus in logarithmic scale versus amplification for different input frequencies are given in Figures 6.2 through 6.5 (see Appendi-ces D-1 and D-2 for detail). These figures clearly indicate that for the certain range of smaller bulk modulus, the amplification is maximum but remains constant whereas after some threshold values it starts decreasing as is shown by the first bending of the curve. There exist another bending, towards the area of higher bulk modulus, for which the amplification is minimum and remains constant for the higher range of bulk modulus from that second threshold value. This behaviour is shown by all the sets of input frequency and slope heights though the magnitude of amplification is different. Moreover, a similar relationship is obtained for the higher slope angle of 35.5º though the magnitude of amplifica-tion is slightly different (Figures 6.4 and 6.5). It is clearly seen from these curves that magnitude of amplification cannot be generalized in terms of the material properties alone but is determined by the combination of input frequency and the slope geometry. However, there is a clear trend of amplifica-tion for the different values of bulk modulus. The nature of the relationship between bulk modulus and amplification is described in the following sections.


53

Bulk modulus vs. amplification for different slope heights (slope angle 25.5 degree)

1

1.5

2

2.5

3

1 10 100 1000 10000Bulk modulus (Mpa)

Am

plifi

catio

n

slope height 10 m slope height 20 m slope height 25 mslope height 30 m slope height 31 m slope height 35 mslope height 40 m

Figure 6.2. Bulk modulus versus amplification for frequency of 3 Hz (with slope angle 25.5 degree).

Bulk modulus Vs. amplification for different slope heights (slope angle 25.5 degree)

02468

1012141618

1 10 100 1000 10000Bulk modulus (MPa)

Am

plifi

catio

n

Slope height 10 m slope height 20 m slope height 25 mslope height 30 m slope height 35 m slope height 40 m



54

Bulk modulus versus amplification for different slope heights with 3 Hz frequency (slope angle 35.5 degree)

0

0.5

1

1.5

2

2.5

3

1 10 100 1000 10000Bulk modulus (Mpa)

Am

plifi

catio

n



Bulk modulus Vs. amplification for different slope heights with 15 Hz input frequency (slope angle 35.5 degree)

02468

1012141618

1 10 100 1000 10000Bulk modulus (Mpa)

Am

plifi

catio

n

slope height 10 m slope height 20 m slope height 25 mslope height 30 m slope height 35 m slope height 40 m



55

6.2.1 General shape of function

As seen in the plot of the bulk modulus versus amplification, the general shape of the function to show the relation between bulk modulus and amplification can be represented schematically as shown in Figure 6.6. There are clearly two threshold values of bulk modulus from where the amplification either starts decreasing or remains constant. Figure 6.6. Schematic diagram showing general shape of the function for bulk modulus versus ampli-

fication Initially for the lower values of bulk modulus, the amplification is maximum which remains constant for some particular value of bulk modulus. In general, the first threshold value range from 500-1000 MPa from where the amplification starts decreasing. For the higher bulk modulus values, amplification starts decreasing approximately linearly on a log scale until for almost 5000 MPa giving a clear slope in the curve after which amplification remains constant giving a second threshold value. This behav-iour could be due to the combining effect of input parameters like slope angle, shear modulus and ma-terial density. The pattern (qualitative relationship) is recognized with all the combinations of input frequency, slope height, and the slope angle. If it were due to the effect of input frequency and slope height, the pattern would have been different for the different combinations of input frequency and slope height. However, the magnitude (quantitative relationship) of maximum and minimum amplifi-cation as well as the steepness of the slope in the bulk modulus versus amplification plot is seen to be the function of slope height and frequency of the input signal. As for example: For the slope height of 40 m, the maximum amplifications with the input frequency of 10 Hz is 12 and that due to 3 Hz is 2.9 (though all other input parameters are same). The influence of input frequency and slope height is de-scribed in the following sections. In general, the function for the bulk modulus versus amplification can be given by the following equations: If Log (K) < Log (K1), A = Amax (6.1) If Log (K1)<Log (K)<Log (K2) then

A

Log (K) Log (K1) Log (K2)

Amin

Amax slope


56

1max max min

2 1

( ) ( ) *( )( ) ( )

Log K Log KA A A ALog K Log K

−= − −

− (6.2)

If Log (K) > Log (K2), A = Amin (6.3) where, K is input bulk modulus, K1 is first threshold bulk modulus, K2 is second threshold bulk modulus (see Figure 6.6), A is any amplification, Amax is maximum amplification and Amin is minimum amplification for the particular combination of slope height and frequency. The influence of any resonance and harmonic effects on the plot of the bulk modulus versus amplifica-tion is not observed, which is the typical feature for the variation with shear modulus (described in the coming sections). Probably, the influence of bulk modulus in the overall amplification of the slope is very less compared to the shear modulus and the property of input signals. In addition, basically shear wave velocity rather than the P-wave velocity are investigated and bulk modulus influence in particu-lar P-wave velocity. However, it is not fully understood why the amplifications for the certain range of bulk modulus values remains constant. The plot of the bulk modulus versus maximum and minimum amplifications for different frequency and slope heights is shown in Figures 6.7 through 6.10. It is clear from the figures that the bandwidth for the maximum to minimum amplifications increases as the frequency of input signal is increased. For example: for the input frequency of 3 Hz, the maximum and minimum amplifications are 2.9 and 1.8 respectively. The difference between the maximum and minimum amplification is thus 1.1. On the other hand, the difference between the maximum and minimum amplification for the input frequency of 5 Hz, 10 Hz and 15 Hz are 1.4, 10, and 12.5 respectively.

Bulk modulus Vs. Maximum and minimum amplifications for frequency 3 Hz

0

1

2

3

4

1 10 100 1000 10000

Bulk modulus (MPa)

Am

plifi

catio

n

minimum amplification, slope height 10 mmaximum amplification, slope height 40 m

Figure 6.7. Bulk modulus versus maximum and minimum amplifications for the input frequency of 3 Hz, slope angle 25.5 degree.


57

Bulk modulus Vs. maximum and minimum amplification for frequency 5 Hz

0

1

2

3

4

1 10 100 1000 10000Bulk modulus (MPa)

Am

plifi

catio

n


Figure 6.8. Bulk modulus versus maximum and minimum amplifications for the input frequencies of 5

Hz, slope angle 25.5 degree.


02468

101214

1 10 100 1000 10000Bulk modulus (MPa)

Am

plifi

catio

n


Figure 6.9. Bulk modulus versus maximum and minimum amplifications for the input frequencies of

10 Hz, slope angle 25.5 degree.


58


0

5

10

15

20

1 10 100 1000 10000Bulk modulus (MPa)

Am

plifi

catio

n


Figure 6.10. Bulk modulus versus maximum and minimum amplifications for the input frequency of 15 Hz, slope angle 25.5 degree.

It is also seen from the plot that the threshold value of bulk modulus from which amplification starts decreasing is the function of slope height and the input frequency.

6.2.2 Influence of slope height and input frequency

As shown in Figures 6.2 through 6.5, the magnitude of maximum amplification is different for differ-ent slope heights. However, it is the function of slope height and input frequency together (see Table 6.2 for detail) because it is seen that some slopes are amplified more by one input frequency and other slopes with different heights are amplified more by another input frequency. For example: with a slope angle of 25.5º, for the input frequency of 10 Hz, a 40 m slope height gives the maximum amplification whereas for the input frequency of 15 Hz, a 20 m slope height gives the maximum amplification (see Figures 6.11 and 6.12). A similar relation is obtained for a slope angle of 35.5º (Figure 6.13 and 6.14). The influence of slope angle on the overall amplification is not very clear, but it seems that the higher slope angles influence the lower input frequency signals more than the higher. The data however is not sufficient to compare and said exactly about the relationship because the calculation is performed only for two slope angles, 25.5º and 35.5º (see also Appendix D-3). In general, the amplification for higher slope height is maximum with the lower input frequency whereas that for the lower slope heights is maximum with the higher input frequency. It can be ex-plained by the theoretical concept that the taller objects will have the greater influence of lower fre-quencies whereas the shorter objects are more affected by the higher frequency signals (Day, 2002, also see Section 2.4.1). Moreover, it can be explained by the resonance frequency of the slope. When the resonance frequency of the slope coincides with the frequency of input signals, larger amplification occurs. The maximum amplification due to 5 Hz frequency is smaller than the minimum amplification due to 15 Hz frequency as can be seen in Figures 6.11 and 6.12, that could be due to the higher damp-ing used in FLAC calculation for higher frequency signals.


59

Slope height Vs. maximum amplification for different input frequencies, slope angle 25.5 degree

1.5

2

2.5

3

3.5

4

5 15 25 35 45

Slope height (m)

Max

imum

am

plifi

catio

n

frequency 3 Hz frequency 5 Hz

Figure 6.11. Variation of amplification with slope height for different input frequencies with slope

angle of 25.5º, input frequency 3 and 5 Hz.

Slope height Vs. maximum amplification for different input frequencies, slope angle 25.5 degree

02468

1012141618

5 15 25 35 45Slope height (m)

Max

imum

am

plifi

catio

n





60

Table 6.2. Maximum amplifications for the combination of slope height and input frequency.

Slope height (m)

Wavelength (m)

Ratio of slope height to wavelength

Frequency (Hz)

Maximum amplification

Slope angle of 25.5 de-gree

Slope angle of 35.5 de-gree

10 745.36 0.01 3 1.80 1.8020 745.36 0.03 3 2.30 2.0025 745.36 0.03 3 2.30 2.5030 745.36 0.04 3 1.90 2.0035 745.36 0.05 3 2.30 2.2040 745.36 0.05 3 2.90 2.5010 447.21 0.02 5 2.30 2.4020 447.21 0.04 5 2.50 2.5025 447.21 0.06 5 2.30 2.2030 447.21 0.07 5 3.50 3.2035 447.21 0.08 5 2.50 2.2040 447.21 0.09 5 2.50 3.5010 223.61 0.04 10 2.00 2.1020 223.61 0.09 10 5.20 5.2025 223.61 0.11 10 10.00 7.5030 223.61 0.13 10 12.00 12.0035 223.61 0.16 10 7.80 7.8040 223.61 0.18 10 12.00 14.0010 149.07 0.07 15 7.00 7.5020 149.07 0.13 15 17.00 17.0025 149.07 0.17 15 9.50 9.5030 149.07 0.20 15 6.00 6.0035 149.07 0.23 15 10.00 9.8040 149.07 0.27 15 4.50 4.20

Another important feature shown by these calculation results is the harmonic effect. A harmonic is a signal or wave whose frequency is an integral (whole number) multiple of the original frequency. This can also refer to the ratio of the frequency of such a signal or wave to the frequency of the reference signal or wave. These harmonic frequencies may generate standing waves (discussed in section 6.3.1), which controls the amplifying in slopes. In other words, if w represents the wavelength of the wave, the second harmonic has a wavelength of w/2, the third harmonic has a wavelength of w/3 and so on. This effect can be observed in table 6.2 also. For example for the slope height of 30 m, wavelength of 223.61 m and 447.21 m both give higher amplifications, which are multiples of each other. As shown by Figures 6.11 through 6.14 (see also Table 6.2), the peaks of amplification repeat within a certain interval of slope heights for the range of frequencies. The slope height versus amplification curves reflects harmonic effect because the fundamental frequency for a particular slope height might be the harmonic frequency for another slope height or vice versa. For the combination of that particu-lar slope height and the frequency, standing waves are generated. For example: in Figure 6.13, with the input signals of 3 Hz frequency, slope height of 25 m gives the first peak with maximum amplification, but another peak repeated at the slope height of 40 m (if we calculate for the slope height of 50 m, it might give even bigger peak and then starts decreasing). From the Table 6.2, it is clear that slope height to wavelength ratio in the range of 0.07 to 0.23 is most vulnerable for ampli-


61

height to wavelength ratio in the range of 0.07 to 0.23 is most vulnerable for amplification (slope height to wavelength ratio of 0.07 to 0.20 also seen to be vulnerable in another calculation, as can be seen in Figure 6.24 in Section 6.3.1).

Slope height versus maximum amplification for different input frequencies, slope angle 35.5 degree

1.51.7

1.92.1

2.32.5

2.72.9

3.13.3

0 10 20 30 40 50

Slope height (m)

Max

imum

Am

plifi

catio

n




Slope height versus maximum amplification for different input frequencies, slope angle 35.5 degree

1.53.55.57.59.5

11.513.515.517.519.5

0 10 20 30 40 50

Slope height (m)

Max

imum

Am

plifi

catio

n





62

Slope height Vs. max amplification for different slope angles (input frequency of 3 Hz)

1.5

2

2.5

3

3.5

4

0 10 20 30 40 50 60 70

Slope height (m)

Max

imum

am

plifi

catio

n

slope angle 25.5 degree slope angle 35.5 degree

Figure 6.15. Variation of maximum amplification with slope height for different slope angles. It is remarkable to notice a harmonic effect in the plot of the slope height versus maximum amplifica-tion for 3 Hz frequency (Figure 6.15). There are two peaks for the maximum amplifications due to slope heights. It is quite possible that the second greater peak which is around 45 m slope heights is due to the resonance effect and the first smaller peak for the slope height of 25 m is the harmonic ef-fect. The pattern of the similar plot for other input frequencies is also the same (see Appendix D-3). The plot of the frequency versus maximum amplifications for different slope heights is shown in Fig-ure 6.16 & Figure 6.17 for the slope angle of 25.5º and in Figure 6.18 & Figure 6.19 for the slope an-gle of 35.5º. The range of lower values of frequencies, which give higher amplification, depends on the slope height. The magnitude of maximum amplifications for a particular slope height due to spe-cific frequency is not varied too much due to the change in slope angle as can be seen from the Figure 6.8 for slope angle of 25.5º and Figure 6.9 for slope angle of 35.5º. However, if the slope angle is in-creased to more than the friction angle of the material in the slope, maximum amplification might in-crease substantially before it fails because material cohesion might resist the slope from failing until some critical slope angle. It is evident from Figures 6.16 and 6.17 that for the slope heights of 10 m and 20 m, the maximum amplification increased sharply for more than 10 Hz input frequency. On the other hand, the magnitude of maximum amplification for the slope height of 30 m and 40 m is sharply increased until the frequency of 10 Hz then after it starts decreasing for higher input frequencies. Hence, it can be expected that if the input frequency for the slope heights of 10 m and 20 m is in-creased even more, the amplification might also starts decreasing as is seen for the slope heights of 30 m and 40 m.


63

Frequency Vs. maximum amplification for slope angle of 25.5 degree

0

2

4

6

810

12

14

16

18

0 2 4 6 8 10 12 14 16

Frequency (Hz)

Max

imum

am

plifi

catio

n

slope height 10 m slope height 20 m slope height 25 m

Figure 6.16. Variation of maximum amplification with the input frequencies for slope angle of 25.5º, slope height of 10 m, 20 m and 25 m.


0

2

4

6

8

10

12

14

0 5 10 15 20Frequency (Hz)

Max

imum

am

plifi

catio

n


Figure 6.17. Variation of maximum amplification with the input frequencies for slope angle of 25.5º,

slope height of 30 m, 35 m and 40 m.


64

Frequency Vs. maximum amplification for the slope angle of 35.5 degree

0

2

4

6

8

10

12

14

16

18

0 2 4 6 8 10 12 14 16

Frequency (Hz)

Max

imum

am

plifi

catio

n


Figure 6.18. Variation of maximum amplification with the input frequencies for slope angle of 35.5º, slope height 10 m, 20 m and 25 m.


0

2

4

6

8

10

12

14

16

0 5 10 15 20

Frequency (Hz)

Max

imum

am

plifi

catio

n


Figure 6.19. Variation of maximum amplification with the input frequencies for slope angle of 35.5º, slope height 30 m, 35 m and 40 m.


65

6.2.3 General relationship between slope height, input frequency and amplification

The relationship between the input frequency, slope height, and the amplification as observed from the FLAC calculation is described above in Section 6.2.2. The pattern of this relationship can be general-ized and shown schematically in qualitative terms as well. The most important concept that can be syn-thesized from the discussion in Section 6.2.2 for the relationship between the input frequency, slope heights, and the amplification is that the relationship is periodic. The peak of the curve for the plot of maximum amplification versus input frequency shifts towards left when the slope height is increased and vice versa (see Figure 6.20). Similar type of schematic representation can be done for the maxi-mum amplification versus slope height for different input frequencies (see Figure 6.21). In this dia-gram, clear harmonic effect can be seen shown by the periodic repetition of the peaks in the curve. Generally, the peaks of the amplification curves shift towards left on increasing the input frequency. Figure 6.20. General sketch for the relationship between the maximum amplification and input fre-

quency for different slope heights.

Amax

Frequency

Decreasing slope height


66

Figure 6.21. General sketch to show the relationship between the maximum amplification and slope

heights for different input frequencies.

6.2.4 The rate of change of amplification with bulk modulus

The rate of change of amplification with the bulk modulus is given by the slope of the shape function in between the maximum and minimum values of amplifications. The plot of the slope of the shape function with the input frequency is shown in Figures 6.22 and 6.23. It is remarkable to notice from these figures that the rate of change of amplification with bulk modulus is high due to lower input fre-quency for the slope with higher height whereas reverse is true for the lower slope height. For exam-ple: for the slope height of 10 m and 20 m, the slope of the function is maximum due to 15 Hz fre-quency whereas for the slope heights of 25 m, 30 m and 40 m, it is maximum with the input frequency of 10 Hz. The curves for the higher slope heights are showing that the slope of the function is con-stantly increasing up to certain threshold frequency and then starts decreasing. The threshold fre-quency from which the rate of change in amplification starts decreasing could be controlled by the resonance frequency of the model. It can also be concluded from Figures 6.22 and 6.23 that the rate of change of slope of the shape function is maximum due to lower frequency for the higher slope heights and due to higher frequency for the lower slope heights.

Amax

Slope height

Decreasing frequency


67

Variation of slope of the function with input frequency

0.00

0.50

1.00

1.50

2.00

2.50

0 5 10 15 20

Frequency (Hz)

Slop

e of

func

tion

*


Figure 6.22. Variation of the slope of the shape function with the input frequency for the slope angle

of 25.5º.

Variation of slope of the function with input frequency

0.00

1.00

2.00

3.00

4.00

5.00

0 2 4 6 8 10 12 14 16Frequency (Hz)

Slop

e of

the

func

tion

*

Slope height of 10 m slope height of 20 m slope height of 25 mslope height of 30 m slope height of 35 m slope height of 40 m

Figure 6.23. Variation of the slope of the shape function with the input frequency for the slope angle

of 35.5º.

* Slope of the function denotes for the rate of change of amplification with bulk modulus as indicated in Figure 6.6 in Section 6.2.1.


68

6.3 Shear modulus versus amplification

To investigate the variation of amplification with the shear modulus, the shear modulus is changed and calculations are performed for the different slope heights and the input wavelength whenever neces-sary (see matrix with calculation numbers in Appendix C-9, calculation numbers 674-735). Some of the parameters used for the simulation are tabulated in Table 6.3. The detail of the calculation results and their interpretations are described in the following sections.

Table 6.3. Input parameters used for the numerical calculation to show variation of amplification

with shear modulus Slope angle

Den-sity (kg/m3)

Bulk modulus (GPa)

Shear modulus

Friction angle (º)

Cohesion (MPa)


Input Frequency

Wavelength

25.5º or 35.5º

2000 1 Variable 42.0 6.7 1.58 Variable Variable

6.3.1 Ratio of slope height to wavelength versus amplification

The relation between the slope height and the wavelength of the input signals with the amplification will have great importance in the engineering design of the slope in earthquake prone areas. The ratio of slope height to wavelength is used for this study. For the design earthquake the wavelength cannot be changed whereas the engineering slope height can be designed. Though, it is beyond the present study to answer the questions about what exactly such relationship would be, but the preliminary idea about this relationship is obtained from the plot of the ratio of slope height to wavelength versus am-plification in Figure 6.24. This plot is obtained for different slope heights with an input frequency of 5 Hz and other material properties as shown in Table 6.3 (also see Appendix C-9, calculation numbers 674 to 729). It is comparable with the typical response spectra of frequency versus amplification. It is remarkable to notice in the Figure 6.24 that there is highest amplification for the slope height to wavelength ratio of 0.07 to 0.20 clearly shown by the first big peak (position of peaks is variable for different slope heights) in the graph. There is another smaller peak on the right side after which the amplification diminishes. The general relation of the energy of the system with the shear modulus and the amplitude of the input wave can be used to explain this curve (see Section 6.3.2). In addition, probably the standing waves are generated when the input wave is reflected back from the slope crest and interfere with the upcoming wave from below the crest, resulting constructive interference. The effect of standing waves can be seen by the position of antinodes (denoted by AN) in Figure 6.25, which denotes the points of maximum displacement due to constructive interference and hence the amplitude of the wave is amplified. It is however most imminent for the slope height to wavelength ratio range of 0.07 to 0.20. The periodic peaks are due to the generation of standing waves for other harmonics of the fundamental frequency because standing waves are created within the medium at harmonic frequencies. It is quite evident from Figure 6.24 that the slope height to wavelength ratio is the most important and reasonable parameter to be considered during the design of a slope. It is re-markable to notice in Figure 6.24 that the maximum amplification for 10 m slope height is little bit


69

towards left than for others that could be due to the relation between fundamental frequency and slope height, since the input frequency is constant of 5 Hz.

Ratio of slope height to wavelength Vs. amplification

0

1

2

3

4

5

6

7

0.001 0.01 0.1 1 10Ratio of slope height to wavelength

Max

imum

Am

plifi

catio

n

slope height 10 m slope height 20 mslope height 30 m slope height 40 m

Figure 6.24. Ratio of slope height to wavelength versus amplification for the slope angle of 25.5º and

input frequency of 5 Hz.

Figure 6.25. Formation of standing waves due to interference of waves. N in the figure denotes node and AN denotes anti node (source: Henderson, 1998).

6.3.2 Shear modulus versus amplification

The energy in the system is proportional to the product of shear modulus of the material and the ampli-tude of the input wave (Bullen, 1980 among others). To remain the energy constant, the amplitude of the wave should be increased for the material with lower shear modulus and should be decreased for the material with higher shear modulus. It clearly indicates that the amplitude of the wave is amplified


70

due to material with a lower shear modulus. The shear modulus versus amplification is plotted in Fig-ure 6.26 (the full data is given in Appendix C-9, calculation numbers 674 to 729). This plot indicates that there are two successive peaks of amplifications for all the slope heights. The peaks, however, are at comparatively lower values of shear modulus for the smaller slope heights (for slope heights of 10 m and 20 m, at around 50 MPa) and at higher values of shear modulus for the higher slope heights (for slope heights of 30 m and 40 m, at around 300 MPa). The second highest peaks are at around 1000 MPa of shear modulus for 10 m and 20 m slope heights whereas at around 3000 MPa for 30 m and 40 m slope heights. Before the peak, there exist very low amplification for the lower shear modulus, which could be the result of very low shear wave velocity and maximum attenuation. These peaks could be the cause of generation of standing waves in that par-ticular combination of frequency, shear modulus, and wavelength as discussed in Section 6.3.1. The repeated peaks are the harmonic effects as explained in Sections 6.2.2 and 6.3.1. For the shear modulus more than 5000 MPa, the amplification is minimum and remains almost con-stant, which could be because of the insufficient amount of energy to amplify for the amplitude of the wave. In other words, for higher values of shear modulus, the modulus is so high that to lower the en-ergy, the amplitude could not be amplified.

Shear modulus Vs. amplification for different slope heights

0

1

2

3

4

5

6

7

1 10 100 1000 10000 100000

Shear modulus (MPa)

Max

imum

Am

plifi

catio

n

slope height 10 m slope height 20 mslope height 30 m slope height 40 m

Figure 6.26. Shear modulus versus amplification, slope angle of 25.5º and frequency 5 Hz.

6.3.3 Relation of shear modulus on amplification for different wavelengths


71

It can be easily seen from the relationship between the shear modulus and the wavelength that as the shear modulus increases, the shear wave velocity and hence the wavelength of the wave increases (Equations 6.4 and 6.5, source ITASCA, 2000):

sGC =ρ

……………………………………………………………………………….(Eq. 6.4)

sCf

=λ …………………………………………………………………………………(Eq. 6.5)

where, sC = Shear wave velocity

G = Shear modulus ρ = Mass density λ = Wavelength

f = Frequency

Numerical calculations are performed to know the relationship between the shear modulus and ampli-fication for different input wavelengths (the slope height used for the calculation is 20 m). It is seen that the higher amplification is obtained for the larger wavelengths (Figure 6.27). Also, see Appendix C-10, calculation numbers 730 to 791 for detail. It is also seen from Figure 6.27 that for the lowest values of shear modulus used in the calculation (100 KPa to 10 MPa), the amplification is quite low. It could be explained by the very high attenuation for the lower values of shear modulus. For the higher than 10 MPa shear modulus, amplification starts increasing sharply and the rate of increase for inter-mediate shear modulus is decreased and keep fluctuating little bit. After certain peak values, the ampli-fication starts decreasing (for wavelengths of 10 m and 44.72 m, amplifications are clearly decreasing, similar can be expected for other wavelengths as well if the modulus is kept increased). The magnitude of amplification however is higher due to the wave with higher wavelengths than due to lower wave-lengths. The generation of standing waves resulting in the resonance and harmonic effect is reflected in this plot of shear modulus versus amplification for different wavelengths.


72

Shear modulus Vs. amplification for different wavelengths

00.5

11.5

22.5

33.5

44.5

0.1 1 10 100 1000 10000

Shear modulus (Mpa)

Max

imum

Am

plifi

catio

n

w avelength of 10m w avelength of 14 m

w avelength of 44.72 m w avelength of 63.24 m

Figure 6.27. Shear modulus versus amplification for different input wavelengths (hence varying fre-

quency, slope height of 20 m). Note: The different size and pattern of the lines in the curve do not have any other meaning than leg-end

6.4 Modelling the function to show the variation of amplification with material properties, slope height and earthquake input signals

After knowing the general relationships of various input parameters and the amplification, as described in Sections 6.2 and 6.3, equations describing the function for such relationships are optimised. Due to the time constraints and the complexity of defining the function, not all the input parameters could be included in the function. Analysis is performed to study the variation of the observed amplification with that of the modelled one. As discussed in Sections 6.2 and 6.3, the most important parameters to be considered to define the function for the input parameters versus amplification are the slope height, input frequency, and the wavelength of the input signals. The wavelength of the input signals reflects the influence of the shear modulus of the material in the slope as can be seen from the equation 6.4 in Section 6.3.6. The influ-ence of bulk modulus on amplification is not understood fully, and more research is necessary to find out its exact relationship. Basically, the relationship presented here therefore reflects the relation be-tween the slope heights; shear modulus or wavelength of the signals and the frequency of the input signal.


73

It is seen in Figures 6.20 and 6.21 in Section 6.2.3 that the amplification has polynomial type of func-tion with the frequency and the slope height and the peak values of amplification repeats periodically. The repetition of the peaks is controlled by the fundamental frequency of the slope and its resonance in relation with the input frequency as well as the harmonic frequencies. If the input frequency matches with the fundamental frequency of the slope, or of multiples thereof, the local maximum for amplifica-tion is obtained. The slope height and the wavelength of the input wave are other important parameters to be considered as is explained in Section 6.3.4. The maximum acceleration and hence the amplifica-tion can therefore be said to be the function of fundamental frequency, input frequency, slope height and the wavelength of the input signal. This can be expressed as:

max. 0( ) ( ) ( ) ( )A F f F f F F h= λ …………………………………………………(Eq. 6.6)

where, 0( )F f = Function of fundamental frequency

( )F f = Function of input frequency

( )F λ = Function of wavelength

( )F h = Function of slope height

But as discussed in Section 6.3, the slope height to wavelength ratio is the most practical and logical relationship to be introduced when talking about the relation with amplification. Similarly, if the input frequency is divided by the fundamental frequency, the influence of the input frequency with respect to the fundamental frequency can be obtained. Hence, to model the function, the functions of funda-mental frequency and that of the input frequency is combined and the ratio of input frequency to the fundamental frequency is used instead. The fundamental frequency can be calculated with the eigen-value analysis but it is beyond the scope and limit of this study to do so. Fundamental frequency is estimated from the plots of the slope height versus amplification curves for different input frequencies (see Figures 6.16 and 6.17 in Section 6.2). Similarly, the function of the slope height and the input wavelength is combined and the ratio of slope height to the wavelength is used to model the function. But the frequency and the wavelength are not the independent parameters, hence would be in a sense are repeating if both of them are used. Therefore, instead of wavelength, the shear modulus is used and later on converted to the shear wave velocity. The analysis is performed by optimising the function by using “solver” command in Microsoft Excel. In this study, basically, it is tried to match the observed values of amplification and the values obtained from the function by modelling different functions incorporating the parameters, which are found to be influencing the amplification from this study. The basic goal for defining a function can be summa-rized as follows:

Careful analysis and interpretation of the FLAC simulation results as described in Sections 6.2 and 6.3. Find out the parameters that are influencing amplification of the studied slope and find out the

general trend of the curves in the plot of input parameters versus amplification. Try to incorporate all the parameters found to be influencing the amplification so far as possi-

ble and integrate them together. Model different functions that can represent the general trend of the curves in different input

versus amplification plots.


74

Recommend the matching function with some physical explanation using simple regression analysis.

It should be realized that more complicated equations could give a better matching function but then there will be more degrees of freedom, but not necessarily represent the realistic function. The physi-cal basis for defining the function is the observation results from the study in FLAC and their interpre-tation. The following function (generalized in Equation 6.7 and Equation 6.8 is applied for the analy-sis) is optimised and recommended after looking at the results by modelling different other functions:

2

max. 0 1 2 20 0

( / ) ( / )* *( / ) ( / )

h hA C C Cf f f f

= + +λ λ

…………………………………………(Eq. 6.7)

But /sC f=λ

\ /( / )s

hhC f

=λ

\

2

2

max. 0 1 2 20 0

* ( * )( )* *

( / ) ( / )s s

h f h fV VA C C C

f f f f

⎡ ⎤⎛ ⎞ ⎡ ⎤⎢ ⎥⎜ ⎟ ⎢ ⎥

⎝ ⎠⎢ ⎥ ⎢ ⎥= + +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥

⎣ ⎦⎢ ⎥⎣ ⎦

or, 22

0 0max. 0 1 2 2

* ** *s s

h f h fA C C CV V

= + + ……………………………………..(Eq. 6.8)

where 0 1 2, ,C C C are the parameters obtained from regression analysis

max.A = Maximum amplification

h = Slope height λ = Wavelength of the input signals

0f = Fundamental elastic (eigen) frequency of slope model

f = Input frequency

sV = Shear wave velocity

The detail of the regression co-efficient (R2) and the observed and modelled values of maximum am-plifications is given in Table 6.4. The R2 value is obtained by using the following formula (Equation 6.9) and successive optimisation using “solver” command in Microsoft Excel:

22

2

( )1

( )av

O eR

O O⎡ ⎤−

= − ⎢ ⎥−⎢ ⎥⎣ ⎦

∑∑

……………………………………………………..(Eq. 6.9)


75

where, O = Observed value of amplification e = Estimated or modelled value of amplification avO = Average of the observed value of amplification

Table 6.4. Detail of the input parameters as well as observed and modelled values of amplification

and regression co-efficients.

Slope height

(m)

Shear modulus

(Pa)

Shear velocity, Cs (m/s)

Wavelength (m)

Input frequency

(Hz)

Fundamental frequency (Hz)

Observed maximum

amplification

Modelled maximum

amplification

(R2)

10 1.0E+10 2236.07 745.36 3 15 1.80 1.8120 1.0E+10 2236.07 745.36 3 15 2.30 2.0525 1.0E+10 2236.07 745.36 3 15 2.30 2.52 0.7830 1.0E+10 2236.07 745.36 3 10 1.90 2.0535 1.0E+10 2236.07 745.36 3 10 2.30 2.3440 1.0E+10 2236.07 745.36 3 10 2.90 2.7310 1.0E+10 2236.07 447.21 5 15 2.30 2.3220 1.0E+10 2236.07 447.21 5 15 2.50 2.9025 1.0E+10 2236.07 447.21 5 15 2.30 2.51 0.3630 1.0E+10 2236.07 447.21 5 10 3.50 2.9035 1.0E+10 2236.07 447.21 5 10 2.50 2.6940 1.0E+10 2236.07 447.21 5 10 2.50 2.2810 1.0E+10 2236.07 223.61 10 15 2.00 2.1420 1.0E+10 2236.07 223.61 10 15 5.20 7.9425 1.0E+10 2236.07 223.61 10 15 10.00 10.34 0.6530 1.0E+10 2236.07 223.61 10 10 12.00 7.9435 1.0E+10 2236.07 223.61 10 10 7.80 9.5840 1.0E+10 2236.07 223.61 10 10 12.00 11.0610 1.0E+10 2236.07 149.07 15 15 7.00 6.9020 1.0E+10 2236.07 149.07 15 15 17.00 11.8825 1.0E+10 2236.07 149.07 15 15 9.50 7.94 0.3430 1.0E+10 2236.07 149.07 15 10 6.00 11.8835 1.0E+10 2236.07 149.07 15 10 10.00 9.7340 1.0E+10 2236.07 149.07 15 10 4.50 5.67

The modelled amplification pattern using this function is more realistic with the observed one. As can be seen in Figures 6.28 through 6.31, the pattern for 3 Hz and 5 Hz input frequency is more similar to the observed amplifications. For the rest two frequencies, the pattern is comparable. More importantly, the function gives the periodic maximum and minimum as is seen from the observation in FLAC cal-culation. It is remarkable to note that for the input frequency equal to the fundamental frequency esti-mated by this study, the function also gives the highest amplifications. For example, for the slope height of 40 m, the estimated fundamental frequency is 10 Hz and the recommended function also gives the maximum amplification due to 10 Hz input frequency as shown in Figure 6.30. Similarly, estimation of the fundamental frequency for the slope height of 20 m is 15 Hz, and the input frequency of 15 Hz gives maximum amplification when the recommended function is used (see Figure 6.31).


76

Slope height Vs. observed and modelled values amplifications for input frequency of 3 Hz (slope angle 25.5

degree)

1.501.701.902.102.302.502.702.903.10

0 10 20 30 40 50Slope height (m)

Max

imum

am

plifi

catio

n

Observed data Modelled data

Figure 6.28. Observed versus modelled amplifications for different slope heights (frequency 3 Hz)

Slope height Vs. observed and modelled amplifications for input frequency of 5 Hz (slope angle 25.5 degree)

2.002.202.402.602.803.003.203.403.60

0 10 20 30 40 50

Slope height (m)

Max

imum

am

plifi

catio

n




77


0.002.004.006.008.00

10.0012.0014.00

0 10 20 30 40 50Slope height (m)

Max

imum

am

plifi

catio

n




0.00

5.00

10.00

15.00

20.00

0 10 20 30 40 50Slope height (m)

Max

imum

am

plifi

catio

n


Figure 6.31. Observed versus modelled amplifications for different slope heights (frequency 15 Hz) The constant parameters determined from regression analysis are tabulated in Table 6.5. It is also in-teresting to see that for the input frequency of 3 Hz, the regression parameters are different from those for the other three input frequencies. For example: C0 value is positive for 3 Hz input frequency whereas that for the rest are negative. The parameters C1 and C2 also have different signs for 3 Hz fre-quencies than rest of the input frequencies (see Table 6.5). It might be interesting to see the effect of the input frequency between 3 and 5 Hz, but could not be done during this study period due to limited time. It can be seen from the residual values that the function defined here overestimate the amplifica-tion values in most of the cases indicated by the negative residuals.


78

Table 6.5. Regression parameters for different input frequencies. Input frequency

(Hz) Regression parameters

C0 C1 C2 3 2.54 -18.04 106.83 5 -0.05 48.73 -199.58 10 -5.03 116.97 -151.00 15 -15.24 458.10 -1907.52

Though, it needs further research and more time to formulate the exact function for different input pa-rameters versus amplification, the function proposed here will at least give some idea about how the variation looks like. Provided more physical explanation is known, the function can be more accu-rately defined. One should be aware that the function recommended here does not include the influ-ence of all the material properties including bulk modulus and slope angle. It might be possible to in-clude all those parameters but due to time constraint for the research, it is not possible to do so during this study. It should also be known that the function defined here is for some particular combinations of slope geometry, material properties and earthquake input signals, and might not necessarily be true for all the cases. Hence, the function cannot be generalized for all the cases.


79

7. Conclusion and recommendation

The finite difference numerical program FLAC was used to study the variation of amplification with different input parameters including the frequency and wavelength of the input signals. Results of 956 numerical calculations for a total of four slope models are used for this study. Out of four models, three are used just to investigate the preliminary response of them in terms of the displacement, veloc-ity, and the acceleration histories. For those three models, the variation parameters are only the mate-rial properties (not the input signals) including the material cohesion, internal friction angle, and the slope angle. The displacement, velocity, and the acceleration histories are measured at the crest of the slope ranging from the edge of the crest of the slope to different distances from the edge. The fourth model is used to find out the true relationships between the input parameters and the output in terms of amplification. The variable parameters for this model are the input frequency, wavelength, material properties including shear & bulk modulus and the slope heights. Important findings of the first three models are already described in Section 5.7. Precisely speaking, the first three models reveal that the displacement, velocity, and accelerations are generally maximum at the edge of the slope crest than away from it. It is also seen from those models that the magnitude of the ground response is strongly dependent on the slope angle, material cohesion as well as friction an-gle, but the ground response obtained for the slope angles more than the friction angle might not give accurate and realistic result (after slope fails). It is also found that the higher the slope angle, the higher the amplification in general, and the higher the material cohesion, the lower the amplification due to the same input signals. If the slope material has higher friction angle, the seismic amplification will be less in general. For steeper slopes, the slope starts to fail if the friction angle and material cohesion are very low. The variation of amplification with the friction angle however is not very high.

The fourth model show that there is strong relationship of the input frequency, wavelength and the slope height on overall amplifications. The input frequency and the wavelength are natural phenome-non and cannot be controlled, but what can be controlled is the slope height. If the magnitude of the anticipated (design) earthquake is known beforehand, the engineering slopes can be designed accord-ingly by analysing the relationships as obtained from this research. Moreover, the existing natural slopes can also be modified to minimize the effect of the expected earthquake to those particular slopes. The important finding of this research can be summarized as given below:

There is significant influence of the earthquake input signals (frequency and wavelength of the wave) and site conditions of the ground as reported by previous researchers. Significant resonance is obtained if the natural frequency of the slope coincides with the input

frequency of the seismic waves. Maximum horizontal amplification of 17 is obtained for limestone slope with 20 m height,

when input signal of 15 Hz frequency. This extremely high amplification could be due to the generation of standing waves when the waves reflected from the slope crest interfere with the upcoming input signals.


80

The peaks of amplification repeat periodically showing clearly the importance of harmonic ef-fect. In general, the shorter slopes are amplified most by the higher frequency and the higher slopes

are amplified most by the lower frequency. The amplification due to the input signals with lower frequency (3 Hz and 5 Hz, maximum horizontal amplification of around 3.5) however is significantly lower than that due to higher frequency (10 and 15 Hz, maximum horizontal ac-celeration up to 17). Overall speaking, the magnitude of maximum horizontal amplification is greater for the shorter slopes with high frequency than the other way round. It is found extremely difficult to separate the effect of each of the material properties individu-

ally. However, extremely lower values of shear modulus (up to 10 MPa) do not show amplifi-cation at all; rather they are mostly showing attenuation. For a shear modulus of more than 10 MPa, amplification first increases sharply and after some extreme peak value, it starts decreas-ing. However, it is also showing harmonic effect. The influence of bulk modulus is not fully understood but with bulk modulus, amplification does not clearly show any resonance or har-monic effect. Nevertheless, the rate of change of amplification with the bulk modulus is fre-quency dependent. For the higher slope heights, the rate of change of amplification with bulk modulus is maximum with the lower input frequency, whereas the reverse is true for the lower slope heights. The slope height to wavelength ratio of 0.07 to 0.23 are extremely vulnerable for amplifica-

tion, as all the slope heights are showing maximum amplification at that particular range (maximum horizontal amplification up to 6.5). This type of relationship is extremely valuable for the seismic microzonation study in GIS environment as digital elevation models can be used for finding the slope characteristics of the terrain, and the hazard maps can be produced for the anticipated earthquake. An empirical function is recommended to show the relation of slope height, shear wave veloc-

ity and the input frequency on amplification. By comparing the time spent to get the results of the present research as well as the result itself, fol-lowing fundamental recommendations are made for the future research:

The results obtained from this research are to be validated by applying it in the real slopes with real earthquake input signals. The results will be clearer if similar type of numerical modelling is performed for more slope

geometries and input signals (for example: wide range of slope heights and input frequencies). The relationship of the bulk modulus versus amplification could be truly explained if more de-

tail study is carried out by spending more time. Similarly, the relationship between the slope angles and the amplification can be fully understood if research is carried out for a larger range of slope angles. The usefulness of the present study is for seismic microzonation and can be tested by applying

these results within GIS for past earthquakes and establishing the damage along slopes.


81

References

Abramson, L. W., Lee, T. S., Sharma, S. and Boyce, G. M., 1996. Slope stability and stabilization methods, John Wiley & Sons, Inc. Aki, K., 1993. Local site effects on weak and strong ground motions. Tectonophysics 218 (1993), 93-111 pp. Aki, K., 1988. Local site effects on ground motion. In J. Lawrence Von Thun, editor, Earthquake En-gineering and Soil Dynamics. II: Recent Advances in Ground motion evaluation; ASCE, 1988; 103-155 pp. Aki, K., Richards P.G., 1980. Quantitative seismology, theory and methods, W.H. Freeman and Co., San Francisco, USA, 1 and 2, 932 pp. Athanasopoulos, G.A., Pelekis, P.C., and Leonidou, E.A., 1998. Effects of surface topography and soil conditions on the seismic ground response – including liquefaction in the Egion (Greece) 15/6/1995 earthquake. In: Proceedings of the Eleventh European Conferences on Earthquake Engineering, Ab-stract volume., 525-525 pp. Bard, P.Y., and Thomas, J.R., 2000. Wave propagation in complex geological structures and their ef-fects on strong ground motion. In Wave motion in Earthquake Engineering, Kausel, E., and Manolis, G (eds), International series advances in earthquake engineering, WIT: Boston, 1999, 37-95 pp. Bielak, J., and Bao, H., 1998. Ground motion modelling using 3D finite element methods; The effects of Surface Geology on seismic motion, vol 1, Irikura, Kudo, Okada & Sasatani (eds), 1998 Balkema, Rotterdam, ISBN, 121-133 pp. Boore, D.M., 1972. A note on the effect of simple topography on seismic SH waves, Bull. Seism. Soc. Am. 62, 275-284 pp. Bouchon, M., 1973. Effect of topography on surface motion, Bull. Seism. Soc. Am. 63 (3), 615-632 pp. Brown, G. E., 2001. Division of Civil and Mechanical Systems, Jr. Network for earthquake Engi-neering Simulation (NEES) website: http://www.eng.nsf.gov/nees/nrcreport/Chapter2.pdf Bullen, K.E., 1980. An introduction to the theory of Seismology, Cambridge University Press, third edition.


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Castro, E., 1999. Topographic site characteristics and damage pattern of the January 25th 1999 earth-quake in Armenia-Colombia. – The Netherlands, M.Sc. Thesis, ITC Delft, 84 pp. Coggan, J.S., Stead, D., and Eyre J.M., 1998. Evaluation of techniques for quarry slope stability as-sessment. Trans. Instit. Min. Metall. - Sect. B, 107: B139-B147. Davis, L.L., and West, L.R., 1973. Observed effects of topography on ground motion, Bulletin of the Seismological Society of America, Vol. 63, No. 1, 283-298 pp. Day, R., 2002. Geotechnical Earthquake Engineering Handbook, New York, McGraw-Hill.

Desai, C. S., and Abel, J. F., 1972. Introduction to the Finite Element Method: A Numerical Method for Engineering Analysis, Van Nostrand Reinhold Company, New York. Geli, L., Bard, P.Y., Jullien, B., 1988. The effect of topography on earthquake ground motion: A re-view and new results, Bulletin of Seismological Society of America, 78 (1): 42-63 pp. Hack, R., Arango, M., Castro, E., Leenders, N., Rengers, N., Soeters, R., Rupke, J., Slob, S., van Bemmelen, B., van Westen, C., Montoya, L., Vargas, R., Horn, J., Carree, P., Scarpas, A., Nieuwen-huis, J., Kruse, G., Rosero, F., Serna, J., Duque, A.L., Campos, A., Guzmán, J., 2000. Rapid inventory of earthquake damage (RIED), Assessment of the damage of the Quindío Earthquake in Armenia and Pereira, Colombia. – Unpublished report, R. Hack (Ed.), March 2000, ITC Delft, The Netherlands, 141 pp. Havenith, H.B., Vanini, M., Jongmans, D., & Faccioli, E., 2003. Initiation of earthquake-induced slope failure: influence of topographical and other site specific amplification effects, Journal of Seismology 7: 2003, 397–412 pp. Henderson, T., 1998. The Physics Classroom, open file of Glenbrook South Physics resources, Glen-view, Illinois, USA. Hu, Y-X., Liu, S-C., and Dong, W., 1996. Earthquake Engineering, E & FN Spon, an imprint of Chapman & Hall, 2-6 Boundary Row, London SE1 8HN, UK. ITASCA, 2000. FLAC 4.0 manuals, Minnessotta, ITASCA Consulting Group, Inc. Jing, L., 2003. A review of techniques, advances and outstanding issues in numerical modelling for rock mechanics and rock engineering, International Journal of Rock Mechanics & Mining Sciences 40, 2003, 283-353 pp. Kramer, S.L., 1996. Geotechnical Earthquake Engineering, Prentice Hall, Upper Saddle River, New Jersey. Leenders, N., 2000. Three-dimensional dynamic modelling of earthquake tremors, Delft Memoirs of the centre of Engineering geology in the Netherlands, No. 200, M.Sc. thesis, TU Delft, 64 pp.


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Loria, C. S., 2003. Numerical Assessment of influence of the earthquakes on irregular topographies- analysis of Colombia, 1999 and El Salvador, 2001 earthquakes; ITC 2003, M.Sc. thesis, 1-141 pp. Paolucci, R., 2002. Amplification of earthquake ground motion by steep topographic irregularities, Earthquake Engineering and Structural dynamics, 2002; 31:1831-1853 pp. Rey, P., and Mueller, D., 2001. Earthquake Seismology, lecture note, University of Sydney, In: Ex-ploratorium, http://www.es.usyd.edu.au/geology/people/staff/prey/Teaching/Geol- 2001GPHS/Waves. Sánchez-Sesma, F.J., Campillo, M., 1993. Topographic effects for incident P, SV, and Rayleigh waves, Tectonophysics, 218 (1-3): 113-125. Stead, D., Eberhardt, E., Coggan, J., and Benko, B., 2001. Advanced numerical techniques in rock slope stability analysis – applications and limitations, landslides – Causes, Impacts and Countermea-sures, 17-21 June, Davos, Switzerland, pp 615-624. Takenaka, H., Furumura, T., and Fujiwara, H., 1998. Recent developments in numerical methods for ground motion simulation, The effects of Surface Geology on seismic Motion, Recent progress and new horizon on ESG study, vol 1, Irikura, Kudo, Okada & Sasatani (eds); 1998 Balkema, Rotterdam, ISBN 90 5809 030 2, pp 91-101. The National Academies Press, 2003. Preventing Earthquake Disasters: The Grand Challenge in Earthquake Engineering: A Research Agenda for the Network for Earthquake Engineering Simulation (NEES), 172pp.


84

Appendices


85

Appendix A-1: Script example used for the calculation (script for calculation no. 684, Appendix C-9)

; Simple dynamic model config dynamic g 100,50 model mohr prop dens 2000 shear 1e9 bulk 1e9 prop coh 6.7e6 fric 42 ten 1.58e6 gen 0,0 0,20 100,20 100,0 i=1,101 j=1,21 gen same 86,50 100,50 same i=23,101 j=20,51 m n i=1,23 j=20,51 ; displacement boundary conditions fix x i=1 j=1,51 fix x i=101 j=1,51 fix y j=1 ; apply gravity set grav=9.81 set dyn=off solve ; displacement history of slope hist ydis i=100 j=51 hist unbal ; save initial state save sl1.sav title static displacement history of slope 1 shear 1e9 height 30 m plot hold his 1 plot hold his 2 ; rest sl1.sav hist reset ; def wave ; sinusoidal wave : acc = 1 m/sec2, freq = 5 Hz, duration = 0.25 sec freq = 5 wave = 1.0 * sin(2.0*pi*freq*dytime) if dytime > 0.25 then wave = 0.0

(contd.) Appendix A-1 (contd.)


86

end_if end apply ff ; apply xquiet j=1 apply yquiet j=1 ; apply xacc=1.0 hist=wave j=1 apply yvel=0.0 j=1 set dyn on ; set damp rayl 0.05 2.5 set dytime=0.0 ini xvel=0 yvel=0 xdis=0 ydis=0 hist reset hist dytime hist xvel i=86 j=50 hist yvel i=86 j=50 hist xvel i=90 j=50 hist yvel i=90 j=50 hist xvel i=95 j=50 hist yvel i=95 j=50 hist xvel i=100 j=50 hist yvel i=100 j=50 hist xdis i=86 j=50 hist ydis i=86 j=50 hist xdis i=90 j=50 hist ydis i=90 j=50 hist xdis i=95 j=50 hist ydis i=95 j=50 hist xdis i=100 j=50 hist ydis i=100 j=50 hist unbal hist xacc i=86 j=50 hist yacc i=86 j=50 hist xacc i=90 j=50 hist yacc i=90 j=50 hist xacc i=95 j=50 hist yacc i=95 j=50 hist xacc i=100 j=50 hist yacc i=100 j=50 solve dytime = 0.5

(contd.) Appendix A-1 (contd.)


87

save stage3.sav ; state at 0.5 sec set pcx on set autoname on title Dynamic analysis of slope 1 shear 1e9 for slope height 30 m plot hold grid vel copy plot hold hist 2 3 v 1 plot hold hist 4 5 6 v 1 plot hold hist 7 8 9 v 1 plot hold hist 10 11 v 1 plot hold hist 12 13 14 v 1 plot hold hist 15 16 17 v 1 plot hold hist 18 v 1 plot hold hist 19 20 v 1 plot hold hist 21 22 v 1 plot hold hist 23 24 v 1 plot hold hist 25 26 v 1 copy ret


88

Appendix A-2: Sample example of the screen capture of calculation results in FLAC (1)

In this figure, the slope starts failing (rotational failure) because the slope angle is too high and the ma-terial cohesion is too low. This example (slope model 1) is for the slope angle of 90º, friction angle of 27.8º and the material cohesion of 270 Pa. Other properties include shear modulus of 7 GPa, bulk modulus of 27 GPa, material density of 2000 kg/m3 and tensile strength of 1.17 MPa. The slope is 60 m wide and 20 m high. The maximum velocity vector is remarkably high (4.121 m/s).


89

Appendix A-3: Sample example of the screen capture of calculation results in FLAC (2)

This is the sample example of the screen capture of FLAC calculation. The slope height is 20 m, slope angle is 25.5 degree, and input frequency of 5 Hz is applied. Other material properties include shear modulus and bulk modulus both of 1 GPa, material cohesion 6.7 MPa, friction angle of 42º, mass den-sity of 2000 kg/m3 and tensile strength of 1.58 MPa.


90

Appendix A-3 (contd.): Sample example of the screen capture of calculation results in FLAC (2).


91

Appendix B-1: The input and output parameters for slope model 1

Calc. No *

Slope angle (°)

Friction angle (°)

Cohesion (Pa)

Y acc. 1(m/s2)

X acc. 2

(m/s2) X disp. 3 (m)

Y disp. 4 (m)

X-velocity (m/s)

Y-velocity (m/s)


1 25.50 42.00 6.70E+06 0.80 1.80 1.55E-02 1.00E-05 6.20E-02 5.00E-03 3.183E-022 29.70 42.00 6.70E+06 0.82 1.80 1.55E-02 1.00E-05 6.30E-02 5.00E-03 3.183E-023 38.60 42.00 6.70E+06 1.00 2.00 1.56E-02 1.00E-05 6.40E-02 5.00E-03 3.183E-024 45.00 42.00 6.70E+06 0.90 2.10 1.56E-02 1.00E-05 6.40E-02 5.00E-03 3.184E-025 53.00 42.00 6.70E+06 1.20 2.00 1.56E-02 1.00E-05 6.40E-02 6.00E-03 3.182E-026 63.00 42.00 6.70E+06 1.20 2.10 1.57E-02 1.00E-05 6.40E-02 3.00E-03 3.185E-027 75.90 42.00 6.70E+06 1.20 2.20 1.57E-02 1.00E-05 6.50E-02 6.00E-03 3.182E-028 81.00 42.00 6.70E+06 1.50 2.20 1.57E-02 1.00E-04 6.60E-02 8.00E-03 3.184E-029 90.00 42.00 6.70E+06 2.00 2.20 1.57E-02 1.00E-04 6.80E-02 1.00E-02 3.180E-02

10 25.50 5.00 6.70E+06 0.85 2.50 1.50E-02 1.00E-04 6.20E-02 5.00E-03 3.183E-0211 25.50 10.00 6.70E+06 0.85 2.50 1.50E-02 1.00E-04 6.20E-02 5.00E-03 3.183E-0212 25.50 15.00 6.70E+06 0.85 2.40 1.50E-02 1.00E-04 6.20E-02 5.00E-03 3.183E-0213 25.50 20.00 6.70E+06 0.85 2.40 1.50E-02 1.00E-04 6.20E-02 5.00E-03 3.183E-0214 25.50 25.00 6.70E+06 0.85 2.40 1.50E-02 1.00E-04 6.20E-02 5.00E-03 3.183E-0215 25.50 30.00 6.70E+06 0.82 2.30 1.45E-02 1.00E-04 6.00E-02 4.50E-03 3.183E-0216 25.50 35.00 6.70E+06 0.82 2.30 1.45E-02 1.00E-04 6.00E-02 4.50E-03 3.183E-0217 25.50 40.00 6.70E+06 0.82 2.30 1.45E-02 1.00E-04 6.00E-02 4.50E-03 3.183E-0218 25.50 60.00 6.70E+06 0.82 2.30 1.45E-02 1.00E-04 6.00E-02 4.50E-03 3.183E-0219 90.00 5.00 6.70E+06 1.50 2.20 1.50E-02 3.00E-04 6.70E-02 9.00E-03 3.180E-0220 90.00 10.00 6.70E+06 1.50 2.10 1.50E-02 3.00E-04 6.70E-02 9.00E-03 3.180E-0221 90.00 30.00 6.70E+06 1.50 2.10 1.50E-02 3.00E-04 6.70E-02 9.00E-03 3.180E-0222 25.50 42.00 6.70E+01 7.00 2.40 1.50E-02 6.00E-04 6.20E-02 1.00E-02 3.183E-0223 25.50 42.00 6.70E+02 6.10 2.20 1.50E-02 5.00E-04 6.20E-02 9.00E-03 3.183E-0224 25.50 42.00 6.70E+03 4.20 2.20 1.50E-02 3.00E-04 6.20E-02 4.00E-03 3.183E-0225 45.00 42.00 1.00E+01 6.00 3.90 1.50E-02 8.00E-04 6.30E-02 1.10E-02 1.023E+0026 45.00 42.00 6.70E+01 6.60 4.20 1.50E-02 8.00E-04 6.30E-02 1.00E-02 9.367E-0127 45.00 42.00 6.70E+02 9.00 3.50 1.50E-02 6.00E-04 6.20E-02 1.00E-02 8.416E-0128 45.00 42.00 6.70E+03 4.80 5.80 1.50E-02 5.00E-04 6.30E-02 1.10E-02 3.191E-0229 45.00 42.00 6.70E+04 0.90 2.00 1.50E-02 1.00E-04 6.30E-02 5.00E-03 3.184E-02

* Calculation number 1 Vertical acceleration 2 Horizontal acceleration 3 Horizontal displacement 4 Vertical displacement


92

Appendix B-1: The input and output parameters for slope model 1 (contd.) Calc. No *

Slope angle (°)

Friction angle (°)

Cohesion (Pa)

Y acc. 1(m/s2)

X acc. 2 (m/s2)

X disp. 3 (m)

Y disp. 4 (m)

X-velocity (m/s)

Y-velocity (m/s)


30 90.00 42.00 6.70E+03 9.00 3.00 2.20E-01 6.00E-01 9.00E-01 2.39E+00 2.903E+0031 90.00 42.00 6.70E+04 1.50 2.20 1.50E-02 1.00E-04 6.60E-02 9.00E-03 3.180E-0232 90.00 42.00 6.70E+05 1.50 2.20 1.50E-02 1.00E-04 6.50E-02 8.00E-03 3.180E-0233 75.90 42.00 1.00E+02 12.00 4.10 6.00E-01 2.20E-01 9.00E-01 2.40E+00 3.334E+0034 75.90 42.00 6.70E+02 15.00 6.00 2.00E-01 6.00E-01 8.00E-01 2.90E+00 3.167E+0035 75.90 42.00 6.70E+03 5.00 4.00 1.80E-01 5.00E-01 7.90E-01 2.00E+00 2.227E+0036 75.90 42.00 6.70E+04 1.30 2.10 1.50E-02 1.00E-04 6.40E-02 6.00E-03 3.182E-0237 25.50 30.00 6.70E+05 0.80 2.30 1.50E-02 1.00E-04 6.20E-02 5.00E-03 3.183E-0238 38.60 30.00 6.70E+05 0.90 2.20 1.50E-02 1.00E-04 6.20E-02 5.00E-03 3.182E-0239 53.00 30.00 6.70E+05 1.10 2.00 1.50E-02 1.00E-04 6.30E-02 4.00E-03 3.182E-0240 75.90 30.00 6.70E+05 1.25 2.00 1.50E-02 1.00E-04 6.40E-02 6.00E-03 3.182E-0241 81.00 30.00 6.70E+05 1.50 2.00 1.50E-02 5.00E-04 6.40E-02 8.00E-03 3.184E-0242 90.00 30.00 6.70E+05 1.50 2.00 1.50E-02 5.00E-04 6.60E-02 9.50E-03 3.180E-0243 90.00 20.00 6.70E+04 2.20 1.40 4.00E-03 2.00E-02 4.00E-02 7.00E-02 6.835E-0244 25.50 20.00 6.70E+04 1.60 2.20 1.50E-02 3.00E-04 6.40E-02 4.00E-03 3.183E-0245 38.60 20.00 6.70E+04 2.00 2.00 1.50E-02 3.00E-04 6.40E-02 4.00E-03 3.182E-0246 53.00 20.00 6.70E+04 1.60 1.90 1.50E-02 5.00E-04 6.40E-02 3.00E-03 3.182E-0247 75.90 20.00 6.70E+04 1.50 1.80 1.40E-02 1.80E-03 6.40E-02 1.00E-02 3.182E-0248 81.00 20.00 6.70E+04 1.50 1.60 1.30E-02 2.50E-03 6.20E-02 1.50E-02 3.184E-02



93


Calc. No *

Slope angle (°)

Friction angle (°)

Cohesion (Pa)

Y acc. 1(m/s2)

X acc. 2(m/s2)

X disp. 3 (m)

Y-disp. 4 (m)

X-velocity (m/s)

Y-velocity (m/s)


49 30.50 42.00 6.70E+06 0.80 1.30 1.45E-02 1.00E-05 6.20E-02 2.00E-03 3.774E-0250 35.50 42.00 6.70E+06 0.80 1.30 1.45E-02 1.00E-04 6.30E-02 1.00E-03 3.773E-0251 42.30 42.00 6.70E+06 0.90 1.30 1.45E-02 1.00E-04 6.30E-02 2.00E-03 3.777E-0252 55.00 42.00 6.70E+06 0.85 1.40 1.45E-02 1.00E-04 6.40E-02 2.00E-03 3.834E-0253 68.20 42.00 6.70E+06 0.80 1.40 1.50E-02 1.00E-04 6.40E-02 1.00E-03 3.885E-0254 78.70 42.00 6.70E+06 1.20 1.50 1.50E-02 1.00E-04 6.40E-02 2.00E-03 3.931E-0255 84.30 42.00 6.70E+06 1.70 1.50 1.50E-02 1.00E-04 6.40E-02 3.00E-03 3.957E-0256 90.00 42.00 6.70E+06 1.00 1.50 1.50E-02 1.00E-04 6.50E-02 3.00E-03 3.983E-0257 30.50 5.00 6.70E+06 0.85 1.50 1.50E-02 1.00E-04 6.40E-02 1.00E-03 3.774E-0258 30.50 10.00 6.70E+06 0.85 1.50 1.50E-02 1.00E-04 6.40E-02 1.00E-03 3.774E-0259 30.50 15.00 6.70E+06 0.82 1.50 1.50E-02 1.00E-04 6.40E-02 1.00E-03 3.774E-0260 30.50 20.00 6.70E+06 0.81 1.50 1.50E-02 1.00E-04 6.40E-02 1.00E-03 3.774E-0261 30.50 25.00 6.70E+06 0.81 1.50 1.50E-02 1.00E-04 6.40E-02 1.00E-03 3.774E-0262 30.50 30.00 6.70E+06 0.80 1.50 1.45E-02 8.00E-05 6.20E-02 8.00E-04 3.774E-0263 30.50 35.00 6.70E+06 0.81 1.50 1.45E-02 8.00E-05 6.20E-02 8.00E-04 3.774E-0264 30.50 40.00 6.70E+06 0.80 1.50 1.45E-02 8.00E-05 6.20E-02 8.00E-04 3.774E-0265 55.00 5.00 6.70E+06 0.80 1.40 1.50E-02 1.00E-04 6.50E-02 1.00E-03 3.834E-0266 55.00 10.00 6.70E+06 0.80 1.40 1.50E-02 1.00E-04 6.50E-02 1.00E-03 3.834E-0267 55.00 25.00 6.70E+06 0.80 1.30 1.50E-02 1.00E-04 6.50E-02 1.00E-03 3.834E-0268 55.00 35.00 6.70E+06 0.80 1.30 1.45E-02 8.00E-05 6.20E-02 8.00E-04 3.834E-0269 55.00 40.00 6.70E+06 0.80 1.30 1.45E-02 8.00E-05 6.20E-02 8.00E-04 3.834E-0270 78.70 5.00 6.70E+06 1.10 1.40 1.45E-02 8.00E-05 6.20E-02 8.00E-04 3.931E-0271 78.70 30.00 6.70E+06 1.20 1.30 1.45E-02 8.00E-05 6.20E-02 8.00E-04 3.931E-0272 78.70 60.00 6.70E+06 1.10 1.30 1.45E-02 8.00E-05 6.20E-02 8.00E-04 3.931E-0273 90.00 10.00 6.70E+06 1.10 1.50 1.50E-02 1.00E-04 6.40E-02 1.00E-03 3.983E-0274 90.00 15.00 6.70E+06 1.10 1.30 1.50E-02 1.00E-04 6.40E-02 1.00E-03 3.983E-0275 90.00 20.00 6.70E+06 1.00 1.30 1.50E-02 1.00E-04 6.40E-02 1.00E-03 3.983E-0276 90.00 25.00 6.70E+06 0.90 1.10 1.50E-02 1.00E-04 6.40E-02 1.00E-03 3.983E-0277 90.00 35.00 6.70E+06 0.90 1.10 1.50E-02 1.00E-04 6.40E-02 1.00E-03 3.983E-02



94


Slope angle (°)

Friction angle (°)

Cohesion (Pa)

Y acc. 1(m/s2)

X acc. 2(m/s2)

X disp. 3 (m)

Y disp. 4 (m)

X-velocity (m/s)

Y-velocity (m/s)

Maximum velocity

78 42.30 42.00 6.70E+01 3.50 2.50 1.42E-02 8.00E-05 6.40E-02 2.00E-03 3.836E-0279 42.30 42.00 6.70E+02 2.60 1.60 1.42E-02 8.00E-05 6.40E-02 2.00E-03 3.793E-0280 42.30 42.00 6.70E+03 0.90 1.60 1.42E-02 8.00E-05 6.40E-02 2.00E-03 3.775E-0281 42.30 42.00 6.70E+04 0.82 1.40 1.40E-02 6.00E-05 6.20E-02 1.00E-03 3.773E-0282 42.30 42.00 6.70E+05 0.82 1.40 1.40E-02 6.00E-05 6.20E-02 1.00E-03 3.773E-0283 84.30 42.00 6.70E+01 6.50 3.00 2.50E-01 7.00E-01 1.20E+00 2.60E+00 3.063E+0084 84.30 42.00 6.70E+02 5.80 2.90 2.90E-01 6.00E-01 1.00E+00 2.50E+00 2.832E+0085 84.30 42.00 6.70E+03 3.00 1.90 1.50E-01 2.70E-01 1.10E+00 6.00E-01 1.325E+0086 84.30 42.00 6.70E+04 1.60 1.40 1.42E-02 1.00E-05 6.40E-02 5.00E-03 3.957E-0287 84.30 42.00 6.70E+05 1.60 1.30 1.42E-02 1.00E-05 6.40E-02 4.00E-03 3.957E-0288 30.50 42.00 1.00E+01 3.00 1.80 1.45E-02 1.00E-05 6.50E-02 2.00E-03 3.814E-0289 30.50 42.00 6.70E+01 2.50 1.70 1.45E-02 1.00E-05 6.50E-02 2.00E-03 3.820E-0290 30.50 42.00 6.70E+02 2.20 1.70 1.45E-02 1.00E-05 6.50E-02 2.00E-03 3.803E-0291 30.50 42.00 6.70E+03 1.00 1.50 1.40E-02 1.00E-05 6.40E-02 1.00E-03 3.777E-0292 30.50 42.00 6.70E+04 0.80 1.45 1.40E-02 1.00E-05 6.40E-02 1.00E-03 3.774E-0293 30.50 42.00 6.70E+05 0.80 1.45 1.40E-02 1.00E-05 6.20E-02 8.00E-04 3.774E-0294 68.20 42.00 1.00E+01 4.00 2.00 1.60E-01 2.40E-01 6.00E-01 1.00E+00 2.180E+0095 68.20 42.00 6.70E+01 4.00 2.00 1.60E-01 2.40E-01 6.00E-01 1.00E+00 2.151E+0096 68.20 42.00 6.70E+02 4.00 2.00 1.50E-01 2.50E-01 6.00E-01 1.10E+00 1.955E+0097 68.20 42.00 6.70E+03 2.20 1.40 5.50E-02 1.00E-01 2.20E-01 4.00E-01 4.163E-0198 68.20 42.00 6.70E+04 0.80 1.40 1.45E-02 1.00E-05 6.40E-02 1.00E-03 3.885E-0299 68.20 42.00 6.70E+05 0.80 1.40 1.45E-02 1.00E-05 6.40E-02 1.00E-03 3.885E-02

100 90.00 42.00 6.70E+02 6.00 3.00 3.00E-01 6.00E-01 1.00E+00 2.50E+00 3.133E+00101 90.00 42.00 6.70E+03 3.60 2.00 2.00E-01 3.00E-01 8.00E-01 1.40E+00 1.652E+00102 90.00 42.00 1.00E+04 4.20 1.60 1.30E-01 2.20E-01 5.00E-01 8.50E-01 1.031E+00103 90.00 42.00 6.70E+04 1.00 1.50 1.45E-02 1.00E-05 6.40E-02 2.00E-03 3.983E-02104 90.00 42.00 6.70E+05 1.00 1.50 1.45E-02 1.00E-05 6.40E-02 2.00E-03 3.983E-02



95


Calc. No *

Slope angle (°)

Friction angle (°)

Cohesion (Pa)

Y acc.1 (m/s2)

X acc. 2 (m/s2)

X disp. 3 (m)

Y disp. 4 (m)

X-velocity (m/s)

Y-velocity (m/s)


105 25.50 27.80 2.70E+07 0.80 2.10 1.45E-02 1.00E-04 8.00E-02 5.00E-03 3.366E-02106 38.60 27.80 2.70E+07 0.80 2.50 1.42E-02 1.00E-04 8.00E-02 6.00E-03 3.390E-02107 53.00 27.80 2.70E+07 1.20 2.50 1.42E-02 1.00E-04 8.00E-02 6.00E-03 3.491E-02108 75.90 27.80 2.70E+07 1.20 2.50 1.45E-02 1.00E-04 8.60E-02 7.00E-03 3.620E-02109 81.00 27.80 2.70E+07 1.20 2.50 1.45E-02 5.00E-04 8.50E-02 7.00E-03 3.646E-02110 90.00 27.80 2.70E+07 1.20 2.30 1.45E-02 2.00E-04 8.60E-02 9.00E-03 3.685E-02111 25.50 5.00 2.70E+07 0.80 2.10 1.42E-02 1.00E-04 7.80E-02 6.00E-03 3.366E-02112 25.50 55.00 2.70E+07 0.76 2.10 1.40E-02 8.00E-05 7.80E-02 5.00E-03 3.366E-02113 53.00 5.00 2.70E+07 1.20 2.50 1.50E-02 2.00E-04 8.00E-02 5.00E-03 3.491E-02114 53.00 55.00 2.70E+07 1.00 2.50 1.40E-02 8.00E-05 7.50E-02 4.80E-03 3.491E-02115 75.90 5.00 2.70E+07 1.20 2.30 1.42E-02 1.00E-04 8.00E-02 5.00E-03 3.620E-02116 75.90 10.00 2.70E+07 1.10 2.20 1.42E-02 1.00E-04 8.00E-02 5.00E-03 3.620E-02117 75.90 15.00 2.70E+07 1.10 2.20 1.42E-02 1.00E-04 8.00E-02 5.00E-03 3.620E-02118 75.90 55.00 2.70E+07 1.00 2.00 1.40E-02 8.00E-05 7.00E-02 3.00E-03 3.620E-02119 90.00 5.00 2.70E+07 1.30 2.30 1.42E-02 1.00E-04 8.00E-02 1.00E-02 3.685E-02120 90.00 10.00 2.70E+07 1.20 2.20 1.40E-02 8.00E-05 7.00E-02 8.00E-04 3.685E-02121 25.50 27.80 2.70E+01 5.00 3.00 1.50E-02 8.00E-04 8.00E-02 1.00E-02 8.772E-02122 25.50 27.80 2.70E+02 4.00 3.00 1.50E-02 8.00E-04 8.00E-02 1.00E-02 3.605E-02123 25.50 27.80 2.70E+03 4.00 3.20 1.50E-02 8.00E-04 7.80E-02 8.00E-03 3.589E-02124 25.50 27.80 2.70E+04 3.50 4.00 1.50E-02 4.00E-04 7.80E-02 7.00E-03 3.545E-02125 25.50 27.80 2.70E+05 0.82 2.20 1.42E-02 1.00E-04 7.80E-02 6.00E-03 3.366E-02126 75.90 27.80 2.70E+01 7.00 2.70 2.50E-01 7.00E-01 1.00E+00 2.70E+00 3.699E+00127 75.90 27.80 2.70E+02 7.50 2.70 2.10E-01 7.00E-01 8.00E-01 2.70E+00 3.596E+00128 75.90 27.80 2.70E+03 6.30 3.70 2.20E-01 7.00E-01 8.00E-01 2.60E+00 3.216E+00129 75.90 27.80 2.70E+04 6.00 2.10 1.20E-01 2.40E-01 4.50E-01 9.00E-01 1.143E+00130 75.90 27.80 2.70E+05 1.30 2.40 1.45E-02 5.00E-05 8.00E-02 4.00E-03 3.620E-02131 90.00 27.80 2.70E+02 9.00 2.50 1.30E-01 8.20E-01 6.00E-01 3.30E+00 4.121E+00132 90.00 27.80 2.70E+03 8.00 2.50 2.00E-01 8.00E-01 7.00E-01 3.30E+00 3.828E+00133 90.00 27.80 2.70E+04 4.20 2.50 1.60E+00 3.90E-01 6.50E-01 1.54E+00 1.841E+00134 90.00 27.80 2.70E+05 1.30 2.30 1.50E-02 3.00E-04 8.10E-02 9.00E-03 3.685E-02



96


Slope angle (°)

Friction angle (°)

Cohesion (Pa)

Y acc. 1

(m/s2) X acc.2 (m/s2)

X-disp. 3 (m)

Y-disp. 4 (m)

X-velocity (m/s)

Y-velocity (m/s)


135 29.70 27.80 2.70E+07 0.85 2.00 1.42E-02 1.00E-04 8.00E-02 8.00E-03 3.405E-02136 45.00 27.80 2.70E+07 1.00 2.50 1.42E-02 1.00E-04 8.00E-02 8.00E-03 3.425E-02137 63.00 27.80 2.70E+07 1.20 2.50 1.45E-02 1.00E-04 8.10E-02 6.00E-03 3.556E-02138 25.50 10.00 2.70E+07 0.85 2.20 1.42E-02 1.00E-04 7.80E-02 6.00E-03 3.366E-02139 25.50 15.00 2.70E+07 0.82 2.20 1.42E-02 1.00E-04 7.50E-02 6.00E-03 3.366E-02140 25.50 20.00 2.70E+07 0.81 2.10 1.42E-02 1.00E-04 7.50E-02 6.00E-03 3.366E-02141 25.50 25.00 2.70E+07 0.81 2.10 1.40E-02 9.00E-05 7.30E-02 5.00E-03 3.366E-02142 25.50 35.00 2.70E+07 0.80 2.00 1.40E-02 9.00E-05 7.30E-02 5.00E-03 3.366E-02143 53.00 10.00 2.70E+07 1.20 2.50 1.42E-02 1.00E-04 7.80E-02 6.00E-03 3.491E-02144 53.00 15.00 2.70E+07 1.20 2.50 1.41E-02 1.00E-04 8.00E-02 6.00E-03 3.491E-02145 53.00 20.00 2.70E+07 1.10 2.40 1.41E-02 1.00E-04 8.00E-02 6.00E-03 3.491E-02146 53.00 25.00 2.70E+07 1.00 2.40 1.40E-02 1.00E-04 8.00E-02 6.00E-03 3.491E-02147 53.00 35.00 2.70E+07 1.00 2.30 1.40E-02 1.00E-04 8.00E-02 5.00E-03 3.491E-02148 75.90 20.00 2.70E+07 1.10 2.50 1.42E-02 1.00E-04 8.00E-02 5.00E-03 3.620E-02149 75.90 25.00 2.70E+07 1.00 2.50 1.42E-02 1.00E-04 8.00E-02 5.00E-03 3.620E-02150 75.90 35.00 2.70E+07 1.00 2.30 1.40E-02 1.00E-04 7.80E-02 4.50E-03 3.620E-02151 90.00 10.00 2.70E+07 1.10 2.50 1.42E-02 1.00E-04 8.10E-02 9.00E-03 3.685E-02152 90.00 15.00 2.70E+07 1.10 2.20 1.42E-02 1.00E-04 8.10E-02 9.00E-03 3.685E-02153 90.00 20.00 2.70E+07 1.10 2.20 1.42E-02 1.00E-04 8.10E-02 9.00E-03 3.685E-02154 90.00 25.00 2.70E+07 1.00 2.00 1.40E-02 8.00E-05 7.80E-02 8.00E-03 3.685E-02155 90.00 35.00 2.70E+07 0.95 2.00 1.40E-02 8.00E-05 7.80E-02 8.00E-03 3.685E-02156 38.60 27.80 2.70E+01 6.00 2.50 1.00E-02 5.00E-03 7.00E-02 4.00E-02 1.336E+00157 38.60 27.80 2.70E+02 5.00 2.20 1.20E-02 3.60E-03 6.50E-02 3.00E-02 1.214E+00158 38.60 27.80 2.70E+03 4.20 2.80 1.20E-02 3.00E-03 6.20E-02 3.00E-02 8.608E-01159 38.60 27.80 2.70E+04 3.50 2.50 1.20E-02 3.00E-04 8.00E-02 8.00E-03 3.566E-02160 38.60 27.80 2.70E+05 0.80 2.50 1.20E-02 1.00E-04 8.00E-02 6.00E-03 3.390E-02161 53.00 27.80 2.70E+01 6.50 2.00 1.10E-01 1.80E-01 5.00E-01 7.20E-01 2.500E+00162 53.00 27.80 2.70E+02 6.20 2.00 1.00E-01 1.80E-01 4.50E-01 6.50E-01 2.322E+00163 53.00 27.80 2.70E+03 6.50 1.60 8.00E-02 1.35E-01 3.20E-01 5.00E-01 1.921E+00164 53.00 27.80 2.70E+04 4.00 1.60 1.00E-02 6.80E-03 5.00E-02 4.80E-02 3.631E-02165 53.00 27.80 2.70E+05 1.20 1.60 1.42E-02 1.00E-04 8.00E-02 5.00E-03 3.491E-02



97

Appendix C-1: Calculation matrix used for simulation (changes in bulk modulus, frequency 3 Hz).

Calculation number

Slope angle (º)

Slope height (m)

Shear Modulus (MPa)

Bulk Modulus (MPa)

Tensile Strength (MPa)

Density (kg/m3)

Cohesion (MPa)

Friction angle (º)

Frequency (Hz)

Ampli-fication

1 25.5 10 10000 10 1.58 2000 6.7 42 3 1.80002 25.5 10 10000 50 1.58 2000 6.7 42 3 1.80003 25.5 10 10000 100 1.58 2000 6.7 42 3 1.80004 25.5 10 10000 200 1.58 2000 6.7 42 3 1.80005 25.5 10 10000 500 1.58 2000 6.7 42 3 1.80006 25.5 10 10000 1000 1.58 2000 6.7 42 3 1.60007 25.5 10 10000 1500 1.58 2000 6.7 42 3 1.60008 25.5 10 10000 2000 1.58 2000 6.7 42 3 1.60009 25.5 10 10000 3000 1.58 2000 6.7 42 3 1.6000

10 25.5 10 10000 4000 1.58 2000 6.7 42 3 1.600011 25.5 10 10000 5000 1.58 2000 6.7 42 3 1.600012 25.5 10 10000 10000 1.58 2000 6.7 42 3 1.600013 25.5 20 10000 10 1.58 2000 6.7 42 3 2.300014 25.5 20 10000 50 1.58 2000 6.7 42 3 2.300015 25.5 20 10000 100 1.58 2000 6.7 42 3 2.300016 25.5 20 10000 200 1.58 2000 6.7 42 3 2.300017 25.5 20 10000 500 1.58 2000 6.7 42 3 2.300018 25.5 20 10000 1000 1.58 2000 6.7 42 3 2.300019 25.5 20 10000 1500 1.58 2000 6.7 42 3 2.100020 25.5 20 10000 2000 1.58 2000 6.7 42 3 2.100021 25.5 20 10000 3000 1.58 2000 6.7 42 3 2.100022 25.5 20 10000 4000 1.58 2000 6.7 42 3 2.100023 25.5 20 10000 5000 1.58 2000 6.7 42 3 2.000024 25.5 20 10000 8000 1.58 2000 6.7 42 3 2.000025 25.5 20 10000 10000 1.58 2000 6.7 42 3 2.000026 25.5 25 10000 10 1.58 2000 6.7 42 3 2.300027 25.5 25 10000 50 1.58 2000 6.7 42 3 2.300028 25.5 25 10000 100 1.58 2000 6.7 42 3 2.300029 25.5 25 10000 200 1.58 2000 6.7 42 3 2.300030 25.5 25 10000 500 1.58 2000 6.7 42 3 2.300031 25.5 25 10000 1000 1.58 2000 6.7 42 3 2.200032 25.5 25 10000 1500 1.58 2000 6.7 42 3 2.200033 25.5 25 10000 2000 1.58 2000 6.7 42 3 2.200034 25.5 25 10000 3000 1.58 2000 6.7 42 3 2.200035 25.5 25 10000 4000 1.58 2000 6.7 42 3 2.200036 25.5 25 10000 5000 1.58 2000 6.7 42 3 1.800037 25.5 25 10000 8000 1.58 2000 6.7 42 3 1.800038 25.5 25 10000 10000 1.58 2000 6.7 42 3 1.800039 25.5 30 10000 10 1.58 2000 6.7 42 3 1.900040 25.5 30 10000 50 1.58 2000 6.7 42 3 1.900041 25.5 30 10000 100 1.58 2000 6.7 42 3 1.900042 25.5 30 10000 200 1.58 2000 6.7 42 3 1.900043 25.5 30 10000 500 1.58 2000 6.7 42 3 1.800044 25.5 30 10000 1000 1.58 2000 6.7 42 3 1.800045 25.5 30 10000 1500 1.58 2000 6.7 42 3 1.7000


98

Appendix C-1: Calculation matrix used for simulation (changes in bulk modulus, frequency 3 Hz): (contd.)

Calculation number

Slope angle (º)

Slope height (m)

Shear Modulus (MPa)

Bulk Modulus (MPa)


Density (kg/m3)

Cohesion (MPa)

Friction angle (º)

Frequency (Hz)

Amplifi-cation

46 25.5 30 10000 2000 1.58 2000 6.7 42 3 1.700047 25.5 30 10000 3000 1.58 2000 6.7 42 3 1.700048 25.5 30 10000 4000 1.58 2000 6.7 42 3 1.600049 25.5 30 10000 5000 1.58 2000 6.7 42 3 1.600050 25.5 30 10000 8000 1.58 2000 6.7 42 3 1.500051 25.5 30 10000 10000 1.58 2000 6.7 42 3 1.500052 25.5 31 10000 10 1.58 2000 6.7 42 3 1.900053 25.5 31 10000 50 1.58 2000 6.7 42 3 1.900054 25.5 31 10000 100 1.58 2000 6.7 42 3 1.900055 25.5 31 10000 200 1.58 2000 6.7 42 3 1.900056 25.5 31 10000 500 1.58 2000 6.7 42 3 1.900057 25.5 31 10000 1000 1.58 2000 6.7 42 3 1.700058 25.5 31 10000 1500 1.58 2000 6.7 42 3 1.700059 25.5 31 10000 2000 1.58 2000 6.7 42 3 1.700060 25.5 31 10000 3000 1.58 2000 6.7 42 3 1.700061 25.5 31 10000 4000 1.58 2000 6.7 42 3 1.600062 25.5 31 10000 5000 1.58 2000 6.7 42 3 1.600063 25.5 31 10000 8000 1.58 2000 6.7 42 3 1.500064 25.5 31 10000 10000 1.58 2000 6.7 42 3 1.500065 25.5 35 10000 10 1.58 2000 6.7 42 3 2.300066 25.5 35 10000 50 1.58 2000 6.7 42 3 2.300067 25.5 35 10000 100 1.58 2000 6.7 42 3 2.300068 25.5 35 10000 200 1.58 2000 6.7 42 3 2.300069 25.5 35 10000 500 1.58 2000 6.7 42 3 2.300070 25.5 35 10000 1000 1.58 2000 6.7 42 3 2.200071 25.5 35 10000 1500 1.58 2000 6.7 42 3 2.200072 25.5 35 10000 2000 1.58 2000 6.7 42 3 2.200073 25.5 35 10000 3000 1.58 2000 6.7 42 3 2.200074 25.5 35 10000 4000 1.58 2000 6.7 42 3 2.200075 25.5 35 10000 5000 1.58 2000 6.7 42 3 2.200076 25.5 35 10000 8000 1.58 2000 6.7 42 3 2.000077 25.5 35 10000 10000 1.58 2000 6.7 42 3 2.000078 25.5 40 10000 10 1.58 2000 6.7 42 3 2.900079 25.5 40 10000 50 1.58 2000 6.7 42 3 2.900080 25.5 40 10000 100 1.58 2000 6.7 42 3 2.900081 25.5 40 10000 200 1.58 2000 6.7 42 3 2.900082 25.5 40 10000 500 1.58 2000 6.7 42 3 2.700083 25.5 40 10000 1000 1.58 2000 6.7 42 3 2.500084 25.5 40 10000 1500 1.58 2000 6.7 42 3 2.500085 25.5 40 10000 2000 1.58 2000 6.7 42 3 2.300086 25.5 40 10000 3000 1.58 2000 6.7 42 3 2.300087 25.5 40 10000 4000 1.58 2000 6.7 42 3 2.300088 25.5 40 10000 5000 1.58 2000 6.7 42 3 2.300089 25.5 40 10000 8000 1.58 2000 6.7 42 3 2.300090 25.5 40 10000 10000 1.58 2000 6.7 42 3 2.300091 25.5 45 10000 10 1.58 2000 6.7 42 3 3.200092 25.5 50 10000 10 1.58 2000 6.7 42 3 3.0000


99


Calculation number

Slope angle (º)

Slope height (m)

Shear Modulus (MPa)

Bulk Modulus (MPa)


Density (kg/m3)

Cohesion (MPa)

Friction angle (º)

Frequency (Hz)

Amplifi-cation

93 25.5 10 10000 10 1.58 2000 6.7 42 5 2.300094 25.5 10 10000 50 1.58 2000 6.7 42 5 2.300095 25.5 10 10000 100 1.58 2000 6.7 42 5 2.300096 25.5 10 10000 200 1.58 2000 6.7 42 5 2.200097 25.5 10 10000 500 1.58 2000 6.7 42 5 2.200098 25.5 10 10000 1000 1.58 2000 6.7 42 5 2.100099 25.5 10 10000 1500 1.58 2000 6.7 42 5 2.1000

100 25.5 10 10000 2000 1.58 2000 6.7 42 5 2.0000101 25.5 10 10000 3000 1.58 2000 6.7 42 5 2.0000102 25.5 10 10000 4000 1.58 2000 6.7 42 5 2.0000103 25.5 10 10000 5000 1.58 2000 6.7 42 5 1.7000104 25.5 10 10000 10000 1.58 2000 6.7 42 5 1.7000105 25.5 20 10000 10 1.58 2000 6.7 42 5 2.5000106 25.5 20 10000 50 1.58 2000 6.7 42 5 2.5000107 25.5 20 10000 100 1.58 2000 6.7 42 5 2.5000108 25.5 20 10000 200 1.58 2000 6.7 42 5 2.5000109 25.5 20 10000 500 1.58 2000 6.7 42 5 2.5000110 25.5 20 10000 1000 1.58 2000 6.7 42 5 2.2000111 25.5 20 10000 1500 1.58 2000 6.7 42 5 2.1000112 25.5 20 10000 2000 1.58 2000 6.7 42 5 2.0000113 25.5 20 10000 3000 1.58 2000 6.7 42 5 2.0000114 25.5 20 10000 4000 1.58 2000 6.7 42 5 1.8000115 25.5 20 10000 5000 1.58 2000 6.7 42 5 1.8000116 25.5 20 10000 8000 1.58 2000 6.7 42 5 1.7000117 25.5 20 10000 10000 1.58 2000 6.7 42 5 1.7000118 25.5 25 10000 10 1.58 2000 6.7 42 5 2.3000119 25.5 25 10000 50 1.58 2000 6.7 42 5 2.3000120 25.5 25 10000 100 1.58 2000 6.7 42 5 2.3000121 25.5 25 10000 200 1.58 2000 6.7 42 5 2.3000122 25.5 25 10000 500 1.58 2000 6.7 42 5 2.3000123 25.5 25 10000 1000 1.58 2000 6.7 42 5 2.3000124 25.5 25 10000 1500 1.58 2000 6.7 42 5 2.3000125 25.5 25 10000 2000 1.58 2000 6.7 42 5 2.3000126 25.5 25 10000 3000 1.58 2000 6.7 42 5 2.0000127 25.5 25 10000 4000 1.58 2000 6.7 42 5 2.0000128 25.5 25 10000 5000 1.58 2000 6.7 42 5 2.0000129 25.5 25 10000 8000 1.58 2000 6.7 42 5 2.0000130 25.5 25 10000 10000 1.58 2000 6.7 42 5 2.0000131 25.5 30 10000 10 1.58 2000 6.7 42 5 3.5000132 25.5 30 10000 50 1.58 2000 6.7 42 5 3.5000133 25.5 30 10000 100 1.58 2000 6.7 42 5 3.5000134 25.5 30 10000 200 1.58 2000 6.7 42 5 3.5000135 25.5 30 10000 500 1.58 2000 6.7 42 5 3.5000136 25.5 30 10000 1000 1.58 2000 6.7 42 5 3.5000137 25.5 30 10000 1500 1.58 2000 6.7 42 5 3.3000


100

Appendix C-2: Calculation matrix used for simulation (changes in bulk modulus, frequency 5 Hz) (contd.)

Calculation number

Slope angle (º)

Slope height (m)

Shear Modulus (MPa)

Bulk Modulus (MPa)


Density (kg/m3)

Cohesion (MPa)

Friction angle (º)

Frequency (Hz)

Ampli-fication

138 25.5 30 10000 2000 1.58 2000 6.7 42 5 3.2000139 25.5 30 10000 3000 1.58 2000 6.7 42 5 3.2000140 25.5 30 10000 4000 1.58 2000 6.7 42 5 2.8000141 25.5 30 10000 5000 1.58 2000 6.7 42 5 2.8000142 25.5 30 10000 8000 1.58 2000 6.7 42 5 2.3000143 25.5 30 10000 10000 1.58 2000 6.7 42 5 2.3000144 25.5 31 10000 10 1.58 2000 6.7 42 5 3.5000145 25.5 31 10000 50 1.58 2000 6.7 42 5 3.5000146 25.5 31 10000 100 1.58 2000 6.7 42 5 3.5000147 25.5 31 10000 200 1.58 2000 6.7 42 5 3.5000148 25.5 31 10000 500 1.58 2000 6.7 42 5 3.5000149 25.5 31 10000 1000 1.58 2000 6.7 42 5 3.5000150 25.5 31 10000 1500 1.58 2000 6.7 42 5 3.3000151 25.5 31 10000 2000 1.58 2000 6.7 42 5 3.2000152 25.5 31 10000 3000 1.58 2000 6.7 42 5 3.2000153 25.5 31 10000 4000 1.58 2000 6.7 42 5 3.0000154 25.5 31 10000 5000 1.58 2000 6.7 42 5 2.8000155 25.5 31 10000 8000 1.58 2000 6.7 42 5 2.3000156 25.5 31 10000 10000 1.58 2000 6.7 42 5 2.3000157 25.5 35 10000 10 1.58 2000 6.7 42 5 2.5000158 25.5 35 10000 50 1.58 2000 6.7 42 5 2.5000159 25.5 35 10000 100 1.58 2000 6.7 42 5 2.5000160 25.5 35 10000 200 1.58 2000 6.7 42 5 2.5000161 25.5 35 10000 500 1.58 2000 6.7 42 5 2.5000162 25.5 35 10000 1000 1.58 2000 6.7 42 5 2.4000163 25.5 35 10000 1500 1.58 2000 6.7 42 5 2.3000164 25.5 35 10000 2000 1.58 2000 6.7 42 5 2.3000165 25.5 35 10000 3000 1.58 2000 6.7 42 5 2.3000166 25.5 35 10000 4000 1.58 2000 6.7 42 5 2.3000167 25.5 35 10000 5000 1.58 2000 6.7 42 5 2.2000168 25.5 35 10000 8000 1.58 2000 6.7 42 5 2.2000169 25.5 35 10000 10000 1.58 2000 6.7 42 5 2.2000170 25.5 40 10000 10 1.58 2000 6.7 42 5 2.5000171 25.5 40 10000 50 1.58 2000 6.7 42 5 2.5000172 25.5 40 10000 100 1.58 2000 6.7 42 5 2.5000173 25.5 40 10000 200 1.58 2000 6.7 42 5 2.5000174 25.5 40 10000 500 1.58 2000 6.7 42 5 2.5000175 25.5 40 10000 1000 1.58 2000 6.7 42 5 2.5000176 25.5 40 10000 1500 1.58 2000 6.7 42 5 2.3000177 25.5 40 10000 2000 1.58 2000 6.7 42 5 2.3000178 25.5 40 10000 3000 1.58 2000 6.7 42 5 2.3000179 25.5 40 10000 4000 1.58 2000 6.7 42 5 2.3000180 25.5 40 10000 5000 1.58 2000 6.7 42 5 2.2000181 25.5 40 10000 8000 1.58 2000 6.7 42 5 2.1000182 25.5 40 10000 10000 1.58 2000 6.7 42 5 2.0000


101


Calculation number

Slope angle (º)

Slope height (m)

Shear Modulus (MPa)

Bulk Modulus (MPa)


Density (kg/m3)

Cohesion (MPa)

Friction angle (º)

Frequency (Hz)

Amplifi-cation

183 25.5 10 10000 10 1.58 2000 6.7 42 10 2.0000184 25.5 10 10000 50 1.58 2000 6.7 42 10 2.0000185 25.5 10 10000 100 1.58 2000 6.7 42 10 2.0000186 25.5 10 10000 200 1.58 2000 6.7 42 10 2.0000187 25.5 10 10000 500 1.58 2000 6.7 42 10 2.0000188 25.5 10 10000 1000 1.58 2000 6.7 42 10 2.0000189 25.5 10 10000 1500 1.58 2000 6.7 42 10 2.0000190 25.5 10 10000 2000 1.58 2000 6.7 42 10 2.0000191 25.5 10 10000 3000 1.58 2000 6.7 42 10 2.0000192 25.5 10 10000 4000 1.58 2000 6.7 42 10 2.0000193 25.5 10 10000 5000 1.58 2000 6.7 42 10 2.0000194 25.5 10 10000 10000 1.58 2000 6.7 42 10 2.0000195 25.5 20 10000 10 1.58 2000 6.7 42 10 5.2000196 25.5 20 10000 50 1.58 2000 6.7 42 10 5.2000197 25.5 20 10000 100 1.58 2000 6.7 42 10 5.2000198 25.5 20 10000 200 1.58 2000 6.7 42 10 5.2000199 25.5 20 10000 500 1.58 2000 6.7 42 10 5.0000200 25.5 20 10000 1000 1.58 2000 6.7 42 10 4.8000201 25.5 20 10000 1500 1.58 2000 6.7 42 10 4.8000202 25.5 20 10000 2000 1.58 2000 6.7 42 10 4.8000203 25.5 20 10000 3000 1.58 2000 6.7 42 10 4.8000204 25.5 20 10000 4000 1.58 2000 6.7 42 10 4.5000205 25.5 20 10000 5000 1.58 2000 6.7 42 10 4.5000206 25.5 20 10000 8000 1.58 2000 6.7 42 10 4.2000207 25.5 20 10000 10000 1.58 2000 6.7 42 10 4.2000208 25.5 25 10000 10 1.58 2000 6.7 42 10 10.0000209 25.5 25 10000 50 1.58 2000 6.7 42 10 10.0000210 25.5 25 10000 100 1.58 2000 6.7 42 10 10.0000211 25.5 25 10000 200 1.58 2000 6.7 42 10 10.0000212 25.5 25 10000 500 1.58 2000 6.7 42 10 10.0000213 25.5 25 10000 1000 1.58 2000 6.7 42 10 7.0000214 25.5 25 10000 1500 1.58 2000 6.7 42 10 7.0000215 25.5 25 10000 2000 1.58 2000 6.7 42 10 7.0000216 25.5 25 10000 3000 1.58 2000 6.7 42 10 7.0000217 25.5 25 10000 4000 1.58 2000 6.7 42 10 6.8000218 25.5 25 10000 5000 1.58 2000 6.7 42 10 6.8000219 25.5 25 10000 8000 1.58 2000 6.7 42 10 6.2000220 25.5 25 10000 10000 1.58 2000 6.7 42 10 6.2000221 25.5 30 10000 10 1.58 2000 6.7 42 10 12.0000222 25.5 30 10000 50 1.58 2000 6.7 42 10 12.0000223 25.5 30 10000 100 1.58 2000 6.7 42 10 12.0000224 25.5 30 10000 200 1.58 2000 6.7 42 10 12.0000225 25.5 30 10000 500 1.58 2000 6.7 42 10 12.0000226 25.5 30 10000 1000 1.58 2000 6.7 42 10 10.0000


102


Calculation number

Slope angle (º)

Slope height (m)

Shear Modulus (Mpa)

Bulk Modulus (MPa)


Density (kg/m3)

Cohesion (MPa)

Friction angle (º)

Fre-quency (Hz)

Ampli-fication

227 25.5 30 10000 1500 1.58 2000 6.7 42 10 10.000228 25.5 30 10000 2000 1.58 2000 6.7 42 10 10.000229 25.5 30 10000 3000 1.58 2000 6.7 42 10 10.000230 25.5 30 10000 4000 1.58 2000 6.7 42 10 10.000231 25.5 30 10000 5000 1.58 2000 6.7 42 10 10.000232 25.5 30 10000 8000 1.58 2000 6.7 42 10 9.000233 25.5 30 10000 10000 1.58 2000 6.7 42 10 9.000234 25.5 31 10000 10 1.58 2000 6.7 42 10 12.000235 25.5 31 10000 50 1.58 2000 6.7 42 10 12.000236 25.5 31 10000 100 1.58 2000 6.7 42 10 12.000237 25.5 31 10000 200 1.58 2000 6.7 42 10 12.000238 25.5 31 10000 500 1.58 2000 6.7 42 10 12.000239 25.5 31 10000 1000 1.58 2000 6.7 42 10 10.000240 25.5 31 10000 1500 1.58 2000 6.7 42 10 10.000241 25.5 31 10000 2000 1.58 2000 6.7 42 10 10.000242 25.5 31 10000 3000 1.58 2000 6.7 42 10 10.000243 25.5 31 10000 4000 1.58 2000 6.7 42 10 10.000244 25.5 31 10000 5000 1.58 2000 6.7 42 10 9.000245 25.5 31 10000 8000 1.58 2000 6.7 42 10 9.000246 25.5 31 10000 10000 1.58 2000 6.7 42 10 9.000247 25.5 35 10000 10 1.58 2000 6.7 42 10 7.800248 25.5 35 10000 50 1.58 2000 6.7 42 10 7.800249 25.5 35 10000 100 1.58 2000 6.7 42 10 7.800250 25.5 35 10000 200 1.58 2000 6.7 42 10 7.800251 25.5 35 10000 500 1.58 2000 6.7 42 10 7.800252 25.5 35 10000 1000 1.58 2000 6.7 42 10 7.800253 25.5 35 10000 1500 1.58 2000 6.7 42 10 7.200254 25.5 35 10000 2000 1.58 2000 6.7 42 10 7.200255 25.5 35 10000 3000 1.58 2000 6.7 42 10 7.200256 25.5 35 10000 4000 1.58 2000 6.7 42 10 7.200257 25.5 35 10000 5000 1.58 2000 6.7 42 10 7.200258 25.5 35 10000 8000 1.58 2000 6.7 42 10 6.100259 25.5 35 10000 10000 1.58 2000 6.7 42 10 6.100260 25.5 40 10000 10 1.58 2000 6.7 42 10 12.000261 25.5 40 10000 50 1.58 2000 6.7 42 10 12.000262 25.5 40 10000 100 1.58 2000 6.7 42 10 12.000263 25.5 40 10000 200 1.58 2000 6.7 42 10 12.000264 25.5 40 10000 500 1.58 2000 6.7 42 10 12.000265 25.5 40 10000 1000 1.58 2000 6.7 42 10 11.000266 25.5 40 10000 1500 1.58 2000 6.7 42 10 11.000267 25.5 40 10000 2000 1.58 2000 6.7 42 10 11.000268 25.5 40 10000 3000 1.58 2000 6.7 42 10 11.000269 25.5 40 10000 4000 1.58 2000 6.7 42 10 11.000270 25.5 40 10000 5000 1.58 2000 6.7 42 10 10.000271 25.5 40 10000 8000 1.58 2000 6.7 42 10 10.000272 25.5 40 10000 10000 1.58 2000 6.7 42 10 10.000


103


Calculation number

Slope angle (º)

Slope height (m)

Shear Modulus (MPa)

Bulk Modulus (MPa)


Density (kg/m3)

Cohesion (MPa)

Friction angle (º)

Fre-quency (Hz)

Amplification

273 25.5 10 10000 10 1.58 2000 6.7 42 15 7.0000274 25.5 10 10000 50 1.58 2000 6.7 42 15 7.0000275 25.5 10 10000 100 1.58 2000 6.7 42 15 7.0000276 25.5 10 10000 200 1.58 2000 6.7 42 15 7.0000277 25.5 10 10000 500 1.58 2000 6.7 42 15 7.0000278 25.5 10 10000 1000 1.58 2000 6.7 42 15 7.0000279 25.5 10 10000 1500 1.58 2000 6.7 42 15 6.7000280 25.5 10 10000 2000 1.58 2000 6.7 42 15 6.7000281 25.5 10 10000 3000 1.58 2000 6.7 42 15 6.7000282 25.5 10 10000 4000 1.58 2000 6.7 42 15 6.7000283 25.5 10 10000 5000 1.58 2000 6.7 42 15 6.2000284 25.5 10 10000 10000 1.58 2000 6.7 42 15 6.0000285 25.5 20 10000 10 1.58 2000 6.7 42 15 17.000286 25.5 20 10000 50 1.58 2000 6.7 42 15 17.000287 25.5 20 10000 100 1.58 2000 6.7 42 15 17.000288 25.5 20 10000 200 1.58 2000 6.7 42 15 17.000289 25.5 20 10000 500 1.58 2000 6.7 42 15 16.000290 25.5 20 10000 1000 1.58 2000 6.7 42 15 15.000291 25.5 20 10000 1500 1.58 2000 6.7 42 15 15.000292 25.5 20 10000 2000 1.58 2000 6.7 42 15 15.000293 25.5 20 10000 3000 1.58 2000 6.7 42 15 15.000294 25.5 20 10000 4000 1.58 2000 6.7 42 15 15.000295 25.5 20 10000 5000 1.58 2000 6.7 42 15 15.000296 25.5 20 10000 8000 1.58 2000 6.7 42 15 13.000297 25.5 20 10000 10000 1.58 2000 6.7 42 15 13.000298 25.5 25 10000 10 1.58 2000 6.7 42 15 9.5000299 25.5 25 10000 50 1.58 2000 6.7 42 15 9.5000300 25.5 25 10000 100 1.58 2000 6.7 42 15 9.5000301 25.5 25 10000 200 1.58 2000 6.7 42 15 9.5000302 25.5 25 10000 500 1.58 2000 6.7 42 15 9.5000303 25.5 25 10000 1000 1.58 2000 6.7 42 15 9.0000304 25.5 25 10000 1500 1.58 2000 6.7 42 15 9.0000305 25.5 25 10000 2000 1.58 2000 6.7 42 15 9.0000306 25.5 25 10000 3000 1.58 2000 6.7 42 15 3.5000307 25.5 25 10000 4000 1.58 2000 6.7 42 15 3.5000308 25.5 25 10000 5000 1.58 2000 6.7 42 15 3.5000309 25.5 25 10000 8000 1.58 2000 6.7 42 15 7.0000310 25.5 25 10000 10000 1.58 2000 6.7 42 15 7.0000311 25.5 30 10000 10 1.58 2000 6.7 42 15 6.0000312 25.5 30 10000 50 1.58 2000 6.7 42 15 6.0000313 25.5 30 10000 100 1.58 2000 6.7 42 15 6.0000314 25.5 30 10000 200 1.58 2000 6.7 42 15 6.0000315 25.5 30 10000 500 1.58 2000 6.7 42 15 6.0000316 25.5 30 10000 1000 1.58 2000 6.7 42 15 5.8000


104


Calculation number

Slope angle (º)

Slope height (m)

Shear Modulus (MPa)

Bulk Modulus (MPa)


Density (kg/m3)

Cohesion (MPa)

Friction angle (º)

Frequency (Hz)

Amplifica-tion

317 25.5 30 10000 1500 1.58 2000 6.7 42 15 5.8000318 25.5 30 10000 2000 1.58 2000 6.7 42 15 5.8000319 25.5 30 10000 3000 1.58 2000 6.7 42 15 5.8000320 25.5 30 10000 4000 1.58 2000 6.7 42 15 5.8000321 25.5 30 10000 5000 1.58 2000 6.7 42 15 5.0000322 25.5 30 10000 8000 1.58 2000 6.7 42 15 5.0000323 25.5 30 10000 10000 1.58 2000 6.7 42 15 4.5000324 25.5 31 10000 10 1.58 2000 6.7 42 15 6.0000325 25.5 31 10000 50 1.58 2000 6.7 42 15 6.0000326 25.5 31 10000 100 1.58 2000 6.7 42 15 6.0000327 25.5 31 10000 200 1.58 2000 6.7 42 15 6.0000328 25.5 31 10000 500 1.58 2000 6.7 42 15 6.0000329 25.5 31 10000 1000 1.58 2000 6.7 42 15 5.8000330 25.5 31 10000 1500 1.58 2000 6.7 42 15 5.8000331 25.5 31 10000 2000 1.58 2000 6.7 42 15 5.8000332 25.5 31 10000 3000 1.58 2000 6.7 42 15 5.8000333 25.5 31 10000 4000 1.58 2000 6.7 42 15 5.8000334 25.5 31 10000 5000 1.58 2000 6.7 42 15 5.8000335 25.5 31 10000 8000 1.58 2000 6.7 42 15 4.5000336 25.5 31 10000 10000 1.58 2000 6.7 42 15 4.5000337 25.5 35 10000 10 1.58 2000 6.7 42 15 10.0000338 25.5 35 10000 50 1.58 2000 6.7 42 15 10.0000339 25.5 35 10000 100 1.58 2000 6.7 42 15 10.0000340 25.5 35 10000 200 1.58 2000 6.7 42 15 10.0000341 25.5 35 10000 500 1.58 2000 6.7 42 15 10.0000342 25.5 35 10000 1000 1.58 2000 6.7 42 15 9.5000343 25.5 35 10000 1500 1.58 2000 6.7 42 15 9.5000344 25.5 35 10000 2000 1.58 2000 6.7 42 15 9.5000345 25.5 35 10000 3000 1.58 2000 6.7 42 15 8.2000346 25.5 35 10000 4000 1.58 2000 6.7 42 15 8.2000347 25.5 35 10000 5000 1.58 2000 6.7 42 15 8.2000348 25.5 35 10000 8000 1.58 2000 6.7 42 15 7.4000349 25.5 35 10000 10000 1.58 2000 6.7 42 15 7.4000350 25.5 40 10000 10 1.58 2000 6.7 42 15 4.5000351 25.5 40 10000 50 1.58 2000 6.7 42 15 4.5000352 25.5 40 10000 100 1.58 2000 6.7 42 15 4.5000353 25.5 40 10000 200 1.58 2000 6.7 42 15 4.5000354 25.5 40 10000 500 1.58 2000 6.7 42 15 4.5000355 25.5 40 10000 1000 1.58 2000 6.7 42 15 4.2000356 25.5 40 10000 1500 1.58 2000 6.7 42 15 4.2000357 25.5 40 10000 2000 1.58 2000 6.7 42 15 4.2000358 25.5 40 10000 3000 1.58 2000 6.7 42 15 4.2000359 25.5 40 10000 4000 1.58 2000 6.7 42 15 3.9000360 25.5 40 10000 5000 1.58 2000 6.7 42 15 3.9000361 25.5 40 10000 8000 1.58 2000 6.7 42 15 3.5000362 25.5 40 10000 10000 1.58 2000 6.7 42 15 3.5000


105

Appendix C-5: Calculation matrix used for simulation (changes in bulk modulus, slope angle 35.5° and frequency 3 Hz).

Calculation number

Slope angle (º)

Slope height (m)

Shear Modulus (MPa)

Bulk Modulus (MPa)


Density (kg/m3)

Cohesion (MPa)

Friction angle (º)

Frequency (Hz)

Amplifi-cation

363 35.5 10 10000 10 1.58 2000 6.7 42 3 1.8000364 35.5 10 10000 50 1.58 2000 6.7 42 3 1.8000365 35.5 10 10000 100 1.58 2000 6.7 42 3 1.8000366 35.5 10 10000 200 1.58 2000 6.7 42 3 1.8000367 35.5 10 10000 500 1.58 2000 6.7 42 3 1.8000368 35.5 10 10000 1000 1.58 2000 6.7 42 3 1.8000369 35.5 10 10000 1500 1.58 2000 6.7 42 3 1.7000370 35.5 10 10000 2000 1.58 2000 6.7 42 3 1.7000371 35.5 10 10000 3000 1.58 2000 6.7 42 3 1.7000372 35.5 10 10000 4000 1.58 2000 6.7 42 3 1.7000373 35.5 10 10000 5000 1.58 2000 6.7 42 3 1.6000374 35.5 10 10000 10000 1.58 2000 6.7 42 3 1.6000375 35.5 20 10000 10 1.58 2000 6.7 42 3 2.0000376 35.5 20 10000 50 1.58 2000 6.7 42 3 2.0000377 35.5 20 10000 100 1.58 2000 6.7 42 3 2.0000378 35.5 20 10000 200 1.58 2000 6.7 42 3 2.0000379 35.5 20 10000 500 1.58 2000 6.7 42 3 2.0000380 35.5 20 10000 1000 1.58 2000 6.7 42 3 1.9000381 35.5 20 10000 1500 1.58 2000 6.7 42 3 1.9000382 35.5 20 10000 2000 1.58 2000 6.7 42 3 1.9000383 35.5 20 10000 3000 1.58 2000 6.7 42 3 1.9000384 35.5 20 10000 4000 1.58 2000 6.7 42 3 1.9000385 35.5 20 10000 5000 1.58 2000 6.7 42 3 1.9000386 35.5 20 10000 8000 1.58 2000 6.7 42 3 1.8000387 35.5 20 10000 10000 1.58 2000 6.7 42 3 1.8000388 35.5 25 10000 10 1.58 2000 6.7 42 3 2.5000389 35.5 25 10000 50 1.58 2000 6.7 42 3 2.5000390 35.5 25 10000 100 1.58 2000 6.7 42 3 2.5000391 35.5 25 10000 200 1.58 2000 6.7 42 3 2.5000392 35.5 25 10000 500 1.58 2000 6.7 42 3 2.5000393 35.5 25 10000 1000 1.58 2000 6.7 42 3 2.5000394 35.5 25 10000 1500 1.58 2000 6.7 42 3 2.2000395 35.5 25 10000 2000 1.58 2000 6.7 42 3 2.2000396 35.5 25 10000 3000 1.58 2000 6.7 42 3 2.2000397 35.5 25 10000 4000 1.58 2000 6.7 42 3 2.0000398 35.5 25 10000 5000 1.58 2000 6.7 42 3 2.0000399 35.5 25 10000 8000 1.58 2000 6.7 42 3 1.7000400 35.5 25 10000 10000 1.58 2000 6.7 42 3 1.7000


106

Appendix C-5 (contd.): Calculation matrix used for simulation (changes in bulk modulus, slope

angle 35.5° frequency 3 Hz). Calculation number

Slope angle (º)

Slope height (m)

Shear Modulus (MPa)

Bulk Modulus (MPa)


Density (kg/m3)

Cohesion (MPa)

Friction angle (º)

Fre-quency (Hz)

Ampli-fication

401 35.5 30 10000 10 1.58 2000 6.7 42 3 2.0000402 35.5 30 10000 50 1.58 2000 6.7 42 3 2.0000403 35.5 30 10000 100 1.58 2000 6.7 42 3 2.0000404 35.5 30 10000 200 1.58 2000 6.7 42 3 2.0000405 35.5 30 10000 500 1.58 2000 6.7 42 3 2.0000406 35.5 30 10000 1000 1.58 2000 6.7 42 3 1.8000407 35.5 30 10000 1500 1.58 2000 6.7 42 3 1.8000408 35.5 30 10000 2000 1.58 2000 6.7 42 3 1.8000409 35.5 30 10000 3000 1.58 2000 6.7 42 3 1.7000410 35.5 30 10000 4000 1.58 2000 6.7 42 3 1.7000411 35.5 30 10000 5000 1.58 2000 6.7 42 3 1.5000412 35.5 30 10000 8000 1.58 2000 6.7 42 3 1.5000413 35.5 30 10000 10000 1.58 2000 6.7 42 3 1.5000414 35.5 35 10000 10 1.58 2000 6.7 42 3 2.2000415 35.5 35 10000 50 1.58 2000 6.7 42 3 2.2000416 35.5 35 10000 100 1.58 2000 6.7 42 3 2.2000417 35.5 35 10000 200 1.58 2000 6.7 42 3 2.2000418 35.5 35 10000 500 1.58 2000 6.7 42 3 2.2000419 35.5 35 10000 1000 1.58 2000 6.7 42 3 2.2000420 35.5 35 10000 1500 1.58 2000 6.7 42 3 2.2000421 35.5 35 10000 2000 1.58 2000 6.7 42 3 2.0000422 35.5 35 10000 3000 1.58 2000 6.7 42 3 2.0000423 35.5 35 10000 4000 1.58 2000 6.7 42 3 2.0000424 35.5 35 10000 5000 1.58 2000 6.7 42 3 1.8000425 35.5 35 10000 8000 1.58 2000 6.7 42 3 1.8000426 35.5 35 10000 10000 1.58 2000 6.7 42 3 1.8000427 35.5 40 10000 10 1.58 2000 6.7 42 3 2.5000428 35.5 40 10000 50 1.58 2000 6.7 42 3 2.5000429 35.5 40 10000 100 1.58 2000 6.7 42 3 2.5000430 35.5 40 10000 200 1.58 2000 6.7 42 3 2.5000431 35.5 40 10000 500 1.58 2000 6.7 42 3 2.5000432 35.5 40 10000 1000 1.58 2000 6.7 42 3 2.5000433 35.5 40 10000 1500 1.58 2000 6.7 42 3 2.5000434 35.5 40 10000 2000 1.58 2000 6.7 42 3 2.5000435 35.5 40 10000 3000 1.58 2000 6.7 42 3 2.5000436 35.5 40 10000 4000 1.58 2000 6.7 42 3 2.3000437 35.5 40 10000 5000 1.58 2000 6.7 42 3 2.3000438 35.5 40 10000 8000 1.58 2000 6.7 42 3 2.3000439 35.5 40 10000 10000 1.58 2000 6.7 42 3 2.3000440 35.5 45 10000 10 1.58 2000 6.7 42 3 3.6000441 35.5 50 10000 10 1.58 2000 6.7 42 3 3.1000442 35.5 60 10000 10 1.58 2000 6.7 42 3 2.4000


107


Calculation number

Slope angle (º)

Slope height (m)

Shear Modulus (MPa)

Bulk Modulus (MPa)


Density (kg/m3)

Cohesion (MPa)

Friction angle (º)

Frequency (Hz)

Amplifi-cation

443 35.5 10 10000 10 1.58 2000 6.7 42 5 2.4000444 35.5 10 10000 50 1.58 2000 6.7 42 5 2.4000445 35.5 10 10000 100 1.58 2000 6.7 42 5 2.4000446 35.5 10 10000 200 1.58 2000 6.7 42 5 2.4000447 35.5 10 10000 500 1.58 2000 6.7 42 5 2.4000448 35.5 10 10000 1000 1.58 2000 6.7 42 5 2.4000449 35.5 10 10000 1500 1.58 2000 6.7 42 5 2.1000450 35.5 10 10000 2000 1.58 2000 6.7 42 5 2.1000451 35.5 10 10000 3000 1.58 2000 6.7 42 5 2.1000452 35.5 10 10000 4000 1.58 2000 6.7 42 5 1.8000453 35.5 10 10000 5000 1.58 2000 6.7 42 5 1.8000454 35.5 10 10000 10000 1.58 2000 6.7 42 5 1.8000455 35.5 20 10000 10 1.58 2000 6.7 42 5 2.5000456 35.5 20 10000 50 1.58 2000 6.7 42 5 2.5000457 35.5 20 10000 100 1.58 2000 6.7 42 5 2.5000458 35.5 20 10000 200 1.58 2000 6.7 42 5 2.5000459 35.5 20 10000 500 1.58 2000 6.7 42 5 2.5000460 35.5 20 10000 1000 1.58 2000 6.7 42 5 2.1000461 35.5 20 10000 1500 1.58 2000 6.7 42 5 2.1000462 35.5 20 10000 2000 1.58 2000 6.7 42 5 2.1000463 35.5 20 10000 3000 1.58 2000 6.7 42 5 1.9000464 35.5 20 10000 4000 1.58 2000 6.7 42 5 1.9000465 35.5 20 10000 5000 1.58 2000 6.7 42 5 1.9000466 35.5 20 10000 8000 1.58 2000 6.7 42 5 1.7000467 35.5 20 10000 10000 1.58 2000 6.7 42 5 1.7000468 35.5 25 10000 10 1.58 2000 6.7 42 5 2.2000469 35.5 25 10000 50 1.58 2000 6.7 42 5 2.2000470 35.5 25 10000 100 1.58 2000 6.7 42 5 2.2000471 35.5 25 10000 200 1.58 2000 6.7 42 5 2.2000472 35.5 25 10000 500 1.58 2000 6.7 42 5 2.2000473 35.5 25 10000 1000 1.58 2000 6.7 42 5 2.2000474 35.5 25 10000 1500 1.58 2000 6.7 42 5 2.3000475 35.5 25 10000 2000 1.58 2000 6.7 42 5 2.3000476 35.5 25 10000 3000 1.58 2000 6.7 42 5 2.1000477 35.5 25 10000 4000 1.58 2000 6.7 42 5 2.1000478 35.5 25 10000 5000 1.58 2000 6.7 42 5 2.1000479 35.5 25 10000 8000 1.58 2000 6.7 42 5 2.0000480 35.5 25 10000 10000 1.58 2000 6.7 42 5 2.0000


108

Appendix C-6: Calculation matrix used for simulation (changes in bulk modulus, slope angle 35.5° frequency 5 Hz) (contd.)

Calculation number

Slope angle (º)

Slope height (m)

Shear Modulus (MPa)

Bulk Modulus (MPa)


Density (kg/m3)

Cohesion (MPa)

Friction angle (º)

Frequency (Hz)

Amplifi-cation

481 35.5 30 10000 10 1.58 2000 6.7 42 5 3.2000482 35.5 30 10000 50 1.58 2000 6.7 42 5 3.2000483 35.5 30 10000 100 1.58 2000 6.7 42 5 3.2000484 35.5 30 10000 200 1.58 2000 6.7 42 5 3.2000485 35.5 30 10000 500 1.58 2000 6.7 42 5 3.2000486 35.5 30 10000 1000 1.58 2000 6.7 42 5 3.0000487 35.5 30 10000 1500 1.58 2000 6.7 42 5 3.0000488 35.5 30 10000 2000 1.58 2000 6.7 42 5 3.0000489 35.5 30 10000 3000 1.58 2000 6.7 42 5 3.0000490 35.5 30 10000 4000 1.58 2000 6.7 42 5 2.8000491 35.5 30 10000 5000 1.58 2000 6.7 42 5 2.8000492 35.5 30 10000 8000 1.58 2000 6.7 42 5 2.4000493 35.5 30 10000 10000 1.58 2000 6.7 42 5 2.4000494 35.5 35 10000 10 1.58 2000 6.7 42 5 2.2000495 35.5 35 10000 50 1.58 2000 6.7 42 5 2.2000496 35.5 35 10000 100 1.58 2000 6.7 42 5 2.2000497 35.5 35 10000 200 1.58 2000 6.7 42 5 2.2000498 35.5 35 10000 500 1.58 2000 6.7 42 5 2.2000499 35.5 35 10000 1000 1.58 2000 6.7 42 5 2.2000500 35.5 35 10000 1500 1.58 2000 6.7 42 5 2.1000501 35.5 35 10000 2000 1.58 2000 6.7 42 5 2.1000502 35.5 35 10000 3000 1.58 2000 6.7 42 5 2.1000503 35.5 35 10000 4000 1.58 2000 6.7 42 5 2.1000504 35.5 35 10000 5000 1.58 2000 6.7 42 5 2.1000505 35.5 35 10000 8000 1.58 2000 6.7 42 5 2.0000506 35.5 35 10000 10000 1.58 2000 6.7 42 5 2.0000507 35.5 40 10000 10 1.58 2000 6.7 42 5 2.5000508 35.5 40 10000 50 1.58 2000 6.7 42 5 2.5000509 35.5 40 10000 100 1.58 2000 6.7 42 5 2.5000510 35.5 40 10000 200 1.58 2000 6.7 42 5 2.5000511 35.5 40 10000 500 1.58 2000 6.7 42 5 2.5000512 35.5 40 10000 1000 1.58 2000 6.7 42 5 2.5000513 35.5 40 10000 1500 1.58 2000 6.7 42 5 2.3000514 35.5 40 10000 2000 1.58 2000 6.7 42 5 2.3000515 35.5 40 10000 3000 1.58 2000 6.7 42 5 2.3000516 35.5 40 10000 4000 1.58 2000 6.7 42 5 2.3000517 35.5 40 10000 5000 1.58 2000 6.7 42 5 2.3000518 35.5 40 10000 8000 1.58 2000 6.7 42 5 2.0000519 35.5 40 10000 10000 1.58 2000 6.7 42 5 2.0000


109


Calculation number

Slope angle (º)

Slope height (m)

Shear Modulus (MPa)

Bulk Modulus (MPa)


Density (kg/m3)

Cohesion (MPa)

Friction angle (º)

Frequency (Hz)

Amplifi-cation

520 35.5 10 10000 10 1.58 2000 6.7 42 10 2.1000521 35.5 10 10000 50 1.58 2000 6.7 42 10 2.1000522 35.5 10 10000 100 1.58 2000 6.7 42 10 2.1000523 35.5 10 10000 200 1.58 2000 6.7 42 10 2.1000524 35.5 10 10000 500 1.58 2000 6.7 42 10 2.1000525 35.5 10 10000 1000 1.58 2000 6.7 42 10 2.1000526 35.5 10 10000 1500 1.58 2000 6.7 42 10 2.1000527 35.5 10 10000 2000 1.58 2000 6.7 42 10 2.1000528 35.5 10 10000 3000 1.58 2000 6.7 42 10 2.1000529 35.5 10 10000 4000 1.58 2000 6.7 42 10 2.1000530 35.5 10 10000 5000 1.58 2000 6.7 42 10 2.1000531 35.5 10 10000 10000 1.58 2000 6.7 42 10 2.1000532 35.5 20 10000 10 1.58 2000 6.7 42 10 5.2000533 35.5 20 10000 50 1.58 2000 6.7 42 10 5.2000534 35.5 20 10000 100 1.58 2000 6.7 42 10 5.2000535 35.5 20 10000 200 1.58 2000 6.7 42 10 5.2000536 35.5 20 10000 500 1.58 2000 6.7 42 10 5.2000537 35.5 20 10000 1000 1.58 2000 6.7 42 10 5.2000538 35.5 20 10000 1500 1.58 2000 6.7 42 10 4.8000539 35.5 20 10000 2000 1.58 2000 6.7 42 10 4.8000540 35.5 20 10000 3000 1.58 2000 6.7 42 10 4.8000541 35.5 20 10000 4000 1.58 2000 6.7 42 10 4.8000542 35.5 20 10000 5000 1.58 2000 6.7 42 10 4.8000543 35.5 20 10000 8000 1.58 2000 6.7 42 10 4.5000544 35.5 20 10000 10000 1.58 2000 6.7 42 10 4.5000545 35.5 25 10000 10 1.58 2000 6.7 42 10 7.5000546 35.5 25 10000 50 1.58 2000 6.7 42 10 7.5000547 35.5 25 10000 100 1.58 2000 6.7 42 10 7.5000548 35.5 25 10000 200 1.58 2000 6.7 42 10 7.5000549 35.5 25 10000 500 1.58 2000 6.7 42 10 7.5000550 35.5 25 10000 1000 1.58 2000 6.7 42 10 7.5000551 35.5 25 10000 1500 1.58 2000 6.7 42 10 7.0000552 35.5 25 10000 2000 1.58 2000 6.7 42 10 7.0000553 35.5 25 10000 3000 1.58 2000 6.7 42 10 7.0000554 35.5 25 10000 4000 1.58 2000 6.7 42 10 6.8000555 35.5 25 10000 5000 1.58 2000 6.7 42 10 6.8000556 35.5 25 10000 8000 1.58 2000 6.7 42 10 6.5000557 35.5 25 10000 10000 1.58 2000 6.7 42 10 6.5000


110

Appendix C-7: Calculation matrix used for simulation (changes in bulk modulus, slope angle

35.5° frequency 10 Hz) (contd.)

Calculation number

Slope angle (º)

Slope height (m)

Shear Modulus (MPa)

Bulk Modulus (MPa)


Density (kg/m3)

Cohesion (MPa)

Friction angle (º)

Frequency (Hz)

Amplifi-cation

558 35.5 30 10000 10 1.58 2000 6.7 42 10 12.0000559 35.5 30 10000 50 1.58 2000 6.7 42 10 12.0000560 35.5 30 10000 100 1.58 2000 6.7 42 10 12.0000561 35.5 30 10000 200 1.58 2000 6.7 42 10 12.0000562 35.5 30 10000 500 1.58 2000 6.7 42 10 12.0000563 35.5 30 10000 1000 1.58 2000 6.7 42 10 11.5000564 35.5 30 10000 1500 1.58 2000 6.7 42 10 11.5000565 35.5 30 10000 2000 1.58 2000 6.7 42 10 11.5000566 35.5 30 10000 3000 1.58 2000 6.7 42 10 10.5000567 35.5 30 10000 4000 1.58 2000 6.7 42 10 10.5000568 35.5 30 10000 5000 1.58 2000 6.7 42 10 10.0000569 35.5 30 10000 8000 1.58 2000 6.7 42 10 10.0000570 35.5 30 10000 10000 1.58 2000 6.7 42 10 10.0000571 35.5 35 10000 10 1.58 2000 6.7 42 10 7.8000572 35.5 35 10000 50 1.58 2000 6.7 42 10 7.8000573 35.5 35 10000 100 1.58 2000 6.7 42 10 7.8000574 35.5 35 10000 200 1.58 2000 6.7 42 10 7.8000575 35.5 35 10000 500 1.58 2000 6.7 42 10 7.8000576 35.5 35 10000 1000 1.58 2000 6.7 42 10 7.2000577 35.5 35 10000 1500 1.58 2000 6.7 42 10 7.2000578 35.5 35 10000 2000 1.58 2000 6.7 42 10 7.2000579 35.5 35 10000 3000 1.58 2000 6.7 42 10 7.2000580 35.5 35 10000 4000 1.58 2000 6.7 42 10 7.2000581 35.5 35 10000 5000 1.58 2000 6.7 42 10 7.0000582 35.5 35 10000 8000 1.58 2000 6.7 42 10 6.5000583 35.5 35 10000 10000 1.58 2000 6.7 42 10 6.5000584 35.5 40 10000 10 1.58 2000 6.7 42 10 14.0000585 35.5 40 10000 50 1.58 2000 6.7 42 10 14.0000586 35.5 40 10000 100 1.58 2000 6.7 42 10 14.0000587 35.5 40 10000 200 1.58 2000 6.7 42 10 14.0000588 35.5 40 10000 500 1.58 2000 6.7 42 10 14.0000589 35.5 40 10000 1000 1.58 2000 6.7 42 10 13.0000590 35.5 40 10000 1500 1.58 2000 6.7 42 10 13.0000591 35.5 40 10000 2000 1.58 2000 6.7 42 10 13.0000592 35.5 40 10000 3000 1.58 2000 6.7 42 10 11.0000593 35.5 40 10000 4000 1.58 2000 6.7 42 10 11.0000594 35.5 40 10000 5000 1.58 2000 6.7 42 10 10.0000595 35.5 40 10000 8000 1.58 2000 6.7 42 10 10.0000596 35.5 40 10000 10000 1.58 2000 6.7 42 10 10.0000


111


Calculation number

Slope angle (º)

Slope height (m)

Shear Modulus (MPa)

Bulk Modulus (MPa)


Density (kg/m3)

Cohesion (MPa)

Friction angle (º)

Frequency (Hz)

Amplifi-cation

597 35.5 10 10000 10 1.58 2000 6.7 42 15 7.5000598 35.5 10 10000 50 1.58 2000 6.7 42 15 7.5000599 35.5 10 10000 100 1.58 2000 6.7 42 15 7.5000600 35.5 10 10000 200 1.58 2000 6.7 42 15 7.5000601 35.5 10 10000 500 1.58 2000 6.7 42 15 7.5000602 35.5 10 10000 1000 1.58 2000 6.7 42 15 7.5000603 35.5 10 10000 1500 1.58 2000 6.7 42 15 7.2000604 35.5 10 10000 2000 1.58 2000 6.7 42 15 7.2000605 35.5 10 10000 3000 1.58 2000 6.7 42 15 7.0000606 35.5 10 10000 4000 1.58 2000 6.7 42 15 7.0000607 35.5 10 10000 5000 1.58 2000 6.7 42 15 6.5000608 35.5 10 10000 10000 1.58 2000 6.7 42 15 6.5000609 35.5 20 10000 10 1.58 2000 6.7 42 15 17.0000610 35.5 20 10000 50 1.58 2000 6.7 42 15 17.0000611 35.5 20 10000 100 1.58 2000 6.7 42 15 17.0000612 35.5 20 10000 200 1.58 2000 6.7 42 15 17.0000613 35.5 20 10000 500 1.58 2000 6.7 42 15 17.0000614 35.5 20 10000 1000 1.58 2000 6.7 42 15 17.0000615 35.5 20 10000 1500 1.58 2000 6.7 42 15 17.0000616 35.5 20 10000 2000 1.58 2000 6.7 42 15 15.0000617 35.5 20 10000 3000 1.58 2000 6.7 42 15 15.0000618 35.5 20 10000 4000 1.58 2000 6.7 42 15 15.0000619 35.5 20 10000 5000 1.58 2000 6.7 42 15 14.0000620 35.5 20 10000 8000 1.58 2000 6.7 42 15 14.0000621 35.5 20 10000 10000 1.58 2000 6.7 42 15 14.0000622 35.5 25 10000 10 1.58 2000 6.7 42 15 9.5000623 35.5 25 10000 50 1.58 2000 6.7 42 15 9.5000624 35.5 25 10000 100 1.58 2000 6.7 42 15 9.5000625 35.5 25 10000 200 1.58 2000 6.7 42 15 9.5000626 35.5 25 10000 500 1.58 2000 6.7 42 15 9.5000627 35.5 25 10000 1000 1.58 2000 6.7 42 15 8.5000628 35.5 25 10000 1500 1.58 2000 6.7 42 15 8.5000629 35.5 25 10000 2000 1.58 2000 6.7 42 15 8.5000630 35.5 25 10000 3000 1.58 2000 6.7 42 15 8.5000631 35.5 25 10000 4000 1.58 2000 6.7 42 15 7.5000632 35.5 25 10000 5000 1.58 2000 6.7 42 15 7.5000633 35.5 25 10000 8000 1.58 2000 6.7 42 15 7.5000634 35.5 25 10000 10000 1.58 2000 6.7 42 15 7.5000


112

Appendix C-8: Calculation matrix used for simulation (changes in bulk modulus, slope angle 35.5° frequency 15 Hz) (contd.) Calculation number

Slope angle (º)

Slope height (m)

Shear Modulus (MPa)

Bulk Modulus (MPa)


Density (kg/m3)

Cohesion (MPa)

Friction angle (º)

Frequency (Hz)

Ampli-fication

635 35.5 30 10000 10 1.58 2000 6.7 42 15 6.0000636 35.5 30 10000 50 1.58 2000 6.7 42 15 6.0000637 35.5 30 10000 100 1.58 2000 6.7 42 15 6.0000638 35.5 30 10000 200 1.58 2000 6.7 42 15 6.0000639 35.5 30 10000 500 1.58 2000 6.7 42 15 6.0000640 35.5 30 10000 1000 1.58 2000 6.7 42 15 6.0000641 35.5 30 10000 1500 1.58 2000 6.7 42 15 5.6000642 35.5 30 10000 2000 1.58 2000 6.7 42 15 5.6000643 35.5 30 10000 3000 1.58 2000 6.7 42 15 5.6000644 35.5 30 10000 4000 1.58 2000 6.7 42 15 5.6000645 35.5 30 10000 5000 1.58 2000 6.7 42 15 5.0000646 35.5 30 10000 8000 1.58 2000 6.7 42 15 5.0000647 35.5 30 10000 10000 1.58 2000 6.7 42 15 5.0000648 35.5 35 10000 10 1.58 2000 6.7 42 15 9.8000649 35.5 35 10000 50 1.58 2000 6.7 42 15 9.8000650 35.5 35 10000 100 1.58 2000 6.7 42 15 9.8000651 35.5 35 10000 200 1.58 2000 6.7 42 15 9.8000652 35.5 35 10000 500 1.58 2000 6.7 42 15 9.8000653 35.5 35 10000 1000 1.58 2000 6.7 42 15 9.0000654 35.5 35 10000 1500 1.58 2000 6.7 42 15 9.0000655 35.5 35 10000 2000 1.58 2000 6.7 42 15 9.0000656 35.5 35 10000 3000 1.58 2000 6.7 42 15 7.8000657 35.5 35 10000 4000 1.58 2000 6.7 42 15 7.8000658 35.5 35 10000 5000 1.58 2000 6.7 42 15 7.5000659 35.5 35 10000 8000 1.58 2000 6.7 42 15 7.0000660 35.5 35 10000 10000 1.58 2000 6.7 42 15 7.0000661 35.5 40 10000 10 1.58 2000 6.7 42 15 4.2000662 35.5 40 10000 50 1.58 2000 6.7 42 15 4.2000663 35.5 40 10000 100 1.58 2000 6.7 42 15 4.2000664 35.5 40 10000 200 1.58 2000 6.7 42 15 4.2000665 35.5 40 10000 500 1.58 2000 6.7 42 15 4.2000666 35.5 40 10000 1000 1.58 2000 6.7 42 15 4.2000667 35.5 40 10000 1500 1.58 2000 6.7 42 15 4.0000668 35.5 40 10000 2000 1.58 2000 6.7 42 15 4.0000669 35.5 40 10000 3000 1.58 2000 6.7 42 15 4.0000670 35.5 40 10000 4000 1.58 2000 6.7 42 15 3.8000671 35.5 40 10000 5000 1.58 2000 6.7 42 15 3.8000672 35.5 40 10000 8000 1.58 2000 6.7 42 15 3.5000673 35.5 40 10000 10000 1.58 2000 6.7 42 15 3.5000


113

Appendix C-9: Calculation matrix used for simulation (changes in shear modulus)

Calculation number

slope angle (º)

Slope height (m)

Shear modulus(MPa)

Bulk modulus(MPa)

Tensile strength(MPa)

Mass Density(kg/m3)

Cohesion (MPa)

Friction angle (º)

Frequency (Hz)

Amplifi-cation

674 25.5 10 5 1000 1.58 2000 6.7 42 5 0.005675 25.5 10 10 1000 1.58 2000 6.7 42 5 2.1676 25.5 10 50 1000 1.58 2000 6.7 42 5 2.5677 25.5 10 100 1000 1.58 2000 6.7 42 5 2.5678 25.5 10 200 1000 1.58 2000 6.7 42 5 2.2679 25.5 10 300 1000 1.58 2000 6.7 42 5 2.4680 25.5 10 400 1000 1.58 2000 6.7 42 5 4681 25.5 10 1000 1000 1.58 2000 6.7 42 5 6682 25.5 10 3000 1000 1.58 2000 6.7 42 5 2.4683 25.5 10 6000 1000 1.58 2000 6.7 42 5 2.3684 25.5 10 50000 1000 1.58 2000 6.7 42 5 1.7685 25.5 10 100000 1000 1.58 2000 6.7 42 5 1.7686 25.5 20 0.1 1000 1.58 2000 6.7 42 5 0.17687 25.5 20 0.5 1000 1.58 2000 6.7 42 5 0.0025688 25.5 20 1 1000 1.58 2000 6.7 42 5 0.002689 25.5 20 5 1000 1.58 2000 6.7 42 5 0.0032690 25.5 20 10 1000 1.58 2000 6.7 42 5 0.02691 25.5 20 50 1000 1.58 2000 6.7 42 5 2.6692 25.5 20 100 1000 1.58 2000 6.7 42 5 2.5693 25.5 20 200 1000 1.58 2000 6.7 42 5 2.4694 25.5 20 300 1000 1.58 2000 6.7 42 5 2.4695 25.5 20 400 1000 1.58 2000 6.7 42 5 2.2696 25.5 20 1000 1000 1.58 2000 6.7 42 5 5.5697 25.5 20 3000 1000 1.58 2000 6.7 42 5 4.1698 25.5 20 6000 1000 1.58 2000 6.7 42 5 3.2699 25.5 20 50000 1000 1.58 2000 6.7 42 5 2.3700 25.5 20 75000 1000 1.58 2000 6.7 42 5 2.3701 25.5 20 100000 1000 1.58 2000 6.7 42 5 2.1


114

Appendix C-9: Calculation matrix used for simulation (changes in shear modulus) (contd.) Calculation number

slope angle (º)

Slope height (m)

Shear modulus(MPa)

Bulk modulus(MPa)

Tensile strength(MPa)

Mass Density(kg/m3)

Cohesion (MPa)

Friction angle (º)

Frequency (Hz)

Ampli-fication

702 25.5 30 5 1000 1.58 2000 6.7 42 5 0.0018703 25.5 30 10 1000 1.58 2000 6.7 42 5 0.018704 25.5 30 50 1000 1.58 2000 6.7 42 5 2.5705 25.5 30 100 1000 1.58 2000 6.7 42 5 2.5706 25.5 30 200 1000 1.58 2000 6.7 42 5 2.5707 25.5 30 300 1000 1.58 2000 6.7 42 5 3708 25.5 30 400 1000 1.58 2000 6.7 42 5 2.3709 25.5 30 1000 1000 1.58 2000 6.7 42 5 3.2710 25.5 30 3000 1000 1.58 2000 6.7 42 5 6711 25.5 30 6000 1000 1.58 2000 6.7 42 5 2.2712 25.5 30 50000 1000 1.58 2000 6.7 42 5 2713 25.5 30 75000 1000 1.58 2000 6.7 42 5 2714 25.5 30 100000 1000 1.58 2000 6.7 42 5 2715 25.5 40 0.1 1000 1.58 2000 6.7 42 5 2.5716 25.5 40 0.5 1000 1.58 2000 6.7 42 5 0.08717 25.5 40 1 1000 1.58 2000 6.7 42 5 0.005718 25.5 40 5 1000 1.58 2000 6.7 42 5 0.002719 25.5 40 10 1000 1.58 2000 6.7 42 5 0.002720 25.5 40 50 1000 1.58 2000 6.7 42 5 2.6721 25.5 40 100 1000 1.58 2000 6.7 42 5 2.6722 25.5 40 200 1000 1.58 2000 6.7 42 5 3.1723 25.5 40 300 1000 1.58 2000 6.7 42 5 3.1724 25.5 40 400 1000 1.58 2000 6.7 42 5 3725 25.5 40 1000 1000 1.58 2000 6.7 42 5 2.3726 25.5 40 3000 1000 1.58 2000 6.7 42 5 6.5727 25.5 40 6000 1000 1.58 2000 6.7 42 5 5.5728 25.5 40 50000 1000 1.58 2000 6.7 42 5 2.2729 25.5 40 100000 1000 1.58 2000 6.7 42 5 2


115

Appendix C-10: Calculation matrix used for simulation (changes in wavelength).

Calc. No Slope

angle (º)

Slope-height (m)

Shear Modulus (MPa)

Bulk Modulus(MPa)


Density (kg/m3)

Cohesion (MPa)

Friction angle (º)

Frequency (Hz)

Wavelength(m)

Amplification

730 25.5 20 0.1 1000 1.58 2000 6.7 42 0.71 10.00 0.1370731 25.5 20 0.5 1000 1.58 2000 6.7 42 1.58 10.00 0.1890732 25.5 20 1 1000 1.58 2000 6.7 42 2.24 10.00 0.1470733 25.5 20 5 1000 1.58 2000 6.7 42 5.00 10.00 0.0032734 25.5 20 10 1000 1.58 2000 6.7 42 7.07 10.00 0.0800735 25.5 20 20 1000 1.58 2000 6.7 42 10.00 10.00 1.4000736 25.5 20 40 1000 1.58 2000 6.7 42 14.14 10.00 1.5000737 25.5 20 50 1000 1.58 2000 6.7 42 15.81 10.00 1.6000738 25.5 20 70 1000 1.58 2000 6.7 42 18.71 10.00 1.5000739 25.5 20 90 1000 1.58 2000 6.7 42 21.21 10.00 1.5000740 25.5 20 100 1000 1.58 2000 6.7 42 22.36 10.00 1.8000741 25.5 20 150 1000 1.58 2000 6.7 42 27.39 10.00 1.8000742 25.5 20 200 1000 1.58 2000 6.7 42 31.62 10.00 1.6000743 25.5 20 0.1 1000 1.58 2000 6.7 42 0.51 14.00 0.2500744 25.5 20 0.5 1000 1.58 2000 6.7 42 1.13 14.00 0.0026745 25.5 20 1 1000 1.58 2000 6.7 42 1.60 14.00 0.0021746 25.5 20 5 1000 1.58 2000 6.7 42 3.57 14.00 0.0033747 25.5 20 10 1000 1.58 2000 6.7 42 5.05 14.00 0.0200748 25.5 20 20 1000 1.58 2000 6.7 42 7.14 14.00 1.8000749 25.5 20 40 1000 1.58 2000 6.7 42 10.10 14.00 2.5000750 25.5 20 50 1000 1.58 2000 6.7 42 11.29 14.00 2.5000751 25.5 20 70 1000 1.58 2000 6.7 42 13.36 14.00 2.5000752 25.5 20 90 1000 1.58 2000 6.7 42 15.15 14.00 2.5000753 25.5 20 100 1000 1.58 2000 6.7 42 15.97 14.00 2.6000754 25.5 20 200 1000 1.58 2000 6.7 42 22.59 14.00 2.6000755 25.5 20 300 1000 1.58 2000 6.7 42 27.66 14.00 3.2000756 25.5 20 350 1000 1.58 2000 6.7 42 29.88 14.00 3.1000757 25.5 20 400 1000 1.58 2000 6.7 42 31.94 14.00 2.5000758 25.5 20 450 1000 1.58 2000 6.7 42 33.88 14.00 2.0000


116

Appendix C-10: Calculation matrix used for simulation (changes in wavelength) (contd.) Calc. No Slope

angle (º)

Slope-height (m)

Shear Modulus (MPa)

Bulk Modulus(MPa)


Density (kg/m3)

Cohesion (MPa)

Friction angle (º)

Frequency (Hz)

Wave-length

Amplifica-tion

759 25.5 20 0.1 1000 1.58 2000 6.7 42 0.16 44.72 0.2600760 25.5 20 0.5 1000 1.58 2000 6.7 42 0.35 44.72 0.0025761 25.5 20 1 1000 1.58 2000 6.7 42 0.50 44.72 0.0018762 25.5 20 5 1000 1.58 2000 6.7 42 1.12 44.72 0.0033763 25.5 20 10 1000 1.58 2000 6.7 42 1.58 44.72 0.0150764 25.5 20 15 1000 1.58 2000 6.7 42 1.94 44.72 0.9500765 25.5 20 20 1000 1.58 2000 6.7 42 2.24 44.72 2.4000766 25.5 20 40 1000 1.58 2000 6.7 42 3.16 44.72 2.5000767 25.5 20 50 1000 1.58 2000 6.7 42 3.54 44.72 2.5000768 25.5 20 70 1000 1.58 2000 6.7 42 4.18 44.72 2.5000769 25.5 20 90 1000 1.58 2000 6.7 42 4.74 44.72 2.5000770 25.5 20 100 1000 1.58 2000 6.7 42 5.00 44.72 2.5000771 25.5 20 200 1000 1.58 2000 6.7 42 7.07 44.72 2.8000772 25.5 20 300 1000 1.58 2000 6.7 42 8.66 44.72 2.8000773 25.5 20 600 1000 1.58 2000 6.7 42 12.25 44.72 2.5000774 25.5 20 800 1000 1.58 2000 6.7 42 14.14 44.72 2.5000775 25.5 20 1000 1000 1.58 2000 6.7 42 15.81 44.72 2.5000776 25.5 20 2000 1000 1.58 2000 6.7 42 22.36 44.72 2.4000777 25.5 20 3000 1000 1.58 2000 6.7 42 27.39 44.72 2.3000778 25.5 20 0.5 1000 1.58 2000 6.7 42 0.25 63.24 0.0024779 25.5 20 1 1000 1.58 2000 6.7 42 0.35 63.24 0.0018780 25.5 20 10 1000 1.58 2000 6.7 42 1.12 63.24 0.0100781 25.5 20 40 1000 1.58 2000 6.7 42 2.24 63.24 2.3000782 25.5 20 70 1000 1.58 2000 6.7 42 2.96 63.24 2.3000783 25.5 20 100 1000 1.58 2000 6.7 42 3.54 63.24 2.3000784 25.5 20 300 1000 1.58 2000 6.7 42 6.12 63.24 2.5000785 25.5 20 600 1000 1.58 2000 6.7 42 8.66 63.24 2.5000786 25.5 20 800 1000 1.58 2000 6.7 42 10.00 63.24 3.5000787 25.5 20 1000 1000 1.58 2000 6.7 42 11.18 63.24 3.5000788 25.5 20 2000 1000 1.58 2000 6.7 42 15.81 63.24 3.3000789 25.5 20 3000 1000 1.58 2000 6.7 42 19.37 63.24 3.7000790 25.5 20 4000 1000 1.58 2000 6.7 42 22.36 63.24 3.8000791 25.5 20 6000 1000 1.58 2000 6.7 42 27.39 63.24 3.9000


117

Appendix D-1: Bulk modulus versus maximum amplification for different slope heights, input frequencies and slope angles (full data can be seen on calculation numbers 1 to 673 in appendices C-1 to C-8).

(1) Calculation numbers 1-92, full data in appendix C-1

Bulk modulus vs. amplification for different slope heights (slope angle 25.5 degree, frequency 3 Hz)

1

1.5

2

2.5

3

1 10 100 1000 10000Bulk modulus (Mpa)

Max

imum

Am

plifi

catio

n

slope height 10 m slope height 20 m slope height 25 mslope height 30 m slope height 31 m slope height 35 mslope height 40 m


Bulk modulus Vs. Maximum amplification for different slope heights (slope angle 25.5 degree, frequency 5 Hz)

1.5

2

2.5

3

3.5

4

1 10 100 1000 10000Bulk modulus (Mpa)

Max

imum

Am

plifi

catio

n



118

Appendix D-1: Bulk modulus versus maximum amplification for different slope heights, input frequencies, and slope angles (contd.)


Bulk modulus vs. amplification for different slope heights (slope angle 25.5 degree, frequency 10 Hz)

02468

101214

1 10 100 1000 10000Bulk modulus (MPa)

Max

imum

Am

plifi

catio

n

slope height 10 m slope height 20 m slope height 25 m slope height 30 mslope height 31 m slope height 35 m slope height 40 m


Bulk modulus Vs. amplification for different slope heights (slope angle 25.5 degree, frequency 15 Hz)

02468

1012141618

1 10 100 1000 10000Bulk modulus (MPa)

Max

imum

Am

plifi

catio

n



119

Appendix D-2: Bulk modulus versus maximum amplification for different slope heights, input frequencies and slope angles (slope angle 35.5º)


Bulk modulus versus amplification for different slope heights with 3 Hz frequency (slope angle 35.5 degree)

0

0.5

1

1.5

2

2.5

3

1 10 100 1000 10000Bulk modulus (MPa)

Max

imum

Am

plifi

catio

n



Bulk modulus versus amplification for different slope heights with 5 Hz input frequency (slope angle 35.5

degree)

00.5

11.5

22.5

33.5

1 10 100 1000 10000Bulk modulus (Mpa)

Max

imum

Am

plifi

catio

n

slope height of 10 m slope height of 20 m slope height of 25 mslope height of 30 m slope height of 35 m slope height of 40 m


120

Appendix D-2: Bulk modulus versus maximum amplification for different slope heights, input frequencies and slope angles (slope angle 35.5º) (contd.)


Bulk modulus Versus amplification for different slope heights with 10 Hz frequency (slope angle 35.5 degree)

0

2

4

6

8

10

12

14

16

1 10 100 1000 10000Bulk modulus (MPa)

Max

imum

Am

plifi

catio

n



Bulk modulus Vs. amplification for different slope heights with 15 Hz input frequency (slope angle 35.5 degree)

02468

1012141618

1 10 100 1000 10000Bulk modulus (Mpa)

Max

imum

Am

plifi

catio

n



121

Appendix D-3: Comparison of maximum amplification for the slope angle of 25.5º and 35.5º from the plot of slope height versus maximum amplification for different input frequencies

(1) Full data from calculation numbers 1-92 in appendix C-1 for slope angle of 25.5º and 363-442 in appendix C-5 for slope angle of 35.5º

Slope height Vs. max amplification for different slope angles (input frequency of 3 Hz)

1.5

2

2.5

3

3.5

4

0 10 20 30 40 50 60 70Slope height (m)

Max

imum

am

plifi

catio

n

slope angle 25.5 degree slope angle 35.5 degree


Slope height Vs. Max. amplification for diffreent slope angles (input frequency 5 Hz)

1.5

2

2.5

3

3.5

4

0 10 20 30 40 50Slope height (m)

Max

imum

am

plifi

catio

n

Slope angle of 25.5 degree Slope angle of 35.5 degree


122

Appendix D-3: Comparison of maximum amplification for the slope angle of 25.5º and 35.5º from the plot of slope height versus maximum amplification for different input frequencies (contd.)


Slope height Vs. max. amplification for different slope angles (input frequency 10 Hz)

02468

10121416

0 10 20 30 40 50Slope height (m)

Max

imum

am

plifi

catio

n



Slope height Vs. max amplification for different slope angles (input frequency 15 Hz)

0

5

10

15

20

0 10 20 30 40 50Slope height (m)

Max

imum

am

plifi

catio

n


Documents

Empirical relations for earthquake response of€¦ · sincere thanks and appreciation to Prof. Dr. Bishal Nath Upreti, Tribhuvan University in Nepal for the great support and encouragement