two and three dimensional stability analyses for soil and rock … · two and three dimensional...

Two and Three Dimensional Stability Analyses

for Soil and Rock Slopes

by

An-Jui Li

A thesis is submitted for the degree of

Doctor of Philosophy

at

The University of Western Australia

School of Civil and Resources Engineering

August 2009

I

Declaration

“I hereby certify that the work embodied in this Thesis is the result of original research and has not been submitted for a higher degree to any other University or Institute”

……………………………..

An-Jui Li

August 2009

II

Abstract Slope stability assessments are classical problems for geotechnical engineers. The

predictions of slope stability in soil or rock masses play an important role when

designing for dams, roads, tunnels, excavations, open pit mines and other engineering

structures.

Stability charts continue to be used by engineers as preliminary design tools and by

educators for training purposes. However, the majority of the existing chart solutions

assume the slope problem is semi-infinite (plane-strain) in length. It is commonly

believed that this assumption is conservative for design, but non-conservative when a

back-analysis is performed. In order to obtain a more economical design or more precise

parameters from a back-analysis, it is therefore important to quantify three dimensional

boundary effects on slope stability. A significant aim of this research is to look more

closely at the effect of three dimensions when predicting slope stability.

In engineering practice, the limit equilibrium method (LEM) is the most popular

approach for estimating the slope stability. It is well known that the solution obtained

from the limit equilibrium method is not rigorous, because neither static nor kinematic

admissibility conditions are satisfied. In addition, assumptions are made regarding inter

slice forces for a two dimensional case and inter-column forces for a three dimensional

case in order to find a solution. Therefore, a number of more theoretically rigorous

numerical methods have been used in this research when studying 2D and 3D slope

problems.

In this thesis, the results of a comprehensive numerical study into the failure

mechanisms of soil and rock slopes are presented. Consideration is given to the wide

range of parameters that influence slope stability. The aim of this research is to better

understand slope failure mechanisms and to develop rigorous stability solutions that can

be used by design engineers. The study is unique in that two distinctly different

numerical methods have been used in tandem to determine the ultimate stability of

slopes, namely the upper and lower bound theorems of limit analysis and the

displacement finite element method. The limit equilibrium method is also employed for

comparison purposes. A comparison of the results from each technique provides an

opportunity to validate the findings and gives a rigorous evaluation of slope stability.

III

Acknowledgements First of all, I would like to express my gratitude to my supervisor, Dr. Richard Merifield

for all his support, time and invaluable guidance in this study. He has brought me to the

special Centre for Offshore Foundations Systems (COFS) and also consistently

provided feedback on my writing, which greatly improved my academic writing skills.

I am indebted to my co-supervisors, A/Prof. Andrei Lyamin (The University of

Newcastle) and Prof. Mark Cassidy for their assistance of numerical modelling and

review comments during this research.

I specially want to thank Prof. H. D. Lin and Prof. C. Y. Ou (National Taiwan

University of Science and Technology) and Dr. B. C. Benson Hsiung (National

Kaohsiung University of Applied Sciences). Their recommendation provided me with

the opportunity to study for PhD.

I also would like to thank all staff, visitors and postgraduate students in the School of

Civil and Resource Engineering and COFS for their friendship, especially to K. K. Lee,

Han Eng Low, Edmond Tang, Hugo Acosta-Martinez, Shinji Taenaka, Dr. Hongjie

Zhou, Ying Wang, Ming Wang, Vickie Kong and Qinyuan Jiang and Kar Lu Teh for

their encouragement and diverse discussions and help over the past 2 years.

Finally, my parents, aunts, uncle and sisters, thank you for your love and support

throughout the years. Last but not least, I would like to say ‘thank you’ to my wife,

Vivian, who has always been encouraging and supportive with love and great passion

throughout the period of my studies.

IV

Contents

DECLARATION ......................................................................................... I

ABSTRACT ................................................................................................ II

ACKNOWLEDGEMENTS ..................................................................... III

CONTENTS .............................................................................................. IV

CHAPTER 1 INTRODUCTION ........................................................... 1-1

1.1 INTRODUCTION ..................................................................................... 1-1

1.2 RESEARCH OBJECTIVES ......................................................................... 1-1

1.2.1 Three dimensional (3D) slope stability ......................................... 1-1

1.2.2 Application of limit analysis to slope stability .............................. 1-2

1.2.3 Stability charts for engineering ..................................................... 1-3

1.2.4 Rock slope stability using yield criteria for rock masses .............. 1-4

1.3 THESIS OUTLINE .................................................................................... 1-4

1.4 PUBLICATIONS ...................................................................................... 1-5

CHAPTER 2 LITERATURE REVIEW ............................................... 2-1

2.1 INTRODUCTION ..................................................................................... 2-1

2.2 CURRENT SLOPE STABILITY DESIGN APPROACHES ................................ 2-2

2.2.1 Limit equilibrium method (LEM) .................................................. 2-2

2.2.2 Limit analysis ................................................................................ 2-3

2.2.3 Numerical modelling ..................................................................... 2-3

2.2.4 Empirical design ........................................................................... 2-4

2.2.5 Physical model tests ...................................................................... 2-4

2.2.6 Probabilistic methods ................................................................... 2-5

2.2.7 Limitations..................................................................................... 2-6

2.3 FAILURE MODES AND FAILURE MECHANISMS FOR SLOPES .................... 2-7

2.3.1 Observed failure modes ................................................................ 2-7

2.3.2 Failure mechanisms inferred from field observations .................. 2-9

2.4 PREVIOUS SLOPE STABILITY INVESTIGATIONS IN SOILS ...................... 2-11

2.4.1 Physical model tests .................................................................... 2-11

V

2.4.2 Limit equilibrium analysis .......................................................... 2-12

2.4.3 Finite element analysis ............................................................... 2-14

2.4.4 Limit analysis .............................................................................. 2-15

2.4.5 Other investigations .................................................................... 2-16

2.5 PREVIOUS SLOPE STABILITY INVESTIGATIONS IN ROCK MASSES ........ 2-16

2.5.1 Physical model tests .................................................................... 2-17

2.5.2 Investigations based on the limit equilibrium method ................ 2-18

2.5.3 Investigations based on the numerical analysis ......................... 2-19

2.5.4 Investigations based on limit analysis theorems ........................ 2-20

2.5.5 Other investigations .................................................................... 2-20

2.6 PREVIOUS SLOPE STABILITY INVESTIGATIONS BASED ON THE PSEUDO

STATIC (PS) METHOD ..................................................................................... 2-21

2.7 EMPIRICAL FAILURE CRITERIA FOR ROCK MASSES ............................. 2-23

2.7.1 The generalised Hoek-Brown failure criterion .......................... 2-23

2.7.2 Mohr-Coulomb criterion ............................................................ 2-25

2.7.3 Douglas criterion ........................................................................ 2-26

2.7.4 Other empirical criteria for rock masses ................................... 2-27

2.8 SUMMARY........................................................................................... 2-30

CHAPTER 3 NUMERICAL FORMULATIONS ................................. 3-1

3.1 INTRODUCTION ..................................................................................... 3-1

3.2 NUMERICAL METHODS IN GEOMECHANICS .......................................... 3-1

3.3 THEORY OF LIMIT ANALYSIS ................................................................ 3-4

3.3.1 The assumption of perfect plasticity ............................................. 3-5

3.3.2 The stability postulate of Drucker ................................................ 3-6

3.3.3 Yield criterion ............................................................................... 3-7

3.3.4 Flow rule ....................................................................................... 3-8

3.3.5 Small deformations and equation of virtual work ........................ 3-9

3.3.6 The limit theorems ...................................................................... 3-10

3.4 LOWER BOUND FINITE ELEMENT LIMIT ANALYSIS FORMULATION ..... 3-11

3.4.1 Constraints from equilibrium conditions .................................... 3-12

3.4.2 Constraints from stress boundary conditions ............................. 3-15

VI

3.4.3 Constraints from yield conditions ............................................... 3-16

3.4.4 Formation of the objective function ............................................ 3-17

3.4.5 Lower bound nonlinear programming problem ......................... 3-17

3.5 UPPER BOUND FINITE ELEMENT LIMIT ANALYSIS FORMULATION ....... 3-18

3.5.1 Constraints from plastic flow in continuum ................................ 3-18

3.5.2 Constraints from yield condition ................................................. 3-19

3.5.3 Constraints due to plastic shearing in discontinuities ................ 3-19

3.5.4 Constraints due to velocity boundary conditions ........................ 3-21

3.5.5 Formation of objective function: Power dissipation in continuum 3-

22

3.5.6 Upper bound nonlinear programming problem ......................... 3-22

3.6 LIMIT ANALYSIS IMPLEMENTATION OF THE HOEK-BROWN FAILURE

CRITERION ...................................................................................................... 3-23

3.7 DISPLACEMENT FINITE ELEMENT METHOD (DFEM) ........................... 3-25

3.8 LIMIT EQUILIBRIUM METHOD .............................................................. 3-26

3.8.1 Introduction ................................................................................. 3-26

3.8.2 Equivalent Mohr-Coulomb parameters in SLIDE ...................... 3-27

3.9 CONCLUSION ....................................................................................... 3-29

CHAPTER 4 SLOPE PROBLEM DEFINITION AND NUMERICAL

MODELLING .......................................................................................... 4-1

4.1 INTRODUCTION ..................................................................................... 4-1

4.2 PLANE STRAIN LIMIT ANALYSIS MODELLING ........................................ 4-2

4.2.1 Mesh details .................................................................................. 4-2

4.2.2 Boundary conditions ..................................................................... 4-3

4.3 THREE DIMENSIONAL LIMIT ANALYSIS MODELLING ............................. 4-3

4.4 DISPLACEMENT FINITE ELEMENT MODELLING ...................................... 4-4

4.4.1 Mesh arrangement ........................................................................ 4-4

4.4.2 Initial stress conditions and optimization of slope failure ............ 4-5

4.5 SUMMARY ............................................................................................. 4-6

VII

CHAPTER 5 SLOPE STABILITY OF PURELY COHESIVE CLAYS

.................................................................................................................... 5-1

5.1 INTRODUCTION ..................................................................................... 5-1

5.2 SOLUTIONS OF HOMOGENEOUS UNDRAINED SLOPES ............................ 5-2

5.2.1 Numerical limit analysis solutions ............................................... 5-2

5.2.2 Solutions based on limit equilibrium method ............................... 5-5

5.2.3 Displacement finite element results (FEM) .................................. 5-6

5.3 SOLUTIONS OF INHOMOGENEOUS UNDRAINED SLOPES ........................ 5-8

5.3.1 3D limit analysis results for cut slopes ......................................... 5-9

5.3.2 3D limit analysis results for natural slopes ................................ 5-12

5.4 SUMMARY AND CONCLUSIONS ........................................................... 5-14

CHAPTER 6 SLOPE STABILITY OF COHESIVE-FRICTIONAL

SOIL .......................................................................................................... 6-1

6.1 INTRODUCTION ..................................................................................... 6-1

6.2 NUMERICAL LIMIT ANALYSIS SOLUTIONS ............................................ 6-2

6.2.1 Stability charts for cohesive-frictional soil slopes ....................... 6-2

6.2.2 Application example ..................................................................... 6-3

6.3 ANALYTICAL SOLUTIONS ..................................................................... 6-4

6.4 DISPLACEMENT FINITE ELEMENT SOLUTIONS ...................................... 6-4

6.4.1 Chart solutions based on displacement finite element analysis ... 6-4

6.4.2 Comparisons with the strength reduction method (SRM) ............ 6-6

6.5 CONCLUSIONS ...................................................................................... 6-7

CHAPTER 7 STATIC STABILITY OF UNIFORM ROCK AND

ROCKFILL SLOPES .............................................................................. 7-1

7.1 INTRODUCTION ..................................................................................... 7-1

7.2 PROBLEM DEFINITION ........................................................................... 7-1

7.3 NUMERICAL LIMIT ANALYSIS SOLUTIONS ............................................ 7-2

7.3.1 Chart solutions ............................................................................. 7-2

7.3.2 Analytical solutions ...................................................................... 7-3

VIII

7.3.3 Comparisons of the tangential method and the numerical limit

analysis solutions ......................................................................................... 7-5

7.3.4 Application example ...................................................................... 7-6

7.4 LIMIT ANALYSIS SOLUTIONS FOR ROCKFILL SLOPES ............................. 7-6

7.5 LIMIT EQUILIBRIUM SOLUTIONS ............................................................ 7-9

7.5.1 Comparisons of the generalized Hoek-Brown model and the Mohr-

Coulomb model .......................................................................................... 7-10

7.5.2 Modification of the equivalent Mohr-Coulomb parameters ....... 7-11

7.6 CONCLUSIONS ..................................................................................... 7-13

CHAPTER 8 SEISMIC STABILITY OF HOMOGENEOUS ROCK

SLOPES .................................................................................................... 8-1

8.1 INTRODUCTION ..................................................................................... 8-1

8.2 LIMIT ANALYSIS SOLUTIONS ................................................................. 8-1

8.2.1 Chart solutions .............................................................................. 8-1

8.2.2 Analytical solutions ....................................................................... 8-3


analysis solutions ......................................................................................... 8-4

8.3 LIMIT EQUILIBRIUM SOLUTIONS ............................................................ 8-6

8.3.1 Comparison of chart solutions between the numerical finite element

limit analysis and limit equilibrium analysis ............................................... 8-6

8.3.2 Investigation of stability numbers increasing with increasing mi . 8-7

8.4 CONCLUSIONS ....................................................................................... 8-8

CHAPTER 9 DISTURBANCE FACTOR EFFECTS ON THE

STATIC ROCK SLOPE STABILITY .................................................. 9-1

9.1 INTRODUCTION ..................................................................................... 9-1

9.2 NUMERICAL LIMIT ANALYSIS SOLUTIONS FOR HOMOGENEOUS

DISTURBED ROCK SLOPES ................................................................................. 9-2

9.2.1 Stability numbers ........................................................................... 9-2

9.2.2 Failure surfaces ............................................................................ 9-6

9.2.3 Application example ...................................................................... 9-7

IX

9.3 NUMERICAL LIMIT ANALYSIS SOLUTIONS FOR INHOMOGENEOUS

DISTURBED ROCK SLOPES .............................................................................. 9-10

9.3.1 Stability numbers ........................................................................ 9-10

9.3.2 Failure surfaces .......................................................................... 9-11

9.4 CONCLUSIONS .................................................................................... 9-12

CHAPTER 10 CONCLUDING REMARKS ....................................... 10-1

10.1 SUMMARY........................................................................................... 10-1

10.2 THE STABILITY OF 2D AND 3D SLOPES IN SOIL .................................. 10-1

10.3 THE STABILITY OF 2D ROCK SLOPES .................................................. 10-2

10.4 RECOMMEDATIONS FOR FURTHER WORK ........................................... 10-3

10.4.1 Pore pressure effects .................................................................. 10-3

10.4.2 Three dimensional (3D) chart solutions for rock slopes ............ 10-4

10.4.3 Slope failure controlled by structural orientations .................... 10-4

10.4.4 Vertical seismic coefficient ......................................................... 10-4

REFERENCES

Two and Three Dimensional Stability Analyses for Soil and Rock Slopes

The University of Western Australia Centre for Offshore Foundation Systems

1-1

CHAPTER 1 INTRODUCTION

1.1 INTRODUCTION

Predicting the stability of soil and rock slopes is a classical problem for geotechnical

engineers and also plays an important role when designing for embankments, dams,

roads, tunnel and other engineering structures. Many researchers have focused on

assessing the stability of slopes (Taylor (1948), Morgenstern (1963), Fredlund and

Krahn (1977), Hoek and Bray (1981), and Goodman and Kieffer (2000)). However, the

problem still presents a significant challenge to designers.

This thesis is concerned with the stability of two dimensional (2D) and three

dimensional (3D) soil and rock slopes by using the upper and lower bound theorems of

limit analysis. More conventional displacement finite element analyses will also be

performed using commercially available software for comparison and verification

purposes. The primary aim of this research project is to apply recently developed 3D

limit analysis formulations to better understand 3D slope behaviour and to develop

rigorous stability solutions that can be used by design engineers. The study will be

unique in that a number of distinctly different numerical methods, namely the upper and

lower bound theorems of limit analysis and the conventional displacement finite

element method (DFEM), have been used in many cases to solve the same problems. In

addition, the popular limit equilibrium method is also employed. A comparison of the

results from each technique provides an opportunity to validate the findings and gives a

rigorous evaluation of slope stability.

The purpose of this Chapter is to outline the objectives of this research and provide an

overview of the thesis.

1.2 RESEARCH OBJECTIVES

1.2.1 Three dimensional (3D) slope stability

For current slope design, two dimensional analyses have been widely accepted as they

are thought to yield a conservative estimate for the factor of safety. Clearly, not all

slopes are infinitely wide and 3D effects influence the stability of most, if not all,



1-2

slopes. There are a range of field conditions under which a 3D method of analysis

would be more appropriate to use. For example, slopes which are curved in plan or

subjected to concentrated surcharge loads, slopes where the potential failure surface is

constrained by physical boundaries, and slopes where non-homogeneities in material

parameters occur.

It is hoped this study will finally provide engineers with a better understanding of 3D

slope stability failure mechanisms and provide a means of estimating the preliminary

stability using simple design charts. These design charts will incorporate all the

necessary parameters in order to estimate the factor of safety for 2D and/or 3D purely

cohesive, cohesive-frictional and rock slopes.

1.2.2 Application of limit analysis to slope stability

Although the limit theorems provide a simple and useful way of analysing the stability

of geotechnical structures, they have not been widely applied to the 3D slope stability

problem. Publications dealing with this subject in three dimensions are limited and are

available in the works of Giger and Krizek (1975), Giger and Krizek (1976),

Michalowski (1989), Donald and Chen (1997), Chen et al. (2001b), Chen et al. (2001a)

and Farzaneh and Askari (2003). One thing in common with the 3D limit analysis

studies to date is that they have all been based on the upper bound method. In addition,

they have not been widely applied to 3D slope stability. The limit equilibrium method is

generally the most widely used in practice for slope stability due to its simplicity and

generality. However, the accuracy of the method is often questioned, particularly for

slope and retaining wall analyses, due to the underlying assumptions that it makes. The

method requires a failure mechanism to be assumed which may consist of plane,

circular or log spiral shaped surfaces. It is also necessary to make sufficient assumptions

regarding the stress distribution along the failure surface such that the overall stability

of the assumed mechanism can be solved by simple statics.

In contrast to the limit equilibrium method, the upper and lower bound methods of limit

analysis provide rigorous bounds on the collapse load. Since the solutions bracket the

exact ultimate load from above and below, they provide an in-built error indicator that is

invaluable in practice. Although some attempt has been made to apply the upper bound

method of limit analysis to the 3D slope problem (Michalowski (1989) and Farzaneh



1-3

and Askari (2003)), most of the work is based on analytical approaches in which the

failure mass is divided into several blocks with simplified slip surface shapes such as

straight or logarithmic lines. The often complex geometry of the surface of the slope is

usually simplified to a plane described by straight lines. The material is assumed to be

homogeneous and a factor of safety is obtained that is only applicable for the defined

failure mechanism. Understandably, such simplifications have limited the application of

these methods to practical problems. Fortunately, the upper and lower bound

formulations developed by Lyamin and Sloan (2002a), Lyamin and Sloan (2002b) and

Krabbenhoft et al. (2005) do not have any constraining simplifications and are

convenient tools for performing numerical limit analysis. Very importantly, unlike the

approximate methods used in previous studies, an excellent indication of the failure

mechanism can be obtained from these formulations without any assumptions being

made in advance. The implication of this for slope stability analyses is profound, i.e.

this ensures the critical factor of safety will be found in all cases.

In light of the above discussion, a primary objective of this thesis is to apply recently

developed 3D limit analysis formulations to better understand 3D slope behaviour and

to develop rigorous stability solutions that can be used by design engineers.

1.2.3 Stability charts for engineering

Stability charts for soil slopes were first produced by Taylor (1948) and they continue to

be used extensively as design tools and draw the attention of many investigators

(Morgenstern (1963), Zanbak (1983), Michalowski (2002), Siad (2003), and Baker et al.

(2006)). However, there are no widely accepted three dimensional stability analysis

solutions for soil and rock slopes available for practicing geotechnical engineers

currently. One aim of this research is to produce stability charts that can be used by

practicing engineers similar to those currently used regularly for 2D slope stability

evaluation. The basic accuracy achievable with slope stability charts is as good as the

accuracy with which slope geometry, unit weights, shear strengths, and pore pressures

can be defined in many cases. It is often argued that stability charts are limited to simple

conditions and approximations are therefore necessary if they are to be applied to real

conditions. Nevertheless, if the necessary approximations are made judiciously, accurate

results can be obtained more quickly with slope stability charts than by using a



1-4

computer program. A very effective procedure is to perform preliminary analyses using

charts, and final analyses using computer software.

1.2.4 Rock slope stability using yield criteria for rock masses

Currently, most of geotechnical software is written in terms of the Mohr-Coulomb

failure criterion. As a consequence the stability of rock slopes is regularly performed in

terms of Mohr-Coulomb cohesion and friction. However, it is not known how

accurately rock slope stability can be estimated using such a criterion. Many criteria

(Hoek and Brown (1980a), Yudhbir et al. (1983), Sheorey et al. (1989) and

Ramamurthy (1995)) have been developed that seek to capture the important elements

of measured rock strengths or seek to modify theoretical approaches to accommodate

experimental evidence. However, the Hoek-Brown failure criterion is virtually the only

nonlinear criterion used by practicing engineers (Mostyn and Douglas (2000)).

For the investigations of rock slope stability in the thesis, the nonlinear yield criterion

proposed by Hoek et al. (2002) has been implemented in the numerical limit analysis

methods (Lyamin and Sloan (2002a), Lyamin and Sloan (2002b) and Krabbenhoft et al.

(2005)). In addition, the recently proposed Douglas failure criterion (Douglas (2002)) is

also incorporated into the study for comparison purposes. More details of the above

nonlinear failure criteria are introduced in Chapter 2

1.3 THESIS OUTLINE

As an overview, the research presented in this Thesis can be divided into five principal

areas:

1. The stability of purely cohesive soil slopes.

2. The stability of cohesive-frictional soil slopes.

3. The static stability of rock slopes.

4. The stability of rock slopes under seismic loadings.

5. The investigation of rock mass disturbance on the stability of rock slopes.



1-5

The structure of the thesis reflects the five main topics listed above. Initially, Chapter 2

provides a background to subsequent Chapters by presenting a summary of numerical

research into the stability of soil and rock slopes.

Chapter 3 provides background to selected aspects of classical plasticity and discusses

the numerical formulations used in detail. In Chapter 4, more precise details are given as

to how a slope stability problem is studied using numerical formulations. This includes

a discussion of the finite element mesh arrangements. In addition, more details on the

consideration and limitation of selected model and method will be described.

Chapter 5 to Chapter 9 constitute the main portion of the thesis and present the results

obtained from the numerical studies for a wide range of slope stability problems. A

separate Chapter is provided for each slope stability analysis based on the type of slope

(soil or rock), loading conditions (undrained, drained and seismic force) and rock mass

disturbance. Where possible, a comparison is made between the results obtained in the

current study and existing solutions.

1.4 PUBLICATIONS

Publications based on this thesis are as follows:

Li, A.J., Merifield, R.S., and Lyamin, A.V. 2008. Stability charts for rock slopes

based on Hoek-Brown failure criterion. International Journal of Rock

Mechanics & Mining Sciences, 45(5): 689-700.

Li, A. J., Lyamin, A. V., Merifield, R. S., (2009), “Seismic rock slope stability

charts based on limit analysis methods,” Computers and Geotechnics, Vol.

36(1~2), p.135~p.148.

Li, A.J., and Merifield, R.S 2007. Rock slope stability assessment based on limit

analysis. In International Symposium on rock slope stability in open pit mining

and civil engineering. Edited by Y. Potvin. Perth, pp. 527-532.



1-6



2-1

CHAPTER 2 LITERATURE REVIEW

2.1 INTRODUCTION

A summary of research into two and three dimensional slope stability analyses using the

limit equilibrium method and finite element method is presented. A comprehensive

overview on the topic of slope stability in soils is given by Duncan (1996).

Slope stability is a common issue for geotechnical engineers and also plays an important

role when designing for dams, roads, tunnels and other engineering structures. Chart

solutions used for preliminary short-term estimates of slope stability for undrained

saturated clay slopes were firstly considered and produced by Taylor (1937). Chart

solutions continue to be useful tools for engineers, and continue to draw the attention of

many investigators (Morgenstern (1963), Cousins (1978), Hoek and Bray (1981),

Leshchinsky and San (1994), and Baker et al. (2006)) in the past decades.

There are two main areas of investigation in this thesis, namely the slope stability in soil

and in rock masses, and thus the brief summary of existing research herein has been

separated based on this distinction. No attempt is made to present a complete

bibliography of all research, but rather a more selective overall summary of research

with greatest relevance to the thesis is presented. For example, the stability of reinforced

soil slopes is also a widely investigated problem, but discussion is limited solely to the

slopes which are without reinforcement.

Generally speaking, current design practices for the slope stability largely rely on a

factor of safety ( F ) that is obtained from the conventional limit equilibrium method

(LEM) or displacement finite element method (DFEM). In addition, it is well known

that the solution obtained from the limit equilibrium method is not rigorous, because

either static or kinematic admissibility conditions are unsatisfied. In contrast, very few

numerical analyses have been performed to evaluate the slope stability based on the

limit theorems. In particular, there are few studies available based on the method of

limit analysis.



2-2

2.2 CURRENT SLOPE STABILITY DESIGN APPROACHES

In general, limit equilibrium analysis is relatively simple and easy to use. Numerical

models have become very popular in recent years, much due to the ease with which

sensitivity analyses and parameter studies can be conducted. Empirical design methods

rely on precedent, but could be, and are often, combined with other analysis methods.

Physical model tests are seldom used today for design purposes, but deserve to be

mentioned since they have contributed to a better understanding of possible failure

modes in rock slopes. There are also design methods which can be used to predict the

risk of a slope failure such as probability analysis.

It is necessary to point out that simplifications are necessary in all design methods. The

assumed failure mechanisms are more or less crude approximations of the actual failure

mechanism. Certain assumptions are also made regarding the slope geometry and the

loads acting on the slope. Assumptions are a requirement because otherwise the design

methods would be overwhelmingly complex and nearly impossible to use rationally.

What marks a robust design method is that the necessary assumptions have very little

influence on the end result. In this section, the current design methods of slope stability

are briefly introduced.

2.2.1 Limit equilibrium method (LEM)

For the simplest form of limit equilibrium analysis, only the equilibrium of forces is

satisfied. The sum of the forces acting to induce sliding of parts of the slope is

compared with the sum of the forces available to resist failure. The ratio between these

two sums is defined as the factor of safety, F (Equation (2.1)).

actionsdriving

actionsresistingF (2.1)

This simple definition of the safety factor can be interpreted in many ways. It could be

expressed in terms of loads, forces, moments etc. The merit of a safety factor is that the

stability of a slope can be quantified by a number. According to Equation (2.1), a safety

factor of less than 1.0 indicates that failure is possible. If there are several potential

failure modes or different failure surfaces which have a calculated safety factor less than

1.0, this indicates that the slope can fail.



2-3

From the studies of many previous investigators (Taylor (1937), Bell (1966), Gibson

and Morgenstern (1962) and Yu et al. (1998)), the slope under a critical condition can

be expressed by a non-dimensional factor, namely stability numbers in this study.

However, there are several existing forms of the stability numbers which are suitable for

different slope materials or loading conditions etc. About the stability numbers used,

more details are described in Chapter 5 to Chapter 9.

2.2.2 Limit analysis

For an exact solution, simultaneous (fulfillment) consideration of the conditions of

equilibrium and compatibility in the slope is required. This includes the differential

equations of equilibrium, the strain compatibility equations, the constitutive equations

for the material and the boundary conditions of the problem. Most of the numerical

methods in use need to make some assumptions to deal with slope stability problem.

This recognition has lead to the development of limit analysis which is a simplified and

relatively rigorous method based on the concepts of the classical theory of plasticity.

Numerical upper and lower bound limit formulations recently developed by Lyamin and

Sloan (2002a), Lyamin and Sloan (2002b) and Krabbenhoft et al. (2005) do not require

to determine the failure surface in advance. By optimising statically admissible stress

fields and kinematically admissible velocity fields, it is possible to bracket the collapse

load from above and below. More details of the upper and lower techniques are

introduced in Chapter 3 and Chapter 4.

2.2.3 Numerical modelling

With numerical modelling, the boundary conditions of the problem, the differential

equations of equilibrium, the constitutive equations for the material, and the strain

compatibility equations are all satisfied in problems discrete equivalent (model). One of

the major benefits of numerical modelling is that both the stress and the displacements

in a body subjected to external loads and imposed boundary conditions can be

calculated. Furthermore, various constitutive relations can be employed (anisotropic,

plastic etc.). There are no restrictions regarding the number of different materials in a

model (other than computation time). Numerical models can also handle complex slope

geometries better than analytical or limit equilibrium methods. Another interesting

feature of many commercially available programs is that they can model groundwater



2-4

flow and the coupled effects between stress and groundwater pressure developing in a

slope.

Today, there are a vast number of different numerical methods available such as the

finite element method (FEM) based on a continuum model and the distinct element

method based on a discontinuum model. In continuum models, the displacement field

will always be continuous. The location of the failure surface can only be judged by the

concentration of shear strain in the model. No actual failure surface discontinuity is

formed and it can thus be difficult to continue to analyse the behavior of the slope after

the first failure surface has formed. In a discontinuum computer code, discontinuities

are included into the basic model geometry already from the start of calculation. The

locations of known pre-existing discontinuities are generally required as an input before

the analysis is begun.

2.2.4 Empirical design

Empirical design often forms a part of the routine design process for slope stability

assessment. As an example, an early attempt toward a systematic grouping of empirical

data was presented by Lutton (1970). Data from the steepest and highest slope in a

specific open pit mine were gathered from several mines, and the slope height was

plotted against the slope angle. This was further developed by Hoek and Bray (1981) by

adding more cases (Figure 2.1).

The dotted line in Figure 2.1 represents an estimated upper limit for stable slopes. The

higher the slope, the lower the slope angle must be to maintain stability. However, for

the higher slopes the angle becomes almost constant which might lead to the conclusion

that there is a lower limit to the required slope angle. In reality, this is probably an effect

of having too little data for higher slopes. Furthermore, the shape and location of the

design curve appears to be chosen somewhat arbitrarily, judging from the cases in

Figure 2.1, since there are several unstable slopes located below the design curve (on

the safe side).

2.2.5 Physical model tests

Physical model tests have developed within the field of geomechanics basically because

of the difficulties and costs associated with full scale testing in the field. Model tests



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provide the means of simulating the conditions of an actual slope in a controlled

environment, where parameters can be more easily varied and their effect on the

stability of the slope studied. They also provide the opportunity of testing up to, and

beyond, the point of failure, something which can be cumbersome in the field. Model

tests are perhaps not a true design method since it is not possible to calculate a slope

angle directly from the results. For this, several tests with varying slope angles would

need to be carried out to make comparisons. On the other hand, physical models have

been very successful in that they have dramatically increased the knowledge and

understanding of the possible failure modes in rock slopes

Three different types of model tests can be distinguished. In the first type of tests, a

model material is used in a down-scaled slope model. Loading is applied only by the

gravity forces developed from the self-weight of the model material. In the second

group of model tests, larger loads are applied to a model using conventional testing

machines in a laboratory. Uniaxial, biaxial or triaxial loading can be applied. The third

group of model tests is centrifuge testing. Here, increased body forces are applied by

rotating the model horizontally in a high speed centrifuge, thus generating centrifugal

forces in the sample.

2.2.6 Probabilistic methods

The basis for probabilistic design methods is the recognition that the factors which

govern slope stability all exhibit some natural variation. Ideally, this variation should be

accounted for in the design method. Using a deterministic approach (typically LEM),

this is only possible by means of a sensitivity analysis. Although a sensitivity analysis

can yield a good qualitative understanding of which factors are most important for a

specific rock slope, such an analysis cannot quantify the actual chance of a slope failure.

In a probabilistic design method, the stochastic nature of the input parameters is

included and the resulting chance, or probability, of failure is calculated. Dealing with

probabilities of failure rather than safety factors means that a finite chance of failure is

always acknowledged, although it can be very small. This is more realistic than stating

that a slope with a certain factor of safety is perfectly stable. Also, a quantitative

description of the failure probability can be used in a risk analysis and linked to

economical decision criteria.



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2.2.7 Limitations

A very brief description of current design approaches for slope stability is shown above.

However, it is important that users understand the assumptions and limitations before

using a design method. The relative merits and shortcomings of the currently existing

design methods are summarized as follows:

1. Limit equilibrium methods include the drawbacks which are the assumption of

(1) the soil or rock masses behaving as a rigid material, and (2) the shear

strength being mobilized at the same time along the entire failure surface.

2. By using both of the upper and lower bound limit analysis, the true failure load

can be bounded. However, the displacement of the slope cannot be predicted.

3. In numerical modelling, standard commercial software for performing rock

mechanics analyses do not allow fracture propagation through intact material,

and new developments in this field have not yet reached full maturity for

practical applications in slope design. Sjöberg (1999) found that it was not

possible to simulate smaller block sizes, as the models were very (computer)

memory-consuming and took a long time to run. His study also indicated that a

reduced block or element size alone might not be sufficient to increase the

ability of the rock mass to fracture.

4. Empirical design charts, such as that shown in Figure 2.1, only provide general

design advice. Because of limited data, the establishment of more detailed

design rules is not possible.

5. Although physical model tests can be useful for determining fundamental failure

mechanisms and for the verification of analytical and numerical methods, they

are not a true design method for simulating the correct loading conditions and

accurately modelling rock mass properties. Centrifuge testing of rocks also

requires somewhat larger model dimensions, compared to soil testing in order to

include discontinuities in the model. Larger model dimensions require a

centrifuge which, besides a high acceleration, also can handle a large mass.

Unfortunately, these two objectives are not easily met simultaneously.



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6. Probabilistic methods require large amounts of input data and assumptions

regarding the distribution functions. Also, probabilistic design methods are

mostly based on the LEM and are thus subject to the same limitations as LEM.

Cost integration into design methods can to some extent be accomplished using

probabilistic methods. However, the vast amount of input data required has

rendered these cost-benefit-methods difficult to use in practical applications.

2.3 FAILURE MODES AND FAILURE MECHANISMS FOR

SLOPES

Based on the geological structure and the stress state of soil and rock masses, certain

slope failure modes appear to be more likely than others. In the following Sections, the

main failure modes in soil and rock slopes will be described. A more detailed

description of the governing failure mechanisms for typically occurring slope failures is

given as well as those based on field observations.

2.3.1 Observed failure modes

Rotation Shear Failure

The failure modes displayed in Figure 2.2 are rotational shear failures. These are

sometimes referred to as circular failures (Hoek and Bray (1981)) which implies that

failure takes place along a circular arc. The condition for a rotational shear failure is

thus that the individual particles in a soil or rock mass should be very small compared to

the size of the slope, and that these particles are not interlocked as a result of their shape

(Hoek and Bray (1981)).

Rotational shear failure generally occurs in soil slopes. The issue of whether a rock

mass can be considered as heavily fractured is thus mostly a matter of scale. Rotational

shear failure in a large scale slope would probably primarily involve failure along pre-

existing discontinuities with perhaps some portions of the failure surface going through

intact rock. Also, rotation and translation of individual blocks in the rock mass would

help to create a failure surface. The resulting failure surface would follow a curved path.

In Figure 2.2, the failure surfaces are drawn to be relatively deep, but they could also be

shallower. There may also be combinations of plane failure, step path failures



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(introduced in the following) and rotational shear failures, with or without tension

cracks at the slope crest. Only 2D representations of the failure surface are illustrated in

Figure 2.2. In reality, rotational shear failure is a three dimensional phenomenon and the

resulting failure surface will be bowl or spoon-shaped (Figure 2.3).

Plane Shear Failure

As shown in Figure 2.4, the failure surface of plane shear failure could be a single

discontinuity (plane failure), two discontinuities intersecting each other (wedge failure)

or a combination of several discontinuities connected together (step path and step wedge

failures). A common feature of most failure modes is the formation of a tension crack at

the slope crest.

The failure modes depicted in Figure 2.4 are, with the exception of wedge failure, two

dimensional representations. For failure to occur, release surfaces must be present to

define the rock block moving in the lateral direction. Alternatively, step path failure

may be a true three dimensional failure in which combinations of discontinuities define

the failure surface in all three dimensions.

Crushing, Buckling and Toppling Failure

Characteristic for these types of failures is that a successive breakdown of the rock slope

occurs. Failure can initiate by crushing of the slope toe, which in turn causes load

transfer to adjacent areas which may fail (Figure 2.5). Obviously, the orientations of

discontinuities and in situ stresses in relation to the rock strength are important factors

governing this failure mode.

The presence of discontinuities in the rock mass can result in several secondary modes

of failure once crushing of the toe has occurred. Large blocks and wedges, or

assemblages of many smaller blocks can be relieved and a combination of block flow

and plane shear failure can develop. An associated form of failure is toppling failure

(Hoek and Bray (1981)). Toppling refers to overturning of columns of rock formed by

steeply dipping discontinuities and sub horizontal cross joints. It could also be initiated

by crushing of the slope toe, which is termed secondary toppling.

Buckling failure could develop in slopes with long, continuous bedding planes or joints

oriented parallel to the slope face. Crushing failure of the toe or plane shear failure



2-9

along cross joints help to initiate slab failure, but slab failure could also be initiated by

hydrostatic uplift due to high groundwater levels. Buckling could develop if the axial

stresses on the rock slab are high and the slab is very thin in relation to its length.

2.3.2 Failure mechanisms inferred from field observations

Shape and Location of Failure Surface

For plane shear failures, the mechanism of failure is relatively simple to explain. Failure

will initiate when the supporting rock at the location where the discontinuity daylights

the slope is removed by excavation. The shape and location of the failure surface is

determined solely by the location and orientation of discontinuities. The dominant

failure mechanism is shear failure as the shear strength of the discontinuity is exceeded.

The mechanisms of rotational shear failure are especially interesting to consider for

large scale slopes. Considerable amount of work on this failure mode has been done in

the field of soil mechanics. It is thus natural to review some of this work and see how

this applies to rock slopes. As was discussed earlier, a very high slope can almost be

considered as a granular material, thus having several similarities with a typical soil. An

important difference, though, is that sandy soils are mostly frictional materials and

clayey soils are mostly cohesive materials, whereas a rock mass probably exhibits both

an effective friction angle and an effective cohesion. One must therefore be careful

when translating the experience gained on the behavior of soil slopes to the behavior of

rock slopes.

The location of the failure surface is determined by the relation between the friction

angle and cohesion (Spencer (1967)). In a purely frictional material, such as sand, the

failure surface is more shallow and daylights at the toe of the slope, whereas in a purely

cohesive soil, the failure surface always tends to be more deep seated. The depth of the

failure surface also relies on the difference between the slope angle and the friction

angle; larger differences lead to a more deep seated failure. For cohesive soil slopes,

observations have also shown that it is more common that the failure surface daylights

below the toe of the slope.

Failure Initiation and Propagation



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Common for both soil and rock slopes is the fact that the failure surface cannot develop

at the same instant throughout the slope. There must be a progressive mechanism of

failure development eventually leading to the full collapse of the slope. The failure

development has been difficult to quantify even for homogeneous soils. Progressive

failure is defined here as the successive development of a failure surface in a slope

through stress redistribution and loss of shear strength of the material. Failure caused by

a decrease in the strength properties with time and associated creep movements could

instead be termed delayed failures (Skempton and Hutchinson (1969)), and are of less

importance for open pit rock slopes which have a limited life.

The mechanism for progressive failure in slopes is that the peak shear strength (Figure

2.6) is exceeded at one point in the slope, resulting in a stress redistribution due to the

lower residual strength of the material. This stress redistribution causes nearby points to

yield which results in further stress redistribution and so the process continues. Failure

can therefore develop for slopes which would appear to be stable when considering only

the peak strength, but where local failure can occur. A progressive failure behaviour

similar to that of soils, could also be envisioned for rock slopes which exhibit brittle and

strain-softening behavior, but there are much fewer observations to substantiate this.

For such a slope (under drained conditions), tension cracks will be the first sign of

failure. Once tension cracks initiated at the crest, this portion of the slope is free to

move, and thus act to increase the load on the lower portion of the slope. Overall failure

occurs when the loads on the middle portion of the failure surface exceed the shear

strength.

Progressive Failure in rock slopes

As known, rock is a much more brittle material than clay which implies that the above

mechanism is not entirely applicable to rock slopes. Müller (1966) discussed some

aspects of progressive failure in rock slopes and concluded that progressive failure in

rock would first involve failure along pre-existing discontinuities but that failure

through the intact rock bridged between pre-existing joints would also contribute to the

failure development. The resulting failure surface would thus be composed mainly of

pre-existing discontinuities but with some portions of initially intact rock. The shape



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and location of the failure surface would be determined by the stresses acting on the

slope, the slope geometry, and the overall rock mass strength.

The question of where failure initiates is also of importance. Although the first signs of

instability in an open pit often are tension cracks at the slope crest, this does not imply

that failure must initiate at this point. For a material in which a substantial part of the

shear strength stems from friction (such as rocks) the failure surface will pass through

the toe (Piteau and Martin (1982)). In open pit mining, where the toe of the slope is

excavated continuously as the pit is being deepened, it is also more likely that

successive failures initiate at the toe. The failure surface can be almost fully developed

before any tension cracks occur at the slope crest. Movements at the toe of the slope

could be small due to the acting confinement. The slope near the toe could also move

more or less as a rigid block in the early stages of failure, thus making it difficult to

visually observe such displacements (Chowdhury (1995)).

In addition, toppling failure has also been observed in natural slopes (Zischinsky

(1966)). This failure is characterized by slowly moving (time-dependent), very large

rock masses, exhibiting signs of toppling failure near the surface but with a more deep

seated shear zone along which sliding occurs. A common denominator for these failures

appears to be that the rock mass exhibits bedding or strong foliation.

2.4 PREVIOUS SLOPE STABILITY INVESTIGATIONS IN

SOILS


Centrifuge systems use physical scaling laws to match the model and prototype

behaviour and can be used to study slope stability. These investigations are based on

generating soil stress fields which are in proportion to the size of the slopes. The stress

field itself is induced through centrifugal force, as the name suggests. Based on the

centrifuge modelling, Resnick and Znidarčić (1990) investigated the effects of

horizontal drains on slope stability. Good agreement between the predicted and

observed slip surfaces was obtained. In the study of Taboada-Urtuzuástegui and Dorby

(1998), centrifuge model tests were employed to study the liquefaction and earthquake-

induced lateral spreading of shallow slopes in sand. It was observed by Taboada-



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Urtuzuástegui and Dorby (1998) that the excess pore pressure decreases both during and

after shaking as the slope angle ( ) increases.

Chen and Liu (2007) adopted two assemblies of cylindrical aluminium rods to simulate

sand particles in laboratory tilting box tests. The simulations from the distinct element

method (DEM) agreed with the laboratory test results where the failure pattern of a dry

slope largely shows a slip plane parallel to the slope surface, but circular slip surface in

a moist slope. Olivares and Damiano (2007) utilised an instrumented flume to examine

the failure mechanism of flowslides. It was summarised that the flowslides mechanism

has the highest probability of occurrence in a steep slope. Three conditions are

necessary for development of this mechanism: susceptibility of the soils to static

liquefaction; attainment of fully saturated condition at the onset of instability; and a

slow enough rate of excess pore pressure dissipation compared to the rate of slope

movement.

In order to investigate the stability of a cut slope experiencing natural pore pressure

recovery, the study of Cooper et al. (1998) raised the pore pressure in a controlled

manner so as to induce a deep-seated failure. It was found that the water pressure

recovery does induce the progressive failure of cut slopes. In their study, the failures

took place rapidly at the toe and crest of the slope, and then extended into the slope as

pore pressures increased. In addition, the observed displacement increased continually

up to the point of collapse, illustrating the progressive reduction in the average

mobilized shear strength along the slip surface with continuing displacement.

Although a number of laboratory and field tests have examined the stability of slopes

for a range of problem variables, there were no chart solutions provided from a review

of past experimental results.

2.4.2 Limit equilibrium analysis

Duncan (1996) (Table 2.1) and Chang (2002) reviewed the main aspects of publications

dealing with 3D limit equilibrium approaches. 3D stability analyses based on the limit

equilibrium method have been performed by Baligh and Azzouz (1975), Hovland

(1977), Chen and Chameau (1982), Ugai (1985), Leshchinsky and Baker (1986), Xing

(1987), Ugai and Hosobori (1988), Gens et al. (1988), Hungr (1987), Hungr et al.

(1989), Lam and Fredlund (1993), and Chang (2002). The majority of 3D methods



2-13

proposed in these studies are based on extensions of Bishop’s Simplified, Spencer’s, or

Morgenstern and Price’s original 2D limit equilibrium slice methods. That is,

differences between each study arise due to the arbitrary assumptions made regarding

inter-column forces. The failure mass is divided into a number of columns with vertical

interfaces and the conditions for static equilibrium are used to find the factor of safety

after making assumptions about the forces on adjacent columns.

Chang (2002) considered force equilibrium for individual blocks and the overall system

in a 3D limit equilibrium analysis. Huang and Tsai (2000) and Huang et al. (2002) took

into account force and/or moment limit equilibrium in two orthogonal directions to

analyse the 3D stability of a potential failure mass. Although the considerations are

more reasonable and thorough, the newly obtained factor of safety does not change

significantly, compared to the previously presented results. In addition, Zhu (2001)

employed numerical limit equilibrium analysis to approximate the critical slip surfaces

where initial trial surfaces are not required and no restrictions are imposed on the shape

of slip surfaces. Unfortunately, stability charts for preliminary design use were not

provided.

Regarding chart solutions based on the LEM, Gens et al. (1988) produced a

comprehensive set of stability charts for 3D purely cohesive soil slopes. The case

records presented in their study showed that the difference in the slope stability

assessment between two and three dimensional analysis can ranges from 3% to 30% and

average 13.9%. This difference is comparable in importance with the corrections

commonly made with regard to undrained shear strength ( uc ), and in back analysis, may

be unsafe. Jiang and Yamagami (2006) proposed chart solutions for cohesive-frictional

slopes. In their study, both simple slopes and long slopes were accounted for. It was

found by Jiang and Yamagami (2006) a long slope has a lager factor of safety than a

simple slope for a given cohesion ( 'c ) and friction angle ( ' ).

Baker et al. (2006) adopted the pseudo static (PS) method in limit equilibrium analysis

and proposed 2D seismic chart solutions for cohesive-frictional soil slopes. Their

investigation focused on the effects of the critical PS coefficient on the slope stability

for a range of geometries, friction angle ( ' ) and stability number ( 'N c H ). This



2-14

form of stability number is the same as that adopted by Gens et al. (1988) which was

proposed by Taylor (1937).

Recently, Chen and Chameau (1982) developed a 3D limit equilibrium method and

found the factor of safety from a 3D analysis is smaller than that from a 2D analysis.

Later, Cavounidis (1987) proved the statement made by Chen and Chameau (1982) is

incorrect. Cavounidis (1987) also highlighted that the 3D factor of safety of a slope is

always greater than 2D factor for the same slope.

Stark and Eid (1998) reviewed three commercially available limit equilibrium based

computer programs in their attempts to analyse several landslide case histories and

concluded that the factor of safety is poorly estimated by this software because of their

limitations in describing geometry, material properties and/or the analytical methods.

2.4.3 Finite element analysis

As pointed out by Duncan (1996), the FEM is a general-purpose method which can be

used to calculate stresses, movements, pore pressure and other characteristics of earth

masses during construction (Zheng et al. (2005) and Lane and Griffiths (2000)) without

previously assuming the potential sliding surface. In particular, Potts et al. (1997) used

FEM to examine the failure mechanism for the delayed collapse of a cut slope in stiff

clays. In addition, Troncone (2005) incorporated the soil stain-softening behaviour into

the elasto-viscoplastic constitutive model and found that the strain- softening behaviour

plays an important role in the slope progressive failure.

In order to estimate the slope stability and obtain its factor of safety by using the finite

element method, the strength reduction method (SRM) is widely used (Griffiths and

Lane (1999), Zheng et al. (2006), Manzari and Nour (2000), and Hoek et al. (2000)).

Based on the SRM, Manzari and Nour (2000) investigated the soil dilatancy effect on

the slope stability analysis. It was found by Manzari and Nour (2000) that the effect of

soil dilatancy on the stability number ( 'N c H ) may become increasingly important

as the friction angle ( ' ) increases. The presence of a soft band with frictional material

investigated by Cheng et al. (2007) showed that the factor of safety is very sensitive to

the size of the elements, the tolerance of the analysis and the number of iterations



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allowed. They also suggested that the LEM should be used for this special case to check

the solutions from the SRM.

Griffiths and Lane (1999) and Griffiths and Marquez (2007) examined 2D and 3D slope

stability respectively. They demonstrated that utilising the SRM in finite element

analysis can obtain rational safety factors. Moreover, the potentially critical conditions

under rapid drawdown for partial submerged slopes have been investigated and

identified by Lane and Griffiths (2000) by utilising finite element analysis.

Based on the SRM, Li (2006) indicated that the difference in the stability evaluation

between using a coarse and fine mesh is around 2%. Hwang et al. (2002) observed that

the critical slip surface determined by the simplified Bishop’s analysis compare well

with the failure surface plotted by using the mobilized friction angle contours from the

finite element analysis of an excavated slope. In addition, the difference in the factors of

safety ( F ) between SRM and LEM was found to be insignificant by Baker et al. (2006)

and Psarropoulos and Tsompanakis (2008).

2.4.4 Limit analysis


of geotechnical structures, they have not been widely applied to 3D slope stability

problem. Currently, most the slope stability evaluations using the limit analysis are

based on the upper bound method alone (Michalowski (1989), Farzaneh and Askari

(2003), Chen et al. (2005), Michalowski (1997), Michalowski (2002), and Viratjandr

and Michalowski (2006)). Major contributions to soil slope stability analysis were

presented by Michalowski and his co-worker who investigated the 3D slope stability

influenced by footing load on the slope crest (Michalowski (1989)) and provided sets of

stability charts for cohesive-frictional slopes which took seismic loadings and pore

pressure into account (Michalowski (2002), Viratjandr and Michalowski (2006)).

However, it should be noticed that by utilising the upper or lower bound method alone

or in isolation, the true solution can not be bracketed.

By using both the lower and upper bound analyses to estimate slope stability, Yu et al.

(1998), Kim et al. (1999) and Loukidis et al. (2003) proposed sets of stability charts for

inhomogeneous cohesive soil slopes and cohesive-frictional soil slopes subjected to



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pore pressure and seismic loadings. All of these studies focused solely on investigating

the stability of 2D slopes.

2.4.5 Other investigations

In addition to the above mentioned methods, the finite difference method and

probability analysis are also used in current soil slope stability assessments. Based on

the finite difference method (FDM), Chugh (2003) made a comparison of the effect of

boundary conditions when simulating slope stability cases. It was noticed that the

displacement of end walls should be assumed as zero for three orthogonal directions

when modelling the field conditions of 3D slopes. Shou and Wang (2003) applied

probability analysis in conjunction with the pseudo static method to a particular case

study. In their investigation, the risk of the slope under the effects of seismic force and

high water level was estimated.

2.5 PREVIOUS SLOPE STABILITY INVESTIGATIONS IN

ROCK MASSES

It is well known that the strength of jointed rock masses is notoriously difficult to assess.

Generally speaking, rock masses are inhomogeneous, discontinuous media composed of

rock material and naturally occurring discontinuities such as joints, fractures and

bedding planes. These features make any analysis very difficult using simple theoretical

solutions, like the limit equilibrium method. Moreover, without including special

interface or joint elements, the displacement finite element method is not suitable for

analysing rock masses with fractures and discontinuities.

To overcome the problem of estimating rock slope strength and stability governed by

the complicated failure mechanisms, Jaeger (1971) and Goodman and Kieffer (2000)

have outlined several simple methods and emphasized their limitations. In addition,

many criteria have been proposed for estimating rock strength (Hoek and Brown

(1980a), Yu et al. (2002), Grasselli and Egger (2003), Sheorey (1997), and Yudhbir et al.

(1983)). Currently, one widely accepted approach to estimating rock mass strength is

the Hoek-Brown failure criterion (Hoek and Brown (1980a) and Hoek et al. (2002)). As

pointed out by Merifield et al. (2006), the Hoek-Brown failure criterion is one of the

few non-linear criteria used by practising engineers to estimate rock mass strength. This



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yield criterion is also employed in this thesis for the investigations of rock slopes. More

details of the Hoek-Brown failure criterion will be described in Section 2.7.1.


Some of the most interesting model tests with respect to possible failure mechanisms in

large scale slopes are those carried out by Ladanyi and Archambault (Ladanyi and

Archambault (1970) and Ladanyi and Archambault (1972)). In the tests with

discontinuous joints, two types of failures were observed. The first type was shear

failure along a well defined failure surface, and the second type of formation was of a

shear zone. The big difference, however, was that shear failure did not occur along the

interfaces of the concrete bricks, but instead appeared as shear failure through the intact

material.

Einstein et al. (1970) carried out model tests using a mix of gypsum plaster, water and

celite. The intention was to simulate a brittle rock of relatively high strength, such as

granite and quartzite. Under triaxial stress loadings, an interesting conclusion which

could be drawn from these tests was that the confining stress strongly affected the

failure mechanisms in the samples. For low confining stress (less than 10 MPa), failure

occurred along the pre-existing joints, but for higher confining stress, failure occurred

mainly through the intact material. The test results also indicated that although failure

occurred through the intact material, the overall strength of the jointed samples was

lower than the intact strength of the model material.

Based on centrifuge tests, several different joint configurations were tested by Stacey

(1973). The results showed that failure occurred as sliding along pre-existing joints, but

failure through the intact model material was not observed in the tests. The above

studies reported some variation in the results from different model tests. This fact can

probably be explained by the differences in model material and loading conditions. The

loading conditions vary from gravitational loading alone to biaxial and triaxial loading.

A review of the literature reveals that there are very few small scale experimental results

presented for rock slopes. This is likely to be because modelling of hard rock could

require a large capacity centrifuge equipment, as highlighted by Stewart et al. (1994). In

the study of Stewart et al. (1994), centrifuge modelling is used to investigate rock slope

failure mechanisms. The collapse mechanism evident in the model compared well with



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flexural toppling failure observed in the field. Furthermore, it was found that the rock

masses with high stiffness would fail in a brittle fashion.

Adhikary et al. (1994) produced a set of stability charts for rock slopes due to flexural

toppling failures. They were based on the results from centrifuge tests and the limiting

equilibrium method. However, the chart solutions for the case of joint angles of less

than 20 - 25 were found to overestimate the rock slope stability by Adhikary and

Dyskin (2007). In addition, Adhikary and Dyskin (2007) indicated that the fractures

could be observed from the toe and (1) propagate instantaneously back into the slope in

the case of high joint friction angle and (2) propagate progressively back into the slope

in the case of low joint friction angle.

2.5.2 Investigations based on the limit equilibrium method

Chen et al. (2003) conducted a series of back-analyses for a case history to investigate

rock slope stability under earthquake loadings. They found that using the reduction

factor of 2/3 for peak strength parameters can reasonably simulate the slope condition in

the field. These peak strength parameters were obtained from laboratory testing in terms

of 'c and ' . Moreover, they indicated that the vertical ground acceleration was an

important factor for inducing rockslide under near field conditions.

Sonmez et al. (1998) utilised back analysis of slope failures to obtain rock slope

strength parameters. In their study, the applicability of rock mass classification, and a

practical procedure of estimating the mobilised shear strength based on the Hoek-Brown

yield criterion were explained. They concluded that the shear strength determination is

very difficult for jointed rock masses, particularly due to the scale effect.

Based on the back-analysis of the failure mechanism, Day and Seery (2007) highlighted

that a major geological structure is the key factor that controls slope failure. Therefore,

it can not be ignored that the slip surface follows the structural features in slope stability

analysis. Harman et al. (2007) adopted an assumed case and considered the effects of

permeability on the rock slope stability. The factor of safety was found to increase with

increasing permeability. Due to the fact that many rock types have a low permeability, it

is shown in their study that the rock slopes should have no pore pressure and hydraulic

continuity, if as intact rock.



2-19

Currently, practising engineers typically use the stability charts of Hoek and Bray (1981)

when attempting to predict the stability of rock slopes. These chart solutions take the

water table in to account and are suited to uniform rock and rockfill slopes. In addition,

Zanbak (1983) proposed a set of stability charts for rock slopes susceptible to toppling

based on the limit equilibrium theorem. However, the conventional Mohr-Coulomb

parameters ( 'c and ' ) of rock masses or block interfaces are required as input for these

two sets of chart solutions.

2.5.3 Investigations based on the numerical analysis

Based on the numerical analysis, Buhan et al. (2002) found that the final results of a

stability analysis may be influenced by scale-effects in rock masses. Previous

investigations (Hoek et al. (2000), Wang et al. (2003), Eberhardt et al. (2004), and Stead

et al. (2006)) of progressive failures and/or safety factor assessment of rock slopes have

used a range of numerical methods. These include the continuum methods (finite

element method and the finite difference method), the discontinuum methods (distinct

element and discontinuous deformation analysis), and finite-/discrete-element codes. In

particular, the study of Elmo et al. (2007) modelled a large scale open pit mine in 2D

and 3D analyses by using finite-/discrete-element codes. It was acknowledged that 3D

large scale analysis of the fracturing process is currently limited by the memory and

processing capacity of computer hardware.

The finite difference method (FDM) was employed by Stewart et al. (1994) to

investigate rock slope stability. In their study, the stain-softening was found to be an

important factor in some slope stability situations. The pattern of stress concentration

around the toe of the slope for the cases of frictionless joints was obtained from the

finite element analysis of Adhikary et al. (1995). Hence, the failure mechanism is

progressive failure for the slope with frictionless joints. Furthermore, they also pointed

out that the frictional sliding along the joints tends to redistribute the moment stresses

rather evenly over a large area. This area extends inside from the toe of the slope and

thus, the slope fails instantaneously. From the study of Adhikary and Dyskin (2007), it

was found that the joint friction plays a major role for the mechanism of toppling failure.

However, the joint cohesion does not have a similar effect on the failure mechanism.



2-20

2.5.4 Investigations based on limit analysis theorems

Siad (2003) produced 2D charts based on the upper bound approach that can be used for

rock slopes with earthquake effects. A range of parameters is considered in this study

which includes slope angle, joint inclination, shear strength of rock masses and joints

etc. In addition, 3D rock slope stability was investigated by Chen et al. (2001a). In their

study, the critical failure mode can be found by optimisation routines, however the

failure surface still needs to be assumed in advance. Moreover, the solutions presented

in the above studies require conventional Mohr-Coulomb soil parameters, cohesion ( 'c )

and friction angle ( ' ), as input.

Collins et al. (1988), Drescher and Christopoulos (1988) and Yang et al. (2004a)

adopted tangential strength parameters ( tc and t ) from the planes of nonlinear

failure criteria to estimate the slope stability. After the study of Yang et al. (2004b), the

latest version of the Hoek-Brown failure criterion is employed to conduct slope stability

analyses. The effects of the seismic loadings (Yang et al. (2004b) and Yang (2007)) and

pore pressure (Yang and Zou (2006)) on the rock slope stability were considered. As far

as the author is aware, the studies of Yang et al. (2004b), Yang et al. (2004b) and Yang

and Zou (2006) represent the only attempt at providing slope stability factors for

estimating rock slope stability.

2.5.5 Other investigations

In order to find the rock slope potential failure key-group and estimate the probability of

failure, the probabilistic analytical method was employed by Yarahmadi Bafghi and

Verdel (2005) and Hack et al. (2003). It should be noted that the Slope Stability

Probability Classification proposed by Hack et al. (2003) does not require cohesion and

friction as input for rock slope stability evaluations. Moreover, reliability analysis was

employed by Wang et al. (2000) and Hack et al. (2007) to predict the failure risk of rock

slopes and investigate the influence of earthquakes on rock slope stability.



2-21

2.6 PREVIOUS SLOPE STABILITY INVESTIGATIONS

BASED ON THE PSEUDO STATIC (PS) METHOD

In order to estimate rock slope stability under earthquake effects, Seed (1979)

recommended the adopted seismic coefficient versus an earthquake Richter’s magnitude

based on the PS analysis. Luo et al. (2004) found that ground water may significantly

reduce slope stability during earthquake excitation where the obtained maximum

seismic coefficient changes by up to 60%. Sepúlveda et al. (2005a) indicated that the

topographic amplification effects such as slope orientation and seismic wavelength may

influence the rock slope stability assessment. In the case study of Chen et al. (2003), the

vertical ground acceleration was found to be an important factor leading to rockslide

under near field conditions.

Newmark (1965) applied and extended the PS method to evaluate the ground movement

induced by an earthquake. This approach has been accepted and extensively used to

study earthquake triggered landslides and rockslides (Sepúlveda et al. (2005b), Huang et

al. (2001) and Ling and Cheng (1997)). Pradel et al. (2005) in particular, obtained a

good agreement of slope crest displacement between the calculated and observed

results. In their study, the strength parameters used in analyses are determined by

repeated direct shear testing and back analysis.

The PS approach has been applied in number of investigations (Newmark (1965), Seed

(1979), Baker et al. (2006), Ling et al. (1997), and Loukidis et al. (2003)), mainly due to

its simplicity. In particular, Baker et al. (2006) and Loukidis et al. (2003) have adopted

the PS method in limit equilibrium analysis and limit analysis respectively to provide

chart solutions for soil slopes. By using complicated dynamic response analysis coupled

with appropriate constitutive laws, a more precise seismic evaluation for slopes can be

obtained. However, the PS method is still applicable and recommended as a screening

procedure to identify any requirement for more sophisticated dynamic analyses.

Although the PS approach has a number of limitations, as highlighted by Cotecchia

(1987) and Kramer (1996), the methodology is considered to be generally conservative,

and is the one most often used in current practice.

In general, the seismic coefficients are determined from experience by using the

maximum horizontal acceleration or peak ground acceleration of a design earthquake. It



2-22

should be noted that, current design using PS analysis is often based on a horizontal

seismic coefficient ( hk ). Therefore, this study is primarily focused on investigating the

earthquake effects on rock slope stability by using a range of horizontal seismic

coefficients. With reference to the magnitude of hk , Seed (1979) suggested that the PS

method is applicable in assessing the performance of embankments constructed of

materials which do not suffer significant strength loss during earthquakes. It is

recommended to utilise 1.0k for earthquakes of Richter’s magnitude 6.5, and

15.0k for earthquakes of Richter’s magnitude 8.5. For both cases, a safety factor

15.1F is required for design.

The suggestion proposed by Hynes-Griffin and Franklin (1984) is one of the widely

used and accepted methods for determining an appropriate value of hk . They

recommended that a PS analysis can be used for preliminary evaluation of slope

stability, where a seismic coefficient equal to one-half the measured bedrock

acceleration is adopted. Provided the obtained factor of safety is greater than 1.0, the

slope design can be accepted. For factors of safety of less than 1, Hynes-Griffin and

Franklin (1984) suggested that a more thorough numerical analysis need to be

performed. However, because the magnitude of hk is related to the measured bedrock

acceleration as discussed above, the PS method may not account for the site

amplification induced by the underlain stratum (Bessason and Kaynia (2002)) or

topography (Sepúlveda et al. (2005b)) etc.

In order to select an appropriate PS coefficient for a given site, a diagram (Figure 2.7)

summarized by the California Division of Mines and Geology (1997) provides the

recommendations in regards to the seismic coefficient ( hk ), versus a required factor of

safety. From this diagram, it can be seen that the recommended hk values do not exceed

0.375. Therefore, the range of the seismic coefficients adopted in the present study will

be between 0.0hk and 375.0hk .



2-23

2.7 EMPIRICAL FAILURE CRITERIA FOR ROCK MASSES

2.7.1 The generalised Hoek-Brown failure criterion

In this Section, details of the latest version of the Hoek-Brown yield criterion (Hoek et

al. (2002)) are discussed. The Hoek-Brown failure criterion for rock masses was first

described in Hoek and Brown (1980a) and has been subsequently updated in 1983, 1988,

1992, 1995, 1997, 2001 and 2002. A brief history of its development can be found in

Hoek and Marinos (2007).

After Hoek et al. (2002), the Hoek-Brown failure criterion can be expressed as the

following equations:

sm

cibci

'3'

3'1 (2.2)

where

D

GSImm ib 1428

100exp

(2.3)

D

GSIs

39

100exp (2.4)

3

2015

6

1

2

1ee

GSI (2.5)

bm is a reduction of im . im is the value of the Hoek-Brown constant which can be

obtained from triaxial test or found in Wyllie and Mah (2004). s and are constants

which depend upon the rock mass characteristics, and ci is the uniaxial compressive

strength of the intact rock pieces.

The GSI was introduced because Bieniawski’s rock mass rating RMR system

(Bieniaski (1976)) and the Q-system (Barton (2002)) were deemed to be unsuitable for

poor rock masses. The GSI ranges from about 10, for extremely poor rock masses, to

100 for intact rock. In addition, the GSI classification system is based upon the

assumption that the rock mass contains sufficient number of ‘‘randomly’’ oriented

discontinuities such that it behaves as an isotropic mass. In other words, the behaviour



2-24

of the rock mass is independent of the direction of the applied loads. Therefore, it is

clear that the GSI system should not be applied to those rock masses in which there is a

clearly defined dominant structural orientation that will lead to highly anisotropic

mechanical behaviour.

The parameter D is a factor that depends on the degree of disturbance. The suggested

value of disturbance factor is 0D for undisturbed in situ rock masses and 1D for

disturbed rock mass properties. The magnitude of the disturbance factor is affected by

blast damage and stress relief due to overburden removal.

The uniaxial compressive strength is obtained by setting 03 in (2.2), giving

scic (2.6)

and the tensile strength is

b

cit m

s (2.7)

Although the empirical failure criterion proposed by Hoek and Brown (Hoek and Brown

(1980a)) is widely accepted as a means of estimating the strength of rock masses, it

assumes the material is isotropic. This implies that the Hoek-Brown yield criterion is

unsuitable for slope stability problems where shear failures are governed by a

preferential direction imposed by a singular discontinuity set or combination of several

discontinuity sets. Examples include sliding over inclined bedding planes, toppling due

to near-vertical discontinuity, or wedge failure over intersecting discontinuity planes.

Because the Hoek-Brown criterion is one of the few nonlinear criteria widely accepted,

it is appropriate to use for the problems of isotropic rock slopes.

The Hoek-Brown criterion is based on the Geological Strength Index GSI

classification system. This comes from the assumption that the rock mass contains

sufficient number of ‘‘randomly’’ oriented discontinuities to behave as an isotropic

mass. Therefore, the behaviour of the rock mass is independent of the direction of the

applied loads. The GSI system should not be applied to rock masses in which there is a

clearly defined and dominant structural orientation that will display the highly

anisotropic mechanical behaviour. In addition, it is also inappropriate to assign GSI



2-25

values to excavated faces in strong hard rock with a few discontinuities spaced at

distances of similar magnitude to the dimensions of slope under consideration. In such

cases the stability of the slope will be controlled by the three dimensional geometry of

the intersecting discontinuities and the free faces created by the excavation.

The stability charts presented in Chapter 7 to Chapter 9 are therefore subject to the same

limitations that underpin the Hoek-Brown yield criterion itself. Further details of the

applicability and limitations of the GSI system can be found in Marinos et al. (2005).

An explanation of the applicability of Hoek-Brown criterion when analysing rock slope

stability is displayed in Figure 2.8. After Hoek (1983), for the same rock properties

throughout the slope, rock masses can be classified into three structural groups, namely

GROUP I, GROUP II, and GROUP III. Figure 2.8 shows the transition from an

isotropic intact rock (GROUP I), through a highly anisotropic rock mass (GROUP II),

to a heavily jointed rock mass (GROUP III). In this study, the rock masses of all slopes

have been assumed as either intact or heavily jointed rocks as GROUP I and GROUP

III so that the Hoek-Brown failure criterion is applicable.

2.7.2 Mohr-Coulomb criterion

Since most geotechnical engineering software is still written in terms of the Mohr-

Coulomb failure criterion, it is necessary for practising engineers to determine

equivalent friction angles and cohesive strengths for each rock mass and stress range. In

the context of this thesis, the solutions obtained by using equivalent Mohr-Coulomb

parameters can be compared directly with the solutions by using the native Hoek-Brown

failure criterion.

Figure 2.9 is an illustration of the Hoek-Brown criterion and equivalent Mohr-Coulomb

envelope. Because the equivalent Mohr-Coulomb envelope is a straight line, it can not

fit the Hoek-Brown curve completely. If we divide Figure 2.9 into three zones, namely

REGION 1, REGION 2, and REGION 3, it can be seen that when rock stress

conditions fall in REGION 1 and REGION 3, using equivalent Mohr-Coulomb

parameters may overestimate the ultimate shear strength when compared to the Hoek-

Brown curve. Regarding the fitting process, more details can be found in Hoek et al.

(2002) where the process involves balancing the areas above and below the Mohr-



2-26

Coulomb plot over a range of minor principal stress values. This results in the following

equations for friction angle and cohesive strength:

216121

1211'

3

1'3

'3'

nbb

nbnbci

msm

msmsc (2.8)

1'3

1'31'

6212

6sin

nbb

nbb

msm

msm (2.9)

where cin 'max33 .

It should be noted that the value of 'max3 has to be determined for each particular

problem. For slope stability problems, Hoek et al. (2002) suggest 'max3 can be

estimated by the following equation:

91.0'

'

'max3 72.0

Hcm

cm

(2.10)

in which H is the height of the slope and is the material unit weight. For the stress

range, 4'3 cit , the compressive strength of the rock mass '

cm can be

determined as:

212

484 1' smsmsm bbb

cicm (2.11)

2.7.3 Douglas criterion

Douglas (2002) proposed a failure criterion for estimating the shear strength of weak

rock masses at low stress level (e.g. rockfill dams) based on a large number of

experimental tests. The data consists of 4507 test results from 511 sets obtained from

the literature and original laboratory test reports. It should be noted that, for the Douglas

criterion, the rock mass disturbance is considered by observing rock mass surface

condition which is incorporated into GSI evaluations.



2-27

For the Douglas criterion, the basic form of the shear strength equation remains

unchanged from the Hoek-Brown criterion (Equation (2.2)). For intact rock im m and

i . The magnitudes of im and i were presented as:

cii

ti

m

(2.12)

1.2

0.41 exp 7i

im

(2.13)

where ti is tensile strength of intact rock. Estimation of bm , b and bs can be made

using following equations:

max 1002.5

ib

GSIm

m

(2.14)

75 300.9 exp b

b i ii

m

m

(2.15)

85exp

min 15

1b

GSIs

(2.16)

As highlighted by Douglas (2002), this criterion can still be used to estimate cohesion

and friction angle because the form of the Hoek-Brown yield criterion (Equation (2.2))

remains unchanged.

2.7.4 Other empirical criteria for rock masses

Yudhbir criterion (Bieniawski)

The Yudhbir criterion (Yudhbir et al. (1983)) was developed for encompassing the

whole range of conditions varying from intact rock to highly jointed rock, which covers

the brittle to ductile behaviour range. The criterion for rock masses was written in the

more general form



2-28

''31

c c

A B

(2.17)

where A is a dimensionless parameter whose value depends on the rock mass quality.

A is equal to 1 for intact rock and equal to 0 for totally disintegrated rock masses. B is

a rock material constant and depends on the rock type. The value of which is

independent of rock type and rock mass quality was suggested to be 0.65 by Yudhbir

criterion (Yudhbir et al. (1983)). B has low values for soft rocks and high values in the

case of hard rocks (Table 2.2).

The value of A was back-fitted with Equation (2.17), using 0.65 and appropriate

value according to Table 2.2. The values of A fitted with a specific B are obtained in

Table 2.3.

The parameter A is correlated to Q-system (Barton (2002)) and the Rock Mass Rating

(Bieniaski (1976)) as follows

0.0176A Q (2.18)

and

100exp 7.65

100

RMRA

(2.19)

To get better results for A and B , Bieniaski and Kalamaras (1993) suggested that these

should both be varied with RMR . They proposed the criterion in Table 2.4, specifically

for coal seams with 0.6.

Criterion of Sheorey et al.

After Sheorey (1997), the criterion of Sheorey et al. (1989) shown in Equation (2.20)

was linked with RMR values. Relations (2.21) - (2.23) were recommended

31 1

mb

cmtm

(2.20)

76 100exp

20cm c

RMR

(2.21)



2-29

76 100exp

27tm t

RMR

(2.22)

76 100 , 95RMRm mb b b (2.23)

The following procedure should be used to determine the 76RMR value:

1. For 1876 RMR , use the 1976 version of the RMR system.

2. For 1876 RMR , determine the Q -value and use Bieniawski’s relation

44ln9 ' QRMR (2.24)

where

a

r

n J

J

J

RQDQ '

nJ , rJ and aJ are the parameters of Q-system (Barton (2002)) which represent joint

number, joint roughness number and joint alternation number, respectively.

Ramamurthy (1995) criterion for jointed rock

Ramamurthy (1995) proposed the following expressions for the shear strength of rock

masses.

ja

cjjB

3331

(2.25)

where

fccj J008.0exp

c

cjij

BB

037.2exp13.0

c

cjij aa



2-30

nr

JJ n

f

Both the r and n parameters are graded values given in tables while the joint frequency

is a field observation and estimation parameter, which receives the same value as in the

observation. Values of n for different joint inclination are given in Table 2.5. The

parameter for joint strength ( tanr ) is the coefficient of friction shown in Table 2.6.

Based on Ramamurthy’s suggestion, the values of 32ia and 313.1 tciiB can

be estimated by using the Brazilian test.

2.8 SUMMARY

A number of conclusions can be made from the forgoing review into the investigations

of soil and rock slope stability:

1. There are two significant limitations in the LEM: (1) A large number of

assumptions have to be introduced to render the problem statically determinate,

including assuming the shape of the failure mechanism a priori, and (2) The

method generally produces a set of non-linear simultaneous equations and

therefore an iterative procedure is necessary to obtain a solution unless further

simplifications are introduced. As a consequence, the inherent limitations of the

limit equilibrium analysis still remain in the 3D solutions. In view of the

underlying assumptions that it makes, the accuracy of limit equilibrium

approach is often questioned, even being the generally used method to estimate

slope stability.

2. The majority of past research has been focused on case studies where slope

failure has already occurred. For these cases, the determination of the residual

strength and the dissipation of the excess pore pressure are the main subjects of

investigation. Therefore, a back analysis can be performed to obtain the factor of

safety in order to evaluate the slope stability for further design purposes.

Although those studies are invaluable for identifying the failure modes, the

results may be very site-specific. This implies that the approach is still to a large

extent empirical. It is difficult to generalise site-specific cases, and thus there are

few comprehensive parametric studies for soil and rock slopes.



2-31

3. Very few rigorous numerical studies have been undertaken to determine the

stability of slopes. Most methods of analysis are based upon the initial

assumption of a particular failure mode (limit equilibrium method and upper

bound limit analysis). Given that few attempts have been made to accurately

monitor internal soil deformations under laboratory conditions, the validity of

the assumed failure mechanisms remain largely unproven. A rigorous numerical

study of slope stability using advanced numerical methods is clearly needed.

4. Except for chart solutions for 3D slopes in homogeneous purely cohesive soil,

no attempt has been made for 3D slopes in non-homogeneous purely cohesive

soil and cohesive-frictional soil, even though they are common field

characteristics. A full 3D study of slope stability in this thesis would quantify

3D boundary effect and lead to an economical design.

5. Although a number of researchers have proposed sets of stability charts for rock

slopes, the author is unaware of any investigations directly based on the specific

criteria for rock masses.



2-32

Table 2.1 3D slope stability by LEM after Duncan (1996)

Table 2.2 Typical value of parameter B (Yudhbir et al. (1983))

Rock type Tuff, Shale

Limestone

Siltstone,

Mudstone

Quartzite,

Sandstone Dolerite

Norite, Granite,

Quartzdiorite, Chert

B 2 3 4 5



2-33

Table 2.3 Parameter A and B for the Yudhbir et al. (1983) yield criterion

Description A B

Model specimen:

intact 1.0 1.9

crushed ( 31.65t m ) 0.3 1.9

crushed ( 31.25t m ) 0.1 1.9

Indian limestone 1.0 1.93

Westerly granite

intact 1.0 4.9

broken

stat VI 0.25 4.9

stat VII 0.075 4.9

stat VIII 0.0 4.9

Phra Wihan sandstone

Lopburi 1.0 6.0

Butitam 1.0 4.0



2-34

Table 2.4 RMR -dependent rock mass failure criteria for coal seams after Bieniaski and

Kalamaras (1993)

Equation Parameters

0.6

31 4c c

A

100

exp , 414

RMRA B

0.6

31

c c

A B

20exp

52

RMRB

100exp

14

RMRA

0.6

31 1 4cm cm

100exp

24cm c

RMR

Table 2.5 Simplified rock mass rating (Ramamurthy (1995))



2-35

Table 2.6 Adjustment factors for in-situ rating components (Ramamurthy (1995))



2-36

Figure 2.1 Slope height versus slope angle relation for hard rock slopes after Hoek and

Bray (1981)



2-37

Figure 2.2 Rotational shear failures and combinations of rotational shear and plane

shear failures

Figure 2.3 Three dimensional failure geometry of rotational shear failure



2-38

Figure 2.4 Combinations of discontinuities forming a failure surface



2-39

Figure 2.5 Crushing, toppling and buckling failures

Crushing failure

Buckling failure

Toppling failure



2-40

Residual strength

Peak strength

Shear stress

Shear displacement

Figure 2.6 Peak and residual shear strength

1.00 1.05 1.10 1.15 1.20 1.25 1.300.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

M 6.5

Seed (1979)

Recommended Pseudo-Static Safety Factor

Pse

udo-

Sta

tic c

oeff

icie

nt, k

h

Hynes & Franklin (1984) M 8.25

Figure 2.7 Design recommendations for pseudo-static analysis



2-41

GROUP IIIGROUP II

JOINTED ROCK MASS

SEVERAL DISCONTINUITIES

TWO DISCONTINUITIES

SINGLE DISCONTINUITIES

INTACT ROCK

GROUP I

Jointed Rock

ci GSI, m

i,

Figure 2.8 Applicability of the Hoek-Brown failure criterion for slope stability

problems

c' , ' p c' underestimated

' overestimated

p overestimated

REGION 3REGION 2

She

ar s

tres

s (

)

Normal stress ()

Hoek-Brown Mohr-Coulomb (best fit)

REGION 1

c' overestimated

' underestimated

p overestimated

Figure 2.9 Hoek-Brown and equivalent Mohr-Coulomb criteria



2-42



3-1

CHAPTER 3 NUMERICAL FORMULATIONS

3.1 INTRODUCTION

In the thesis, four different numerical methods have been used to determine the stability

of soil and rock slopes. These are the upper and lower bound limit analysis methods,

displacement finite element method, and limit equilibrium method. Initially, this

Chapter will provide some background by briefly discussing these techniques along

with a number of other methods available for analysing problems in geomechanics. It is

hoped that this will provide an insight into the advantages and disadvantages of the

numerical approaches currently in use by engineers.

The upper and lower bound methods of limit analysis have been used extensively in the

thesis and a thorough discussion is appropriate. Although many engineers are now

familiar with the finite element concept, fewer have a detailed knowledge of limit

analysis. Consequently, the middle part of this Chapter provides some background to

selected aspects of classical plasticity, including the limit analysis theorems.

In the remaining sections of this Chapter, the nonlinear implementation of the lower and

upper bound theorem by Lyamin and Sloan (2002a), Lyamin and Sloan (2002b) and

Krabbenhoft et al. (2005) will be presented.

3.2 NUMERICAL METHODS IN GEOMECHANICS

At present, there are a number of techniques available for use by geotechnical

practitioners and researchers when analysing geotechnical problems. These techniques

include:

Limit equilibrium method (LEM)

Slip line method/method of characteristics

Limit analysis

Displacement finite element method (DFEM)



3-2

The fundamental requirements for an exact theoretical solution need to be mentioned.

This will provide a framework in which the different methods of analysis may then be

compared.

In general, there are three basic conditions needed for the solution of a boundary value

problem in the mechanics of deformable solids: the stress equilibrium equations, the

stress-strain relations and the compatibility equations relating strain to displacement.

When performing plastic analysis, an infinity of stress states will satisfy the stress

boundary conditions, the equilibrium equations and also the yield criterion alone, and an

infinite number of displacement modes will be compatible with a continuous distortion

of the continuum and satisfy the displacement boundary conditions. Here, as in the

theory of elasticity, use has to be made of the stress-strain relations to determine

whether the stress and displacement states correspond to each other and if a unique

solution exists. In incremental plasticity, there are generally three phases in solid body

behaviour as the loads are applied: an initial elastic response, an intermediate stage of

contained plastic flow and, finally, a state of collapse due to unrestricted plastic flow.

The complete solution which satisfies all the just mentioned needed criteria can be

cumbersome. To enable more easily obtained solutions, approximations are generally

introduced by either relaxing certain constraints required for a complete solution, or by

making mathematical approximations. The first three methods listed above fall into this

category.

Historically, the bulk of analysis in geotechnical engineering has been carried out using

the first two techniques considered. The limit equilibrium method is generally the most

widely used in practice due to its simplicity and generality. The method can be used to

deal with problems with complicated boundary conditions, soil properties and loading

conditions. However, the accuracy of the method is often questioned, particularly for

slope and retaining wall analyses, due to the underlying assumptions that it makes. The

method requires a failure mechanism to be assumed which may consist of plane,

circular or logspiral shaped surfaces. In addition, the method gives no consideration to

soil kinematics and thus compatibility requirements are ignored. The example of the

application of limit equilibrium method, is the Coulomb’s earth pressure theory. A

solution, in this case for the active/passive thrust on the wall, can be obtained by

satisfying overall equilibrium in terms of the stresses and forces.



3-3

Whilst the slip-line method has the advantage of being mathematically rigorous, it is

notoriously difficult to apply to problems with complex geometries or complicated

loadings. A further shortcoming is that the boundary conditions need to be treated

specifically for each problem, thus making it impossible to develop a general purpose

computer code which can analyse a broad range of cases. Despite these limitations, slip-

line analysis has provided many fundamental solutions that are used routinely in

geotechnical engineering practice.

Although both the limit equilibrium method and slip-line method are considered to give

satisfactory results for a wide range of problems, they only satisfy the requirements for

a valid solution in a limited sense. The ability of the limit equilibrium and slip-line

methods to satisfy the fundamental requirements, and provide design information, is

summarised in Table 3.1 after Potts and Zdravkovic (1999).

The displacement finite element technique is now widely used for predicting the load-

deformation response, and hence collapse, of geotechnical structures. Table 3.1

indicates this method satisfies all the theoretical requirements for a valid solution. This

technique can deal with complicated loadings, excavation and deposition sequences,

geometries of arbitrary shape, anisotropy, layered deposits and complex stress-strain

relationships.

Clearly there is a place for methods which determine the limit load directly without the

need to trace the complete load-deformation history. This is the niche for the limit

theorems which are able to simplify the problem yet still make a definite statement

about the collapse load. By ignoring the fundamental requirements of compatibility

(lower bound) and equilibrium (upper bound), these theorems can be used to bracket the

true collapse load from above and below. The upper bound theorem is based on the

notion of a kinematically admissible velocity field, while the lower bound theorem is

based on the notion of a statically admissible stress field. A kinematically admissible

velocity field is simply a failure mechanism in which the velocities (displacement

increments) satisfy both the velocity boundary conditions and the flow rule, whilst a

statically admissible stress field is one which satisfies equilibrium, the stress boundary

conditions, and the yield criterion.



3-4

From a design perspective, each method of analysis has its advantages and

disadvantages; which method is adopted will depend largely on the problem at hand and

the design requirements. With the exception of the displacement finite element method,

all the methods discussed above fail to satisfy at least one of the fundamental

requirements of a valid solution and therefore can only provide limited design

information. A distinct disadvantage of the limit equilibrium, slip line and limit analysis

methods is that they do not provide information on displacements under working

conditions. However, given that collapse is catastrophic, it is easy to understand why

simple methods for predicting the stability of geomechanics problems are still of

primary importance in geotechnical design.

In the thesis, the theorems of limit analysis, the displacement finite element method and

limit equilibrium method will be used to estimate the ultimate slope stability for a wide

range of problems. This will provide a rigorous and thorough insight into the ultimate

stability of soil and rock slopes.

3.3 THEORY OF LIMIT ANALYSIS

One of the most important developments in plasticity theory was undoubtedly the

establishment of the upper and lower bound theorems of limit analysis by Drucker,

Greenberg and Prager in 1952 (Drucker et al. (1952)). It was, however, recognised

before this work that these theorems are deducible from the work principles published

in Hill (1950), and the earliest reference to them can be found in Gvozdev (1936).

The methods of limit analysis assume a perfectly plastic model with an associated flow

rule. The latter, which is also known as the normality principle, implies that the plastic

strain rates are normal to the yield surface and is central to the derivation of the two

limit theorems. Within the framework of these assumptions, limit analysis is rigorous

and the solution techniques are, in some instances, much simpler than those which are

based on incremental plasticity. In the context of soil mechanics, the use of an

associated flow rule with a Tresca yield criterion is appropriate for undrained

deformation, where the soil deforms at constant volume. For drained deformation,

however, the use of an associated flow rule with a Mohr-Coulomb soil model usually

gives excessive volume changes during plastic failure. Consequently, it is often thought

that the limit theorems are inappropriate tools for predicting drained collapse loads. As



3-5

pointed out by Davis (1968), this conclusion is true only for a restricted number of cases

where the field of plastic flow is subject to strong kinematic constraints (such as flow in

a silo). For problems involving semi-infinite boundaries and a free surface, which occur

frequently in soil mechanics, the precise form of the flow rule does not usually have a

marked effect on the collapse load, and the limit theorems are very useful tools for

performing drained stability analysis.


of geotechnical structures, they have not been widely applied to the problem of slope

stability. Due to the inherent difficulty in manually constructing statically admissible

(lower bound) stress fields and kinematically admissible (upper bound) velocity fields,

simple hand solutions rarely bracket the slope stability to sufficient accuracy. A major

aim of the thesis is to take full advantage of the ability of recent numerical formulations

of the limit theorems to bracket the actual collapse load accurately from above and

below. For all applications in the thesis, the lower and upper bounds are computed using

the nonlinear numerical techniques developed by Lyamin and Sloan (2002a), Lyamin

and Sloan (2002b) and Krabbenhoft et al. (2005).

In view of the uncertainties inherent in many engineering problems, and the essential

role of judgement in their solution, it follows that the approximate nature of limit

analysis is not a severe handicap. The major source of error in this method arises from

the assumptions that are made about the behaviour of the real material, which often

exhibits some degree of work softening or hardening and a non-associated flow rule.

Since these assumptions determine the range of validity of the theory, it is appropriate

to summarise them in detail.

3.3.1 The assumption of perfect plasticity

Figure 3.1 shows some typical stress-strain diagrams for deformable solids. The stress-

strain behaviour of overconsolidated clays and dense sands, under conditions of direct

shear, is typically characterised by an initial (roughly) linear response, a peak stress, and

a softening to a residual stress (Figure 3.1(a)). For normally consolidated clays and

loose sands, the degree of softening is less pronounced and may be replaced by an

asymptotic approach to the limiting strength. Under uniaxial tension, a typical stress-

strain diagram for a metal comprises a linear elastic response followed by a strain



3-6

hardening path, as shown in Figure 3.1(b). In an elastic perfectly plastic model, the

strain softening or hardening features of the stress-strain diagram are ignored and the

behaviour is approximated by the two dashed lines shown in Figure 3.1. A material

which exhibits the property of continuing plastic flow at constant stress is often said to

be perfectly plastic.

Note that the material strength used in limit analysis may be chosen to represent the

average strength over an appropriate range of strain. This increases the validity of the

perfectly plastic assumption and permits realistic estimates to be made of the collapse

load. As the choice of strength is not an absolute one, it may be determined in

accordance with the most significant features of the problem to be solved.

3.3.2 The stability postulate of Drucker

Consider the symbolic uniaxial stress curves in Figure 3.2. According to Drucker et al.

(1952), there are two different classes of material behaviour:

(1) The material is classed as stable if the stress is uniquely determined from the

strain, and vice versa (Figure 3.2(a), (b) and (c)). For materials of this type, a

stress increment always does positive work starting from any point along the

stress-strain curve.

(2) The material is classed as unstable if the stress is not a unique function of the

strain (Figure 3.2(e)) or, conversely, if the strain is not a unique function of the

stress (Figure 3.2(d)). For these materials, a stress increment can do negative

work over some parts of the stress-strain curve.

Using this simple type of uniaxial stress-strain behaviour, Drucker generalised his

concept of a material stability according to

0 0pij ij ij

where 0ij are the initial stresses which are in equilibrium with the system of applied

forces, ij are the final stresses after the additional external forces have been added,

and pij are the plastic strain rates. Alternatively, Drucker’s stability postulate may be

written in the vector form



3-7

0 0T p (3.1)

where the current and initial stress vectors are

Tzxyzxyzyx ,,,,,

T

zxyzxyzyx0000000 ,,,,,

and the plastic strain rate vector is

, , , , ,Tp p p p p p p

x y z xy yz zx

3.3.3 Yield criterion

For a perfectly plastic material, the yield function, f , depends only on the set of stress

components ij and not on the strain components ij . Consequently, the yield function

is static in stress space and plastic flow occurs when

0ijf

By definition, stress states for which 0ijf are excluded while 0ijf implies

elastic behaviour.

The term yield surface is used to emphasise the fact that up to nine components of the

stress tensor ij may be taken as coordinate axes. In practice, it is helpful to visualise a

state of stress in nine-dimensional stress space as a point in a two dimensional plot, as

shown in Figure 3.3.

If the material is isotropic, plastic yielding depends only on the magnitudes of the three

principal stresses and not on their directions. Any yield criterion can thus be expressed

in the form

0,, 321 JJJf

where 1J , 2J and 3J are the first three invariants of the stress tensor ij .



3-8

3.3.4 Flow rule

In discussing the kinematics of plastic flow, we cannot say anything about the total

plastic strain pij because the magnitude of uncontained plastic flow is unlimited. Due to

this fact, it is necessary to describe the process of plastic flow in terms of the strain

rates, ij , rather than the actual strains ij . For a perfectly plastic material, the total

strain rate ij is assumed to be composed of elastic and plastic parts according to

e pij ij ij (3.2)

where the eij are related to the stress rates ij through Hooke’s law. For an isotropic

perfectly plastic material, it is assumed that the axes of the principal plastic strain rates

will coincide with the axes of the principal stresses. As shown in Figure 3.3, it is

convenient to use the axes of a yield surface plot to simultaneously represent plastic

strain rates as well as stresses, with each axis of ij being an axis of the corresponding

plastic strain rate component of pij .

For any stress increment ( 0 ) and plastic strain rate p , Drucker’s stability

postulate is satisfied provided the yield surface is convex and

pij

ij

f

(3.3)

where is a non-negative scalar known as the plastic multiplier rate. Because Equation

(3.3) implies that the plastic strains are normal to the yield surface, this type of flow rule

is often said to be associated (with the yield surface) and obey the normality principle.

To prove that convexity and normality are sufficient to guarantee satisfaction of

Drucker’s postulate, we note, with reference to Figure 3.3, that Equation (3.3) is

equivalent to

0 0 0

22 2 2cos cos 0

T p p f (3.4)

where Tzxyzxyzyx fffffff ,,,,, is the

gradient to the yield surface and is the angle between the two vectors ( 0 ) and



3-9

p . Since 0 for normality, Equation (3.4) implies that the geometry of the yield

surface is such that 0cos . This condition, which imposes the constraint that must

not be larger than 90 , is automatically satisfied for a convex yield surface which

contains the initial and final stress states.

Once the plastic multiplier rate is known, the plastic strain rates pij can be obtained

from Equation (3.3) and the total strain rate can be computed using Equation (3.2).

3.3.5 Small deformations and equation of virtual work

The equation of virtual work deals with two separate and unrelated sets of variables,

those defining an equilibrium stress field and those defining a compatible deformation

field, and may be expressed as

i i i i ij ijA V VT u dA Fu dV dV (3.5)

In the above, the integration is over the surface area A and volume V of the body and

the tensor ij is any set of stresses, real or otherwise, that is in equilibrium with the

body forces iF and the external surface tractions iT . Referring to Figure 3.4(a), an

equilibrium stress field must satisfy the following equations:

jiji nT (at surface points)

0

j

j

ij Fx

(at interior points)

jiij

where in is the unit outward normal to a surface element at any point.

Of the remaining terms in Equation (3.5), the strain rate ij represents any set of strains,

compatible with the real or imagined (virtual) displacement rate iu , which arises from

the application of iF and iT . Referring to Figure 3.4(b), a compatible deformation field

must satisfy the compatibility relation:



3-10

2 jiij

j i

uu

x x

It is important to emphasise that neither the equilibrium stress field, nor the compatible

deformation field, needs to correspond to the actual state. Moreover, these fields do not

need to be related to one another.

3.3.6 The limit theorems

In what follows, the limit load is defined as the plastic collapse load applied to a body

having ideal properties. Using the rate form of the virtual work equation, it can be

shown that all deformation at collapse is purely plastic (see, for example, Chen (1975)).

This feature implies that the elastic properties play no part in collapse, and is used to

establish the limit theorems.

The lower bound limit theorem of Drucker et al. (1952) may be stated as follows:

If a stress distribution sij can be found which satisfies equilibrium, balances the

applied tractions iT on the boundary TA , and does not violate the yield condition so

that 0sijf , then the tractions iT and body forces iF will be less than, or equal to,

the actual tractions and body forces that cause collapse.

The upper bound limit theorem of Drucker et al. (1952) may be stated as follows:

If a compatible plastic deformation field ( ,pk pkij iu ) can be found which satisfies the

velocity boundary condition 0piu on the boundary uA and the normality condition

pkij ijf , then the tractions iT and body forces iF determined by equating the

rate of work of the external forces

T

pk pki i i iA V

T u dA Fu dV (3.6)

to the rate of internal dissipation

V

pkij

pkij

pkij dVD (3.7)

will be greater than, or equal to, the actual tractions and body forces.



3-11

3.4 LOWER BOUND FINITE ELEMENT LIMIT ANALYSIS

FORMULATION

The use of finite elements and linear programming to compute rigorous lower bounds

for soil mechanics problems appears to have been first proposed by Lysmer (1970).

More recently, Lyamin and Sloan (2002a) used nonlinear programming in the lower

bound analysis. Because of its modest memory demands, this type of lower bound

formulation can be run easily on a low-end workstation or a desktop computer.

As the finite element lower bound method is not widely known, and is often confused

with the conventional displacement finite element technique, an outline of its

formulation will now be given using a two dimensional (2D) example. The sign

conventions for the stresses, with tension taken as positive, are shown in Figure 3.5.

Three types of elements are employed, as depicted in Figure 3.6, and each of these

permit the stresses to vary linearly according to

3

1

i

ixiix N (3.8)

3

1

i

iyiiy N

(3.9)

3

1

i

ixyiixy N

(3.10)

where iN are linear shape functions and ( xyiyixi ,, ) are nodal stresses. The iN may

be expressed in terms of the nodal coordinates ( ii yx , ) according to

AyxxyyxyxN 2322323321 (3.11)

AyxxyyxyxN 2133131132 (3.12)

AyxxyyxyxN 2211212213 (3.13)

where

2332 xxx 3223 yyy



3-12

3113 xxx 1331 yyy

1221 xxx 2112 yyy

and

313223132 yxyxA

is twice the triangle area. Note that the rectangular and triangular extension elements,

which enable a statically admissible stress field to be obtained for a semi-infinite

domain, are based on the same linear expansion as the 3-noded triangle.

Unlike the more familiar types of elements used in the displacement finite element

method, each node is unique to a single element in the lower bound mesh and several

nodes may share the same coordinates.

To broaden the range of stress fields that are available to a particular grid, statically

admissible stress discontinuities are permitted at all edges that are shared by adjacent

elements, including those edges that are shared by adjacent extension elements.

A rigorous lower bound on the exact collapse load is ensured by insisting that the

stresses obey equilibrium and satisfy both the stress boundary conditions and the yield

criterion. Each of these requirements imposes a separate set of constraints on the nodal

stresses. It should be noted that the lower bound technique described above is also

suitable for three dimensional (3D) problems.

3.4.1 Constraints from equilibrium conditions

With the sign convention of Figure 3.7, the equilibrium conditions for plane strain are

xyx

x y

(3.14)

xyxyy (3.15)

where can be of 0 or a portion of . is a portion of implies that the horizontal

force is a portion of vertical force such as for example used in pseudo-static method.



3-13

Substituting Equations (3.8) - (3.10) into the equilibrium Equations (3.14) and (3.15),

we see that the nodal stresses for each element are subject to two equilibrium constraints

of the form

bxa 1 (3.16)

where

122131132332

2112133132231 000

000

yxyxyx

xyxyxya

333111 ,,,,,, xyyxxyyxTx

,1 Tb

For each rectangular extension element, three additional equalities are necessary to

extend the linear stress distribution to the fourth node. These equalities are

2314 xxxx

2314 yyyy

2314 xyxyxyxy

and may be written as

02 xa (3.17)

where

IIIIa 2

100

010

001

I




3-14

The fourth node of the rectangular extension element is essentially a dummy node but is

necessary to permit semi-infinite stress discontinuities between adjacent extension

elements.

A stress discontinuity is statically admissible if the shear and normal stresses acting on

the discontinuity plane are continuous (only the tangential stress may jump). With

reference to Figure 3.5, the normal and shear stresses acting on a plane inclined at an

angle to the x-axis are given by the stress transformation equations

xyyxn 2sincossin 22 (3.18)

xyxy 2cos2sin2

1 (3.19)

A typical stress discontinuity between adjacent triangles is shown in Figure 3.7. It is

defined by the nodal pairs (1,2) and (3,4), where the nodes in each pair have identical

coordinates. Since the stresses in our model are assumed to vary linearly, the

equilibrium condition is met by forcing all pairs of nodes on opposite sides of the

discontinuity to have equal shear and normal stresses.

With reference to Figure 3.7, these constraints may be written as

21 nn 43 nn (3.20)

21 43 (3.21)

where the subscripts denote node numbers. Substituting Equations (3.18) and (3.19) into

Equations (3.20) and (3.21), the discontinuity equilibrium conditions become

03 xa (3.22)

where

TT

TTa

00

003



3-15

2cos2sin2

12sin

2

1

2sincossin 22

T


3.4.2 Constraints from stress boundary conditions

To enforce prescribed boundary conditions, it is necessary to impose additional equality

constraints on the nodal stresses. If the normal and shear stresses at the ends of a

boundary segment are specified to be (q1, t1) and (q2, t2), as shown in Figure 3.8, then

it is sufficient to impose the conditions

11 qn 22 qn (3.23)

11 t 22 t (3.24)

since the stresses are only permitted to vary linearly along an element edge.

Substituting the stress transformation Equations (3.18) and (3.19) into (3.23) and (3.24)

leads to four equalities of the general form

04 xa (3.25)

where

T

Ta

0

04

22114 ,,, tqtqbT

222111 ,,,,, xyyxxyyxTx

Note that Equation (3.25) can also be applied to an extension element to ensure that the

stress boundary conditions are satisfied everywhere along a semi-infinite edge.

In cases involving a uniform (but unknown) applied surface traction, it is necessary to

place additional constraints on the unknown stresses which are of the form



3-16

21 nn 21

Substituting the stress transformation equations, these conditions lead to two equalities

of the form

05 xa (3.26)

where

TTa 5

222111 ,,,,, xyyxxyyxTx

This type of constraint is required in the analysis of flexible foundations, for example,

and strictly speaking is a constraint on the applied loads.

3.4.3 Constraints from yield conditions

A key feature of lower bound formulation in Lyamin and Sloan (2002a) is the use of

nonlinear programming (NLP). For yield functions which have singularities in their

derivatives, such as Tresca and Mohr-Coulomb criteria, it is necessary to adopt a

smooth approximation of the original yield surface.

Figure 3.9(a) is an example of a composite yield function where a conventional (non-

smooth) Tresca criterion is combined with a von Mises cylinder to round the corners in

the octahedral plane. Another example, Figure 3.9(b), shows the use of a simple plane to

cut the apex off a cone-like yield surface. This type of cut-off is often used for

modelling no-tension materials such as rock, and leads to a cup-shaped surface. It

should be noted that these combinations can be different for different parts of the

discretized body.

Provided the stresses vary linearly, the yield condition is satisfied throughout an

element if it is satisfied at all its nodes. This implies that the stresses at all N nodes in

the finite element model must satisfy the following inequality:

0, 1, ,lijf l N (3.27)



3-17

Thus, in total, the yield conditions give rise to N non-linear inequality constraints

(considering composite yield criteria as one constraint) on the nodal stresses.

3.4.4 Formation of the objective function

For many plane strain geotechnical problems, we seek a statically admissible stress field

which maximises an integral of the normal stress n over some part of the boundary.

Denoting the out-of-plane thickness by h , these integrals are typically of the form

s nu dshQ (3.28)

where uQ represents the collapse load. For the case of Equation (3.28), the integration

can be performed analytically and after substitution of the stress transformation

equations, the collapse load uQ may be written as

xcQ Tu (3.29)

where Tc is known as the objective function vector since it defines the quantity which

is to be optimised.

Once the elemental constraint matrices and objective function coefficients have been

found using Equations (3.16), (3.17), (3.22), (3.25), (3.26), (3.27) and (3.29), the

various terms may be assembled to furnish the lower bound nonlinear programming

problem.

3.4.5 Lower bound nonlinear programming problem

After assembling the various objective function coefficients and equality constraints for

the mesh, and imposing the nonlinear yield inequalities on each node, the lower bound

formulation of Lyamin and Sloan (2002a) leads to a nonlinear programming problem of

the form

Nif

bAtoSubject

cMaximize

j

T

,,1,0

(3.30)

where c is a vector of objective function coefficients, is a vector of problem

unknowns (nodal stresses and possibly element unit weights), Tc is the collapse load,



3-18

A is a matrix of equality constraint coefficients, b is a vector of coefficients, if is the

yield function for node i , and N is the number of nodes.

3.5 UPPER BOUND FINITE ELEMENT LIMIT ANALYSIS

FORMULATION

An upper bound on the exact slope stability can be obtained by modelling a

kinematically admissible velocity field. To be kinematically admissible, a velocity field

must satisfy the set of constraints imposed by compatibility, the velocity boundary

conditions and the flow rule. By equating the power dissipated internally by plastic

yielding within the soil mass and sliding of the velocity discontinuities and the power

dissipated by the external loads, we can obtain a strict upper bound on the true limit

load. In linear upper bound formulation of Sloan and Kleeman (1995), the direction of

shearing of each velocity discontinuity is found automatically and need not be specified

a priori. A good indication of the failure mechanism can therefore be obtained without

any assumptions being made in advance. After Lyamin and Sloan (2002b), the nonlinear

programming (NLP) has been incorporated into the upper bound technique.

The three-noded triangle used in the upper bound formulation is shown in Figure 3.10.

Each node has two velocity components and each element has 3 stress components

( xyyx ,, ).

Within a triangle, the velocities are assumed to vary linearly according to

3

1

i

iiiuNu

(3.31)

3

1

i

iiivNv

(3.32)

where ii vu , are nodal velocities in the x - and y - directions respectively and iN are

linear shape functions defined by Equation (3.11) to Equation (3.13).

3.5.1 Constraints from plastic flow in continuum

To be kinematically admissible, and thus provide a rigorous upper bound on the exact

collapse load, the velocity field must satisfy the set of constraints imposed by an



3-19

associated flow rule. For plane strain deformation of a rigid plastic soil, the associated

flow rule is of the form

xx

u F

x

yy

v F

y

xyxy

v u F

x y

where 0 is a plastic multiplier rate and tensile strains are taken as positive. These

equations, together with the boundary conditions and flow rule relations for the velocity

discontinuities, define a kinematically admissible velocity field.

3.5.2 Constraints from yield condition

The only requirement for the stresses in the upper bound formulation is that they satisfy

the yield condition. For a perfectly plastic solid, it is thus given

0ijf (3.33)

As the element stresses are assumed to be constant, there is only one yield condition per

finite element. This implies that the stresses in the finite element model must satisfy the

following inequality:

0, 1, ,ij

ef e E (3.34)

0,f j J (3.35)

for all E elements. Thus, in total, the yield conditions give rise to E non-linear

inequality constraints on the element stresses.

3.5.3 Constraints due to plastic shearing in discontinuities

After Krabbenhoft et al. (2005), an assembly of triangular elements connected at the

nodes by a two-element patch of infinitely thin elements are adopted to present velocity

discontinuities (Figure 3.11).

Each triangle is a constant stress-linear velocity element. Then the velocities in triangle

k, l, m vary according to



3-20

mmllkk

mmllkk

vNvNvNv

uNuNuNu

(3.36)

where

, , .2 2 2

k k k l l l m m m

k l m

a b x c y a b x c y a b x c yN N N

(3.37)

in which

, ,

, ,

, ,

k l m m l k l m k m l

l m k k m l m k l k m

m k l l k m k l m l k

a x y x y b y y c x x



(3.38)

and where is the area of triangle. The compatibility matrix B is given by

0 0 01 1 1

0 0 02

k l m

k l m k l m k l m

k l mk l m

b b bc c c

c c cb b bB B B B B B B B

(3.39)

and the power dissipated in any triangular element regardless of its area calculated from

d d

W σε σBu σBu (3.40)

Considering now the case of infinitely thin element with side lm being collapsed we will

find that kB 0 and l mB B resulting in the compatibility matrix

lm lm B 0 B B (3.41)

where lB is replaced by lmB for notation convenience. It is readily seen that the strain

rate in element k, l, m in this case can be expressed in terms of differences between

velocities (velocity jumps) at nodes l and m.

lmlm ε Bu B u (3.42)

The yield criteria are applied in exactly the same way as for the continuum elements.

Since the state of stress is constant within each element, discontinuity elements are

treated no differently to the continuum elements.



3-21

3.5.4 Constraints due to velocity boundary conditions

To be kinematically admissible, the computed velocity field must satisfy the prescribed

boundary conditions. Consider a node i on a boundary which is inclined at an angle

to the x -axis. For the general case, where the boundary is subject to a prescribed

tangential velocity u and a prescribed normal velocity v , the nodal velocity

components ( iu , iv ) must satisfy the equalities

cos sin

sin cosi

i

u u

v v

These constraints may be expressed in matrix from as

3131 bxa (3.43)

where

cossin

sincos31a

vubT ,3 , iiT vux ,1

The above type of velocity boundary condition may be used to define the “loading”

caused by a stiff structure, such as a rigid strip footing or retaining wall.

For problems where part of the body is loaded by a uniform normal pressure, such as a

flexible strip footing, it is often convenient to impose constraints on the surface normal

velocities of the form

QvdSS

(3.44)

In the above, Q is a prescribed rate of flow of material across the boundary S and is

typically set to unity. This type of constraint, when substituted into the power expended

by the external loads, permits an applied uniform pressure to be minimised directly.

Since the velocities vary linearly, Equation (3.44) may be expressed in terms of the

nodal velocities according to



3-22

Qluuvvedges

1212211221 sincos2

1

where 12l and 12 denote the length and inclination of each segment on S and each

segment is defined by the end nodes (1,2). This boundary condition may be written in

matrix form as

4141 bxa (3.45)

,cos,sin,cos,sin2

1121212121212121241 lllla

Qb 4 , ,,,, 22111 vuvuxT

3.5.5 Formation of objective function: Power dissipation in continuum

A key feature of the formulation is that plastic flow may occur in both the continuum

and the velocity discontinuities. The total power dissipated in these modes constitutes

the objective function and is expressed in terms of the unknowns. Within each triangle,

the power dissipated by the plastic stresses is given by

c x x y y xy xyAp dA

Since the plastic multiplier rates are constrained, it follows that the power dissipated in

each triangle is always nonnegative.

3.5.6 Upper bound nonlinear programming problem

Since Krabbenhoft et al. (2005) proved that the power dissipation and flow rule do not

depend on element areas (triangular 1, 2, A and B) in Figure 3.11. The problem of

finding a kinematically admissible velocity field which minimizes the internal power

dissipation may be stated as



3-23

Ejf

Ejf

Ej

fBu

bAutoSubject

uonucBuQMinimize

j

jj

j

E

jj

TT

,1,0

,1,0

,1,01

(3.46)

where B is global compatibility matrix, c is a vector of objective function coefficients

for the velocities, N is the number of nodes in the mesh, A is a matrix of equality

constraint coefficients for the velocities, jf are yield functions, j are non-

negative multipliers, and u and are problem unknowns.

3.6 LIMIT ANALYSIS IMPLEMENTATION OF THE HOEK-

BROWN FAILURE CRITERION

In a similar manner to the Mohr-Coulomb failure envelope, the Hoek-Brown yield

surface has apex and corner singularities in stress space. The direct computation of the

derivatives at these locations, which are required for the non-linear programming (NLP)

solver, becomes impossible. This issue can be resolved using three different approaches;

namely, global smoothing, local smoothing and multi-surface representation (which

includes both a priori and dynamic linearisation). As the current study is limited to the

case of plain strain conditions, the corners are automatically avoided and the only

singularity which needs to be dealt with is the apex of the yield surface. The easiest

options to implement are a simple tension cutoff (which is a multi-surface technique) or

a quasihyperbolic approximation (which is a global smoothing technique).

The latter approach is adopted in this thesis, as a similar method has been previously

employed by Abbo and Sloan (1995) for smoothing the Mohr-Coulomb yield criteria.

The prefix “quasi” is used here because the Hoek-Brown yield surface is already curved

in the meridional plane and the suggested approximation is not a pure hyperbolic one. A

brief description of the procedure is provides as follows.

From Merifield et al. (2006), the Hoek-Brown yield function given in Equation (2.2) is

expressed in terms of stress invariants as



3-24

2 2 1HBf J g J h I

(3.47)

where 1I is the first stress invariant, 2J is the deviatoric stress invariant and is the

Lode angle related to the third stress invariant 3J , whereas parameters and , and

function g and h are given by the following expressions:

2cosg (3.48)

1 sincos

3b cih m

(3.49)

1b cim (3.50)

1cis (3.51)

Next, quasi-hyperbolic smoothing is applied by permuting 2J with a small term

according to

22 2J J (3.52)

on condition that is related to tensile strength of material by the rule

min , | 0 0 0g h

(3.53)

The constants and must be chosen to balance the efficiency of the NLP solver

against the accuracy of the representation of the original yield surface. The values used

in the current study are 610 and 110

The resulting approximation of the Hoek-Brown yield criterion can be written as

2 2 1ˆ ˆ

HBf J g J h I

(3.54)

and is now a smooth and convex function in the meridional plane. An illustration of the

original and smoothed Hoek–Brown curves in the 1 2,I J plane for zero y is given

in Figure 3.13. It should be noted that the difference between the smooth approximation



3-25

and the original yield surface has been greatly exaggerated in this figure by selecting

values of and that are much larger than what was actually adopted. The original

and smoothed yield surfaces are almost indistinguishable when the actual values of

and are used.

3.7 DISPLACEMENT FINITE ELEMENT METHOD (DFEM)

The use of the finite element method is now widespread amongst researchers and

practitioners. Theoretically, the finite element technique can deal with complicated

loadings, excavation and deposition sequences, geometries of arbitrary shape,

anisotropy, layered deposits and complex stress-strain relationships.

The choice of modelling tool deserves some explanation. There are a vast number of

numerical methods available today, each with its own strengths and weaknesses.

Besides the general prerequisites, such as the ability to handle different slope

geometries, it is essential that the modelling tool allow the simulation of 2D and 3D

slope failures.

In this thesis, the commercial displacement finite element software, ABAQUS, was

employed to make comparisons to the solutions obtained from the numerical upper and

lower bound limit analysis. For the ABAQUS analyses, the soils have been modelled as

a linearly elastic-perfectly plastic (Figure 3.1) isotropic material with the Mohr-

Coulomb failure criterion (Figure 3.14). The yield envelope has been defined in

Equation (3.55).

' 'tanc (3.55)

Generally speaking, the soil masses can be seen as continuous material, and therefore

continuum model is well suited to analyse the stability of soil slopes. In addition, it is

possible to simulate shear band in the model if the number of elements in the mesh is

large enough. The location of the shear band corresponds to the location of the failure

surface in the material. Unfortunately, one of the major difficulties with continuum

models is the correct representation of the shear bands in a material. The mesh used in

the models can affect both the orientation of the shear bands and the band thickness.

Most commercially available programs cannot simulate the real band thickness very



3-26

well, whereas the problem with correct orientation and location of the shear band can be

overcome to some extent with careful mesh generation.

3.8 LIMIT EQUILIBRIUM METHOD

3.8.1 Introduction

Hoek and Brown (1980a) proposed a method for obtaining estimates of the strength of

jointed rock masses, based upon an assessment of the interlocking of rock blocks and

the condition of the surfaces between these blocks. This method was modified over the

years in order to meet the needs of users who were applying it to the problems. The

generalised Hoek-Brown failure criterion for jointed rock masses is defined by Equation

(2.2). The Mohr envelope, relating normal and shear stresses, can be determined by the

method proposed by Hoek and Brown (1980b). In this approach, Equation (2.2) is used

to generate a series of triaxial test values, simulating full scale field tests, and a

statistical curve fitting process is used to derive an equivalent Mohr envelope defined by

the equation

' B

n tmci

ci

A

(3.56)

where A and B are material constants, 'n is the normal effective stress, and tm is the

tensile strength of the rock mass.

This tensile strength, which reflects the interlocking of the rock particles when they are

not free to dilate, is given by:

2 42ci

tm b bm m s

(3.57)

In order to use the Hoek-Brown criterion for estimating the strength and deformability

of jointed rock masses, three ‘properties’ of the rock mass have to be estimated. These

are

1. The uniaxial compressive strength ci of the intact rock pieces.

2. The value of the Hoek-Brown constant im for these intact rock pieces.



3-27

3. The value of the Geological Strength Index (GSI ) for the rock mass.

In this thesis, limit equilibrium analysis in conjunction with Bishop’s simplified method

(Bishop (1955)) is employed to make comparisons with the numerical upper and lower

bound limit analysis for rock slopes. The used limit equilibrium software, SLIDE,

developed by Rocscience (2005) based on the Mohr-Coulomb yield or the generalised

Hoek-Brown criterion is described in the following sections.

3.8.2 Equivalent Mohr-Coulomb parameters in SLIDE

Because most geotechnical software is written in terms of the Mohr-Coulomb failure

criterion, the rock mass strength defined by the cohesion 'c and the angle of friction '

is required as input. The linear relationship between the major and minor principal

stresses, '1 and '

3 , for the Mohr-Coulomb criterion is

' '1 3cm k (3.58)

where k is the slope of line relating '1 and '

3 . The values of 'c and ' can be

calculated from

' 1sin

1

k

k

(3.59)

'''

1 sin

2coscm

c

(3.60)

There is no direct correlation between Equation (3.58) and the non-linear Hoek-Brown

criterion defined by Equation (2.2). Consequently, determination of the values of 'c and

' for a rock mass that has been evaluated as a Hoek-Brown material is a difficult

problem.

As highlighted by Hoek (2000), the most rigorous approach available, for the original

Hoek-Brown criterion, is that developed by Dr J.W. Bray and reported by Hoek (1983).

For any point on a surface of concern in an analysis such as a slope stability calculation,

the effective normal stress is calculated using an appropriate stress analysis technique.

The shear strength developed at that value of effective normal stress is then calculated

from the equations given in Hoek and Brown (1997). The difficulty in applying this



3-28

approach in practice is that most of the geotechnical software currently available

assumes for constant rather than dependent on effective normal stress values of 'c and

' .

The most practical solution is to treat the problem as an analysis of a set of full-scale

triaxial strength tests. The results of such tests are simulated by using the Hoek-Brown

(Equation (2.2)) to generate a series of triaxial test values. Equation (3.58) is then fitted

to these test results by linear regression analysis and the values of 'c and ' are

determined from Equations (3.59) and (3.60). The steps required to determine the

parameters A , B , 'c and ' are given below.

The relationship between the normal and shear stresses can be expressed in terms of the

corresponding principal effective stresses as suggested by Balmer (1952).

' ' ' ' ' '' 1 3 1 3 1 3

' '1 3

1

2 2 1n

d d

d d

(3.61)

' '1 3' '

1 3 ' '1 3 1

d d

d d

(3.62)

where

1' ' '1 3 31 b b cid d m m s

(3.63)

The equivalent Mohr envelop, defined by Equation (3.56), may be written in the form:

logY A BX (3.64)

where

logci

Y

, '

log n tm

ci

X

(3.65)

Using tm calculated from Equation (3.57) and a range of values and 'n calculated

from Equations (3.61) and (3.62), the magnitudes of A and B are determined by linear

regression where:



3-29

22

XY X Y TB

X X T

(3.66)

10 ^A Y T B X T (3.67)

where T is the total number of data pairs included in the regression analysis.

Hoek (2000) indicated that the most critical step in this process is the selection of the

range of '3 values. For a Mohr envelop defined by Equation (3.56), the friction angle

( 'i ) and cohesive strength ( '

ic ) for a specified normal stress 'ni is given by Equations

(3.68) and (3.69) respectively.

1'' arctan

B

ni tmi

ci

AB

(3.68)

' ' 'tani ni ic (3.69)

In limit equilibrium analysis, the software SLIDE will calculate a set of instantaneous

equivalent Mohr-Coulomb parameters for each slice based on the above method when

the Hoek-Brown criterion is selected. Therefore, the cohesion ( 'c ) and the friction angle

( ' ) will vary along any given slip surface. By calculating equivalent Mohr-Coulomb

parameters in this way, a more accurate representation of the curved nature of the Hoek-

Brown criterion in n space is obtained. However, when the Mohr-Coulomb

criterion is used, the cohesion ( 'c ) and friction angle ( ' ) are constant along any given

slip surface and are independent of the normal stress as expected. More details on how

the parameters are actually calculated can be found in Hoek (2000).

3.9 CONCLUSION

Brief details of the numerical limit analysis procedures used in the thesis have been

presented. These include the nonlinear finite element implementations of the upper and

lower bound theorems provided by Lyamin and Sloan (2002a), Lyamin and Sloan

(2002b) and Krabbenhoft et al. (2005). Several of the advantages and disadvantages of

each method of analysis have been highlighted.



3-30

Each of the numerical methods presented in this Chapter have been used to estimate the

slope stability for a wide range of problems. This is in contrast to past numerical studies

which typically present results from a single method of analysis. Comparing the results

obtained from several methods provides an opportunity to not only validate the findings

from each numerical procedure, but also provide a truly rigorous evaluation of slope

stability.



3-31

Table 3.1 Comparison of existing methods of analysis

Method of

Analysis

Solutions Requirements Design information

Stress

equilibrium Compatibility

Constitutive

behaviour Stability Displacements

Limit equilibrium (P)

Rigid-plastic

Slip-line method (P)

Rigid-plastic

Limit analysis

-Lower bound

- Upper bound

Perfectly-plastic

Displacement

finite element

Any

(P): Partial



3-32

Work softening

PerfectlyplasticPeak

Strain

Stress

Work hardening

Perfectly plastic

Strain

Stress

(a) Soils (b) Metals

Figure 3.1 Stress-strain relationships for ideal and real materials

(e)(d)

(c)(b)

(a)

Figure 3.2 Stable and unstable stress-strain curves: (a), (b) and (c) stable materials with

0 ; (d) and (e) unstable materials with 0



3-33

Smooth (unique gradient)

f (ij) = 0

Coner (non-unique gradient)

elasticf (

ij) < 0

Figure 3.3 Pictorial representation of yield surface and flow rules

Ti =

ijn

j

ni

AT

Au

Au

(a) Equilibrium Stress field (b) Compatible deformation field

Figure 3.4 Stress and deformation fields in the equation of virtual work

p

ij

.

p

ijij

.

,

ij

0ij

0ijij

ij

p

ijf

..

iF jiij

ij

ij Fx

0

iu.

iu.

i

j

j

iij

x

u

x

u

..

.

2



3-34

xy

xy

y

y

x

x

n

y

x

Figure 3.5 Stress sign convention

x1

, y1

, xy1

)

x2

, y2

, xy2

)x3

, y3

, xy3

)

3-noded triangular element

3-noded triangularextension element

direction ofextension

4-noded rectangularextension element

direction ofextension

1

3

2

43

2 1

3

2

1x

y

Figure 3.6 Elements for finite element lower bound analysis



3-35

x2

, y2

, xy2

)

x1

, y1

, xy1

)

x4

, y4

, xy4

)

n

y

x

2

1

3

x3,

y3,

xy3)

4

Figure 3.7 Stress discontinuity

prescribed shearstresses

prescribed normalstresses

q2

q1

t2

t1

x2

, y2

, xy2

)

x1

, y1

, xy1

)

n2

n2

y

x

1

2n1

n1

Figure 3.8 Stress boundary conditions



3-36

Figure 3.9 Internal linearization of Mohr-Coulomb yield function

u4, v

4)

uv

2

1

3

4

x, u

y, v u1, v

1)

u2, v

2)

u3, v

3)

Figure 3.10 Geometry of velocity discontinuity



3-37

Figure 3.11 Interpretation of discontinuous upper bound formulation in terms of

stresses

Figure 3.12 Discontinuity as a patch of interconnected thin elements

66 , vu

55 , vu

33 , vu

22 , vu

44 , vu

11 , vu

ux,

vy,

x

y

m

i

l

k

lu

lv



3-38

Figure 3.13 Meridian plane section of Hoek-Brown yield surface and its smooth

approximation



3-39

c

(b) plane

(b) Principle stress space

Figure 3.14 Mohr-Coulomb yield criterion

231

m

231

s

,

1

3



3-40



4-1

CHAPTER 4 SLOPE PROBLEM DEFINITION AND

NUMERICAL MODELLING

4.1 INTRODUCTION

One of the major outcomes of the research presented in this thesis are stability charts for

soil and rock slopes. The proposed stability charts are based on the Mohr-Coulomb and

the Hoek-Brown (Hoek et al. (2002)) failure criteria, respectively. Using these criteria,

factors that affect the stability of soil and rock slopes include soil strength ( 'c and ' ) or

rock strength ( ci , GSI , im and D ) and slope geometry.

Figure 4.1 shows a 3D illustration of the slope stability problem analysed. For a given

geometry ( , HL and Hd ), cohesion ( 'c , uc ) and friction angle ( ' , 0u ) are the

soil strength parameters when Mohr-Coulomb failure criterion is used. In addition, the

uniaxial compressive strength of rock mass ( ci ), geological strength index ( GSI ),

intact rock yield parameter ( im ) and disturbance factor ( D ) are the rock mass strength

parameters of the Hoek-Brown failure criterion for the rock slope analyses. An

overview of the slope problems investigated in the thesis is shown in Table 4.1 in which

the variables considered and the methods used are included. The definitions and

descriptions of the dimensionless parameters used in Table 4.1 can be found in the

Chapter relevant to that slope type.

For slope stability under earthquake loading, the conventional pseudo-static (PS)

approach is employed as it is still a widely accepted means for evaluating slope stability

under seismic loadings (Seed (1979), Chen et al. (2003) and Shou and Wang (2003)). In

a PS analysis, earthquake effects are simplified as horizontal and/or vertical seismic

coefficients ( hk and vk ). The magnitude of the coefficients is expressed in terms of a

percentage of gravity acceleration.

In the following sections, details of how the slope problem was modelled by the

numerical upper and lower bound limit analysis and the finite element analysis methods

are provided.



4-2

4.2 PLANE STRAIN LIMIT ANALYSIS MODELLING

4.2.1 Mesh details

Previous numerical studies (Sloan et al. (1990) and Yu et al. (1998)) using the

formulations of Sloan (1988) and Sloan and Kleeman (1995) have provided several

important guidelines for mesh generation. These indicated that the successful mesh

generation typically proceeds ensuring:

a) The overall mesh dimensions are adequate to contain the computed stress field

(lower bound) or velocity/plastic field (upper bound).

b) There is an adequate concentration of elements within critical regions.

The adaptive limit analysis techniques developed by Lyamin and Sloan (2002a),

Lyamin and Sloan (2002b) and Krabbenhoft et al. (2005) also follow the above

suggestions when generating upper and lower bound meshes. It should be noted that an

engineering appreciation of each problem will be advantageous when generating finite

element meshes. However, there are some points which need to be considered in a slope

stability analysis. Firstly, a greater concentration of elements should be provided in

areas where high stress gradients (lower bound), or high velocity gradients (upper

bound) are likely to occur. For the problem of slope stability, these regions are directly

related to the slope inclinations ( ) and the strength parameters of soil and rock masses.

Secondly, where possible, elements with severely distorted geometries should be

avoided. This is particular for some specific upper bound analyses, where such elements

can have a significant effect on the observed mechanism and collapse load.

In accordance with the above discussion, the final finite element mesh arrangements

(both upper and lower bound) were selected only after considerable refinements had

been made. The process of mesh optimisation followed an iterative procedure, and the

final selected mesh characteristics were those that were found to either minimise the

upper bound, or maximise the lower bound solution. This will have the desirable effect

of reducing the gap between two solutions, hence bracketing the actual collapse load

more closely.



4-3

4.2.2 Boundary conditions

For the slope stability problem considered in this thesis, soil unit weight is used to

define an objective function. The typical finite element meshes and boundary conditions

of the upper and lower bound limit analysis are shown in Figure 4.2(a) and (b),

respectively. For purely cohesive homogeneous soil slopes it was found that the depth

factor ( Hd ) can play an important role as it acts as a boundary to any slip surfaces.

Ratios of Hd between 1 and 5 have been taken into account and the lateral extent of

the mesh is that required to fully contain the plastic zones.

4.3 THREE DIMENSIONAL LIMIT ANALYSIS MODELLING

Currently, there are no widely accepted three dimensional stability analysis solutions for

soil and rock slopes available for practicing geotechnical engineers. In most cases it is

not feasible to perform a full displacement finite element analysis and as such the three

dimensional effects of the slope in question are often ignored which can lead to unsafe

solutions. In the back analyses of shear strengths, for example, neglecting the 3D effects

will lead to values that are too high, and therefore affecting any further stability

assessments performed with these data. As stated previously, one aim of this study is to

produce 3D stability charts that can be used by practicing engineers, extending those

currently used regularly for 2D slope stability evaluation.

A simplified representation of the upper and lower bound mesh arrangements and

boundary conditions used to analyse the 3D slope problem is given in Figure 4.3. These

meshes are typical for all analysed cases, but vary from case to case in some aspects.

Similar rules of mesh arrangement to the 2D limit analysis were considered. The overall

mesh dimensions were adjusted so that a statically admissible stress field for the lower

bound analysis and kinematically admissible the velocity/plastic field for the upper

bound analysis were maintained.

By taking symmetry into account, the overall problem size can be reduced. For a 3D

slope stability analysis, symmetry implies that x-z dimensions (Figure 4.3) can be

extended by a distance in the y direction to simulate the 3D condition. This requires the

values of the HL ratio to be adjusted to model a slope with various geometries for a

given and Hd . In Figure 4.3, the boundaries of domains coinciding with the planes



4-4

of symmetry are subject to the appropriate velocity and stress boundary conditions. For

3D slope analyses in this thesis, an HL ratio ranging from 1 to 5 was considered. This

was assumed adequate as Chugh (2003) observed that the difference between 2D and

3D safety factors tend to lose significance when 5HL .

4.4 DISPLACEMENT FINITE ELEMENT MODELLING

In this thesis, more conventional displacement finite element analysis will also be

performed using the commercially available software (ABAQUS) (Hibbitt et al. (2001)).

This allows for comparison and verification of the limit equilibrium and limit analysis

solutions. It should be noted that the finite element method (FEM) can be used to

compute displacements and stresses caused by applied loads. However, it does not

provide a value for the overall factor of safety without additional processing of the

computed stresses. A description of the mesh arrangement used and how slope failure

was determined is introduced in the following Sections.

4.4.1 Mesh arrangement

In general, when constructing a finite element mesh, the size and number of elements

depend largely on the material behaviour, since this influences the final solution. For a

linear material the procedure is relatively simple and only the zones where unknowns

vary rapidly need special attention. In order to obtain the best solutions, these zones

require a refined mesh containing smaller elements. The situation is more complex for

general nonlinear material behaviour since the final solution may depend, for example,

on the previous loading history. However, based on the results obtained from the FEM

estimates in this thesis, using a 6-node modified quadratic plane strain triangle for 2D

cases and 10-node modified quadratic tetrahedron for 3D cases will give reasonable

results, and elements with distorted geometries should be avoided.

Typical meshes for the 2D and 3D problem of slope stability are shown in Figure 4.4

where 1Hd . It can be seen that finer meshes are adopted in the region where the

slide would occur. The boundary conditions utilised in ABAQUS are the same as for the

upper bound limit analysis. The numbers of elements and nodes will be adjusted to be

adequate for the problem. Note that all meshes need to cover the zones of plastic

shearing and the observed displacement fields.



4-5

4.4.2 Initial stress conditions and optimization of slope failure

For slope stability problems, gravity loading is one of the key points that needs to be

taken into account. In the ABAQUS analyses, the vertical stress is simply hv

(where h is the depth below the surface) and the horizontal stress can be calculated

automatically by applying the gravity loading to the soil mass. The linear elastic-

perfectly plastic isotropic material model and Mohr-Coulomb failure criterion are

employed. It should be noted that the soil obeys the associated flow rule under

undrained loading and non-associated flow rule under drained condition. For the drained

cases, the dilation angle ( ) is assumed slightly smaller than the friction angle ( ' ) for

numerical reasons.

In general, the strength reduction method (SRM) (Griffiths and Lane (1999), Zheng et al.

(2006) and Hoek et al. (2000)) is one of widely used approaches to obtain the factor of

safety for a slope stability assessment using the FEM. However, it was found by Yu et

al. (1998) that optimisation of the stability number by unit weight ( ) or cohesion ( 'c )

achieves the same final result (for a given slope inclination ( ) and friction angle ( ' )).

Therefore, either the unit weight or the cohesion can be used as the means of obtaining

the factor of safety.

The process for obtaining accurate slope stability numbers is shown in Figure 4.5 for a

purely cohesive slope where H is the slope height and z is the vertical displacement

of the slope crest, as shown by point C in Figure 4.4. After applying the gravity loading

to the soil mass of the slope, it can be observed in Figure 4.5 that the downward vertical

displacement of point C increases significantly as the cohesion is reduced. This

displacement will increase rapidly at a certain point indicating slope collapse.

The analysis is performed by utilising the parametric study method in ABAQUS where

a small decrement of the cohesion ( 'c ) or increment of the unit weight ( ) is used to

obtain and observe the final vertical displacement of point C. The turning point (Point A)

of the optimised curve is the critical value which is used to define slope failure in the

FEM analyses and is used to calculate the slope stability number.



4-6

4.5 SUMMARY

In this Chapter, the slope stability problems in two and three dimensions have been

defined. It should be stated again that one of the aims of this study is to apply the

numerical upper and lower bound techniques to investigate the stability of 2D and 3D

slopes in soil and rock masses. The FEM and LEM are utilised for comparison purposes

for soil and rock slopes respectively. Details of the techniques used have been

mentioned in this Chapter. In addition, a summary of analysed problems versus used

techniques is shown in Table 4.1.



4-7

Table 4.1 Summary of analyses performed in this thesis

Slope type Variables Dimensionless parameter

used in this thesis

Methods used in analysis of

this thesis

Homogeneous slope in

purely cohesive soil

7.5 90 (UB and LB)

7.5 45 (FEM)

1HL

51Hd

h uN c HF UB, LB, FEM

Non-homogeneous cut

slope in purely cohesive

soil

10

15 90

1HL

21Hd

0uN HF c

0c uHF c UB, LB

UB = Upper bound LB = Lower bound FEM = Finite element method (ABAQUS), LEM = Limit equilibrium method (SLIDE)



4-8

Table 4.1 (continued)


used in this thesis


this thesis

Non-homogeneous natural

slope in purely cohesive

soil

10

7515

1HL

21Hd

0uN HF c

0c uHF c UB, LB

Homogeneous slope in

cohesive-frictional soil

353 (UB and LB)

10 35 (FEM)

7515

1HL

' 'tanc H

'tanF UB, LB, FEM



4-9



used in this thesis


this thesis

Natural rock slope under

static loadings

7515

10010 GSI

355 im

FHN ci

FGSI UB, LB, LEM

Natural rock slope under

seismic loadings

7515

10010 GSI

355 im

3.01.0 hk

FHN ci

FGSI UB, LB, LEM



4-10



used in this thesis


this thesis

Cut rock slope under static

loadings

7515

10010 GSI

355 im

0.10.0 D

FHN ci

FGSI UB, LB, LEM



4-11

N.B: L = for plane strain

L

Soil

cu ,

u=0 or c','

Toe

Rigid Base

d

Jointed Rock

ciGSI,m

iD

H

Figure 4.1 Problem configuration for 3D limit analysis modelling



4-12

u = v = 0

u =

v =

0

u = v =

0

(a) Upper bound

n = = 0 n

=

= 0

n = = 0

(b) Lower bound

Figure 4.2 Typical two dimensional finite element meshes and boundary conditions

used in limit analysis



4-13

u = v = w = 0(Upper bound)u = v = w = 0(Upper bound)

u = v = w = 0(Upper bound)

u = v = w = 0(Upper bound)

y, vx, u

z, w

v = 0Symmetric face(Upper bound)

(a) Upper bound

u = v = w = 0

n = = 0

(Lower bound)

y, vx, u

z, w

= 0Symmetric face(Lower bound)

(b) Lower bound

Figure 4.3 Typical three dimensional finite element limit analysis meshes and boundary

conditions



4-14

(a) 2D

(b) 3D

Figure 4.4 Typically used mesh in ABAQUS

C

C

Symmetric face



4-15

-0.04

-0.03

-0.02

-0.01

0.000.8 0.9 1.0 1.1 1.2

x

z

-z

cu ,

u=0

C

Unstable

cu / H

Poi

nt C

z / H

)A

Stable

H

Figure 4.5 Illustration of FEM slope failure optimisation



4-16



5-1

CHAPTER 5 SLOPE STABILITY OF PURELY COHESIVE

CLAYS

5.1 INTRODUCTION

In this Chapter, the stability of two dimensional and three dimensional homogeneous

and inhomogeneous purely cohesive slopes under undrained conditions is presented.

The typical 3D slope geometries for the problem of this Chapter are shown in Figure 5.1

where the x-z dimension can be extended by a distance in the y direction to simulate the

simplified 3D boundary. In general, the slope failure mode can be divided into three

types; 1) face-failure; 2) toe-failure and; 3) base-failure, as shown in Figure 5.1. In order

to describe the cross-section of base-failure, the parameter n in conjunction with the

failure surface of a slope will be employed to discuss and compare the presented results

in the following sections.

In general, the strength of cohesive soil is determined by the undrained shear strength

( uc ). In this study, uc is assumed constant throughout the slope or increasing with depth,

referred to as the homogeneous and inhomogeneous undrained slopes respectively. In

this study, a range of slope inclination ( ), depth factor ( Hd ) and HL ratios are

considered. The soil is modelled by the Mohr-Coulomb yield criterion with zero friction

angle due to the purely cohesive soil under undrained loading conditions. In order to

decrease the total number of elements, symmetry is exploited for the 3D cases. As

shown in Figure 5.1, the applied stress and velocity boundary conditions are given to

simulate the fixed and symmetric faces in the upper and lower bound analyses.

The stability of slopes in purely cohesive undrained clay is usually expressed in terms of

a dimensionless stability number in the following form

h uN c HF (5.1)

where hN is the stability number, is the unit weight, H is the slope height and F is

the safety factor of the slope. This form of stability number was firstly proposed by

Taylor (1937). It should be stressed that the upper bound stability number is smaller

than the lower bound as you optimise and it is in the denominater of Equation (5.1).



5-2

5.2 SOLUTIONS OF HOMOGENEOUS UNDRAINED

SLOPES

5.2.1 Numerical limit analysis solutions

Figure 5.2 shows a comparison of stability numbers between the solutions of the

numerical limit analysis methods and LEM for homogeneous cohesive undrained

slopes. The results from the LEM in Figure 5.2 were produced by Taylor (1948). It can

be seen that the stability numbers are bounded within a small range by the upper and

lower bound solutions. In addition, the trends in stability numbers obtained from these

methods are the same in that the stability number ( hN ) increases with increasing Hd

and decreasing .

The 2D and 3D chart solutions for homogeneous cohesive undrained slopes obtained

from the numerical upper and lower bound analysis are displayed in Figure 5.3 to

Figure 5.8 for a range of slope angles ( ), depth factors ( Hd ) and HL ratios. It is

noted that the upper and lower bound limit analysis solutions bracket a range of stability

numbers ( hN ) to within 5 to 9 % for 3D cases and 2.5 % for 2D cases. No

particular trend of the greatest difference in the upper and lower bound solutions

occurring was observed.

For the larger Hd ratios and 5L H , the line of the stability numbers should be flat

for 45 . However, the obtained results do not plot exactly as a flat line due to mesh-

dependency. Except when 1Hd , hN in Figure 5.4 to Figure 5.7 represent average

values of for the limit analysis solutions when 45 . It should be stated that the

difference in hN between the average and originally obtained values is less than

2.5 %.

As expected, the stability number hN increases when and the HL ratio increase.

For a given and Hd , hN achieves a maximum value when HL . This implies

that the factor of safety will reduce with increasing HL ratio. As known, the plain

strain analysis does not consider the resistance provided by the two curved ends of the

slip surface. The boundary resistance from these two curved ends can be seen as 3D end



5-3

boundary effects which makes the slope more stable. While increasing the HL ratio,

the contributions of resistances provided by these two curved ends decrease which

means that 3D end boundary effect reduces. Therefore, using 2D stability numbers will

lead to a more conservative slope design.

It should be noted that the magnitude of n is not shown in Figure 5.3 to Figure 5.7. This

is due to the fact that the plastic zones obtained from the upper bound analyses are less

precise than those from LEM or FEM. Using a finer mesh in the 3D analyses may help

but would make computations more time-consuming. Therefore defining an accurate n

value was found to be difficult. In addition, Gens et al. (1988) observed that there are

divergences between the actual and predicted n values.

Figure 5.8 presents the stability numbers ( hN ) obtained from the upper and lower

bound limit analysis for vertical slopes ( 90 ). Figure 5.8 can be used for estimating

the stability of shallow excavated slopes without retaining walls and props. Due to the

fact that the vertical slope is shallow, the soil properties can be seen as uniform. As

shown in Figure 5.8, the depth factor ( Hd ) does not play an important role for vertical

slopes. It is shown in Figure 5.8 that the stability number hN increases with HL ratio

increasing as well.

Figure 5.9 and Figure 5.10 show the 2D upper bound plastic zones for 2Hd and

5Hd respectively. It can be seen that the major failure mode is base-failure. The

transition of the failure modes is shown in Figure 5.11 where the failure mode can be

observed to change from base-failure to toe-failure as increases. In this part of the

study, all analyses indicate that base-failure is the primary failure type for purely

cohesive homogeneous slopes when 60 . This implies that the slip surfaces occur

from the slope crest and pass below the toe of the slope. On the other hand, from the 2D

plastic zones in Figure 5.11, the Hd ratio is found to have almost no effect on stability

numbers for 75 . Therefore, all chart solutions are flat lines for 75 in Figure

5.3 to Figure 5.8. It should be stressed that all of the propagated slip surfaces in Figure

5.9 and Figure 5.10 have reached the rigid bottom base. It can be concluded that a rigid

layer underneath the slope controls the slip surface for a 2D uniform undrained slope

with 60 .



5-4

For 15 and 5HL , the 3D plastic zones obtained from the upper bound limit

analysis for various depth factors ( Hd ) are shown in Figure 5.12. It can be observed

that the slip surface distance between the top and bottom increase slightly from the

rough face to the smooth face (symmetric face). The dominant 3D slope failure mode

for homogeneous undrained slopes is still of base-failure. In general, the depth of slip

surface increases with depth factor ( Hd ). However, it should be noted that the failure

surface for 5HL (Figure 5.12(c)) does not touch the rigid layer. Compared with the

2D case (Figure 5.10(c)), the depth of slip surface is found to be shallower. Figure 5.13

shows the failure surfaces for 5.22 , 5HL and the various Hd values.

Compared to Figure 5.12, it is found that the slip surface does not touch the rigid

bottom layer when 4HL . It implies that the boundary effect of the depth factor

( Hd ) reduces with increasing slope inclination for 3D homogeneous undrained slopes.

On the other hand, Hd plays a more important role for a slope with lower slope angle.

Figure 5.12(b) and Figure 5.14 show that the depth of slip surface varies when the ratio

of HL is changing. As expected, the depth of failure surface decreases with a

reduction of HL ratio. Again this is due to the 3D end boundary effect increasing. The

depth of slip surface changes significantly when the ratio of HL varies between 1 and

5. A transition of 3D failure mode is displayed in Figure 5.12(a), Figure 5.13(a) and

Figure 5.15. The failure mode is observed to change from base-failure to toe-failure

with increasing. Observations of plastic zones in Figure 5.11(a) and Figure 5.15(b)

demonstrate that 3D end boundary effects may influence the depth of the slip surface.

A comparison of the equivalent 2D and 3D cases can be made by investigating the

factor of safety ratio DD FF 23 for the same slope angle ( ), depth factor ( Hd ), slope

height ( H ), unit weight ( ) and undrained shear strength ( uc ). The ratio DD FF 23 is

also simply the inverse ratio of the stability numbers DhDh NN 32 . Figure 5.16

shows the average of the upper and lower bound ratios of DD FF 23 for various depth

factors ( Hd ) and slope angles ( ). In this figure, the magnitude of DD FF 23 denotes

the degree in which the 2D analysis underestimated the slope stability. It should be



5-5

acknowledged that the true ratio of DD FF 23 has been bracketed by the numerical upper

and lower bound analysis within a range of 5.7 %.

Referring to Figure 5.16, the ratio of DD FF 23 is found to increase with increasing

Hd , decreasing and decreasing HL . The comparisons of DD FF 23 between

7.5 and 45 show that slopes with higher depth factors ( Hd ) and the lower

slope angle ( ) result in more significant underestimates in the factor of safety. In

particular, for 7.5 and 5Hd , the ratio of DD FF 23 shows 3D factor of safety is

approximately 4.05 times to the 2D factor of safety from the upper and lower bound

solutions. Therefore, a very conservative design would be obtained by using 2D

solutions when the slope has a low slope angle ( ). However, the DD FF 23 ratio of

7.5 changes more significantly than that of 45 while Hd increases from 1

to 5. As discussed above, the difference in the depth of slip surface between various

HL ratios is less significant for a slope with higher slope angle. The change of the

DD FF 23 ratio should be influenced by the depth change of slip surface. Therefore, the

phenomenon that the DD FF 23 ratio of 7.5 changes more significantly than that of

45 for various Hd values could be due to the boundary effect of the depth factor

( Hd ) reducing with increasing slope inclination for 3D slopes.

It should be noticed in Figure 5.16 for 75 that the DD FF 23 ratio is almost

unchanging when 2Hd . Again this shows that the Hd ratio does not play an

important role for the cases with 75 . Based on the solutions presented in Figure

5.16, it is shown obviously that the factor of safety from a 3D analysis will be greater

than that from a 2D analysis for homogeneous undrained slopes. Therefore, the

statement made by Chen and Chameau (1982) is not valid for homogeneous undrained

slopes.

5.2.2 Solutions based on limit equilibrium method

Gens et al. (1988) presented a set of three dimensional stability charts based on the

conventional limit equilibrium analysis for homogeneous, isotropic purely cohesive

slopes. These provide a useful benchmark on the estimates obtained from the numerical



5-6

limit analysis approach and therefore in this Chapter, the chart solutions of Gens et al.

(1988) will be presented for comparative purposes.

The comparisons of stability number ( hN ) between the numerical limit analysis and the

limit equilibrium method (LEM) are displayed in Figure 5.2 and Figure 5.17 where

most solutions from LEM are found to fall within the upper and lower bound results.

The stability charts in Figure 5.18 are obtained based on the limit equilibrium analysis.

Except for the presented symbols, all lines in Figure 5.18 were originally presented by

Gens et al. (1988). The dash line was obtained by assuming the failure mode as toe-

failure.

5.2.3 Displacement finite element results (FEM)

In this Section, numerical estimates of stability number for 2D and 3D homogeneous

undrained slopes are obtained using the commercial displacement finite element method

(ABAQUS). These stability numbers are then be compared to those obtained using the

finite element upper and lower bound limit analysis theorems presented in the previous

Section.

As discussed in Section 2.4.3, very few displacement finite element analyses have been

performed to produce the chart solutions for homogeneous undrained slopes. A full

description of examining slope stability by using the commercial displacement finite

element method (ABAQUS) can be found in Section 4.4.

The comparisons of 2D and 3D stability numbers between the results of the FEM and

LEM solutions of Gens et al. (1988) are displayed in Figure 5.18. The influence of

various parameters on stability numbers for various slope angles ( 5.7 , 15 , 5.22 ,

30 , and 45 ) is shown. It is indicated that the most of stability numbers are larger than

those obtained by Gens et al. (1988) from the LEM, except a few points with depth

factors of 1Hd . The range of difference is between 1% and 10% and the largest

value of difference occurs with slope inclination 45 and 5HL (Figure

5.18(d)). These differences could be induced by the assumptions in limit equilibrium

analyses. From the comparisons of the stability numbers, it is shown that the stability

charts obtained by Gens et al. (1988) overestimate the factor of safety by around 10%.

This overestimation increases slightly with increasing slope inclination ( ) and HL



5-7

ratio. For the back analysis of a failed slope in practise, this difference may be important

and cannot be neglected. Therefore, corrections to the safety factor obtained are

required when the 3D stability charts based on the limit equilibrium analysis are used.

In addition, for low slope angles ( 5.7 ), the difference in stability numbers of Gens

et al. (1988) (LEM) and the FEM increases with depth factor ( Hd ).

Figure 5.2 and Figure 5.17 also show the stability number from the FEM compared to

those from the numerical upper and lower bound limit analysis. It can be found that

most of the FEM results are bracketed by the bounding solutions. The stability numbers

of the FEM are much closer to the results of the lower bound than those of the upper

bound. This phenomenon will be discussed in Section 6.4.1.

The ratio of safety factors ( DD FF 23 ) obtained from ABAQUS are shown in Figure

5.19. The plotted graphs explicitly demonstrate that the factors of safety from 3D

analysis are always larger than those from 2D analysis. It is found in Figure 5.19 that

the largest DD FF 23 ratio occurred at 1HL . In addition, the maximum DD FF 23

ratio for various slope inclinations ranges from 1.4-1.8 for 1Hd to 1.6-3.9 for

5Hd . For the 5.7 case, the value of DD FF 23 ratio varies much more

significantly with Hd than that of 15 . Hence, the 3D effect on the gentle slopes

is significant and cannot be ignored in analysis or design. Comparing FEM with the

numerical limit analysis solutions (Figure 5.16), the magnitudes of DD FF 23 ratio are

remarkably close to the lower bound results.

Figure 5.20 and Figure 5.21 display the plastic zones on symmetric faces for the 3D

FEM solutions. These plastic zones can be used to define the failure surfaces of the

slopes. The figures show one hard layer below the undrained clay material. Therefore

the failure surfaces can not propagate through this rigid stratum. The shapes of the

plastic zones indicate that the failure surfaces are cylindrical for undrained clay

materials. This failure mechanism occurs in all of the cases considered in this Section.

As mentioned in Section 5.1 and Figure 5.1, the parameter n is used to describe the

base-failure and compare with the presented results. The calculated magnitudes of n

here are around 0.8 for Figure 5.20 and 1.15 for Figure 5.21 respectively. These are



5-8

compared to the solutions of Gens et al. (1988) in Figure 5.18(b) and (d). The

magnitudes of n are found to be similar.

5.3 SOLUTIONS OF INHOMOGENEOUS UNDRAINED

SLOPES

The undrained shear strength profile assumed for inhomogeneous undrained slopes is

displayed in Figure 5.22. In general, the shear strength uc may increase linearly with

depth as is the case in normally consolidated clays (Gibson and Morgenstern (1962)).

Therefore, uc is assumed to increase linearly with depth according to Equation (5.2) in

which 0uc is the undrained shear strength at the slope top.

0( )u uc z c z (5.2)

where is the increasing rate of the undrained shear strength with depth and z is the

depth from the top of the slope.

Two types of strength profiles have been analysed in this thesis. For cut slopes, the

contour of the undrained shear strength is assumed horizontal (Figure 5.22(a)). For

natural slopes, the contour of the undrained shear strength is assumed parallel to the

slope surface (Figure 5.22(b)). This latter type of undrained shear strength profile in

Figure 5.22(b) may exist in onshore and offshore natural slopes. Equation (5.2) can be

used to represent the increment of the undrained shear strength for both cut and natural

slopes. The only discrepancy is in the distribution of uc which has the same magnitude

at the top of the slope, but a different magnitude on the inclined face and at the toe of

the slope.

In order to compare the results obtained in this thesis to the existing results of Yu et al.

(1998), two dimensionless parameters as shown in Equation (5.3) and Equation (5.4) are

defined. These two equations were proposed by Yu et al. (1998) to account for the

effect of increasing strength with depth. Equation (5.3) can be seen as the stability

number for inhomogeneous undrained slopes.

0uN HF c (5.3)



5-9

0c uHF c (5.4)

The above definition for the stability number is somewhat different to that defined for

homogeneous slopes (Equation (5.1)). As a consequence, it is expected that the upper

bound stability number will be larger than the lower bound stability number, as apposed

to the reverse which was observed for homogeneous slopes.

5.3.1 3D limit analysis results for cut slopes

Figure 5.23 to Figure 5.28 display the limit analysis stability numbers N for 3D cut

slopes. They show N increasing with decreasing and d H and increasing c . The

obtained 3D stability numbers ( N ) in Figure 5.23 to Figure 5.28 are bounded by the

numerical upper and lower bound solutions within 8 %. It is interesting to note that

N increases almost linearly with the dimensionless parameter c .

Yu et al. (1998) presented a set of two dimensional stability charts based on the upper

and lower bound limit analysis for simple slopes relevant to excavations and man-made

fills built on soil. In their studies, the shear strength profile is the same as illustrated

Figure 5.22(a). The 2D solutions shown in Figure 5.23 to Figure 5.27 were presented by

Yu et al. (1998). In Figure 5.23 to Figure 5.28, it can be observed that the difference

between the 2D (Yu et al. (1998)) and 3D stability numbers ( N ) decreases with

increasing HL ratio, as the 3D end boundary effect decreases. Based on the

comprehensive observation for all analytical results, this range of difference changes

from around 30%-60% to 8%-25% when the ratio of HL increases from 1 to 5. It

should be stressed in Figure 5.28 that the chart solutions are not presented for various

d H ratio as its effect is insignificant for vertical cuts ( 90 ).

It is found in Figure 5.29(a) that the difference in N between the 2D and 3D upper

bound solutions increases slightly with increasing c and decreasing HL . A similar

trend also occurs in the lower bound solutions (Figure 5.29(b)). This implies that, for

inhomogeneous undrained slopes, the increasing strength with depth has a more

significant effect on the stability numbers for the slope with a lower HL ratio.



5-10

Figure 5.30 presents the effect of slope angle ( ) on the stability number using the

upper bound solutions for 5HL and different values of depth factor ( Hd ) and c .

The comparisons between lines of 0.1c (cut slope) and 0.0c show that the

effects of slope angle on the stability number is more significant for the slopes with a

high value of c and a low depth factor ( Hd ). A similar trend was observed by Yu et

al. (1998) for 2D slopes. Moreover, the difference in stability numbers between

0.0c and 0.1c (cut slope) is found to decrease with increasing. This

indicates that the effect of c on the gentle slopes is more significant than that on the

steep slopes.

Figure 5.31 displays several of the upper bound plastic zones for 0.1c , 2d H ,

5HL and various slope inclinations. The depth of failure surface increases with a

reduction of the slope angle. In addition, it can be observed that the failure mode

transfers gradually from base-failure to toe-failure when increases from 30 (Figure

5.31 (a)) to 60 (Figure 5.31 (c)). As expected, for most numerical results in this

Chapter, the depth of slip surface for the inhomogeneous undrained slopes is found to

be shallower than that for the homogeneous undrained slopes. It means that the depth

factor boundary effect plays a more important role for the homogeneous undrained

slopes.

Application example for a cut slope

In order to make comparisons of the factor of safety between the newly proposed 3D

chart solutions and the 2D chart solutions presented by Yu et al. (1998), the same

example from Yu et al. (1998) is used in this study. This example is a cut slope

excavated in a normally consolidated clay. The slope descriptions are as follows: the

slope inclination 60 , the height of the slope is H 12m, the depth factor is

5.1Hd , and the soil unit weight is 35.18 mkN . The undrained shear strength of

the soil on the top of the slope surface is 0 40uc kPa and the rate of the increasing

undrained shear strength with depth is estimated as mkPa5.1 .



5-11

A procedure for obtaining the factor of safety by using the chart solutions presented in

this study can be summarized in the following stages.

1. From the slope descriptions, the non-dimensional parameter 0uH c and

cN are calculated as 55.540125.18 and

0 0 18.5 1.5 12.33c u uN HF c HF c , respectively.

2. For 60 and 5.1Hd , the chart solutions shown in Figure 5.26(b) are

employed to determine the safety factor.

3. In Figure 5.26(b), a straight line passing through origin with a gradient

33.12 is plotted. This straight line intersects with four curves, which are

the 2D and 3D chart solutions of the numerical limit analysis.

4. The 2D upper and lower bound stability numbers of Yu et al. (1998) are

N 6.8 and 7.8 for the lower bound and upper bound respectively, and

therefore the factors of safety are 0uF N H c 1.23 and 1.41. To

account for the effect of HL ratio on the safety factor, the factors of safety for

3D slopes are calculated, with details shown in Table 5.1.

In Table 5.1, the stability numbers ( N ) and the calculated safety factors from the upper

and lower bound chart solutions for different HL ratios are provided. The average of

the upper and lower bound safety factors are 1.7, 1.55 and 1.52 for 2HL , 3HL

and 5HL respectively. The safety factors of the 3D solutions are around 1.15-1.30

times greater than those of the safety factors of the 2D solutions of Yu et al. (1998).

This demonstrates that the factor of safety obtained from 3D analysis should be always

larger or equal to that obtained from 2D analysis. This observation is different from the

results of Chen and Chameau (1982). Therefore, using a 2D solution is conservative for

design and non-conservative for the back analysis of slope stability. In addition, from

Table 5.1, the difference between the upper and lower bound factors of safety for this

application example is found to be around 9 %. This difference decreases slightly

when the ratio of HL increases.



5-12

5.3.2 3D limit analysis results for natural slopes

Figure 5.32 to Figure 5.36 show the 2D and 3D results for the purely cohesive natural

slopes. The obtained 2D and 3D stability numbers ( N ) in Figure 5.32 to Figure 5.36

are bracketed by the numerical upper and lower bound solutions within 5 % and

10 % respectively. Moreover, it can be seen in Figure 5.37 that the stability numbers

for the natural slopes are the same as the stability numbers for the cut slopes when

0.0c . This is to be expected as 0.0c is simply the homogeneous slope case.

From Figure 5.37, the difference in stability number is found to increase with increasing

c .

Referring to Figure 5.32 to Figure 5.36, similar trends to the stability charts for cut

slopes are observed. That is, N increases with a reduction of and Hd and

increases almost linearly with the dimensionless parameter c . In addition, it can be

observed that the difference between the 2D and 3D stability numbers ( N ) decreases

with increasing HL ratio due to decreasing 3D end boundary effect. From the

observation for all obtained results, this difference can change from around 25%-60% to

2%-17% when the ratio of HL increases from 1 to 5. Compared with the results of the

cut slope in Section 5.3.1, the range is slightly smaller for 5HL . It implies that the

3D end boundary effect decreases more significantly with increasing HL ratio for the

natural slopes.

Figure 5.30 also presents the effect of slope angle ( ) on the stability number for

natural slopes based on the upper bound solutions. This effect is found to be more

significant for the slopes with a high value of c and a low depth factor ( Hd ). It is

apparent that such trend exists in both the cut and natural slopes. In Figure 5.30(b), the

difference in stability numbers between 0.0c and 0.1c (natural slope) is found

to decrease with increasing, however this phenomenon is not obvious when 1Hd .

Based on the observations of all numerical results for natural slopes, it can be stated that

the failure mode transfers from base-failure to the toe-failure gradually as increases.

Also, the depth factor ( Hd ) has a small effect on the obtained stability numbers for the



5-13

natural slopes as long as Hd is greater than 2, as was the case for cut slopes

investigated in Section 5.3.1

Application example for a natural slope

In order to investigate the difference in the factor of safety between the chart solutions

of the cut and natural slopes, the same example as in Section 5.3.1 is employed in this

Section. The only exception is that the example here is a normally consolidated natural

slope so that the soil profiles are different, as shown in Figure 5.22. From the

description of the slope in Section 5.3.1, 33.12 is known. In Figure 5.35(b), a

straight line passing through origin with a gradient 33.12 is plotted. This straight

line will intersect four curves which are the 2D and 3D chart solutions of the numerical

limit analysis.

Table 5.2 shows the stability numbers ( N ) and factors of safety obtained from the

solutions in Figure 5.35(b). As expected, the factor of safety from a 3D analysis will be

approaching gradually the factor of safety from a 2D plane strain analysis when the ratio

of HL increases. The average of upper and lower bound safety factors are 1.48, 1.36,

1.27 and 1.11 for 2HL , 3HL , 5HL and HL respectively.

Comparisons with the results of the cut slope (Table 5.1) are made by using cut naturalF F

shown in Table 5.2. It is found that using chart solutions in Figure 5.23 to Figure 5.27 to

evaluate the stability of a natural slope may result in overestimating its factor of safety

by up to 10%-20%. This can be seen the stability charts of Figure 5.23 to Figure 5.27

are not safe and suited to dealing with the natural slope problems.

Referring to Table 5.2, the safety factors from the 3D solutions are around 1.04-1.44

times greater than safety factors from the 2D solutions. The difference between the

upper and lower bound factors of safety for this application example is found to be

around 10 %. It should be note that the difference in the lower bound factor of safety

between the 2D case and the case with 5HL is less than 5%. It means that the 3D

boundary end effect on the slope stability is significantly smaller and almost unnoticed

when 5HL .



5-14

5.4 SUMMARY AND CONCLUSIONS

Although several studies (Hovland (1977) and Seed et al. (1990)) have indicated that the

factor of safety from a 3D analysis will be greater than that from a 2D analysis, this fact

should be stressed again in this Chapter. It has been proved based on the comparisons

between 2D and 3D safety factors, and therefore the 3D estimates of Chen and

Chameau (1982) are not correct.

In addition, Chugh (2003) analysed a sample problem and pointed out that the

difference between 2D and 3D safety factors tends to lose significance when 5HL .

However, results of this study showed that the DD FF 23 ratio as high as 1.76 for the

undrained uniform slopes, 1.15 for the undrained cut slopes and 1.04 for the undrained

natural slopes when 5HL . The difference between the 2D and 3D factors of safety

estimates is greater than 15% which would be important and is not negligible for the

back analysis of a failed slope in practise. Moreover, this range of difference is greater

than the results of Gens et al. (1988) which range between 3%-30% with the average of

13.9% based on the case records.

The statement made by Chugh (2003) was based on the results of frictional soil slopes

and is not applicable to the purely cohesive slopes. Therefore, engineers need to apply

2D solutions with caution.

Three dimensional stability charts for homogeneous and inhomogeneous purely

cohesive slopes have been proposed in this Chapter. Based on the results presented, the

following conclusions can be made:

1. Using the numerical upper and lower bound techniques, a range of stability

numbers ( hN and N ) have been bounded within 10 % or better for all cases

considered. For homogeneous undrained slopes, the upper and lower bound limit

analysis solutions bracket the ratio of DD FF 23 within 5.7 %. Based on the

results of the application example, the difference between the upper and lower

bound factors of safety is found to be around 10 % for non-homogeneous

undrained slopes.



5-15

2. For both the 3D homogeneous and inhomogeneous undrained slopes with a

slope angle ( 30 ), the primary failure mode is that of base-failure which

changes gradually to toe-failure with increasing .

3. The depth factor ( Hd ) boundary effect is found to reduce with increasing slope

inclination for the 3D homogeneous undrained slopes. In addition, the stability

numbers ( N ) of the 3D inhomogeneous undrained slopes are almost unchanged

when 2Hd .

4. For the 3D inhomogeneous undrained slopes, it is found that the effect of c on

the stability numbers is more significant for the slopes with a lower slope angle

or HL ratio, and the effect of slope angle on the stability number is more

significant for the slopes with a high value of c and a low depth factor ( Hd ).



5-16

Table 5.1 Safety factors for the cut slope example problem

0 5.55uH c 2HL 3HL 5HL 2D

UB LB UB LB UB LB UB LB

N 10.25 8.6 9.3 7.9 9 7.8 7.8 6.8

0uF N H c 1.85 1.55 1.68 1.42 1.62 1.41 1.41 1.23

Average F 1.7 1.55 1.52 1.32

Table 5.2 Safety factors for the natural slope example problem

0 5.55uH c 2HL 3HL 5HL 2D

UB LB UB LB UB LB UB LB

N 9.1 7.3 8.25 6.85 7.75 6.25 6.3 6

0uF N H c 1.64 1.32 1.49 1.23 1.40 1.13 1.14 1.08

Average F 1.48 1.36 1.27 1.11

cut naturalF F 1.15 1.14 1.20 1.19



5-17

Figure 5.1 Problem configuration for homogeneous slopes in purely cohesive soil

Fixed face

u = v = w = 0 (Upper bound)


d


Symmetric face

τ = 0 (Lower bound)

v = 0 (Upper bound)


x, u y, v

z, w

L/2

H

β σn = τ = 0 (Lower bound)

σn = τ = 0 (Lower bound)


Mode of Failure:

F = Face failure

T = Toe failure

B = Base failure



5-18

1 2 3 4 50.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

Depth factor (d / H)

Upper bound LEM (Taylor) Lower bound FEM (ABAQUS)

Nh =

c u/H

F= 30

= 7.5

Less stable

L / H =

Figure 5.2 Comparisons of 2D stability numbers between the numerical limit analysis,

LEM and FEM



5-19

1 2 3 4 5

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

0.24= 75

= 60


= 7.5

= 15

= 22.5= 30

Nh =

cu/

HF

= 45

H

Less stable

(a) Lower bound

1 2 3 4 5

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

= 75

= 60


= 7.5

= 15= 22.5

= 30

Nh =

cu/

HF

= 45

H

Less stable

(b) Upper bound

Figure 5.3 Two dimensional limit analysis solutions of stability numbers for

homogeneous undrained slopes ( HL )



5-20

1 2 3 4 50.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16= 75

= 60

= 7.5

= 15

= 22.5

= 30

Nh =

cu/

HF


= 45

H

Less stable

(a) Lower bound

1 2 3 4 50.02

0.04

0.06

0.08

0.10

0.12

0.14 = 75

= 60


= 7.5

= 15

= 22.5

= 30

Nh =

cu/

HF

= 45

H

Less stable

(b) Upper bound

Figure 5.4 Three dimensional limit analysis solutions of stability numbers for

homogeneous undrained slopes ( 1HL )



5-21

1 2 3 4 5

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

= 75

= 60


= 7.5

= 15

= 22.5

= 30

Nh =

cu/

HF

= 45

H

Less stable

(a) Lower bound

1 2 3 4 5

0.04

0.06

0.08

0.10

0.12

0.14

0.16


= 7.5

= 15

= 22.5

= 30

Nh =

cu/

HF

= 45

H

= 60

= 75

Less stable

(b) Upper bound





5-22

1 2 3 4 50.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20


= 7.5

= 15

= 22.5

= 30

Nh =

cu/

HF

= 45

H

= 60

= 75

Less stable

(a) Lower bound

1 2 3 4 5

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18


= 7.5

= 15

= 22.5

= 30

Nh =

cu/

HF

= 45

H

= 60

= 75

Less stable

(b) Upper bound





5-23

1 2 3 4 50.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22


= 7.5

= 15

= 22.5

= 30

Nh =

cu/

HF

= 45

H

= 60

= 75L

ess stable

(a) Lower bound

1 2 3 4 50.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20


= 7.5

= 15

= 22.5

= 30

Nh =

cu/

HF

= 45

H

= 60

= 75

Less stable

(b) Upper bound





5-24

2 40.10

0.12

0.14

0.16

0.18

0.20

0.22

0.24

0.26

0.28

L / H = 5

L / H = 2

L / H = 3

H

L / H =

L / H = 1

Less stableN

h = c

u/H

F


(a) Lower bound

1 2 3 4 50.10

0.12

0.14

0.16

0.18

0.20

0.22

0.24

0.26

0.28

L / H = 5

L / H = 2

L / H = 3

H

L / H =

L / H = 1

Less stableN

h = c

u/H

F


(b) Upper bound


homogeneous undrained slopes ( 90 )



5-25

(a) 30

(b) 5.22

(c) 15

Figure 5.9 2D upper bound plastic zones for 2Hd



5-26

(a) 30

(b) 5.22

(c) 15

Figure 5.10 2D upper bound plastic zones for 5Hd



5-27

(a) 60

(b) 75

(c) 90

Figure 5.11 2D upper bound plastic zones for various slope angles ( 2Hd )



5-28

(a) 2Hd

(b) 4Hd

(c) 5Hd

Figure 5.12 3D upper bound plastic zones for various Hd ( 15 and 5HL )

Symmetric face



5-29

(a) 2Hd

(b) 4Hd

(c) 5Hd

Figure 5.13 3D plastic zones for various Hd ( 5.22 and 5HL )

Symmetric face



5-30

(a) 3HL

(b) 1HL

Figure 5.14 3D plastic zones for various HL ratios ( 15 and 4Hd )

Symmetric face



5-31

(a) 45

(b) 60

(c) 75

Figure 5.15 3D plastic zones for various slope angles ( 2Hd and 5HL )

Symmetric face



5-32

1 2 3 4 51 .0

1 .5

2 .0

2 .5

3 .0

3 .5

4 .0

4 .5

F3D

/ F

2D d / H = 1 d / H = 2 d / H = 3 d / H = 5

L / H

7 .5

1 2 3 4 51 .0

1 .5

2 .0

2 .5

d / H = 1 d / H = 2 d / H = 3 d / H = 5

2 2 .5

1 2 3 4 51 .0

1 .5

2 .0

2 .5

F3D

/ F

2D

d / H = 1 d / H = 2 d / H = 3 d / H = 5

L / H

3 0

1 2 3 4 51 .0

1 .5

2 .0

d / H = 1 d / H = 2 d / H = 3 d / H = 5

4 5

1 2 3 4 51 .0

1 .2

1 .4

1 .6

1 .8

2 .0

F3D

/ F

2D

d / H = 1 d / H = 2 d / H = 3 d / H = 5

L / H

6 0

1 2 3 4 51 .0

1 .2

1 .4

1 .6

1 .8

2 .0

d / H = 1 d / H = 2 d / H = 3 d / H = 5

7 5

Figure 5.16 Factor of safety ratio of DD FF 23 (limit analysis)



5-33

1 2 3 4 50.02

0.04

0.06

0.08

0.10

0.12

L / H = 1


= 7.5

= 30

Nh =

cu/

HF

Upper bound LEM (Gens et al.) Lower bound FEM (ABAQUS)

Less stable

1 2 3 4 50.06

0.08

0.10

0.12

0.14

0.16

0.18

L / H = 3


Upper bound LEM (Gens et al.) Lower bound FEM (ABAQUS)

= 15

= 45

Nh =

Su/

HF

Less stable

Figure 5.17 Comparisons of 3D stability numbers between the numerical limit analysis,

LEM and FEM



5-34

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

1 1.5 2 2.5 3 3.5 4 4.5 5

Gens et al.(LEM)

Gens et al.(toe-failure)

β=7.5

β=15

β=22.5

β=30

β=45

Dcrit

n = 0.25 n = 0

n = 0

N=

c u / H

F

Depth Factor (D)

= 45

= 7.5

= 22.5

= 15

= 30

(a) 1HL

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

1 1.5 2 2.5 3 3.5 4 4.5 5

Gens et al.(LEM)


β=7.5

β=15

β=22.5

β=30

β=45

n = 0

n = 0

n = 0.5

Dcrit

N=

c u / H

F

Depth Factor (D)

= 45

= 30

= 22.5

= 15

= 7.5

(b) 2HL

Figure 5.18 Comparison of three dimensional stability numbers between FEM and

LEM (Gens et al. (1988))



5-35

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

1 1.5 2 2.5 3 3.5 4 4.5 5

Gens et al.(LEM)


β=7.5

β=15

β=22.5

β=30

β=45

Dcrit

n = 0.5

n = 0

n = 0

n = 1

= 45

= 30

= 22.5

= 15

= 7.5

Depth Factor (D)

N=

c u / H

F

(c) 3HL

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

1 1.5 2 2.5 3 3.5 4 4.5 5

Gens et al.(LEM)


β=7.5

β=15

β=22.5

β=30

β=45

Dcritn = 0.5

n = 0

n = 0

n = 1

n = 1.5

= 45

= 30

= 22.5

= 15

= 7.5

Depth Factor (D)

N=

c u / H

F

(d) 5HL

Figure 5.18 (continued) Comparison of three dimensional stability numbers between

FEM and LEM (Gens et al. (1988))



5-36

0.05

0.07

0.09

0.11

0.13

0.15

0.17

0.19

1 1.5 2 2.5 3 3.5 4 4.5 5

Gens et al.(LEM)


β=7.5

β=15

β=22.5

β=30

β=45

n = 0

n = 1

n = 2

n = 3

N=

c u / H

F

Depth Factor (D)

= 45

= 30

= 22.5

= 15

= 7.5

(e) HL

Figure 5.18 (continued) Comparison of three dimensional stability numbers between

FEM and LEM (Gens et al. (1988))



5-37

1 2 3 4 51.0

1.5

2.0

2.5

= 45

= 30

= 22.5

= 15

= 7.5

F3D

/ F

2D

L / H

(a) 1Hd

1 2 3 4 51.0

1.5

2.0

2.5

3.0

3.5

4.0

F3D

/ F

2D

L / H

= 45

= 30

= 22.5

= 15

= 7.5

(b) 5Hd

Figure 5.19 Ratios of DD FF 23 for various Hd (FEM)



5-38

Figure 5.20 Plastic zone for the case 30 , 2Hd and 2HL

Figure 5.21 Plastic zone for the case 5.22 , 2Hd and 5HL

H nH

H nH



5-39

z

cu(z) = c

u0 + z

cu0

Toe

Rigid Base

d

1

H

(a) Cut slope

z

cu(z) = c

u0 + z

cu0

Toe

Rigid Base

d

1

H

(b) Natural slope

Figure 5.22 The analysed strength profile for inhomogeneous undrained slopes



5-40

0

4

8

12

16

20

24

28

32

360.0 0.2 0.4 0.6 0.8 1.0

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

24

28

32

36

UB

LB UB

LB

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

28

32

36

L / H = 5L / H = 3

UB

LB

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

28

32

36

3D 2D (Yu et al.)

UBLB

UBLB

More stable

(a) 1Hd

0

4

8

12

16

20

24

28

32

360.0 0.2 0.4 0.6 0.8 1.0

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

24

28

32

36

UB

LBUB

LB

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

28

32

36

L / H = 5L / H = 3

UB

LB

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

28

32

36

3D 2D (Yu et al.)

UBLB

UBLB

More stable

(b) 5.1Hd

Figure 5.23 Limit analysis solutions of stability numbers for inhomogeneous undrained

cut slopes ( 15 )



5-41

0

4

8

12

16

20

24

28

32

360.0 0.2 0.4 0.6 0.8 1.0

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

24

28

32

36

UB

LBUB

LB

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

28

32

36

L / H = 5L / H = 3

UB

LB

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

28

32

36

3D 2D (Yu et al.)

UBLB

UBLB

More stable

(c) 2Hd

Figure 5.23 (continued) Limit analysis solutions of stability numbers for

inhomogeneous undrained cut slopes ( 15 )



5-42

0

4

8

12

16

20

240.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

24

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LBUB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UB

LB UBLB

(a) 1Hd

0

4

8

12

16

20

240.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

24

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LB UB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UBLB UB

LB

(b) 5.1Hd


cut slopes ( 30 )



5-43

0

4

8

12

16

20

240.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

24

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LB UB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UBLB UB

LB

(c) 2Hd





5-44

0

4

8

12

16

200.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LB UB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UBLB UB

LB

(a) 1Hd

0

4

8

12

16

20

240.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

24

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LB UB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UBLB UB

LB

(b) 5.1Hd


cut slopes ( 45 )



5-45

0

4

8

12

16

20

240.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

24

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LB UB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UBLB UB

LB

(c) 2Hd





5-46

0

4

8

12

160.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LB UB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UB

LB

UB

LB

(a) 1Hd

0

4

8

12

160.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

/ 1

/ 1 /

1

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LB UB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UB

LB

UB

LB

(b) 5.1Hd


cut slopes ( 60 )



5-47

0

4

8

12

160.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LB UB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UB

LB

UB

LB

(c) 2Hd





5-48

0

4

8

120.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LBUB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UB

LB

UB

LB

(a) 1Hd

0

4

8

120.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LBUB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UB

LB

UB

LB

(b) 5.1Hd


cut slopes ( 75 )



5-49

0

4

8

120.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LBUB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UB

LB

UB

LB

(c) 2Hd



0

4

8

120.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LBUB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UB

LBUBLB


cut slopes ( 90 )



5-50

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

L / H =

3D 2D (Yu et al.)

=

H

F /

c u0

cp

= HF / cu0

L / H = 1, 2, 3, 5

(a) Upper bound

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

L / H =

3D 2D (Yu et al.)

=

H

F /

c u0

cp

= HF / cu0

L / H = 1, 2, 3, 5

(b) Lower bound

Figure 5.29 Comparisons of stability numbers for different magnitudes of HL

( 45 and 5.1Hd )



5-51

15 30 45 60 750

5

10

15

20

25

30

cp

= 1.0 (Cut slope)

cp

= 1.0 (Natural slope)

cp

= 0.0

=

H

F /

s u0

Slope angle ()

(a) 1Hd

15 30 45 60 750

5

10

15

20

25

cp

= 1.0 (Cut slope)

cp

= 1.0 (Natural slope)

cp

= 0.0

=

H

F /

c u0

Slope angle ()

(b) 2Hd

Figure 5.30 Effect of slope angle on stability number based on the upper bound

solutions ( 5HL )



5-52

(a) 30

(b) 45

(c) 60

Figure 5.31 3D upper bound plastic zones for various 0.1c , 2Hd and 5HL

Symmetric face



5-53

0

4

8

12

16

20

24

28

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

24

28

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

28

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

28

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LBUB

LB

L / H = 5L / H = 3

UB

LB

3D 2D

UB

LB UBLB

(a) 1Hd

0

4

8

12

16

20

24

28

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

24

28

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

28

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

28

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LB UB

LB

L / H = 5L / H = 3

UB

LB

3D 2D

UBLB UB

LB

(b) 5.1Hd


natural slopes ( 15 )



5-54

0

4

8

12

16

20

24

28

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

24

28

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

28

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

28

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LB UB

LB

L / H = 5L / H = 3

UB

LB

3D 2D

UBLB UB

LB

(c) 2Hd


inhomogeneous undrained natural slopes ( 15 )



5-55

0

4

8

12

16

200.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LBUB

LB

L / H = 5L / H = 3

UB

LB

3D 2D

UBLB UB

LB

(a) 1Hd

0

4

8

12

16

200.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LBUB

LB

L / H = 5L / H = 3

UB

LB

3D 2D

UBLB

UBLB

(b) 5.1Hd





5-56

0

4

8

12

16

200.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LBUB

LB

L / H = 5L / H = 3

UB

LB

3D 2D

UBLB

UBLB

(c) 2Hd





5-57

0

4

8

12

160.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LBUB

LB

L / H = 5L / H = 3

UB

LB

3D 2D

UB

LB

UBLB

(a) 1Hd

0

4

8

12

160.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LBUB

LB

L / H = 5L / H = 3

UB

LB

3D 2D

UB

LBUB

LB

(b) 5.1Hd





5-58

0

4

8

12

160.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LBUB

LB

L / H = 5L / H = 3

UB

LB

3D 2D

UB

LB

UB

LB

(c) 2Hd





5-59

0

4

8

12

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LBUB

LB

L / H = 5L / H = 3

UB

LB

3D 2D

UB

LB

UB

LB

(a) 1Hd

0

4

8

12

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

/ =12.331

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

/ =12.331

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LB

UB

LB

L / H = 5L / H = 3

UB

LB

3D 2D

UB

LB

UB

LB

/ =12.331

(b) 5.1Hd


slopes ( 60 )



5-60

0

4

8

12

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LB

UB

LB

L / H = 5L / H = 3

UB

LB

3D 2D

UB

LB

UB

LB

(c) 2Hd


inhomogeneous undrained slopes ( 60 )



5-61

0

4

8

120.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LBUB

LB

L / H = 5L / H = 3

UB

LB

3D 2D

UB

LB

UB

LB

(a) 1Hd

0

4

8

120.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LB

UB

LB

L / H = 5L / H = 3

UB

LB

3D 2D

UB

LB

UB

LB

(b) 5.1Hd


slopes ( 75 )



5-62

0

4

8

120.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

More stable

L / H = 2L / H = 1

=

H

F /

cu0

cp

= HF / cu0

UB

LB

UB

LB

L / H = 5L / H = 3

UB

LB

3D 2D

UB

LB

UB

LB

(c) 2Hd


inhomogeneous undrained slopes ( 75 )

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

L / H = 2

3D (Cut slope) 3D (Natural slope)

UB

LB

UB

LB

=

H

F /

c u0

cp

= HF / cu0

Figure 5.37 Comparisons of the stability number N between the cut and natural

slopes ( 2HL and 45 )



6-1

CHAPTER 6 SLOPE STABILITY OF COHESIVE-

FRICTIONAL SOIL

6.1 INTRODUCTION

In this Chapter, results of the investigation of two and three dimensional cohesive-

frictional soil slopes under drained conditions are presented. The typical 3D slope

geometries for the problem analysed are shown in Figure 6.1 together with the applied

stress and velocity boundary conditions to simulate the fixed and symmetric end faces

in the upper and lower bound analyses. A depth factor of 1Hd is adopted in

calculations unless stated otherwise. This is because for almost all considered cohesive-

frictional soil slopes the critical failure surface tends to pass through the slope toe

(Taylor (1948) and Chen (1975)) meaning that domain discretization is not required

below toe level. The exceptions are possible for slopes with very low slope angles (ie

15 ) and unrealistically low friction angle. A range of friction angles have been

considered and corresponding stability charts have been produced.

The overall aim of this Chapter is to provide simple to use stability chart solutions for

preliminary stability esitimates of cohesive-frictional soil slopes. These solutions are

obtained by using numerical finite element upper and lower bound techniques as

discussed in previous chapters and presented in dimensionless manner as specified by

Equation (6.1) and 'tanF . This form of stability chart was originally proposed by

Bell (1966) and employed later by others (i.e. Michalowski (2002)). It should be noted

that the stability charts developed in terms of cannot be used for soil slopes with zero

internal friction angle, as the expression in Equation (6.1) becomes singular in this case.

'

'

tan

H

c (6.1)



6-2

6.2 NUMERICAL LIMIT ANALYSIS SOLUTIONS

6.2.1 Stability charts for cohesive-frictional soil slopes

Figure 6.2 to Figure 6.6 show stability charts for cohesive-frictional soil slopes obtained

by numerical upper and lower bound limit analyses for cases with 1Hd , different

slope angles ( ) and HL ratios. Figure 6.2(b) shows the solutions by zooming in the

blue region in Figure 6.2(a). Based on the same pattern, (b) and (d) in Figure 6.3 to

Figure 6.6 are also displayed. From these figures, the depth factor ( Hd ) is found to

have only a small effect on the chart solutions as is almost unchanged for a given

'tanF and . The presented results indicate that the upper and lower bound

solutions generally bracket the exact value of 'tanF within 10 %. And this gap

shrinks rapidly when the slope angle decreases.

From Figure 6.2 to Figure 6.6, it can be noticed that, for a given 'tanF , the

dimensionless parameter increases with increasing HL and slope angle ( ). For a

given , the difference in 'tanF for any two slope angles can provide a ratio of

safety factors. For example, it can be found from Figure 6.2(a) that decreasing the slope

angle from 75 to 60 can increase the factor of safety by more than 15% for

0.1 and 20' .

Figure 6.7 displays an alternative form of stability charts where factor of safety is

presented as a function of HL . Users of these charts only need to know the slope

geometry and soil strength parameters ( 'c and ' ), and then 'tanF can be estimated

for a given ratio of HL .

Figure 6.8 displays the 2D plastic zones obtained from the upper bound limit analysis

solutions for various friction angles ( ' ). It can be observed that the depth of the slip

surface increases with the reduction of the friction angle. This trend is also found to be

valid for 3D cohesive-frictional soil slopes. Figure 6.9 shows the 3D failure surfaces for

25' , 45 and different ratios of HL . Comparing to 2D case of 25' and

45 shown in Figure 6.8, the depth of slip surfaces on the symmetric face is quite



6-3

similar. Therefore, 3D end boundary effects on the maximum depth of slip surface are

found to be insignificant.

6.2.2 Application example

In this section a demonstration is given how stability charts presented in Figure 6.2 to

Figure 6.6 can be used to determine the factor of safety for a given soil slope with

known geometry and soil strength.

The example slope has the following parameters: 116.0' Hc , friction angle 15'

and slope angle 60 . Therefore, the calculated is 433.0tan '' Hc . This

assumed slope was studied by Leshchinsky et al. (1985) and examined by Hungr et al.

(1989), Huang et al. (2002) and Xie et al. (2006). The obtained two dimensional factor

of safety is 1 and three dimensional factors of safety from these studies varied are

between 1.18 and 1.25. As is known, 'tanF can be found by using the chart

solutions in Figure 6.2 to Figure 6.6(c) or (d) for different ratios of HL .

Table 6.1 shows the calculated safety factors based on the numerical upper and lower

bound limit analysis solutions. It can be seen that the difference between the upper and

lower bound solutions is within 12%. As expected, the factor of safety increases when

HL ratio decreases for both the upper and lower bound values. The upper bound

results in Table 6.1 show that using the solution from a 2D analysis may underestimate

the slope stability by up to 40% for the case of 1HL . The comparison of 2D safety

factors obtained in this study with those presented by Leshchinsky et al. (1985) where

1F indicates that two sets of results are almost the same, with the difference being

just 2 percent.

In addition, it is found in Table 6.1 that, for this assumed case, the 3D end boundary

effects on the factor of safety are less than 10% for both the upper and lower bound

solutions when 5HL . It is similar to the finding of Chugh (2003). This implies that,

when 5HL , the effect of HL ratio is insignificant for cohesive-frictional slopes.

Therefore, it would be reasonable in this case to adopt solutions from 2D analyses.



6-4

6.3 ANALYTICAL SOLUTIONS

Currently, stability charts for cohesive-frictional soil slopes in the form of and

'tanF are presented only by Bell (1966) and Michalowski (2002). It should be

stressed, however, that these studies were limited to the two dimensional plane strain

cases based on the LEM and upper bound limit analysis respectively. Figure 6.10 shows

the comparisons of the numerical upper and lower bound solutions obtained in present

research work with the upper bound results by Michalowski (2002) for various slope

inclinations. It should be stated that 2Hd is adopted in the numerical upper and

lower bound analyses in order to compare with Michalowski (2002), and therefore the

effects of a bottom rigid base on the slope stability are avoided.

Figure 6.10(b) is obtained by zooming in the blue region in Figure 6.10(a). In Figure

6.10, it can be seen that 'tanF is closely bracketed by the upper and lower bound

solutions. The trends of the numerical bounding results are similar to the solutions of

Michalowski (2002). Moreover, the two upper bound solutions are remarkably close to

each other for most of cases. This implies that the assumed mechanism in Michalowski

(2002) is obviously very close to the true collapse mechanism, particularly for the cases

where the numerical upper bound solutions are above the Michalowski’s results. It

should be stated that the only way to perhaps improve the numerical upper and lower

bound methods is by adaptive remeshing if it were present in the formulations. From a

comparison between Figure 6.10 and Figure 6.2, it can be found that depth factor ( Hd )

has limited effect on chart solutions as is almost constant for a given 'tanF and .

6.4 DISPLACEMENT FINITE ELEMENT SOLUTIONS

6.4.1 Chart solutions based on displacement finite element analysis

In this Section, results from commercial displacement finite element software

(ABAQUS) are employed to examine and make comparisons with chart solutions from

the numerical bounding methods.

Square symbols in Figure 6.2(a), Figure 6.3 to Figure 6.6 ((a) and (c)) are used to denote

results from analyses using the displacement finite element method (DFEM). It is

observed that most FEM solutions fall within the range between the upper and lower



6-5

bound solutions. They are closer to the lower bound results, which is a similar trend to

observations presented for purely cohesive slopes in Section 5.2.3. However, this

phenomenon differs from the presented results for anchors in Merifield et al. (2005) and

Merifield and Sloan (2006) where the results from the displacement finite element

analyses are generally close to those from the upper bound analyses.

The above phenomenon obtained is worthy of more discussion. The relation of the FEM

and numerical limit analysis solutions could be a function of how the slope failure is

determined in the finite element analyses. It can be seen in Figure 4.5 that the turning

point, A, was used to define slope failure in the optimization. But this obtained failure

load (Point A) might not represent the true failure point for the whole slope which

means that the true failure might occur when Hc is located on the unstable side in

Figure 4.5. It should be noted that the vertical displacement observation of the slope

crest is a widely accepted and used method (Manzari and Nour (2000) and Hoek et al.

(2000)) to determine slope failure. Moreover, Griffiths and Lane (1999) highlighted that

there is no exact method for determining the true failure load for slope stability

problems in displacement finite element analyses. Based on the FEM results presented

in this thesis, it should be acknowledged here that the failure determination employed in

ABAQUS analyses can be seen as a conservative method due the solutions plotting

close to the lower bound results.

Referring to Figure 6.2 to Figure 6.6, the effect of water pore pressure on slope stability

is demonstrated for the case of 60 . For finite element analyses, the water table is

on the slope surface which is simulated by using hydrostatic conditions. Therefore, the

slopes are considered to be fully saturated. As expected, pore pressure may reduce the

factor of safety of a slope. It can be seen from Figure 6.2(a) that values of fully

saturated slopes with 60 are closer to those of dry slopes with 75 .

As an example of pore pressure effect on slope stability, the case study shown in

Section 6.2.2 and Table 6.1 is considered next. For 433.0tan '' Hc and

15' , the 2D factor of safety is approximately equal to 0.86 (as can be found from

Figure 6.2(a)). Comparing with the solution of dry slope where 1F , the safety factor

is smaller by 14% which is a significant figure. The pore pressure effects are not the



6-6

main topic of this thesis, therefore, it is suggested that more detailed studies be

performed in the future.

Figure 6.11 shows the 3D plastic zones obtained from the ABAQUS analyses for slopes

with 45 , 3HL and different friction angles ( ' ). Similar to what has been

observed for 2D cases this figure shows that the slip surface depth increases with the

friction angle decreasing. However, no effect on the depth of the failure surface can be

reported for various HL ratios. Comparing Figure 6.9 and Figure 6.11 for 25'

and 3HL , it can be noted that the shape and the extension of upper bound plastic

zones compare well for limit analysis and FEM solutions, especially on symmetric faces.

6.4.2 Comparisons with the strength reduction method (SRM)

As shown in the previous section, stability numbers found by limit analysis methods

and by FEM are quite similar. Whether the strength reduction method (SRM) can obtain

the factor of safety which is close to chart solutions provided in this Chapter is now

explored. According to strength reduction method, the factor of safety ( F ) of a slope is

defined as the number by which the original shear strength parameters must be reduced

in order to bring the slope to the point of failure. The factored shear strength parameters

'fc and '

f , are therefore given by:

F

cc f

'' (6.2)

Ff

'' tan

arctan

(6.3)

To find the true safety factor, it is necessary to initiate a systematic search for the value

of F that will just cause the slope to fail. This is achieved by using a sequence of user-

specified F values.

A plane strain slope with kPac 4' , 20' , 320 mkN and 45 is

considered as an example case to examine the presented chart solutions. Based on these

parameters, the factor 5495.020tan1204 (Equation (6.1)) can be

calculated. Using this value and following the charts in Figure 6.2(a) one can obtain



6-7

'tanF equal to 5.25. Hence, the safety factor 91.120tan25.5 F . Figure 6.12

on the other hand illustrates the factor of safety obtained by SRM method, as defined by

Equation (6.2) and Equation (6.3). According to this figure the SRM value of safety

factor ( F ) is equal to 1.93 which is close to the value obtained using chart solutions

based on limit analysis and FEM.

As discussed in Section 4.4.2 the stability charts are obtained by varying either cohesion

( c ) or unit weight ( ). It has been demonstrated here that there is no difference

between these two options and SRM approach as the same factors of safety have been

obtained in both cases (up to chart reading accuracy).

6.5 CONCLUSIONS

In this Chapter, a set of three dimensional stability charts was proposed for cohesive-

frictional soil slopes. These chart solutions are developed using numerical upper and

lower bound limit analysis methods. Based on results obtained, the following

conclusions can be made:

1. The upper and lower bound limit analysis solutions bracket the exact value of

'tanF within 10 % for stability assessment of all cohesive-frictional soil

slopes. It was found that using the two dimensional solutions to evaluate the

stability of three dimensional slopes may underestimate (or overestimate in

back-analysis) factors of safety by up to 40%. For cases with 5HL , the three

dimensional end boundary effects on the factor of safety are observed to be less

than 10%.

2. Preliminary investigations indicate that the presence of pore pressure may

decrease the factor of safety by up to 14%, which cannot be ignored. Hence, it is

worth investigating the pore pressure effects on the stability of cohesive

frictional slopes in further studies.

3. The displacement finite element solutions fall within the range bounded by the

results of upper and lower bound limit analyses and are closer to the lower

bound solutions. This is similar to results obtained for homogeneous undrained



6-8

slopes presented in Section 5.2.3. Therefore, results for slope stability obtained

using the FEM are conservative solutions.



6-9

Table 6.1 Safety factors for the application example

433.0tan'' Hc and 268.015tan

'tanF

(UB) F (UB) 'tanF

(LB) F (LB) Diff (%)

1HL 5.35 1.44 4.75 1.27 11.8

2HL 4.55 1.22 4.05 1.09 10.7

3HL 4.25 1.14 3.95 1.06 7

5HL 4.2 1.13 3.8 1.02 9.8

D2 3.8 1.02 3.7 0.99 2.9

LB = Lower Bound, UB = Upper Bound



6-10

Figure 6.1 Configuration for simple 3D homogeneous slopes

0.0 0.5 1.0 1.5 2.0 2.5 3.00

2

4

6

8

10

12

14

Upper bound Lower bound FEM (dry) FEM (fully saturated)

=60

60

30

7545

=1

5

F /

tan

'

c' /Htan'0.0 0.1 0.2 0.3 0.4 0.5

0

1

2

3

4

5

60

30

Upper bound Lower bound

75

45

F /

tan

'

c' /Htan'

(a) (b)

Figure 6.2 Stability charts for cohesive-frictional slopes ( D2 )

u = v = w = 0 (Upper bound) Fixed face


d

Symmetric face

τ = 0 (Lower bound)

v = 0 (Upper bound)

L/2

H

β

x, u y, v

z, w




0.5495



6-11

0.0 0.5 1.0 1.5 2.00

2

4

6

8

10

12

14

Upper bound Lower bound FEM

75

45

=1

5

F /

tan

'

c' /Htan'0.0 0.1 0.2 0.3 0.4 0.5

0

1

2

3

4

5


75

45

F /

tan

'

c' /Htan'

(a) (b)

0.0 0.5 1.0 1.5 2.00

2

4

6

8

10

12

14


=60

60

F /

tan

'

c' /Htan'

30

0.0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

60


30

F /

tan

'

c' /Htan'

(c) (d)

Figure 6.3 Stability charts for cohesive-frictional slopes ( 1HL )



6-12

0.0 0.5 1.0 1.5 2.0 2.50

2

4

6

8

10

12

14


F /

tan(

c /Htan(

75

45

=1

5

0.0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5


75

45

F /

tan

'

c' /Htan'

(a) (b)

0.0 0.5 1.0 1.5 2.0 2.50

2

4

6

8

10

12

14


=60

6030

F /

tan

'

c' /Htan'0.0 0.1 0.2 0.3 0.4 0.5

0

1

2

3

4

5

60


30

F /

tan

'

c' /Htan'

(c) (d)




6-13

0.0 0.5 1.0 1.5 2.0 2.5 3.00

2

4

6

8

10

12

14


7545

=1

5

F /

tan

'

c' /Htan'0.0 0.1 0.2 0.3 0.4 0.5

0

1

2

3

4

5


75

45

F /

tan

'

c' /Htan'

(a) (b)

0.0 0.5 1.0 1.5 2.0 2.50

2

4

6

8

10

12

14


=60

60

30

F /

tan

'

c' /Htan'0.0 0.1 0.2 0.3 0.4 0.5

0

1

2

3

4

5

60


30

F /

tan

'

c' /Htan'

(c) (d)




6-14

=1

5

0.0 0.5 1.0 1.5 2.0 2.5 3.00

2

4

6

8

10

12

14


75

45

F /

tan

'

c' /Htan'0.0 0.1 0.2 0.3 0.4 0.5

0

1

2

3

4

5


75

45

F /

tan

'

c' /Htan'

(a) (b)

0.0 0.5 1.0 1.5 2.0 2.50

2

4

6

8

10

12

14

60

30


=60

F /

tan

'

c' /Htan'0.0 0.1 0.2 0.3 0.4 0.5

0

1

2

3

4

560


30

F /

tan

'

c' /Htan'

(c) (d)




6-15

0.0 0.5 1.0 1.50

2

4

6

8

10

12

14


2DL / H =3

F /

tan

'

c' /Htan'

L / H =

1

(a) 30

0.0 0.5 1.0 1.5 2.0 2.50

2

4

6

8

10

12

14


2D

L / H =3

L / H =

1

F /

tan

'

c' /Htan'

(b) 60

Figure 6.7 Stability charts for various HL ratios



6-16

45 60

Figure 6.8 2D plastic zones from upper bound limit analyses for different friction

angles ( ' )

15'

25'

35'



6-17

(a) 1HL

(b) 3HL

Figure 6.9 3D plastic zones from upper bound limit analyses for different HL ratios

( 45 and 25' )

Symmetric face



6-18

0.0 0.5 1.0 1.5 2.0 2.50

2

4

6

8

10

12

14

60

45

Upper bound Lower bound Michalowski (2002)

=30

F /

tan

'

c /Htan '

(a)

0.0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

F /

tan

'

c /Htan '

Upper bound Lower bound Michalowski (2002)

60

45

=30

(b) Figure 6.10 Comparisons between upper and lower bound solutions and solutions by

Michalowski (2002) for 2D slopes



6-19

(a) 15

(b) 25

Figure 6.11 3D plastic zones from FEM analyses for different friction angles ( 45

and 3HL )



6-20

-0.16

-0.08

0.00

1.6 1.7 1.8 1.9 2.0 2.1 2.2

F = 1.93

z

Factor of safety (F)

z

/ H

H

Figure 6.12 Factor of safety versus normalised displacement.



7-1

CHAPTER 7 STATIC STABILITY OF UNIFORM ROCK

AND ROCKFILL SLOPES

7.1 INTRODUCTION

In this Chapter, results of the static stability of rock slopes governed by the Hoek-

Brown (Hoek et al. (2002)), Douglas (2002) and Mohr-Coulomb yield criteria are

presented. These yield criteria were discussed in Section 2.7. In the case of the Hoek-

Brown failure criterion, rock strength parameters GSI and im have been varied, whilst

the disturbance factor was assumed to be zero ( 0D ) to model a rock mass of natural

slopes without disturbance.

7.2 PROBLEM DEFINITION

The general configuration of the problem to be analysed is shown in Figure 7.1, where

the jointed rock mass has an intact uniaxial compressive strength ci , geological

strength index GSI , intact rock yield parameter im , and unit weight . In practise, the

rock weight can be estimated from core samples and ci and im are obtained from

either triaxial test results or from the tables proposed in Hoek (2000). Several

approaches can be used to evaluate GSI , as outlined by Hoek (2000). These include

using table solutions or estimating GSI by using the rock mass rating (RMR) (Bieniaski

(1976)). Excavated slope and tunnel faces are probably the most reliable source of

information for GSI estimates. Hoek and Brown (Hoek and Brown (1997)) also pointed

out that small adjustments of GSI can be used to incorporate the effects of surface

weathering. Greater detail on how to best estimate the Hoek-Brown material parameters

can be found in Hoek and Brown (1997), Hoek (2000) and Wyllie and Mah (2004).

In this study, all the quantities are assumed constant throughout the slope. In the limit

analyses, for a given slope geometry ( H , ) and rock mass ( ci , GSI , im ), the upper

and lower bound solutions can be optimised with respect to the unit weight ( ). In this

study, slope inclinations of 60,45,30,15 , and 75 are analysed. The effect of

depth factor ( Hd ) was found to be insignificant. With the exception of the case where



7-2

15 , all analyses indicated the primary failure mode was one where the slip line

passed through the toe of the slope. Results are presented in terms of two dimensionless

stability numbers, which are defined in Equations (7.1) and (7.2).

FHN ci

(7.1)

GSIN

F (7.2)

where F is the safety factor of the slope. It should be noted that these two definitions of

the factor of safety are not equivalent. Equation (7.1) will effectively provide a factor of

safety on the uniaxial compressive strength while equation (7.2) will provide a factor of

safety on the values of GSI. Unless stated otherwise, the factors of safety obtained in

this thesis are found using equation (7.1) and equation(7.2) is provided as an alternative

if necessary.

7.3 NUMERICAL LIMIT ANALYSIS SOLUTIONS

7.3.1 Chart solutions

Figure 7.2 to Figure 7.6 present stability charts from the numerical upper and lower

bound formulations for angles of 15 - 75 for a range of GSI and im . The stability

number N was defined in Equation (7.1). In Figure 7.4, it is apparent that the upper and

lower bound results bracket a narrow range of stability numbers N for 10GSI , so an

average value from the bound solutions could be adopted for simplicity. In fact, it was

found that, for all the analyses performed, the range between upper and lower bound

stability numbers was always less than 5 %. The only exception to this observation

occurs for the cases of 45 and low GSI values, where the range is around 9 %.

Therefore, average values of the stability number N have been adopted and presented

unless stated otherwise. The parameter N can be seen to decrease as the value of GSI

or im increases.

Figure 7.7 and Figure 7.8 show an alternative form of stability charts presented, as a

function of the slope angle ( ). The users only need to estimate GSI and im for the



7-3

rock mass, and then the stability number can be estimated for a given slope angle. For

the same rock slope material, the differences in stability number between various slope

angles can provide a ratio of safety factor. For example, it can be found that decreasing

slope angles from 75 to 60 for 80GSI can increase the factor of safety by

more than 50%.

Referring to the above results, for any given rock mass ( ci , GSI , im ) and unit weight

of the material , the obtained stability number can be used to determine the critical

height of slopes. In addition, the charts indicate that the stability number N increases

with increasing slope angle for a given GSI and im . As the factor of safety ( F ) is

proportional to the inverse of N , this indicates that F as expected decreases as

increases.

Figure 7.9 provides the users with the safety factor evaluation based on GSI values

(Equation (7.2)). For a given Hci , im and slope inclination ( ), the required

minimum GSI can be estimated for a factor safety equal to one ( 1F ). The procedure

on how to use the above stability charts will be introduced in detail in Section 7.3.4. It

should be noted that the stability charts in Figure 7.9 are based on the classification of

rock masses (GSI ).This implies that obtained factor of safety can be also verified by

RMR system (Bieniaski (1976)) or Q-system (Barton (2002)). In order to convert GSI

values to the RMR system or Q-system, Equations (7.3) and (2.24) can be used.

5 RMRGSI (7.3)

Figure 7.10 displays several of the observed upper bound plastic zones for different

slope angles. The depth of failure surface increases with the reduction of the slope

angle. But such variation is hardly noticed when the slope angle 45 . As shown in

Figure 7.11, for a given GSI , it was found that the depth of plastic zone is almost

unchanged with increasing im .

7.3.2 Analytical solutions

Currently, only stability factors presented by Yang et al. (2004a) and Yang et al. (2004b)

are proposed for estimating rock slope stability based on the latest version of the Hoek-



7-4

Brown yield criterion (Hoek et al. (2002)). As shown in Figure 7.12, Yang and co-

authors used the tangential technique proposed by Yang et al. (2004b) in conjunction

with the assumed failure mechanism proposed by Chen (1975). The optimised height of

a slope with Hoek-Brown rock strength parameters can be obtained using tangential

parameters ( tc and t ). Yang et al. (2004b) proposed that, for a given friction angle

( t ), tc can be expressed as:

tbt

tb

t

b

t

t

tbt

ci

t

m

sm

m

mc

tansin2

)sin1(

sin1

tan

sin2

)sin1(

2

cos

)11(

)1(

(7.4)

where s , bm and are from the latest version of Hoek-Brown yield criterion shown in

Equation (2.2). Therefore, the new tangential Mohr-Coulomb yield criterion can be

expressed as:

ci

tt

ci

n

ci

c

tan (7.5)

To transfer the yield surface from the cinci plane to the major and minor

principal stress plane cici 31 , the following equation can be utilised:

ci

ci

ci

c

31

sin1

sin1

sin1

cos2

(7.6)

where c and are the cohesion and friction angle in the n plane. It should be

noted that t in Equation (7.5) is unknown. It is determined by the optimisation of the

smallest slope height, which is obtained as presented by Chen (1975):

)(sintan)(exp)(sin

)(

1tan)(2exp

tan)(sin2

sin

00

87654321

0'

'

thh

h

tht

t ffffkffff

cH

(7.7)

Equation (7.7) has been formulated in FORTRAN and optimised in this study using the

Hooke-Jeeves algorithm which is a function of three variables, namely h , 0 and '



7-5

for specifying the assumed failure mechanism and t for specifying the location of the

tangential point. More details on how the parameters ( h , 0 and ' ) are determined

and the definitions of the parameters 81 ff can be found in Chen (1975) and Yang et

al. (2004b).

Figure 7.5 displays the stability factors for 10GSI , 30, 50 and 70 (square symbols) of

Yang et al. (2004b) obtained by using Equation (7.1). As far as the author is aware, the

studies of Yang et al. (2004a) and Yang et al. (2004b) represent the only attempt at

providing stability factors for rock slopes. The method of Yang et al. (2004b) will be

used extensively in subsequent section for comparison purposes.


analysis solutions

A comparison of the average upper and lower bound solutions against the results of

tangential method from Yang et al. (2004b) is displayed in Figure 7.3 to Figure 7.6. It is

found in Figure 7.5 that the stability numbers of Yang et al. (2004b) may be larger

( 10GSI ), equal ( 30GSI ) or smaller ( 50GSI ) than the average upper and lower

bound solutions. As discussed by Yang et al. (2004a), the limit load computed from

tangential method will be an upper bound on the actual limit load. This is because the

tangential line circumscribes the actual yield surface. Therefore, the stability numbers of

tangential method should be equal or smaller than the bound solutions presented here. In

Figure 7.5, the tangential method results of Yang et al. (2004b) are found to be

unreasonable for lower GSI because the stability numbers exceed the solutions of the

numerical upper bound on N .

In attempt to resolve the above mentioned contradiction, the tangential technique is

utilised to examine the stability factors of Yang et al. (2004b). The cross symbols in

Figure 7.3 to Figure 7.6 are the stability numbers ( 10GSI , 50 and 100) obtained in

this study using the tangential method proposed by Yang et al. (2004b). It is observed

that the cross symbols are obviously smaller than those from the numerical upper and

lower bound method. Therefore, it is suggested that there must be errors in the upper

bound analyses using the tangential technique done by Yang et al. (2004b). Referring to

Figure 7.3 to Figure 7.6, the difference in N between the tangential method



7-6

(implemented here) and the average bound solutions is found to increase slightly as

GSI or slope angle ( ) increases.


The stability charts illustrated in Figure 7.2 to Figure 7.6 provide an efficient method to

determine the factor of safety ( F ) for a rock slope. The following example is of a slope

constructed in a very poor quality rock mass. It has the following parameters: the slope

angle 60 , the height of the slope mH 50 , the intact uniaxial compressive

strength MPaci 10 , geological strength index 30GSI , intact rock yield parameter

8im , and unit weight of rock mass 323 mkN . With this information, the safety

factor ( F ) of this rock slope can be obtained using the following procedure:

From the values of ci , and H , a dimensionless parameter 502310000 Hci

is calculated to be 8.7.

In Figure 7.5, 6.4 FHN ci .

The factor of safety can be calculated as 9.16.47.8 F .

Alternatively, using the safety factor assessment based on GSI in Figure 7.9(d), the

obtained 18FGSI (approximation).

The factor of safety then can be calculated as 7.11830 F .

Although the above obtained factors of safety, 9.1F and 1.7 imply the slope is stable,

it should be noted that they have different meanings due to their definitions of safety

factors are based on Equations (7.1) and (7.2) respectively. For this example, the factor

of safety on the uniaxial compressive strength is 1.9 whereas the factor of safety on the

value of GSI is 1.7.

7.4 LIMIT ANALYSIS SOLUTIONS FOR ROCKFILL

SLOPES

As discussed in Section 2.7.3, Douglas (2002) found the Hoek-Brown failure criterion

to underestimate the shear strength of poor rock masses (rockfill). Therefore, a failure



7-7

criterion was proposed for rockfill slopes based on a large number of experimental

results, as outlined in Section 2.7.3.

Figure 7.13 shows the average stability numbers for various slope angles ( 30 - 75 )

based on the Douglas criterion. It should be noted that these chart solutions are bounded

by the upper and lower bound analysis within 9 % which is similar to the accuracy of

results obtained with the Hoek-Brown failure criterion. As expected, the stability

numbers increase with an increase of slope inclination ( ). Compared with the charts

solutions in Figure 7.2 to Figure 7.6, the trends and magnitudes of average stability

numbers in Figure 7.13 were found to be different. The tangential method of Yang et al.

(2004b) was employed here with the Douglas criterion (Douglas (2002)) to examine the

solutions from the numerical limit analysis. As illustrated by triangular symbols, the

stability numbers of the tangential method are lower than those of the average upper and

lower bound limit analysis. This is reasonable as the tangential method is based on the

upper bound theorem alone, and therefore, the higher values of the limit load

correspond to the lower values of stability factor.

Figure 7.13 shows that the stability numbers decrease with increasing im for steep

slopes ( 60 and 75 ). This trend is similar to the results in Figure 7.5 and Figure

7.6, however the magnitudes are different. The magnitudes of the stability numbers

follow the difference between Hoek-Brown and Douglas yield surfaces. It implies that

the Douglas criterion is “smaller” than the Hoek-Brown criterion for high GSI ,

“similar” for medium GSI and “larger” for low GSI , as shown in Figure 7.14. It should

be noted that the definitions of im are different for the Hoek-Brown and Douglas failure

criteria. im is material constant for Hoek-Brown yield criterion. However, it is the ratio

of tici (Equation (2.12)) for the Douglas failure criterion.

Referring to Figure 7.13, it is found that the average lines are flat between 5im and

25im for 10GSI and between 5im and 10im for 20GSI . This is due to

the native form of Douglas failure criterion given by expression (2.14) where 5.2bm

is required. For the case of 30 (Figure 7.13(a)), the average stability numbers were

found to increase with im increasing for 25im . This is also due to the native form of



7-8

Douglas failure criterion, as shown in Figure 7.15 where the yield surface for 40im

may be “larger”, “equal” or “smaller” than those for 10im and 3im at different

levels of stress conditions ( cc 31 ). For 25im , most stress conditions

obtained from the lower bound slip surface fall in the region with higher

cc 31 where the rock mass with higher im has the “smaller” yield surface. In

addition, attention should be paid in Figure 7.13(a) to the cases with 50GSI as the

rock slope with lower GSI may have a lower stability number. This means that a larger

safety factor can be obtained. As discussed above, it is induced by the native form of

Douglas failure criterion. However, it is difficult to judge whether these trends may

exist in rockfill slopes. To examine this question, more experimental studies are

required in the future.

Figure 7.16 displays that, for given rock mass properties, the stress states from the lower

bound plastic zone for the slope with higher scatter at the lower level of

cc 31 where the rock mass with higher GSI or im generally has the larger

yield surface. This explains why the trends of the stability chart solutions in Figure 7.13

for 60 and 75 are totally different from those for 30 and 45 .

In Figure 7.13(a), the stability number for 10GSI and 30im does not differ from

that of 40GSI and 30im significantly. This can be explained using Figure 7.17

where it is shown that the stress states at collapse of these two rock slopes are quite

similar. In addition, it can be seen in Figure 7.17 that the yield surface of the rock mass

with 40GSI and 30im is smaller than that of 10GSI and 30im at a higher

stress level ( 5.23 c ). This is the explanation of the fact that in Figure 7.13(a) the

rock masses with lower GSI ( 20GSI ) and im may have a lower stability number as

the slope fails at higher stresses which implies that the larger factor of safety may be

obtained.

Referring to Figure 7.13, the magnitudes of the stability numbers for gentle slope

( 30 and 45 ) are found to be similar for a given GSI ( GSI 30-100) and im

between 15 and 35. This is due to the fact that the stress states for these two cases are

located near the intersection of two yield curves shown by point A in Figure 7.15.



7-9

Comparing to the chart solutions based on the Hoek-Brown failure criterion (Figure 7.4

to Figure 7.6), it is shown that using the Douglas criterion results in different solutions.

It should be mentioned here that the Hoek-Brown failure criterion was proposed for

natural or cut rock slopes, while the Douglas criterion is suitable for rockfill slopes (i.e.

dams). This may explain the discrepancy in obtained results. Due to the different

purposes of the Hoek-Brown and Douglas yield criteria, it was found not appropriate to

make more detailed comparisons. It is suggested that plots in Figure 7.4 to Figure 7.6

are used for natural or cut rock slopes and those in Figure 7.13 are used for rockfill

slopes.

7.5 LIMIT EQUILIBRIUM SOLUTIONS

In general, rock slope stability is more often analysed using the limit equilibrium

method and equivalent Mohr-Coulomb parameters as determined by Equation (2.8) and

Equation (2.9). With this being the case, an obvious question is how do the limit

equilibrium results using equivalent Mohr-Coulomb parameters compare to the limit

analysis results using the Hoek-Brown criterion. In order to make this comparison, the

commercial limit equilibrium software SLIDE (Rocscience (2005)) and Bishop’s

simplified method (Bishop (1955)) have been used. The software SLIDE can perform a

slope analysis using the Mohr-Coulomb yield or the generalised Hoek-Brown criterion.

When the Mohr-Coulomb criterion is used, the cohesion ( c ) and friction angle ( ) are

constant along any given slip surface and are independent of the normal stress as

expected. However, when the Hoek-Brown criterion is selected, the software will

calculate a set of instantaneous equivalent Mohr-Coulomb parameters when analysing

the slope based on the normal stress at the base of each individual slice. More details on

how the parameters are actually calculated can be found in Hoek (2000). Therefore, the

cohesion ( c ) and the friction angle ( ) will vary along any given slip surface. By

calculating equivalent Mohr-Coulomb parameters in this way, a more accurate

representation of the curved nature of the Hoek-Brown criterion in - n space is

obtained.



7-10

7.5.1 Comparisons of the generalized Hoek-Brown model and the Mohr-

Coulomb model

Referring to the Figure 7.4 to Figure 7.6, the triangular points shown represent the

stability numbers obtained from the limit equilibrium method (SLIDE) based on Hoek-

Brown strength parameters. It can be found that these points are remarkably close to the

average lines of the limit analysis solutions and most of them locate between the upper

and lower bound solutions.

For the given materials and geometrical properties of the slope, the finite element lower

bound analysis will provide the optimum unit weight ( ) such that collapse has just

occurred (i.e. Factor of safety 1F ). A critical non-dimensional parameter

critci H can then be defined for the subsequent SLIDE analyses. In Table 7.1, the

safety factor ( 1F ) and ( 2F ) are obtained using the Hoek-Brown criterion and the

Mohr-Coulomb criterion in SLIDE, respectively. Both of these analyses are based on

equivalent Mohr-Coulomb parameters with the only difference being how these

parameters are calculated (as discussed above).

The comparisons of the safety factors F , 1F and 2F are shown in Table 7.1 where the

largest difference between F and 1F and F and 2F are about 4% and 64%,

respectively. This shows that the results of SLIDE analyses using the Hoek-Brown

model compare well with the results of the lower bound limit analyses. In contrast the

results of SLIDE analyses using the Mohr-Coulomb model do not compare favourably

with the lower bound results. From Table 7.1, it can be found that using the Mohr-

Coulomb model may lead to significant overestimations of safety factors, particularly

for steep slopes. The average difference between F and 2F for 60 and 75

was found to be 16.8% and 34.3% respectively. For all the cases, the average

overestimation is 12.8%. It should be stressed that, a high estimation of safety factor

will induce a non-conservative design. It was found that using the Hoek-Brown model

in SLIDE will produce a failure mechanism in good agreement with the upper bound

mechanism. The same could not be said when using the Mohr-Coulomb model. For

30 , both of the two above models achieve similar failure surfaces which agree

well with the upper bound plastic zone. In almost all cases, a toe-failure mode was

observed, the only exception is the case of 15 (base-failure).



7-11

7.5.2 Modification of the equivalent Mohr-Coulomb parameters

In order to determine the source of overestimations in factors of safety ( 2F ) for steep

slopes, the stress conditions on each slice from the SLIDE limit equilibrium analyses

were observed more closely. It was found that, for steep slopes, the stress conditions of

the slices along the failure plane tend to be located in REGION 1 (Figure 2.9) where

the shape of the Hoek-Brown and Mohr-Coulomb strength criterions differ the greatest.

In this region, at the same normal stress, the ultimate shear strength using the Hoek-

Brown criterion is smaller than that of the Mohr-Coulomb criterion. Therefore, it is

reasonable to conclude that using the equivalent Mohr-Coulomb parameters will

provide a higher estimate of the safety factor.

From the results of this study, it appears that the equivalent parameters ( c and )

obtained from Equation (2.8) to Equation (2.11) will lead to an unconservative factor of

safety estimate, particular for steep slopes where 45 . In order to improve the

estimate of 2F , it becomes apparent a better estimate of 'max3 , and therefore a

different form of Equation (2.10), is required.

To determine a more appropriate value of 'max3 to be used in Equation (2.8) and

Equation (2.9), a similar study as performed by Hoek et al. (2002) is conducted. In these

studies, Bishop’s simplified method and SLIDE is used to analyse the cases in Table

7.1. For a factor of safety of 1, the relationship between Hcm ' and ''max3 cm is as

illustrated in Figure 7.18 and Figure 7.19. In this research, a fit of only one equation

incorporating all data to replace Equation (2.10) was found to be unsatisfactory. Instead

separate equations are presented for what is defined as steep slopes 45 and gentle

slopes 45 as Equation (7.8) and Equation (7.9) respectively.

07.1'

'

'max3 2.0

Hcm

cm

(Steep slope 45 ) (7.8)

23.1'

'

'max3 41.0

Hcm

cm

(Gentle slope 45 ) (7.9)



7-12

It can be seen in Figure 7.18 and Figure 7.19 the newly fitted Equation (7.8) for steep

slopes plots below the original Equation (2.10) and the newly proposed Equation (7.9)

for gentle slopes plots above the original Equation (2.10). For this reason, it is apparent

that one curve fit is not suitable for all slope angles.

In Table 7.1, the safety factors 3F and 4F are obtained from SLIDE using Mohr-

Coulomb parameters which are calculated by estimating 'max3 from Equation (7.8) and

Equation (7.9). Comparing 3F and 4F with 2F , shows that for steep slope, the safety

factors estimates are much improved. A summary of the results in Table 7.1 shows that,

using newly proposed equations to calculate the equivalent Mohr-coulomb parameters,

the largest difference of safety factor has decreased from 64% to 21% and the average

difference has reduced from 12.8% to 3.4%. Thus, it can be concluded that using the

modified Equation (7.8) and Equation (7.9) will provide better results of safety factors

which are on average only 3.4% higher than the lower bound results. The newly

proposed Equation (7.8) and Equation (7.9) are both applicable in estimating 'max3 for

45 cases. The results show that the difference in safety factor between these two

equations is less than 8%. This would be acceptable for preliminary assessment of rock

slope stability.

Figure 7.20 displays the upper bound plastic zones compared with failure surfaces

obtained using SLIDE with different strength parameters from Equation (7.8) and

Equation (7.9). 1F , 2F , and 3F denote the safety factors obtained from using the

Hoek-Brown ( ci , GSI , im , D ), the original equivalent Mohr-Coulomb (proposed in

Hoek et al. (2002)), and the new equivalent Mohr-Coulomb (proposed in this Chapter)

strength parameters, respectively. Moreover, the slip surfaces obtained from the

tangential method are also displayed in Figure 7.20.

It is shown in Figure 7.20 that using the original estimated Mohr-Coulomb parameters

in analyses gives poor assessment of the stability and predictions of failure surfaces for

steep slopes. On the other hand, by using the new proposed equivalent Mohr-Coulomb

parameters the predicted failure mechanism compares more favourably to the upper

bound mechanism and the factor of safety is much improved. The slip surfaces from

tangential method almost located between those obtained using the original estimated



7-13

and newly proposed Mohr-Coulomb parameters. However, it is known in Section 7.3.3

that the factors of safety are overestimated based on this method.

7.6 CONCLUSIONS

Stability charts based on the Hoek-Brown and Douglas failure criteria have been

presented using formulations of the upper and lower bound limit theorems. These chart

solutions can be used for estimating rock slope stability for preliminary design. It is

important that users understand the assumptions and limitations before using these new

rock slope stability charts. In particular, it should be noted that the chart solutions

proposed in this Chapter are applicable to isotropic rock or rock masses only. Regarding

the results of this study, the following conclusions can be made:

1. The general mode of failure for rock slopes was observed to be of the toe-failure

type, except for the case of 15 , where a base-failure type was observed.

2. The accuracy of using equivalent Mohr-Coulomb parameters for the rock mass

in a traditional limit equilibrium method of slice analysis has been investigated.

It was found that the factor of safety can be overestimated by up to 64% for

steep slopes if existing guidelines for equivalent parameter determination are

used. In order to improve the factor of safety estimate, two modified equations

for steep and gentle slopes have been proposed. These equations are

modifications of those originally proposed by Hoek et al. (2002). When they are

used to determine equivalent Mohr-Coulomb parameters that are subsequently

used in a method of slice analysis, the factor of safety estimate is much

improved and is at most 21% above the limit analysis result.

3. It was found that a limit equilibrium method of slice analysis can be used in

conjunction with equivalent Mohr-Coulomb parameters to produce factor of

safety estimates close to the limit analysis results, provided modifications are

made to the underlying formulations. Such modifications have been made in the

software SLIDE where a set of equivalent Mohr-Coulomb parameters are

calculated at the base of each individual slice. This approach predicts factors of

safety remarkably close to the limit analysis solutions that are based on the

native form of the Hoek-Brown criterion.



7-14

Table 7.1 Comparisons of safety factors between the Hoek-Brown strength parameters and the equivalent Mohr-Coulomb parameters

LIMIT ANALYSIS - LOWER BOUND

SLIDE - Limit Equilibrium using equivalent Mohr-Coulomb Parameters

Nonlinear Hoek-Brown

Nonlinear Hoek-Brown Eqs. (2.8),(2.9) & (2.10) Eqs. (2.8),(2.9) & (7.8) Eqs. (2.8),(2.9) & (7.9) Linear Mohr-Coulomb Linear Mohr-Coulomb Linear Mohr-Coulomb

β GSI mi critci H F F1 %Diff F2 %Diff F3 %Diff F4 %Diff

75 100 5 0.360 1 0.963 -3.7% 1.008 1% 1.028 3% - - 75 100 15 0.278 1 0.999 -0.1% 1.164 16% 1.042 4% - - 75 100 25 0.228 1 1.002 0.2% 1.218 22% 1.079 8% - - 75 100 35 0.194 1 1.004 0.4% 1.286 29% 1.112 11% - - 75 70 5 1.703 1 0.988 -1.2% 1.081 8% 1.025 2% - - 75 70 15 1.169 1 1.002 0.2% 1.287 29% 1.081 8% - - 75 70 25 0.890 1 1.005 0.5% 1.35 35% 1.124 12% - - 75 70 35 0.717 1 1.016 1.6% 1.394 39% 1.156 16% - - 75 50 5 4.980 1 0.997 -0.3% 1.154 15% 1.036 4% - - 75 50 15 2.988 1 1.004 0.4% 1.336 34% 1.119 12% - - 75 50 25 2.156 1 1.018 1.8% 1.425 43% 1.148 15% - - 75 50 35 1.668 1 1.024 2.4% 1.45 45% 1.174 17% - - 75 30 5 15.011 1 1.001 0.1% 1.248 25% 1.047 5% - - 75 30 15 8.576 1 1.016 1.6% 1.459 46% 1.136 14% - - 75 30 25 5.824 1 1.025 2.5% 1.51 51% 1.173 17% - - 75 30 35 4.327 1 1.033 3.3% 1.516 52% 1.194 19% - - 75 10 5 93.721 1 1.004 0.4% 1.224 22% 1.018 2% - - 75 10 15 53.362 1 1.023 2.3% 1.504 50% 1.126 13% - -



7-15







75 10 25 35.186 1 1.035 3.5% 1.605 61% 1.185 19% - - 75 10 35 24.994 1 1.046 4.6% 1.642 64% 1.21 21% - - 60 100 5 0.232 1 1.001 0.1% 1.033 3% 1.043 4% - - 60 100 15 0.130 1 1.004 0.4% 1.114 11% 1.026 3% - - 60 100 25 0.088 1 1.004 0.4% 1.146 15% 1.035 3% - - 60 100 35 0.066 1 1.004 0.4% 1.141 14% 1.04 4% - - 60 70 5 0.946 1 1.013 1.3% 1.059 6% 1.024 2% - - 60 70 15 0.435 1 1.004 0.4% 1.143 14% 1.033 3% - - 60 70 25 0.276 1 1.004 0.4% 1.161 16% 1.043 4% - - 60 70 35 0.200 1 1.005 0.5% 1.183 18% 1.047 5% - - 60 50 5 2.337 1 1.005 0.5% 1.124 12% 1.026 3% - - 60 50 15 0.953 1 1.004 0.4% 1.171 17% 1.036 4% - - 60 50 25 0.584 1 1.008 0.8% 1.176 18% 1.046 5% - - 60 50 35 0.419 1 1.009 0.9% 1.172 17% 1.049 5% - - 60 30 5 6.439 1 1.009 0.9% 1.15 15% 1.023 2% - - 60 30 15 2.317 1 1.009 0.9% 1.197 20% 1.044 4% - - 60 30 25 1.356 1 1.01 1.0% 1.201 20% 1.049 5% - - 60 30 35 0.945 1 1.011 1.1% 1.23 23% 1.051 5% - -



7-16







60 10 5 38.926 1 1.004 0.4% 1.183 18% 1.013 1% - - 60 10 15 11.734 1 1.013 1.3% 1.257 26% 1.048 5% - - 60 10 25 5.928 1 1.017 1.7% 1.261 26% 1.054 5% - - 60 10 35 3.729 1 1.018 1.8% 1.258 26% 1.059 6% - - 45 100 5 0.135 1 1 0.0% 1.008 1% 1.022 2% 1.027 3% 45 100 15 0.058 1 1.005 0.5% 1.041 4% 1.003 0% 1.086 9% 45 100 25 0.036 1 1.012 1.2% 1.047 5% 1.003 0% 1.11 11% 45 100 35 0.026 1 1.015 1.5% 1.06 6% 1.005 0% 1.126 13% 45 70 5 0.469 1 1.001 0.1% 1.038 4% 1.001 0% 1.055 5% 45 70 15 0.176 1 1.012 1.2% 1.08 8% 1.002 0% 1.098 10% 45 70 25 0.108 1 1.017 1.7% 1.06 6% 1.007 1% 1.113 11% 45 70 35 0.077 1 1.019 1.9% 1.061 6% 1.009 1% 1.123 12% 45 50 5 1.046 1 1.004 0.4% 1.045 4% 1.001 0% 1.063 6% 45 50 15 0.369 1 1.009 0.9% 1.065 6% 1.004 0% 1.098 10% 45 50 25 0.222 1 1.02 2.0% 1.066 7% 1.01 1% 1.11 11% 45 50 35 0.158 1 1.021 2.1% 1.044 4% 1.011 1% 1.118 12% 45 30 5 2.593 1 1.011 1.1% 1.066 7% 0.999 0% 1.06 6% 45 30 15 0.829 1 1.018 1.8% 1.07 7% 1.007 1% 1.094 9%



7-17







45 30 25 0.480 1 1.021 2.1% 1.076 8% 1.01 1% 1.11 11% 45 30 35 0.334 1 1.024 2.4% 1.085 9% 1.011 1% 1.118 12% 45 10 5 13.585 1 1.014 1.4% 1.087 9% 1 0% 1.039 4% 45 10 15 3.155 1 1.023 2.3% 1.106 11% 1.005 0% 1.08 8% 45 10 25 1.552 1 1.023 2.3% 1.107 11% 1.009 1% 1.103 10% 45 10 35 0.969 1 1.026 2.6% 1.079 8% 1.01 1% 1.115 12% 30 100 5 0.070 1 1.014 1.4% 0.988 -1% - - 1 0% 30 100 15 0.026 1 1.02 2.0% 0.999 0% - - 1.024 2% 30 100 25 0.016 1 1.023 2.3% 1.003 0% - - 1.036 4% 30 100 35 0.011 1 1.024 2.4% 1.007 1% - - 1.044 4% 30 70 5 0.218 1 1.018 1.8% 0.985 -2% - - 1.011 1% 30 70 15 0.075 1 1.023 2.3% 0.996 0% - - 1.028 3% 30 70 25 0.045 1 1.024 2.4% 1.004 0% - - 1.035 3% 30 70 35 0.032 1 1.025 2.5% 1.01 1% - - 1.04 4% 30 50 5 0.461 1 1.02 2.0% 0.993 -1% - - 1.014 1% 30 50 15 0.153 1 1.024 2.4% 1.003 0% - - 1.026 3% 30 50 25 0.091 1 1.025 2.5% 1.024 2% - - 1.032 3% 30 50 35 0.065 1 1.026 2.6% 1.008 1% - - 1.036 4%



7-18







30 30 5 1.057 1 1.022 2.2% 1.001 0% - - 1.012 1% 30 30 15 0.323 1 1.026 2.6% 1.003 0% - - 1.026 3% 30 30 25 0.185 1 1.026 2.6% 1.005 0% - - 1.031 3% 30 30 35 0.129 1 1.027 2.7% 1.004 0% - - 1.035 3% 30 10 5 4.363 1 1.023 2.3% 1.002 0% - - 1.006 1% 30 10 15 0.943 1 1.025 2.5% 1.007 1% - - 1.023 2% 30 10 25 0.460 1 1.026 2.6% 0.996 0% - - 1.033 3% 30 10 35 0.286 1 1.026 2.6% 1.004 0% - - 1.04 4% 10 100 5 0.026 1 1.009 0.9% 1.067 7% - - 1 0% 10 100 15 0.009 1 1.011 1.1% 1.079 8% - - 0.987 -1% 10 100 25 0.005 1 1.011 1.1% 1.091 9% - - 0.985 -2% 10 100 35 0.004 1 1.012 1.2% 1.094 9% - - 0.986 -1% 10 70 5 0.078 1 1.01 1.0% 1.069 7% - - 0.994 -1% 10 70 15 0.026 1 1.01 1.0% 1.087 9% - - 0.987 -1% 10 70 25 0.015 1 1.011 1.1% 1.091 9% - - 0.985 -2% 10 70 35 0.011 1 1.011 1.1% 1.094 9% - - 0.985 -2% 10 50 5 0.158 1 1.01 1.0% 1.067 7% - - 0.996 0% 10 50 15 0.052 1 1.01 1.0% 1.055 5% - - 0.989 -1%



7-19







10 50 25 0.031 1 1.011 1.1% 1.081 8% - - 0.986 -1% 10 50 35 0.022 1 1.011 1.1% 1.084 8% - - 0.985 -2% 10 30 5 0.334 1 1.01 1.0% 1.05 5% - - 0.997 0% 10 30 15 0.101 1 1.011 1.1% 1.068 7% - - 0.99 -1% 10 30 25 0.058 1 1.011 1.1% 1.072 7% - - 0.988 -1% 10 30 35 0.040 1 1.011 1.1% 1.075 8% - - 0.986 -1% 10 10 5 0.994 1 1.012 1.2% 1.036 4% - - 0.994 -1% 10 10 15 0.211 1 1.013 1.3% 1.039 4% - - 0.985 -2% 10 10 25 0.103 1 1.013 1.3% 1.041 4% - - 0.985 -2% 10 10 35 0.064 1 1.013 1.3% 1.032 3% - - 0.986 -1%



7-20

Toe

Rigid Base

d

Jointed Rock

ciGSI,m

i

H

Figure 7.1 Problem configuration for simple homogeneous slopes



7-21

5 10 15 20 25 30 351E-3

0.01

0.1

1

10

N= ci

/H

F Average SLIDE-Hoek-Brown Model

GSI=50

GSI=100

GSI=10

H

= 15

mi

Increasing Stability

Figure 7.2 Average finite element limit analysis solutions for stability numbers

( 15 )



7-22

5 10 15 20 25 30 351E-3

0.01

0.1

1

10 = 30

Average SLIDE-Hoek-Brown Model Tangential Method

GSI=50

GSI=100

GSI=10

N= ci

/H

F

H

mi



( 30 )



7-23

5 10 15 20 25 30 350.01

0.1

1

10

100

Average Lower bound Upper bound SLIDE-Hoek-Brown Model Tangential Method

GSI=50

GSI=100

GSI=10

mi

N= ci

/H

F

= 45

H



( 45 )



7-24

5 10 15 20 25 30 350.01

0.1

1

10

100

Average SLIDE-Hoek-Brown Model Tangential Method (this study) Tangential Method (Yang et al.)

GSI=50

GSI=100

GSI=10

= 60

H

N= ci

/H

F

mi



( 60 )



7-25

5 10 15 20 25 30 350.01

0.1

1

10

100


GSI=50

GSI=100

GSI=10

N= ci

/H

F

= 75

mi

H



( 75 )



7-26

5 10 15 20 25 30 351E-3

0.01

0.1

1

= 75 = 60 = 45 = 30 = 15

N=

ci/

HF

mi

GSI=100

H

5 10 15 20 25 30 35

0.01

0.1

1

= 75 = 60 = 45 = 30 = 15

N= ci

/H

F

mi

GSI=80

H

5 10 15 20 25 30 350.01

0.1

1

= 75 = 60 = 45 = 30 = 15

N= ci

/H

F

GSI=60

mi

H


( GSI 100, 80 and 60)



7-27

5 10 15 20 25 30 350.01

0.1

1

10

= 75 = 60 = 45 = 30 = 15

mi

N= ci

/H

F

GSI=40

H

5 10 15 20 25 30 350.01

0.1

1

10

= 75 = 60 = 45 = 30 = 15

mi

N=

ci/

HF

GSI=20

H

5 10 15 20 25 30 35

0.1

1

10

100

= 75 = 60 = 45 = 30 = 15

mi

H

GSI=10

N=

ci/

HF


( GSI 40, 20 and 10)



7-28

20 40 60 80 100

0.01

0.1

1

mi = 35

mi = 5

= 15

ci/

H

N=GSI / F

H

20 40 60 80 100

0.01

0.1

1

= 30

mi = 35

mi = 5

ci/

H

N=GSI / F

H

(a) 15 (b) 30

20 40 60 80 1000.01

0.1

1

10

= 45

mi = 35

mi = 5

ci/

H

N=GSI / F

H

20 40 60 80 100

0.1

1

10

= 60

mi = 35

mi = 5

ci/

H

N=GSI / F

H

(c) 45 (d) 60

Figure 7.9 Factor of safety assessment based on GSI



7-29

20 40 60 80 1000.1

1

10

100 = 75

mi = 35

mi = 5

ci/

H

N=GSI / F

H

(e) 75

Figure 7.9 (continued) Factor of safety assessment based on GSI



7-30

Figure 7.10 Upper bound plastic zones for different slope angles ( 70GSI and

15im )

45

30

15



7-31

Figure 7.11 Upper bound plastic zones for different im values ( 60 and

70GSI )

5im 15im

25im 35im



7-32

ct

She

ar s

tres

s (

)

Normal stress (n)

t

(a) Tangential method

(b) Logarithmic spiral failure mechanism Chen (1975)

Figure 7.12 Illustration of adopted tangential method and failure mechanism (Yang et

al. (2004b))

H

ho

hk

vk



7-33

5 10 15 20 25 30 351E-3

0.01

0.1

1

GSI = 100

GSI = 30-100 GSI = 20 GSI = 10 Tangential method (GSI = 50)

N= ci

/H

F

= 30

mi

H

GSI = 30

(a) 30

5 10 15 20 25 30 350.01

0.1

1

10


mi

N= ci

/H

F

= 45HGSI = 100

GSI = 30

(b) 45

Figure 7.13 Average upper and lower bound solutions based on Douglas failure

criterion



7-34

5 10 15 20 25 30 35

0.1

1

10

100

mi


N= ci

/H

F

= 60HGSI = 100

GSI = 30

(c) 60

5 10 15 20 25 30 350.1

1

10

100

N= ci

/H

F

mi


= 75H

GSI = 100

GSI = 30

(d) 75

Figure 7.13 (continued)



7-35

Douglas

Figure 7.14 Comparison of Douglas criterion and Hoek-Brown criterion for 40im

Figure 7.15 Illustration of yield envelopes for different im on cc 31 plane

Higher stress level

Lower stress level

A

s

m

cicici

'3

'3

'1



7-36

0 1 2 3 4 5 60

2

4

6

8

10

12

Douglas criterion (GSI=40, m

i=10)

/ ci

/

ci

Figure 7.16 Stress conditions for different slope angles ( 40GSI and 10im )

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.00

2

4

6

8

10

12


i=30)

GSI=40, mi=30


i=30)

GSI=10, mi=30

/ ci

/

ci

Figure 7.17 Comparison of yield surfaces for different GSI



7-37

0.01 0.1 1

0.01

0.1

1

SLIDE results

'

3max'

cm'

cmH-1.07

'

3max'

cm'

cmH-0.91

Ratio of '

cmH

Rat

io o

f

' 3max

' cm

Figure 7.18 Relationship for the calculation of 'max3 between equivalent Mohr-

Coulomb and Hoek-Brown parameters for steep slopes ( 45 )



7-38

1E-3 0.01 0.1

1

10

100

SLIDE results

'

3max'

cm'

cmH-1.23

'

3max'

cm'

cmH-0.91

Ratio of '

cmH

Rat

io o

f

' 3max

' cm

Figure 7.19 Relationship for the calculation of 'max3 between equivalent Mohr-

Coulomb and Hoek-Brown parameters for gentle slopes ( 45 )



7-39

Tangential method

F3=1.047

F2=1.183

F1=1.005

F3=1.156

F2=1.394

F1=1.016

Figure 7.20 Comparison between upper bound plastic zones and failure surfaces from

cases with different strength parameters ( 70GSI and 35im )



7-40



8-1

CHAPTER 8 SEISMIC STABILITY OF HOMOGENEOUS

ROCK SLOPES

8.1 INTRODUCTION

In this Chapter, the seismic stability of rock slopes governed by the Hoek-Brown failure

criterion (Hoek et al. (2002)) will be investigated by applying the finite element upper

and lower bound techniques with the aim of providing seismic stability charts for rock

slopes. The seismic effects on rock slope stability are, therefore simulated using the

pseudo static (PS) method. The PS analyses in this study do not account for the effects

of pore pressure, and the strength of rock masses is assumed to be unaffected during

earthquake excitation.

For the purpose of comparison, the limit equilibrium method will then be used based on

the equivalent Mohr-Coulomb parameters for the rock and the results will be plotted

against the solutions obtained from the numerical limit analysis approaches. The

stability number as given by Equation (7.1) is adopted in this Chapter.

The general configuration of the problem to be analysed is shown in Figure 8.1 where

the jointed rock mass has an intact uniaxial compressive strength ci , geological

strength index GSI , intact rock yield parameter im , and unit weight . All the

quantities are assumed constant throughout the slope. In the limit analyses, the seismic

force is assumed as a horizontal internal body force and its magnitude is represented by

the horizontal seismic coefficient ( hk ). The direction of Wkh , is considered to be

positive when acting outward with respect to the slope (see Figure 8.1). In this Chapter,

slope inclinations of 60,45,30 and 75 are analysed.

8.2 LIMIT ANALYSIS SOLUTIONS

8.2.1 Chart solutions

Figure 8.2 to Figure 8.4 present three sets of stability charts obtained from the numerical

upper and lower bound formulations with horizontal seismic coefficients of hk 0.1,

0.2 and 0.3, respectively. The magnitude of hk in current design codes is generally



8-2

within this range. For all analyses performed in this study, the maximum difference

between the bound solutions was found to be less than 9 %. As a consequence,

average values of the upper and lower bound stability numbers will be used in the

following discussions.

Referring to Figure 8.2 to Figure 8.4, it can be observed that the stability number N

decreases when GSI or im increases, as expected. Remembering from Equation (7.1),

the stability number N is proportional to the inverse of the factor of safety F . A lower

factor of safety correspond to a higher stability number and visa-versa. The direction of

increasing stability is shown by the arrow in Figure 8.2(a). A decrease in stability

number with increasing GSI or im is not unexpected because, based on the definition

of the Hoek-Brown yield criterion, the larger magnitudes of GSI or im signify that the

rock masses have greater overall strength for any given normal stress. However, one

exception to this trend is shown in Figure 8.4(d) where N increases slightly with

increasing im . This phenomenon will be discussed in more detail below.

The chart solutions can also be presented in an alternative form which is a function of

the slope angle ( ) as shown in Figure 8.5 where N can be seen to increase when the

inclination of a slope increases. For a given slope inclination, the stability number can

be obtained by estimating GSI and im . For the same rock mass properties of a slope,

the difference in stability numbers between various slope angles can provide the

corresponding variation in factor of safety. For instance, it can be observed in Figure 8.5

that decreasing slope angle from 75 to 60 can increase the safety factors by

%100%50 for 5im . This trend is similar to the results of Chapter 7 for the static

slopes in which reducing the slope angle was found to increase the factor of safety by

more than 50%.

Figure 8.6 and Figure 8.7 display the stability numbers from the PS analyses compared

to the results of these static analyses. The average lines shown in these figures with

1.0hk represent the stability numbers obtained when the PS force acts toward the

rock slope face. Understandably, when the PS force acts toward the slope, this was

found to increase the stability of the slope as seen in Figure 8.7. In general, the rock

slopes and the potential epicentres such as faults or volcanos can scatter around in



8-3

seismically active region, so adopting the more critical Wkh direction (away from the

slope) in the analyses is more appropriate from a design perspective.

Referring to Figure 8.6 and Figure 8.7, it can be found that N increases with increasing

hk . This means that the factors of safety will be smaller as the earthquake loading

increases. In addition, from these figures, the factor of safety is found to decrease by

30% ( 5im ) or more when hk increases by 0.1 for all slope angles.

Figure 8.8 to Figure 8.10 are the stability charts of GSI base estimates (Equation (7.2)).

It can be seen in Figure 8.10(d) that the chart solutions are close to each other for

different im values. This is due to the fact that the average limit analysis results are

almost independent of im , as shown in Figure 8.4(d). As stated above, it was found that

a slope with the lower im has a larger value of N , for the cases of 75 and

3.0hk . Therefore, the line for 5im in Figure 8.10(d) is underneath the line of

35im which is different from other results shown in Figure 8.8 to Figure 8.10.

Figure 8.11 shows several of the observed upper bound plastic zones for different GSI .

In general, the modes of failure consisted of shallow toe type mechanisms for all

analyses. It can be noticed that the depth of slip surface increases only slightly with

increasing GSI . But this phenomenon is not observed when the slope angle 45 . A

similar trend is found where the depth of slip surface increases slightly with the

reduction of im . Moreover, Figure 8.12 indicates that the depth of the plastic zones is

almost unchanged for various seismic coefficients. This means that the shape of the

potential failure surface is almost independent of the magnitude of hk , for a given

geometry and rock strength parameters.

8.2.2 Analytical solutions

As mentioned in Chapter 7, only the stability factors presented by Yang et al. (2004a)

and Yang et al. (2004b) are available for estimating rock slope stability. As shown in

Figure 7.12, Yang and co-authors used the tangential technique proposed by Yang et al.

(2004b) in conjunction with the assumed failure mechanism proposed by Chen (1975).

The optimised height of a slope with Hoek-Brown rock strength parameters can be



8-4

obtained using tangential parameters ( tc and t ). This method will be used to compare

the current limit analysis solutions to the tangential technique in the following sections.

The square symbols displayed in Figure 8.2(c) and Figure 8.3(c) are the stability factors

of Yang et al. (2004b) presented by using Equation (7.1) for 10GSI , 30, 50 and 70.

Comparing with the chart solutions in Figure 7.5, the only difference is that Figure 8.2(c)

and Figure 8.3(c) account for the seismic effects by using the PS method.


analysis solutions

A comparison of the average upper and lower bound solutions against the results of

tangential method from Yang et al. (2004b) is displayed in Figure 8.2(c) and Figure

8.3(c). It is found that the stability numbers of Yang et al. (2004b) may be larger

( 10GSI ), equal ( 30GSI ) or smaller ( 50GSI and 70) than the average upper

and lower bound solutions which is similar to the trend of the static case in Chapter 7.

As pointed out by Yang et al. (2004a), the limit load computed from tangential method

will be an upper bound on the actual limit load as the tangential line circumscribe the

actual yield surface. Therefore, the stability numbers of tangential method should be

equal or smaller than our bound solutions.

To investigate the stability factors proposed by Yang et al. (2004b), the tangential

method of Yang et al. (2004b) is adopted here to check the average solutions of the

numerical upper and lower bound analysis. However, it is shown that most results of

using the tangential technique are obviously smaller than the average solutions of the

upper and lower bound analysis. In Figure 8.2(c) and Figure 8.3(c), the tangential

method results of Yang et al. (2004b) are found to be unreasonable for lower GSI due

to the stability numbers exceeding the solutions of even lower bound analysis.

More comparisons of the stability number between the average solutions from the upper

and lower bound limit analysis and the results from the tangential technique proposed

by Yang et al. (2004b) are displayed in Figure 8.2 and Figure 8.3. The tangential upper

bound solutions are shown as cross symbols by using Equation (7.1) for GSI 10, 50

and 100. It can be observed that most of the cross symbols plot below the newly

obtained average lower and upper bound solutions. This means that using the tangential

approach may overestimate the factor of safety for the rock slope.



8-5

The difference in stability numbers between the average bounding solutions and the

results of the tangential method is found to increase sharply when GSI and slope

inclination increase. For the case of 75 and 100GSI (Figure 8.2(d) and Figure

8.3(d)), the difference can be by up to 80% which is too significant and not to be

ignored. Referring to these two figures, the value of N obtained using the tangential

method with 50GSI is found to be smaller than the average lower and upper bound

solutions with 60GSI . In particular, for larger im ( 25im ), they are close to the

average bound solutions with 70GSI . It was found that the stability number estimate

of tangential technique tends to be unconservative, particularly for larger GSI .

In order to determine the source of overestimation in the factor of safety when using the

tangential method, the stress state at failure from the numerical lower bound results is

observed more closely. The stress points in the vicinity of failure zone have been

extracted and the location of each point on the Hoek-Brown yield envelope can be

observed in Figure 8.13 for 75 and 1.0hk . For comparison purposes, linear

regression and Equation (7.6) are also employed here to obtain the magnitudes of t

and citc from the lower bound solutions.

Table 8.1 displays optimised t and citc values from the tangential approach.

Compared with the lower bound linear regression values, the differences in t and

citc are significant, except for the case of 10GSI . Due to the fact that t and

citc are almost the same for 10GSI and 10im , the difference in stability

numbers between these two cases is small, as shown in Figure 8.4(d).

Referring to Figure 8.13, based on Equation (7.6) and Table 8.1, the yield surface

adopted in tangential method can be plotted in the cici 31 plane. The

difference between the tangential method yield envelope and native Hoek-Brown yield

envelope is found to increase significantly with increasing GSI . This explains the

difference in stability numbers between the average bounding solutions and the results

of tangential method, as described previously. For the case of 50GSI and 100, it is

apparent from Figure 8.13 that the yield envelope from the lower bound solutions plots

below that from the results using the tangential technique. In particular, for 100GSI ,



8-6

the yield envelope of the tangential method is significantly above the Hoek-Brown

failure envelope which will lead to a larger shear strength and thus a larger factor of

safety. This explains the large differences observed in Figure 8.2 and Figure 8.3 and the

unconservative nature of the tangential technique for these cases.

The predicted slip surfaces by using the tangential method (bold line) are compared

with the upper bound plastic zones in Figure 8.11 and Figure 8.12. It can be observed

that the difference between the obtained slip surfaces increases significantly with

increasing GSI , which agrees with the difference in the yield surfaces shown in Figure

8.13. In Figure 8.12, by using the tangential method, the variation of slip surface is also

found to be insignificant for various hk .

8.3 LIMIT EQUILIBRIUM SOLUTIONS

8.3.1 Comparison of chart solutions between the numerical finite element

limit analysis and limit equilibrium analysis

In this Chapter, the commercial limit equilibrium software SLIDE (Rocscience (2005))

and Bishop’s simplified method (Bishop (1955)) have been employed to make

comparisons with the average lower and upper bound solutions.

Referring to Figure 8.2 to Figure 8.4, the displayed triangular points are the stability

numbers obtained using the Hoek-Brown strength parameters based on the limit

equilibrium analysis. It can be seen that most of the results from SLIDE are remarkably

close to the average lines of the upper and lower bound limit analysis solutions.

However, the exception to this observation can be found in the case of 75 , shown

in Figure 8.2(d), Figure 8.3(d) and Figure 8.4(d). In these figures, for lower GSI values

( 50GSI ), the stability numbers obtained using the LEM have been underestimated by

18%-30%, compared to the average bound solutions. Although comparisons between

the upper bound and LEM show an underestimation of between 10%-22%, the

difference in stability numbers between the limit analysis and the LEM is still quite

significant. Due to the fact that true stability numbers are being bounded by the upper

and lower bound limit analysis solutions, results obtained from the LEM tend to be un-

conservative. This means that, for rock slope of high slope inclination ( 75 ), unsafe



8-7

factors will be obtained by using the PS limit equilibrium analyses. It should be stated

that limit analysis does not necessarily give true solutions for a non-associativity.

The dashed lines in Figure 8.11 and Figure 8.12 are the slip surfaces obtained from the

limit equilibrium analyses (SLIDE). It can be observed that the difference in predictions

of failure surfaces between the numerical upper bound analysis and limit equilibrium

methods is insignificant.

8.3.2 Investigation of stability numbers increasing with increasing mi

As mentioned above, Figure 8.4(d) indicated that for high slope angles ( 75 ) and

high lateral coefficients ( 3.0hk ), the stability number N was found to actually

increase slightly with increasing im . This implies that the stability of the slope is

essentially independent of the material shear strength. This observation requires a more

thorough investigation.

It is evident from Figure 8.4(d) that the LEM results do not exhibit the same response

compared to the numerical bounds for 75 and 0.3hk . Therefore, additional limit

equilibrium analyses for slopes with 75 and 0.4hk have been performed (using

SLIDE) with the results illustrated in Figure 8.14. Now we can observe that the stability

number N increases with im increasing when 4.0hk is adopted. This trend is similar

to the results of the bounding methods shown in Figure 8.4(d) where 3.0hk .

With this being the case we can conclude that the observed phenomenon is real and

needs further investigation. In doing so, the lower bound stress conditions in the region

of the failure plane were extracted and observed more closely. The obtained information

is displayed in Figure 8.15 along with the Hoek-Brown yield envelope for each rock

material. When 100GSI , 35im and 0.0hk , most of stress points extracted along

the slip surface, and therefore much of the slip surface length itself, are in a state of

compression. In contrast, for the case of 100GSI , 35im , and 3.0hk (square

symbols), most of the stress points extracted along the slip surface fall in the region with

small cici 31 and are in tension. This observation is also true when

100GSI , 5im , and 3.0hk (triangular symbols). This suggests that the tensile



8-8

strength of the material will tend to dictate the overall stability. It is found in Figure

8.15 that when 5im the tensile strength is larger than that for 35im . This means

that when collapse of the rock slope is due to tensile failure, rock masses with a smaller

im can provide high strength and therefore are more stable. This would explain the

decrease in stability number N with increasing im seen in Figure 8.4(d). It should be

noted that (Figure 8.15), for the rock masses with lower GSI , even the tensile strength

is relative small, it still plays an important role. Therefore, the phenomenon that stability

number N increases slightly with increasing im exists as well.

The observed stress conditions on the base of each slice in limit equilibrium analysis

using SLIDE are shown in Figure 8.16 for 75 , 35im , 100GSI and 4.0hk .

The square points represent the stresses of each slice for this case. It can be clearly seen

that these stresses locate intensively in the region with small cin and comparing to

the case with 5im , the shear stress is actually lower. This means that, based on the

Hoek-Brown model, the shear strength of rock masses with 5im and 35im will be

similar for small ratios of cin . While considering the case with 0.0hk , larger

cin was observed on the base of the slices, where shear strength for 5im is

obviously less than that of 35im .

It was found that, under the strong earthquake loading, rock slopes tend to fail due to

tensile stresses. It should be noted that this phenomenon only occurs for steep slopes

combined with a high seismic coefficient. However, as the tensile strength of rocks or

rock masses can be small, a rock slope stability based design is recommended to avoid

this situation.

8.4 CONCLUSIONS

Using the numerical upper and lower bound techniques, the seismic rock slope stability

have been presented as chart solutions. These stability charts, including the earthquake

effects, are based on the Hoek-Brown failure criterion and can be used for estimating

seismic rock slope stability in the initial design phase. This study follows the general

consideration of the earthquake effects which only takes the horizontal seismic



8-9

coefficient ( hk ) into account. A range of hk magnitudes was presented, which is

consistent with most design codes. Based on this study, the following conclusions can

be made:

1. The stability numbers ( N ) have been bounded using Pseudo Static (PS) upper

bound and lower bound solutions within 9 % or better for all considered cases.

2. The stability analysis of rock slopes using the limit equilibrium method was

found to overestimate the factors of safety for the cases with higher slope angles

and lower GSI . Comparing with the upper bound solutions alone, this

overestimate can be as high as 22% for the steep slopes (i.e 75 ) with

GSI values less than approximately 50.

3. When the horizontal seismic coefficient ( hk ) increases by a factor of 0.1, the

safety factor of a rock slopes may decrease by more than 30%. But reducing the

angle of slope by 15 can increase the safety factor by at least 50%.

4. By using the tangential method proposed by Yang et al. (2004b), the

overestimates of rock slope stability increase with GSI increasing and can be up

to 80%. In addition, an inaccurate prediction of slip surface would be obtained.

5. It was found that for the cases with high slope angle and significant seismic

coefficient the stability numbers for rock slopes obeying Hoek-Brown yield

criterion will increase with increasing im . The reason for this was due to the

tensile nature of overall failure for these specific cases.

6. Due to the fact that rocks and rock masses are not good materials when it comes

to providing tensile strength, the design for a rock slope should avoid the

development of tensile stresses.



8-10

Table 8.1 Comparisons of tangential Mohr-Coulomb parameters for various quality

rocks ( 75 and 1.0hk )

GSI im Tangential method Lower bound linear regression

t citc t citc

10 10 60.7 0.000314 61.74 0.000315

50 10 48.51 0.021347 59.31 0.00701

100 10 20.038 1.56842 45.72 0.1796



8-11

khW

WToe

Rigid Base

Jointed Rock

ciGSI,m

i

H d

Figure 8.1 Problem configuration for simple homogeneous slopes



8-12

5 10 15 20 25 30 35

0.01

0.1

1

10


= 30, kh = 0.1

N= ci

/H

F

GSI=10

GSI=50

GSI=100

mi

H


(a)


( 1.0hk )



8-13

5 10 15 20 25 30 350.01

0.1

1

10

100

mi

Average SLIDE-Hoek-Brown Model Tangential method

= 45, kh = 0.1

GSI=10

GSI=50

GSI=100

N= ci

/H

F

H


(b)

Figure 8.2 (continued) Average finite element limit analysis solutions for stability

numbers ( 1.0hk )



8-14

5 10 15 20 25 30 350.01

0.1

1

10

100


= 60, kh = 0.1

GSI=10

GSI=50

GSI=100

H

N= ci

/H

F

mi


(c)


numbers ( 1.0hk )



8-15

5 10 15 20 25 30 350.01

0.1

1

10

100


mi

= 75, kh = 0.1

GSI=50

GSI=100

GSI=10

N= ci

/H

F

H


(d)


numbers ( 1.0hk )



8-16

5 10 15 20 25 30 350.01

0.1

1

10


mi

= 30, kh = 0.2

GSI=10

GSI=50

GSI=100

N= ci

/H

F

H


(a)


( 2.0hk )



8-17

5 10 15 20 25 30 350.01

0.1

1

10

100

Average SLIDE-Hoek-Brown Model Tangential method

mi

= 45, kh = 0.2

N

= ci

/H

F

GSI=10

GSI=50

GSI=100

H


(b)


numbers ( 2.0hk )



8-18

5 10 15 20 25 30 35

0.1

1

10

100

1000


= 60, kh = 0.2

GSI=10

GSI=50

GSI=100

H

N= ci

/H

F

mi


(c)


numbers ( 2.0hk )



8-19

5 10 15 20 25 30 350.01

0.1

1

10

100

1000 Average SLIDE-Hoek-Brown Model Tangential Method

mi

= 75, kh = 0.2

GSI=50

GSI=10

GSI=100

N= ci

/H

F

H


(d)


numbers ( 2.0hk )



8-20

5 10 15 20 25 30 350.01

0.1

1

10

100

mi

Average SLIDE-Hoek-Brown Model

= 30, kh = 0.3

GSI=100

GSI=50

GSI=10

N= ci

/H

F

H


(a)

Figure 8.4 Average finite element limit analysis solutions of stability numbers

( 3.0hk )



8-21

5 10 15 20 25 30 35

0.1

1

10

100

mi


= 45, kh = 0.3

GSI=10

GSI=50

GSI=100

N= ci

/H

F

H


(b)

Figure 8.4 (continued) Average finite element limit analysis solutions of stability

numbers ( 3.0hk )



8-22

5 10 15 20 25 30 350.1

1

10

100

mi


= 60, kh = 0.3

N

= ci

/H

F

GSI=10

GSI=50

GSI=100H


(c)


numbers ( 3.0hk )



8-23

5 10 15 20 25 30 350.1

1

10

100

1000

mi


= 75, kh = 0.3

GSI=10

GSI=50

GSI=100

N= ci

/H

F

H


(d)


numbers ( 3.0hk )



8-24

5 10 15 20 25 30 350.01

0.1

1

10

= 75 = 60 = 45 = 30

GSI=80, kh = 0.1

N= ci

/H

F

mi

H

5 10 15 20 25 30 35

0.1

1

10

N= ci

/H

F

mi

GSI=50, kh = 0.2

= 75 = 60 = 45 = 30

H

5 10 15 20 25 30 35

1

10

100

N= ci

/H

F

mi

GSI=20, kh = 0.3

= 75 = 60 = 45 = 30

H

Figure 8.5 Average finite element limit analysis solutions of stability numbers under

pseudo static earthquake loading



8-25

5 10 15 20 25 30 35

0.01

0.1

kh = -0.1

kh = 0.3

kh = 0.2

kh = 0.1

kh = 0.0

=30, GSI =100

H

N= ci

/H

F

mi

Decreasing Stability

5 10 15 20 25 30 350.01

0.1

1

kh = -0.1

=45, GSI =70

N= ci

/H

F

mi

kh = 0.0

kh = 0.1

kh = 0.2

kh = 0.3

H


Figure 8.6 Comparisons of stability numbers between the static and pseudo static

analyses ( 30 and 45 )



8-26

5 10 15 20 25 30 35

0.1

1

10

kh = -0.1

=60, GSI =50

N= ci

/H

F

mi

kh = 0.0

kh = 0.1

kh = 0.2

kh = 0.3

H


5 10 15 20 25 30 351

10

100

kh = -0.1

N

= ci

/H

F

=75, GSI =20

kh = 0.3

kh = 0.2

kh = 0.1

kh = 0.0

mi

H


Figure 8.7 Comparisons of stability numbers between the static and pseudo static

analyses ( 60 and 75 )



8-27

20 40 60 80 1000.01

0.1

1

10= 30, k

h = 0.1

mi = 35

mi = 5

N= ci

/H

GSI / F

H

20 40 60 80 100

0.1

1

10

= 45, kh = 0.1

mi = 35

mi = 5

N= ci

/H

GSI / F

H

(a) 30 (b) 45

20 40 60 80 100

0.1

1

10

= 60, kh = 0.1

mi = 35

mi = 5

N= ci

/H

GSI / F

H

20 40 60 80 100

1

10

100

= 75, kh = 0.1

mi = 35

mi = 5

N= ci

/H

GSI / F

H

(c) 60 (d) 75

Figure 8.8 Factor of safety assessment based on GSI ( 1.0hk )



8-28

20 40 60 80 100

0.1

1

10

= 30, kh = 0.2

mi = 35

mi = 5

N= ci

/H

GSI / F

H

20 40 60 80 100

0.1

1

10

= 45, kh = 0.2

mi = 35

mi = 5

N= ci

/H

GSI / F

H

(a) 30 (b) 45

20 40 60 80 1000.1

1

10

100= 60, k

h = 0.2

mi = 35

mi = 5

N= ci

/H

GSI / F

H

20 40 60 80 100

1

10

100

= 75, kh = 0.2

mi = 35

mi = 5

N= ci

/H

GSI / F

H

(c) 60 (d) 75




8-29

20 40 60 80 100

0.1

1

10

= 30, kh = 0.3

mi = 35

mi = 5

N= ci

/H

GSI / F

H

20 40 60 80 100

0.1

1

10

= 45, kh = 0.3

mi = 35

mi = 5

N= ci

/H

GSI / F

H

(a) 30 (b) 45

20 40 60 80 100

1

10

100

= 60, kh = 0.2

mi = 35

mi = 5

N= ci

/H

GSI / F

H

20 40 60 80 100

1

10

100

= 75, kh = 0.3

mi = 35

mi = 5

N= ci

/H

GSI / F

H

(c) 60 (d) 75




8-30

Tangential method SLIDE

Figure 8.11 Comparisons between the upper bound plastic zones and failure surfaces of

the tangential method for various GSI ( 1.0hk , 10im and 75 )

10GSI

50GSI

100GSI



8-31

Tangential method SLIDE

Figure 8.12 Comparisons between the upper bound plastic zones and failure surfaces of

the tangential method for various hk ( 50GSI , 15im and 60 )

1.0hk

2.0hk

3.0hk



8-32

-0.0005 0.0000 0.0005 0.00100.00

0.01

ci

ci

Hoek-Brown (GSI=10, mi=10)

Tangential method Lower bound (k

h=0.1)

Lower bound fitted yield surface

ci=15.784 (

ci) + 0.0025

-0.01 0.00 0.01 0.02 0.03 0.04 0.050.0

0.1

0.2

0.3

0.4

0.5



h=0.1)

Fitted yield surface

ci=13.276 (

ci) + 0.0511

ci

ci

-0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0



h=0.1)

Lower Bound Fitted yield surface

ci=6.0417 (

ci) + 0.8826

ci

ci

Figure 8.13 Comparisons of yield surfaces between numerical lower bound and

tangential method ( 75 )



8-33

5 10 15 20 25 30 35

1

10

100

GSI=70

GSI=30

SLIDE-Hoek-Brown Model

mi

N= ci

/H

F

= 75, kh = 0.4

GSI=10

GSI=50

GSI=100

H

Figure 8.14 Stability numbers from the limit equilibrium analyses ( 4.0hk )



8-34

-0.2 -0.1 0.0 0.1 0.2 0.30.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

/ ci

/

GSI=100 mi=35, Lower bound (k

h = 0.0)


h = 0.3)


h = 0.0)


h = 0.3)

GSI=100 mi=35

GSI=100 mi=5

GSI=50 mi=5

Tensile Compressive

Figure 8.15 The Hoek-Brown failure criterion for variable GSI and im values and

observed lower bound results

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.0

0.2

0.4

0.6

0.8

1.0

1.2

/

ci

n

ci

Tensile Compressive

GSI=100 mi=35 GSI=100 m

i=5

GSI=100 mi=35, SLIDE (k

h = 0.0)


h = 0.4)


h = 0.4)

Figure 8.16 The observed stress conditions along the slip plane from the limit

equilibrium analyses



9-1

CHAPTER 9 DISTURBANCE FACTOR EFFECTS ON

THE STATIC ROCK SLOPE STABILITY

9.1 INTRODUCTION

In this Chapter, the effect of disturbance factor ( D ) on the static stability of rock slopes

is reported and compared with the results in Chapter 7 for undisturbed rock slopes

( 0.0D ). The disturbance factor which is one of the parameters in the Hoek-Brown

failure criterion (Equation (2.3) and Equation (2.4)) that represents the degree of

disturbance of the rock mass. It ranges from 0 for undisturbed in situ rock masses to 1

for completely disturbed rock mass properties.

Based on Hoek et al. (2002), for small scale rock slope blasting, 7.0D and 0.1D

were suggested for good blasting and poor blasting, respectively, due to the stress relief

caused by disturbance. In addition, for very large open pit mines, 0.1D was

recommended because of significant disturbance induced by heavy production blasting

and stress relief from overburden removal. For mechanical excavation, 7.0D was

suggested. From the above recommendations proposed by Hoek et al. (2002), the

importance of incorporating the degree of disturbance of the rock masses into the

stability analysis of man-made fill and cut rock slopes can be investigated.

The slope geometry analysed in this Chapter for the homogeneous rock slopes is shown

in Figure 9.1(a) where the jointed rock mass has the Hoek-Brown strength parameters

( ci , GSI , im ) and D is either 0.7 or 1.0.

As pointed out by Marinos et al. (2005), it is appropriate to simulate a distribution of

disturbance factor that decreases as the distance from the surface increases. The

previous investigations of Chen and Liu (1990) found that the primary influence zone of

blast damage surrounds the excavation perimeter to a depth of around 2m. However,

Marinos et al. (2005) also indicated that, for very large open pit mine slopes which

involve many tons of explosives, blast damage had been observed up to 100 m or more

behind the excavated slope face. These influence zones of disturbance can be seen to

change significantly and are highly related to the quality of controlled blasting and the

scale of overburden removal. Fortunately, Hoek and Karzulovic (2000) recommended a



9-2

range for the potential damaged zone based upon their experience. Moreover, the initial

magnitudes of disturbance factor are also suggested and can be found in Hoek et al.

(2002).

To investigate the effects of the various disturbance factor distributions in a slope on the

stability number, this study adopts a simple variation of the disturbance factor that

decreases linearly with increasing depth. The shape of the primary damage zone is

determined based on Hoek and Karzulovic (2000). In addition, the extent of the

disturbed rock mass is assumed to have a dimension equivalent to the slope height ( H ),

calculated from the slope inclined surface as shown in Figure 9.1(b) where the

parameter, 0D , needs to be determined. The rock mass in other sections of the slope are

treated as undisturbed, and therefore the disturbance factor D for these sections are

taken as zero. In view of the above assumptions, the contour of disturbance factor will

be parallel to the slope surface (Figure 9.1(b)) and the rock masses of the slope are

inhomogeneous.

There is no intention in this Chapter to define or quantify the exact influence zone and

the true disturbance distribution. The purpose of this part of study is to examine and

understand the various disturbance factor effects on the rock slope stability by

comparing stability numbers. To obtain more precise estimates of the rock slope

stability, as suggested by Marinos et al. (2005), it would require the rock mass to be

divided into a number of zones and to assign decreasing values of D to successive

zones appropriately with the distance from the face. In order to make comparisons

between the disturbed and undisturbed rock slope stability, the stability number

(Equation (7.1)) is adopted.

9.2 NUMERICAL LIMIT ANALYSIS SOLUTIONS FOR

HOMOGENEOUS DISTURBED ROCK SLOPES

9.2.1 Stability numbers

Figure 9.2 and Figure 9.3 present two sets of rock slope stability charts obtained from

the numerical upper and lower bound methods for disturbance factors of 7.0D and

0.1D respectively. In Figure 9.2(a), it can be seen that the true stability numbers ( N )

defined in Equation (7.1) are bounded by the limit analysis solutions from above and



9-3

below for 10GSI . Thus, an average value of the stability numbers is adopted for

simplicity. It should be noted that the bounding methods have bracketed the true

stability numbers within 8 % or better. Referring to Figure 9.2 and Figure 9.3, the

stability number N is found to increase when GSI or im decreases. This trend was

also observed for the cases with D 0.0 in Chapter 7. The triangular symbols shown in

Figure 9.2 and Figure 9.3, which represent the stability numbers obtained from limit

equilibrium analyses (SLIDE), are remarkably close to the average lines of the bound

solutions. The difference in stability number between the average limit analysis

solutions and the limit equilibrium analyses is less than 8%.

Figure 9.4 and Figure 9.5 provide an alternative form of safety factor assessment based

on Equation (7.2). The designers can utilise both of the above proposed stability charts

to determine the factor of safety. It should be noted that using the chart solutions in

Figure 9.4 and Figure 9.5 may obtain different factor of safety from using those in

Figure 9.2 and Figure 9.3 which has been examined in Section 7.3.4.

The comparisons of stability numbers between the different disturbance factors for a

range of slope angles ( 75,45,15 ) are shown in Figure 9.6 to Figure 9.8. The

stability numbers are found to increase as the disturbance factors increase for a given

ci , GSI and im . This trend indicates that the rock masses with a smaller level of

disturbance ( D ) are more stable, which is to be expected.

This is also demonstrated by Figure 9.9, which displays Hoek-Brown envelopes for

45 , 50GSI , 10im and three different disturbance factors. The labels shown

in Figure 9.9 are the stress points from the plastic zones of the lower bound limit

analysis. These points always fell on or slightly inside the yield surfaces. The fact that

the lower disturbance factor results in the “larger” yield surface explains the increasing

stability numbers with a reduction of D .

Referring to Figure 9.6 to Figure 9.8, the difference in the stability numbers between

0.0D and 0.1D is found to increase with a reduction of GSI . For example, in

Figure 9.8, the ratio of stability numbers for the case of 0.0D and 5im compared

to 0.1D is around 1.33 when 90GSI and 25.16 when 10GSI . Even compared

with the stability numbers for D 0.7, the ratio of stability numbers for 0.0D to



9-4

7.0D is between 1.18 and 6.78. This means that, for a slope inclination of 75 ,

using the chart solutions without disturbance ( 0.0D ) to estimate the stability of a

disturbed rock slope with 7.0D will produce a larger unconservative safety factor. If

the disturbance factor effects are ignored, the safety factor for the rock slope will be

overestimated, and an unconservative solution will be obtained. It is obvious that

determining the disturbance factor value carefully is necessary as it may lead to

significant difference in the stability number assessment.

Looking at Figure 9.6 to Figure 9.8 one can observe that the stability numbers for

10GSI and 0.0D can be either larger, equal or lower than those of 50GSI and

0.1D for various slope angles. In order to determine the source of this phenomenon,

the native forms of the Hoek-Brown failure criterion and the slip surfaces from the

lower bound analyses have been observed. Figure 9.10 shows the Hoek-Brown yield

surfaces for 50GSI , 10im , 0.1D and 10GSI , 10im , 0.0D which are

the solid line and the dashed line, respectively. It can be seen in Figure 9.10(a) that the

dashed line is slightly lower than the solid line at low levels of ci 3 . With increasing

ci 3 , the two lines converge, then after point A displayed in Figure 9.10, the dashed

line is higher than the solid line. Figure 9.10 shows that the difference between these

two yield surfaces is not obvious. The points shown in Figure 9.10 are the stress

conditions obtained form the plastic zones of the numerical lower bound solutions for

different slope angles and strength parameters. In Figure 9.10(a), it can be seen that all

cross symbols fell at the lower levels of normal stresses, where the dashed line is below

the solid line for the case of 45 . This means that the rock mass with 10GSI ,

10im , 0.0D yields at lower load than that with 50GSI , 10im , 0.1D

(which is denoted by star symbols in Figure 9.10(a)). This has been demonstrated in

Figure 9.7, where the stability number for 10GSI , 10im , 0.0D is larger than

that for 50GSI , 10im , 0.1D . On the other hand, when most of the stress points

from the lower bound plastic zone are located in the region with higher levels of stress,

the rock mass with 50GSI , 10im , 0.1D will yield at lower level of loading, as

seen in Figure 9.10(b) where the majority of triangular points located below the square



9-5

points. Thus, this explains the fact that the stability number in Figure 9.6 for 50GSI ,

10im , 0.1D is larger than that for 10GSI , 10im , 0.0D .

It should be noted that, for 100GSI , the disturbance factor has no effect on the

stability numbers. The reason is due to the native form of the Hoek-Brown yield surface

which is determined by Equation (2.2) to Equation (2.5). When 100GSI is put in

Equation (2.3) and Equation (2.4), no matter what magnitude D is, the parameters, bm

and s , are still the same. Thus, the yield surface for the Hoek-Brown failure criterion

does not change for different disturbance factors when 100GSI .

As pointed out by Hoek et al. (2002), slope cutting may cause a certain degree of

disturbance to the rock mass due to the stress relief. Figure 9.11 presents the average

finite element limit analysis solutions of the stability numbers for various slope angles

and disturbance factors. In Figure 9.11 the lines of 0.0D are from the results of

undisturbed rock slopes in Chapter 7. For 90GSI and 5im , decreasing the slope

inclination from 75 to 60 can increase the factors of safety by up to 46% and

20% for 7.0D and 0.1D , respectively. However, using the chart solutions in

Chapter 7 where 0.0D , the increment of safety factors can be by more than 50 for

the same GSI and im . This implies that the factor of safety for a cut slope may be

overestimated by utilising stability charts without considering rock disturbance.

Comparing to the solutions in Chapter 7, it is apparent that the rock slope factor of

safety reduces because of considering the effects from the disturbance factor which are

induced by the slope cutting.

Referring to Figure 9.11, when cutting a slope from 75 to 60 , the final

average stability numbers for different GSI values can be either “larger”, “equal” or

“smaller” than the initial ones which are the solid lines. These phenomena are also due

to the native forms of the Hoek-Brown failure criterion and the stress distributions at

collapse, as illustrated in Figure 9.10. It should be noted that, for 10GSI and 5im

(Figure 9.11), slope cutting will increase the stability numbers by up to 1.4 or 16.1 times

to the original stability number when D is considered as 0.7 or 1.0. In other words, this

means that decreasing slope angles may reduce the rock slope stability in certain cases.



9-6

Even for 50GSI and 5im , a reduction of slope inclination can mean the stability

numbers increase between 1.2 and 2.5 times to the original stability number.

From the above discussions, it has been indicated that the disturbance factor has

significant influence on the rock slope stability evaluations, particularly for the poorer

quality rocks (low GSI ). Therefore, the rock mass properties have to be considered

carefully if decreasing the slope inclination is required in design. Furthermore, as

highlighted by Hoek et al. (2002), the values of the disturbance factors need to be

applied with caution. The importance of the disturbance factor effects can be seen, and

thus the attention should be paid on selecting its value appropriately for estimating the

stability of cut rock slopes.

9.2.2 Failure surfaces

Figure 9.12 presents the upper bound plastic zones for 10im , 0.1D and the

various GSI . It can be found that, for gentle slopes ( 30 ), the depth of slip surfaces

increases slightly as GSI increases. However, this phenomenon is relatively small for

the steeper slopes ( 45 and 60 ). In addition, the effects of the parameters im

and D , were found to be insignificant on the failure surface shape. The observed plastic

zones are almost unchanged for different D . Therefore, it can be concluded that, for

homogeneous slopes, the disturbance factors are found to have no significant influence

on the rock slope slip surface size.

Based on the upper bound limit analysis results of 0.0D in Chapter 7 and this

Chapter for the homogeneous rock slopes, the potential influence zones of slip surface

can be determined. Figure 9.13 shows the relation between the maximum distance of the

influence zone behind the slope crest in relation to the slope height for different slope

inclinations, which are determined from the comprehensive observations of all the upper

bound plastic zones. It was observed in Figure 9.13 that the maximum distance of the

influence zone decreases with an increasing slope angle. However, this relation does not

change significantly for slope angles of 45 . In addition, for a given slope angle,

the maximum distance behind the slope crest was found to occur generally when the

rock slope has high GSI and low im . Figure 9.13 provides a reference for engineers.



9-7

An important construction recommendation should be not to build within this influence

zone near the slope crest.


In this Section, several case studies are examined in order to evaluate the effects of the

disturbance factor on the rock slope stability estimation. The following examples

include the assumed and practical cases.

Case 1: Example of assumed case employed in Section 7.3.4

From Section 7.3.4, Hci is known as 8.7. In Figure 9.2(d) and Figure 9.3(d),

FHN ci 17 and 50 for the homogeneous rock masses with 7.0D and 1.0,

respectively. Therefore, the factor of safety can be calculated as 51.0177.8 F for

7.0D and 17.0507.8 F for 0.1D . This means that the rock masses of the

slope for both 7.0D and 1.0 are unstable. Comparing these numbers to the case

where 0.0D from Section 7.3.4 where the safety factor is 1.9, the factor of safety

was found to diminish considerably when D is changed from 0 to 0.7. Thus, it can be

concluded that the disturbance factor has a significant effect on the stability of rock

slopes with low GSI .

Case 2: Slope failure in closely jointed rock mass in barite open pit mine

The rock slope was located at Baskoyak barite open pit mine, in western Anatolia,

which was employed to perform a back analysis by Sonmez and Ulusay (1999) where

the analysed failure surface satisfied factors of safety of unity. Due to the heavily

jointed nature of the schist, the rock mass was assumed as homogeneous and isotropic.

The mean unit weight and uniaxial compressive strength of the heavily broken part of

the schist are 322.2kN m and 5.2MPa , respectively. Other parameters required for

chart solutions can be obtained in Sonmez and Ulusay (1999) and Sonmez et al. (2003)

where 20H m , 34 7im and 16GSI . It should be noted that the chart

employed for evaluating GSI was modified by Sonmez and Ulusay (1999) using

Surface Condition Rating (SCR). However, this chart is still similar to that proposed by

Hoek and Brown (1997). As indicated by Sonmez and Ulusay (1999), no sign of

groundwater was encountered through the geotechnical and previously drilled boreholes



9-8

and on the pit benches. Thus, the pit slopes were treated as dry for stability assessments.

Since the overburden material and the ore are removed by excavators without any

blasting, the disturbance factor 0.7D is adopted, as suggested by Hoek et al. (2002).

Based on the information above, 11.7ci H . By using the line with 10GSI and

20GSI in Figure 9.2(b), the stability number 13ciN HF was obtained for

16GSI and 7im . Therefore, 11.7 13 0.9F is obtained. This shows obviously

that the rock slope is unstable. A comparison of the slip area between the present upper

bound and that given by Sonmez and Ulusay (1999) solutions is displayed in Figure

9.14, where it can be observed that both solutions provide similar failure shapes.

Case 3: Slope instability in coal mine in western Turkey

This example of rock slope instability originates from the Kisrakdere open pit mine

located at Soma lignite basin, western Turkey. The necessary data collected by Sonmez

and Ulusay (1999) shows the geometry of the failed slope in which a single thin coal

seam with a thickness of 4.5 m is overlain by a sequence of compact marl and soft clay

beds about 10 m of thickness. The observations of slope surfaces and available records

indicated that the groundwater was below the failed marly rock mass, and the coal seam

acted as an aquifer. The marly rock with a uniaxial compressive strength of 40MPa and

9.04im has a carbonate content more than its clay content. The observed actual slip

surface was of circular shape and passed through the compact marl rock mass and along

the clay bed, above the coal seam. Three main joint sets are moderately and closely

spaced, and bedding planes in the marly sequence resulted in a jointed rock mass. Other

parameters required for chart solutions obtained from Sonmez and Ulusay (1999)

displayed 80H m , 60 and 37GSI . An approximate unit weight 321kN m

is adopted based on the information of material properties employed by Sonmez et al.

(1998). In addition, Sonmez et al. (1998) mentioned that the method of excavation used

for this case is blasting, and thus 1.0D .

For this case, 23.8ci H is obtained. The stability number was estimated to be

21ciN HF for 40GSI and 9im using plots in Figure 9.3(d), and therefore

23.8 21 1.13F . Although the factor of safety ( F ) indicates the slope is just safe

from failure, the author has to stress that the values of F obtained from the stability



9-9

charts in this case are approximate as uniform slope inclination and not exactly

matching GSI value have been used for stability number estimation. Figure 9.15 shows

that the plastic zone obtained from the upper bound analysis is not as extended as the

actual one reported by Sonmez and Ulusay (1999), but indicates correctly the failed

zone. However, it can be seen that there are also some small scale plastic zones

occurring at the upper slope benches above the major failure. The velocities in Figure

9.15 also indicate that this slope is very close to instability. It should be noted that, the

upper and lower bound used in this thesis are not required to assume the failure

mechanism in advance. Therefore, the obtained slip surface should be the most critical

one.

Case 4: Other case studies

Table 9.1 summarises the case studies obtained from Douglas (2002) for rock slopes in

mines. Each case represents a particular open pit mine. Each mine may have several

stable/unstable slopes in the data base, which are are denoted by a, b, c etc. Douglas

(2002) has assessed these cases by using GSI based system. The magnitudes of im

shown in Table 9.1 are estimated using suggestion of Hoek (2000). In view of the

absence of unit weight information in Douglas (2002), 325kN m is assumed for all

cases in Table 9.1.

The stability charts for 0.7D are employed firstly to assess the above cases as no

excavation method descriptions could be found in Douglas (2002). Table 9.2 displays

the factors of safety estimated by using the chart solutions for 0.7D (Figure 9.2). For

truly stable cases (2a, 2b, 2c, 3, 4, 5a and 5b), the estimated safety factors confirm

reliable slope stability. However, for Case 1a it should be noted that the obtained factor

of safety is less than 1 for 40GSI and greater than 1 for 50GSI . It implies that the

slope is under critical conditions, even if it is declared as stable. For the failed cases (1c

and 1e), the estimations also provide quite good agreement. Table 9.2 indicates that

these slopes are unsafe if using the lower bound estimation of GSI value for stability

assessment.

It should be noted in Table 9.2 that the evaluations do not show the instability for the

failed cases (1b, 1d and 6). It will be interesting to know whether these cases may fail

by using the chart solution for 1.0D (Figure 9.3). The rock slope stability



9-10

assessments based on the solutions for 1.0D are shown in Table 9.3 where the

estimated factors of safety reduce. The results show that Case 1b, Case 1d and Case 6

may fail with higher rock mass disturbance ( 1.0D ) than that with 0.7D . Table 9.3

also indicates the rock slope of Case 4 which had the highest safety factor with 0.7D

is still stable with 1.0D .

9.3 NUMERICAL LIMIT ANALYSIS SOLUTIONS FOR

INHOMOGENEOUS DISTURBED ROCK SLOPES

9.3.1 Stability numbers

In view of the uncertainties in the disturbance factor value distribution over the

damaged zone, the complete chart solutions with a range of parameters will not be

presented in this thesis. However, in order to understand the influence of the various

disturbance factors, it is still worth investigating the rock slope stability in the case of

varying D (Figure 9.1 (b)) by comparing stability numbers.

Based on the assumptions made earlier in this Chapter about linear varying disturbance

factor with the distance from the surface of the slope, the corresponding average

stability numbers have been obtained and plotted in Figure 9.16. It is evident that these

lines ( 0.10 D ) fall within the lines of 7.0D and 1.0D which are the results of

homogeneous rock slopes, as shown in Figure 9.16. The average stability numbers for

0.10 D are smaller than those for 1.0D with the difference between these two cases

ranging from 1% to 40% which increases when , im or GSI decreaes. It should be

noted that the difference in stability numbers between disturbed and undisturbed rock

slopes is still significant, even when D was considered to decrease as the distance from

the face increases.

Figure 9.17 displays the variation in stability number after change in slope angle from

75 to 45 or 60 based on the varying disturbance factor distribution

assumed in this part of the study (Figure 9.1(b)). Compared to the results in Figure 9.11,

the average stability numbers for reduced slope angles and varying D are still either

larger or smaller than the initial ones which are the solid lines of 75 and 0.0D

in Figure 9.17. This is even though an analysis based on this approach can obtain



9-11

greater safety factors of the cut rock slopes than that based on constant D when

0DD . The observed stability number variation is also related to the magnitude of

GSI . The comparison between Figure 9.11 and Figure 9.17 shows that the difference in

stability numbers for pair 0.10.10 DD can be as much as 50% for the cut slope

with 60 , 10GSI and 5im . This means that, using the varying D to assess the

rock slope stability may result in a drop in stability number by up to 50%. Therefore, it

can be concluded that using constant and varying disturbance factor distributions may

lead to very different estimates of the rock slope stability.

9.3.2 Failure surfaces

The upper bound plastic zones for different cases of varying disturbance factor are

shown in Figure 9.18. It can be noticed that the failure surfaces obtained from these

analyses are shallower than those from the homogeneous cases ( D constant). The top

part of observed failure surfaces moves from the slope crest closer to inclined face, but

the failure modes are still of toe-failure. However, this trend is not obvious for greater

values of GSI ( 60GSI ).

In general, the rock slope stability is highly related to the material strength in the area of

the potential slip surface. As the majority of obtained modes of failure being of toe-type

failure (for 30 ), it can be assumed that the value of disturbance factor of rock

masses in Region 2 (Figure 9.1(a)) should have an insignificant effect on slope stability.

These assumptions are based on the recommendation from Marinos et al. (2005), where

the disturbance factor is considered to decrease as the distance from the face increases.

However, with this being the case the question is how do the disturbance factors in

Region 2 influence the stability number. In order to answer this question, the rock mass

of 1Hd (Region 2) is assumed to be either 1.0 or 0.0 and the distribution of D

within 1Hd (Region 1) was kept as before. This assumption might be not realistic,

however, it is only used here to investigate whether the disturbance factor in Region 2

still influence the rock slope stability or not.

The rock mass properties and geometry adopted in this investigation are 50GSI ,

20im and 45 . The average value of stability number from the bounding

methods is computed as 1.12 for both the rock mass with 0.0D and 0.1D in



9-12

Region 2. Comparing to the solution shown in Figure 9.16, the stability number is

almost unchanged. Therefore, the degree of disturbance of the rock masses in Region 2

has no significant effect on the rock slope stability for given ci , GSI , im and

30 .

9.4 CONCLUSIONS

In this Chapter, the stability charts for a range of disturbance factors have been

proposed using the numerical upper and lower bound methods. The stability numbers

for the cut rock slopes have been confidently bracketed by the numerical upper and

lower bound solutions within a narrow range ( 8 % or better). These chart solutions are

based on the latest Hoek-Brown failure criterion and the recommendations for the

disturbance factor values from Hoek et al. (2002). Based on the results of case studies,

the accuracy of stability charts produced in this thesis is examined and its applicability

is verified. It should be stressed that the Hoek-Brown failure criterion is suited to the

intact rock or heavily jointed rock masses which are isotropic. Therefore, it is important

to know this limitation while using the stability charts.

The parametric study results show that the disturbance factors have significant effects

on the stability numbers. The chart solutions for 0.0D will overestimate the rock

slope stability of disturbed rock slopes. This overestimation can be up to several

hundred percent and increases with D increasing or GSI decreasing. As highlighted by

Hoek et al. (2002), the disturbance factor should be determined with caution and the

recommended magnitudes could serve only as a starting point which can be used for the

initial assessment. If by observation and/or measurement of the excavation a better

estimate of disturbance factor can be obtained, then it must be adjusted.

By reducing slope inclination, it was found that the stability numbers may be larger,

equal or smaller than those of the originally undisturbed rock slopes ( 0.0D ), as

illustrated in Figure 9.11. This variation is highly related to the Geological Strength

Index (GSI ). The stability numbers increase more significantly after slope cutting if the

rock properties are with low GSI . Hence, for poor quality rock masses, the disturbance

factor was found to be the most important parameter which affects the accuracy of the



9-13

rock slope stability evaluation. Therefore, cutting the slope angle might not be an

effective way to increase the stability.

Based on the latest Hoek-Brown failure criterion, the potential influence zones of slope

slip surfaces have been proposed for homogeneous rock masses by using the limit

analysis solutions. The dimensions of these zones (e.g. distance behind the slope crest)

can be seen as a reference for use in practical design.

As the result of this numerical investigation it can be concluded that the properties of

the blast damaged rock mass will affect the stability significantly, as discussed by Hoek

and Karzulovic (2000). It was shown that by considering the rock mass as either

disturbed or undisturbed, very different slope stability estimates can be obtained. Even

if the disturbance factor is considered to decrease with distance from the face, the

obtained stability numbers are still significantly different from the results of the

undisturbed rock slopes. If the damaged zones and their D values of the rock masses

are unknown, it is recommended to use more conservative design practices. The more

conservative chart solutions have been proposed in Figure 9.2 and Figure 9.3. However,

to obtain more accurate magnitudes of disturbance factor in damaged zones, more

research and observations are required.



9-14

Table 9.1 Summary of slope data from case studies

Geology Mine

case

Height

( m )

Slope angle

( )

GSI

interval im Stable

Saprolite/Basalt 1a 70 49 40-50 17 Yes

Saprolite/Basalt 1b 41 50 40-50 17 No

Saprolite/Basalt 1c 41 55 40-50 17 No

Saprolite/Basalt 1d 46 49 40-50 17 No

Saprolite/Basalt 1e 57 50 40-50 17 No

Saprolite/Basalt 2a 58 50 50-60 17 Yes

Saprolite/Basalt 2b 60 48 50-60 17 Yes

Saprolite/Basalt 2c 60 52 50-60 17 Yes

Mudstone/Siltstone 3 38 39 50-60 9 Yes

Breccia 4 200 65 70-80 18 Yes

Siltstone 5a 157 48 60-70 9 Yes

Siltstone 5b 60 53 60-70 9 Yes

Siltstone 6 110 48 40-50 9 No



9-15

Table 9.2 Assessments based on the chart solutions ( 0.7D )

Numerical Assessment

Mine case ci ( MPa ) ci H Stable Safety factor ( F ) Stable

1a 3 1.71 Yes 40GSI 0.73F No

50GSI 1.45F Yes

1b 3 2.93 No 40GSI 1.33F Yes

50GSI 2.66F Yes

1c 3 2.93 No 40GSI 0.98F No

50GSI 1.63F Yes

1d 3 2.61 No 40GSI 23.1F No

50GSI 45.2F Yes

1e 3 2.11 No 40GSI 0.96F No

50GSI 1.92F Yes

2a 5 3.45 Yes 50GSI 3.14F Yes

60GSI 5.75F Yes

2b 5 3.33 Yes 50GSI 3.03F Yes

60GSI 5.55F Yes

2c 5 3.33 Yes 50GSI 3.03F Yes

60GSI 55.4F Yes

3 5 5.26 Yes 50GSI 4.38F Yes

60GSI 7.3F Yes

4 150 30 Yes 70GSI 35.3F Yes

80GSI 60F Yes



9-16




5a 23 5.86 Yes 60GSI 5.33F Yes

70GSI 8.37F Yes

5b 23 15.3 Yes 60GSI 12.75F Yes

70GSI 21.86F Yes

6 25 9.1 No 40GSI 2.2F Yes

50GSI 4.33F Yes

Table 9.3 Assessments based on the chart solutions ( 1.0D )



1b 3 2.93 No 40GSI 5.0F No

50GSI 25.1F Yes

1d 3 2.61 No 40GSI 44.0F No

50GSI 13.1F Yes

4 150 30 Yes 70GSI 16.7F Yes

80GSI 37.5F Yes

6 25 9.1 No 40GSI 0.9F No

50GSI 2.12F Yes



9-17

Toe

Rigid Base

d

Jointed Rock

ciGSI ,m

iD,

H

Region 2

Region 1

(a) Analysed slope geometry

d

2

D0

D0

D =

0.0

D = 0.0Toe

Rigid Base

H

D0

H

1

1

1

(b) Contours of disturbance factor

Figure 9.1 Problem definition for disturbed slopes



9-18

5 10 15 20 25 30 351E-3

0.01

0.1

1

10

Average Lower bound Upper bound SLIDE-Hoek-Brown Model

= 15, D = 0.7

GSI=10

GSI=50N= ci

/H

F

mi

GSI=100H


(a)

Figure 9.2 Average finite element limit analysis solutions of stability numbers for

disturbed slopes ( D 0.7)



9-19

5 10 15 20 25 30 35

0.01

0.1

1

10

100

= 30, D = 0.7


N= ci

/H

F

GSI=10

GSI=50

GSI=100

mi

H


(b)


numbers for disturbed slopes ( D 0.7)



9-20

5 10 15 20 25 30 350.01

0.1

1

10

100

= 45, D = 0.7

mi


GSI=10

GSI=50

GSI=100

N= ci

/H

F

H


(c)





9-21

5 10 15 20 25 30 35

0.1

1

10

100

= 60, D = 0.7

mi


GSI=10

GSI=50

GSI=100

N= ci

/H

F

H


(d)





9-22

5 10 15 20 25 30 35

0.1

1

10

100

1000

mi


= 75, D = 0.7

GSI=10

GSI=50

GSI=100

N= ci

/H

F

H


(e)





9-23

5 10 15 20 25 30 351E-3

0.01

0.1

1

10

100


H

N= ci

/H

F

= 15, D = 1.0

GSI=10

GSI=50

GSI=100

mi


(a)


disturbed slopes ( D 1.0)



9-24

5 10 15 20 25 30 35

0.01

0.1

1

10

100

= 30, D = 1.0

mi


GSI=10

GSI=50

GSI=100

N= ci

/H

F

H


(b)





9-25

5 10 15 20 25 30 350.01

0.1

1

10

100

1000

= 45, D = 1.0

mi


N

= ci

/H

F

GSI=10

GSI=50

GSI=100H


(c)





9-26

5 10 15 20 25 30 35

0.1

1

10

100

1000

= 60, D = 1.0

mi


GSI=10

GSI=50

GSI=100

N= ci

/H

F

H


(d)





9-27

5 10 15 20 25 30 35

0.1

1

10

100

1000

= 75, D = 1.0

mi


GSI=50

GSI=10

GSI=100

N= ci

/H

F

H


(e)





9-28

20 40 60 80 100

0.01

0.1

1

10

= 15, D = 0.7

mi = 35

mi = 5

N= ci

/H

GSI / F

H

20 40 60 80 100

0.01

0.1

1

10

= 30, D = 0.7

mi = 35

mi = 5

N= ci

/H

GSI / F

H

(a) 15 (b) 30

20 40 60 80 1000.01

0.1

1

10

100

= 45, D = 0.7

mi = 35

mi = 5

N= ci

/H

GSI / F

H

20 40 60 80 100

0.1

1

10

100

= 60, D = 0.7

mi = 35

mi = 5

N= ci

/H

GSI / F

H

(c) 45 (d) 60

Figure 9.4 Factor of safety assessment based on GSI for disturbed slopes ( 7.0D )



9-29

20 40 60 80 1000.1

1

10

100

= 75, D = 0.7

mi = 35

mi = 5

N= ci

/H

GSI / F

H

(e) 75

Figure 9.4 (continued) Factor of safety assessment based on GSI for disturbed slopes

( 7.0D )



9-30

20 40 60 80 100

0.01

0.1

1

10

100= 15, D = 1.0

mi = 35

mi = 5

N= ci

/H

GSI / F

H

20 40 60 80 1000.01

0.1

1

10

100

= 30, D = 1.0

mi = 35

mi = 5

N= ci

/H

GSI / F

H

(a) 15 (b) 30

20 40 60 80 1000.01

0.1

1

10

100

1000= 45, D = 1.0

mi = 35

mi = 5

N= ci

/H

GSI / F

H

20 40 60 80 100

0.1

1

10

100

1000

= 60, D = 1.0

mi = 35

mi = 5

N= ci

/H

GSI / F

H

(c) 45 (d) 60

Figure 9.5 Factor of safety assessment based on GSI for disturbed slopes ( 0.1D )



9-31

20 40 60 80 1000.1

1

10

100

1000

= 75, D = 1.0

mi = 35

mi = 5

N= ci

/H

GSI / F

H

(e) 75

Figure 9.5 (continued) Factor of safety assessment based on GSI for disturbed slopes

( 0.1D )



9-32

5 10 15 20 25 30 35

0.01

0.1

1

10

100 GSI=90 GSI=50 GSI=10

H

mi

N= ci

/H

F

= 15

D = 0.7

D = 1.0

D = 0.0

D = 0.7

D = 1.0

D = 0.0

D = 0.7

D = 1.0

D = 0.0


Figure 9.6 Comparisons of stability numbers for different disturbance factors ( 15 )



9-33

5 10 15 20 25 30 350.01

0.1

1

10

100

1000

GSI=90 GSI=50 GSI=10

H

N= ci

/H

F

= 45

mi

D = 0.7D = 1.0

D = 0.0

D = 0.7

D = 1.0

D = 0.0

D = 0.7

D = 1.0

D = 0.0





9-34

5 10 15 20 25 30 350.1

1

10

100

1000

10000

= 75

D = 0.7D = 1.0

D = 0.0

GSI=90 GSI=50 GSI=10

H

mi

N= ci

/H

F

D = 0.7

D = 1.0

D = 0.0

D = 0.7

D = 1.0

D = 0.0





9-35

0.00 0.05 0.10 0.15 0.20 0.25 0.300.0

0.2

0.4

0.6

0.8

1.0

Lower bound (D = 0.0) Lower bound (D = 0.7) Lower bound (D = 1.0)

GSI = 50, mi =10

D = 1.0

D = 0.7M

ajor

pri

nci

pa

l str

ess

(M

Pa

)

Minor principal stress (MPa)

D = 0.0

Figure 9.9 The forms of the Hoek-Brown failure criterion with different disturbance

factors ( 45 )



9-36

0.00 0.05 0.10 0.15 0.20 0.25 0.300.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

GSI =50, mi =10, D = 1.0

GSI =10, mi =10, D = 0.0

Lower bound ( = 15) GSI =10, m

i =10, D = 0.0


i =10, D = 0.0


i =10, D = 1.0


i =10, D = 1.0

/ ci

/

ci

A

(a)

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0 GSI =50, m

i =10, D = 1.0

GSI =10, mi =10, D = 0.0


i =10, D = 0.0


i =10, D = 0.0


i =10, D = 1.0


i =10, D = 1.0

A

/ ci

/

ci

(b)

Figure 9.10 The forms of the Hoek-Brown failure criterion for various strength

parameters



9-37

5 10 15 20 25 30 350.01

0.1

1

= 45

= 60

= 45

= 60

= 75

D = 0.0 D = 0.7 D = 1.0

mi

N= ci

/H

F

GSI = 90

H

5 10 15 20 25 30 35

1

10

= 45

= 60

= 45

= 60

= 75

D = 0.0 D = 0.7 D = 1.0

H

GSI = 50

N= ci

/H

F

mi

5 10 15 20 25 30 35

10

100

1000

N= ci

/H

F

D = 0.0 D = 0.7 D = 1.0

GSI = 10

mi

H = 45

= 60

= 45

= 60

= 75


different slope angles and disturbance factors ( GSI 90, 50 and 10)



9-38

10GSI 80GSI

Figure 9.12 Upper bound plastic zones for different GSI ( 0.1D and 10im )

45

30

60



9-39

Figure 9.13 Lateral extent of failure surfaces for 100GSI and 5im

H1.1

H55.0

H5.0

30

15

45



9-40

Figure 9.13 (continued) Lateral extent of failure surfaces for 100GSI and 5im

H5.0

H5.0

60

75



9-41

(a) Upper bound solution

(b) Solution of

Figure 9.14 Case study 1



9-42

Figure 9.15 Case study 2



9-43

5 10 15 20 25 30 350.01

0.1

1

10= 45

GSI = 50

GSI = 90

D = 0.7 D = 1.0 Varying D

0 = 1.0

H

N= ci

/H

F

mi

5 10 15 20 25 30 350.1

1

10

H

mi

N= ci

/H

F

= 75

D = 0.7 D = 1.0 Varying D

0 = 1.0

GSI = 50

GSI = 90

Figure 9.16 Comparison of average finite element limit analysis solutions for stability

numbers for constant and varyings disturbance factors ( 45 and 75 )



9-44

5 10 15 20 25 30 350.01

0.1

1

= 45

= 60

= 75

D = 0.0 Various D

0 = 1.0

mi

N= ci

/H

F

GSI = 90

H

5 10 15 20 25 30 35

1

10

D = 0.0 Various D

0 = 1.0

= 45

= 60

= 75

H

GSI = 50

N= ci

/H

F

mi

5 10 15 20 25 30 35

10

100

1000

D = 0.0 Various D

0 = 1.0

N= ci

/H

F

GSI = 10

mi

H

= 45

= 60

= 75

Figure 9.17 Average finite element limit analysis solutions for stability numbers for

different slope angles and D distributions of a slope ( GSI 90, 50 and 10)



9-45

0.1D 0.10 D

Figure 9.18 Upper bound plastic zones for different distributions of D ( 10GSI and

10im )

45

60



9-46



10-1

CHAPTER 10 CONCLUDING REMARKS

10.1 SUMMARY

For geotechnical engineers, predicting the stability of slopes is a routine task. The most

widely used approach is limit equilibrium analysis, but it is often questioned in view of

its inherently arbitrary assumptions. In contrast, the limit theorems of plasticity are

theoretically rigorous and a specific purpose of this research was to use numerical

formulations of the limit theorems to predict slope stability in soil and rock masses.

So far, very few studies have applied both the upper and the lower bound methods to the

two dimensional (2D) and three dimensional (3D) slope stability problems although the

limit theorems provide a simple and useful way of analysing the stability of

geotechnical structures. Thus, a significant proportion of the analyses presented in the

thesis have been performed using the finite element formulations of the upper and lower

bound theorems. Furthermore, the more conventional displacement finite element

method and limit equilibrium method are also employed for comparison purposes.

The aim of this study was to provide a better understanding of two dimensional and

three dimensional stability problems in soil and rock slopes and to present 2D and 3D

stability chart solutions that could be used for preliminary design purposes. A

comparison of the results obtained using several numerical methods enabled the

findings to be validated and provide a truly rigorous evaluation of slope stability.

Broadly speaking, the work presented in this thesis can be divided into two distinct

areas; namely, (1) the investigation of 2D and 3D slope stability in purely cohesive and

cohesive-frictional soil; and (2) the investigation of 2D rock slopes under static and

seismic conditions. A summary of the results along with a critical insight into the

current and future work is provided in the following sections.

10.2 THE STABILITY OF 2D AND 3D SLOPES IN SOIL

In Chapter 5 and Chapter 6, the stability of 2D and 3D slopes in purely cohesive and

cohesive-frictional soil was examined. For purely cohesive cases both homogeneous

and non-homogeneous soil profiles with undrained shear strength increasing linearly



10-2

with depth were considered. By including a wide range of slope types, geometries and

strength parameters, simple parametric equations have been provided. Using these

equations the slope stability can be estimated fast and reliably for preliminary design

purposes.

Table 10.1 shows a summary of stability numbers bracketed by the upper and lower

bound results. For most cases, the exact factor of safety is found to be predicted within

5 % for two dimensional plane strain analyses and 10 % for three dimensional

analyses. Moreover, most results obtained from the displacement finite element method

fall between the upper and lower bound solutions and are close to the lower bound

results.

The primary failure mechanisms obtained were also discussed in this thesis. For purely

cohesive slopes in homogeneous soil, the primary failure mode is of base-failure when

the slope angle 45 . The slip surface generally extends to the bottom firm layer, and

therefore the depth of the failure surface increases when depth factor ( Hd ) increases.

However, for purely cohesive slopes in homogeneous soil with 45 , the primary

failure mode is of toe-failure. Though, some exceptions which are of base-failure can be

found when HL ratio is relatively large ( HL ).

For purely cohesive slopes in inhomogeneous soils, the effect of the depth factor ( Hd )

on both 2D and 3D stability is insignificant if Hd is greater than 2. In addition, the

toe-failure is a primary mode of failure for most of the inhomogeneous undrained slopes

and cohesive-frictional slopes. For simplicity of design of the cohesive-frictional slopes,

2D analyses are recommended to replace 3D analyses when 5HL as the three

dimensional end boundary effects on the factor of safety then do not exceed 10%.

10.3 THE STABILITY OF 2D ROCK SLOPES

In view of the difficulty of estimating rock mass strength, assessing the rock slope

stability is an essential problem in current geotechnical engineering. The majority of

rock mass stability evaluations have been made using Hoek-Brown yield criterion

(Hoek et al. (2002). This criterion has a genuinely nonlinear shape, and therefore most

of studies so far have used “equivalent” Mohr-Coulomb strength parameters, c and ,



10-3

to simplify the analysis. In contrast, the rock masses stability assessments made in this

thesis are based on native nonlinear form of Hoek-Brown yield criterion.

In Chapter 7 and Chapter 8, the static and seismic stability of natural rock slopes has

been examined where the simulation of earthquake effects is based on the quasi-static

method for plane strain cases. The exact solutions of stability numbers are bracketed

within 10 % for all cases by the numerical bounding methods (Table 10.2). The

comparisons of safety factors indicate that using “equivalent” parameters, c and , to

evaluate the stability of rock slopes may significantly overestimate the safety factor,

especially for steep slopes. However, most commercial software is still written in terms

of traditional Mohr-Coulomb soil parameters. Therefore, along with providing rock

slope stability charts, this thesis also propses two modified equations to estimate the

“equivalent” parameters ( c and ) more accurately for shallow and steep slopes,

respectively.

In the course of this study it was found that, under the strong earthquake loading, rock

slopes tend to fail due to tensile stresses. For the rock slope stability based design,

reliance on the tensile strength of rock masses should be avoided.

The rock disturbance due to blasting or overburden removal was part of investigations

attempted in this thesis. It was found that the primary failure mode of rock slopes is of

toe-failure ( 30 ). The case studies verified that the disturbance factor suggested by

Hoek et al. (2002) is ideally suitable for preliminary design. Furthermore, the

disturbance factor ( D ) does not influence the depth of the slip surface for homogeneous

rock slopes. Therefore, the primary failure zones can be found and they were quantified

for different slope inclinations in Chapter 9.

10.4 RECOMMEDATIONS FOR FURTHER WORK

10.4.1 Pore pressure effects

Although the numerical upper and lower bound limit analyses are useful tools for

evaluation of slope stability, the pore pressure functions are not yet employed in the

formulations used in this thesis. For soil and rock slope design consideration, pore

pressure needs to be included as it can have significant effects on the slope stability,



10-4

particularly in wet climates. The first priority in future work using the limit theorems,

therefore should be given to the investigation of soil and rock slope stability with pore

pressure effects.

10.4.2 Three dimensional (3D) chart solutions for rock slopes

The stability charts presented in Chapter 7 to Chapter 9 are limited to two dimensional

rock slopes. Because of limitations of available limit analysis code and computer

resources, it was not possible to conduct 3D analyses well with nonlinear yield surfaces.

Advances in computational methods and equipment may help to improve the situation,

and therefore, accurate 3D chart solutions would be possible to obtain using the upper

and lower bound limit analysis.

10.4.3 Slope failure controlled by structural orientations

The thesis considered only isotropic rock masses governed by Hoek-Brown yield

criterion. This implies that presented chart solutions are limited to the slope stability

problems where shear failures are not governed by a preferential direction imposed by a

singular discontinuity set or combination of several discontinuity sets (e.g. sliding over

inclined bedding planes, toppling due to near-vertical discontinuity, or wedge failure

over intersecting discontinuity planes).

For the problems where the slope failure is controlled by structural orientations, it is

possible to utilise the unique features of the limit analysis formulations which allows the

different yield criteria to be used for the solid domains and for the discontinuities

between them. Therefore, the solutions obtained were found to be influenced by many

factors such as the strength of joints, mesh density, element shape, orientations of joints,

etc. In further studies, it is important to sort out the effects induced by each factor.

10.4.4 Vertical seismic coefficient

The vertical ground acceleration is one of the key factors leading to rockslides near the

epicentre (Chen et al. (2003) and Sepúlveda et al. (2005b)). Consequently, extension of

current method to account for vertical seismic coefficient ( vk ) is seen as a next

important step. Another potential study could be related to the assessment of the

displacement under earthquake loading based on the Newmark’s method, which

requires the estimation of the yield acceleration for each slope. Hence, examining the



10-5

variation in yield seismic coefficient with a change in rock mass strength parameters

( ci , GSI , im and D ) would be a valuable addition.



10-6

Table 10.1 Summary of the bounded factor of safety for soil slopes

Slope type Limits of the upper and lower bound

Homogeneous slope in purely

cohesive soil hN ( 9 )

Non-homogeneous cut slope in

purely cohesive soil inN ( 8 )

Non-homogeneous natural slope in

purely cohesive soil inN ( 10 )

Homogeneous slope in cohesive-

frictional soil tanF ( 10 )

Table 10.2 Summary of the bounded factor of safety for rock slopes

Slope type Limits of the upper and lower bound

Natural rock slope under static

loadings N ( 9 )

Natural rock slope under seismic

loadings N ( 9 )

Cut rock slope under static

loadings N ( 8 )



R-1

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