two and three dimensional stability analyses for soil...
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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
8.2.3 Comparisons of the tangential method and the numerical limit
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
X
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,
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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.
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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.
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
1-6
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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.
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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.
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
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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
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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
2.4.1 Physical model tests
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
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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
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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
Although the limit theorems provide a simple and useful way of analysing the stability
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.
2.5.1 Physical model tests
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.
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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.
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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.
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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
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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 .
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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
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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
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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-
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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.
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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
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''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)
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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
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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.
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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.
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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
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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
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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))
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Table 2.6 Adjustment factors for in-situ rating components (Ramamurthy (1995))
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Figure 2.1 Slope height versus slope angle relation for hard rock slopes after Hoek and
Bray (1981)
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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
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Figure 2.4 Combinations of discontinuities forming a failure surface
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Figure 2.5 Crushing, toppling and buckling failures
Crushing failure
Buckling failure
Toppling failure
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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
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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
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Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
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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)
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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.
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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.
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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
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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.
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 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
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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
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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 .
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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
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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:
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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.
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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
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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.
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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
444111 ,,,,,, xyyxxyyxTx
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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
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2cos2sin2
12sin
2
1
2sincossin 22
T
444111 ,,,,,, xyyxxyyxTx
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
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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)
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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,
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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
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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
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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
a x y x y b y y c x x
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.
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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
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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
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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
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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
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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
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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.
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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
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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:
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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.
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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.
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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
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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
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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
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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
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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
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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
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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
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Figure 3.13 Meridian plane section of Hoek-Brown yield surface and its smooth
approximation
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c
(b) plane
(b) Principle stress space
Figure 3.14 Mohr-Coulomb yield criterion
231
m
231
s
,
1
3
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Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
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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.
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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.
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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
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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.
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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.
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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.
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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)
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Table 4.1 (continued)
Slope type Variables Dimensionless parameter
used in this thesis
Methods used in analysis of
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
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Table 4.1 (continued)
Slope type Variables Dimensionless parameter
used in this thesis
Methods used in analysis of
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
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Table 4.1 (continued)
Slope type Variables Dimensionless parameter
used in this thesis
Methods used in analysis of
this thesis
Cut rock slope under static
loadings
7515
10010 GSI
355 im
0.10.0 D
FHN ci
FGSI UB, LB, LEM
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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
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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
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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
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(a) 2D
(b) 3D
Figure 4.4 Typically used mesh in ABAQUS
C
C
Symmetric face
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-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
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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).
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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
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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 .
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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
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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
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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
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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
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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)
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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.
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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 .
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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.
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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
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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 .
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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.
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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 ).
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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
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Figure 5.1 Problem configuration for homogeneous slopes in purely cohesive soil
Fixed face
u = v = w = 0 (Upper bound)
u = v = w = 0 (Upper bound)
d
u = v = w = 0 (Upper bound)
Symmetric face
τ = 0 (Lower bound)
v = 0 (Upper bound)
u = v = w = 0 (Upper bound)
x, u y, v
z, w
L/2
H
β σn = τ = 0 (Lower bound)
σn = τ = 0 (Lower bound)
σn = τ = 0 (Lower bound)
Mode of Failure:
F = Face failure
T = Toe failure
B = Base failure
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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
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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
Depth factor (d / H)
= 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
Depth factor (d / H)
= 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 )
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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
Depth factor (d / H)
= 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
Depth factor (d / H)
= 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 )
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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
Depth factor (d / H)
= 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
Depth factor (d / H)
= 7.5
= 15
= 22.5
= 30
Nh =
cu/
HF
= 45
H
= 60
= 75
Less stable
(b) Upper bound
Figure 5.5 Three dimensional limit analysis solutions of stability numbers for
homogeneous undrained slopes ( 2HL )
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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)
= 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
Depth factor (d / H)
= 7.5
= 15
= 22.5
= 30
Nh =
cu/
HF
= 45
H
= 60
= 75
Less stable
(b) Upper bound
Figure 5.6 Three dimensional limit analysis solutions of stability numbers for
homogeneous undrained slopes ( 3HL )
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1 2 3 4 50.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
Depth factor (d / H)
= 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
Depth factor (d / H)
= 7.5
= 15
= 22.5
= 30
Nh =
cu/
HF
= 45
H
= 60
= 75
Less stable
(b) Upper bound
Figure 5.7 Three dimensional limit analysis solutions of stability numbers for
homogeneous undrained slopes ( 5HL )
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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
Depth factor (d / H)
(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
Depth factor (d / H)
(b) Upper bound
Figure 5.8 Three dimensional limit analysis solutions of stability numbers for
homogeneous undrained slopes ( 90 )
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(a) 30
(b) 5.22
(c) 15
Figure 5.9 2D upper bound plastic zones for 2Hd
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(a) 30
(b) 5.22
(c) 15
Figure 5.10 2D upper bound plastic zones for 5Hd
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(a) 60
(b) 75
(c) 90
Figure 5.11 2D upper bound plastic zones for various slope angles ( 2Hd )
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(a) 2Hd
(b) 4Hd
(c) 5Hd
Figure 5.12 3D upper bound plastic zones for various Hd ( 15 and 5HL )
Symmetric face
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(a) 2Hd
(b) 4Hd
(c) 5Hd
Figure 5.13 3D plastic zones for various Hd ( 5.22 and 5HL )
Symmetric face
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(a) 3HL
(b) 1HL
Figure 5.14 3D plastic zones for various HL ratios ( 15 and 4Hd )
Symmetric face
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(a) 45
(b) 60
(c) 75
Figure 5.15 3D plastic zones for various slope angles ( 2Hd and 5HL )
Symmetric face
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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)
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1 2 3 4 50.02
0.04
0.06
0.08
0.10
0.12
L / H = 1
Depth factor (d / H)
= 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
Depth factor (d / H)
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
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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)
Gens et al.(toe-failure)
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))
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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)
Gens et al.(toe-failure)
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)
Gens et al.(toe-failure)
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))
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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)
Gens et al.(toe-failure)
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))
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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)
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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
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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
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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 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Figure 5.24 Limit analysis solutions of stability numbers for inhomogeneous undrained
cut slopes ( 30 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Figure 5.24 (continued) Limit analysis solutions of stability numbers for
inhomogeneous undrained cut slopes ( 30 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Figure 5.25 Limit analysis solutions of stability numbers for inhomogeneous undrained
cut slopes ( 45 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Figure 5.25 (continued) Limit analysis solutions of stability numbers for
inhomogeneous undrained cut slopes ( 45 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Figure 5.26 Limit analysis solutions of stability numbers for inhomogeneous undrained
cut slopes ( 60 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Figure 5.26 (continued) Limit analysis solutions of stability numbers for
inhomogeneous undrained cut slopes ( 60 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Figure 5.27 Limit analysis solutions of stability numbers for inhomogeneous undrained
cut slopes ( 75 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Figure 5.27 (continued) Limit analysis solutions of stability numbers for
inhomogeneous undrained cut slopes ( 75 )
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
Figure 5.28 Limit analysis solutions of stability numbers for inhomogeneous undrained
cut slopes ( 90 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Figure 5.32 Limit analysis solutions of stability numbers for inhomogeneous undrained
natural slopes ( 15 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Figure 5.32 (continued) Limit analysis solutions of stability numbers for
inhomogeneous undrained natural slopes ( 15 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Figure 5.33 Limit analysis solutions of stability numbers for inhomogeneous undrained
natural slopes ( 30 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Figure 5.33 (continued) Limit analysis solutions of stability numbers for
inhomogeneous undrained natural slopes ( 30 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Figure 5.34 Limit analysis solutions of stability numbers for inhomogeneous undrained
natural slopes ( 45 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Figure 5.34 (continued) Limit analysis solutions of stability numbers for
inhomogeneous undrained natural slopes ( 45 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Figure 5.35 Limit analysis solutions of stability numbers for inhomogeneous undrained
slopes ( 60 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Figure 5.35 (continued) Limit analysis solutions of stability numbers for
inhomogeneous undrained slopes ( 60 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Figure 5.36 Limit analysis solutions of stability numbers for inhomogeneous undrained
slopes ( 75 )
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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
Figure 5.36 (continued) Limit analysis solutions of stability numbers for
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 )
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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)
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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
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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.
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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
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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
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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
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'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
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slopes presented in Section 5.2.3. Therefore, results for slope stability obtained
using the FEM are conservative solutions.
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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
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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
u = v = w = 0 (Upper bound)
d
Symmetric face
τ = 0 (Lower bound)
v = 0 (Upper bound)
L/2
H
β
x, u y, v
z, w
σn = τ = 0 (Lower bound)
σn = τ = 0 (Lower bound)
u = v = w = 0 (Upper bound)
0.5495
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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
Upper bound Lower bound
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
Upper bound Lower bound FEM (dry) FEM (fully saturated)
=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
Upper bound Lower bound
30
F /
tan
'
c' /Htan'
(c) (d)
Figure 6.3 Stability charts for cohesive-frictional slopes ( 1HL )
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0.0 0.5 1.0 1.5 2.0 2.50
2
4
6
8
10
12
14
Upper bound Lower bound FEM
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
Upper bound Lower bound
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
Upper bound Lower bound FEM (dry) FEM (fully saturated)
=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
Upper bound Lower bound
30
F /
tan
'
c' /Htan'
(c) (d)
Figure 6.4 Stability charts for cohesive-frictional slopes ( 2HL )
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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
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
Upper bound Lower bound
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
Upper bound Lower bound FEM (dry) FEM (fully saturated)
=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
Upper bound Lower bound
30
F /
tan
'
c' /Htan'
(c) (d)
Figure 6.5 Stability charts for cohesive-frictional slopes ( 3HL )
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=1
5
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
75
45
F /
tan
'
c' /Htan'0.0 0.1 0.2 0.3 0.4 0.5
0
1
2
3
4
5
Upper bound Lower bound
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
Upper bound Lower bound FEM (dry) FEM (fully saturated)
=60
F /
tan
'
c' /Htan'0.0 0.1 0.2 0.3 0.4 0.5
0
1
2
3
4
560
Upper bound Lower bound
30
F /
tan
'
c' /Htan'
(c) (d)
Figure 6.6 Stability charts for cohesive-frictional slopes ( 5HL )
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0.0 0.5 1.0 1.50
2
4
6
8
10
12
14
Upper bound Lower bound
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
Upper bound Lower bound
2D
L / H =3
L / H =
1
F /
tan
'
c' /Htan'
(b) 60
Figure 6.7 Stability charts for various HL ratios
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45 60
Figure 6.8 2D plastic zones from upper bound limit analyses for different friction
angles ( ' )
15'
25'
35'
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(a) 1HL
(b) 3HL
Figure 6.9 3D plastic zones from upper bound limit analyses for different HL ratios
( 45 and 25' )
Symmetric face
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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
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(a) 15
(b) 25
Figure 6.11 3D plastic zones from FEM analyses for different friction angles ( 45
and 3HL )
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-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.
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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
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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
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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-
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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 '
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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.
7.3.3 Comparisons of the tangential method and the numerical limit
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
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(implemented here) and the average bound solutions is found to increase slightly as
GSI or slope angle ( ) increases.
7.3.4 Application example
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
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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
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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.
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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.
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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).
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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)
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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
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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.
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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% - -
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Table 7.1 (continued)
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 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% - -
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Table 7.1 (continued)
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
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%
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
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Table 7.1 (continued)
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
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%
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
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Table 7.1 (continued)
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
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%
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
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Table 7.1 (continued)
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
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%
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
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7-20
Toe
Rigid Base
d
Jointed Rock
ciGSI,m
i
H
Figure 7.1 Problem configuration for simple homogeneous slopes
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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 )
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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
Increasing Stability
Figure 7.3 Average finite element limit analysis solutions for stability numbers
( 30 )
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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
Increasing Stability
Figure 7.4 Average finite element limit analysis solutions for stability numbers
( 45 )
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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
Increasing Stability
Figure 7.5 Average finite element limit analysis solutions for stability numbers
( 60 )
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5 10 15 20 25 30 350.01
0.1
1
10
100
Average SLIDE-Hoek-Brown Model Tangential Method
GSI=50
GSI=100
GSI=10
N= ci
/H
F
= 75
mi
H
Increasing Stability
Figure 7.6 Average finite element limit analysis solutions for stability numbers
( 75 )
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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
Figure 7.7 Average finite element limit analysis solutions for stability numbers
( GSI 100, 80 and 60)
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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
Figure 7.8 Average finite element limit analysis solutions for stability numbers
( GSI 40, 20 and 10)
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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
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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
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Figure 7.10 Upper bound plastic zones for different slope angles ( 70GSI and
15im )
45
30
15
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Figure 7.11 Upper bound plastic zones for different im values ( 60 and
70GSI )
5im 15im
25im 35im
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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
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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
GSI = 30-100 GSI = 20 GSI = 10 Tangential method (GSI = 50)
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
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5 10 15 20 25 30 35
0.1
1
10
100
mi
GSI = 30-100 GSI = 20 GSI = 10 Tangential method (GSI = 50)
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
GSI = 30-100 GSI = 20 GSI = 10 Tangential method (GSI = 50)
= 75H
GSI = 100
GSI = 30
(d) 75
Figure 7.13 (continued)
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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
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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
Douglas criterion (GSI=40, m
i=30)
GSI=40, mi=30
Douglas criterion (GSI=10, m
i=30)
GSI=10, mi=30
/ ci
/
ci
Figure 7.17 Comparison of yield surfaces for different GSI
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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 )
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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 )
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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 )
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Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
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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
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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
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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
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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.
8.2.3 Comparisons of the tangential method and the numerical limit
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.
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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 ,
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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.
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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
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8-11
khW
WToe
Rigid Base
Jointed Rock
ciGSI,m
i
H d
Figure 8.1 Problem configuration for simple homogeneous slopes
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8-12
5 10 15 20 25 30 35
0.01
0.1
1
10
Average SLIDE-Hoek-Brown Model Tangential Method
= 30, kh = 0.1
N= ci
/H
F
GSI=10
GSI=50
GSI=100
mi
H
Increasing Stability
(a)
Figure 8.2 Average finite element limit analysis solutions for stability numbers
( 1.0hk )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Increasing Stability
(b)
Figure 8.2 (continued) Average finite element limit analysis solutions for stability
numbers ( 1.0hk )
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8-14
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.)
= 60, kh = 0.1
GSI=10
GSI=50
GSI=100
H
N= ci
/H
F
mi
Increasing Stability
(c)
Figure 8.2 (continued) Average finite element limit analysis solutions for stability
numbers ( 1.0hk )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
8-15
5 10 15 20 25 30 350.01
0.1
1
10
100
Average SLIDE-Hoek-Brown Model Tangential Method
mi
= 75, kh = 0.1
GSI=50
GSI=100
GSI=10
N= ci
/H
F
H
Increasing Stability
(d)
Figure 8.2 (continued) Average finite element limit analysis solutions for stability
numbers ( 1.0hk )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
8-16
5 10 15 20 25 30 350.01
0.1
1
10
Average SLIDE-Hoek-Brown Model Tangential Method
mi
= 30, kh = 0.2
GSI=10
GSI=50
GSI=100
N= ci
/H
F
H
Increasing Stability
(a)
Figure 8.3 Average finite element limit analysis solutions for stability numbers
( 2.0hk )
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The University of Western Australia Centre for Offshore Foundation Systems
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
Increasing Stability
(b)
Figure 8.3 (continued) Average finite element limit analysis solutions for stability
numbers ( 2.0hk )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
8-18
5 10 15 20 25 30 35
0.1
1
10
100
1000
Average SLIDE-Hoek-Brown Model Tangential Method (this study) Tangential Method (Yang et al.)
= 60, kh = 0.2
GSI=10
GSI=50
GSI=100
H
N= ci
/H
F
mi
Increasing Stability
(c)
Figure 8.3 (continued) Average finite element limit analysis solutions for stability
numbers ( 2.0hk )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Increasing Stability
(d)
Figure 8.3 (continued) Average finite element limit analysis solutions for stability
numbers ( 2.0hk )
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The University of Western Australia Centre for Offshore Foundation Systems
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
Increasing Stability
(a)
Figure 8.4 Average finite element limit analysis solutions of stability numbers
( 3.0hk )
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8-21
5 10 15 20 25 30 35
0.1
1
10
100
mi
Average SLIDE-Hoek-Brown Model
= 45, kh = 0.3
GSI=10
GSI=50
GSI=100
N= ci
/H
F
H
Increasing Stability
(b)
Figure 8.4 (continued) Average finite element limit analysis solutions of stability
numbers ( 3.0hk )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
8-22
5 10 15 20 25 30 350.1
1
10
100
mi
Average SLIDE-Hoek-Brown Model
= 60, kh = 0.3
N
= ci
/H
F
GSI=10
GSI=50
GSI=100H
Increasing Stability
(c)
Figure 8.4 (continued) Average finite element limit analysis solutions of stability
numbers ( 3.0hk )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
8-23
5 10 15 20 25 30 350.1
1
10
100
1000
mi
Average SLIDE-Hoek-Brown Model
= 75, kh = 0.3
GSI=10
GSI=50
GSI=100
N= ci
/H
F
H
Increasing Stability
(d)
Figure 8.4 (continued) Average finite element limit analysis solutions of stability
numbers ( 3.0hk )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
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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
Decreasing Stability
Figure 8.6 Comparisons of stability numbers between the static and pseudo static
analyses ( 30 and 45 )
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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
Decreasing Stability
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
Decreasing Stability
Figure 8.7 Comparisons of stability numbers between the static and pseudo static
analyses ( 60 and 75 )
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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 )
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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
Figure 8.9 Factor of safety assessment based on GSI ( 2.0hk )
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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
Figure 8.10 Factor of safety assessment based on GSI ( 3.0hk )
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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
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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
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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
Hoek-Brown (GSI=100, mi=10)
Tangential method Lower bound (k
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
Hoek-Brown (GSI=100, mi=10)
Tangential method Lower bound (k
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 )
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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 )
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-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)
GSI=100 mi=35, Lower bound (k
h = 0.3)
GSI=100 mi=5, Lower bound (k
h = 0.0)
GSI=100 mi=5, Lower bound (k
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)
GSI=100 mi=35, SLIDE (k
h = 0.4)
GSI=100 mi=5, SLIDE (k
h = 0.4)
Figure 8.16 The observed stress conditions along the slip plane from the limit
equilibrium analyses
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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
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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
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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
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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
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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.
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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.
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An important construction recommendation should be not to build within this influence
zone near the slope crest.
9.2.3 Application example
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
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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
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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
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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
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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
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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
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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.
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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
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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
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Table 9.2 (continued)
Numerical Assessment
Mine case ci ( MPa ) ci H Stable Safety factor ( F ) Stable
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 )
Numerical Assessment
Mine case ci ( MPa ) ci H Stable Safety factor ( F ) Stable
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
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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
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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
Increasing Stability
(a)
Figure 9.2 Average finite element limit analysis solutions of stability numbers for
disturbed slopes ( D 0.7)
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
9-19
5 10 15 20 25 30 35
0.01
0.1
1
10
100
= 30, D = 0.7
Average SLIDE-Hoek-Brown Model
N= ci
/H
F
GSI=10
GSI=50
GSI=100
mi
H
Increasing Stability
(b)
Figure 9.2 (continued) Average finite element limit analysis solutions of stability
numbers for disturbed slopes ( D 0.7)
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
9-20
5 10 15 20 25 30 350.01
0.1
1
10
100
= 45, D = 0.7
mi
Average SLIDE-Hoek-Brown Model
GSI=10
GSI=50
GSI=100
N= ci
/H
F
H
Increasing Stability
(c)
Figure 9.2 (continued) Average finite element limit analysis solutions of stability
numbers for disturbed slopes ( D 0.7)
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
9-21
5 10 15 20 25 30 35
0.1
1
10
100
= 60, D = 0.7
mi
Average SLIDE-Hoek-Brown Model
GSI=10
GSI=50
GSI=100
N= ci
/H
F
H
Increasing Stability
(d)
Figure 9.2 (continued) Average finite element limit analysis solutions of stability
numbers for disturbed slopes ( D 0.7)
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
9-22
5 10 15 20 25 30 35
0.1
1
10
100
1000
mi
Average SLIDE-Hoek-Brown Model
= 75, D = 0.7
GSI=10
GSI=50
GSI=100
N= ci
/H
F
H
Increasing Stability
(e)
Figure 9.2 (continued) Average finite element limit analysis solutions of stability
numbers for disturbed slopes ( D 0.7)
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
9-23
5 10 15 20 25 30 351E-3
0.01
0.1
1
10
100
Average SLIDE-Hoek-Brown Model
H
N= ci
/H
F
= 15, D = 1.0
GSI=10
GSI=50
GSI=100
mi
Increasing Stability
(a)
Figure 9.3 Average finite element limit analysis solutions of stability numbers for
disturbed slopes ( D 1.0)
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
9-24
5 10 15 20 25 30 35
0.01
0.1
1
10
100
= 30, D = 1.0
mi
Average SLIDE-Hoek-Brown Model
GSI=10
GSI=50
GSI=100
N= ci
/H
F
H
Increasing Stability
(b)
Figure 9.3 (continued) Average finite element limit analysis solutions of stability
numbers for disturbed slopes ( D 1.0)
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
9-25
5 10 15 20 25 30 350.01
0.1
1
10
100
1000
= 45, D = 1.0
mi
Average SLIDE-Hoek-Brown Model
N
= ci
/H
F
GSI=10
GSI=50
GSI=100H
Increasing Stability
(c)
Figure 9.3 (continued) Average finite element limit analysis solutions of stability
numbers for disturbed slopes ( D 1.0)
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
9-26
5 10 15 20 25 30 35
0.1
1
10
100
1000
= 60, D = 1.0
mi
Average SLIDE-Hoek-Brown Model
GSI=10
GSI=50
GSI=100
N= ci
/H
F
H
Increasing Stability
(d)
Figure 9.3 (continued) Average finite element limit analysis solutions of stability
numbers for disturbed slopes ( D 1.0)
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
9-27
5 10 15 20 25 30 35
0.1
1
10
100
1000
= 75, D = 1.0
mi
Average SLIDE-Hoek-Brown Model
GSI=50
GSI=10
GSI=100
N= ci
/H
F
H
Increasing Stability
(e)
Figure 9.3 (continued) Average finite element limit analysis solutions of stability
numbers for disturbed slopes ( D 1.0)
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Decreasing Stability
Figure 9.6 Comparisons of stability numbers for different disturbance factors ( 15 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Decreasing Stability
Figure 9.7 Comparisons of stability numbers for different disturbance factors ( 45 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Decreasing Stability
Figure 9.8 Comparisons of stability numbers for different disturbance factors ( 75 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Lower bound ( = 45) GSI =10, m
i =10, D = 0.0
Lower bound ( = 15) GSI =50, m
i =10, D = 1.0
Lower bound ( = 45) GSI =50, m
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
Lower bound ( = 15) GSI =10, m
i =10, D = 0.0
Lower bound ( = 45) GSI =10, m
i =10, D = 0.0
Lower bound ( = 15) GSI =50, m
i =10, D = 1.0
Lower bound ( = 45) GSI =50, m
i =10, D = 1.0
A
/ ci
/
ci
(b)
Figure 9.10 The forms of the Hoek-Brown failure criterion for various strength
parameters
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Figure 9.11 Average finite element limit analysis solutions of stability numbers for
different slope angles and disturbance factors ( GSI 90, 50 and 10)
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
9-38
10GSI 80GSI
Figure 9.12 Upper bound plastic zones for different GSI ( 0.1D and 10im )
45
30
60
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
9-39
Figure 9.13 Lateral extent of failure surfaces for 100GSI and 5im
H1.1
H55.0
H5.0
30
15
45
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
9-40
Figure 9.13 (continued) Lateral extent of failure surfaces for 100GSI and 5im
H5.0
H5.0
60
75
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
9-41
(a) Upper bound solution
(b) Solution of
Figure 9.14 Case study 1
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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Figure 9.15 Case study 2
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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)
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
9-45
0.1D 0.10 D
Figure 9.18 Upper bound plastic zones for different distributions of D ( 10GSI and
10im )
45
60
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
9-46
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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 ,
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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,
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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.
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
The University of Western Australia Centre for Offshore Foundation Systems
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 )
Two and Three Dimensional Stability Analyses for Soil and Rock Slopes
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R-1
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