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Volume 1
Stability of Geotechnical Structures:
Theoretical and Numerical Analysis
Authored By
Y.M. Cheng
Department of Civil and Environmental Engineering,
Hong Kong Polytechnic University, Hong Kong
H. Wong
LTDS (UMR 5513), ENTPE, Université de Lyon, France
C.J. Leo
School of Computing, Engineering and Mathematics,
University of Western Sydney, Locked Bag 1797,
Penrith NSW 2751, Australia
&
C.K. Lau
Fong On Geotechnics Limited, Hong Kong
Frontiers in Civil Engineering
Stability Analysis of Geotechnical Structures: Theoretical and Numerical Approaches
Volume # 1
Authors: Y.M. Cheng, H. Wong, C.J. Leo & C.K. Lau
ISSN (Online): 2468-4708
ISSN: Print: 2468-4694
ISBN (eBook): 978-1-68108-303-2
ISBN (Print): 978-1-68108-304-9
© 2016, Bentham eBooks imprint.
Published by Bentham Science Publishers – Sharjah, UAE. All Rights Reserved.
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CONTENTS
PREFACE ........................................................................................................................................................................... i
ACKNOWLEDGEMENTS ...................................................................................................................................... ii
CONFLICT OF INTEREST .................................................................................................................................... i i
CHAPTER 1 INTRODUCTION ................................................................................................................................... 3
1.1. INTRODUCTION .............................................................................................................................................. 3
1.2. BACKGROUND ................................................................................................................................................. 4
1.3. CLOSED-FORM SOLUTIONS ....................................................................................................................... 6
1.4. ENGINEERING JUDGMENT ......................................................................................................................... 8
1.5. LAYOUT OF THIS BOOK ............................................................................................................................... 9
REFERENCES .......................................................................................................................................................... 9
CHAPTER 2 UPPER AND LOWER BOUND APPROACHES ............................................................................. 12
2.1. INTRODUCTION .............................................................................................................................................12
2.2. MATERIAL STRENGTH AND ITS MATHEMATICAL DESCRIPTION .............................................14
2.3. PROBLEM SETTING AND OBJECTIVE. ...................................................................................................16
2.3.1. Geometry and Loadings ..........................................................................................................................16
2.3.2. Objective of Analysis: Domain of Supportable Loads. ...........................................................................20
2.4. LOWER BOUND OR STATIC APPROACH ..............................................................................................21
2.4.1. Statically Admissible Stress Fields .........................................................................................................21
2.4.2. Plastically Admissible Stress Fields .......................................................................................................23
2.4.3. The Static Theorem .................................................................................................................................23
2.4.4. Example of Applications: Stability of a Long Wall Mining Tunnel ......................................................25
2.4.5. Physical Interpretation of the Two Bounds ............................................................................................30
2.4.6. Improvement of the Static Bounds .........................................................................................................32
2.5. UPPER BOUND OR KINEMATIC APPROACH .......................................................................................32
2.5.1. Virtual Work Theorem ............................................................................................................................33
2.5.2. The Maximum Resisting Power and the π-Functions.............................................................................37
Example 1 of πv-function: a Circular Domain .........................................................................................38
2.5.3. π-functions in Practical Applications ......................................................................................................40
2.5.3.1. Criterion of Tresca .......................................................................................................................41
2.5.3.2. Criterion of Mohr-Coulomb .........................................................................................................42
2.5.4. The Kinematic Theorem .........................................................................................................................43
2.5.5. Example of Applications: Stability of a Long Wall Mining Tunnel ......................................................45
2.6. FURTHER WORKED EXAMPLES .............................................................................................................48
2.6.1. Stability of a Shallow Foundation on a Weightless Cohesive Soil ........................................................48
2.6.1.1. Static Approach ............................................................................................................................49
2.6.1.2. Kinematic Approach .....................................................................................................................53
2.6.2. Stability of a Vertical Slope in a Cohesive-frictional Soil Mass ............................................................60
2.6.3. Failure Surfaces in the Form of Logarithmic Spirals in Cohesive-frictional Soils ................................63
2.7. CONVEXITY OF THE DOMAIN OF SUPPORTABLE LOADS K .........................................................64
2.8. CONSTRUCTION OF DOMAIN K BY CONSIDERING RADIAL VECTORS ....................................65
2.9. CONCLUSION .................................................................................................................................................67
REFERENCES ........................................................................................................................................................68
CHAPTER 3 SLIP LINE, LIMIT EQUILIBRIUM AND LIMIT ANALYSIS METHODS ................................ .70
3.1. INTRODUCTION ............................................................................................................................................70
3.2. SLIP-LINE METHOD .....................................................................................................................................71
3.2.1. Slip-line Method for Plane Strain Problem .............................................................................................73
3.2.2. Slip Line Analysis for Axi-symmetric Problem .....................................................................................90
3.2.3. Discussion on Slip Line Analysis ...........................................................................................................99
3.3. INTRODUCTION TO LIMIT EQUILIBRIUM METHOD ......................................................................101
3.3.1. Definition of the Factor of Safety for Slope Stability Analysis ...........................................................102
3.3.2. Formulation of Limit Equilibrium Methods .........................................................................................103
3.3.3. Interslice Force Function .......................................................................................................................108
3.3.4. Discussion on the Interslice Force Function .........................................................................................118
3.4. UNIFICATION OF BEARING CAPACITY, LATERAL EARTH PRESSURE AND SLOPE
STABILITY PROBLEMS ...........................................................................................................................131
3.4.1. Discussion on Unification of Stability Analysis Methods ....................................................................154
3.5. LIMIT ANALYSIS METHOD .....................................................................................................................156
3.5.1. Lower Bound Approach ........................................................................................................................157
3.5.2. Upper Bound Approach ........................................................................................................................159
3.6. OVERALL DISCUSSION .............................................................................................................................173
ACKNOWLEDGEMENTS ..................................................................................................................................175
REFERENCES ......................................................................................................................................................175
CHAPTER 4 NUMERICAL METHODS - FINITE ELEMENT AND DISTINCT ELEMENT METHODS 183
4.1. PLANE STRAIN AND PLANE STRESS FOR CONTINUOUS PROBLEMS ......................................184
4.1.1. Plane Strain............................................................................................................................................185
4.1.2. Plane Stress............................................................................................................................................186
4.1.3. Relationship between Plane Strain and Plane Stress ............................................................................188
4.2. Fundamentals of FEM ..............................................................................................................................189
4.2.1. Principle of Virtual Displacement ................................................................................................190
4.2.2. Principle of Minimum Potential Energy (PMPE) ........................................................................191
4.3. General Expressions and Implementation Procedure of FEM ................................................................192
4.3.1. Discretization of Domain ......................................................................................................................192
4.3.2. Interpolation or Displacement Model ...................................................................................................193
4.3.3. Stiffness Equilibrium Equation (SEE) of FEM Derived from PMPE ..................................................195
4.3.4. Derivation of Element Stiffness Matrices (ESM) .................................................................................197
4.3.5. Assembling of ESMs and ENLMs ........................................................................................................198
4.3.6. Isoparametric Element and Numerical Integration ...............................................................................198
4.3.7. Derivative and Integral Transformation ................................................................................................201
4.4. DISTINCT ELEMENT METHOD ..............................................................................................................203
4.4.1. General Formulation of DEM ...............................................................................................................207
4.4.2. The Force-displacement Law ................................................................................................................207
4.4.3. Law of Motion .......................................................................................................................................208
4.4.3. Measuring Logic ...................................................................................................................................210
4.4.4. Contact Constitutive Models .................................................................................................................211
4.4.5. Model Generation ..................................................................................................................................212
4.5. FINITE ELEMENT ANALYSIS OF SLOPE STABILITY PROBLEMS ..............................................214
4.5.1. Three-dimensional Slope Stability Analysis by Finite Element ...........................................................215
4.5.2. Failure Mechanism of Curvilinear Slope ..............................................................................................217
4.5.3. 3D Intercolumn Force Function on Plane Slope by Stress Analysis ....................................................223
4.5.4. Curvature Effect on the Internal Force Distribution .............................................................................225
4.5.5. Stability of locally loaded slope with curvature ...................................................................................232
4.6. DISTINCT ELEMENT ANALYSIS OF 3D SLOPE .................................................................................233
4.6.1. DEM Analysis of 3D Slope with Curvature .........................................................................................233
4.6.2. Laboratory and 3D DEM Analysis of Failure Mechanism of Slope under External Surcharge ..........242
4.7. CONCLUSION ...............................................................................................................................................260
ACKNOWLEDGEMENTS ..................................................................................................................................260
REFERENCES ......................................................................................................................................................260
CHAPTER 5 NUMERICAL TECHNIQUES FOR SOLUTION OF STABILITY PROBLEMS .....................264
5.1. LIMIT EQUILIBRIUM ................................................................................................................................264
5.1.1. Critical Slip Surface ..............................................................................................................................266
5.2. DISPLACEMENT FINITE-ELEMENT ANALYSIS ................................................................................268
5.3. FINITE ELEMENT LIMIT ANALYSIS ....................................................................................................271
5.4. LOWER BOUNDFINITE ELEMENTS ......................................................................................................271
5.4.1. Continuum and Discontinuity Equilibrium ...........................................................................................273
5.4.2. Boundary Conditions ............................................................................................................................277
5.4.3. Yield Conditions, Linear Programming ................................................................................................278
5.4.4. Yield Conditions, Nonlinear Programming ..........................................................................................281
Objective function ...................................................................................................................................281
5.4.5. Lower Bound Optimisation Problem ....................................................................................................282
5.4.6. Solution Procedures ..............................................................................................................................283
Linear Programming ...............................................................................................................................283
Nonlinear Programming .........................................................................................................................286
5.5. UPPER BOUNDFINITE ELEMENTS ........................................................................................................286
5.5.1. Objective Function ................................................................................................................................291
5.5.2. Discontinuity Flow Rule .......................................................................................................................292
5.5.3. Prescribed Boundary Velocities ............................................................................................................295
5.3.4. Loadings ................................................................................................................................................296
5.5.5. Upper Bound Optimisation Problem ....................................................................................................297
Non-linear Programming ........................................................................................................................297
5.6. APPLICATIONS ............................................................................................................................................298
5.6.1. Finite element Lower Bound Limit Analysis using Linear Programming ...........................................298
5.6.2. Finite Element Upper Bound Limit Analysis using Linear Programming ...........................................300
5.6.3. Finite Element Lower Bound Limit Analysis using Nonlinear Programming .....................................303
5.7. CONCLUSION ...............................................................................................................................................306
ACKNOWLEDGEMENTS. .................................................................................................................................306
REFERENCES ......................................................................................................................................................306
CHAPTER 6 APPLICATIONS OF PLASTICITY THEORY AND LIMIT ANALYSIS TO THE BEARING
CAPACITY OF SHALLOW FOUNDATIONS .........................................................................................................310
6.1. OUTLINE ........................................................................................................................................................310
6.2. INTRODUCTION ..........................................................................................................................................312
6.2.1. The Method of Characteristics ..............................................................................................................320
6.2.2. Method of Computation ........................................................................................................................324
6.2.3. Principle of Superposition .....................................................................................................................328
6.2.4. Calculation of Values for Nq and Nγ for Smooth and Rough Bases ...................................................332
6.2.5. Shape Effects .........................................................................................................................................338
6.3. EXTENDED PRANDTL EQUATION WITH VARIABLE Ø ..................................................................340
6.3.1. Validation of Stress-rotation Equation ..................................................................................................344
6.3.2. The method of Characteristic with Variable Ø .....................................................................................346
6.3.3. Ø Against s: Data for Typical Sands .....................................................................................................349
6.3.4. Validation of the Method of Characteristics with Variable Ø ..............................................................350
6.4. PRACTICAL APPLICATIONS ...................................................................................................................355
Material Acquisition ........................................................................................................................................362
Sample/Model Preparation ..............................................................................................................................363
6.4.1. Triaxial Test Programme .......................................................................................................................364
1-g Model Tests .......................................................................................................................................368
6.4.2. Penetration Effects ................................................................................................................................370
1-g Tests: Discussion ..............................................................................................................................375
Centrifuge model tests .............................................................................................................................380
6.4.3. Centrifuge Tests: Interpretation ............................................................................................................381
Centrifuge Tests: Discussion...................................................................................................................382
6.5. CONCLUSION ...............................................................................................................................................383
NOTATION ...........................................................................................................................................................386
REFERENCES ......................................................................................................................................................388
SUBJECT INDEX .................................................................................................................................................... 392
i
PREFACE
Due to the difficulty to define the initial stress and boundary conditions, the loading paths as
well as the constitutive model of geomaterials, stability analysis has always been a very
important discipline in geotechnical engineering. Towards this, engineers will assess the
ultimate conditions where the strength of the system is fully mobilized. The facture of failure
or the collapse load will then be assessed based on the ultimate analysis which is considered to
be unaffected by the initial conditions of the system.
This book will introduce the fundamental concepts and applications of plasticity theory, limit
equilibrium, and limit analysis in geotechnical engineering. These concepts will be illustrated
using analytical examples whenever possible in order to enhance understanding at a
fundamental level and can also be used to make preliminary estimates of geotechnical stability.
In parallel, suitable numerical methods and advanced computational tools will be introduced
for the engineers to solve theoretical and geotechnical problems of practical interests which
require greater detailed consideration.
This book deals with the challenging subject matter in a systematic fashion, from a theoretical
standpoint to practice in the real world. For this reason, the book is divided into 3 parts. In part
1, the fundamental concepts in plasticity, limit equilibrium, limit analysis and instability for
geomaterials are presented as a first step in introducing readers as the theoretical basis.
Analytical and semi-analytical solutions are then discussed in Part 2, with liberal use of
illustrative examples, as a further step to shed insights and reinforce the underlying principles
embodied in the theory. Finally in Part 3, examples utilising advanced computational tools like
the finite element and discrete element methods are covered for the purpose of elucidating the
complexity of dealing with stability problems of the real world using numerical approaches.
A particular feature of this book is that it stresses the rigorous formulation as much as the
computational techniques to tackle stability problems. It is noted that the solution of these
problems is far from trivial. The search of failure load and the corresponding failure mechanism
involve the constrained optimisation of discontinuous objective functions containing multiple
optimum points. In short, this book is an attempt to present within a single volume the
fundamentals as well as the practical developments of stability analysis in geotechnical
engineering in an easily accessible manner. Most of the materials are either based on the
research works from the authors or the teaching materials to the postgraduate students.
This book is aimed at researchers and engineers working in the field of g e otechnical
