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1/93 mus-m4

 

A PRACTICAL APPROACH TO COMBINED PROBABILISTIC ANALYSIS OF
SLOPE STABILITY AND SEEPAGE PROBLEMS

By

Ahmed Mohamed Hassan

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree

DOCTOR OF PHILOSOPHY

Department of Civil and Environmental Engineering

1998

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ABSTRACT

A PRACTICAL APPROACH TO COMBINED PROBABILISTIC ANALYSIS OF
SLOPE STABILITY AND SEEPAGE PROBLEMS

By

Ahmed Mohamed Hassan

This research is concerned with studying issues and providing tools to improve
some of the aspects required to perform consistent and accurate reliability assessments of
embankments. Probabilistic slope stability and probabilistic seepage analyses reported in
the literature have generally been applied independently to special, simple problems. This
study focused on the assessment of how practicing engineers could best use and combine
practical, existing, analysis tools in a probabilistic framework that handles complex
problems with a simple approach rather than simple problems with a complex approach.

One aspect investigated in this research is the effect of different deterministic and
probabilistic models on reliability analysis. The effect of adopting a lognormal
distribution for strength parameters was also investigated. The results showed that, for the
problems considered, the shape of slip surface and the assumption of lognormally
distributed soil parameters have a significant influence on embankment reliability. The
effect of both deterministic and probabilistic methods were found to be not significant.

The uncertainty in pore pressure for end—of-construction analysis was studied in
terms of the variability in pore pressure coefficient (ru). This parameter was modeled as

spatial random variable and as a perfectly correlated random variable with both total and

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systematic variance. The two cases were shown to provide reasonable reliability results
that compare well to the case of considering the variability in an actual ru grid values.

A practical and simple algorithm to locate the critical probabilistic slip surface
was developed. This approach is based on systematically examining combinations of soil
parameters other than the design (mean) values. The approach was examined and applied
to an extensive parametric study on simple problems as well as several case-studies. It
proved to provide a practical tool to locate the critical probabilistic surface. The algorithm
also has the advantage of being applicable within the context of existing available slope
stability computer programs. In many of the problems analyzed, the reliability index
associated with the critical probabilistic surface was significantly lower than that
associated with the critical deterministic surface.

The uncertainty in pore pressure u due to spatial variability of hydraulic
conductivity k was investigated by assuming k as a spatially correlated random variable.
The iterative finite-difference relaxation method was applied using spreadsheets both to
generate spatially correlated values of k and to solve the steady-state flow equation. The
spreadsheets provided simple tools and proved to produce reasonable results with
considerably reduced computation effort. The study included the effect of the coefficient
of variation of k, the scale of fluctuation of k, and the anisotropy in the correlation

structure of k on the variability of u.

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ACKNOWLEDGMENTS

The author wishes to express his sincere gratitude to Dr. Thomas F. Wolff for his
encouragement, support and advice throughout this study and his timely helpful reviews
during preparation of the thesis. His time, effort, and suggestions greatly helped me to
improve the quality of this work. Gratitude is also due to Dr. Ronald Harichandran, Dr.
Neeraj Buch, and Dr. Grahame Larson for their kind advice and valuable suggestions.
Thanks also to Dr. Fayek Hassona for his encouragement and support.

Data and some computer models used in this study were fiom a research effort
sponsored by the US. Army Corps of Engineers. The Egyptian Ministry of Education
provided funds for the first two years of this research. The Department of Civil and
Environmental Engineering, Michigan State University, also provided financial support
which helped in the completion of this study. My deep thanks in this regard to Dr.
William Taylor, Dr. Thomas F. Wolff, and Dr. Ronald Harichandran.

Acknowledgment is given to Abeer Nasr for her helpful assistance throughout the
course of this study. Many of the figures and computer runs were prepared by her in
addition to helping in word processing.

Finally, and most important, thanks go to my wife Abeer and my daughter Lina
for their extreme patience during the time required to complete this study. Their

encouragement and support are greatly appreciated. I also will never forget the support

and guidance provided by four great people I lost: my parents and my two brothers: Dr.

Ibrahim Hassan and Dr. Ismail Hassan.

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TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................ XI
LIST OF FIGURES ....................................................................................................... XIV
LIST OF SYMBOLS AND ABBREVIATIONS .......................................................... XIX
Chapter 1
INTRODUCTION ............................................................................................................... 1
Chapter 2
BASIC CONCEPTS OF PROBABILITY ANALYSIS ...................................................... 8
2.1 General ..................................................................................................................... 8
2.2 Random Variables .................................................................................................... 8
2.3 Moments of Random Variables ............................................................................... 9
2.4 Probability Distributions ........................................................................................ 12
2.4.1 General ............................................................................................................ 12
2.4.2 The Normal Distribution ................................................................................. 13
2.4.3 The Lognormal Distribution ........................................................................... 15
2.5 The Capacity-Demand Model ................................................................................ 16
2.6 The Reliability Index ............................................................................................. 21
2.7 Reliability Analysis ................................................................................................ 23
2.7.1 General ............................................................................................................ 23
2.7.2 The Monte Carlo Simulation .......................................................................... 24
2.7.3 The Mean-value First Order Second Moment Method (MFOSM) ................. 27
2.7.4 The Point Estimate Method (PEM) ................................................................. 32
2.8 Uncertainty in Soil Properties ................................................................................ 35
2.8.1 General ............................................................................................................. 35
2.8.2 Spatial Correlation ........................................................................................... 37
2.9 Probabilistic Versus Factor of Safety Approach ..................................................... 43
Chapter 3
BASIC CONCEPTS OF SLOPE STABILITY ANALYSIS ............................................ 46
3.1 General ................................................................................................................... 46
3.2 Modified Swedish Method ..................................................................................... 49
3.3 Simplified Bishop Method ..................................................................................... 50
3.4 Mogenstem-Price Method ..................................................................................... 50

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3.5 Spencer’s Method .................................................................................................. 52

3.6 UTEXAS3 Computer Program .............................................................................. 54
3.7 Loading Conditions for Embankments, Modeling Strength Parameters ............... 56
3.7.1 General ............................................................................................................ 56
3.7.2 End-of-Construction Conditions ..................................................................... 56
3.7.3 Steady Seepage Conditions ............................................................................. 58
3.7.4 Partial Pool Conditions ................................................................................... 58
CHAPTER 4
PRACTICAL CONSIDERATIONS FOR TOTAL STRESS ANALYSIS
OF END-OF-CONSTRUCTION CONDITIONS ............................................................. 59
4.1 General ................................................................................................................... 59
4.2 Case Studies ........................................................................................................... 61
4.2.1 General ............................................................................................................ 61
4.2.2 Cannon Dam .................................................................................................. 61
4.2.3 Shelbyville Dam .............................................................................................. 63
4.3 Strength Parameters ............................................................................................... 65
4.3.1 General ............................................................................................................ 65
4.3.1 Strength Parameters for Cannon Dam ............................................................ 67
4.3.2 Strength Parameters for Shelbyville Dam ....................................................... 68
4.4 Slope Stability Analysis ......................................................................................... 69
4.5 Reliability Analysis ................................................................................................ 75
4.5.1 Mean-Value First Order Second Moment Method (MFOSM) ....................... 75
4.5.2 Point Estimate Method (PEM) ........................................................................ 79
4.5.3 Lognormally Distributed Soil Parameters ...................................................... 80
4.6 Summary ............................................................................................................. 85
Chapter 5
PRACTICAL CONSIDERATIONS FOR EFFECTIVE STRESS ANALYSIS
OF END-OF-CONSTRUCTION CONDITIONS ............................................................. 87
5.1 General ................................................................................................................... 87
5.2 Previous Investigations .......................................................................................... 88
5.3 Modeling Uncertainty in ru .................................................................................... 90
5.3.1 General ............................................................................................................ 90
5.3.2 Case A: Average ru Value, ru Deterministic .................................................... 93
5.3.2 Case B: Actual r“ Values Grid, ru Deterministic ............................................. 93
5.3.3 Case C: Average ru Value, ru Perfectly Correlated With E[ru] and 0'[ru] ......... 95
5.3.4 Case D: Average ru Value, ru Perfectly Correlated With Fig] and o[ru] ..... 95
5.3.5 Case E: Average ru Value, ru Spatially Correlated .......................................... 96
5.3.6 Case F: Actual ru Values Grid, r,JI Rrandom Values ..................................... 102
5.4 Scale of Fluctuation ............................................................................................. 107
5.5 Analysis of Shelbyville Dam ............................................................................... 115
5.6 Total Stress Versus Effective Stress Conditions .................................................. 119
5.6 Summary ............................................................................................................... 123

VIII

 

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SEARCH ALGORITHM FOR MINIMUM RELIABILITY INDEX ............................. 125
6.1 General ................................................................................................................. 125
6.2 Previous Work ..................................................................................................... 126
6.3 The Proposed Approach ....................................................................................... 126
6.4 Application to Case Studies ................................................................................. 131

6.4.1 General .......................................................................................................... 131
6.4.2 Cannon Darn ................................................................................................. 131
6.4.3 Analysis of Bois Brule Levee ....................................................................... 136
6.4.3.1 Structure Description ............................................................................ 136
6.4.3.2 Seepage Analysis .................................................................................. 136
6.4.3.3 Slope Stability Analysis ........................................................................ 138
6.4.3.4 Reliability Analysis ............................................................................... 141
6.5 Improvement of The Method ................................................................................ 146
6.5.1 General .......................................................................................................... 146
6.5.2 Cannon Dam, Modified Swedish Method .................................................... 147
6.5.2.1 Considering Only Surfaces of Lower Values of Strength Parameters

(u - o) .................................................................................................... 147

6.5.2.2 Considering a Lower Value of Strength Parameters of u-0.50
(m = 0.50) .............................................................................................. 149

6.5.2.3 Considering a Lower Value of Strength Parameters of 1.1-1.5 0'
(m = 1.5) ................................................................................................ 149
6.5.2.4 Searching Around The Minimum-B Surface ........................................ 152
6.4.3 Analysis of Cannon Dam, Simplified Bishop Method ................................. 152
6.4.4 Parametric Studies ........................................................................................ 156
6.4.4.1 Application of The Proposed Search Method ...................................... 156
6.4.4.2 Systematic Search ................................................................................. 159
6.4.4.3 Setting More Than one Parameter to a Lower Value ............................ 164
6.6 Effect of Changing The Parameter m .................................................................. 164
6.6 Discussion ............................................................................................................ 166
6.7 Further Applications ............................................................................................ 167
6.7.1 General .......................................................................................................... 167
6.7.2 Analysis of a Homogeneous Slope .............................................................. 167
6.7.3 Analysis of Shelbyville Dam ........................................................................ 171
6.7.4 Analysis of Congress Street Open Cut .......................................................... 173
6.7.5 Non-Circular Failure Surfaces ...................................................................... 175
6.7.6 Application to AF OSM (Hasofer-Lind) ........................................................ 180
6.8 Summary .............................................................................................................. 181

CHAPTER 7

PROBABILISTIC STEADY-STATE SEEPAGE ANALYSIS ...................................... 185
7.1 General ................................................................................................................. 185
7.2 Spatial Variability of Hydraulic Conductivity ..................................................... 186
7.3 Modeling The Spatial Variability of k ................................................................. 187

IX

 

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7.3.1 Approaches to Probabilistic Analysis ........................................................... 187

7.3.2 The Nearest Neighbor Method ...................................................................... 188
7.3.3 Applying Nearest-Neighbor Process By Relaxation Method ....................... 192
7.4 Steady-State Seepage Analysis ............................................................................ 199
7.4.1 General .......................................................................................................... 199
7.4.2 Relaxation Method ........................................................................................ 200
7.4.2.1 General .................................................................................................. 200
7.4.2.2 Case of Homogeneous Flow Domain (constant k) ............................... 200
7.4.2.3 Case of Non-Homogeneous Flow Domain ........................................... 205
7.4.2.4 The Phreatic Line .................................................................................. 207
7.5 Applications ......................................................................................................... 215
7.5.1 General .......................................................................................................... 215
7.5.2 Effect of The Coefficient of Variation of k ................................................... 215
7.5.3 Effect of Scale of Fluctuation ....................................................................... 225
7.5.4 The Anisotropic Correlation Structure Case ................................................. 227
7.5.5 Anisotropy in k Values ................................................................................. 232
7.6 Application on Slope Stability Analysis ................................................................ 232
7.6.1 General .......................................................................................................... 232
7.6.2 Analysis of a Typical Levee .......................................................................... 233
7.6.3 Analysis of Bois Brule Levee ....................................................................... 234
7.7 Summary .............................................................................................................. 234
Chapter 8
SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS ................................... 236
8.1 Summary .............................................................................................................. 236
8.2 Conclusions .......................................................................................................... 239
8.2.1 Deterministic and Probabilistic Models, and Parameters Distribution ......... 239
8.2.2 Uncertainty in Pore Pressure For End-of-Construction Analysis ................. 240
8.2.3 Locating The Critical Probabilistic Surface .................................................. 242
8.2.4 Uncertainty in Pore Pressure Due to Spatial Variability of k ...................... 243
8.3 Recommendations ................................................................................................ 244
Appendix A Example Calculation of an Autocovariance Function ................................ 246
Appendix B A Typical Spreadsheet For Reliability Analysis Using MFOSM .............. 247
Appendix C A Typical Spreadsheet For Reliability Analysis Using PEM .................... 248
Appendix D Calculation of Hasofer-Lind Reliabilty Index ............................................ 249
LIST OF REFERENCES ................................................................................................. 251

 

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LIST OF TABLES

Table 1.1 Some Aspects of Probabilistic Slope Stability Studies Reported in

the Literature .................................................................................................... 3
Table 4.1 Cannon Dam, Total Stress Soil Parameters ................................................... 67
Table 4.2 Shelbyville Dam, Total Stress Soil Parameters ............................................ 68
Table 4.3 Factor of Safety Corresponding to Different Slope Stability Methods ......... 70
Table 4.4 MFOSM, [3 Values Corresponding to Different Detenninistic Models ........ 77
Table 4.5 PEM, B Values Corresponding to Different Deterministic Models .............. 79

Table 4.6 Comparison Between MFOSM, PEM, and AF OSM Analysis Results ......... 80

Table 4.7 Cannon Dam, Shear Parameters Transformed
from Lognormal to Normal Distribution ....................................................... 83

Table 4.8 Shelbyville Dam, Total Strength Soil Parameters Transformed
from Logormal to Normal Distribution ......................................................... 83

Table 4.9 Cannon Dam, Lognormally Transformed Strength Parameters
Reliability Index Corresponding to Different Deterministic Methods .......... 84

Table 4.10 Shelbyville Dam, Lognormally Transformed Strength Parameters
Reliability Index Corresponding to Different Deterministic Methods ........ 84

Table 5.1 Vallecito Dam, Moments of ru ..................................................................... 100

Table 5.2 Vallecito Dam, Reliability Analysis Results
Random Variables: 4)’ and ru ........................................................................ 105

Table 5.3 Vallecito Dam, Reliability Analysis
Random Variables: c', d)’, and ru .................................................................. 107

Table 5.4 Shelbyville Dam, Effective Strength Soil Parameters
for End of Construction Conditions ............................................................. 115

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Table 5.5 Shelbyville Dam, Effective Transformed Parameters ................................. 115

Table 5.6 Shelbyville Dam, Moments of Pore Pressure Ratio, ru ................................ 116
Table 5.7 Shelbyville Dam, End-of-Construction Conditions

Results of Effective-Stress Reliability Analysis .......................................... 118
Table 5.8 Shelbyville Dam, End-of-Construction Conditions

Results of Total-Stress and Effective-Stress Reliability Analysis ............... 121
Table 6.1 Cannon Dam, List of Circular Slip Surfaces

Associated with Different Combinations of Shear Parameters ................... 132
Table 6.2 Bois Brule Levee, Moments of Soil Parameters .......................................... 138
Table 6.3 Different Slip Surfaces Found For Bois Brule Levee .................................. 140
Table 6.4 Bois Brule Levee, Reliability Index for Different Surfaces ........................ 142
Table 6.5 Bois Brule Levee, Values of BPS, Bf, and [3min ............................................. 144
Table 6.6 Cannon Dam, Values of BPS, [3,«, and [3mm ................................................... 156

Table 6.7 Example Homogeneous Slope (after Li and Lumb (1987))
Moments of Soil Parameters ........................................................................ 169

Table 6.8 A Homogeneous Slope (after Li & Lumb (1987))

B for Different Failure Surfaces ................................................................... 170
Table 6.9 Shelbyville Dam, Results of Search Process ............................................... 171
Table 6.10 Comparison of Bf, [31:5, and 13min. for Non-Circular Surfaces ...................... 175

Table 6.11 Value of Mean-Value FOSM and Hasofer-Lind Reliability Indices ......... 181

Table 7.1 Comparison Between Pore Pressures For Homogeneous and
Non-Homogeneous k ................................................................................... 214

Table 7.2 Scale of Fluctuation Corresponding to Values of Vk
ax=orz=0.90,b=5ft ............................................................................... 216

Table 7.3 Statistical Moments of Pore Pressure for Different Cases .......................... 219

Table 7.4 Scale of Fluctuation Corresponding to Different Values of or
k = LN[O.25,0.18] ........................................................................................ 225

Table 7.5 Statistical Moments of Pore Pressure for Different Values of a ................. 228

XII

 

Table 7.

Table 7.

Table 7.6 Comparison of Isotropic and Anisotropic Cases ......................................... 231

Table 7.7 A Typical Levee, Results of Reliability Analysis ....................................... 233

XIII

LIST OF FIGURES

Figure 2.1 pdf and CDF for Normal Distribution ............................................................. 14
Figure 2.2 The Capacity-Demand Model ......................................................................... 17
Figure 2.3 pdf of Safety Margin (SM) .............................................................................. 20
Figure 2.4 pdf of The Factor of Safety (FS) .................................................................... 20
Figure 2.5 Generating Random Variables of a Given Distribution .................................. 26
Figure 2.6 Reliability Index for Two Variables

(After Hasofer and Lind, 1974) ........................................................................ 30
Figure 2.7 The Point Estimate Method ............................................................................. 33
Figure 2.8 Variation of Soil Properties ....... 36
Figure 2.9 Random Field and Moving Average ............................................................... 38
Figure 2.10 Mathematical Representation of Autocovariance Function .......................... 40

Figure 2.11 Two Cases of Different Distribution of FS

(After Christian et. al., 1994) ........................................................................ 44
Figure 3.1 Forces on a Slice ............................................................................................. 48
Figure 3.2 Modified Swedish Methods ............................................................................. 48
Figure 3.3 Simplified Bishop Method .............................................................................. 51
Figure 3.4 Morgenstem-Price Method .............................................................................. 51
Figure 3.5 Spencer’s Method ............................................................................................ 53
Figure 4.1 Cross-Section of Cannon Dam ........................................................................ 62

XIV

 

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Figure 4.2 Cross-Section of Shelbyville Dam .................................................................. 64

Figure 4.3 Characterizing Strength Data, Method 1(After Wolff et. al., 1995) ................ 66
Figure 4.4 Cannon Dam, Circular Slip Surfaces ............................................................... 71
Figure 4.5 Cannon Dam, Non- Circular Failure Surfaces ................................................. 72
Figure 4.6 Shelbyville Dam, Circular Failure Surfaces .................................................... 73
Figure 4.7 Shelbyvilee Dam, Non-Circular Surfaces ....................................................... 74
Figure 4.8 Transformation of Parameter “c” .................................................................... 82

Figure 5.1 Vallecito Dam, End-of—Construction Distribution of ru

(After Bishop and Morgenstem, 1960) ............................................................ 92
Figure 5.2 Vallecito Dam, Grid of ru Values ................................................................... 94
Figure 5.3 Vallecito Dam, Grid of ru Values and Corresponding Horizontal ACF ......... 98
Figure 5.4 Vallecito Dam, Autocovariance Function For ru Values ................................ 99
Figure 5.5 Vallecito Dam, Example of Calculating the Reduced Variance .................... 103
Figure 5.6 Vallecito Dam, Effect of Scale of Fluctuation on B ...................................... 109

Figure 5.7 Vallecito Dam, Effect of Scale of Fluctuation on Reliability Index
(For Different Modeling of ru ) .................................................................... 110

Figure 5.8 Vallecito Dam, Effect of Vru on Reliability Index for
Different Value of Scale of Fluctuation, c’ = constant .................................. 112

Figure 5.9 Vallecito Dam, Effect of Vru on Reliability Index for
Different Value of Scale of Fluctuation, c' = random .................................. 113

Figure 5.10 Vallecito Dam, Efiect of Vru on Reliability Index for
Scale Fluctuation = 30 ft ............................................................................. 114

Figure 5.11 Shelbyville Dam, ru Measurement in The Vicinity of Failure
Surface ......................................................................................................... 117

Figure 5.12 Effect of Scale of Fluctuation on Reliability Index
Scale Different Failure Surface Length (LE) ................................................ 120

XV

 

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For Total and Effective Strength Conditions ............................................... 122
Figure 6.1 Proposed Search Process ............................................................................... 128
Figure 6.2 Cannon Dam, End-of-Construction, Spencer Circular
Failure Surfaces A Through I ....................................................................... 133
Figure 6.3 Cannon Dam, End of Construction
Circular Critical Deterministic and Probabilistic Surfaces ............................ 135
Figure 6.4 Cross-Section of Bois Brule Levee ............................................................... 137
Figure 6.5 Bois Brule Levee, Critical Circular Surfaces, Spencer Method .................... 143

Figure 6.6 Bois Brule Levee, Critical Deterministic and Probabilistic Surfaces
Deterministic Method: Spencer Circular ..................................................... 145

Figure 6.7 Cannon Dam, Modified Swedish Method, Slip Surfaces

Obtained by Search Process, m = 1.0 .......................................................... 148
Figure 6.8 Cannon Dam, Slip Surfaces Obtained by Search Process
End of Construction, Modified Swedish Method, m = 0.50 ......................... 150
Figure 6.9 Cannon Dam, Slip Circles Obtained by Search Process
End of Construction, Modified Swedish Method, m = 1.50 ........................ 151
Figure 6.10 Cannon Dam, Search Around The Minimum- B Surface
End of Construction, Modified Swedish Method ........................................ 153
Figure 6.11 Cannon Dam, Slip Surfaces Obtained by Search Process,
Simplified Bishop Method ........................................................................... 154
Figure 6.12 Cannon Dam, Search Around The Minimum-B Surface
Simplified Bishop Method ........................................................................... 155
Figure 6.13 Cannon Dam, Critical Probabilistic Surfaces
For Different Deterministic Models ............................................................. 157
Figure 6.14 Cross-Section and Soil Parameters For a Typical Earth Slope ................... 158
Figure 6.15 Cases A and B ............................................................................................. 160
Figure 6.16 Cases C and D ............................................................................................. 160

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Figure 6.17 Cases E and F .............................................................................................. 161

Figure 6.18 CasesGandH ............................................................................................. 161
Figure 6.19 Cases I and J ................................................................................................ 162
Figure 6.20 Cases K and L .............................................................................................. 162
Figure 6.21 Cases M and N, ............................................................................................ 163
Figure 6.22 Case M, Systematic Search ......................................................................... 163
Figure 6.23 Case A, Testing Other Parameter Combinations ........................................ 163
Figure 6.24 Effect of The Parameter m on Bmin .............................................................. 165

Figure 6.25 A Homogeneous Slope, After Li and Lumb (1987)
Location of Critical Surfaces ....................................................................... 168

Figure 6.26 Shelbyville Dam, Slip Surfaces Obtained by

The Search Process, Spencer’s Method ....................................................... 172
Figure 6.27 Congress Street Cut, Slip Surfaces Obtained by

Search Process, Spencer’s Method .............................................................. 174
Figure 6.28 Cannon dam, Search Process For Non-Circular Surfaces ........................... 176
Figure 6.29 Bois Brule Levee, Search Process For Non-Circular Surfaces .................... 177

Figure 6.30 Cannon Dam, End-of-Construction
Non-Circular Critical Deterministic and Probabilistic Surfaces .................. 178

Figure 6.31 Bois Brule Levee, Non-Circular Critical
Deterministic and Probabilistic Surfaces ..................................................... 179

Figure 7.1 Schematic Representation of Nearest-Neighbor Grid ................................... 189

Figure 7.2 Probability Distribution of k Values Generated
By Nearest-Neighbor Method, or = 0.90 ....................................................... 195

Figure 7.3 Probability Distribution of k Values Generated
By Nearest-Neighbor Method, or = 0.99 ....................................................... 195

XVII

 

Figure 7.4 AC

Figure 7.4 ACF Obtained by N.N. Solved by Both
Relaxation and Matrix Inversion, or = 0.90 ................................................... 196

Figure 7.5 ACF Obtained by N.N. Solved by Both

Relaxation and Matrix Inversion, or = 0.99 ................................................... 197
Figure 7.6 Schematic Representation of a Relaxation Grid ............................................ 202
Figure 7.7 Relaxation Grid at Line AB ........................................................................... 202
Figure 7.8 Typical Cross-Section of A Pervious Levee .................................................. 208
Figure 7.9 Approaching The Phreatic Line by Relaxation Method ................................ 210
Figure 7.10 Comparison Between Phreatic Lines Obtained by CSEEP

and Relaxation Method ................................................................................ 211
Figure 7.11 Non-Homogeneous Levee ........................................................................... 212
Figure 7.12 Grid For Generating k Values ..................................................................... 213
Figure 7.13 Probability Distribution of Some Generated k Values ................................ 217
Figure 7.14 ACF For Cases A Through D ...................................................................... 218
Figure 7.15 Effectoka on Vu For Points 1 Through 4 ................................................. 220
Figure 7.16 Effectoka on Vu For Points 5 and 6 ........................................................ 221
Figure 7.17 Effectoka on Vu For Points 7 Through 10 ............................................... 222
Figure 7.18 Case A, Probability Distribution of Pore Pressure at

Points 1 Through 10 ..................................................................................... 224
Figure 7.19 ACF For Different Values of or, k = LN[O.25, 0.18] ................................... 226
Figure 7.20 Effect of 8 on Vu For Constant and Variable Vk ......................................... 229
Figure 7.21 ACF For an Anisotropic k Generation ........................................................ 230

XVIII

 

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LIST OF SYMBOLS AND ABBREVIATIONS

ROMAN LETTERS

C Capacity

Cov (x,y) Covariance of x and y

D Demand

E Interslice force on side of slice
Ew Water force on side of slice
E[X] Expected value of X

Pr(X) Probability of the event X

P,, P, Point concentration used in point estimate method
U Water force on base of slice

Vx Coefficient of variation of X
Var(X) Variance of X

W Weight of slice

X General random variable

Y General random variable

c Cohesion

f(x) Probability density function of x
g(x) General function of x

XIX

GREEK
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or

11 Number of soil parameters
GREEK LETTERS

(upper case)

I"(x) Variance reduction function
I'I Product
2 Summation

(lower case)

or Inclination of slice base from horizontal
B Reliability index

5 Scale of fluctuation

7 Unit weight

px Mean value of X

p Correlation coefficient

ox Standard deviation of X

r Shear stress

(p Friction angle

Abbreviations

ACF Autocovariance function

AFOSM Advanced first—order second-moment method
CDF Cumulative density function

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Mean-value first-order second-moment method
Factor of safety

Point estimate method

Probability density function

Pounds per square foot

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Consolidated undrained

Consolidated drained

Safety Margin

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Chapter 1

INTRODUCTION

Although probabilistic slope stability analyses methods have been under
development over 20 years, there has recently been increased interest in applying such
methods in practice. The US. Corps of Engineers, for example, now applies probabilistic
cost-benefit analysis in the evaluation of existing water-resource structures for
rehabilitation purposes (Wolff, 1996). As practical applications increase, improvements
are required to simplify the analyses and develop tools to enable practicing engineers to
perform complete slope-reliability assessment.

Performing meaningful and consistent reliability analysis for embankment slope
stability is difficult, especially in practice, because of modeling issues and interpretation
problems. The modeling issues are related to the fact that there are many deterministic
and probabilistic models that can be applied and combined in reliability analysis. The
results may depend on the deterministic and/or probabilistic model used. Locating the
critical probabilistic surface (surface of minimum reliability index) is also a significant
modeling problem. The interpretation problems are concerned with the effect of spatial
variability in strength parameters, pore pressure, and hydraulic conductivity. The
probability distribution of strength parameters (e.g. normal or lognormal distribution)
also influences slope-reliability. Some of the aspects that should be considered while

performing slope-reliability analysis may be listed as follows:

0 Location (and difference) of “critical” slip surface for deterministic and probabilistic
models.

0 Probability distribution of strength parameters.

0 Spatial variability in strength parameters.

0 Spatial variability in pore pressure ratio, ru, (end-of-construction conditions).

0 Spatial variability in hydraulic conductivity (steady-state seepage conditions).

Each of these aspects and their relative role in slope reliability assessment is considered

in the research reported herein as will be discussed later in this chapter.

In order to summarize how other researchers have dealt with the aspects
previously listed, a summary of some previous investigations is provided in Table 1.1.
This review shows that a number of researchers have proposed techniques and reported
studies for probabilistic slope stability analysis; however, no one has, generally, made a
comprehensive evaluation of the effect of different deterministic and probabilistic models
on the reliability analysis. It was noted also that probabilistic slope stability and
probabilistic seepage analyses have usually been done separately and often have been
applied to special, simple problems.

It may be noted that most of the investigations listed in Table 1.1 considered
either the slip surface of minimum factor of safety or an arbitrary surface. The only
research which investigated the surface of minimum reliability index is Li and Lumb
(1987). However, their application was a simple homogeneous slope and nothing was

mentioned about how that surface was located.

 

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Variability or uncertainty in pore pressure was not addressed in most studies listed
in Table 1.1. However, Li and Lumb (1987) considered pore pressure ratio, ru, as a single
random variable (without spatial correlation) with a judgmental coefficient of variation of
20 %. Bergado and Anderson (1985) considered variability in hydraulic conductivity and
modeled its effect as a variation in the phreatic line applied only to cohesionless soil.

The research reported herein critically studied the aspects previously listed with a
view towards identifying comprehensive improvements in performing probabilistic slope
stability analysis. It focused on assessing how practicing engineers could best use and
combine practical, largely—existing, analysis tools in a probabilistic framework that
handles complex problems with simple approaches rather than handling simple problems
with complex approaches. In doing so, the effect of applying different deterministic and
probabilistic methods, spatial variability in pore pressure, and assessment of good
estimation of minimum “critica ” reliability index were investigated. This thesis is
organized in several parts:

Chapter 2 reviews some basic concepts of probabilistic stability analysis. Basic
concepts of reliability and probability are defined, described, and summarized. A brief
discussion of spatial correlation of random variables is also provided.

Chapter 3 summarizes three deterministic models for slope stability analysis:
Spencer’s method, the Simplified Bishop method, and Modified Swedish Method. These
are commonly used for reliability assessment for dams and other embankments. Chapter
3 also provides a description of UTEXAS3; the computer program used for slope stability

analysis in this research.

Chapter 4 is concerned with short-term or end-of-construction slope stability
analysis modeled using total stress parameters. The influence of applying different
deterministic and probabilistic methods in the analysis is investigated, as is the effect of
modeling soil parameters with high coefficients of variation as normally and lognormally
distributed.

In Chapter 5, effective stress analyses of end-of—construction conditions are
performed introducing uncertainty in pore pressure, modeled in terms of pore pressure
ratio ru. This ratio was modeled as a random field (spatially correlated random variable)
and its possible effect on slope reliability was explored.

In Chapter 6, an algorithm to locate the critical slip surface of minimum
reliability index is developed. This search technique is described and applied for some
case studies. To test the algorithm and evaluate differences in reliability index calculated
for both the critical probabilistic and the critical deterministic surfaces; different types of
earth structures were considered including dams, a levee, and a homogeneous slope. A
parametric study was also performed on a typical two-layer slope. The analyses included
various loading conditions: total and effective stress analysis for end-of-construction, and
effective stress analysis for steady seepage conditions.

The effect of spatial variability of hydraulic conductivity is studied in Chapter 7.
A spreadsheet-based procedure using the finite difference method to generate spatially
correlated values of hydraulic conductivity was developed and applied to steady-state
seepage analysis throughout a pervious levee. The steady-state flow equation was also

solved by the finite difference relaxation method. The effect of spatial variability of

hydraulic conductivity on the distribution of pore pressure throughout the embankment
was assessed. An application to slope stability analysis was then provided to examine the
influence of variability of pore pressure on the slope reliability.

A summary, conclusions, and recommendations are provided in Chapter 8

Chapter 2

BASIC CONCEPTS OF PROBABILITY ANALYSIS

2.1 General

In this chapter some basic concepts of probability and reliability analysis are
presented. Uncertainty in soil parameters is discussed, including spatial and systematic
variability. The reliability index is introduced as a measure of reliability. Three
probabilistic models; Monte Carlo simulation, Taylor’s series method and the point
estimate method are also briefly presented. A comparison between the probabilistic
approach and the factor of safety approach is provided. These models are described in this
chapter to provide a background for the analyses performed in Chapter 4 which will deal
with probabilistic analysis performed by different probabilistic models.

It may be noted that the materials described in this chapter are fimdamental tools
which may be recognized in many references in the literature. The main references on
which this chapter depends are Vanmarcke (1977), Whitman (1984), Ang and Tang
(1984), Harr (1987), and Wolff (1994).They are summarized herein for the convenience

of the reader.

2.2 Random Variables
Random variables are quantities having uncertain magnitude, i.e. not precisely
fixed. Instead, they assume a range of values for which relative likelihood is given by a

probability distribution. Almost all variables used in geotechnical engineering analysis

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can be considered as random variables. Sources of uncertainty include errors in test
results, sample disturbance, unsatisfactory simulation of field conditions, spatial
variability, etc. as will be discussed later. However, some variables have relatively
insignificant uncertainty and may be considered not random in order to reduce
computational effort, e.g. soil density in slope stability analysis. In this concept, with the
aid of probability theory one can evaluate the relative significance of various
uncertainties, and hence reasonably decide whether to represent variables as random or

deterministic (non- random).
2.3 Moments of Random Variables

Mean Value: The mean value, ux, is the summation of values sampled from a

random variable X divided by the number of the values N:

i=1 ' (2.1)

Expected Value: The expected value, E[X] is the integration of the values of a random
variable multiplied by their likelihood of occurrence; for discrete random variables

integration is done by summation. Hence:

N
E[X]: IXfx(X)dx z 2 Xip(xi) (2.2)

10

where f,( (X) is the probability density fimction of X (for continuous random variables)
and p(Xi) is the probability of occurrence of the value Xi (for discrete random variables).
The mean u, provides an unbiased estimation of the expected value for discrete random
variables and they are numerically the same.

Variance: The variance, Var[X], is the weighted average (expected value) of squared
deviations from the mean, where each value is weighted (multiplied) by its probability
density. The variance can be calculated for a sample of finite size by squaring the
difference between each value and the mean, summing these squares, and dividing the

summation by the sample size, N:

ikxi -Hx)2]
Varlxl=E[(X-ux)2]= J(X-l~lx)2f(X)dx= I=I

 

N (2.3)

Note that in the above equation the summation over N element gives the variance of a
population of exactly N elements. In engineering analysis a sample of size N is usually
used to estimate the statistical properties of the entire population of items. The unbiased
estimate of variance of the entire population using a finite sample is obtained by replacing

beN-l:

2N:[(Xi - P'x)2]

V = i=1
ar[X] N_ 1 (2.4)

 

11
Standard Deviation: The standard deviation, ox , is the square root of the variance. It has
the same units as the expected value and measures the scatter or dispersion of a random

variable from the mean (expected value):

a x = 1lVar[X] (2.5)

Coefficient of Variation: The coefficient of variation, Vx, is the ratio of standard deviation
to the expected value. It is dimensionless and provides a convenient measure of

uncertainty inherent in a random variable:

 

x1 0 o (2.6)

Covariance: Random variables encountered in geotechnical engineering may vary
correlated to each other. The covariance of two random variables X and Y measures this

correlation:

Cov[X, Y] = E[(X - ux )(Y — u Y )] (2.7)

for continuous random variables:

C8v[x, Y] = H (x - ux)(Y — u,,)dydx (2.8)

12

for discrete random variables:
1 N
COV[X:Y]=EZ(Xi _“X)(Yi "Hv) (2-9)
i=1

The correlation coeflicient, p , gives a non-dimensional measure of correlation:

= C0\{ X, Y]

O'xc'v

px.v (2-10)

Many random variables encountered in engineering applications may have no
correlation, i.e. they are independent, the correlation coefficient takes the value of zero in

this case.

2.4 Probability Distributions
2.4.1 General

The probability density function, PDF, or f(X) describes the likelihood of that a
random variable, X, takes a particular value. If the random variable can take all possible
values within its limits; the pdf is continuous. The area under the pdf is unity. The
probability that a random variable takes a value between two particular values X1 and X2

is the area under pdf between these two values:

13

X2
Pr(X, < x < x,) = j fx(X)dx (2.11)
Xl

The cumulative distribution function, CDF, or F (X) measures the integral of the
probability density fimction from minus infinity to a particular value X and it gives the

probability that the variable will have a value less than or equal to X.

X

FX (X) = j r, (X)dx (2.12)

-€0

Figure 2.1 shows the pdf and the CDF.

2.4.2 The Normal Distribution
The normal or Gaussian distribution is the most commonly used probability

distribution. It is defined in terms of the mean, 1.1., and the standard deviation, 6., as:

 

_ 1 (x-u)2
f,(X)_ 8J2? exp[———262 ] (2.13)

where u and o are the parameters of the distribution which is shortly denoted as N (up).
If u=0 and 0:1 the normal distribution is called the standard normal distribution, N(O,1)

which can be used to find probabilities associated with other normal distributions by

14

f(X)

 

 

 

 

 

X
(o) Probothy Densuy Funcfion, pdf
1
3?
U.’
0
X

(b) Conufloflve Denshy Funcfion, CDF

Figure 2.1 pdf and CDF for Normal Distribution

15
transformation. Friction angle, and density are examples of soil properties that are
generally assumed normally distributed for their small coefficient of variation. Generally,
the normal distribution may be a good model for random variables with known mean and

variance and with small coefficient of variation (Vx s 30 %).

2.4.3 The Lognormal Distribution
If the normal logarithm of a random variable X (In X) is normally distributed, then

X is lognormally distributed. The density fiinction is then given by:

___1__ -1 w
fX(X)_XoY\/Eexp[ 2[ 0y ]] (2.14)

where, Y=ln X,
CY z GlnX : Vln(1+ Vii) ,

E[Y] = E[ln x] = 1n 11x}?

The lognormal model is often adopted where the random variables assume only non-
negative values and the coefficient of variation is high. Soil permeability or hydraulic
conductivity is the most common geotechnical property that is taken to follow a

lognormal distribution due its relatively high coefficient of variation.

16
2.5 The Capacity-Demand Model

The capacity-demand model is a probabilistic method to characterize reliability in
engineering design. The designer usually seeks to determine the capacity, C, of the
system to meet or exceed specific demand, D, requirements. Examples of the capacity
versus demand are allowable strength versus acting stress in structural design and
resisting moment versus driving moment in slope stability analysis. It is obvious that the
determination of capacity and demand of an engineering system is not a simple process.
Uncertainties are inherent and estimation and experienced engineering judgment are
essential.

If the probability distribution of both the capacity and demand are available as
shown in Figure 2.2-A and both C and D are independent, and assuming the actual
demand is x, then the shaded area indicates the probability that the capacity is less than x
which implies that failure will occur. In this case (D = x) the probability of failure is

equal to shaded area :

Pr( f) = jfc(x)dx = FC (x) (2.15)

In fact, D = x is associated with probability fD (x), integrating over all values of D yields:

1w): ]FC(x)f,,(x)dx (2.16)

17

0")
0")

Cpaacfiy

   

Demand

 

 

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a. pdf for Capacity and Demand

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Demand Cpaacfiy

 

oveflop C!
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n")
u-fi

Demand Cpaacfiy

 

 

X

ovenap
k——4

 

c. Decreasing The Variability in C and/or D

Figure 2.2 The Capacity-Demand Model

18

It is clear that the probability of failure decreases as the overlap distance decreases. This

can be achieved by:

Increasing the separation between the two curves as shown in Figure 2.2-B. This may
be expressed as increasing the ratio uc / #1) which is referred to as the mean factor of

safety, F S, where:

FS = (2.17)

blfl

The relative position of the two curves may also be measured by the difference (uc -

up) which is the mean safety margin, SM, where:

SM=C- D (2.18)

Decreasing the variability in C and D or in other words decreasing the degree of

dispersion in the pdf of C and D. as shown in Figure 2.2-C.

The capacity-demand model may be formulated in terms of the safety margin,

SM. As C and D are random fimctions; SM is a random function with a pdf fsm (SM). In

this case :

Pr( f) = Pr(SM < 0) p (2.19)

19

As shown in Figure 2.3.
0
Pr(f) = I f,,, (SM)dSM = FSM (0) (2.20)

In terms of the factor of safety; F S is a random variable with pdf fps (F S) as

shown in Figure 2.4.

Pr(f) = Pr(FS <1) (2.21)
Pr(f) = jf,,(Fs)dFs = F,,(1) (2.22)

The factor of safety FS and the safety margin SM are called performance
fimctions. The case of SM = O is defined as the limit state which may also be expressed as
the condition of FS = 1. The reliability, R, is probability of SM > 0 or F8 > 1. From

probability theory; the probability of failure and reliability sum to unity:

R = 1— Pr( f) (2.23)

 

20

g(sm)

AfiOSM

 

fSM(SM)

Area=Pr(f)

 

 

Figure 2.3 pdf of The Safety Margin (SM)

(gas)

(1%.

fFS(FS)

Area=Pr(f)g

 

 

ll

0 1'0 lu’rs FS

Figure 2.4 pdf of The Factor of Safety (FS)

21

2.6 The Reliability Index

In the previous section the performance function was defined as a function of
random variables. To determine the reliability or the probability of failure, the pdf of the
performance function must be evaluated. This requires jointly integrating the
performance function over the pdf’s of the random variables. The des of the random
variables are not generally well defined in practical cases and the performance function
may be implicit, especially in geotechnical applications. Hence, evaluating the pdf of the
performance function is often not possible fi'om a practical point of view. Instead, the
moments of the performance function are estimated by the method of moments as will be
explained later and the reliability is expressed in terms of the reliability index, [3. The
reliability index provides a good comparative measure of reliability; structures of higher
[3 are considered more reliable than those of lower [3. The reliability index also provide a
method of calculating the probability of failure by assuming the shape of the probability
distribution of the performance function as will be discussed later.

In terms of the safety margin, as shown in Figure 2.3, the reliability index is
defined as follows: if the capacity C and the demand D are both normally distributed, the

reliability index, [3, is given by:

[3 = ”fl (2.24)
65M
B = m (2.25)

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In terms of the factor of safety (FS), as shown in Figure 2.4, the reliability index is
the difference between the expected value of the performance function and the limit state

value divided by the standard deviation of the performance function:

— 1
B = ——“F5 (2.26)
01:3

The above equation defines the reliability index as the number of standard deviations
separating the expected value of the factor of safety from its limit state of 1.0, or in other
words provides a way of normalizing the factor of safety with respect to its standard
deviation. In this definition the density function of the factor of safety is implied to be
normal. However, it is advantageous to assume the pdf of FS as lognormal; since the
lognormal distribution has a lower value of zero and upper bound of infinity (always
positive) and so does the factor of safety. Hence, the performance function is taken as In

FS instead of FS. In this case the reliability index is given by:

B = M (2.27)

Gln rs

Beside being a good comparative measure of reliability, the reliability index may
be used to calculate the probability of failure keeping in mind that the calculated
probability of failure should be looked at as a relative, not absolute, value and can be used

as a comparative quantity. The shape of the density function of the factor of safety in this

23
case is not known but should be reasonably assumed. The reliability R and the probability

of failure Pr(f) can then be calculated as follows:

R = (13(5) (2.28)
Pr(f)=1—R=1—<I>(B) (2.29)

where (D denotes the standard normal cumulative distribution firnction, CDF, of the
performance function (F S). Tables and spreadsheets are available to determine the

function (D.

2.7 Reliability Analysis
2.7.1 General

In geotechnical engineering one usually deals with functions of random variables
which may contain two or more random variables. The factor of safety of soil slopes, for
example, may be a function of cohesion, fiiction, density, pore water pressure and slope
geometry. To analyze the reliability of a slope, the probability distribution of the function
of random variables (factor of safety) should be evaluated considering the reliability of
the random variables (soil properties and slope geometry). It should be noted that not all
the parameters encountered in the analysis are necessarily taken as random variables. For
example, soil density is usually considered a deterministic value due to its relatively

small uncertainty.

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24

Several methods have been developed to evaluate or estimate the probability
distribution of functions of random variables. These methods form two categories: exact
methods and the method of moments. The Monte Carlo Simulation method belongs to the
so-called exact methods which require that the probability distribution of all the
component variable be known (Harr (1987)). Other methods commonly used. are the
Mean-value First Order Second Moment method (MFSOM), the Advanced First Order
Second Moment (AF OSM), and the Point Estimate Method (PEM). These methods
belong to the method of moments and are used to estimate the probabilistic moment of
the function of random variables in terms of the moments of the random variables. The

above-mentioned methods are briefly described below.

2.7.2 The Monte Carlo Simulation

In the Monte Carlo simulation method, real events are simulated by simple
events. For example an event of two equal probability outcomes may be simulated by
tossing a fair coin. Also rolling a 6-face die may be used to simulate a random uniformly
distributed (with equal probability) integer number between 1 and 6. These events should
be repeated until sufficient simulation of the modeled event is achieved. Each trial may
be represented as an experiment. The method is described by Rubinstein (1981), Ang and
Tang (1984), and Harr (1987). This concept could be applied in evaluating the pdf of a
function of random variables. If Y if a function of random variables X1, X2, ...XN a
random value for each variable is selected from its assumed pdf and the function Y is

computed using these values. This process is repeated a large number of times

25
(simulations). The expected value and variance of the function Y is then calculated from
the output of the simulation. The required number of simulations depend on the
variability of the input and output parameters and the degree of accuracy desired.

A key to applying Monte Carlo simulation is the generation of uniformly
distributed random numbers between 0 and l, U (0,1). There are many functions which
make this generation and some are available in computer software and many calculators.
This generation, U (0,1) , can be extended to any particular range, U (a,b), by the

following:

U(a, b) = (b — a)U(0,1) + a (2.30)

A generation of random numbers fitting a given probability distribution can also be done.
As shown in Figure 2.5, the ordinate of the cumulative probability distribution function
may be considered as a uniform random variate, U (0,1) . First, a number from this range,
I, is generated such that F (X) = r. Then, the corresponding value, X, will be the desired
random number of the given distribution. Several trials (experiments) should be
performed to ensure sufficient representation of the distribution of the original random
variable. Random values of a specific probability distribution can be generated using
computer programs such as EXCELTM and @RISKTM which is the method adopted in
Chapter 7 of the present study.

Functions of random variables could be simulated by generating random numbers

with the same probability distribution for each variable as mentioned above. Hence the

26

r, (X)

 

(a) The Required Distibution

F, (X) ll

r=Fl (X)

 

 

 

 

(b) Generation Procedures

Figure 2.5 Generating Random Variables

of 0 Given Distribution

27
probability distribution fimction for all variables should be initially known, which is one
of the disadvantages of the Monte Carlo simulation. Another disadvantage is that it
requires a large number of trials and computation effort, especially when the variables are
correlated. Modeling spatial variability of the hydraulic conductivity is one of the most
common application of MCS in the literature (e.g. Smith and Freeze, 1979; and Bergado

and Anderson, 1985).

2.7.3 The Mean-value First Order Second Moment Method (MFOSM)

Due to lack of data, the probability distribution of the capacity and the demand is
not known. Hence, an approximate method of calculating the reliability or the probability
of failure is needed. One of these methods is the First Order-Second Moment method,
FOSM, in which the reliability is calculated based on the first and second moments of the
design values; i.e., the expected value and the standard deviation.

The design variables, X,, of an engineering system are related together by a

performance fimction, g (X), such as the factor of safety, FS, as follows:

FS= g(X,,X,,...,X,,) (2.31)

Where g (X,) is a function representing an engineering model; e.g., F S-1 or In F S from
the simplified Bishop method of slices in slope stability analysis. It is obvious that the
safe state is represented by: g(Xi) > 0, failure state is : g(Xi) < 0, the limit state is : g(Xi) =

O. The limit state is an n-dimensional surface that may be called the “failure surface”.

28
Now, to calculate the probability of failure the joint pdf of the design variable, fx (Xi),

Two difficulties are associated with the use of Equation (2.32) ( Shinozuka, 1983):

1. Constructing the joint pdf of design variable is impractical since data are scarce and
insufficient.

2. The volume integration over an irregular region is practically impossible to carry out
analytically and quite costly to perform numerically.

The above mentioned difficulties emphasizes the need of first-order second-moment

approximation in reliability analysis. The approach is well recognized in the literature:

Cornell (1971), Hasofer and Lind (1974), Ang and Conrnell (1974), Ditlevsen (1979),

and others. The method is explained as following:

First the reduced variable is introduced:

—— (2.33)

Note that the variables, X, , are assumed uncorrelated; however, if not Hasofer and Lind
(1974) presented an orthogonal transformation of correlated variables to uncorrelated

ones. The safe, failure, and limit states may be represented in the space of the reduced

29

variables, in case of two variables, X1 and X2, these states are shown in Figure 2.6.
Hasofer and Lind (1974) showed that the distance from the origin to the limit state
(failure surface) must be greater than the reliability index [3. Shinozuka (1983) showed
that the point on the failure surface with minimum distance to the origin is the most
probable failure point and this distance could be a measure of the reliability index.

Expanding the performance function, g(Xi) in a Taylor series at the most probable
failure point X. and keeping only the first-order terms gives the so-called Hasofer-Lind

reliability index, BHL, as:

= 2 (2.34)

Where, (fig—J indicates that the partial derivatives are performed at most probable

failure point (x‘).

It should be noted that an iterative procedure is required to locate the most probable
failure point (X‘) and 5111. (see Hasofer and Lind, 1974; Ang and Tang, 1984; Parkinson,
1987; Chowdhury and Xu, 1994). The Hasofer-Lind reliability has the advantage of being
invariant, i.e. independent of the form of performance function hence, allowing
comparing values of BHL regardless of the format of performance functions they are

derived from. It is more commonly applied in structural engineering than in geotechnical

30

\

\Failure Surface, C(Xi)=O

safe region, g(Xi)>O

BHL

 

 

 

Figure 2.6 Reliability Index For Two Variables
(After Hasofer and Lind, 1974)

31
engineering. However, it was applied to probabilistic slope stability analysis by Li and
Lumb (1987), and Chowdhury and Xu (1994).
A practical more common reliability method for slope stability is the MFOSM. If
the performance function is expanded at the mean value of the design variables, er , and
keeping only first-order terms, the conventional reliability index, B, for uncorrelated

variables is given by:

,3 = is. = g(”x' ) 2 (2.35)
68 N 6g 2

 

 

where the partial derivatives, [66%) , are evaluated at the mean point (11x1).

Because the performance function is usually implicit in slope stability analysis; the partial
derivatives are approximated by divided differences (Wolff, 1994; Christian et. al., 1994;
Wolff et. al., 1996) i.e. by calculating the change in g(Xi) due to changing each variable
by an increment above and below its mean value. The US. Corps of Engineers adopted
an increment of one standard deviation which is also applied in the present research. In
this case the MFOSM requires 2n + 1 evaluation of the performance function where n is
the number of random variable. For two random variables (c and (p), five runs of a slope
stability program are required.

For correlated random variables, the reliability index is given by:

32

B = 2 1411)..) (2.36)
a 6 6
J2EE?) 0'3“ +22—as(g—i—é§g;Cov(Xi,Xj)

2.7.4 The Point Estimate Method (PEM)

To approximately estimate moments of functions of random variables; the Point
Estimate Method was developed by Rosenblueth (1975, 1981), and Evans (1962, 1972).
As shown in Figure 2.7 a random variable, X, is represented by two point estimates X,
and X_ with two probability concentration P,. and P_, These two estimates are used to
simulate the probability distribution of the random variable. For independent random
variables of symmetrical distribution, the two point estimates are chosen as the expected

value plus or minus one standard deviation:

x, = 13[x] + ox (2.37)

X_ =EIX]—ox (2.38)
and the probability concentrations are:

P, = P. = 050 (2.39)

33

\

__p_ __ __... _ __ _ __
Y+ \Estimated

 

Y=Y(X)

?
) MY) A fx(X) i 17",
\
Y-" _ ”‘ /f£ R \E

 

Figure 2.7 The Point Estimate Method

34
For functions of random variables, the fiinction is evaluated for all possible combination
of point estimates of the independent variables (2N for N variables). The moment is
calculated as the summation of all these combination each weighted by the product of the
associated probability concentrations, for example the case of two random variables, X1,

X2, is given by:

51W] = P,,Y(x1+ ,x,,)M + P,_Y(x,, ,x,_)M + P_,Y(x,_,x,,)M + 1>__Y(x,_,x,_)M

(2.40)

The expected value is determined by substituting M = 1 and the variance is determined by

substituting M = 2 to get E[Yz] applying the following formula:
Var[Y] = [at Y2] —(15[Y])2 (2.41)

For correlated random variables, the joint probability concentrations are related to

the correlation coefficient. For two correlated random variables:

P... =P--= —1 + p (2.42)
4
1- p

P = P = — (2.43)

35
For three or more correlated random variables the products of probability concentration
are given by Rosenblueth (1975).
It is clear that the MFOSM method requires fewer evaluations of the performance
function than the PEM when number of random variables is more than two (2N+1 versus

2N). A comparison of the two methods is provided in Chapter 4.

2.8 Uncertainty in Soil Properties
2.8.1 General

Soil properties are generally uncertain values and, hence, are treated as random
variables. Sources of uncertainty include inherent spatial variability and systematic error.
Spatial variability arises from the fact that soil properties change from point to point. The
systematic error arises from the statistical error in the computed mean value of the
property due to the limited number of tests performed and bias in the measurements due
to excessive sample disturbance. To illustrate the difference between systematic and
spatial variability; the soil property Z may be looked at as having two components: trend

and residuals, as follows:

z(x) = T(X) + g(x) (2.44)

where X is the distance, T(X) is the trend component , and c (X) requests the residuals

that fluctuate around the trend (see Figure 2.8). Spatial variability describes the scatter of

36

 

 

 

Figure 2.8 Variation of Soil Properties

ll

37
the soil property around the mean trend and the systematic error is the uncertainty in the
location of mean trend itself (Christian, et. al., 1994) .
The variance of a random variable is, in turn, divided into spatial and systematic

variance as follows:

2 2 2
Var total = Var spatial + Var systematic (2.45)

The systematic variance is given by

Var{ E[ru]} = Varl ru] / n (2.46)

where n is the number of measurements. Hence the systematic error is governed by the

number and accuracy of measurements

2.8.2 Spatial Correlation

The values of a random variable may be spatially correlated, i.e. be a function of
distance. In this case the random variable is called a stochastic process. The value of the
random variables at one point is correlated to its value at nearby point. This correlation is
governed by an autocovariance function, ACF. To deal with spatial variability; the
random variable is assumed statistically homogeneous or stationary, Figure 2.9.

Stationarity requires that the random variable, Z, fluctuate around a constant mean, i.e. no

38

Z(x)

averagmg dstance
b T
_L 5
NA barLe
‘\<=7L’ \\$J/T‘

X

 

mean
m

 

 

Figure 2.9 Random Field and Moving Average

39
trend, as shown in Figure 2.9. For a stationary random variable, the covariance between

any two points X, and xm is only a function of the vector, L, separating this pair of points.

9123)] = p, = m (2.47)

Co»{xi ,x,,,] = E{[z(x,) — m] [2(x,,,) — 111]} = Cov[T] (2.48)

where m = constant,

If T = 0 then Equation (2.48) reduces to the variance of the random variable:
Cov[ 0] = 62 (2.49)

The autocovariance is calculated by summing the covariance of all pair of points

separated by T along the direction x:
Auto cov ariance[T] = l2(Z(x,) — m)(Z(xM ) — m) (2.50)
n

where, n = number of pair of points separated by L

The ACF is then calculated by dividing the autocovariance by 62; an example of
calculating the ACF is given in Appendix A. The ACF takes the value of 1.0 at T = O and
decays as T increases. Many mathematical forms could be fitted to ACF. Negative

exponential and square exponential are the most commonly used forms in the literature,

40

A.
7

exp (-T/<5)

ACF

 

 

“ll

(0) Negoflve Exponenflal

exp <—<r/o )2)

ACF

 

.4

(b) Negaflve Squared Exponenfial

Figure 2.10 Mathematical Representation of The

Autocovariance Function

41
Figure 2.10. The autocovariance distance (6) is the distance at which the autocovariance
fiinction yields to the inverse of natural logarithms (p = l/e). It is a characteristic distance
within which soil properties shows relatively strong correlation It may be also be defined
as the area under the ACF p from zero to infinity and is sometimes referred to as the

Integral Scale.

w

5 = j p(1:)dt (2.51)

0

where 1 indicates separation distance.

Vanmarcke (1977 and 1983) presented the concept of reducing the variance of the
random variable due to averaging it over a given distance (moving average). The variance
of the averaged random variable is less than its value at a point. This is due to low
values being compensated by larger ones over the averaging distance, T. This reduction
depends on the degree of decay of the ACF. It may be noted that the expected value is not

affected by the averaging process for a stationary random variable.

Var[ z,(x)] = a: = r2(T)c’, (2.52)

F2 (T) is the variance function of Z(x), which measures the reduction of the point

variance 0'2 due to spatial averaging over a distance T. It is related to the ACF p by:

42

F2(T) = —l— :[3’p(x, — x2 )dxldx2 = % 3(1-%)p(1)d13 (2.53)

T2
for a simple negative exponential ACF :

p('c) = e""”’ (2.54)

r2 (T) = 431% — 1 + a“) (2.55)

Other formulae may be derived for different ACF forms. However, Vanmarcke (1983)
showed that regardless of the form of ACF, all mathematical forms of F2 approach the

approximate expression:

F2(T) = 1 for T < 5 (2.56)

for T > 8 (2.57)

This expression is adopted in the present work due to its simplicity and convenience with
type of problems included. The square root of the covariance function (F) is called the
reduction factor.

It should be noticed that the above mentioned variance reduction due to averaging

process must be applied to spatial variance only and hence:

43

0 (Z) = [F(L)° :patial + 6 :ystematic ] “2 (2.58)

However, the two types of variances are not always well recognized and hence soil
variability is lumped in only one source of uncertainty (spatial or systematic). This
depends on the size of the problem, the value of the scale of fluctuation and the size of

sampling layout.

2.9 Probabilistic Versus Factor of Safety Approach

In deterministic analysis each soil property is usually modeled as single value,
typically the mean value or a design value lower the mean. No information about the
“goodness” or the certainty of this single value is included in the calculation. The results
are expressed in terms of the factor of safety. For slope stability problems, several failure
surfaces are be examined and the surface of the minimum factor of safety is considered
the most critical one, which is not consistent with the fact that the slope may fail at some
less reliable surface. Also to compare two embankments, the factor of safety does not
offer the best way of comparison since an embankment of more reliable material should
be considered better than another one of less reliable material for equal values of the
factor of safety. To make this point clear two cases are considered (Christian et. al.
(1994)). Case A represents an embankment of E[FS] = 1.2, 0'[FS] = 0.10, and [3 = 2.0
whereas case B represents an embankment in which E[FS] = 1.5, o[FS] = 0.50 and [3 =
1.0. It is clear from Figure 2.11 that case A, which has a smaller factor of safety, is much

more reliable than case B. This is due to the smaller uncertainty induced in case A

44

$2: :6 no c2220 8:3
m... co 8:39.35 2885 B .380 2: __.N 83m:

m a om mm om 2

ll

 

3 m
m.O nub
3:3.

 

 

 

 

 

45
expressed in terms of a lower standard deviation of the factor of safety and a higher
reliability index. This concept is also significant for an embankment consisting of two or
more materials. The surface of lower reliability index exists or takes its longest way in the
most unreliable material, i.e. the material of which soil parameters experience much
uncertainty. This surface is not necessarily the surface of minimum factor of safety as
will be studied in detail in Chapter 6.

From the above discussion, it may be concluded that the probabilistic approach
contains more information about soil parameters than the factor of safety approach.
Statistical properties such as the coefficient of variation and the scale of fluctuation
should be taken into consideration. This provides a better way of comparing different
designs and/or case of loading. It should be noted that probabilistic approach is not a
substitute of engineering judgment. Engineers performing probabilistic analyses may
need their judgment more than when doing deterministic analyses since practicing

probabilistic analysis may help quantify judgment.

Chapter 3

BASIC CONCEPTS OF SLOPE STABILITY ANALYSIS

3.1 General

A part of the present study (Chapter 4) is to investigate the effect of applying
difi‘erent deterministic models on the results of slope reliability analysis. In this chapter
methods of slope stability analysis adopted in this study are summarized. These slope
stability methods are applied using the computer program UTEXAS3 (Edsris and Wright,
1993) which is also described in this chapter. The chapter also provides a description of
the different loading conditions considered for analysis of embankments and the modeling
of strength parameters for these conditions.

Fellenius (1927) defined the factor of safety as the ratio of available shear strength

If to required shear strength ‘CI

135:3 (3.1)

In other words, it is the factor by which the shear strength parameters must be reduced
before the slope is brought into a state of limiting equilibrium (Bishop (1960)). Based on

the Coulomb failure criteria:

t=c+o tand) (3.2)

46

47

Hence, in terms of total stress:

 

= —— + ——---— 3.3
I F8 F8 ( )
and in terms of effective stress:
t'=E—+O “ml (34)
F8 F8

The forces acting on a typical slice of a slope are shown in Figure 3.1 where,

W is the slice weight

Z is the interslice force
E is the horizontal component of interslice force
X is the vertical component of interslice force

P AW is the side water force

 

S is the developed shear force acting along slice base:
3 = £1 + N “ml (35)
F8 F8
where,
P' is the effective normal force

U is the water force on the of slice base

48 .

 

 

 

Figure 3.1 Forces on 0 Slice

'I F
c / S S
pdono/FS
Ei+1 6H1
iQEiX,
Z+1 +1 W .

 

 

7 9%/

 

Figure 3.2 Modified Swedish Method

49
It is obvious that the problem is indeterminate. To obtain a determinate solution
assumptions are made which vary by method. These assumptions usually deal with
interslice forces by ignoring them, assuming them horizontal, assuming them parallel, or
assuming a relationship between their vertical and horizontal components.

The factor of safety is determined (minimized) by iteration. First a trial factor of
safety is assumed, then developed strengths are calculated and the sliding mass is checked
for equilibrium. The process is repeated until equilibrium is obtained. The factor of safety
for the slip surface which has the lowest factor of safety in equilibrium is taken as the
factor of safety of the slope. The different slope stability methods are discussed in the
following sections. The Morgenstem-Price method, although not applied in the present

study, is introduced because Spencer’s method is a variant of it.

3.2 The Modified Swedish Method

In the Modified Swedish method all side forces are assumed parallel and the
inclination is given by the user (see Figure 3.2). This inclination is assumed to be equal to
the average slope of the embankment according to US. Army Corps of Engineers (1970).
This assumption gives a high factor of safety. However, other inclinations are frequently
assumed in practice. A horizontal inclination gives a low factor of safety. The procedure
satisfies force equilibrium in both vertical and horizontal directions for individual slices.
Moment equilibrium is not considered. The factor of safety is calculated by a trial and
error solution. Successive assumption of factor of safety is done until equilibrium of

forces is satisfied. The method is sometimes sensitive to side force inclination. However,

50
the procedure adopted here is to assume a side force inclination close to that obtained by

Spencer’s method.

3.3 The Simplified Bishop Method

Bishop (1955) presented a rigorous solution including both components of side
forces, X and E. This method was simplified (Bishop and Morgenstren, 1960) by omitting
the vertical side force, X, from consideration. Vertical force equilibrium is satisfied for
each slice and then a moment equilibrium equation is solved for the whole sliding mass.

The resulting factor of safety is given by (see Figure 3.3):

 

1 , , seca
FS-WZ Cid-(W ul)tan¢ 1+tanatantl)’ (3.6)
FS

Because F.S. appears in both sides of Equation 3.6, it is solved iteratively; F S on the right
hand side is assumed and F8 on the left hand side is calculated. The two values are
compared and process is repeated until convergence is achieved. The method is limited to

circular surfaces.

3.4 The Mogenstem-Price Method
The Morgenstem-Price method (Morgenstern and Price, 1965) is considered one

of the “rigorous” methods in which all equilibrium conditions are satisfied. It is an

5|

\ Cvi/FS

 

 

 

 

Figure 3.3 Simplified Bishop Method

 

 

 

 

} \
r -'\~_-Z |
M T1152! 1 iw ‘3' F3 _fl\-r
l ‘ i T Ei1.1/—~,.l unions Fs
i i ZSSKJX‘“ \ 7\
i \ I “l i l \0'/ f
.3. S\}\T5 l E ' \ "'1 i” /
i T? N;' \\ ”l | /. Q"
All i gx\l\ ., / 3’ “/1
3‘" I U] l. 71- +i K!
/ l
l 1 ' 1
i l i
i‘ xb % l
X21

 

Figure 3.4 Morgenstern-Price Method

52
application of the method of slices. Any shape of slip surface can be analyzed. A function
relating horizontal component, E, and vertical component, X, of the interslice force (see

Figure 3.4) is assumed:

x = M0013 (3.7)

Moment and two force equilibrium (in both P and S directions) are evaluated. An analysis
starts with guessed values of 2» and F S and they are iteratively modified until equilibrium
is achieved. The function relating interslice forces f(x) must be assumed in the analysis.
Typical forms are constant (f(x) = 1) and half-sine, (Li and Lumb, 1987). However,
Morgenstern and Price (1965) showed that the factor of safety does not appear to be very
sensitive to the form of this function. For a circular slip surface of a homogeneous slope,
they showed that the factor of safety is insensitive to varying the relationship between the
internal forces. In probabilistic analysis framework, Li and Lumb (1987) showed that the
Hasofer-Lind reliability index calculated for a homogeneous slope is not sensitive to the

interslice force fimction.

3.5 Spencer’s Method

Spencer’s method (Spencer, 1967) is a special case of Morgenstem-Price method.
Assuming all side forces are parallel, the method satisfies both force and moment
equilibrium (Figure 3.5). Two factors of safety are obtained based on both force and

moment equilibrium. The interslice force inclination is varied until the two factors of

 

S3

c'I/FS

piano/rs

V

 

 

 

U/

Figure 3.5 Spencer’s Method

54
safety are equal. The slip surface may be circular or non-circular. The method requires
computer calculation. Solution is achieved iteratively by successive assumptions for the

factor of safety and side force inclination until both force and moment equilibrium are

satisfied.

3.6 UTEXAS3 Computer Program

Slope stability analysis in the present study is performed by the computer program
UTEXAS3 (Edris and Wright, 1993). The program offers the user an Opportunity to use
any or all of the following methods:
0 Spencer’s method for both circular and non-circular failure surfaces.

0 The simplified Bishop method.

0 Corps of Engineer’s method (modified Swedish) for circular or non-circular failure
surfaces.

0 Lowe and Karafiath’s method

Only the first three methods are used in the present research. The program calculates the
factor of safety for a given slip surface or searches for a surface of minimum factor of
safety. In the search process, the user provides the program with information about a
starting surface and the program proceeds from this surface to locate the surface of
minimum factor of safety. However, it was found that the location of the surface of
minimum factor of safety is sometimes sensitive to the starting surface which indicates
that search process should be performed with special care. Several starting surfaces

should be tried to obtain the minimum factor of safety.

 

55
Data required by the program includes geometry of embankment, material
identification, and material properties (density, cohesion, friction angle and pore
pressure). Pore pressure can be modeled in one of the following options:
Option 1 No pore pressures are to be used, i.e., total stresses are to be used, or the pore
pressure are equal to zero.

Option 2 The pore water pressure is constant throughout the given material; the given

value of pore pressure is then input.
Option 3 Pore pressure is expressed by a constant value of the pore pressure ratio ru

where:

r = -— (3.8)

where, u is the pore pressure at any point and yh is the corresponding total vertical stress.
A value of ru should be input in this case.

Option 4 Pore pressure is defined by a piezometric line. Coordinates of the piezometric
line must be provided. Pore pressures below the piezometric line are assumed positive.
Above the piezometric line pore pressure is assumed negative. Normally the program sets
any negative pore pressure to zero unless specified by the user.

Option 5 Pore pressures are computed by interpolating pore pressure from an irregular

“grid” of pore pressure values which is given by the user.

56
Option 6 Pore pressures are computed by interpolation from an irregular grid of pore

pressure ratio, r“, values.

All of the above options, except for Option 2, were used to model pore pressure in the
present study.

A graphical presentation of results is provided by the program. A hard copy of the
embankment geometry and slip surfaces has been prepared by AutoCADR (Lockhart and
Reagh, 1995) for different cases considered in the present research. UTEXAS3 provides

other special features which are not considered herein such as soil reinforcement and

earthquake analysis.

3.7 Loading Conditions for Embankments, Modeling Strength Parameters
3.7.1 General

Some of the loading conditions for which a water-retaining embankment should
be analyzed are presented. Modeling strength parameters for each case is also discussed.
These loading cases are:

0 End-of-construction conditions
0 Steady seepage conditions

0 Partial pool conditions

3.7.2 End-of-Construction Conditions
During construction, high pore pressures may develop in an embankment and/or

its foundation due to weight of overlying materials and compaction equipment, especially

57
for low permeability soils where the rate of dissipation of built-up pore pressure is slow.
Hence, conditions immediately after construction are often critical. This may control the
design and rate of construction for high embankments such as dams. Soil parameters may
be modeled by two approaches:
Total Stress Approach: Unconsolidated undrained, UU or Q tests are recommended in
this case (Lowe, 1967; and US. Army, 1970) The test specimens are compacted to the
density and the water content which will be expected in the embankment. The effect of
pore pressures developed during consolidation and shearing are automatically
incorporated in the total stress analysis since they are implicit in the shear parameters
(Lowe, 1967). In this case it is assumed that Au in the field is approximately equal to Au
in the laboratory and is included in the shear parameters, c and (l).
Eflective Stress Approach: Field compaction-included pore pressures may exceed those
developed in the laboratory tests and hence a more recent trend is to perform effective
stress analysis in which pore pressures are modeled in terms of the pore pressure ratio, ru.
In terms of effective stress, the parameter (3’ represents friction and the cohesion intercept,
c', is very small and could be reasonably neglected. It should be noted that the uncertainty
in d)’ is smaller than in the case of total stress because of eliminating the uncertainty in the
pore pressure response. However, it should be recognized that there is still considerable

uncertainty due to the pore pressure change during shear (Wolff, 1985).

58

3.7.3 Steady Seepage Conditions

Steady seepage checks slow or long-term failure. Analysis is usually performed

for the downstream slope where the critical situation is full pool conditions. The CD or S
test provides the most logical strength to model steady seepage conditions (Lowe, 1967).
However, some other strengths are used in practice such as CU or the average of CD and
CU tests. U.S. Corps of Engineers uses the average of CD and CU tests, or the CD tests
in low normal stress ranges where it is less than the CU strength as a conservative

approach.

3.7.4 Partial Pool Conditions

The partial pool condition is a steady state analysis for the upstream slope where
the full pool may not be the critical situation. The same strength for steady seepage
should be used (CD or S strength). However, CU or R strength may be used to represent
a quick or undrained failure. As a conservative approach, U.S. Corps of Engineers uses
the average of CD and CU strengths or CD tests in low normal stress ranges where it is
less than the CU strength. Although this approach models some incomplete
consolidation, it avoids relying on negative pore pressures developed during shear

(Wolff, 1985).

Chapter 4
PRACTICAL CONSIDERATIONS FOR TOTAL STRESS ANALYSIS

OF END-OF-CONSTRUCTION CONDITIONS

4.1 General

This chapter is concerned with some alternative assumptions generally
encountered in reliability analysis such as: the deterministic model, the probabilistic
model, and the probability distribution of the soil parameters. The results are used to
assess the relative effects of some of these alternative throughout the rest of the present
research.

Regarding the deterministic model, it was noted (see Table 1.1) that in most
previous investigations only one deterministic model was applied, and circular slip
surfaces were the most common approach in the analyses (especially, the Simplified
Bishop method). In this study, some of these deterministic methods are compared in
terms of their possible effects on the calculated reliability index and the location of
critical failure surface. Four combinations of deterministic slope stability model and
failure surface shape were considered:

0 Spencer method with circular slip surface (Spencer Circular).
0 Simplified Bishop method (Bishop).
0 Modified Swedish method (Corps).

0 Spencer method with non-circular surface (Spencer Non-Circular)

59

60
These methods were reviewed in Chapter 3. UTEXAS3 was used for the analysis as it
permits running all four methods at the same time and the results can be easily compared.

Regarding the probabilistic model, the analyses include the influence of
probabilistic models on the results of reliability analysis. Probabilistic models were
reviewed in Chapter 2. Two of these models are well recognized in the literature and have
been commonly applied in practice:

0 Mean-value First Order Second Moment Method (MFOSM)

0 Point Estimate Method (PEM)

The third probabilistic model considered in this chapter is the advanced First Order
Second Moment Method (AF OSM). It has the advantage of being not variant but it is not
widely used in practice for slope reliability analysis. It requires much more computational
effort than MFOSM and PEM.

Soil parameters (mainly c and o) are implied to be normally distributed in the
MFOSM method and in the PEM method (when not adjusted for skew). As noted in
Chapter 2, lognormally distributed random variables may be more appropriate in cases
where soil parameters have high coefficients of variation. In this study the concept of
assuming soil parameters lognormally distributed and calculating transformed equivalent
moments (as will be explained later) was compared to the conventional approach of using
the normal moments directly.

It should be noted that the analyses included in this chapter are concerned with total

stress conditions for the case of end—of-construction of earth embankments. Analyses are

61

performed using soil parameters of UU tests or CU tests with total stresses. The pore

pressure change due to shear is implicit in the shear parameters.

4.2 Case Studies
4.2.1 General

Two embankments were analyzed:
0 Cannon Darn
o Shelbyville Dam
It was noted in Chapter 3 that the end-of-construction condition is more critical for high
embankments (dams); hence, the two embankments chosen are dams. An important factor
affected the selection of these two dams is availability of data and measurements. The
two dams were designed and constructed under the direction of US. Army Corps of

Engineers. A description of each of the two dams is provided below.

4.2.2 Cannon Dam

The structure is located on the Salt River in northeastern Missouri, USA. (Wolff,
1985). In addition to flood control, the project provides recreation, water supply, fish and
wildlife conservation, and hydropower. The dam has a 1000 ft (305 m) long earth
embankment, a gated concrete spillway section and a concrete powerhouse. The cross-
section of the earth-fill embankment is 139 ft (42.4 m) high and is shown in Figure 4.1.
The embankment consists of two compacted clay zones, Phase I and Phase II,

constructed over a 15 ft. thick sand foundation which overlays a thick limestone stratum.

62

E00 .8550 do cozoomlmmEQ 7v 830E

m m <m 2 m _ u
s\..vs-\.\\u.\.\\\.\.ovu.\.\\u.s..v...\.\\s.\.\\.u.\.cvk.\.ov.r\..vs.\.\\.~.\.\\.P\.\\k.\.\\s.\.o.\s.\.\\s.\.\\>\.%\s.\.o.\55¢»..on...\..vh.\.\\.r\\vs.Acesk.\..v.r\.\\\.\.ov\.\.\\s.\RVB.Asss.«.95on5.\.c.\u.\.ovu.\.\\\.\.0v§.\kv5.5..v5\.o~\%.\.%\\.\..v\.\..
Aooev
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Phase I contains a cutoff trench through the sand foundation down to limestone. A sand
filter is placed in the Phase II clay. The Phase I fill was compacted more wet of optimum
than originally intended, leading to an adjustment in the placement and compaction of
Phase 11 (see Wolff (1985)). The structure was previously analyzed by Wolff (1985), and

Wolff, et. al. (1995).

4.2.3 Shelbyville Dam

The structure is located is located on the Kaskaskia River in central Illinois,
USA. about 100 miles (160 km) northeast of St. Louis, Missouri (Humphrey and
Leonards, 1985). The St. Louis District, U.S. Corps of Engineers was responsible for
design and construction supervision. The dam was constructed for the purpose of flood
control in addition to secondary uses for recreation, fish and wildlife conservation,
downstream water quality control, and water supply. The dam is a combined earth-fill
and concrete gravity structure with a total length of 3,392 ft (1,034 m) The main earth
embankment cross-section is shown in Figure 4.2. Its maximum height is 108 fi (33 m)
and consists of a homogeneous section with upstream and downstream berms, an inclined
chimney drain and a horizontal drainage blanket. The embankment is constructed over a
thin sand layer resting on a firm rock foundation. A cutoff trench extends through sand to
the rock foundation. The structure was of interest due to a slide during construction and

was previously analyzed by Humpher and Leonards (1985) and Wolff (1985).

64

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4.3 Strength Parameters
4.3.1 General

Laboratory shear tests on soil generally provide results in the form of pairs of c
and (l) (or c’ and ti"). Several methods may be used to characterize random strength
functions , i.e. provide statistical moments (mean, u, standard deviation, a, correlation
coefficient, p) working from routine test data. A review and critical analysis of these
methods is given by Wolff, et. al. (1995). This includes eleven methods of estimating
moments of c and (I) from sets of soil strength data. Only Method 1 will be shortly
described here. A more detailed analysis is provided by Ur-Rasul (1995).

Paired values of c and cl) are routinely reported based on a visual best-fit linear
strength envelope tangent to Mohr’s circles for triaxial tests or through the o, I data
points for direct shear tests. In method 1, conventional statistical analysis on the resulting
set of c, 4) pairs is performed to obtain uc, 114,, cc, 0,, as shown in Figure 4.3. The method
is considered to be simple to apply, and to preserve engineering judgment. Bad test
results can be easily recognized and deleted. However, bias may result from personal
error in selecting pairs of c and ¢~

Soil strength parameters used in the present study are mainly based on the above

mentioned method (Wolff, et. al., 1995) together with other data available in the

literature.

66

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4.3.1 Strength Parameters for Cannon Dam

Four soil parameters were taken as random variables; c1 and (it, for Phase I Clay,
and c2 and 92 for Phase II clay. Note that UU strength envelopes for partially saturated
compacted clays curve downwards at low confining stresses; in practice, curved
envelopes are approximated by straight lines giving non-zero (1) parameters for undrained
strength. The statistical moments for the undrained strength of theses parameter are given
in Table 4.1 (Wolff et a1, 1995). These values were determined from a regression analysis
of Q (UU) test results performed on specimens trimmed from block samples of the fill

(record samples) .

Table 4.1 Cannon Dam, Total Stress Soil Parameters

 

 

Material Parameter Mean Standard C oeflicient of Correlation
Deviation Variation Coeflicient
Phase I fill c, 2460 psf 1230 50 % +0.10
(pl 85° 85° 100 %
Phase II fill c2 3000 psf 1650 55 % -0.55
(p2 15° 9° 60 %

 

 

It can be noted that the coefficient of variation is high, especially in the Phase I clay,
which suggests that adapting the normal distribution may lead to some practical modeling

problems previously described.

68
4.3.2 Strength Parameters for Shelbyville Dam
The total strength parameters for end of construction conditions are given in Table
4.2 (Humphrey and Leonards, 1985). These were similarly obtained from the results of Q
(UU) tests performed on specimens taken from “record samples” of the embankment. The
UU tests were performed on unsaturated specimens at their in-situ moisture content. No
correlation coefficient information was available and, hence, shear parameters were

considered independent (zero correlation coefficient).

Table 4.2 Shelbyville Dam, Total Stress Soil Parameters

 

 

 

Material Parameter Mean Standard C oefi‘icient of
Deviation Variation
Embankment c, 2360 psf 778 33 %
(pl 96" 7.7° 80 %
Sand Foundation (p2 32° 2° 6.25 %

 

 

It should be noted that no information was available on the spatial correlation for
the strength parameters of the two embankments considered. The present analyses were
limited to studying the effect of deterministic models, probabilistic models, and
parameter distributions on the reliability index and, hence, no reduction of variance due

to spatial correlation was considered.

69
4.4 Slope Stability Analysis
Slope stability analyses were performed using UTEXAS3 computer program,

(Edris and Wright, 1993), described in Chapter 3. Before performing probabilistic

analysis, a deterministic analysis was performed to locate the surface of minimum factor
of safety. This surface will be referred to as the critical deterministic surface throughout
the present study. It may be noted that this surface is located with the soil parameters are
set at their mean values. As discussed in Chapter 2, probability methods require using
different combinations of parameters to calculate the partial derivatives numerically. In
Chapter 6, parameter combinations weaker than the mean will be used to locate the
critical probabilistic surface (of minimum [3).

For circular surfaces, UTEXAS3 has its own search algorithm which starts with a
user-defined initial surface, proceeds to change the either the center or the radius of the
surface in a systematic manner, and each time calculates a factor of safety. The process
ends when a minimum factor of safety is found, or a certain number of iterations is
reached. A similar search process is used by the program for non-circular surfaces in
which a user-defined surface is also first assigned. The program then systematically
changes the coordinates of the points connecting the surface segments and calculates a
factor of safety each time until a minimum one is reached. The process for both circular
and non-circular surfaces depends on the initial surface specified to the program and
hence several initial surfaces should be examined until a minimum factor of safety is
achieved. For the two embankments considered, different initial slip surfaces were

specified to the program and the search process was repeated until a surface of minimum

70

factor of safety was located corresponding to each deterministic model analyzed using
mean strength values. Figure 4.4 shows critical circular surfaces for Cannon Dam
corresponding to Spencer, Bishop, and Modified Swedish methods. The non-circular
critical surface is shown in Figure 4.5. It may be noted that the three circular slip surfaces
are very close together although they correspond to different slope stability methods. For
Shelbyville Dam, Figure 4.6 shows three circular critical surfaces corresponding to
Spencer, Bishop and Modified Swedish methods, and the non-circular critical surface is
shown in Figure 4.7. The three circular surfaces are also very close together as found for
Cannon Darn, which may indicate that the critical deterministic surface is not
significantly sensitive to the deterministic model.

The values of factor of safety for each case are given in the following Table 4.3

Table 4.3 Factor of Safety Corresponding to Different Slope Stability Methods

 

 

Structure Spencer Bishop Corps Spencer
Circular Circular Circular Non-Circular
Cannon Dam 2.775 2.753 2.789 2.647
Shelbyville Dam 3.115 3.118 3.117 3.071

 

 

It could be noted that, for each embankment, values of factor of safety are very close
together and the influence of the deterministic model on the factor of safety and the

location of the critical deterministic surface is not significant. The non-circular slip

71

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surface gives the smallest factor of safety for both embankments. Further non-circular

analysis will be provided in Chapter 6.

4.5 Reliability Analysis
4.5.1 Mean-Value First Order Second Moment Method (MFOSM)

MFOSM finite difference method based on Taylor series approximation was
described in Chapter 2. In this method the variance of the factor of safety is determined
by expanding it in a Taylor series at the point represented by the mean values of shear
parameters. The method was applied to the two considered dams. For Cannon dam, four
random variables were considered: cohesion and friction of both Phase I and Phase II clay

embankments (CI, (in, c2, ¢2)- In this case:

E[FS] = 11.. = F801. .11..) (4.1)

2 2
6FS 6FS dFS 6F S
Var FS = 02 = [—] 02 +(—] a2 +2 [——)0 o 4.2
[ ] F5 XL aci Cr a¢i 91 2 OCi a¢i Cr ¢1p¢h¢1 ( )
where i = 1, 2 and p is the correlation coefficient between ci and 4)]. No correlation was
assumed between the parameters of the different types of soils.

As noted in Chapter 2, partial derivatives were calculated numerically by

evaluating the performance function over a small increment which is taken here as one

76
standard deviation following the Corps’ guidance (US. Army, 1992). In this case, for

example:

an FS... - FS.._

4.3
6cl 20' ( )

 

91

where F Scl+ is the factor of safety obtained by considering the upper value of c, (Ilcr +
06,), and FSC1_ is the factor of safety obtained by considering the lower value of 01(llc1 -
0,1) with all other variables kept at their mean values. The factor of safety was assumed
lognormally distributed, as it is bounded by zero and one, and the reliability index is

given by:

lnFS
t3 = H—J (4.4)

OlnFS

where E[ln FS] and amps are calculated from E[FS] and 01:5 as described in Chapter 2.

The probability of failure, Pr(t), is then given by (see Section 2.6):

Pr(f) = 1— 0(0) = <r>(—p) (4.5)

where (D (~B) is the cumulative distribution function of the standard normal distribution

evaluated at -B. The calculations are performed by a spreadsheet (Wolff et. al., 1995).

77

For each probabilistic analysis of Cannon Dam, nine slope stability analyses were
performed: one considering mean values of all variables (Equation 4.1), and eight other
considering upper and lower values of each variable (Equation 4.2). The reliability index,
B, was calculated for the critical surface of minimum factor of safety. However, it will be
shown later in Chapter 6 that some other slip surface may have a reliability index smaller
than that calculated at the critical surface of minimum F.S. A typical spreadsheet showing
the details of reliability analysis of Cannon Dam by MFOSM is given in Appendix B.

For Shelbyville Dam, three random variables were considered: cohesion and
friction of the clay embankment (c1, 0,), and friction of the sand foundation (4)2). The
parameters were considered independent (correlation coefficient = 0). Hence seven slope
stability analyses were required. The reliability index was calculated, also, for the critical
surface of minimum factor of safety.

The reliability indices and probabilities of failure for each dam corresponding to

different slope stability methods are summarized in Table 4.4.

Table 4.4 MFOSM, B Values Corresponding to Different Deterministic Models

 

 

Structure Spencer Bishop Corps Spencer Non-
Circular Circular Circular Circular
Cannon Dam 10.853 10.356 10.279 7.028

Shelbyville Dam 3.397 3.398 3.432 3.351

 

 

78
It may be noted that no significant difference in reliability index values was found
among all circular surfaces at a given structure. The non-circular surface gives a relatively
lower reliability index. The difference in the reliability index between circular and non-
circular surfaces is much more significant in case of Cannon Dam than in case of
Shelbyville Dam. On the other hand, analysis of Bois Brule Levee (see Chapter 6) for the

case of flood conditions showed that the Corps method gave the smallest reliability index

among the four considered methods. These findings may indicate that in some cases the
method of slope of slope stability (deterministic model) may have a significant effect on
the results of the reliability analysis. If extensive and consistent reliability analysis is
required, as is the Corps’ practice for prioritizing rehabilitation investment for dams,
several methods should be considered including both circular and non-circular failure
surfaces. However, Spencer’s method as a rigorous method which includes force and
moment equilibrium may be recommended for both circular and non-circular slip
surfaces.

Results in Table 4.4 also indicate that Cannon Dam was significantly more
reliable during after-construction loading than Shelbyville Dam although the latter had a
factor of safety larger than that of the former. This supports the discussion provided in
Section 2.9 regarding the fact that a higher factor of safety does not, necessarily, indicate

a more reliable structure.

79
4.5.2 Point Estimate Method (PEM)

In this section, the PEM method (described in Chapter 2) was used as the
probabilistic model to both Cannon Dam and Shelbyville Dam. The number of slope
stability computer runs required are 2N where N is the number of random variables (16
runs for Cannon Dam and 8 runs for Shelbyville Dam). The runs represent all possible
combinations of lower and upper point estimates values for the considered random
variables. The method was programmed in a spreadsheet (Wolff et. al., 1995). A typical
spreadsheet of PEM for Cannon Dam is shown in Appendix C. The results are

summarized in Table 4.5.

Table 4.5 PEM, B Values Corresponding To Different Deterministic Models

 

 

Structure Spencer Bishop Corps Spencer Non-
Circular Circular Circular Circular
Cannon Dam 11.34 10.83 10.74 7.39
Shelbyville Dam 3.39 3.40 3.43 3.35

 

 

As noted in the MFOSM analyses provided in the previous section, the Spencer non-
circular method gave the smallest B among all methods and again the difference is

significant in the case of Cannon Dam.

80

The AF OSM method was applied to both Cannon Darn (Spencer non-circular)
and Shelbyville (Spencer circular) and the results are compared to MFOSM and PEM in

Table 4.6.

Table 4.6 Comparison Between MFOSM, PEM, and AFOSM Analysis Results

 

Structure Spencer Circular Bishop Circular Corps Circular Spencer Non-Circular
MFOSM PEM AF OSM MFOSM PEM MFOSM PEM MFOSM PEM AF OSM

 

 

Cannon 10.85 11.34 - 10.36 10.83 10.28 10.74 7.03 7.39 *
Dam
Shelbyville 3.40 3.39 1.67 3.40 3.40 3.43 3.43 3.35 3.35
Dam

 

" did not converge

It may be noted that the AFOSM provided a B value significantly lower than MFOSM
and PEM for Shelbyville Dam. The method did not converge for Cannon Dam because
the failure surface passes through the sand layer for which if is taken as a deterministic
value. The failure point could not be reached through the other four parameters (c1, (1)1, c2,
02) because 4) of the sand alone was sufficient to provide a factor of safety greater than

one. In other words, no values of the random variables correspond to the limit state.

4.5.3 Lognormally Distributed Soil Parameters

Although the method of moments does not assume a probability distribution, it
implies a symmetrical distribution. For soil parameters of high coefficient of variation
(more than 30%) the probability distribution cannot be symmetrical given the constraint
that values are non-negative. In extreme cases, a coefficient of variation larger than 100%

will require negative values in the Corps’ MFOSM. A proposed approach (Wolff et. al.,

81

1995) is to assume soil parameters as lognormally distributed and then transform the
values to normally distributed soil parameters c* and ¢* to obtain a symmetrical
distribution as shown in Figure 4.8. An example calculation is provided below.

If e is lognormally distributed with the following mean and standard deviation:

E[cl] = 2460 psf

0[c1] = 1230 psf

o[c,]_1230 —0.50

°* = F.[c,] T 2460 '

 

Then the parameter c* = In C is normally distributed (see Chapter 2) with moments:

 

2
o[1nc,]= ,/1n(1+ v3, = J1n(1+ 0.502) =.4723, E[lnC,]= lnE[c1]—&2"—°'-= 7.696

Inc

The factor of safety and its derivative can be calculated using c* = e . Hence, the

transformed parameters are;

E{cl *] = e7'696 = 2200 psf
CH" = e7.696+.4723 = 3530 p sf

‘31-" = e7.696-.4723 =1372 psf

82

EM

f(C)
+0

 

 

 

C

A) Parameter "c" Assumed Lognormally Distributed

InC—/.\InC+

f(ln C)

 

 

 

 

 

 

 

 

 

In C
b) "In C” is Normally Distributed
c* E[c")
Q C:
C
C) Transformed Parameter "cw”
H C”

Figure 4.8 Transformation of Parameter

83
Hence, the points used to calculate derivatives are not equidistant from the mean in “c*”
space, but are in “In c” space. Furthermore, their distances from the mean are equally
probable. The transformed parameters for Cannon Dam and Shelbyville Dam are given in

Tables 4.7 and 4.8 respectively. Note that values will always be positive, regardless of

coefficient of variation.

Table 4.7 Cannon Dam, Shear Parameters Transformed
from Lognormal to Normal Distribution

 

 

Parameter c, * psf ¢ , * deg. c2* psf 412* deg.
Expected value 2200 6.01 2629 12.86

Upper value 3530 13.82 4394 22.39
Lower value 1370 2.61 1572 7.39

 

 

Table 4.8 Shelbyville Dam, Total Strength Soil Parameters Transformed
from Lognormal to Normal Distribution

 

 

Parameter c ,* psf (I) 1* deg.
Expected value 2241 7.50
Upper value 3091 15.15
Lower value 1625 3 .71

 

 

84
The transformed parameters are then used as input parameters for the MFOSM method as
described in Section 4.51. As no significant difference was observed between the outputs
of MFOSM and PEM; only the former is reported here. The results for Cannon Dam and
Shelbyville Dam are given in Tables 4.9 and 4.10. The previous results obtained by

assuming parameters normally distributed are also given for comparison.

Table 4.9 Cannon Dam, Lognormally Transformed Strength Parameters
Reliability Index Corresponding To different Deterministic Methods

 

 

Soil Parameters Spencer Bishop Corps Spencer Non-
Circular Circular Circular Circular
Normal Parameters 10.853 10.356 10.279 7.028
Lognormal Parameters 10.267 9.987 9.544 5.175

 

 

Table 4.10 Shelbyville Dam, Lognormally Transformed Strength Parameters
Reliability Index Corresponding To different Deterministic Methods

 

 

Structure Spencer Bishop Corps Spencer Non-
Circular Circular Circular Circular
Normal Parameters 3.397 3.398 3.432 3.351
Lognormal Parameters 3.224 3.226 3.259 3.182

 

 

It is clear that the lognormal parameters give lower reliability indices than normal

parameters. The difference is significant in case of Spencer non-circular surface for

85
Cannon Dam (5.18 versus 7.03). Shelbyville Dam was less sensitive to parameter

distribution.

4.6 Summary

Four deterministic models and three probabilistic models are used to evaluate the
reliability of two embankments; Cannon Dam and Shelbyville Dam. The soil parameters
were modeled in two forms: normally distributed and lognormally distributed. The
following findings were observed.

For the cases studied, deterministic models did not have a significant effect on the
results of reliability analysis. The non-circular slip surface provided a more critical B than
the circular failure surfaces for Cannon Dam. If extensive and consistent reliability
analyses are to be performed; several deterministic models should examined. Generally, a
non-circular slip surface should be considered especially in case of layered earth
embankments.

No practical difference was observed between the results of reliability analyses
done by the mean-value First Order Second Moment method (MFOSM) and Point
Estimate Method (PEM). The MFOSM may be preferred due to its simplicity.

The AF OSM has the advantages of being invariant and independent of the form of
the performance function and providing a more conservative B. However, due to the
iterative nature of slope stability problems, use in practice (the thrust of this work) is
cumbersome. F urtherrnore, numerical problems may occur as shown in case of Cannon

Dam. To simplify the analyses in the following chapters which emphasize other

86

questions, the MFOSM method was adopted. However, a further application of AF OSM
is provided in Chapter 6.

The lognormal distribution provided a more appropriate method to model shear

parameters especially when the coefficient of variation is high. However, for low values
of coefficient of variation the lognormal distribution approaches the normal distribution.
Hence, adopting a lognormal distribution for parameters has the following advantages:
0 It is more appropriate for high C.O.V.
o The results are quite similar to that of normal distribution when C.O.V. is low.
0 It provides a conservative solution.

A quite large difference in B may be obtained depending on the deterministic
model and soil parameter distribution. The analyses presented in this chapter gave a B
ranging from 11.34 to 5.13 calculated for the same section of Cannon Dam. Further
analyses in Chapter 6 will show that even a smaller value of B may be obtained for the
same section when searching for the surface of minimum B. Hence, these findings
suggest that intensive reliability analysis may be necessary especially if the results are
related to economical decisions.

Shelbyville Dam was less sensitive to the items investigated here than Cannon
Dam. This may indicate that the sensitivity to deterministic and probabilistic models and

parameter distribution varies from structure to another.

Chapter 5
PRACTICAL CONSIDERATIONS FOR EFFECTIVE STRESS ANALYSIS

OF END-OF-CONSTRUCTION CONDITIONS

5.1 General
An alternative method to model end-of-construction conditions is effective stress
analysis as described in Chapter 3. In terms of effective stresses, soil strength is

represented by the parameters c' and o';

t} =c'+0'tan¢' (5.1)

The cohesion intercept, c', is very small in this case and the uncertainty in 't' is smaller
than that in terms of total stress; however, it should be noted that there is still
considerable uncertainty due to the pore pressure change during shear (Wolff (1985)).
This pore pressure may develop in impervious sections of earth dams during or
immediately after construction due to the weight of overlying fill, and stresses due to
roller pressure which may remain for sometime after placing another layer of fill due to
low permeability. This case may control the design and or the rate of construction,
especially for dams with clay embankment or foundation constructed on the wet side of
optimum moisture content. In this case the rate of stress increase is higher than rate of

pore pressure dissipation due to low soil permeability. The significance, in terms of factor

87

88
of safety, of high average pore pressure throughout the embankment cross-section was
studied by Bishop and Mogenstem (1960) as described in the next section.
In this chapter, the effect of the uncertainty in pore pressure on the reliability of
earth slopes is studied. The effect of the method of modeling ru as a random variable is
also investigated. Pore pressure is expressed in terms of pore pressure ratio, ru, (BishOp

and Morgenstern, 1960) where:

r = —— (5.2)

where, h is the height of soil above the point, and y is the total unit weight of soil. The
effect of the method of modeling ru as a random variable is investigated by examining

different cases as will be described later in this chapter.

5.2 Previous Investigations

In a deterministic slope stability analysis, Bishop and Morgenstern (1960)
compared two methods of modeling ru: using the actual varying distribution of r“, and
considering a single mean value of ru throughout the slope. The difference, in terms of the
factor of safety, was found not significant. Modeling ru as a single mean value has
proven successful in giving values of F S that correspond closely to those obtained for
cases in which r“ varied considerably throughout the section. This significantly simplifies

the modeling of practical problems. However, the method of averaging ru suggested by

89

Bishop and Morgenstern may be the reason of this consistency. They did not simply take
the statistical average of the values of ru. Instead, they divided the embankment into
zones each with an average value of ru. These zones are obtained by dividing the base into
four equal parts starting from the middle of the crest, and the average ru for each zone is
calculated by averaging ru along the center line of each zone. Then , they averaged these
values after weighting each value by its corresponding zone area. This may interpreted as
some kind of “spatially averaging” the values.

In the probabilistic analysis of embankments, little is found in dealing with pore
pressure variability for the end-of-construction case. In analyzing Shelbyville Dam, Wolff
(1985) considered different values of ru assuming a variance of zero. The probability of
failure increased from .026 (1 in 38) to 0.54 (1 in 2) as ru increased from 0.3 to 0.5.
However, he noted that piezometric pore pressure is a random variable and calculated
probabilities of failure are conditional probabilities and depend on the modeled
coefficient of pore pressure.

Li and Lumb (1987) used a simple model in which pore pressure ratio was modeled as a
random variable with a judgmental coefficient of variation of 10%. Pending more
information on the correlation structure of pore pressure the pore pressure ratio was
assumed to be perfectly correlated within the slope, i.e. was regarded as a single variable.
Furthermore, the cross-correlation of pore pressure ratio with other soil properties was
neglected for simplicity. Termant and Calle (1994) considered uncertainty in the
estimation of excess pore pressure in a cohesive soil layer by expressing dissipation of

excess pore pressure in term of the consolidation factor, u. This factor takes values

90
between 0.0 (undrained condition) and 1.0 (complete consolidation). A Beta probability
density function was assumed for this factor to consider it as a random variable. A normal
distribution was considered less appropriate, because the consolidation rate is physically
limited. The results showed that uncertainty in pore pressure had a significant influence

on the probability of failure.

5.3 Modeling Uncertainty in rll
5.3.1 General
The distribution of pore pressure throughout an embankment is not uniform; it
follows a contour pattern in which pore pressure is higher in middle of the embankment
and lower at the edges. This occurs because dissipation of pore pressure (consolidation) is
greater near the edges or near an adjacent pervious boundary (sand drain). Modeling the
variability in the actual distribution of ru in slope stability problems is possible in some
slope stability programs such as UTEXAS3 (Edris and Wright, 1993). However, this may
not be a practical approach in routine probabilistic analysis because:
0 Measurements of end-of-construction pore pressure are often insufficient to
characterize a complete spatial distribution of ru.
0 Modeling a grid of ru values in slope stability problems is a time and cost-intensive
computational effort.
Hence, an alternative method is often used in which ru is modeled as single mean value

throughout the embankment (Wolff, 1985; Li and Lumb, 1987).

91
In order to study the effect of uncertainty in ru on the slope reliability and to
examine the effect of modeling ru as a deterministic value or a single random variable,
different cases are studied herein:
1. Case A: Modeling ru as a deterministic value (taken as an average value throughout
the dam).
2. Case B: Modeling ru as a deterministic value (taken as a grid of actual ru values).

3. Case C: Modeling ru as a perfectly correlated random variable with E[ru] and o[ ru].
4. Case D: Modeling ru as a perfectly correlated random variable with E[fu] and aha].

5. Case E: Modeling ru as a spatially correlated random variable.
6. Case F: Considering the variability in an actual ru values grid.
It may be noted that modeling the variability in the actual r" values grid represents the
ideal solution but it is not a practical one as mentioned before, and it was aimed here to
find a practical modeling of ru that best approximates this ideal solution. To deal with an
actual distribution of ru, the data available from Vallecito Dam, (Bishop and
Morgenstern, 1960) were utilized. Figure 5.1 shows the cross-section of the dam together
with the distribution of ru.

The Spencer method was used in all cases considering a circular failure surface.
The MFOSM reliability analysis was applied using effective stress parameters and
considering first 0' as a random variable with an assumed coefficient of variation of 6.67
%, and then considering both 0’ and c' as random variables with an assumed coefficient

of variation for c‘ of 20 %. The mean values of 0' and c' were taken the same as those

92

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given by Bishop and Mogenstem(1960). The reliability index was calculated for the

critical surface of minimum factor of safety.

5.3.2 Case A: Average ru Value, rll Deterministic

In this case ru was considered as a deterministic value (the C.O.V. was assumed
zero). The value of ru was considered as a constant average value throughout the
embankment as the mean value of the grid shown in Figure 5.2 (E[ru] = .31). It may be
noted that Bishop and Mogenstem (1960) used a value of ru = .23 as an average value
calculated by the method discussed in Section 5.2 whereas the value used here (0.31) is
the statistical mean value of the available grid values.

The only random variable considered here is the fiiction 0’. However, the case of
considering two random variables, c’ and 6’, is studied later in this chapter. The resulting
factor of safety was 1.88 and reliability index was 10.90. These values will be compared

with other cases in the following sections.

5.3.2 Case B: Actual ru Values Grid, rIll Deterministic

In this case ru was considered also as a deterministic value; but the actual grid
values shown in Figure 5.2 were used instead of the mean value used in previous case.
The value of the factor of safety was 1.91 and B value was 10.70. These values are very
close to the values obtained in Case A when using an average value of r,,. Regarding the

factor of safety, the present results are similar to those of Bishop and Mogenstem (1960)

94

moa_0> 3.. :0 0:0 .800 o:om__0> N6 939...

 

 

 

omnoo

 

 

 

0+ 0+ o. 0:. or. or. m.+ aw. mp. m.+ 0..

O+O—.+ mrro. ON... 0N... WNIQ WNt. WN.+ mN+ WN... WNZ.
:4 av. om; mm» on+ mm.mn.+ on. on. mm» on. 4
:. om. mm. mm. oetlnv. me. me. on. no. mnrrrw
_+ mm; on; we; on; on4 on; on” o 4 . on; em.
:4 mm. oer om. on; on. on; on. n. on; out o_.
:4 mm; mm. was om.+ an; o + e) on» out 0:4 Aw
+ 3.5+ 35.... >..+ 5L Dr... .4 >34 >43 >..+ 9+

CC S s U‘ “V CV S CC Cw C

0+ 00.... ON.+ WN.+ OWl. OW? ON.+ 0.?

o. o. o... o 4 om. o... o. F

o+ ou\\wu o+ m

n_+ m.+ .4
mu» o~+ 0+ :4
mallow. :4 F: Ill!
om. ,4 a. q.
F .+ Ce 0+
0 .1
m3<om

 

95

discussed previously in Section 5.2. The values of B are close obviously because ru is a

deterministic value in both cases and F8 values are very close.

5.3.3 Case C: Average ru Value, ru Perfectly Correlated With E[ru] and 0[r,,]

In this case r,I was modeled as a perfectly correlated random variable. In other words,
the spatial variability is not considered (6 is assumed very large compared with failure
surface length) and no reduction is applied to the variance of ru. An expected value E[ru]
of .31 and a standard deviation 0[r,,] of 0.16 were applied. The reliability index in this
case was 3.21 which is significantly lower than the value obtained in Cases A and B by
considering ru as a deterministic value. This indicates that B is very sensitive to the

variability of r,

5.3.4 Case D: Average r,I Value, ru Perfectly Correlated With Fig] and 0[fu]

This is similar to Case C except that the mean value was considered as a random

variable with an expected Elia] and a standard deviation a[fu] given by:

151;]: E{r,] (5.3)

 

ch] = 652:] (5.4)

96

where n is the number of measurements as will be discussed in Section 5.3.5. The values
of E[ru] and 0[ ru] were 0.31 and 0.035 and the resulting B was 8.93 which is significantly
higher than that obtained in Case C. The significance of using the standard error of mean

will be discussed after analyzing Cases E and F in the next two sections.

5.3.5 Case E: Average ru Value, r,l Spatially Correlated

As mentioned before, ru is generally not constant throughout an embankment, it
follows a contour pattern. Plotted contours superimposed on a grid of ru values grid may
help in visualizing ru as a spatially correlated random variable. In order to consider the
spatial variability of ru in the calculation, three steps should be performed:

0 First, an ACF of the ru values should be calculated.

0 Then, the variance of ru should be separated to systematic and spatial variance.

0 Finally, the variance reduction factor, F, is calculated and used to obtained the
reduced variance which is applied in the reliability analysis.

These three steps are described in the following paragraphs.

The ru contours shown in Figure 5.2 were used to generate a grid of ru values. This
grid is also shown in Figure 5.2. These values are used to calculate an estimate of the
possible autocovariance function, ACF, associated with the data. To calculate the
horizontal ACF a set of data was selected from the grid shown in Figure 5.2. This data
was selected from the mid-area of the distribution and has a separation (lag) distance of
20 it. It is reasonably considered a good representation of the entire distribution. This data

set was refined by inserting mid-distance points to make the separation equal to 10 ft. A

97

horizontal ACF was calculated for each row in the grid and then an average ACF was
calculated as shown in Figure 5.3, however all the ACF’s are very close to each other. An
example of calculating the ACF is shown in Appendix A. It was noted that the original
grid (lag = 20 ft) gave almost the same ACF and scale of fluctuation as the refined grid
(lag = 10 ft). As shown in Figure 5.4 a negative square exponential ACF was fitted to the
data. The horizontal correlation distance “scale of fluctuation” was found to be 42 ft. To
calculate a vertical ACF the same procedures were followed. The vertical ACF is shown
also in Figure 5.4. The vertical scale of fluctuation was found to be 15.5 it. This was
considered reasonable compared with the oval shape of the ru contour which spreads
horizontally more than twice vertically. Also, some reported 8 for other soil parameters
are in the same order. Lacasse and Nadim (1996) reported a value of 8x = 16.540 fl. and
82 = 3.5-6.5 ft. for cone penetration resistance of silty clay. Also, DeGroot (1996)
reported a value of 3-10 ft for 82 of clay undrained shear strength determined by field
vane test for which Soulie et. al. (1990) and Chiasson et. al. (1995) also reported values
of about 10 ft., and 6.5 ft. respectively.

The next step in the calculation is to separate the variance of ru into systematic
and spatial variance as described in Chapter 2. The systematic variance is the variance in

estimating the mean and is given by:

V{£[r,]} = V[r,]/n (5.5)

Where n is the number of measurements. The spatial variance is given by:

98

ed
9d
3.0
rd
o

mwd
Nd
Nd
_.d
_.d

Nd
mNd
mNd
mNd
Nd

mNd
md
md
md
mNd

n54‘ 303:0... 0505002000 2:. 0:0 no:_a> 0.. “.0 2.5 .800 ogoo=~> fin 00:2“.

mNd md
mmd vd
vd md
mmd vd

md mmd Vd mvd mvd

mmd vd vd
mvd mvd md
md md md
mvd md dd

0.0. 3.0 00.0 0.0 5.0 «0.0 F .02
~00 0.0 00.0 $0 0.0 2.0 0 $2
.0. $0 03 04.0 00.0 2.0 0 IS.
.0. 00.0 8.0 04.0 00.0 5.0 F 0a0<
’0. 00.0 00.0 5.0 00.0 00.0 F ~a0<
:9 00.0 0.0 00.0 000 00.0 a lo...
00 00 00 00 0a 0. 0 003

”whoa. _Scoutor

vd mvd mvd mvd mvd mvd vd and mad and and and mad md Nd
md md md md md md de vd mmd mad mmd md mNd Nd ed
md md md md md md vd mmd mmd md mNd Nd m..d rd 0
md md md md mvd vd mmd md mNd Nd 3.0 2.0 rd 0 d
mvd mvd med vd mmd md mNd Nd mwd rd 0 o o d o

”00:.m> E .0 25 End ogoo=m>

99

 

(18

(16

     
 

ACF

s . , _
y— ........................................................................... —
x I .

. ‘ I I
~ 3 \ vertical ACF

I
t
O
' 0
I
I
. I
. ‘ I I . '
I . I I . . .
o . ' ‘ '
I I
I ‘
I I .
— ........................................................................... _
' 4 ' . I .
n
' ' . l I .
I . '
I ' I . / ' ' .
I ' Q . I .
I I I l
u ' . a . ‘
o
o

. l . I . l -
lllllillLlllll~1$llllllllll;LlLlllLLlL

(14

 

 

 

O 10 20 30 40 50 60 70 80

Lag (ft.)

Figure 5.4 Vallecito Dam, Autocovariance Functions For ru Values

100

2 2 2
Var spatial = Var total — Var systematic (5 .6)

It may be noted the systematic standard deviation is the same as the standard error of

mean used in Case C. The values are summarized in Table 5.1

Table 5.1 Vallecito Dam, Moments of rll

 

 

Expected Standard Deviations
Value Systematic Spatial Total
0.31 0.035 0.156 0.16

 

 

The number of measurements, 11, used to calculate systematic variance deserves some
attention. This number should not be the total number of points in the grid of data; but it
should be the number of independent samples (Li and Lumb, 1987). To obtain
independent samples, points separated horizontally by 5,. and vertically by 52 are chosen
from the data set shown in Figure 5.3. The resulting number of independent samples was
approximately 21.

The variance reduction factor, F, was calculated according to the procedures
given by Vanmarcke (1977, 1983). The failure surface was divided into two portions Ax

and A2 in the x and 2 directions respectively such that:

L = Ax + A2 (5.7)

101

where L = failure surface length.
The failure surface shown in Figure 5.2 was obtained by performing a slope stability
analysis of the dam using the mean values of the soil parameters and ru and its length was

found to be 353 ft. The correlation distance along the failure surface, 5L , is given by:

E[L] (5.8)
L 5

Then, the reduction factor is given by:

[105(85): forL>8L (5.9)

The reduction factor is used to reduce the spatial variance of a random variable due to

spatial variability as follows:
Var [X] = r2 (L) 52m (5.10)

It may be noted that the systematic variance is not affected by the averaging process and

hence the reduction factor is applied to the spatial variance only and thus:

0' [fa] = [F2 (Lb 2spatial +0 Zs'ystematic]l/2 (5.11)

102
The calculation of the reduced variance for Vallecito Dam is given in Figure 5.5. The
reduced standard deviation, o[ru], was found to be 0.057. This value was then used in the
reliability analysis. The resulting reliability index was 7.21. This value is significantly
higher than that of ru being perfectly correlated in Case C (B = 3.21). This is due to fact
that spatially averaging a random variable reduces its variability because lower values are
compensated by higher ones as discussed in Chapter 2. On the other hand, this value is
close to B when the standard error of the mean is considered in Case D (B = 8.93). This is
because the systematic standard deviation (standard error of mean) is a good estimate of
the reduced standard deviation when the reduction factor is very small (see Equation
5.11). However, it may be argued that from Equation 5.9 the variance reduction factor is
directly related to the scale of fluctuation (8). At the same time, the systematic variance is
also directly related to 6; because large 8 implies a small n and, hence, large osystematic. So,
a larger 6 will lead to a large reduced standard deviation and, also, a large systematic
standard deviation, and vice versa. Hence, the systematic standard deviation will remain a
good estimate of the reduced standard deviation although the former is always smaller

than the latter.

5.3.6 Case F: Actual 1'“ Values Grid, r“ Random Value
In this approach ru is modeled as a grid of points reflecting the variability in ru
from point to point. The grid of ru values shown in Figure 5.2 was used to simulate the

actual distribution of r“. This grid of points was used as input to UTEXAS3 to interpolate

103

108 H

3“
L‘35

324 H

L=Dx+Dz

flu
Ax 324

Ax=266fi
Az=89fi

From Equation 5.8:

_. (89) 1 (266)]
a, = — —+ —— —
353 155 353 42

SL = 29.232 it

From Equation 5.9:

U2
“04%) 0.287
353

From Equation 5.11 and Table 5.1:

O'[l’u]L =[(287)’(.156)2 +(.035)2]"2 =.os7

Figure 5.5 Vallecito Dam, Example of Calculating The Reduced Variance

104

pore pressure values at each slice base as described in Chapter 3. Since the spatial effect
is covered by varying ru from point to point, this grid should experience only the
systematic variance due to measurement error. From Table 5 .1, the systematic coefficient
of variation, C.O.V, for the ru grid values is about 11.3 %. This C.O.V. was applied to all
points simultaneously. In other words, each value of ru was considered a random variable
with C.O.V = 11.3 %. As in the mean-value FOSM method each point has taken its mean,
upper value (p + o), and lower value (p. - o) and the factor of safety was calculated in
each case. The resulting reliability index was 8.37 which is relatively close to the value
obtained by considering ru as spatial random variable in Case E (B = 7.21) and the value
obtained by using the standard error of mean in Case D (B = 8.93). The results of the

above discussed six cases are given in Table 5.2.

105

Table 5.2 Vallecito Dam, Reliability Analysis Results
Random Variables: ¢' and r,|

 

 

 

Case FS [3

A) Average r“l value, ru deterministic, 1.882 10.90
E[ru] = 0.31

B) Actual ru values grid, ru deterministic 1.905 10.70

C) Average ru value, ru perfectly correlated with 1.882 3.21
E[ru] = 0.31, o[ ru] = 0.16

D) Average ru value, ru perfectly correlated with 1.882 8.93
E[‘r'u] = 0.31, o[fu] = 0.035

E) Average ru value, ru spatially correlated with 1.882 7.21
E[ru] = 0.31, o[ ru]L = 0.057

F) Actual ru values grid, random ru values with 1.905 8.37
o’systemafic = 0.035

 

From Table 5.2, it is clear that modeling ru as a perfectly correlated random variable
with E[ru] and o[ru] (Case D) and modeling ru as a spatially correlated random variable
(Case E) gave a reasonable and consistent estimate of the reliability index obtained by the
ideal but impractical solution of considering the variability in an actual grid of ru values
(Case F). This provides a practical method of dealing with uncertainty in pore pressure
and avoids the difficulties associated with considering a distribution of ru varying from
point to point. On the other hand, assuming ru as perfectly correlated with E[ru] and c'[ru]
(Case C) significantly underestimated the reliability index. This is generally because
using the ‘point’ standard deviation without reduction does not account for the fact that

low values at some points are compensated by larger values at other points. Similar

106

results were noted by Li and Lumb (1987) for strength parameters, c' and d)’. To model a
random variable as spatially correlated, two items are required: the autocovariance
function, ACF, and the scale of fluctuation, 6. The forms of ACF are discussed in Chapter
2, however, Li and Lumb ( 1987) noted that the reliability index is not very sensitive to
the shape of ACF especially when the scale of fluctuation is small compared with length
of the failure surface. Also to calculate the standard error of mean (systematic variance)
used in Case C, the number of independent samples is required and, hence, the scale of
fluctuation needs to be known (or estimated).This may introduce the importance of the
scale of fluctuation as a measure of spatial variability which will be discussed in the
following section.

The results shown in Table 5.2 were based on considering only ¢' and ru as
random variables whereas effective cohesion, c', was taken as a deterministic value.
However, the analyses were repeated by considering c' as a random variable of 20 %

C.O.V. The results are shown in Table 5.3.

107

Table 5.3 Vallecito Dam, Reliability Analysis Results
Random Variables: c', ‘1’" and r“

 

 

Case FS B
A) Average ru value, ru deterministic, 1.882 7.62
E[ru] = 0.31
B) Actual ru values grid, ru deterministic 1.905 8.16

C) Average ru value, ru perfectly correlated with 1.882 3.06
E[ru] = 0.31, o[ ru] = 0.16
D) Average ru value, ru perfectly correlated with 1.882 6.84
E[ru] = 0.31, o{fu] = 0.035
E) Average ru value, ru spatially correlated with 1.882 5.97
E[ru] = 0.31, o[ ru]L = 0.057
F) Actual r“ values grid, random r.Jl values with 1.905 6.97

 

osystematic = 0035

 

It may be noted that again Cases D and E provided a reasonable and consistent good
estimate of the reliability index obtained by the explicit model of Case F. The reliability
indices obtained in case of modeling 0' as random variables are, of course, generally

lower than the case of c' as a deterministic value.

5.4 Scale of Fluctuation
As discussed in Chapter 2, the scale of fluctuation of a soil parameter is a
characteristic distance within which the parameter experiences relatively strong

correlation. It is used to calculate the required reduction in the variance of a parameter

ClE
ex
hig

of

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rep]

con.
EXIIC
may
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Show
and 1

(H350

Vaflalh

dillerez

108

due to spatially averaging the parameter over a given length or area. To investigate the
influence of the value of scale of fluctuation, 5, on the reliability index, B, several values
of 8 were used to calculate B for Vallecito Dam by applying the model used in Case E
discussed above. The results shown in Figure 5.6 include two cases: taking c' as a
deterministic value and considering it as a random variable. The results indicated, as
expected, that the reliability index decreases as 8 increases. The degree of decrease of B is
higher for the case of assuming c' as a deterministic value compared with that for the case
of assuming it as a random variable. This is because when the number of random variable
increases, the influence of each variable on B decreases. Figure 5.7 shows the effect of 6
on B for the case of considering c' as a random variable. The figure also shows two lines
representing B when ru is a deterministic value and when ru is a perfectly correlated
random variable. It is clear that the proposed method of considering ru as a spatially
correlated random variable represents an appropriate solution which lies between the two
extreme solutions (ru = deterministic value, and ru = perfectly correlated r.v.). However, it
may be noted that the value of 5 has a considerable effect on B in the case studied.
Another study for Shelbyville Darn will be presented in Section 5.5. Similar results were
shown by Li and Lumb ( 1987) for the effect of the scale of fluctuation of both cohesion
and friction (assumed equal) on B calculated according to Hasofer-Lind definition
(Hasofer and Lind , 1974).

The effect of 8 on B was investigated for different values of the coefficient of
variation in ru. Figure 5.8 shows the relationship between C.O.V of ru and B calculated for

different values of 8 in case of assuming c' as a deterministic value. It could be noted that

109

 

 

    
   

 

 

9 T T r] T T r T I T T T T l’ T T TTI T T T T I T T T T
L ............................................... —0—c'constant .;
815 ~ .
: -)(-— c'random :

L— ............................................. .
8 » 1
~ +
‘15 - ~
r.........................: ........................... l ...................... _
7 e I j - ‘ "
.- X . . I ..
b 3‘5; 3 ' i l
_s ............. ' ......................... ._.
65 - ‘K : ' «
. l- : ~K j - J
l— : x 1 2 4
l- \)(~ I j '1
6 :- -------------------------------------- . --------- N -.- ----------- 2 ----------- —l
_ : ‘x- - : 1
. ' ‘X 4
b1 1 1 1 l 1 l l 1 l J 1 1 1 L 1 1 l 1 l l l 1 l l 1 1 l l-4

 

5.5
0 1 O 20 30 40 50

Scale of Fluctuation (ft.)

0)
O

Figer 5.6 Vallecito Dam, Effect of Scale of Fluctuation on B

110

 

 

 

 

 

12 Lfi T T T I T T T T I T T T T I T T # T I T l l T I T T T T
r; as a deterministic value ‘
10 E
8 :
6 _ E _
4 _ ru as a perfectly correlated random variable 7
r— ——————— + ———————— -t
1 L 1_ l l l l 1 1 l 1 1 1 1 l 1 l l 1 l 1 1 1 l i l _L 1 _l_
2

0 1O 20 30 40 50 60
Scale of Fluctuation (ft)

Figure 5.7 Vallecito Dam, Effect of Scale of Fluctuation on
Reliability Index (For Different Modelling of ru)

111

B is very sensitive to the variability in pore pressure when ru is assumed perfectly
correlated, and becomes less sensitive as 8 decreases. The same effect may be noted for
the modeling of c' as a random variable in Figure 5.9. The sensitivity of B to C.O.V of In
is less in this case as shown in Figure 5 .10 which provides a comparison between the two
cases for 8 = 30 fi. Figures 5.8 and 5.9 also show that effect 8 on B depends to some
extent on the C.O.V.; for small values of C.O.V this effect is not significant. As the
variability increases this effect becomes considerable. However, as a practical approach,
reasonably accurate B values could be obtained by estimating the value of 8 according to
the data available and/or from other data on similar sites with considerable engineering
judgment. This will give a good estimate of B compared with that obtained by
considering ru as a perfectly correlated variable. This is clear in Figures 5.8 and 5.9 since
the curves associated with 8 = 30 ft and 8 = 50 it are quite close compared to that
associated with perfect correlation.

It should be understood, from Equation (5.6), that the influence of the scale of
fluctuation, 8, is not absolute; but it is related to the failure surface length, L, which may
be referred to as the averaging length or problem size. The reduction of variance due to
spatial averaging is more significant when the averaging dimension is large relative to the
scale of fluctuation. A practical concept of “dense sampling” and “sparse sampling” was
introduced by Wolff et. al. (1995) in which the problem size is considered as an effective
factor compared to the size of measurement when considering the uncertainty in the

measured soil properties.

 

112

 

12 errr I T T TTITT T T I TTT T171 TTTITTTTITTTT
I I I

 

4 , : I E Correlated -

—1

 

 

111L1111111111L111lllilllLLllLillL

 

0 1o 20 30 4o 50 60 7o
Vr %
U

Figure 5.8 Vallecito Dam, Effect of Vru on reliability Index for
Different Value of Scale of. Fluctuation, c‘= constant

113

 

 

 

 

 

9 _T T T T I T T T TIT T T T I T T T T I T T r T I T T T T I T T T Td
r ' I 1
- -+

3 T

7

6 E

5 E «

: ; : 51’“ f : 2
.......... 5 hp .|

4 E- 2 3 : ‘sA erfecy: fl
*- . . ' . ' . Correlated ~
~ Wows; r c,¢ : ; .
t. .......... f ........... '. ................................. ' ..................... .1

3 >- ' ' .1
_l 1 1 lLLL 1 111 11111 1 1 11111 11 11 L 111 1 11‘

2
0 10 20 30 40 50 60 70

Vru°/o

Figure 5.9 Vallecito Darn, Effect of Vru on reliability Index for
Different Value of Scale of Fluctuation, c‘= random

114

 

 

   
 

12 .- T T T I T T T T I T T T T I TiT T T r T T T T I T T T T d
9* c 3 . ,

r c' determrnrstrc .

10 :- ------------------------------------------------------------------------ —.:
8 y;-_ _*_ _ g c random 5 ; q
: 3 -“X-- 33‘ 3 .

a "‘ l - . -‘l *
6 _ .......................................................................... J
4 L— ............................................................................ -
I- -l
1 l 1 1 1 1 1 1 1 L L 1 1 1 1 1 1 1 1 1 L 1 1 1 L L L J

 

 

 

Vr%
1.1

Figure 5.10 Vallecito Dam, Effect of Vru on Reliability Index for
Scale Fluctuation = 30 ft.

115
5.5 Analysis of Shelbyville Dam
Total stress probabilistic analyses for end-of-construction conditions for
Shelbyville Dam were described in Chapter 4. In this section effective stress analysis is
performed considering variability in r”. In this case effective S (CD) parameters were
used. These parameters are given in Table 5.4 (Wolff, 1985).

Table 5.4 Shelbyville Dam, Effective Strength Soil Parameters
for End of Construction Conditions

 

 

 

Material Parameter Mean, Standard C oeflicient of
psf Deviation Variation
Embankment c'l 92 110.4 120 %
rp’, 30.5 2.75 0.09 %
Sand Foundation (p'z 32° 2° 6.25 %

 

 

As shown in table, the coefficient of variation for c', is very high. Hence parameters were
assumed lognormally distributed and then transformed to normally distributed as

discussed in Chapter 4. The transformed parameters are shown in Table 5.5.

Table 5.5 Shelbyville Dam, Effective Transformed Parameters

 

 

Parameter c’ , * psf ¢' 1 "‘ degree
Expected value 58.9 30.38
Upper value 151.5 33.23
Lower value 22.9 27.77

 

 

116

The observed critical slip surface occurred in the upper part of clay embankment; hence,
the fi'iction angle of sand will not be considered as a random variable.

Field measurements in the vicinity of the slip surface (US Corps of Engineers,
1974) are shown in Figure 5.11. These measured ru values were used to determine
systematic and spatial variance together with variance reduction factor as described in
Section 5.3.1. The mean value and standard deviation of the measured ru values were
found to be 0.38 and 0.15 respectively. Since there are not enough data points to establish
a uniform grid for calculating an ACF, an assumption is made. A scale of fluctuation
along the failure surface, 8L, was taken a judgmental value of 25 it based on results found

previously for Vallecito Dam. The results are shown in Table 5.6

Table 5.6 Shelbyville Dam, Moments of Pore Pressure Ratio, r“

 

 

Expected Standard Deviation
Value Systematic Spatial Total
0.38 0.048 0.145 0.15

 

 

The embankment was analyzed for four cases:

0 Case C: Average ru value, ru perfectly correlated with E[ru] and o[ru].
0 Case D: Average ru value, ru perfectly correlated with E[f'u] and o'[i'u].

0 Case E: Average rul value, ru spatially correlated.

- Case P: The actual ru grid, ru random with osystcmafic.

117

mootam 95:8 do £505 of. E

EoEoSmooI 3.. .Eoo o___>>n_ozm :6 wood:

 

\/

 

118

Based on a failure surface length of 140.5 ft, the reduction factor was found equal to 0.42;

and the reduced standard deviation was 0.073. A systematic variance of 0.002 (Table 5.6)

was applied to the grid of ru.

The following were fixed during slope stability and reliability analysis:

0 Spencer slope stability method (circular surface).

0 MFOSM.

The factor of safety, FS, reliability index, B, and probability of failure, Pr (f), for the

two cases considered are given in Table 5.7

Table 5.7 Shelbyville Dam, End-of-Construction Conditions
Results of Effective-Stress Reliability Analysis

 

 

Case FS B Pr (0
C) Average ru value, ru perfectly correlated 1.044 -.012 0.505
with E[ru] and o[ru].
D) Average ru value, ru perfectly correlated 1.044 0.11 0.456
with lira] and o[fu].
E) Average ru value, ru spatially correlated. 1.044 0.09 0.465
F) Actual ru values grid with systematic 1.052 0.16 0.436

variance

 

 

The results shown in Table 5.7 indicate that modeling ru as a perfectly correlated random
variable with systematic variance or a spatially correlated random variable with reduced

variance gives a good estimate of the reliability index and the probability of failure which

119

compares well with that obtained by using a more “rigorous” model dealing with the
actual ru grid. On the other hand, modeling ru as single random variable with total
variance gave a value B relatively far from that for the ru grid, which is consistent with
findings obtained for Vallecito Dam in Section 5.3. However, a case of an actual failure
was observed for Shelbyvile Dam (Humphrey and Leonards, 1985). Hence, the B values
and probabilities of failure support the failure conclusion regardless of the model used in
the analysis.

Several values of 8 were used to calculate B to examine their possible dependence.
The results are compared to that of Vallecito Dam in Figure 5.12. It could be noted that
the sensitivity of B to 8 is relatively insignificant in case of Shelbyville Dam compared
with Vallecito Dam. This may emphasize the concept of problem size discussed in
Section 5.4 noting that failure surface length is 140.5 ft for Shelbyville Dam and is 355 ft
for Vallecito Dam. This may be also related to the fact that B approaches zero as FS

approaches unity regardless of the strength.

5.6 Total Stress Versus Effective Stress Conditions
This section provides a comparison between total stress and effective stress
analyses. Shelbyville Dam was analyzed for total strength conditions in Chapter 4 and for

effective strengths conditions in Section 5.5. The results are compared in Table 5.8

120

 

 

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Scale of Fluctuation (ft)

Figure 5.12 Effect of Scale of Fluctuation on Reliability Index
For Different Failure Surface Length (Lf).

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121

Table 5.8 Shelbyville Dam, End-of-Construction Conditions
Results of Total-Stress and Effective-Stress Reliability Analysis

 

 

Strength Conditions FS B Pr ()9
Total Strength Conditions, Q 2.805 3.22 0.000632
(UU) Tests 1 in 1582
Effective Strength Conditions, 1.044 .16 0.436
S (CD) Tests 1 in 2.3

 

 

It is obvious that, for this case, effective-stress analysis which considers variability in
pore pressure gives significantly higher probability of failure than total-stress analysis.
The probability of failure for end-of-construction for Shelbyville (effective-stress
analysis) indicates that failure is more likely to occur, which is consistent with the slide
reported for the dam (Humphery and Leonard, 1985). The location of the failure surface
is also closer to the reported slide compared with that obtained by total stress analysis. As
shown in Figure 5.13, total stress analysis suggested that the failure surface would be
tangent to the sand foundation whereas effective stress analysis suggested a failure
surface in the top area of the embankment which is much closer to the reported slide.
Also the lower value of B in effective-stress analysis compared with total-stress analysis
indicates that using total stress parameters and assuming that the variability of pore
pressure is implied in the variability of c may lead to inaccurate results. This is because

pore pressure in the laboratory may not perfectly simulate that in the field.

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5.6 Summary

The effect of uncertainty in the pore pressure ratio, r“, on end-of—construction
slope stability based on effective stress analysis was investigated. The pore pressure ratio,
ru, was considered as random variable. Uncertainty in pore pressure proved to have a
significant effect on the reliability of earth slopes. Performing reliability analysis without
considering uncertainty in pore pressure may result in an inaccurate reliability index.

The two cases of modeling ru as a perfectly correlated random variable with
systematic variance or as a spatially correlated random variable with reduced variance
provided a reasonable and consistent estimate of the reliability index. They gave results
close to that obtained by modeling variability in an actual grid of ru. For both cases, the
scale of fluctuation should be detemrined or reasonably estimated; because both the
systematic and the reduced variances depend on it. On the other hand, modeling ru as a
perfectly correlated random variable with total variance considerably overestimates the
reliability index.

The effect of scale of fluctuation on the reliability index is generally related to the
dimensions of the failure surface and to coefficient of variation of r“. For relatively small
length of failure surface and small coefficient of variation of ru, this effect is not very
significant. However, as a practical approach, in case of absence of enough data, a good
estimate of reliability may be obtained by assuming a reasonable value of scale of
fluctuation.

For the case of Shelbyville Dam, effective stress analysis provided much

reasonable results of probability of failure and location of failure surface compared with

124
total stress analysis. The effective stress analysis results, in terms of the probability of
failure and slip surface location, are in good agreement with a slide reported for the
embankment (Humphery and Leonard, 1985). This may be because effective stress
analysis usually contain much more information regarding pore pressure variation and

best simulates the field conditions.

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Chapter 6

SEARCH ALGORITHM FOR MINIMUM RELIABILITY INDEX

6.1 General

In probabilistic slope stability analysis, the concept of the relationship between
the surface of minimum factor of safety and that of minimmn reliability index has not
been studied in detail. Probabilistic analyses have often been performed on the critical
deterministic surface (see Table 1.1). The critical deterministic surface (of minimum FS)
and the critical probabilistic surface (of minimum F S) do not, generally, coincide. In any
slope, there are an infinite number of potential surfaces, each with different reliability
indices and, hence, different probabilities of failure. In evaluating probability of failure
of a slope all of these surfaces should be taken into account; this is obviously a very
complex procedure. Ideally, it is desired to have a computer program which searches for
Bmin . However, several slope stability analysis programs that locate only the critical
deterministic surface are well-established in practice. While improvements to these may
eventually be made, it is usefiil to explore what simple approaches may be available. In
this chapter, a simple technique to search for the minimum B surface utilizing an existing
slope stability program, UTEXAS3 (Edris and Wright, 1993) is developed and presented.
This technique is examined and applied to some case studies. First, a general approach is
hypothesized and tested in Sections 6.3 and 6.4.. Then improvements of the proposed

method are considered in Section 6.5.

125

126

6.2 Previous Work

A common approach to determine the reliability of a slope is based on minimizing
the factor of safety over the range of failure surfaces and then calculating the reliability
index, B, corresponding to this critical deterministic surface. Examples of this approach
include Tang et. a1. (1976), Vanmarcke (1977), Sharp et. al. (1981), Chowdhury and A-
Grivas (1982), Chowdhury (1984), Calle (1985), Wolff (1985), Wolff (1994), Li and
White (1987), Chowdhury et. al. (1987), Honjo and Kuroda (1991), Christian et. al.
(1994), and others.

Another approach is to consider specified surfaces of different radii which are not
necessarily associated with neither the critical deterministic surface nor the critical
probabilistic surface. Examples of that include Bergado and Anderson (1985), and
Chowdhury and Xu (1993). This approach is convenient only for the pmpose of
performing parametric studies.

Li and Lumb (1987) located the critical deterministic surface for a simple
homogeneous slope, and then used it as the initial trial surface to search for the surface of
minimum B. However, they did not mention the method of obtaining the minimum B

surface.

6.3 The Proposed Approach
The computer program UTEXAS3 (Edris and Wright, 1993) a typical
comprehensive slope stability program, is designed to locate the critical deterministic

surface for a given slope corresponding to specified soil strength parameter values. The

127

reliability index is conventionally calculated for that surface (BFS); but it may not be the
minimum value of B (Bmin) which may occur on a different surface. It was noted that
varying the soil strength parameter values for a given slope, as is done in several
probabilistic methods to estimate variances of FS while the search routine is enabled, will
result in different failure surfaces. A reliability index could be calculated for these groups
of surfaces (floating surface) and is denoted by Bf(Wolff et. al., 1995). Again, Bf may not
be the minimum B, and it may be argued that it does not have a physical meaning because
it is not calculated for a specific slip surface as required by the MFOSM method.

The proposed approach to locate the critical probabilistic surface is based on
systematically setting different combinations of soil strength parameter values and
searching for the critical probabilistic surface among the surfaces obtained by considering
these combinations. To obtain different parameter value combinations, an approach built
upon the FOSM (Taylor series reliability method, Wolff et. al., 1995) is used. Each
parameter value is changed by adding and then subtracting a specified increment from its
mean value while keeping the rest of parameters at their mean values. In addition to this,
the case of setting all the strength parameters at their expected values is also considered.
The increment was taken as one standard deviation at first, and other increments were
also examined.

An example is shown in Figure 6.1, The soil profile consists of two layers, each
with strength defined by one parameter c and five parameter value combinations are
considered. For each combination a critical slip surface is located. Figure 6.1 shows these

surfaces, where Srs is the critical deterministic surface, and S1 through 8,, are surfaces

128

   

 

 

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129

corresponding to cf ,c,‘ ,c)‘ , and c; respectively while the other parameter is at its mean
value. In other words, surface S 1 is located by searching for the critical deterministic
surface corresponding to the strength parameters: (lic1+0'c1) and (llcz) and surface S2 is
corresponding to (Her-Ger) and (ucz), and so on. Hence, the number of surfaces obtained

is equal to 2N+1, where N is number of soil parameters taken as random variables. For

each surface a factor of safety (FS,) and a reliability index (131) are calculated.
To search for Bmin, the values of B, are compared to BFS which is the reliability

index corresponding to the critical deterministic surface (F S = f(ucl, uc2)). The method is

based on expecting that the surface for which reducing the strength to a low value

achieves the lowest FS is likely to be a good predictor of the minimum B surface. Hence

the lowest B is expected to occur on the surface of lowest PS, The method is summarized

as follows:

0 A set of slip surfaces is obtained by searching out the critical deterministic surfaces,
each corresponding to a different combination of soil parameter values by adding and
subtracting one standard deviation to the mean value of each parameter while keeping

the rest at their mean values.

0 A reliability index (B,) for each of the obtained 2N+l surfaces is determined by fixing
each surface geometry and calculating the variance of FS for that surface numerically
by varying soil parameter values according to MFOSM (Taylor series) reliability
model.

0 The reliability index for the critical deterministic surface (BFS) is then calculated also

by using MFOSM method.

130

o The reliability index for the 2N+1 group of surfaces (Bf) is calculated (see Wolff et.

al. (1995)).

o The lowest B among the values of B, is taken as a measure of the slope reliability
(Bmin) and the surface associated with this value is considered the critical probabilistic
surface.

This approach was tested by applying it to two case studies as will be shown in the next

section. At this moment, each of the 2N+1 surfaces was located by adding and then

subtracting one standard deviation from the expected value of each soil parameter while
keeping the rest at their expected values. In other words the increment of varying each
parameter was taken as one 6. Generally, an increment Inc where “m” is an arbitrary
parameter needs to be used. The question of what is the best and most practical value of

“m” then arises. This issue will be introduced and discussed in Section 6.5 as an

improvement of the present approach. Also, the parameters were varied (above and below

expected values) one at a time, and each time a surface, Si, was obtained. In Section 6.5,

the idea of varying two parameters together will be evaluated. Also, varying parameters

only below expected values will be introduced in Section 5.4 and studied in Section 6.5.

Finally, it may be noted that Bf was first used for the purpose of comparison. However,

as will be discussed in Section 6.4, B, gave an idea about the variability in F S, and hence

the possibility of having a value of Bmin significantly lower than 131:5 as will be discussed

later.

131
6.4 Application to Case Studies
6.4.1 General
The proposed search technique is first tested on two case studies. The analyses are
limited to circular slip surfaces. However a non-circular analysis is provided later in this
chapter. The two case studies are:
0 Cannon Dam

0 Bois Brule Levee

6.4.2 Cannon Dam
Cannon Dam was previously described in Section 4.2. As shown in Figure 4.1, the
structure consists of two zones of compacted clay, Phase I fill and Phase II fill, founded
over layers of sand and limestone. For the end-of-construction condition, shear
parameters of the two clay layers were considered as random variables (c1, (p1, c2, (p2).
Values of statistical moments for these parameters are shown in Table 4.1. The Spencer
slope stability method is used in the analysis.
Using the method outlined in Section 6.3.1, nine different slip surfaces found by
program, marked A through I, were obtained and are listed together with their associated
factor of safety in Table 6.1. These surfaces are shown also in Figure 6.1. It may be noted

that surface C has the lowest factor of safety (PS). This surface is associated with using
the lower value of cohesion (cf ) for the Phase I clay fill.

To find B for each of the nine surfaces, they were then analyzed with fixed

geometry using the MFOSM reliability method. For each surface soil parameter values

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were changed while fixing the failure surface geometry. The values of reliability index,

Bi, for each surface are listed in Table 6.1. From this table the following are noted:

1.

The surface of minimum B coincided with that of minimum F 8, among the nine
surfaces. This supports the hypothesis in Section 6.3; additional examples will be
tested later in this section and the following sections.

The surface of Bmin (Surface C) lies completely in the two clay layers and is tangent
to the sand layer. The program located it when using the lower value of cohesion (cf )

of Phase I clay. The surface having the second-lowest B value (Surface G) was

located when using the lower value of cohesion of Phase II clay (c; ). Although the
Phase II soils are more variant than the Phase 1, Surface G has a higher B than Surface
C; because the effect of a low or uncertain strength is more significant near the base
of a circle and in the passive pressure region near the toe. Hence, this effect was less
significant for Surface G as it goes down to the sand layer. Figure 6.3 shows the
critical probabilistic surface (Bmin = 3.648) and the critical deterministic surface (BFS =
10.853).

A very significant difference between Bmin and l3rs may easily be noted (BFS = 10.853
versus Bmin = 3.648).

The floating surface reliability, Bf = 7.986, is significantly lower than Brs which
indicates that the variability of FS, over the set of surfaces, Si, is significantly greater
than that over the critical deterministic surface. 80, when Bf is significantly higher
than BFS a significant value of Bmin is more likely to be obtained as will be examined

and discussed in the following case study and examples.

 

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6.4.3 Analysis of Bois Brule Levee
6.4.3.1 Structure Description

The Bois Brule Levee is located along the west bank of the middle Mississippi
River in Perry County, Missouri, USA (Wolff et. al., 1995). It was selected because it is a
typical hydraulic structure for which reliability assessment may be required in economic
studies. The levee encloses a rectangular area approximately 15 miles long and 2 miles
wide. The northeastern long side faces the Mississippi River, and the southwestern long

side faces Bois Brule Creek. A cross-section adjacent to Bois Brule pumping station was

analyzed. This cross-section is shown in Figure 6.4.

6.4.3.2 Seepage Analysis

The levee was analyzed for the case of steady seepage conditions under a flood on
the levee to elevation 382.5. A seepage analysis was required to locate the piezometric
lines which would serve as input data for slope stability analysis. Seepage analysis was
performed by using the CSEEP finite element computer program (Tracy, 1973). This
problem contains two types of seepage:

o Underseepage: This is associated with horizontal flow in the sand layer and vertical
flow in the clay blanket. The total head in the sand layer at the base of the clay basket
was then modeled by piezometric line 1 (see Figure 6.4)

0 Through seepage: This is associated with clay embankment and represented by

piezometric line 2 (see Figure 6.4)

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The two piezometric lines were then modeled in slope stability analysis. It may be noted
that the uncertainty in pore pressure was not considered in this analysis; the piezometric
lines were treated as deterministic values. However, variability in steady-seepage pore

pressure is considered in Chapter 7.

6.4.3.3 Slope Stability Analysis

Consolidated-undrained strength parameters were used for the clay materials. All
the shear strength parameters were taken as random variables (c, and 4)] for clay
embankment, 02 and (1)2 for clay blanket, and 03 for sand foundation). The statistical
moments for these parameters (Wolff et. al., 1995) are shown in Table 6.2. The table

shows also the correlation coefficient, p, between c1 and 4)] (+0.5) and between c2 and 4'92

 

 

(+0.7).
Table 6.2 Bois Brule Levee, Moments of Soil Parameters
Parameter Mean Standard Coefi‘icient of Correlation
Deviation Variation Coeflicient
c, 860 psf 600 69.8 % + 0.50
(p, 18° 4.5° 25 %
c2 1000 psf 800 80 % +0.70
(b2 18° 6° 33.3 %
(p3 32° 2° 6.25 %

 

 

139

In this analysis, different deterministic models were used to investigate their effect on the
proposed approach. The following three deterministic models were used in slope stability
analysis:

0 Spencer Method

0 Simplified Bishop Method

0 Modified Swedish Method (US Corps of Engineers Method)

Circular-arc shaped slip surfaces were assumed, however, non-circular surfaces will be
analyzed and presented for both Cannon Dam and Bois Brule Levee. As described in
Section 6.3.1, soil parameters were set to their lower and upper values allowing the
program to search for a critical surface for each combination of shear parameters. Since
there are eleven different combinations of soil parameter values, the program located
eleven slip surfaces for each slope stability analysis method. The resulting slip surfaces
together with each parameter combination and their corresponding factor of safety F8, are
listed in Table 6.3. It may be noted that the surface of lowest F8, for all considered slope
stability methods corresponded to the lower value of cohesion for clay blanket, c2,
(Surface G). The different circular slip surfaces determined using the Spencer method are
shown in Figure 6.5. Most surfaces are close together except Surface G (corresponding to
the lower value for blanket cohesion) and Surface B (corresponding to the upper value of
embankment cohesion). Surface A, corresponding to mean values of all shear parameters,
represents the critical deterministic surface of minimum F S which is usually considered

in calculating B in conventional reliability analysis as discussed before.

 

 

140

Table 6.3 Different Slip Surfaces Found For Bois Brule Levee

 

 

A)Spencer Circular Method:
Strength Parameters Critical Circle
Surface c1 f1 c2 f2 1‘3 x y radius FSi
A 860 18.00 1000 18.00 32.00 70.2 71 90.4 2.552
B 1460 18.00 1000 18.00 32.00 68.6 74.8 96.2 2.770
C 260 18.00 1000 18.00 32.00 71.4 73.6 91.8 2.322
D 860 22.50 1000 18.00 32.00 69.8 70 89.8 2.586
E 860 13.50 1000 18.00 32.00 70.6 72.2 91.4 2.518
F 860 18.00 1800 18.00 32.00 69.4 71.6 95.6 2.960
G 860 18.00 200 18.00 32.00 66.4 66 79.6 1.826
H 860 18.00 1000 24.00 32.00 70.4 70.2 90.8 2.668
I 860 18.00 1000 12.00 32.00 69.8 72.2 90.6 2.436
1 860 18.00 1000 18.00 34.00 70.4 72 90.6 2.643
K 860 18.00 1000 18.00 30.00 69.8 70.2 90.8 2.459

 

 

 

 

 

 

 

 

 

 

B)Simplified Bishop Method:
Strength Parameters Critical Circle
Surface c1 fl c2 f2 B x y radius FSi
A 860 18.00 1000 18.00 32.00 70 70.6 90.2 2.538
B 1460 18.00 1000 18.00 32.00 68.4 74.2 95.8 2.759
C 260 18.00 1000 18.00 32.00 71.2 73.2 91.4 2.304
D 860 22.50 1000 18.00 32.00 69.8 69.2 89.2 2.574
E 860 13.50 1000 18.00 32.00 70.4 71.8 91 2.500
F 860 18.00 1800 18.00 32.00 69.4 71.2 95.4 2.949
G 860 18.00 200 18.00 32.00 66.4 65.8 79.4 1.817
H 860 18.00 1000 24.00 32.00 70.2 69.6 90.2 2.653
I 860 18.00 1000 12.00 32.00 69.6 71 .4 90 2.422
I 860 1 8.00 1000 1 8.00 34.00 70.2 70.4 90 2.627
K 860 18.00 1000 18.00 30.00 69.8 70.2 90.8 2.447

 

 

 

 

C)Modified Swedish Method:
Strength Parameters Critical Circle
Surface c1 f1 c2 12 1‘3 x y radius FSi

 

 

860 18.00 1000 18.00 32.00 67.8 75.2 85.8 2.577
1460 18.00 1000 18.00 32.00 67.6 80.4 99.6 2.937
260 18.00 1000 18.00 32.00 69.6 72.8 83.4 2.187
860 22.50 1000 18.00 32.00 67.4 73.4 84 2.636
860 13.50 1000 18.00 32.00 68.2 76.8 87.4 2.518
860 18.00 1800 18.00 32.00 68.2 76 97.2 3.154
860 18.00 200 18.00 32.00 65.6 68.4 81.8 1.874
860 18.00 1000 24.00 32.00 69 73.2 91.8 2.786
860 18.00 1000 12.00 32.00 67.8 75.8 86.4 2.467
860 18.00 1000 18.00 34.00 68.4 75.6 86.2 2.635
860 18.00 1000 18.00 30.00 67.8 77.6 96.6 2.573

 

 

 

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141
6.4.3.4 Reliability Analysis

As in the case of Cannon Dam, the MFOSM reliability method was applied to
calculate [3 for each of the above-mentioned failure surfaces. Table 6.4 shows, for each
surface, the expected values of factor of safety, E[FS], factor of safety during search, FSi,
reliability index, 0,. The reliability index associated with the critical deterministic surface
(Surface A) is 91:3-

The results indicate that surface G gives the lowest value of B for all slope
stability methods considered. This surface (Surface G) is associated with the lowest value
of F8, as was noted in the analysis of Cannon Dam. This surface corresponded to the
lower value of clay blanket cohesion (see Table 6.3). As shown in Figure 6.5, this surface
does not go down in the sand layer as the other surfaces do, it rather stops tangential to
the thin clay layer. In other words, surface G avoided the soil layer of least variance
(sand foundation) and maximizes length in layers of greater variance (Clay embankment
and blanket). This suggests why this surface is associated with the lowest reliability
index.

The reliability index associated with Surface A is that for the critical deterministic
surface (BPS). The reliability index for floating surface (Bf) is also calculated for each
slope stability method. Table 6.5 gives the values of Brs , 13min, and Bf for the different

methods of slope stability.

142

 

 

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Table 6.5 Bois Brule Levee, Values of BPS, BF, and [3min

 

 

Slop Stability Method BFS Bf 0",,"
Spencer Circular 3.78 3.33 2.32
Simplified Bishop 3.73 3.29 2.31
Modified Swedish 2.97 2.74 2.40

 

 

F rom these results the following may be noted:

0 The surface of lowest value of reliability index (13min) coincides with that of minimum
factor of safety (FSi) among all surfaces obtained by systematically using different
combinations of high and low strength parameter values.

0 The difference between [3min and firs is significant, although it is less significant than
that for the case of Cannon Dam.

o The lowest-B surface is obtained by applying lower value of a strength parameter,
which leads to the modification described in the following section.

Figure 6.6 shows the critical deterministic surface and that the critical probabilistic for the

Spencer method.

 

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6.5 Improvement of The Method
6.5.1 General

In this section some improvement of the method is developed and presented.
Based on the results obtained in Section 6.4, the surface of lowest B was located among
the surfaces obtained by using a lower value of a shear parameter while keeping the rest
at their mean values. Hence, instead of considering all combinations of shear parameter
values the lower value combinations only need to be considered. This reduces the number
of search trials to about half the number considered before.

In Section 6.4, the lower value for any considered strength parameter was taken as
the mean value minus one standard deviation. In general, this lower value, Lx, may be

expressed as:

Lx = 14x "max (6.1)

where m is an arbitrary parameter. It is required to determine the optimum value of m
that best locates the critical probabilistic surface. Three values will be examined for this
parameter: 0.5, 1.0; 1.5. It may be noted that changing each parameter lower value will
be done during the search process only. The usual lower and upper values for the FOSM
method (trio) will still be used in the calculation of [3.

To further test the present approach and to investigate possibility of finding other
surface(s) of lower 0 than the one located by the approach, some surfaces around that

surface are examined. The improvements can be summarized as follows:

147

0 Considering surfaces obtained by using lower values of strength combinations only.
0 Examining different values of the parameter m: 0.5, 1.0, and 1.5.

o Examining surfaces around that of pm.

The above proposals are applied to:

o Cannon Dam: end-of-construction, modified Swedish method.

0 Cannon Dam: end-of-construction, simplified Bishop method

0 Parametric studies.

6.5.2 Cannon Dam, Modified Swedish Method
6.5.2.1 Considering Only Surfaces of Lower Values of Strength Parameters (u - o)
The same analysis done in Section 6.4.2 was repeated using the modified Swedish
method rather than Spencer’s method. Four slip circles were obtained: one by applying
mean values of shear parameters, and three by applying a lower value of each shear
parameter (p. - 0) while keeping the rest of parameters at their mean value. Figure 6.7
shows the slip circles together with their corresponding factor of safety F8, and reliability
index [5,. It can easily be noted that the slip circle of lowest [3, coincides with that of
lowest factor of safety, FSi, (Surface B). As found also in Section 6.4.2, this circle is
located completely in clay layers and does not go down into the sand foundation as other

circles do.

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149

6.5.2.2 Considering a Lower Value of Strength Parameters of u-0.50' (m = 0.50)

Based on the previous discussion in Section 6.5.1, only lower values of strength
parameter combinations need to considered together with the mean values, and the
parameter “m” takes a value of 0.50. In other words, the structure is analyzed for the
cases of using mean values of strength parameters in addition to using a lower value of
mean minus half standard deviation (1.1-0.56) for each parameters while keeping the rest
at their mean values. Figure 6.8 shows the resulting slip circle together with
corresponding F8, and [3, values. The surfaces are located very close together because the
difference in soil parameter values is small. The resulting surfaces are very similar to the
mean-values surface (critical deterministic surface). The lower reliability index in this
case is higher than that in the case of m = 1.0 (see Section 6.5.2.1 and Figure 6.7). This
suggests that adopting an increment of half standard deviation (m = 0.5) during the search

process would not be sufficient to locate a surface of lower [3 than in the case of m = 1.0.

6.5.2.3 Considering a Lower Value of Strength Parameters of u-l.5 o (m = 1.5)

The same procedures were applied here; except that the parameter m took the
value of 1.50, i.e. the lower value became (u - 1.50). The resulting slip surfaces and their
PS, and 0, values are shown in Figure 6.9. The surfaces locations and [3, values are very
similar to these of ordinary lower value (m = 1.0), but the lower 0, is still greater (3.690
versus 3.689). This means that searching in a wider range of parameter value did locate a
surface of lower [3. Other values of m will be considered for Cannon Dam and other

examples later in this chapter (see Figure 6.24).

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152

6.5.2.4 Searching Around The Minimum-B Surface

Some surfaces located above the minimum-B surface (circle B in Figure 6.7) are
considered. The surfaces have been chosen above circle B because surfaces below it will
go into sand layer and hence give higher B (as circles A, C, and D). The considered
surfaces together with corresponding E (FS) and B values are shown in Figure 6.10.
Surfaces G and F are located completely in the Phase II clay which has the largest
coefficient of variation for both cohesion and fiiction. In spite of that these two surfaces
have a higher value B (5.29 and 4.582). Surface H also does not pick up a lower value of
B. This indicates that surface B located by the proposed search process still gives the

minimum B among all tested methods.

6.4.3 Analysis of Cannon Dam, Simplified Bishop Method

Since no gain in accuracy was obtained by changing the lower value of strength
parameters during the search process, only the usual increment of one standard deviation
(m = 1.0) was applied. The four surfaces obtained and the corresponding factor of safety
PS, and reliability index B, are shown in Figure 6.11. As found in the previous cases,
surface B of lowest B, is the surface of lowest F 8,. This surface is also tangent to the sand
foundation which is the most reliable material within the cross-section.

Some other trial surfaces were examined. Figure 6.12 shows these surfaces
together with surface B for comparison. The reliability index B, for each surface is also
shown. No surface other than surface B was found to have a lower value of B. Hence, it

may be noted that the proposed search algorithm proved to provide a good estimate of

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minimum reliability index for an embankment and the choice of deterministic model
(Corps or Bishop) has no practical effect on the results.
Table 6.6 shows a comparison between BFS, Bf, and Bmin of Cannon Dam for all

slope stability methods considered in this section.

Table 6.6 Cannon Dam, Values of BFS, Bo and Bmin

 

 

Slop Stability Method B FS Bf BM."
Spencer Circular 10.85 7.23 3.65
Simplified Bishop 10.36 7.45 3.99
Modified Swedish 10.28 7.08 3.69

 

 

It may easily noted that the difference between BPS and Bmin is very significant for all
methods of SIOpe stability analysis. The values of Bf are lower than the values of BFS for
all cases considered. Figure 6.13 shows the critical probabilistic surfaces for the different

deterministic models considered.

6.4.4 Parametric Studies
6.4.4.1 Application of The Proposed Search Method

To determine the applicability of the proposed method to a more general range of
slope stability problems, a series of parametric studies was performed on a set of two-
layer slopes. The general cross-section is shown in Figure 6.14. The parametric study was

done by specifying various values for the cohesion c and The angle of internal fiiction ¢.

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///////////////////////////////77/////;

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Cross-Section

Soil Parameters:

 

 

Soil 1 Soil 2 Soil 1 Soil 2
Case c1 psf ¢1 deg e2 psf 01 deg Vc1 % V¢t % Vc2 % V412 %
A 800 o 5‘00 12 20 - 0 10
B 800 o 500 12 4o - 20 10
c 800 o o 30 20 - - 10
o 800 o o 30 4o - - 10
E 500 12 800 o 20 1o 20 -
F 500 12 800 o 40 1o 20 -
G 500 12 o 30 20 1o - 10
H 500 12 o 30 4o 20 - 20
1 0 so 800 o - 1o 20 -
.1 o 30 800 o - 20 4o -
K 800 o 800 0 4o - 20 -
1. 800 o 800 o 20 - 4o -
M 800 o 1000 o 40 - 20 -
N 500 12 1000 0 4o 20 1o -

 

 

 

 

 

 

Figure 6.14 Cross-Section and Soil Parameters For a Typical Earth Slope

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159

The coefficient of variation for each strength property was also systematically changed as
shown in Figure 6.14. Pore pressure was assumed zero in all cases. The properties were
chosen for each layer to represent either clay (total stress), c-d) soil (effective stress), or
sand (4) = 0). The same search technique was applied considering m = 1.0. The results are
shown in Figures 6.15 through 6.21. For each case, the figures show F8, and B, for each
located surface. The values BFS, Bf, and Bmin are also shown. These results indicate that:

1. As found previously for Cannon Dam and Bois Brule Levee; the surface of minimum
B coincided with that of lowest FSi for each of the present examples.

2. The difference between BPS and Bmin is significant where Bf is significantly lower than
Brs- This result can be noted also for Cannon Dam and Bois Brule Levee. This
indicates that although Bf is not associated with a particular slip surface and there
may be some controversy over its physical meaning, it may be a efficient indicator of
the possibility of finding a surface of significantly lower B that of the critical

deterministic surface.

6.4.4.2 Systematic Search

A systematic search for different surfaces of different center around the minimum
B surface was also performed for Case M. The process is based on search along a square
grid of centers separated by 2 foot as shown in Figure 6.22. The results are also shown in

Figure 6.22. No surface of lower B was located during this systematic search.

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6.4.4.3 Setting More Than one Parameter to a Lower Value

In all the previously discussed analyses only one strength parameter was set to its
lower value (p - m0) to search for the critical probabilistic surface. However, two
additional combinations of strength parameter values were considered for Case A. One of
these two combinations was specified by setting two strength parameters (c1 and oz) to
their lower values. The other combination was specified by setting three parameters (cl,
c2, and ¢2) to their lower values. The results are shown in Figure 6.23. It could be noted
that setting only one strength parameter to its lower value resulted in the lowest value of
[3min This may be due to the fact that lowering only one parameter in each search process
results in locating the most dominant parameter, i.e., the parameter which has the greatest

influence on B.

6.6 Effect of Changing The Parameter m

As described in Section 6.5 .1, the parameter m represents the number of standard
deviations to be deducted from the mean value of a parameter to obtain the lower value
used in the proposed process. In Sections 6.5.2.2 and 6.5.2.3 two values of m (m = 0.50
and m = 1.50) were examined for Cannon Dam in addition to the case of m = 1.0. In this
section some other values of m were examined for Cannon Dam, Bois Brule Levee,
Example A, and Example N. The m value ranged from 0.50 to 1.75. Figure 6.24 shows a
relationship between m and B for each case. It can be noted that B approaches a minimum
value in the range 0.5 < m < 1.0, and the assumption of m = 1.0 is justified as a practical

value. Values of B are approximately constant for m higher than 1.0.

165

 

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Figure 6.24 Effect of The Parameter m on [3min

166

6.6 Discussion
As described in Chapter 2, the reliability index (for the case of no parameter

correlation) based on the mean-value F OSM model is given by

 

 

_ qu —1 _ FS(uxi)—1 _ FS(ucl,uh,...)—l 6 2
B"'EIS—_ an 2 ‘ Z(Fs,-Fs_) (')
Jig?) 0;. 2

where F S. is the factor of safety associated with applying an upper value (p + 0) of each
soil parameter respectively and FS_ is associated with the lower value (p - 0). [5min is the
minimum reliability index corresponding to any analyzed surface. To locate this surface,
strength parameter values are set to specific combinations and the factor of safety, F 8,, is
calculated for each combination as described in Section 6.3.1. During this search process,
the lowest F8, is associated with the combination of parameters for which the most
significant soil strength parameter is set to a low value within its range of likelihood.
Along the surface obtained by this parameter combination the sum (FS(XL). - FS(XL),)/2
is the greatest among the surfaces associated with other, stronger combinations. In other

words, the variability of the factor of safety along this surface is maximum:

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This makes the B associated with that surface the lowest value among the considered
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When Bf is significantly lower than BFS, this indicates that the variability in F S, is
greater than the variability in F8 along the critical deterministic surface. This indicates

that a significantly low Bmin could be obtained.

6.7 Further Applications
6.7.1 General

Having originally developed the method using two actual embankments and having
checked its validity with the set of parametric studies, it was applied to some additional
cases previously analyzed in the literature. The cases were chosen such that they include a

variety of structures: a homogenous slope, a dam, and a layered open cut.

6.7.2 Analysis of a Homogeneous Slope

A homogeneous slope analyzed previously by Li and Lumb (1987) was examined
by the proposed search method. The geometry of the slope is shown in Figure 6.25. The
soil parameters were taken the same as those presented by Li and Lurnb (1987). These
parameters together with their coefficient of variation are given in the following Table

6.7.

168

Legend
__... Slip Circle with minimum F.S.

—— Slip Circle with minimum

 

 

 

10m

 

 

 

Figure 6.25 A Homogeneous Slope, After Li and Lumb
(1987), Location of Critical Surfaces

 

Sli
dw
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Star
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169

Table 6.7 Example Homogeneous Slope (after Li and Lumb (1987))
Moments of Soil Parameters

 

 

Parameter Mean Coefficient of Variation
c’ 18 kPa 20 %
g 18 kN/m" 5 %
tan ¢' tan 30° 10 %
r“ 0.20 10 %

 

 

Note that pore pressure in this case is modeled by providing a value of pore pressure

coefficient, ru , where:

r = ———- (6.3)

Slope stability analysis was done by Spencer circular method. Five circles were obtained
during the search process (one corresponding to mean values and four corresponding to
lower value of soil parameters) The lower value for density, 7, was set to the mean plus
standard deviation because that should result in a lower factor of safety. Increasing
density will increase driving moment and, hence, decrease PS. The mean value FOSM
reliability method was used to calculate B for each circle. The F8, and Bi, for each slip

surface are shown in Table 6.8

170

Table 6.8 A Homogeneous Slope (after Li and Lumb (1987))
B For Different Failure Surfaces

 

 

Surface FS, [3,.
A 1.334 2.336
1.190 2.293
C 1.271 2.381
D 1.300 2.324
E 1.360 2.336

 

 

The results indicate that the proposed method worked quite well for this case. The surface
of minimum [3, is that of minimum F83. The critical deterministic surface (Surface A)
and the critical probabilistic surface (Surface B), are shown in Figure 6.25. It should be
noted that the surfaces are close together which is consistent with what Li and Lumb
found. The difference in [3 values (2.293 versus 2.336) is not significant as should be
expected since there is only one layer of soil and chances of obtaining combinations of
weak (less reliable) zones are non-existent.

It should be noted that Li and Lumb (1987) obtained a B value of about 2.5 based
on Hasofer-Lind (1974) definition using perfectly correlated soil parameters. This value

compares very well with the values obtained here.

171

6.7.3 Analysis of Shelbyville Dam

Shelbyville Dam was analyzed in Chapter 4 and Chapter 5 for the case of end-of-
construction. Both total and effective strengths were considered. It was noted that the
effective stress analysis gave the most critical reliability index. Hence, only effective
stress analysis was considered here. Three different surfaces (Surfaces B, C, and D) were
obtained in the search process by applying the lower value of cohesion and friction, and
the higher value of pore pressure ratio in addition to the critical deterministic surface
(Surface A) obtained using mean values of all parameters. These four surfaces are shown

in Figure 6.26 and the results of reliability analysis are shown in Table 6.9.

Table 6.9 Shelbyville Dam, Results of Search Process

 

 

Surface F S , [3,. Pr. (f)
A 1 .044 0.087 0.465
B 0.940 0.106 0.458
C 0.967 0.208 0.418
D 0.915 0.090 0.464

 

 

It is clear that all values of factor of safety, reliability index, and probability of failure are
very close together since all surfaces are near failure and, as noted in Chapter 5, B
approaches zero as FS approaches unity. Here, the lowest [3 occurred on the critical
deterministic surface (Surface A) rather than on that of lowest FSi (Surface D). However,

the difference is very small and practically negligible.

172

 

 

 

\ /

Figure 6.26 Shelbyville Dom, Slip Surfaces Obtained by

The Search Process, Spencer’s Method

173

6.7.4 Analysis of Congress Street Open Cut

The Congress Street Cut, Chicago, Illinois, USA, was analyzed by Ireland (1954),
Tang et al (1976), Oka and Wu (1990), and Chowdhury et. a1. (1994). The cross-section
of the cut is shown in Figure 6.27. This slope was chosen because it has many soil layers
and parameters, and has a relatively low coefficient of variation compared to the non-
homogeneous embankments studied before. The soil profile consists of a sand layer and
three clay layers. As modeled by Chowdhury and Xu (1994), the soil parameters of clay
layers were considered random while the sand friction and density were considered
deterministic due to their small contribution to stability in this case. The statistical
moments of the clay layer shear parameters and densities are listed in Figure 6.27.
Spencer’s circular method was applied in the analysis. The lower values of the parameters
(9 different combinations) in addition to the mean-values case were considered. It may be
noted that the upper value (p + 0‘) of density was applied whenever this reduced the factor
of safety, FSi. Due to the small variability of the parameters; the search process resulted
in only three different surfaces (instead of nine). This was because some different
combinations of parameters resulted in the same failure surface. The three failure surfaces
are shown in Figure 6.27 together with the associated F Si and Bi values noting that
Surface 1 represents the critical deterministic surface.

Again, the surface of lowest Bi (Surface 3) coincided with that of lowest F 8,. The
difference between Brs and [5min is not significant and the critical deterministic and
probabilistic surfaces are located very close to each other. This is due to the small

variability in soil parameters especially in Clay 2. Hence, the method did not locate a

Soil Parameters

174

 

 

 

 

 

 

 

 

 

 

      

 

 

 

 

 

Clay 1 Clay 2 Clay 3
Parameter
7(kN/rn’) (0' c‘(kPa) 7(kN/m’) «2' c'(kPa) 7(kN/m’) ¢’ c'(kPa)
M 19.5 5 55 19.5 7 43 20 15 56
a 2.34 1 20.4 2.34 1.5 8.3 3 3 13.2
Surface FSi 5,
1 1.358 2.156
16 2 1.306 2.192
F 3 1.198 2.149
_,2 SAND
E _8 CLAY 1
’- I
<2
’—
< 1-4
a. CLAY 2
_l
.. l
l-o I
CLAY 3
4o 32 24 16 a o —8
L J l l l l l

 

Figure 6.27 Congress Street Cut, Slip Surfaces Obtained

by Search Process, Spencer’s Method

HORIZONTAL DISTANCE (m)

 

175

surface tangent to Clay 3 which might be the case if the variability in Clay 2 is larger than

mama”;

6.7.5 Non-Circular Failure Surfaces

In this section, the method is tested on non-circular failure surfaces. Two
structures were analyzed; Cannon Dam and Bois Brule Levee. Figure 6.28 and 6.29 show
the resulting failure surfaces together with their associated factors of safety F Si. As was
noted for circular surfaces, the minimum B surface is the one with lowest FSi. Figures
6.30 and 6.31 show the critical non-circular deterministic and probabilistic surfaces for
Cannon Dam and Bois Brule Levee respectively. Table 6.10 shows the values of Bf, BFS,

and Bmin for both embankments.

Table 6.10 Comparison of B” BFS, and Bmin For Non-Circular Surfaces

 

 

Structure B, B rs BM,
Cannon Dam 4.189 7.028 2.664
Bois Brule Levee 3.532 3.813 2.527

 

 

The value the floating surface reliability index Bf was significantly lower than BFS for
Cannon Dam, and the difference between Brs and Bmin was also significant. For Bois
Brule, the differences (BPS-Bf) and (BFS-Bmin) were less significant. It may also be noted

that Bmin of the non-circular surface was less than that of the circular surface for Cannon

I76

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180
Dam (2.664 versus 3.648). This was not the case for Bois Brule Levee, (2.527 versus
2.310). This indicates that both circular and non-circular surfaces should be considered in

the analysis.

6.7.6 Application to AFOSM (Hasofer-Lind)

The surface of minimum B was located in the previous sections by a searching
process based on the MFOSM method. In this section the AFOSM (see Chapter 2) was
applied and the Hasofer-Lind reliability index, BHL, was calculated for both the critical
deterministic and critical probabilistic surfaces. The method was applied to Case N of
Typical Examples (Section 6.4.4). This example was chosen because it was one of the
examples which gave a significant difference between Brs and Bmin The performance

function was taken as:

g(x) = FS(x) —1 in the x-space, and

h(z) = FS(z) -1 in the z-space,

where z =

The partial derivatives of the performance function with respect to 2 were calculated
numerically by considering an increment, AZ = 0.10, and the design point and reliability
index converged after 3 iterations as shown in Appendix D. The results are given in Table

6.11

 

 

 

Tat

met
betwec'
B rega
smalls
compl
MFOE

AFOE

6.8 S

 

181

Table 6.11 Value of Mean-Value FOSM and Hasofer-Lind Reliability Indices

 

 

MFOSM AFOSM
13,, 4.764 3.847
pm," 1.961 1.564

 

 

From these results it may be noted that AF OSM method also gave a significant difference
between Brs and Bmin which emphasizes the need of searching for the surface of minimum
B regardless of the reliability method used. It may also be noted that Bmin of AF OSM is
smaller than Bmin of MFOSM. Because AF OSM method requires much more
computational effort, it may be adequate to the perform the search process using the
MFOSM method, and then calculate B for the critical probabilistic surface using the

AFOSM method.

6.8 Summary

A reasonably simple search technique for determining the minimum reliability
index is presented. This technique should assist geotechnical engineers in finding Bmin
using only a deterministic slope stability program. This technique is based on obtaining
different candidate surfaces by applying offset-values of each of the random variables
considered in the analysis while keeping the rest parameters at their mean values. The
case of considering the mean values for all parameters is also considered. This brings the
number of analyzed strength value combinations to N + 1, where N is the number of

random variables. The offset values are taken as u 1- ma, where m is an arbitrary

 

 

param
presa
lower
differ
surfa

estirr

3st

2A

182

parameter. The value of m should be negative for strength parameters, positive for pore

pressure, and positive or negative for density depending on which value will give the

lower FSi. Each combination of soil parameters gives a different slip surface with a

different factor of safety (FS,) and reliability index “31)- Among these surfaces, the

surfaces which gives the lowest value of F 8, usually can be shown to be a very good
estimate of the surface of lowest reliability index (Bmin).

In the calculation of B three different values of reliability index may be obtained
as follows:

1. The BPS value is calculated for the critical deterministic surface which is located by
keeping all soil parameters at the mean value. This value is not generally the lowest
value of B and a significantly lower value could be obtained.

2. A “mixed” measure of slope reliability, using mixed or “floating” surfaces (Bf) is
obtained by using F8, for the lower and upper values of each soil parameter
considered as a random variable. In other word, Bf is not calculated for a specific
surface since each value of F S, is associated with a different surface. Changing the
slip surface obviously changes the form of the performance function which leads to Bf
being of no physical meaning. In spite of that, Bf gives an idea about the variability in
FSi, and hence acts as an indicator as to whether there may be a surface of
significantly lower B than the critical deterministic surface. It was noted that when Bf
is significantly lower than BPS, the value of Bmin is significantly lower than BFS.

3. Bmin is the lowest value of reliability index and could be calculated by the previously

described search technique.

 

 

183
The presented search technique proved to work well in finding a relatively
accurate estimate to the critical probabilistic surface. It was examined over several case
studies including dams, levees, and homogeneous slopes. Different loading conditions
and strength modeling were also considered. These include: total stress analysis of end-
of-construction conditions, effective stress analysis of end-of-construction conditions, and

steady seepage conditions. The numerous analyses presented in this chapter lead to a

number of significant observations:

0 Surfaces found in calculating Bf provide a key to the possible surfaces among
which the critical probabilistic surface may be located.

0 Only surfaces obtained by combinations using values that give lower FSi (u i ma)
for each soil parameter need to be considered in the search process.

0 Taking m = 1.0 appears to be a practical approach.

0 Using parameter combination corresponding to values for two or more parameters did
not locate a critical probabilistic surface, because that did not emphasize the single
dominant parameter that usually picks up the critical surface.

0 Surfaces considered near that of Bmin obtained by the proposed method did not lead to
revision of Bmin

o The difference in the values of B can be significant, especially for stratified
embankments and in case of high coefficient of variation of soil parameters. In these
cases, calculating the reliability index over the critical deterministic surface (BFS) may
significantly overestimate B and hence result in an unconservative probability of

failure.

 

184
o For homogeneous embankments, and/or when the coefficient of variation of soil
parameters is small, the critical deterministic and the critical probabilistic surfaces are
located close to each other. No significant difference between BFS and Bmin was found

in these cases.

Chapter 7

PROBABILISTIC STEADY-STATE SEEPAGE ANALYSIS

7.1 General

In analyzing the reliability of dams and levees against seepage as well as
determining pore pressure for steady-seepage stability analysis, a dominating variable is
the hydraulic conductivity, k, which is spatially variant. This spatial variability influences
the values of pore water pressure throughout the embankment, and hence affects the
stability. The hydraulic conductivity usually has a high coefficient of variation, typically
exceeding 30 percent and in some cases exceeding 100 % (Wolff et. al., 1995). Hence,
hydraulic conductivity is generally modeled in the literature as lognormally distributed
(e.g. Bergado and Anderson, 1985) due to its high coefficient of variation and the fact
that it can only assume positive values.

In this chapter, the problem of modeling the variability of the pore pressure
distribution throughout an embankment due to spatial variability of hydraulic

conductivity is investigated. The cases studied include the effects of the following on

pore pressure distribution:

0 Coefficient of variation of hydraulic conductivity, k,

0 The scale of fluctuation, 5, of the hydraulic conductivity.

0 Anisotropy in the correlation structure of k.

Seepage analysis was performed under the assumption that k is a spatially correlated

random variable. To model k as a spatially correlated random variable, the nearest-

185

 

 

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The

186

neighbor (Bartlett, 1975) was used to generate random values of k. The relaxation method
was utilized in applying the nearest-neighbor method as will be discussed later in this
chapter. The finite difference method solved iteratively (by relaxation method) is used to
solve for steady-state seepage condition through a pervious levee. The relaxation method
provided a practical simple technique which was easily applied by spreadsheets. The
Monte Carlo approach was applied to study the variability and probability distribution of
pore pressure due to the variability of k.

Finally, the effect of variability in pore pressure on slope reliability was
investigated. Two case-studies were considered: a typical levee, and the Bois Brule levee.
Slope reliability was assessed for steady seepage conditions assuming the pore pressure u

as a random variable.

7.2 Spatial Variability of Hydraulic Conductivity

The hydraulic conductivity of soil naturally varies from point to point in an
apparently random fashion. This led to the adoption of geostatistics and the modeling of k
as a random field (Delhomme, 1979; Neuman, 1984; Neuman, and Orr, 1993). To model
a parameter as a random field the following should be specified:
0 A probability distribution, or at least its first two moments; mean and standard

deviation

0 An autocovariance function, ACF
- A scale of fluctuation, 5

These items were presented and discussed in Chapter 2.

 

 

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dis

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Sc

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mc

prc

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187

7.3 Modeling The Spatial Variability of k
7.3.1 Approaches to Probabilistic Analysis

Two techniques for performing probabilistic seepage analysis involving spatial
variability for k are recognized in the literature: Monte Carlo Simulation and the
analytical approach. In the Monte Carlo approach, numerous repeated simulations are
made to solve Laplace’s equation over the field each time using a value of k modeled as a
random field. In the analytical approach, the flow equation is solved taking k as a random
variable and then analytical expressions for the moments of output head are derived. A
major advantage of the Monte Carlo technique over analytical techniques is that the entire
probability density function of head (and hence pore pressure) is estimated whereas the
analytical methods yield only moments (mean and covariance) of output head. The Monte
Carlo technique is associated with a random field generator; i.e. generating spatially
correlated values of hydraulic conductivity. One of the methods used as a random field
generator is the matrix method in which the field is represented only at a number of
discrete points or blocks. The nearest-neighbor method (Bartlett, 1975) is an example of a
matrix method. It was applied to ground water flow (Smith and Freeze, 1979; Smith and
Schwartz, 1980) and was applied by Bergado and Anderson (1985) to steady-state
seepage through earth dams. In the present study, the nearest neighbor method is
modified, coupled with the relaxation method, and applied by spreadsheets. This was
proven to be a practical approach that reduced the computational effort as will be

discussed in the following sections.

 

01
In
at a

(lest:

 

188

7.3.2 The Nearest Neighbor Method

As described by Smith and Freeze (1979), the nearest neighbor method is one of
the methods used to model spatial variability in the hydraulic conductivity of soil as a
statistically homogeneous random field. Statistical homogeneity, noted also as
stationararity in the literature, assumes that the hydraulic conductivity has the same
expected value at every point in random field, i.e. it fluctuates around a constant average
with no trend. The covariance between hydraulic conductivity at any two points depends
only on the vector separating those points and not on their absolute position.

To generate spatially correlated values of k, the flow domain is divided into a
grid of square blocks and the correlation structure is represented by a first-order nearest
neighbor process. For the block i,i (see Figure 7.1) the nearest neighbor process is

modeled by the following equation:

Y1.) = “xvi-1.1 + K+1,j)+ 0t :(YlJ-l + XJH) + 31,} (7.1)
where,
am : is a normal uncorrelated random variable
or, and az: are autoregressive parameters expressing the degree of spatial dependence

of YiJ on its two neighbors
In words, the hydraulic conductivity value at a point is a weighted average of the values
at adjacent points, and a random error term. It may be noted that Equation (7.1) only

describes the nearest neighbor process but the detailed sequences for generating the

189

 

 

Yi—1,J'

Y1+1,J'

 

 

 

Yi,J-1

 

 

 

Figure 7.1 Schematic Representation of

Nearest-Neighbor Grid

2L

X

 

 

desired
holds 1
covari:
values

anisov

where

(see S]

Where

The 1

 

relate
gene

51am

 

190

desired random field are presented later. For a statistically homogeneous meditun, (7.1)
holds for every block within medium. If orx and ory are equal, the medium has an isotropic

covariance structure. If 01,( and orz are not equal, the covariance between conductivity
values depends upon the direction in which it is calculated; i.e. the medium has an

anisotropic covariance structure. In a matrix form Equation (7.1) can be written as:

{Y} =[WllYl +{8} (7.2)

where the matrix [W] is a spatial lag operator of scaled weights wk. which are defined by

(see Smith and Freeze, 1979):

W1] = w”. /r i¢j (7.3)

where
W.” = orx if blocks i and j are contiguous in the x direction,

w‘ij= orz if blocks I and j are contiguous in the z direction,
w.“ = 0 otherwise
r = total number of contiguous blocks surrounding block i.
The matrix [W] indicates which conductivity values in the block system are linearly
related to each other. The value r is required to preserve statistical homogeneity in the

generated sequence. It should be noted that it is required to simulate a predetermined

standard deviation Cy. Starting from the random values {a} with zero mean and standard

de

de

 

for

The

. 1

' l

The
one

lnfir

191

deviation of one, these values are premultiplied by an appropriate factor n to yield the

desired value of Cy. This factor is applied to Equation (7.2) to yield:

{Y} =[WllYl +1118} (7.4)

Equations for this premultiplier factor are derived by Smith and Freeze (1979). Solving

for vector [Y] yields:

{Y}=(lll-lW1)"n{e} as)

The nearest neighbor method may be summarized as follows:

0 First a set of normal uncorrelated values, {a} of zero mean and unit standard
deviation are generated at each point in the field modeled: {a} = N[0,1].

0 These values are filtered by Equation (7.5) and at the same time multiplied by 11 to
result in internally correlated normal random conductivity values: {Y1} = N[O,Cy]

o The desired mean try is then added to the generated values resulting in the internally
correlated values: {Y}= N[py,0'y].

0 Finally these values are transformed to lognormally distributed values of k by k =
EXP (Y) to yield {k} = LN[uk , ck ]

The generated values of k are called a realization of the hydraulic conductivity. It is just

one of an infinite number of possible realizations which could be generated. This set of

infinite number of possible realizations is termed an ensemble. The concepts of a

 

realiz

a 50":

butr
rnay
over
the.
Free
afler
nxnh
duac
rdad
lloxt.

deser

 

 

192
realization and an ensemble in stochastic process theory are equivalent to the concepts of
a sample and a population in statistical theory.
For each realization a sample autocovariance function (ACF) can be calculated,
but need not necessarily correspond to the ensemble ACF because the random component

may not average over the finite sample area. However, the mean sample ACF averaged

over a set of Monte Carlo realizations should closely approximate the theoretical form of
the ACF of the ensemble (Smith, 1978). A loss of correlation was noticed (Smith and
Freeze, 1979) due to lognormal transformation. In general, the nearest-neighbor method
attempts to preserve the ACF of the field, but not as precisely as other generation
methods; e.g. the local average subdivision (Fenton and Vanmarcke, 1990). Another
disadvantage of the nearest-neighbor method is that it requires the inversion of a
relatively large matrix (I-W in Equation 7.5) which means much computational effort.
However, the use of relaxation method significantly simplified the process as will be

described in the next section.

7.3.3 Applying Nearest-Neighbor Process By Relaxation Method

Equation (7.1) can be solved iteratively by the “relaxation method” in which the
node representing each block (cell) takes a value related to its surrounding blocks and
then is iteratively recalculated as the neighboring values change. The recalculation is
repeated until a desired accuracy is obtained or a particular number of iterations is

reached. Generating k values by the relaxation method can be explained as follows:

 

wh

ite:

Tht

des;

and

193
The domain is divided into a set of square blocks (may be rectangular with some
modification).
Normally distributed values of Y are generated with a mean of zero and a standard
deviation of one: N[0,1]. In this study this was done using the @RISKTM computer
program (Palisade Corporation, 1995), an ExcelTM add-in which is capable of
generating data sets of different probability distributions.

These values are then adjusted to fit the form of Equation (7.1):

 

Y1. . = “x (vi,j + Y,,,.j)+ “fl-(Yipl + YW) (7.6)

where r = number of blocks surrounding block i,j. This equation is solved by automatic

iteration by a spreadsheet without need to perform matrix calculations.

The resulting Y1 values are internally correlated with mean try, slightly different than
zero and standard deviation 0Y1 slightly lower than one due to statistical randomness.
As in the nearest neighbor method, values are multiplied by a multiplier n = Oy/ Cy]
to yield the desired value of Cy.

The difference (try-u“) is then added to the values of Y to yield Y = N[py, Cy].

Finally the values are transformed to lognormal: k = EXP (Y) to get K = LN[uk, ak].

The process is performed by an ExcelTM spreadsheet. Starting with Y = N[0,l] and the

desired values try and try; the spreadsheet iteratively solves Equation 7.6, calculates u“

and CY] and according to these values the above mentioned ‘corrections’ are made by the

 

194

program. The goodness-of-fit was checked by the Best-FitTM computer program (Palisade
Corporation, 1995). This approach was found to be both practical and simple and the
computational effort is less than that in the matrix-based nearest-neighbor method.

To compare the relaxation-based to the matrix-based nearest neighbor methods;
the two methods were applied to generate k values in set of 50 blocks, 5 rows by 10
columns. The number of block was chosen relatively small in order to be able to invert a

50x50 matrix. The parameters or,( and at, were taken equal with two cases considered: or =

0.9 and a = 0.99. The distributions of the generated values for each case are shown in
Figures 7.2 and 7.3 from which it can be noted that the relaxation-based method of
generating spatially correlated values of k compares well to the nearest neighbor method.
The ACF was calculated for both methods and plotted in Figures 7.4 and 7.5. It is clear
that the relaxation-based method provides a good fit to an exponential ACF. The ACF for
the case of at = 0.9 decays faster than that for the case of or = 0.99 for both methods. It is
clear that the matrix-based approach preserves the spatial correlation better than the
relaxation-based approach. Another disadvantage of the relaxation-based approach is that
one can not obtain a predetermined shape of ACF although it still can be controlled by
changing the values of or and C.O.V. of k. In other words, one can increase the degree of
spatial correlation (increase the fluctuation distance) by increasing the value of or and/or
decreasing the value of C.O.V. of k.

The aim of the present study, however, is to examine how the spatial variability of
hydraulic conductivity affects the distribution and variation of pore water pressure in the

steady state case. In order to do so, a set of values of k of the predetermined mean and

195

  

 

 

 

 

 

1 -.

 

 

 

l l 0'8] 125 2.4 3.3 4.1

a) Using Matrix Inversion i ib)lUsing Relaxation Method

Figure 7.2 Probability Distribution of k Values Generated by
Nearest Neighbor Method, at = 0.90

      

   

 

 

 

l [

 

 

  

5:0

 

18'5 1.4 2.3 312 4.1510 l 1 o i '2 3 4

2a) Using Matrix Inversion b) Using Relaxation Method

Figure 7.3 Probability Distribution of k Values Generated by
Nearest Neighbor Method, at = 0.99

.ACF

196

 

I I I f I I I I I I I I I I I I I I I I I I I I I I I I r

  
 
 

\ 5 5 § K=LNmaaoam i 4
0.8 — .......... l """"""""" ; """"""""""""""""" : ........... _

0.6
0.4

0.2

    

. Relaxatiojn .
- : = 0 ~~=

 

I ' I -
I . x - I
1 l l l l l 1 1 l l l l l l l l l l l l l l l l 1 l l l l

 

 

0 1 2 3 4 5 6

Lag(unfls)

Figure 7.4 ACF Obtained by N.N. Solved by Both
Relaxation and Matrix Inversion, or = 0.90

ACF

197

 

1 I I I I I I 1 I I 1 T I I ! I I I I I I I f I LI 7* I I I
_ § § k = LN[O.25, 0.17]

0.8 " ' I i I
0.6

0.4

0.2

 

'- i l ' - -\:

O
I I
l l 1 l l l 4 l l I L 1 1 l l l 1 l l J l l l l l 1 L 1 1

 

 

 

o 1 2 3 4 5
Lag (units)

Figure 7.5 ACF Obtained by N.N. Solved by Both
Relaxation and Matrix Inversion, or = 0.99

198

standard deviation can be generated by the relaxation-based method and an ACF could be
calculated for these values. The shape of ACF could be controlled through the values of or
and C.O.V. as mentioned before. A predetermined shape of ACF is not necessary in the
proposed study. Besides, even in the nearest neighbor method one can not exactly obtain
a predetermined ACF but an approximation of it and the way of controlling the shape of
ACF is still by changing the values of or and C.O.V. of k. It can be concluded that the
relaxation-based method can be used to generate k values for the proposed study taking
advantages of being simple and the capability to be programmed in a simple spreadsheet.
A relatively large grid of k can be easily generated without the need to invert a large
matrix, noting that the number of generated values of k considerably affect the accuracy
of calculation. A better description of the distribution of pore pressure could be obtained
through a larger size grid of generated values of k.

It is clear that the block size affects the shape of ACF and scale of fluctuation.
Using a large block size results in a large scale of fluctuation and a long range of decay of
ACF. Smith and Freeze (1979) recommended that the block size should be chosen to
preserve both the scale of fluctuation and spatial decay of ACF. In this study a block size
of 5.0 ft. was found suitable to provide reasonable values of scale of fluctuation

compatible with those found in the literature.

199

7.4 Steady-State Seepage Analysis

7.4.1 General

The equation governing steady-state two-dimensional flow through non-

homogeneous anisotropic media is given by:

1P, fl]+_€_[k .61.]:0 (7.7)
5x

where, k, and k, are the hydraulic conductivity in x and y directions, and h is total head

(pressure head plus elevation head measured from a fixed datum). In a non-homogeneous

material, k, and ky can differ (kx = k,(x,y), and ky = ky(x,y)).

Equation (7.7) can be written as :

62¢! + 624)), =
6x2 ay2

 

0 ~ (7.8)

which is known as Laplace equation where it), = k,h and (by = k,h
For a homogeneous isotropic flow domain, k is constant and Equation (7.8) can be

written as:

 

0 (7-9)

200
The above differential flow equation may be solved graphically by constructing a flow
net or analytically by finite element or finite difference methods. The finite element
method is adopted in the computer program CSEEP (Tracy, 1973) used in this study to
compare and check results. The finite difference method will be presented in the next

section.

7.4.2 Relaxation Method
7.4.2.1 General

The method of “systematic relaxation of constraints” developed by Southwell
(1946) and explained by Capper and Cassie (1953) can be used to solve the flow equation
over a domain in a finite difference form. The method is simple although it is
traditionally laborious and requires computer programming. However, the iterative
capability of spreadsheets and their inherent two-dimensional nature can overcome this
disadvantage. Hence, spreadsheets were used in this study as a practical tool to solve the
Laplace equation by the relaxation method. The case of a homogeneous flow domain will

be first presented and then the non-homogeneous case will be explained.

7.4.2.2 Case of Homogeneous Flow Domain (constant k)

In the present study, the flow domain is by definition non-homogeneous as k is a
random field. However, the case of a homogeneous flow domain will be explained first
and then the non-homogeneous case will be explained. If the cross-section of a typical

embankment is covered with a square grid, the variation of (11 between the nodes of the

 

 

201
grid can be studied in finite difference form. As shown in Figure 7.6 the potential it varies
through the soil from point 1 to point 3 in the x-direction. This variation is assumed to be
at a uniform rate over the short distance between points 1 and 3. The rate of change of it)
with x (i. e. 0(1) / ax) between 1 and 0 is equal to the difference between ([11 and ([10 divided

by the distance between 1 and 0.

9:9- : 9':$°- between 1 and 0;
a x a
Similarly, all? = iii—i between 0 and 3
a x a
hence,
6 d) 1
5;——=;(¢.+¢.—2¢.) (7.10)

The second differential coefficients of o with respect to x is merely the rate of change of

4) divided by the distance “a “over which the rate of change occurs.

62¢ 1
“5‘7:

x (12

(¢1+¢3‘2¢0) (7-11)

and in the y direction:

202

 

 

01 1 0 3

8
tfl—t

 

 

 

 

Figure 7.6 Schematic Representation of 0 Relaxation Grid

 

 

 

01/8]: 1 0 3

 

 

Figure 7.7 Relaxation Grid at Line AB

203

52¢ 1
fly. - ,(¢2 +¢. -2¢.) (7.12)

a

 

Hence, summing (7.1 1) and (7.12) Laplace equation can be written as:

¢1+¢2+¢3+¢4—4¢o=0 (7-13)

which holds at each interior node in the grid. For homogeneous soil, k is constant,

equation (7.13) reduces to:

qm~t'm_

_ hl +h2 +h,+h,
' 4

 

h, (7.14)

Equation 7.14 implies that the total head at any interior node in the grid is simply the
average of the heads of its surrounding nodes and can be represented by the stencil (Al-

Khafaji and Tooley, 1986):

 

l
v2.=.,(0001
a

204

Some modifications must be made for boundaries and the free surface. The boundary
condition for the points along an impervious boundary is 6111 / 6y = 0. For node 0 in

Figure 7.7:

911:: _ =
M 2aft. 1..) 0 (7.15)

Node 2 is an imaginary one and hence hz should be expressed in terms of h.,. That is to

satisfy the no-flow condition,

Substituting these values in equation (7.13):

t». +2¢1 +¢3 -4<l>o = 0 (7.16)

and because k is constant (7.16) reduces to:

h, = ——-——--— (7.17)

which can be represented by:

205

1
who 0 o>

Other modifications may be required for comers and sheet piles (see Al-Khafaji and

Tooley, 1986).

7.4.2.3 Case of Non-Homogeneous Flow Domain

In a non-homogeneous flow domain the values of the hydraulic conductivity k
vary from point to point according to a spatially correlated random field. In this section,
the flow equations derived in the previous section are modified to simulate a non-
homogeneous flow domain. In Figure 7.6, k is assumed constant in the distance between
any two adjacent nodes; a channel of constant hydraulic conductivity between any two
adjacent nodes is assumed. The hydraulic conductivity of that equivalent channel is

assumed equal to the average of the k values at the two end points; for example:

 

k = between points 1 and 0

Following the same argument above and substituting d) = k h in Equation (7.11):

 

 

206

 

 

6;: = 2;, [(k, + k,)(hl — ho)-(k0 + k_,)(h0 - 113)] (7.18)
and:

52

a j: = 2;; [(k2 + k,)(h2 —h,)—(lc0 + k,)(ho —h,)] (7.19)

Summing Equation (7.18) Equation (7. 19) yields to :

h = h,(k0 +k,)+h.,(k0 +k2)+h3(ko +k3)+h,(k0 +k,) (7 20)
° 4ko+k,+k,+k,+k, °

It may be noticed that Equation (7.20) reduces to Equation (7.14) if k is constant.

For an impervious boundary (see Figure 7.7):

 

k k
and hence: i“ 2: 4)(h2 —h,)]=0

Node 2 is an imaginary one and hence h2 should be expressed in terms of h,. That is,

 

 

 

207

and the value (k2 + k4)/2 is the average k between points 2 and 4 and may be reasonably

assumed equal to k0 and hence,
k2 = 2k0 — k,
Substituting these values in equation (7.20):

h __ h1_((k0 + k1)+h,(k0 +k,)+h,(4ko)
° ‘ 6k, +1:l +k,

 

(7.21)

It should be noted that Equation (7.21) reduces to Equation (7.17) if k is constant.
Equations (7.20) and (7.21) provide the capability to model flow in a random field with a
simple spreadsheet. The head at each node is first calculated by either Equation (7.20) or
Equation (7.21) according to the location of the node. Then heads are recalculated

iteratively by relaxation as described previously.

7.4.2.4 The Phreatic Line
The boundary condition at the phreatic line (line CE in Figure 7.8) is 6h/0x +
ahlay = 0 or h = y (total head = elevation head). To locate the phreatic line numerically;

many iterations may be required: first an approximate location is assumed (line (1) in

 

 

208

 

oo>a._ m:o_>can_ < Go cozoomlmmeo .003: wK 6.30:

 

 

 

9
cm
om
ov
cm

00.

 

 

209

Figure 7.9) and the relaxation is performed assuming this line is fixed. Then the phreatic
line is modified according to the obtained heads. In other word resulting heads at some
points are smaller than their elevation head and hence the phreatic line should be
modified to match these points which gives line (2) The modification is repeated until no
significant change in heads is obtained and hence we get the final phreatic line (4). This
line is compared to the resulting phreatic line from a CSEEP run for the same levee. As
shown in Figure 7.10, the two lines are acceptably close to each other. It is clear that for
the case of a random field with variable k the position of phreatic line is not certain.
However, the repeated process to locate the phreatic surface is simple for one run; but for
200 Monte Carlo simulations it would be a massive computational effort. Hence, in order
to take advantage of the simplicity of the relaxation method in performing a large number
of simulations, the phreatic line was fixed at its final position found in the case of
homogeneous embankment. This approximation was tested by applying the relaxation
method for two cases: a homogeneous levee (Figure 7.8) and a non-homogeneous levee
(Figure 7.11). Pore pressure values were calculated at each of points 1 through 10 shown
in Figure 7.12 and the results were compared to the results of CSEEP for the same two
cases. The selected points were chosen in such a way to reasonably reflect variation of
pore pressure throughout the cross-section. The results shown in Table 7.1 indicate that
difference in pore pressure for the proposed method and CSEEP varies from point to
point. For homogenous case the maximum difference among the considered case was
found 7.8 % at Point 9. For non-homogenous case the maximum difference was 8.3 % at

Point 4. In other words, the maximum difference in pore pressure values obtained by

210

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7.5 App
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Freeze, T

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7.5.2 El

pressure
' Cas
1 C35

' Cas

215
relaxation and CSEEP increased from 7.8 % to 8.3 %. Hence, adopting a fixed phreatic
line did not significantly affect the accuracy of the results, and was considered a

reasonable assumption in the present study.

7.5 Applications
7.5.1 General

The Monte Carlo simulation method was applied to study the effect of variability
in hydraulic conductivity on the pore pressure distribution in the pervious levee shown in
Figure 7.8. Generation of spatially correlated values of k was performed by the relaxation
method according to the nearest-neighbor process as described previously. A grid of 11
by 56 points was utilized to generate realizations of k values as shown in Figure 7.12.
The number of Monte Carlo Simulations, MCS, was taken as 200. This was similar to the
values used in the literature when dealing with spatial variability of k (see Smith and
Freeze, 1979; Bergado and Anderson, 19850) and was considered sufficient to describe

the variability of pore pressure due to the variability of k.

7.5.2 Effect of The Coefficient of Variation of k

In order to study the effect of the coefficient of variation of k, Vk, on pore
pressure, 200 sets of k realizations were generated for four different values of Vk each of:
0 Case A: k = LN[O.26,0.13], Vk = 50 %
a Case B: k = LN[O.25,0.15], Vk = 60 %

0 Case C: k = LN[O.25,0.18], Vk = 72 %

f1
3
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ii

 

 

 

o Case
In genera
numberl
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constant
fluctuati

shown 1:

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' The

100]
° The

fr0r

216
a Case D: k = LN[O.20,0.20], Vk = 100 %
In generating k values for the four cases, the following values were fixed: orx = orZ = 0.90,
number of Monte Carlo simulations MCS = 200, block size b = 5 ft. Typical distribution
of generated k values for the different cases are shown in Figure 7.13. The ACF for the
four cases are shown in Figure 7.14. From this figure, although the parameter or is

constant; the shape of ACF and the scale of fluctuation varies with V1,. The scale of

fluctuation increases as Vk decreases. The different values of scale of fluctuation, 6, are I

 

shown in Table 7.2.

1
1’

Table 7.2 Scale of Fluctuation Corresponding to Values of VI,
a,=a,=0.90,b=5ft

 

Vk (%) 50 60 72 10 0
8 (ft) 14.5 13.5 11.75 9.0

 

 

 

The resulting pore pressure at points 1 through 10 for the four cases considered together

with constant k are shown in Table 7.3 and Figures 7.15 through 7.17, from which the

 

following may be noted:

0 The variability of pore pressure depends on the variability in k; Vu increases with the
increase of Vk.

o The variation of pore pressure is not constant throughout the embankment; it changes

from point to point being negligible near the upstream face (Point 1) and significant

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3

 

 

 

 

 

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0.8

0.6

IMSF

0.4

0.2

  
 

218

 

 

—0— Case A, Vk=50%
-)(—- Case B, Vk=60%

—+—- Case C, Vk=72%

- -a - - Case D, Vk=100%

 
    
 
   
    

 

.............................................................

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I I I I I rI I I I I I I I IfffI Ifi I fiI I I l I r
l ' I ' .

 

 

0 1 2 3 4 5
Lag (units)

Figure 7.14 ACF For Cases A Through D

219

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V%

15

10

 

 

 

 

   

  
    

 

 

Figure 7.15 Effect of Vk on Vu For Points 1 Through 4

220
1 l 1 I I l
- : - : - 4‘
. . I:
_ ..... —0—Point1 .................................... 48...} .......... .
-)(—- Point2 ’
—+——Point3 ' ;
“”" --A--Point4 f X """"" ‘
./..: ................................ _
40 50 60 70 80 90 100 110
O
Vk /o

if? 55" ; Tn?

T7

1%

 

 

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Fig

 

 

221

 

 

 
   
 

—e—Point5 ___________ ___________ 3 ......... A ......... A

 

-t-— Point 6

d
\O .
O r_ _.
3 i
> i— d
I
i
6 f.— .......................................................................... _4
__ _
4
b- -1
p— —1

 

 

l
3
;—— «4
3
t
r— -—<
I
L .‘
% . . g
284111111111i111111LLl:1811111111'11E

4o 50 60 7o 80 90 100 110
Vk %

Figure 7.16 Effectoka on Vu For Points 5 and 6

E”

 

V‘%

25

20

15

10

222

 

 

 

‘TYlTjIVIYI'TT YII‘IIIIITI. I}

 

 

—e—Point7 . ' ‘ :
-a— Point8 § 3 g ")2
-<>— Point9 """""" ." """"""

--x--Pmm1o g 3,.v

 

 

‘ . '. ' -
I'11111LL111L1;111111I11111=11].

50 60 70 80 90 1 00

V‘%
k

Figure 7.17 Effectoka on Vu For Points 7 Through 10

 

 

..'_—_———¢—
‘ A ‘ .._
.‘k. . .
r
I

' . w_ 1—‘

‘1“-

223

near the exit (Point 10) due to constraints in boundary conditions. In a typical levee,
the flow channels near the upstream face are wide and the pore pressure gradient is
low; whereas the flow channels are very narrow and the pore pressure gradient is high
near the exit.

0 For Points 7 through 10, Vu and Vk have an approximately linear relationship,
whereas this relationship is not linear for the rest of points. However, point 6 may be
taken as the average of all points (see Figure 7.12). This point is located nearly in the
middle of area through which the critical slope stability failure surface is most likely
to pass.

The probability distribution of pore pressure at the specified points is evaluated for the

200 values of pore pressure obtained at each point. The goodness-of—fit was assessed by

the BestFitTM computer program which fits the data to about 12 different probability

distributions and sorts these distribution according to their goodness-of-fit. It runs three
different tests: K-S, A-D, and Chi-Square test which was used in this study. Pore pressure
at most of the points can be fit to a normal distribution. Figure 7.18 shows the normal
distribution for pore pressure at points 6 and 7 for case A. Although the normal
distribution is not the “best” fit to pore pressure it is preferred because it is a probability
distribution commonly used in geotechnical probability analysis and widely used in many
applications, in addition to its simplicity to apply in slope stability probability analysis.
The probability distribution that best tits at most points was the Weibull distribution.
However, the normal distribution is a special case of the Weibull distribution. It may be

concluded that a normal distribution can be adapted to describe the variability of pore

Wanna-“I’m. 0%!
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kw- - S ‘

l

 

224

. Comparison of Input Distribution and i
Normal(1.05e + 3,38.00)

I 0.013.

 

 

 

 

 

 

 

 

 

0.95 0.98 i .02 1 .06 1 .09 1 .113

~ 1
1 Values in 10‘3 !

 

Eta-‘4.

   
  

 

 

 

 

‘ Comparison of Input Distribution and

i Normal(1.85e + 3,67.57) ‘

‘ 0.010;

i . i

. j .

i . E '

i 0.005; ‘ i

‘ i

i

i 0.000 X . , T Y , , , i
1.68 1.74 1.81 1.87 1.94 2.00 g

1 Values in 103 1

Point? 77 i 7 if 7 7 7

Figure 7.18 Case A, Probability Distribution of Pore Pressure
at Points 6 and 7

225
pressure in embankments due to spatial variability of hydraulic conductivity. For slope
stability analysis, this provides a simple way to deal with variability in pore pressure as

will be discussed later.

7.5.3 Effect of Scale of Fluctuation

In the previous section, the value of on was fixed and the variability of R was
controlled by changing the coefficient of variation of k. To examine the effect of
changing the scale of fluctuation on pore pressure, the value of or was changed while the
C.O.V was kept fixed. Four values of or were used to generate realizations of k with the

same mean and standard deviation (k = LN[O.25,0.18]):

‘-.-:_..'. - 6, .:,
__ L i

0 Case E: or = 0.8

0 Case C: or = 0.9

0 Case F: or = 0.95
0 Case G: or = 0.99

The ACF for these realizations of k is shown in Figure 7.19. It is clear that changing or

 

results in a change of the scale of fluctuation, 8. The values of 8 corresponding to these or

are shown in Table 7.4.

 

Table 7.4 Scale of Fluctuation Corresponding to Different Values of or
k = LN[0.25,0.18]

 

a 0.80 0. 90 0. 95 0. 99
6 (ft) 9.70 11.75 15.50 25.00

 

 

 

 

226

 

 
 
    
   
   

 

—O—0L=0.8
-[3— (1:0.9
-0— 0t=0.95

--)(--0t=0.99

 

 

ACF

 

Figure 7.19 ACF For Different Values of or, k=LN[0.25,0.18]

 

0
v
I
’\ ‘0

 

 

 

227
The resulting statistical moments of pore pressure are shown in Table 7.5. These results
indicate that no significant change in the mean and standard deviation of pore pressure
occurs due to changing the scale of fluctuation when Vk is kept fixed. Vu varies from 5 .91
% to 5.21 % at point 6 as 5 changes from 9.70 it. to 25.00 it. On the other hand, Vu
changed from 3.6 % to 10.13 % as Vk changed from 50 % to 100 % at the same point as
shown in Table 7.3. This can be noted in Figure 7.20 which provides a comparison

between the change of Vu when Vk is constant and when Vk varies from 50 % to 100 %.

7.5.4 The Anisotropic Correlation Structure Case

In the cases previously studied, the hydraulic conductivity field has an isotropic
correlation structure (orx = (11). In this section anisotropically correlated conductivity
realizations are generated according to the following:
ax = 0.99
0tz = 0.80
k = LN[0.25,0.18].
The ACF for this case is shown in Figure 7.21 in which it is clear that the ACF decays
vertically faster than horizontally, indicating stronger spatial correlation in the horizontal
direction. The resulting variation of pore pressure is compared to Case G in which ax =
0.99, Oil = 0.99 and k = LN[0.25,0.18], i.e. Vk is the same for the two cases. The
comparison is shown in Table 7.6 from which it may be noted that no significant
difference in Vu was obtained by considering k as an anisotropically correlated random

field. This may be because Vk was the same for the two cases.

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+ Vk varies

-x-— Vk constant

   

 

 

llllllllllllljjlllljllllllllLLl

 

10 12 14 16 18 20 22 24
5 (1’!)

Figure 7.20 Effect of 5 on Vu For Contant and Vriable Vk

 

0.8

0.6

IVSF

0.4

0.2

230

 

  
   
 

y—

b

b

 

T TT TlTTTTITTTf‘I’TfiTTITTTT[TTfTITTT T

Vertical ACF 8k 5 g
' ' Q : .

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llllllLlliiJLllJLJLlLllllllI‘AJl[4

 

 

1 2 3 4 5 6 7
Lag (units)

Figure 7.21 ACF For an Anisotropic k Generation

 

 

 

 

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232
7.5.5 Anisotropy in k Values
The issue of anisotropy in k values was considered beyond the scope of the
present study; however, it is discussed here for the purpose of completeness. The
hydraulic conductivity is usually assumed anisotropic in geotechnical analysis by taking
the k value in the horizontal direction (kh) greater than that in the vertical direction (kv).
A typical assumption is to consider kh = ckv, where c is a constant. Following the same

assumption, Equation (7.20) can be modified to account for anisotropy as following:

h,(ck0_+ck,)+h2(k0 +k,_)+h3(ck0 +ck3)+h4(k0 + k4)

h =
° 2cko +2k0 +ck, +k2 +ck3 +k4

 

(7.22)

Hence, generated k values are considered as kv, and then multiplied by c to get kh. The
anisotropy in the correlation structure of k may or may not be considered when
considering the anisotropy in k values although, however, the later may imply that the

former should exist.

7.6 Application on Slope Stability Analysis
7.6.1 General ‘

In the previous section the effect of spatial variability of k on variation of pore
pressure for the steady seepage case was investigated. In this section the possible
influence of the variability of pore pressure on the reliability of embankments during

steady seepage conditions is explored.

 

 

 

 

233

7.6.2 Analysis of A Typical Levee

The levee shown in Figure 7.8 was analyzed for slope reliability. Two random
variables were considered: friction angle (4)), and pore pressure (11). The mean value of (t)
was taken as 35° with a standard deviation of 2°. The coefficient of variation of u was
taken for simplicity as a constant value of 10%. This value was based on the values
obtained in Table 7.3 for Point 6 when Vk = 100 %. The value of u at each point was

considered as a random variable with coefficient of variation of 10 %. The mean value of

t

F)" iii‘.‘
_ .17.

u was obtained by performing a seepage analysis using the relaxation method with a

constant value of k (k = 0.25 ft/sec.).

‘,W3‘ 3;; L 9.84 at: m a zen-n.
" " O o

The reliability index was calculated for the case of u as a deterministic value and
then for the case of u as a random variable. [3 was calculated for the critical deterministic

surface. The results are shown in Table 7.7.

Table 7.7 A Typical Levee, Results of Reliability Analysis

 

 

Case [3
Pore Pressure as a Deterministic Value 8.78
Pore Pressure as a Random Variable 7.02

 

 

It can be seen that a considerable difference in B is obtained by considering the variability
in u. Although this difference is not significant in a practical sense, it should taken into

consideration if accurate reliability analysis is needed.

 

234

7.6.3 Analysis of Bois Brule Levee

The case of steady-seepage conditions for Bois Brule levee considered in Chapter
6 was reanalyzed here with u being a random variable (V u = 10 %). B was calculated for
the critical probabilistic surface. The results indicated that no significant difference in B
was obtained. The value of [3 was 2.30 when assuming u as a random variable, whereas it
was 2.32 when u was taken as a deterministic value. This is because the variability in the
soil parameters (cl, 4),, c2, and ¢2) in this problem has a more significant influence on [3

than the variability in u.

7 .7 Summary

The iterative finite-difference relaxation method was 'applied using spreadsheets
to both generate spatially correlated values of k and to solve the steady-state flow
equation. The method is checked and compared with the matrix method and shown to
produce reasonable results with considerably reduced programming effort.

A parametric study is performed to examine the influence of the coefficient of
variability of k, Vk, the scale of fluctuation , 8, and the anisotropic correlation structure
on the variability and probability distribution of pore pressure. The following were noted:
1. The coefficient of variation of k (Vk) is a major contributor to the variation in pore

pressure for the steady seepage through a typical pervious levee.
2. A normal distribution may reasonably be fit to the values of pore pressure at most

points considered in the analysis.

lI-I.

 

 

P

 

 

235

3. The shape of the ACF and the value of the scale of fluctuation (6) have a relatively
small effect on the variability of pore pressure when the statistical moments of k are
kept constant.

4. The influence of anisotropy in the correlation structure of k is relatively small for the
same statistical moments of k.

5. The effect of the variability of pore pressure (u) on slope reliability depends on the
problem type and the variability in the other soil parameters. Two . Two problems
were analyzed and the effect of the uncertainty in u was considerable in one problem

and was negligible in the other.

Chapter 8

SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS

8.1 Summary
Despite considerable published research, there is not yet a consistent chain of
procedures for performing probabilistic slope stability analysis, and the effects of
alternative methods have not been fully explored and compared. A number of such
aspects were investigated in this study. These aspects are considered important and their
effects should be understood if meaningful and accurate reliability analyses of
embankments are to be made. A list of these aspects is given below:
0 The effect of different deterministic and probabilistic models on the reliability index.
0 The effect of probability distribution of strength parameters on the reliability index.
0 The effect of alternative methods to model spatial variability in pore pressure for
short-term, effective-stress analysis.
0 Differences in location and meaning of the critical probabilistic surface and the
critical deterministic surface.
0 Development of a simplified method for modeling the variability of pore pressure due
to spatial variability of hydraulic conductivity.
In studying these aspects, a practical approach was adopted. Simple and practical tools
were developed and adapted to analyze actual complicated problems rather than applying

complicated procedures to simple special-case problems.

236

 

 

 

 

237

To study the effect of deterministic and probabilistic models, four deterministic
models and three probabilistic models were used and compared. The deterministic models
considered were: Spencer’s method for circular slip surfaces, the simplified Bishop
method, the modified Swedish method, and Spencer’s method for non-circular surfaces.
The two probabilistic models were: mean-value first-order second-moment (MFOSM)
method, the point estimate method (PEM), and the advanced first-order second-moment
(AF OSM). Two existing embankments were considered: Cannon Dam, and Shelbyville
Dam. In the same study, the effect of adopting a lognormal distribution for strength
parameters was investigated. It may be noted that the analyses performed here are
considered an extension to those in Wolff et. al. (1995).

The uncertainty in pore pressure for end-of-construction analysis was studied in
terms of the variability in pore pressure ratio ru. This parameter was modeled as a spatial
random variable and as a perfectly correlated random variable. The two cases were
compared to the assumption of considering the variability in an actual grid of ru values.
The latter is excessively cumbersome in application in practice due to the difficulty in the
input of ru at each point of the grid. The study also included the effect of scale of
fluctuation (a measure of spatial correlation structure) with Vallecito Dam and
Shelbyville Dam used as examples.

A practical and simple algorithm to locate the critical probabilistic slip surface
was developed. This approach is based on systematically examining combinations of soil
parameters other than the design (mean) values. These combination were obtained by

setting each soil parameter to a value of u i mo, where p. is the mean value, c is the

—
.9
r-..’
:7

 

 

 

238

standard deviation, and m is an arbitrary constant, with the rest of parameters kept at their
mean value. Each combination provided a factor of safety F S, and a slip surface. The
critical probabilistic slip surface was found to be the one among these surfaces which has
the lowest F 8,. The reliability index associated with this surface is the minimum
reliability index [3min and that associated with the critical deterministic surface was termed
BFS- Another reliability index was calculated using all FSi values combining their mix of
surfaces; this was the reliability index for the floating surface Bf and was used to compare
and interpret the results. The approach was applied to an extensive parametric study on
simple embankments as well as several case-studies and proved to provide a practical tool
to locate the critical probabilistic surface.

The uncertainty in pore pressure u = u(x,y) due to spatial variability of hydraulic
conductivity k = k(x,y) was investigated by assuming k as a spatially correlated random
variable. The iterative finite-difference relaxation method was applied using spreadsheets
both to generate spatially correlated values of k and to solve the steady-state flow
equation. The spreadsheets provided simple tools and were shown to produce reasonable
results with considerably reduced computation effort. The study included the effect of the
coefficient of variation of k, the scale of fluctuation of k, and the anisotropy in the
correlation structure of k on the variability of u and how these may affect the results of

slope reliability analysis.

 

 

1‘

 

8.2

 

m

P"

35

CI"

IC

en

an

ac

f0

Sig

Hc

01h

Ge

ear

MF

239

8.2 Conclusions

The present study has shown that each part of the slope reliability analysis
problem is a case in itself. Sensitivity to the aspects considered varies according to
problem geometry and soil parameters. It is difficult to generalize a certain approach or
assumption to all cases. For example, considering a non-circular failure surface is more

critical for a multi-layer embankment than for a homogeneous one. Nevertheless, this

research is important in providing practical tools and studies that may help in assessing
embankment reliability. Performing complete and accurate reliability analysis is difficult,
and it is the designer’s decision to consider all or some of the aspects of analysis
according to the problem size and importance. In addition to that conclusion, the

following conclusions are drawn.

8.2.1 Deterministic and Probabilistic Models, and Parameters Distribution

The deterministic model (slope stability method) was shown to have no
significant effect on the results of reliability analysis for the problems considered.
However, the influence of the shape of failure surface was significant for Cannon Dam.
The [3 associated with a non-circular slip surface was more critical than that for a circular
one (7.03 versus 10.85) using the same deterministic method (Spencer’s method).
Generally, a non-circular slip surface should be considered, especially in case of layered
earth embankments.

For the problems considered, no practical difference was observed between

MFOSM and PEM reliability results. The MFOSM may be preferred due to its

 

 

 

240

computational simplicity and its ability to show sensitivity to various random variables
through partial variances. The AFOSM has the advantages of being invariant and
providing a more conservative [3. However, it requires more computational effort and
numerical problems may occur as shown in case of Cannon Dam.

The lognormal distribution provided a more appropriate method to model shear
parameters especially for high coefficients of variation. However, for low coefficients of
variation, the lognormal distribution approaches the normal distribution. Hence, using
transformed moments implying a lognormal distribution of parameters has the following
advantages: (1) It is more appropriate for high C.O.V. (2) The results are quite similar to

that of normal distribution when C.O.V. is low. (3) It provides a conservative solution.

The combination of deterministic model (including failure surface shape), and
soil parameter distribution were shown to give a significant difference in 0 values for the
same structure. 0 values calculated for Cannon Dam ranged from 11.34 (using Spencer’s
method with circular slip surface and normal parameters) to 5.13 (using Spencer’s
method with non-circular slip surface and lognormal parameters). These findings suggest

the significance of these two factors and the fact that they should be taken into account,
especially if probabilistic analysis is used to make comparisons among structures for

economical investment decisions.

8.2.2 Uncertainty in Pore Pressure For End-of-Construction Analysis
Two cases of modeling the pore pressure coefficient ru were investigated; as a

perfectly correlated random variable with systematic variance and as a spatially correlated

 

 

ran'
esfi

mo

var
V31

me

she
sur

rek
var
the
the
pro!
dun
esfir
It’po
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day
than

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24]

random variable with reduced variance. These provided reasonable and consistent
estimates of the reliability index. Furthermore, they gave results close to that obtained by
modeling variability in an actual grid of ru. For both cases, the scale of fluctuation should
be determined or reasonably estimated, because both the systematic and the reduced
variances depend on it. On the other hand, modeling ru as a perfectly correlated random
variable with total variance considerably overestimates the reliability index, and
modeling ru as a deterministic value significantly underestimates the reliability index.

For the problems analyzed, the influence of the scale of fluctuation (8) on B was

shown to be dependent on the coefficient of variation of ru (Vru) and length the failure
surface compared to 8. This influence was found significant for a large V,“ and a

relatively long failure surface; and vice versa. However, the effect of 5 and spatial
variability in general increases with the increase of problem size compared to 5, due to
the effect of spatial averaging and the fact that great values compensate small ones. On
the other hand, when 8 is greater than or not significantly smaller than problem size the
problem approaches a perfectly correlated one and the effect of spatial correlation
diminishes. However, as a practical approach, in case of absence of enough data, a good
estimate of B may be obtained by assuming a reasonable value of 5 by looking at values
reported in the literature for other properties for the same soil as noted in Chapter 5 (for
example, 6,, .=_ 40 ft. and 52 5 10 ft. are common reported values for many parameters of
clayey soil). This approximate value of 6 will, however, provide a better estimation of B
than considering ru as a deterministic value or a perfectly correlated random variable as

the results in Chapter 5 indicated.

 

W
-.I Ii-‘J.

 

 

8.23 I

signific
both t]
reliabi'.
existin
numbe
0 Th

in

Sig
0 Or

va
' Ct

sh

CI

SI

CE

242

8.2.3 Locating The Critical Probabilistic Surface

A practical and simple search process was developed and examined over a
significant number of examples and case studies. It proved to provide a good estimate of
both the location of the critical probabilistic surface and the value of the minimum
reliability index Bmin. It also has the advantage of being applicable within the context of
existing available slope stability software. The numerous analyses presented lead to a
number of significant observations:
0 The reliability index based on a floating surface Bf gives an idea about the variability

in FSi, and hence acts as an indicator as to whether there may be a surface of

 

Wm; .\_\.‘.' . .: :o .

significantly lower B than the critical deterministic surface.
0 Only surfaces obtained by parameter combinations containing lower-than-mean
values (p - mo) for each soil parameter need to be considered in the search process.
0 Considering the lower value of each soil parameter as u - o (i.e. taking m = 1) was
shown to be a practical approach.

0 Using parameter combinations corresponding to values for two or more parameters

 

did not locate a critical probabilistic surface, because that approach does not
emphasize the single dominant parameter that usually picks up the critical surface

0 Surfaces which were systematically considered near the obtained critical probabilistic

 

surface did not lead to revision of Bmin.
- The difference between B calculated for the critical deterministic surface (BFS) and B
calculated for the critical probabilistic surface (Bmin) was shown to be significant in

many of the analyzed structures, especially for stratified embankments and in the case

 

where
may
uncor
0 For I
paran
locat:

in the

8.2.4 Ur

variation
seepage
Vu, notir
point to

reasonat

and the :

vElriabili

dEV’iatior

problem

243

where soil parameter values are highly uncertain. In these cases, calculating only Brs
may significantly overestimate the reliability index and hence result in an
unconservative probability of failure.

0 For homogeneous embankments, and/or when the coefficient of variation of soil
parameters is small, the critical deterministic and the critical probabilistic surfaces are
located close to each other. No significant difference between BFS and Bmin was found

in these cases.

8.2.4 Uncertainty in Pore Pressure Due to Spatial Variability of k

When the permeability k is assumed to be a random field, the coefficient of
variation Vk is a major contributor to the variation in pore pressure u for the steady
seepage case through a typical pervious levee. It was shown that the higher Vk the higher
Vu, noting that Vu is not constant throughout the embankment. Rather, it varies from
point to point, being greater near the downstream exit. A normal distribution may
reasonably be fit to the values of pore pressure at most points considered in the analysis.

The shape of the autocovariance function ACF, the value of scale of fluctuation 8,
and the anisotropy in the correlation structure of k showed a relatively small effect on the
variability of pore pressure for the same statistical moments (mean and standard
deviation) of k.

The effect of the variability of pore pressure u on slope reliability depends on the

problem type and the variability in the other soil parameters. Two problems were

 

 

analyz

neglig

8.311

with

emba
differ
the r
come

slope

recon
variar
rando
accou

dealir.

model
coeffi.
COrpg

adOPte

244

analyzed and the effect of the uncertainty in u was considerable in one problem and

negligible in the other.

8.3 Recommendations

For reliability analysis of embankments, it is recommended that Bmin associated
with the critical probabilistic slip surface be calculated especially for layered
embankments for which soil parameters have high coefficients of variation. The
difference between Bmm and BFS is significant in this case and could affect the accuracy of
the reliability analysis results. The algorithm developed in this study provides a
convenient method to locate the critical probabilistic slip surface using existing available
slope stability software.

For the analysis of embankments for end-of-construction conditions, it is
recommended to model ru as a spatially correlated random variable with reduced
variance. An alternative approximate solution is to model ru as a perfectly correlated
random variable with systematic variance. Both approaches provide a practical method to
account for the variability in pore pressure with results close to the ideal solution of
dealing with variability in ru at each point.

If complete and accurate reliability analysis is desired, different deterministic
models and failure surface shapes should be examined. Soil parameters with high
coefficients of variation are recommended to be modeled as lognormally distributed if the
Corps’ method of calculating partial derivatives of parameters for MFOSM method is

adopted.

 

‘futgu- .1. If.
“‘1'.
.‘

 

 

ushn;
dgor

u)cal

accor
pr0\i
prob]

thezr

245

This research provided a method for locating the critical probabilistic slip surface
using an existing slope stability program. Ideally, for further investigation, the proposed
algorithm may be programmed and added to a slope stability program which will be able
to calculate both Brs and Bmin.

The proposed approach of studying spatial variability of k may be expanded to
account for allowing the phreatic line to change for every realization of k. This may
provide a method to model the phreatic line as random variable in slope reliability

problems. Also, accounting for the anisotropy in k values may provide an improvement to

the approach.

a

 

. 8.7
‘._..
s'.~.s3_

 

 

 

APPENDIX A

 

 

Appendix A
Example Calculation of an Autocovariance Function

 

 

 

 

 

 

 

 

 

 

Value. x (xmxxzw (xmxxeru) (xwxxxm) «WWW
0.711 0.088 0.042 0.029 0.014 -0.019 -0.021 «0.007
0.443 0.018 0.012 0.008 -0.008 -0.009 -o.ooe 0.005
0.343 0.008 0.003 -0.004 -0.004 -0.001 0.002 0.000
0.314 0.002 -0.003 -0.003 , -0.001 0.002 0.000 -0.003
0.282 -0.001 -0.001 0.000 0.001 0.000 -0.001 -0.002
0.21 1 0.002 0.001 -0.001 0.000 0.002 0.003 0.003
0.207 0.001 -0.001 0.000 0.002 0.003 0.003 0.005
0.235 0.000 0.000 0.001 0.001 0.001 0.002 0.002
0.278 0.000 -0.001 -0.002 «0.002 -0.003 -0.003 -0.004
0.248 0.000 0.000 0.000 0.000 0.000 0.001 0.001
0.208 0.003 0.003 0.005 0.005 0.008 0.008 0.008
0.178 0.008 0.008 0.009 0.01 1 0.01 1 0.009 0.009
0.174 0.009 0.009 0.01 1 0.011 0.010 0.010 0.008
0.139 0.013 0.017 0.018 0.014 0.014 0.009 0.004
0.134 0.017 0.017 0.015 0.015 0.010 0.004 -0.008
0.104 0.022 0.019 0.019 0.012 0.005 -0.011 0.002
0.105 0.019 0.019 0.012 0.005 -0.010 0.002 0.003
0.122 0.015 0.011 0.004 -0.009 0.002 0.003 0.003
0.125 0.011 0.004 -0.009 0.002 0.003 0.003 0.003
0.188 0.003 -0.008 0.001 0.002 0.002 0.002 -0.001
0.21 8 -0.002 0.000 0.001 0.001 0.001 0.000 0.000
0.323 -0.001 -0.002 -0.002 -0.002 0.001 0.000 -0.001
0.239 0.000 0.000 0.000 0.000 0.000 0.000 0.001
0.229 0.001 0.001 0.000 0.000 0.000 0.001 0.001
0.228 0.001 0.000 0.000 0.000 0.002 0.001 0.001
0.228 0.000 0.000 0.000 0.001 0.001 0.001 0.001
0.288 0.000 0.000 -0.001 -0.001 -0.001 0.000 0.000
0.258 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.234 0.001 0.001 0.001 0.000 0.000 -0.001 «0.001
0.192 0.003 0.003 0.002 0.000 -0.003 -0.005 -0.007
0.203 0.002 0.001 0.000 -0.002 -0.004 -0.008 -0.015
0.201 0.001 0.000 -0.002 -0.004 -0.008 -0.015 -0.014
0.223 0.000 -0.001 -0.002 -0.003 -0.009 -0.008 -0.010
0.258 0.000 0.001 0.001 0.002 0.002 0.002 0.003
0.301 0.004 0.008 0.01 5 0.014 0.017 0.022 0.007
0.340 0.011 0.027 0.025 0.030 0.040 0.012 0.003
0.375 0.038 0.035 0.042 0.058 0.017 0.004 0.004
0.557 0.088 0.103 0.138 0.041 0.010 0.009 -0.007
0.533 0.095 0.127 0.038 0.009 0.008 -0.008 -0.012
0.589 0.153 0.048 0.01 1 0.010 -0.008 -0.015 -0.027
0.703 0.081 0.015 0.013 -0.010 -0.020 -0.038 -0.058
0.387 0.004 0.004 -0.003 -0.008 -0.011 -0.01 7 -0.017
0.284 0.001 -0.001 -0.001 -0.003 -0.004 -0.004 -0.004
0.280 -0.001 -0.001 -0.002 -0.004 -0.004 -0.004 -0.003
0.228 0.001 0.002 0.003 0.003 0.003 0.003 0.003
0.208 0.003 0.008 0.008 0.008 0.005 0.005 0.008
0.171 0.010 0.010 0.010 0.009 0.010 0.012 0.012
0.124 0.018 0.017 0.014 0.018 0.018 0.019 0.01 9
0.123 0.017 0.014 0.018 0.018 0.019 0.020 0.013
0.122 0.015 0.015 0.019 0M0 0.020 0.013
0.138 0.014 0.018 0.017 0.017 0.012
0.127 0.018 0.019 0.019 0.013
0.108 0.022 0.022 0.015
0.100 0.023 0.018
0.099 0.018
0.148

“on 0.251

8.0. 0.14;

00V. 0.020 0.015 0.012 0.009 0.008 0.003 0.001 -0.002
|t_.g o r 2 :l 4 15 o 7 ‘

ACF 1.000 0.784 0.801 0.488 0.389 0.1 41 0.028 -0.078

 

 

 

 

 

 

 

 

 

 

 

 

 

 

246

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247

 

APPENDIX C

 

 

 

 

 

 

 

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248

 

APPENDIX D

 

 

 

Appendix D

Calculation of Hasofer-Lind Reliabilty Index

Vcl = 40 %
Gel = 200 psf

v.1 = 20 %
0.1 = 2.40

30“

Va =10 %
CC; = 100 psf

20 it

g(x) = FS(x) - 1
h(z) = FS(z)-l

+11

_X x ...
Z—‘6—-=>X—O'xl+ux

X

C1=20021+500
¢,=2.4zz+12
Cz= 10023+1000

Critical Probabilistic Surfa_l_c_§:
Iteration l

2“”:0 ,Az=0.10

Run 0: c1 = 500 , 4). = 12°
Run 1: Cl: 520 , 01:1?
Run 2:

Run 3: Cl: 500 , ¢1=12°

0.48
Vh(z‘°’) = 0.12
0

 

|Vh(z‘°’j —_- J(.48)2 + (.12)2 + (0)2 = 0.4948

 

 
  

Soil 1
C, -SOOpsf

¢, -12°

 

SO" 2 CIIIOOO psf
¢2 '0

 

,C2=1000
,Cz=1000
C1=500 ,¢1=12,24° ,C2=1000
,Cz=1010

77//////////////////////////7/////////1

Case N

,f¥3==L770
,FS = 1.818
,FS= 1.782
,FS= 1.770

 

 

.48 -.970
(1(0) _ ‘Zh(l(o)) _ ‘1

 

 

 

 

_ _ V (m _ 4948 .12 = —.2425
|_h(z I . 0 0
v‘°"a‘°’=0
h(g‘°’) = 1.770—1.0- 0.770
(0, —15095
2‘”: z‘°’Ta‘°’+———h(z ) a‘°’=[O+—'77O]= —3774
" IZh(z‘°’ )l .4948 ’ 0
[3' = J(-15095)2 +(.3774)2 +(0)2 = 1556
198.1 psf
5: 11.094 deg.
1000 psf
Summary of Iterations:
Iteration z, Z2 Z3 c; psf ¢, deg. c; psf ,6
1 -151 -0377 0 198.1 11.094 1000 1.556
2 -1515 -0.379 0 197 11.09 1000 1.562
3 -1523 -0357 0 195.4 11.14 1000 1.564

 

 

BHL(min.) = 1.564, Design Point: c1* = 195.4 psf, (01* = 11.l4°, c2* = 1000 psf

Critical Deterministic Surface:

 

 

Summary of Iterations:

Iteration z: 22 z3 c, psf ¢, deg. C; psf ,6
1 -1.962 -0.491 -3. 185 107.6 , 10.822 681.5 3.773
2 -l.807 -O.516 -3.357 138.6 10.762 664.3 3.847
3 -l.807 -O.517 ~3.357 138.6 10.759 664.3 3.847

 

 

HL(FS) = 3.847, Design Point: c1* = 138.6 sf, ¢.* = 10.759°, c2* = 664.3 psf
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LIST OF REFERENCES

 

 

 

REFRENCES

Alonso, E. E. (1976), “Risk Analysis of Slopes and its Application to Slopes in Canadian
Sensitive Clays”, Geotechnique, 26, No. 3, pp. 453-472.

Al-Khafaji, A. W., and Tooley, J. R. (1986) “Computerized Numerical Analysis”, Hbj
College & School Div., June 1986.

Ang, A. H.-S., and Cornell, C. A. (1979) “Reliability Bases of Structural Safety and
Design”, Journal of the Structural Division, ASCE, vol. 100, April 1979, pp. 10777-1769

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D.“

Ang, A. H.-S., and Tang W.H. (1984): Probability Concepts in Engineering Planning and
Design, Volume II : Decision, Risk, and Reliability, John Wiley and Sons, New York.

Bartlett, M. S. (1975): The Statistical Analysis of Spatial Pattern, Chapman and Hall,
London

Bear, J. (1972) “Dynamics of Fluids in Porous Media”, American Elsevier Publishing
Company Inc. ”

Bergado, D. T., Anderson, L. R. (1985)“ Stochastic Analysis of Pore Pressure
Uncertainity For The Probabilistic Assessment of The Safety ‘of Earth Slopes”, Soils And
Foundations, Vol. 25, No. 2, June 1985, pp. 87-105.

Bishop, A. W., (1954), “Use of Pore-Pressure Coefficients in Pactice”, Geotechnique, 4,
pp. 148-152.

 

Bishop, A. W., (1955), “The Use of The Slip Circle in The Stability Analysis of Slopes”,
Geotechnique, 5, pp. 7-17.

Bishop, A. W., and N. Morgenstern, (1960), “Stability Coefficients for Earth Slopes”,
Geotechnique, 10, pp. 129-150.

Calle, E. O. F. (1985), “Probabilistic Analysis Of Stability of Earth Slopes”, Proceedings

of the 11th International Conference on Soil Mechanics and Foundation Engineering,
San Francisco, Vol. 2, pp. 809-812.

251

 

252

Capper, P. L., and Cassie, W. F . (1953) The Mechanics 0s Engineering Soils, McGraw-
Hill.

Chowdhury, RN. (1984), “Risk Estimation for Failure Progression Along a Slip
Surface”, 4th Inernational Symp. on Landslides, Toronto, pp. 381-386.

Chowdhury, RN, and A-Grivas, D., (1982), “Probabilistic Model of Progressive Failure
of Slopes”, Journal of Geotechnical Engineering Diviosn, ASCE, Vol. 108, No. GT6,
June 1982, pp. 803-819.

Chowdhury, R.N., Tang, W. H., and Sidi, I. (1987), “Reliability Model of Progressive
Slope Failure”, Geotechnique, 37, No. 4, pp. 467-481.

Chowdhury, RN, and Xu, D.W. (1993), “Rational Polynomial Technique in Slope-
Reliability Analysis”, Journal of Geotechnical Engineering, ASCE, Vol. 119, No. 12,
December 1993, pp. 1910-1928.

Chowdhury, RN, and Xu, D.W. (1994), “Slope System Reliaibility with General Slip
Surfaces,” Soils and Foundations, Japanese Sociaety of Soil Mechanics and Foundation
Engineering, Vol. 34, No. 3, pp. 99-105, September 1994.

Christian, J. T., Ladd, C. C., and Baecher, G. B., (1994), “Reliability Applied to Slope
Stability Analysis,” Journal of Geotechnical Engineering, ASCE, Vol. 120, No. 12,
December 1994, pp. 2180-2207.

Cornell, C. A. (1971), “First-Order Uncertainity Analysis of Soils Deformation and
Stability”, Proceedings of The Inernational Conference on Applications of Statistics and
Probability to Soil and Structural Engineering”.

DeGroot, D. J ., “Analyzing Spatial Variability of In Situ Soil Properties”, Proceeding of
the ASCE Specialty Conference on Uncertainity in the Geologic Environment, ASCE,
NY, pp.210-238.

DeGroot, D. J., Baecher, G. B. (1993). “Estimating Autocovariance of In-Situ Soil
Properties”, Journal of Geotechnical Engineering, ASCE, Vol. 119, No, 1, Januray 1993,
pp. 147-166.

Delhomme, J. P. (1979): “Spatial Variability and Uncertainity in Groundwater Flow
Parameters: A geostatistical Approach”, Water Resources Research, Vol. 15, No. 2, April
1979, pp. 269-280.

Ditlevsen, O. (1979) “Generalized Second Moment Reliability Index”, Journal of
Structural Mechanics, ASCE, vol 7(4), pp. 435-451.

 

253

Edris, E. V., Jr., and Wright, S. G. (1993), “User’s Guide: UTEXAS3 Slope Stability
Package: Volume 4- User’s Manual, Instruction Report GL-xx-x, US. Army Engineer
Waterways Experiment Station, Vicksburg, MS.

Evans, D. H. (1962), “An Application of Numerical Techniques to Statistical
Tolerancing”, Technometrics, vol. 9.

Evans, D. H. (1972), “An Application of Numerical Integration Techniques to Statistical
Tolerancing, General Distribution”, Technometrics, vol. 14.

Fellenius, W. (1927) Erdstatische Berenungen mit Reibung und Kokaesion, Ernst, Berlin.

F enton, G. A., and Vanmarcke, E. H., “Simulation of Random Fields via Local Average
Subdivision”, Journal of Engineering Mechanics, ASCE, Vol. 116, No. 8, August 1990,
pp. 1733-1749.

Handy, R. L. (1981), “Linearizing Trixial Test Failure Envelopes, ” Geotechnical Testing
Journal, ASTM, vol. 4, 1981, pp. 188-191.

Harr, M. E. (1987), Reliability-Based Design in Civil Engineering, McGraw-Hill.

Hasofer, A. A., and Lind, A. M. (1974), “Exact and Invariant Second- Moment Code
Format,” ASCE Journal of Engineering Mech. Div. vol. 100, No. EMl, February 1974,
pp 111-121.

Honjo, Y., and Kuroda, K. (1991), “A New Look at Fluctuating Geotechnical Data for
Reliability Design”, Soils and Foundations, Vol. 31, No. 1, March 1991, pp. 110-120

Hotz, W. G., and Noell, GA. (1950), “Suggested Method of Test of Internal Friction and
Cohesion Values of Soil and Soil Mixtures by Triaxial Loading,” Procedures for Testing
Soil, ASTM, Philiadelpia, 1950, pp. 301-314 (also reprinted in subsequent editions
through 1970).

Humphery A. M., and G. A. Leonards (1986), “Slide in Upstream Slope of Lake
Shelbyville Dam”, Journal of Geotechnical Engineering, ASCE, vol. 112, No.5, May
1986, pp. 564-577.

Ireland, H. 0., ( 1954), “Stability Analysis of The Congress Street Open Cut in Chicago”,
Geotechnique, 4, pp. 163-168.

Johnson, S. J. (1975): “Analysis and Design Relating to Embankments: A Sate-of-the-Art
Review,” Miscellaneous Paper S- 75-3, US. Army Engineer Waterways Experiment
Station, Vicksburg, MS, 1975, pp. 48.

 

 

254

Lacasse, S. and Nadim, F. (1996), “Uncertainities in Characterizing Soil Properties”,
Proceeding of the ASCE Specialty Conference on Uncertainity in the Geologic
Environment, ASCE, NY, pp. 49-76.

Li, KS, and Lumb, P. (1987), “Probabilistic Design of Slopes,” Canadian Geotechnical
Journal, vol. 24, pp. 520 - 535.

Li, KS, and White, W., (1987), “Reliability Index of Slopes”, Reliability and Risk
Analysis in Civil Engineering, ICASP% 1987, Univertsity of British Columbia,
Vancouver, BC, Canada, May 1987, pp 755-762.

Lockhart, S. D., and Reagh, K. P., (1995) “The AUTODESK Collection, Student
Manual”, Addison-Wesley Publishing Co.

Lowe, J ., III (1967), “Stability Analysis of Embankments” Journal of Geotechnical
Engineering, Division, ASCE, Vol. 93, No.SM4, July 1967, pp. 1-33.

Mantoglou, A., and Wilson, J. L. (1982), “The Turning Bands Method for Simulation of
Random Fields Using Line Generation by a Spectral Method”, Water Resources
Research, Vol. 18, No. 5, October, 1982, pp. 1379-1394.

McGuffey, V., Iori, J, Kyfor, Z., and A-Grivas, D. (1981), “Use of Point Estimates for
Probability Moments in Geotechnical Engineering”, Transportation Research Record,
No. 809, pp. 60-63.

Morgenstern, N. R., Price, M. A., (1965), “The Analysis of The Stability of General Slip
Surface”, Geotechnique, 15, pp. 79-93.

Neuman, S. P. (1984): “Role of Geostatistics in Subsurface Hydrology” Geostatistics for
Natural Resources Characterization, NATO ASI, D. Reidel, Hingham, Mass., pp. 127-
140.

Neuman, S. P., and Orr, S. (1993): “Prediction of Steady State Flow in Nonuniforrn
Geologic Media by Conditinal Moments: Exact Nonlocal F orrnalism, Effective
Conductivities, and Weak Approximation”, Water Resources Research, Vol. 29, No. 2,
February 1993, pp. 341-364.

Oka, Y., and Wu, T. H. (1990), “System Reliability of Slope Stability,” Journal of
Geotechnical Engineering, ASCE, vol. 116, No.8, August, 1990, pp. 1185—1189.

Parkinson, D. B. (1978), “Solution for Second Moment Reliability Index”, Journal of the
Engineering Mechanics Division, ASCE, vol. 104, No. EMS, October 1978, pp 1267-
1275.

 

 

 

255

Palisade Corporation (1995), “ BestF it, Distribution Fitting For Windows, User Guide”

Palisade Corporation (1995), “ @RISK, Risk Analysis and Simulation Add-In for
Microsoft Excel or Lotus 1-2-3, User Guid”

Rosenblueth, E. (1975): “Point Estimates for Probability Moments,” Proc. National
Academy of Science USA, vol. 72, no.10.

Rosenblueth, E. (1981): “Tow-Point Estimates in Probabilities,” Applied Mathmatical
Modeling, vol. 5, October 1981.

Rubinstein, R. Y. (1981): Simulation and the Monte Carlo Method. Wiley, New York.

Schoeneman, M. R.. (1990): “Statistical Description of Triaxial Shear Test Results,”
Geotechnical Testing Journal, ASTM, vol. 13, no. 1, March 1990, pp. 58-62.

Shannon and Wilson, Inc., and Wolff, T. F. (1994), “Probability Models for Geotechnical
Aspects of Navigation Structures,” prepared for St. Louis District, US. Army Corps of
Engineers, in connection with the Upper Mississippi River Study, January, 1994.

Shinozuka, M. (1983): “Basic Analysisof Structural Safety”, Journal of Structural
Engineering, ASCE, vol. 109, No. 3, March 1983, pp. 721-740.

Spencer, E. (1967). “A Method of Analysis of The Stability of Embankments Assuming
Parallel Inter-Slice Forces”, Geotechnique, 17, pp. 11-26.

Smith, L, and Freeze, A. (1979) “Stochastic Analysis of Steady State Groundwater Flow
in a Bounded Domain”, Water Resources Research, V10. 15, No. 6, December 1997, pp.
1 543-1 559.

Smith, L, and Schwartz, F. W. (1980) “Mass Transport 1.A Stochastic Analysis of
Microscopic Dispersion”, Water Resources Research, V10. 16, No. 2, April 1980, pp.
303-313.

Soule, M. Montes, P., and Silvestri, V. (1990), “Modelling Spatial Variability of Soil
Parameters”, Canadian Geotechnical Journal, Vol. 27, pp. 617-630.

Termaat, R. J ., and Calle, E. O. F. (1994), “Short Term Acceptable Risk of Slope Failuer
of Levees”, Proceedings of the 13th International Conference of Soil Mechanics and
Foundation Engineering, New Delhi, India.

Tracy, F. T. (1973) “ A Plane and Axisymetric Finite Element Program for Steady-State
and Transient Seepage Problems”, Miscellaneous Paper K-73-4 US. Army Engineers
Waterways Experiment Station, Vicksburg MS, May 1973.

256

Tang, W. H., Yucemen, M. S., and Ang, A. H.-S. (1976), “Probability-Based Short Term
Design of Soil Slopes”, Canadian Geotechnical Journal, 13, pp 201-215.

US. Army (1970), “Engineering and Design, Stability of Earth and Rockfill Dams,”
Engineer Manual 1110-2-1902, Department of the Army, Corps of Engineers, Office of
the Chieve of Engineers, Washington, Dc.

US. Army (1992), “Reliability Assessment of Navigation Structures,” Engineer
Technical Letter No. 1110-2-532, Departement of the Army, US Army Corps of
Engineers, CECW-ED, 1 May, 1992.

US. Army (1993): “Reliability Assessment of Nevigation Structures: Stability
Assessment of Existing Gravity Structures,” Engineer Technical Letter No. 1110-2-310,

Department of the Army, US Army Corps of Engineers, CECW-ED, 31 December, 1993.

Ur-Rasul, I. (1995), “Characterizing Soil Strength for Probabilistic Analysis Working
From Test Results, A Practical Approach”, MS Thesis, Michigan State University,
December 1995.

Vanmarcke, E. H. (1977a), “Probabilistic Moments of Soil Profiles”, Journal of
Geotechnical Engineering Division, ASCE, Vol. 103, No. GT1 1, November, 1977, pp.
1227-1246.

Vanmarcke, E. H. (1977b), “Reliability of Earth Slopes”, Journal of Geotechnical
Engineering Division, ASCE, Vol. 103, No. GT1 1, November, 1977, pp. 1247-1265.

Whitman, R. V. (1984), “Evaluating Calculated Risk in Geotechnical Engineering”,
Journal of Geotechnical Engineering, ASCE, Vol. 110, No. 2, F ebuary, 1984, pp. 145-
1 87.

Wolff, T. F. (1985), “Analysis and Design of Embankment Dam Slopes: A Probabilistic
Approach,”Ph.D. Dissertation, Purdue University, West Lafayette, IN, May 1985.

Wolff, T. F., and Harr, ME. (1987), “Slope Design for Earth Dams,” in Reliability and
Risk Analysis in Civil Engineering 2, Proceeding of the Fifth International Conference on
Applications of Statistics and Probability in Soil and Stuctural Engineering, Vancouver,
BC, Canada, May, 1987, pp. 725-732.

Wolff, T. F ., (1991),”Embankment Reliability versus Factor of Safety: Before and Afier
Slide Repair,” International Journal for Numerical and Analytical Methods in
Geomechanics, Volume 15, Issue No. 1, January 1991.

Wolff, T. F., and Wang, W. (1992), “Engineering Reliability of Navigation Structures,”
research report, Michigan State University, for U. S. Army Cops of Engineers, January,
1992.

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257

Wolff, T. F., and Wang, W. (1993), “Reliability Analysis of Navigation Structures,” in
Geotecnical Practice in Dam Rehabilitation, Proceeding of the specialty conference
sponsored by the Geotechnical Engineering Division, ASCE, NORTH Carolina State
University, Raleigh, North Carolina, April 25-28, 1993, ASCE Geotechnical Specialty
Publication No. 35, pp. 159-173.

Wolff, TR, (1994), “Evaluating the Reliability of Existing Levees,” A technical report
perpard for US. Army Engineer Waterways Experiment Station, Geotechnical
Laboratory, Vicksburg, MS, September 1994.

Wolff, T. F ., Hassan, A., Khan, R., Ur-Rasu1,I, and Miller, M. (1995), “Geotechnical
Reliability of Dam and Levee Embankments”, A technical report prepared to US. Army
Engineer Waterways Experiment Station Geotechnical Labortory, September 1995.

Wolff, T. F. (1996), “Probabilistic Slope Stability in Theory and Practice”, Proceeding of
the ASCE Specialty Conference on Uncertainity in the Geologic Environment, ASCE,
NY, pp. 419-433.

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