PROBABILISTIC APPROACHES TO SLOPE DESIGN

PROBABILISTIC APPROACHES TO SLOPE DESIGN

KAI SHUN LI, BSc(Eng).

A thesis submitted in the

Department of Civil Engineering,

University College,

University of New South Wales,

Australian Defence Force Academy,

for the degree of

Doctor of Philosophy.

September, 1987

UNIVERSITY OF N.S.W.

2 8 JUNI988LIBRARY

STATEMENT OF ORIGINALITY

I hereby declare that this thesis report is my own work and that, to the best of

my knowledge and belief, it contains no material previously published or written

by another person nor material which to a substantial extent has been accepted

for the award of any other degree or diploma of a university or other institute of

higher learning, except where due acknowledgement is made in the text.

K.S. LI

TABLE OF CONTENTS

Table of Contents.......................................................................................................i

Abstract.................................................................................................................. viii

Acknowledgements................................................................................................. x

Notations and Abbreviations............................................................................... xii

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

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

1.2 SCOPE OF THE PRESENT WORK...........................................................1-4

CHAPTER 2 THEORY OF PROBABILISTIC DESIGN...............................2-1

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

2.2 PERFORMANCE FUNCTION...................................................................2-2

2.3 LEVEL I DESIGN.......................................................................................... 2-4

2.4 LEVEL II DESIGN........................................................................................ 2-12

2.4.1 /?-approach................................................................................................ 2-13

2.4.2 fiHL-approach............................................................................................ 2-14

2.5 LEVEL III DESIGN.....................................................................................2-17

2.6 APPROXIMATE LEVEL III DESIGN.....................................................2-19

2.6.1 Normal tail approximation .....................................................................2-19

i

Table of Contents

2.6.2 Method of PDF fitting.............................................................................2-21

2.6.2.1 Approach A........................................................................................ 2-21

2.6.2.2 Approach B........................................................................................ 2-22

2.6.2.3 Examples............................................................................................ 2-25

2.6.3 Method of probability bound .................................................................. 2-29

CHAPTER 3 PERFORMANCE FUNCTION OF SLOPES........................3-1

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

3.2 BASIC EQUATIONS...................................................................................... 3-6

3.2.1 Limit equilibrium criterion ...................................................................... 3-6

3.2.2 Vertical equilibrium of a slice...................................................................3-8

3.2.3 Horizontal equilibrium of a slice.............................................................. 3-9

3.2.4 Moment equilibrium of a slice...................................................................3-9

3.2.5 Boundary conditions.................................................................................. 3-9

3.2.6 Overall moment equilibrium.....................................................................3-10

3.3 LIMIT EQUILIBRIUM MODELS.............................................................3-12

3.3.1 Taylor’s method........................................................................................ 3-13

3.3.2 Janbu’s method........................................................................................ 3-14

3.3.3 Bishop’s method........................................................................................ 3-15

3.3.4 Lowe and Karafiath’s method.................................................................3-16

3.3.5 Morgenstern and Price’s method............................................................ 3-16

3.3.6 Spencer’s method .................................................................................... 3-17

3.3.7 Bell’s method............................................................................................ 3-18

3.3.3 Method of transmission of unbalanced thrust (TUT).........................3-18

3.3.9 Pan’s method............................................................................................ 3-19

3.4 FORMULATION OF PERFORMANCE FUNCTION 3-20

Table of Contents iii

CHAPTER 4 PROBABILISTIC MODELLING

OF SOIL PROFILES........................

4.1 INTRODUCTION................................................

4.2 HISTORICAL DEVELOPMENT........................

4.3 RANDOM FIELD MODEL................................

4.3.1 Type I soil profile............................................

4.3.2 Type II soil profile............................................

4.3.3 Type III soil profile............................................

4.4 STATISTICS OF SPATIAL AVERAGES . . .

4.4.1 Type I and II soil profiles................................

4.4.1.1 Variance reduction factor for line averages

4.4.1.2 Covariance factor for line averages . . .

4.4.1.3 Variance reduction factor for areal averages

4.4.1.4 Covariance factor of areal averages . . .

4.4.2 Type III soil profiles........................................

4.5 WHITE NOISE PROCESS................................

4.6 COMPOSITE RANDOM PROCESS................

4.7 NUGGET EFFECT............................................

4.8 SAMPLE SPATIAL AVERAGES........................

4.8.1 Type I and II soil profiles................................

4.8.2 Type III soil profiles........................................

4.9 NON-HOMOGENEOUS SOIL PROFILES . .

4.10 ILLUSTRATIVE EXAMPLE............................

4-1

4-1

4-2

4-5

4-7

4-13

4-1?

4-14

4-15

4-16

4-19

4-22

4-24

4-25

4-29

4-30

4-31

4-35

4-36

4-38

4-40

4-41

CHAPTER 5 STRUCTURAL ANALYSIS OF SOIL DATA........................5-1

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

5.2 TYPE I SOIL PROFILES 5-2

Table of Contents iv

5.2.1 Estimation of mean value.......................................................................... 5-2

5.2.2 Estimation of point variance...................................................................... 5-5

5.2.3 Estimation of trend variance.................................................................... 5-12

5.2.4 Estimation of correlation structure........................................................ 5-14

5.2.4.1 Sample ACVF....................................................................................5-15

5.2.4.2 Sample ACF........................................................................................5-22

5.2.4.3 Sample variogram ............................................................................ 5-23

5.2.5 Parameter estimation of autocorrelation models............................5-28

5.2.5.1 Fitting by ‘eye’.................................................................................... 5-28

5.2.5.2 Variance plot .................................................................................... 5-30

5.2.5.3 Curve fitting by least squares ........................................................ 5-39

5.2.6 Effects of regularization............................................................................ 5-41

5.3 TYPE II SOIL PROFILES........................................................................ 5-48

5.3.1 Introduction................................................................................................5-48

5.3.2 A simplified procedure............................................................................ 5-50

5.3.3 Iterative least squares method................................................................ 5-51

5.3.4 Maximum likelihood estimation............................................................ 5-54

5.3.5 Filtering out of the trend component.................................................... 5-59

5.4 TYPE III SOIL PROFILES........................................................................ 5-61

5.5 PLANNING OF A SITE INVESTIGATION............................................ 5-63

CHAPTER 6 PROBABILISTIC DESIGN OF SLOPES ............................6-1

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

6.2 HISTORICAL DEVELOPMENT.................................................................. 6-2

6.3 REVIEW ON EXISTING APPROACHES .............................................. 6-3

6.4 /?-APPROACH................................................................................................ 6-11

6.5 /^-APPROACH 6-15

Table of Contents v

6.6 METHOD OF PDF FITTING....................................................................6-16

6.6.1 Bounds of performance function - Rigorous method ....................... 6-17

u.6.2 Bounds of performance function - Simplified method....................... 6-18

6.6.3 Bounds of performance function for Spencer’s method ................... 6-19

6.7 SYSTEM RELIABILITY OF SLOPES....................................................6-24

6.8 PROBABILISTIC MODELLING OF PORE-WATER PRESSURE . 6-25

6.9 ILLUSTRATIVE EXAMPLES....................................................................6-29

6.9.1 Example 6.1................................................................................................6-33

6.9.1.1 Adjustment of A................................................................................6-33

6.9.1.2 Accuracy of linear approximation for G(Y) ............................6-35

6.9.1.3 Comparison of different approaches........................................... 6-39

6.9.1.4 Influence of interslice force function on Pf.................................... 6-39

6.9.1.5 Influence of the form of ACF on Pf................................................ 6-41

6.9.1.6 Influence of scale of fluctuation........................................................ 6-42

6.9.1.7 Location of Critical Slip Surface.................................................... 6-42

6.9.2 Example 6.2 (Selset landslide)................................................................ 6-44

CHAPTER 7 LOCATION OF CRITICAL SURFACE.................................. 7-1

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

7.2 DEFINITION OF PROBLEM...................................................................... 7-4

7.2.1 Non-circular slip surface .......................................................................... 7-4

7.2.2 Circular slip surface..................................................................................7-6

7.3 SEARCHING PROCEDURE...................................................................... 7-7

7.4 ILLUSTRATIVE EXAMPLES.................................................................... 7-10

7.4.1 Example 7.1................................................................................................7-10

7.4.2 Example 7.2................................................................................................7-14

Table of Contents vi

CHAPTER 8 LIMITATIONS AND SUGGESTIONS...................................... 8-1

CHAPTER 9 CONCLUSIONS.......................................................................... 9-1

REFERENCES ............................................................................................... R-l

APPENDIX A PARTIAL DERIVATIVES OF

PERFORMANCE FUNCTIONS........................................ A-l

A.l COHESION............................................................................................... A-l

A.2 AWi........................................................................................................... A-2

A.3 PORE-WATER PRESSURE................................................................ A-3

A.4 COEFFICIENT OF INTERNAL RESISTANCE................................ A-5

A.5 A Qi........................................................................................................... A-6

A.6 END FORCES ....................................................................................... A-7

APPENDIX B PROOF OF EQUALITY OF p AND ftHL

FOR LINEAR PERFORMANCE FUNCTIONS..................B-l

APPENDIX C FORMULAE FOR VARIANCE REDUCTION

AND COVARIANCE FACTORS ..........................................C-l

APPENDIX D SAMPLING VARIANCE OF VARIANCE PLOT ... D-l

Table of Contents vii

APPENDIX E TECHNIQUES OF RATIONAL APPROXIMATION . . . E-l

E.l SOLVING NON-LINEAR EQUATIONS ..................................................E-l

E.2 OPTIMIZATION..........................................................................................E-3

E.2.1 Univariate function ..................................................................................E-3

E.2.1.1 Theory..................................................................................................E-3

E.2.1.2 Illustrative Example..........................................................................E-8

E.2.2 Multivariate functions..............................................................................E-9

E.2.2.1 Technique of alternating variables..................................................E-9

E.2.2.2 Method of steepest descent............................................................E-ll

APPENDIX F SOIL DATA..................................................................................F-l

ABSTRACT

The implementation of first-order-second-moment approaches of slope design

is discussed. This study features a number of improvements and extensions to the

current approaches.

A new and much simpler solution scheme is developed for Morgenstern and

Price’s method . This enables a c-(f) slope with an arbitrary slip surface to be

analysed using a rigorous stability model.

The random field model which is now generally used for probabilistic charac

terization of soil profiles are extended to cover non-homogeneous slopes. A series

of formulae are also developed by which covariances of the spatial averages along

a general slip surface can be evaluated. A procedure is also developed to take

account of sampling uncertainty of the soil properties.

This study advocates the use of reliability index /3hl defined by Hasofer and

Lind. This index possesses the advantage of ‘invariance’ which is lacking in the

conventional reliability index /? defined in Cornell’s sense. Furthermore, a new

probabilistic approach based on the technique of curve fitting is proposed for the

analysis of slopes. This new approach utilizes the additional information of the

lower and upper bounds of the soil properties to produce a better estimate of the

failure probability.

Implementation of the probabilistic approaches based on reliability index (3,

reliability index (3hl and the method of curve fitting will be illustrated by examples

viii

ABSTRACT ix

and comparison of these three different approaches is also made.

An optimization algorithm is also developed for locating the critical slip sur

face with the maximum failure probability.

ACKNOWLEDGEMENTS

I wish to express my sincere gratitude to Professor Peter Lumb who has

aroused my interests in geotechnical engineering and statistics, and suggested

the need for this study. He also supervised the early part of this work while I

was studying in the then Department of Civil Engineering (now Department of

Civil and Structural Engineering), University of Hong Kong. I am also grateful to

Professor Y.K. Cheung, Head of Department of Civil and Structural Engineering,

University of Hong Kong, for permission to transfer the study from the University

of Hong Kong to the University of New South Wales, Australia.

Special thanks must also be extended to Professor Ian K. Lee, Head of De

partment of Civil Engineering, University College, the University of New South

Wales, who kindly accepted my transfer of study from the University of Hong

Kong and supervised my PhD study in the Department. His concern and support

has also made my study in Australia an enjoyable experience. I am also indebted

to my co-supervisor Mr. Weeks White, Senior Lecturer, Department of Civil En

gineering, University College, the University of New South Wales, with whom I

have had many useful discussions. His constant encouragement and interest in my

work are grateful acknowledged.

Thanks are also due to Dr. J. Petrolito for his kind permission for using his

TgX-macros for type-setting this thesis.

It should be mentioned with gratitude that my study in the Department of

ACKNOWLEDGEMENTS xi

Civil Engineering, University College, the University of New South Wales was

supported by the Dean’s Scholarship without which my study in Australia would

have been impossible.

Last but not the least, I am very fortunate to have an understanding wife,

Melinda, whose constant support from the Northern Hemisphere is deeply felt in

Australia.

NOTATIONS AND ABBREVIATIONS

MATHEMATICAL SYMBOLS

{ }-1 inverse of a matrix

{ }T transpose of a matrix

V gradient operator

LATIN SYMBOLS

B(-) covariance factor for Type I or Type II soil profiles

C(-) autocovariance function (ACVF)

d effective cohesion of a soil

cov{-} covariance operator

D{} covariance factor for Type III soil profiles

E{•} expectation operator

E{ horizontal interslice force

F factor of safety

f(x) interslice force function

fx_(x) probability density function of X

G[X) performance function

Gf(2Q performance function of slopes based on overall horizontal force equi

librium

Gm(X_) performance function of slopes based on overall moment equilibrium

xii

NOTATIONS AND ABBREVIATIONS xiii

/ an integral used in the calculation of the covariance factor of line aver

ages

J an integral used in the calculation of the covariance factor of line aver

ages

L length of a spatial domain

A/ sampling interval

Alt length of the base of a slice

N total number of samples

Nn total number of moving averages of n samples

Ny_ total number of samples pairs having a lag distance of v

Pf failure probability

Pff failure probability inferred from the method of PDF fitting

AP{ external concentrated vertical loads acting on a slice

Pj the ;th term of the generalized polynomial

Pr(E) probability of an event E

Pi total external vertical loads averaged over the width of a slice

AQi external concentrated horizontal loads acting on a slice

r pore-water pressure ratio

s sample standard deviation

s2 sample variance

s2 corrected sample variance

sf sample variance of a property regularized over a length l

sf corrected sample variance of a property regularized over a length /

T{ vertical interslice force acting a slice

t coefficient of internal resistance of a soil

t location of a point in a soil profile

ut pore-water pressure

NOTATIONS AND ABBREVIATIONS xiv

u;V

W

X

x

y{x)yi(z)

yP(x)

yu(x)

z

total thrust exerted on the interslice boundary by pore water

dimension of a spatial domain

white noise intensity

position of the phreatic surface

total external vertical force acting on a slice

set of variables

a value of X

position of slip surface

the lower bound of phreatic surface

the most probable position of phreatic surface

the upper bound of phreatic surface

set of standardized random variables

GREEK SYMBOLS

a a multiplying factor for the calculation of the trend variance

cti a multiplying factor for the calculation of the trend variance of regu

larized soil properties

P reliability index in Cornell’s sense

Phl reliability index in Hasofer and Lind’s sense

r2(-) variance reduction factor v 7 soil density

7(-) semi-variogram

6 scale of fluctuation

e random component

k a soil property

ky spatial average soil property over a domain V

ky sample spatial average

A a multiplier for the interslice force function in Morgenstern and Price’s

NOTATIONS AND ABBREVIATIONS xv

method

// mean value

E2( ) variance reduction factor for Type III soil profiles

a standard deviation

cr2 variance

v lag distance

$(•) cumulative distribution function of a standard normal variate

(f> angle of internal resistance of a soil

SUPERSCRIPTS

estimator of a variable

— sample mean value of a variable

spatial average of a variable

ABBREVIATIONS

ACF autocorrelation function

ACVF autocovariance function

AM autocorrelation matrix

CDF cumulative distribution function

COV coefficient of variation

EDA exploratory data analysis

FOS factor of safety

GLS generalized least squares

GPS generalized procedure of slices

ML maximum likelihoods

MSE mean square errors

PDF probability density function

NOTATIONS AND ABBREVIATIONS xvi

RCOV coefficient of variation based on mean square errors

CHAPTER 1

INTRODUCTION

1.1 GENERAL INTRODUCTION

In slope stability problems, the calculated factor of safety has been used for

decades for assessing the reliability of slopes. It has been known for a long time

that the factor of safety is not a consistent measure of risk since slopes with the

same factor of safety can have widely different levels of reliability depending on

the variability of soil properties. The choice of a suitable factor of safety would

therefore rely heavily upon the engineer’s subjective interpretation of the data.

As far as a codified design is concerned, it is always desirable to have a de

sign criterion which is objective. The use of the probability of failure has been

advocated for this purpose. As the goal of a slope design is to minimize the risk

of failure at the most reasonable cost, the use of failure probability should be

the most objective decision rule. A probabilistic approach also has the following

advantages.

• The interpretation of data can be done using a formal statistical procedure;

• It enables design parameters to be updated when more information becomes

available;

• It enables decision analysis to be performed for choice of a suitable design

scheme and site investigation program;

1-1

1.1. GENERAL INTRODUCTION 1-2

• The consequences of failure can be taken into account so that the expected loss

could be maintained at a small level.

Looking at the advantages of a probabilistic approach, Olsson (1983) expressed

the view that the sooner we get rid of the safety factor the better. Despite its

advantages, the soil engineers have been slow to adopt a probabilistic approach in

slope design or geotechnical design at large. This may be attributed in part to the

lack of familarity of engineers with probabilistic methods. Although a number of

good books have already been published on probabilistic structural design such as

Bolotin (1973), Ghiocel and Lungu (1975), Leporati (1979), Ang and Tang (1984)

and Madsen et a/ (1986), there is no comparable book for the soil engineers.

Although the work by Lumb (1974) is still the most comprehensive and relevant

reference for researchers to date, it seems to be too complex for use by practising

soil engineers.

The advocates of probabilistic approach should also carry some of the blame

for the slow adoption of such an approach. Although the literature on geotechnical

reliability analysis has now been extensive, many of the publications are filled with

misconceptions and the predicted values of failure probability based on incorrect

models are sometimes so high that the factor-of-safety users would simply be scared

away. A very recent work by Kuwahara and Yamamoto (1987) serves as a very

typical example to illustrate this point. Not knowing the importance of variance

reduction arising from spatial averaging of soil properties, they came up with the

predictions of failure probability as shown in Fig. 1.1 for different modes of failure

in a braced excavation. The predicted value of failure probability is so high that

a factor of safety of at least 3 would be required to limit the risk of failure to an

acceptable level. Kuwahara and Yamamoto (1987) went even further to suggest

the following design level of failure probability Pf for different modes of failure.

• Pf — 0.17 for the bending failure of sheet piles;

1.1. GENERAL INTRODUCTION 1-3

Figure 1.1Failure Probability of Braced Excavation (a) Bending Failures of Sheet Piles

(b) Strut Buckling (c) Toe Failure (d) Heaving (after Kuwahara and Yamamoto, 1987)

1.2. SCOPE OF THE PRESENT WORK 1-4

• Pf — 0.16 for the strut buckling;

• Pf — 0.28 for the toe failure and;

• Pj = 0.15 for the base failure by heaving.

If the above design criteria are adhered to, at least one out of five braced

excavations would fail in one of the above four failure modes. If this is what a

probabilistic approach can offer, why would a soil engineer bother to give up a

factor of safety approach when the profession has been living with it for decades,

although often not without the engineer’s own frustration.

1.2 SCOPE OF THE PRESENT WORK

In light of the above discussion, the objectives of the present study are to:

1. point out some of the fallacies and misconceptions prevailing in current ap

proaches and;

2. develop a general probabilistic model which is applicable to c-(f) slopes with

general slip surfaces, incorporating modern developments such as the unified

solution scheme for of the generalized procedure of slices for the formulation of

the performance function and the random field model for the characterization

of soil profiles.

The outline of the work is as follows. Chapter 2 briefly reviews the theory of

probabilistic design. In Chapter 3, a new solution scheme for the generalized pro

cedure of slices is presented. This scheme greatly simplifies the calculations and

enables the probabilistic analysis of slopes to be performed using a rigorous stabil

ity model. Chapter 4 discusses the random field theory which has been extended

to cover non-homogeneous soil profiles. The relevance of sampling uncertainty is

pointed out and a procedure is devised whereby this uncertainty can be accounted

for in the analysis. A series of formulae are also developed to facilitate the calcula-

1.2. SCOPE OF THE PRESENT WORK 1-5

tion of the variances and covariances of spatial averages. Chapter 5 is an overview

of the procedure for estimating the statistical parameters of soil properties. In

Chapter 6, features of the probabilistic approaches to slope design are discussed

and compared. In particular, a new probabibistic approach is proposed whereby

information on the bounds of the soil properties can be incorporated into the an

alysis to produce a sharper estimate of the failure probability. Example problems

are also presented to illustrate the implementation of various approaches. Chap

ter 7 presents an optimization algorithm for locating the most critical slip surface

with the greatest failure probability. The limitations of the present study will be

discussed in Chapter 8. Finally, the main conclusions drawn from this study are

summarized in Chapter 9.

To appraise the fallacies of the current approaches, it is required to have the

pre-requisite knowledge of the concepts of probabilistic design and random field

theory. Because of this, the review of current approaches to slope design are

presented in Chapter 6 and not earlier. Readers familiar with the above concepts

may wish to look at Sections 6.2 and 6.3 before commencing to read Chapter 2.

CHAPTER 2

THEORY OF PROBABILISTIC DESIGN

2.1 INTRODUCTION

The methods of risk analysis can be categorized into three basic levels, namely

Levels I, II and III, depending on the rigour and sophistication of the analysis.

Level III methods refer to the complete analysis in which the random variables

are represented by their joint probability density function (PDF) and the failure

probability is calculated by performing the integration of the joint PDF over the

entire failure domain of the random variables.

Level II methods are approximate probabilistic procedures in which random

variables are characterized by their mean values and variances. The reliability of

the system is expressed in terms of some consistent safety measures such as the

reliability index which can often be related to the failure probability using some

approximate equations.

Level I methods are the most primitive level of risk analysis. Uncertainty of the

problem is lumped into a single index of the factor of safety which is calculated

using the mean values (or some arbitrarily chosen value, e.g. mean minus one

standard deviation) of the soil parameters. The variability of the input parameters

is not considered explicitly in the analysis, but often accounted for in design by

so-called ‘worse case’ evaluations. For instance, use of Fmin = 1.3 for ‘normal’

2-1

2.2. PERFORMANCE FUNCTION 2-2

conditions and 1.1 for situations when the water-table reaches an extreme level.

However, the choice of a suitable value of F for each ‘worse case’ is usually quite

arbitrary. Because of this, the factor of safety does not give a consistent measure

of risk and the choice of which would therefore be based on experience of the

enginers. The so-called ‘local’ experience is usually region specific. The factor of

safety which is adequate for a particular region having a particular soil variability

may not be suitable for other areas with different soil variability.

The state-of-the-art of probabilistic design are well discussed in Ang and Tang

(1984) and Madsen et a1 (1986). This Chapter briefly reviews the three levels of risk

analysis. The merits and shortcomings of various methods will also be critically

discussed.

2.2 PERFORMANCE FUNCTION

In general, the performance or the response of an engineering system can

be depicted explicitly by means of a mathematical expression or implicitly by a

computational procedure such as a computer program. Such a function is called

the performance function or the limit state function G(X) in probabilistic design,

namely

G(X) = G(X1,X2,'",Xl) (2.1)

where X = (X1} X2, • • •, Xi) is the vector of input parameters. The input param

eters can be subdivided into:

1. stochastic parameters, that is parameters which are random in nature, e.g. the

soil strength and;

2. deterministic parameters which are constant in value or parameters whose vari

ability can be neglected in practice.

2.2. PERFORMANCE FUNCTION 2-3

As we are concerned with the probabilistic aspects of the analysis in this work,

the deterministic parameters will not be written out in the expression. In what

follows, X_ will mean the collection of random input parameters.

The performance function is usually formulated in such a way that failure of

the system is signified by G(X’) < 0 and safety by G(X) > 0. Here ‘failure’ and

‘safety’ are taken in the most general sense. For example, the performance function

may describe the settlement of a foundation which is said to have ‘failed’ if the

average settlement has exceeded the allowable value. Furthermore, there may be

more than one performance function for a system. For instance, it is common to

have two performance functions in foundation design - one for the stability and

another for the settlement. The failure probability of the system is given by

P; = Pr(G(X) < 0) (2.2)

The hypersurface, defined by the equation

G(X) = 0 (2.3)

therefore partitions the multi-dimensional (stochastic) parameter space into two

distinct regions, namely the safety region 5 in which G(X) > 0 and the failure

region Jin which G'(X) < 0. Such a surface is called a limit state surface or

boundary.

In analyzing the stability of a slope, it is possible to separate the forces into

two basic components, namely the resisting component R(X') and the disturbing

component S(X) (Li and White, 1987c). R(K) and .S(2Q are functions of the

more fundamental variables such as soil strength, pore-water pressure and density

etc. The performance function can be defined as a safety margin, viz

G{X) = R(X) - S(X) (2.4)

2.3. LEVEL I DESIGN 2-4

As R(2Q and S(X) are positive quantities, the performance function can also be

formulated in the following equivalent formats.

A more detailed discussion of the performance function will be given in Chapter

Fig.2.1 shows a schematic representation of Eqn.2.2. The failure probability

is given numerically by the volume bounded by the probability density function

(PDF) and the R-S plane within the failure region. For convenience, the point

(/?, 5) defined by the mean values of the distribution of R(K) and 5(X) will be

called the centroid of the distribution.

As a general rule in slope design, the further away from the limit state bound

ary the centroid is, the smaller will be the volume PQRS and hence the smaller

will be the failure probability.

2.3 LEVEL I DESIGN

A Level I design is the most primitive procedure of probabilistic design. In this

approach, the factor of safety (FOS), F, is used as a measure of the safety of slopes.

Typical examples include Bishop (1955), Janbu (1973) and Morgenstern and Price

(1965). A review of the existing FOS approaches is given in Li and White (1987c).

There is no common consensus as to how the factor of safety should be defined.

Typically, an overall factor of safety is introduced to the resisting component to

obtain the so-called ‘mobilized’ resistance, Rm()Q, viz

(2.5)

(2.6)

3.

(2.7)


Probability density function of R J S

R > Sor G(R,S)>0Safety Region

volume = P(

R=Sor G(R,S) = 0

Limit State BoundaryR ^S

or G(R,S) <0 Failure Region

Probabilitycontours

R > S

volume = P,

R= S

( b )

Figure 2.1Joint Probability Density Function of R and S

(a) 3-D View (b) Probability Contour


The value of the factor of safety is such that the following design equation is

satisfied.

G(Rn(X),S(X))=0 or G(^p-,S(20) = 0 (2.8)

The slope is deemed to be sufficiently safe if the calculated value of F is greater

than some specified minimum value stipulated in the design code.

R

s R = S

Figure 2.2 Overall Factor of Safety Approach

The basic idea of the overall FOS approach is to ensure that the centroid of the

distribution is sufficiently far away from the limit state boundary. The specification

of a minimum value of F is equivalent to stipulating a minimum distance AC

(Fig.2.2) between the centroid and the limit state boundary. Obviously, the larger

the value of F is, the smaller will be the failure probability.


The reason that an overall FOS is only applied to the resisting component,

although not mentioned by the proponents of the FOS approach, may be justified

by the fact that the variability of the disturbing component, which is mainly due

to the weight of the soil mass, is usually less than that of the resisting component.

Therefore, the distance AC in Fig.2.2 is more important than BC in controlling

the failure probability.

AREA= A

Figure 2.3 Details of a Cohesive Slope

Although the FOS approach is simple to implement and methods are now

available whereby the value of F can be calculated very efficiently to the required

precision (Li and White, 1987a&c), it has several shortcomings which can be

discerned by the simple example of a cohesive slope as shown in Fig.2.3. The

first disadvantage of the FOS approach is the ‘variance’ of the definition of F,

that is, the value of F depends on how F is defined (Hoeg and Murarka, 1974;

Yong, 1967). With the notations given in Fig.2.3, the FOS is usually defined as

\Vldl -W2d2(2.9)


where c is the mean cohesive strength of the soil. However, some engineers prefer

to treat the soil mass W2 as contributing to the stability of the slope (particularly

if W2 includes a toe berm added for stability as indicated by the dotted lines in

Fig.2.3) and define the FOS as

cLR + W2d2W^d[

There can be a substantial difference for the computed value of F whether the

term W2d2 appears in the numerator as part of the resisting moment or in the

denominator as part of the overturning moment. The same argument applies to

the way pore-water pressure is treated in slope stability analysis. The pore-water

pressure term can appear either in the numerator or the denominator depending

on whether it is treated as a loading to the system or as a reduction to the strength

term.

Table 2.1 shows the FOS calculated using Equations 2.9 and 2.10 for some ac

tual slope designs. The difference in F is very significant in some cases. Therefore,

an ‘unsafe’ slope which has a FOS smaller than the specified value in the code may

become a ‘safe’ slope if an alternative definition is used for the calculation of F.

The second undesirable property of the FOS approach is that it is not a

consistent measure of structural safety. Table 2.2 shows the failure probability of

the slope in Fig.2.3 assuming Gaussian distributions and independence of average

shear strength and soil density. Vr and Vs in the table denote respectively the

coefficient of variation (COV) of the resisting and disturbing moment. A wide

range of values of Pj can be obtained for the same value of F. Therefore, specifying

a constant value of FOS cannot ensure a consistent risk level of slopes. As a

corollary, it is impossible to say how much safer a slope becomes as the FOS is

increased.

Fig.2.4 shows the variation of Pf with F assuming that the soil density is


Factor of safety based on

Cases Eqn.2.9 Eqn.2.10

1 1.57 1.22

2 1.70 1.24

3 0.63 0.75

4 0.81 0.87

5 0.74 0.83

6 0.67 0.70

8 2.00 1.67

9 1.03 1.05

10 0.97 0.87

11 1.05 1.22

Table 2.1.Variant Property of Factor of Safety

(after Yong, 1967; Yang, 1982)

constant and the cohesive shear strength is a Gaussian variate with a typical value

of 0.3 for COV. The parameter T2(L) is the variance reduction factor (discussed

in Chapter 4) which reduces the variance of the average strength below that of the

point value of the strength. (This effect arises because of the compensating effects

as a result of spatial averaging. Low values of strength, for instance, in some

locations are compensated by larger values at other locations. In consequence, the

fluctuation of the average strength and hence the variance is smaller). As indicated

in this example, the value of Pj is sensitive to the value of F within the typical


Vr Vs Pf

0.2 0.2 8.3 x icr2

0.2 0.05 5.0 x 10“2

0.1 0.2 2.3 x 10~2

0.1 0.05 7.8 x 10~4

0.05 0.2 9.6 x 10“3

0.05 0.05 1.4 x 10“8

Table 2.2.Variation of Pj with Variability of Soil Property for

a Constant FOS of 1.5 (after Lumb, 1983)

range of design FOS (1.2-1.5) when the variance reduction factor is smaller than

about 0.3 which is not uncommon for real slopes.

A partial FOS approach has been proposed as an alternative to the overall

FOS approach (Hansen, 1967; Lumb, 1970 and Meyerhof, 1970&1984). A larger

FOS is assigned to the variables with greater variability and vice versa. To design

under the partial FOS approach is equivalent to checking whether the point D in

Fig.2.5 defined by (R/Fi, F2-S) where F\ and F2 are the partial factors of safety, is

in the safety domain or not. The study of the partial FOS approach is therefore to

tailor the partial FOSs in such a way that the location of D can effectively control

the failure probability to a small value. Although the partial FOS approach is an

improvement over the overall FOS approach, it cannot eliminate the shortcomings

of the FOS approach. Because of the above drawbacks, the FOS of a slope is not


1.1 1.2 13 1.4 1.5 1.6 1.7 1.8 1.9

Figure 2.4 Variation of Pj with FOS

2.4. LEVEL II DESIGN 2-12

R

Figure 2.5 Partial Factor of Safety Approach

a satisfactory risk measure.

2.4 LEVEL II DESIGN

A Level II design is also commonly known as the first-order-second moment

(FOSM) approach. In this approach, the performance function is linearized by

means of a first order Taylor’s series approximation and the random parameters

are characterized by their first two moments (hence the name).

In the following, two probabilistic approaches within the framework of a Level

II design are discussed. The first one is based on the conventional reliability index

P defined in Cornell's sense, hereafter called the /^-approach. The second one is


based on the reliability index Phl defined in Hasofer and Lind’s sense, hereafter

called the Phl-approach. Both methods will be used in Chapter 6 for analyzing

the reliability of slopes.

2.4.1 /^-approach

Because of the drawbacks of the FOS approach mentioned in the previous

section, Cornell (1969) advocated the use of the reliability index (3 as an alternative

risk format to the conventional FOS. Given a performance function G(20, the

reliability index (3 is defined as

P = llGOG

(2.11)

where hg and oq are respectively the mean and standard deviation of the per

formance function.The reason for using p as a safety measure is based on the

following observation. Defining a new variable Z by

z=c(xy-jiGCTG

the probability of failure can be written as

Pf = Pr(G(X) < 0)

_ p / G(X) ~ ^ Vg \VG <?g

= Pr(Z < -P)—0

=1- ip(z)dz

= n-fi)

(2.13)

where xp(z) and ^(2) are respectively the probability density function (PDF) and

cumulative distribution function (CDF) of Z. As a CDF is always a non-decreasing

function, a one-to-one correspondence exists between the failure probability and


the reliability index. All the uncertainties of the random variables have been

suitably condensed into the single reliability index (3. Provided that the reliability

index for two different slopes are equal, they will have a similar risk level although

the variability of the random variables may be different in the two cases. This is a

great improvement over the FOS approach as a slope with the same FOS can have

widely different risk levels depending on the variability of the input parameters.

Figure 2.6 Probability Density Function of G(X)

Fig.2.6 is a schematic representation of the PDF of G(X). If the value of (3 is

larger, the mean value of G will be further away from the cutoff point G = 0 and

the failure probability will therefore be smaller. However, Eqn.2.11 is a variant

definition - the value of (3 depends on the definition of G()Q.

Table 2.3 shows the reliability indices for different formats of the performance

function. The dependence of {3 on the formats of G(X) is clearly evident.

2.4.2 /3H L-approach

Hasofer and Lind (1974) proposed an alternative invariant definition for the

reliability index. In this format, the random variables X_ are transformed into the

standardized uncorrelated space Z by means of an orthogonal transformation such


Formats 0

R-S F— 1JF’-V'+Vi

#-iF — 1

fSvr + vs

InfIn F

s/''Z + v£

Table 2.3.Formats of G'(X) and Reliability Index /? (F = /?/5)

that

= 0

i>ar{Zi} = 1 (214)

cov{Zi,Zj} = 0 i^j

Examples for transforming correlated variables into uncorrelated variables are

given in Ang and Tang (1984). Hasofer and Lind (1974) defined the reliability

index as the minimum distance between the origin of the Z_ space and the trans

formed limit state surface (i.e. OD in F:g.2.7). Point D is commonly called the

design point. To distinguish the reliability index defined in Hasofer and Lind’s

sense to that defined in Cornell’s sense, the former is denoted as (3hl-

The property of ‘invariance’ for /3hl is clear from its definition. As an exam

ple, let us consider the formats of G(X) given in Table 2.3. For the first format,

the limit state surface is defined by

R-S = 0 (2.15)


volume = R

Figure 2.7Definition of Reliability Index in Hasofer and Lind’s Sense

For the second format, the limit state surface is described by |r — 1 =0, which after

simplication, would also lead to Eqn.2.15. The same is true for the third format.

Since the limit state surfaces for different formats are the same, the minimum

distance of the transformed limit state surface to the origin of the Z_ space and

hence will also be the same.

Therefore, as far as the codified design of slopes is concerned, a suitable min

imum value of (3hl can be specified. The format of G'(X) need not be stipulated

2.5. LEVEL III DESIGN 2-17

as a result of the invariance of /3hl-

2.5 LEVEL III DESIGN

A Level III design is the most complete method of risk analysis. All the

random variables are represented by their joint PDF and the failure probability is

evaluated directly by multiple integrations extended over the entire failure domain,

via

where /x(^) is the joint PDF of X. Although Eqn.2.16 appears to be a simple

expression, the difficulties involved in evaluating the integral are tremendous. For

complex performance function, the derivation of the integration limits correspond

ing to the failure domain G'(X) < 0 is extremely difficult. However, a simple result

exists for the case where the performance function varies monotonically with one

of the random variables over the entire domain X_. This condition is normally

satisfied for slope stability problems. For example, the greater the strength of the

soil, the larger will be the resistance and hence the value of G(X). Without loss

of generality, let us suppose that G(X) is an monotonically increasing function of

X\. It can be shown that Eqn.2.16 can be re-written as (Harbitz, 1983)

where x\ is the value of X\ given the values of G, X2, •••, Xi. If G(X) is a

monotonically decreasing function of Xi, the inner most integral is integrated

from 0 to 00.

(2.16)

Pf = •'37^ dGdx^dxs ■ ■ ■ dxi (2.17)Okj .

Eqn.2.17 can be evaluated using numerical integration, but practicable only

2.5. LEVEL III DESIGN 2-18

when the dimension of X_ is small, say less than 5. Very often, repeated calculations

of Pf are necessary in an engineering analysis. A typical example is perhaps the

location of the critical slip surface of the slope. The value of Pf has to be evaluated

for each trial slip surface. In this case, the use of Eqn.2.17 will becomes very

expensive and exceedingly time consuming.

An alternative approach for evaluating Eqn.2.16 is by mean of simulation.

The Monte Carlo simulation technique is now well known and is commonly used

in reliability analysis of small problems. The accuracy of Monte Carlo simulation,

which is of order 1 /\/N where N is the number of simulations, is measured in the

statistical sense and in terms of the standard deviation of the probability estimate.

Another type of simulation which receives less attention is the number theoretic

methods (Hua and Wang, 1981). The error bound of the probability estimate ob

tained by number theoretic methods, which is of order 1/V, is absolute and can be

estimated at least in theory with the given knowledge of the performance function.

Unlike Monte Carlo simulation, the error bound for number theoretic methods de

pends on the nature of the performance function as well as the dimension of the

problem. Theoretically, the number theoretic methods are asymptotically more

efficient than Monte Carlo simulations. However, preliminary studies indicate

theoretically that number theoretic methods are preferred to Monte Carlo simula

tions only when N is exceedingly large, although a recent application by Goni and

Hadj-Hamou (1987) shows that the number theoretic methods gives more accurate

results than Monte Carlo simulation even for a relatively low value of N (of the

order of 103). Harbitz (1986) has recently proposed a procedure in which Monte

Carlo simulation is used in conjunction with the FCSM analysis to enhance the

efficiency of the simulation.

From a theoretical standpoint, a rigorous Level III design will give the most

accurate answer. In practice, a Level III method cannot be used for slope stability

2.6. APPROXIMATE LEVEL III DESIGN 2-19

analyses. This is because the joint PDF of soil properties is generally not known,

although the marginal distribution of the point property may be estimated with

some degree of certainty. Although assumptions can always be made to obtain an

answer, the value of such an analysis is lost and the tremendous effort mvolved

in the Level III calculations is not warranted. Of course, the Level III procedure

remains the only valid procedure, at least for some simple idealized cases, for

checking the validity of the Level II procedure. In the following, a number of

approximate Level III methods are discussed. These methods incorporate more

information than just the first two moments into the analysis and requires less

computation effort than a rigorous Level III analysis.

2.6 APPROXIMATE LEVEL III DESIGN

A number of approximate approaches has emerged over the past decade or

so, in attempts to provide an approximate solution to Eqn.2.16 and reduce the

computing effort of the equation. These approaches can be divided into three

main categories, namely the technique of Normal tail approximation, the method

of PDF fitting and the method of probability bound.

2.6.1 Normal tail approximation

The technique of Normal tail approximation is sometimes called the advanced-

first-order-second-moment (AFOSM) method. It can be proved (Madsen et al ,

1986) that if the performance function is linear and X follows a joint Gaussian

distribution, the reliability index Phl would be related to the failure probability

t>y

Pf = H-Phl) (2.18)

where <£(•) is the CDF of a standard Gaussian variate. If the performance function


is not highly non-linear and X_ is jointly Gaussian, Eqn.2.18 remains a good ap

proximation. Otherwise, Eqn.2.18 may give a poor answer. The basic idea of the

Normal tail approximation is to transform the non-Gaussian variates into some

kind of ‘equivalent’ Gaussian distributions so that Eqn.2.18 remains a valid ap

proximation. The mathematical formality of the Normal tail distribution is given

in Madsen et a I (1986) and Ditlevsen (1981&1983). Five different approaches of

Normal tail approximation have been proposed so far, namely Paloheimo and Han-

nus (1974), Rackwitz and Fiessler (1978), Chen and Lind (1983), Nishino et a 1 ,

(1984) and Der Kiureghian and Liu (1986).

The first four approaches are developed on the basis that the random vari

ables are independent of each other and each independent variable is represented

by means of an equivalent Gaussian distribution. For non-Gaussian dependent

variables, it is required to transform the variables into independent variables be

fore these approaches can be employed. This can be done by means of Rosenblatt’s

transformation (Rosenblatt, 1952; Hohenbichler and Rackwitz, 1981). However,

this transformation though simple in theory is very troublesome to implement

in practice and is usually highly non-linear. Although these approaches take ac

count of the non-Gaussian variables, they do not necessarily yield a better answer

using Eqn.2.18 due to the increase in non-linearity of the limit state surface. Fur

thermore, the answer is affected by the ordering of the variables taken in the

transformation (Madsen et aI , 1986).

The approach by Der Kiureghian and Liu (1986) is somewhat different to

the above four approaches. Knowing the marginal distributions and coefficient of

correlation of the variables, they devised a procedure for fitting a standard joint

Gaussian distribution to the variables. Der Kiureghian and Liu’s approach differs

from the previous methods in that the former seeks to obtain a joint PDF of X by

an equivalent joint Gaussian distribution whereas the latter aims at representing


each of the transformed independent variables by an ‘equivalent’ Gaussian dis

tribution. The approach by Dec Kiureghian and Liu for fitting a joint Gaussian

distribution appears to be simpler than Rosenblatt’s transformation required in

other approaches. If all the variables are independent, Der Kiureghian and Liu’s

approach will be the same as the method by Rackwitz and Fiessler (1978).

No comparison has yet been carried out to find out which approach of Normal

tail approximation will give the most accurate answer. Illustrative examples of

the Normal tail approximation can be found in Ang and Tang (1984) and Madsen

et a/ (1986). Further discussions and modifications of the AFOSM approach are

given in Veneziano (1974) and Fiessler et a1 (1979).

Der Kiureghian and Liu’s method has been used as one of the techniques for

slope stability analysis by Luckman (1987). Further discussion of this will be given

in Chapter 6.

2.6.2 Method of PDF fitting

The second category of approximate approaches is the method of PDF fitting.

As the name implies, the method of PDF fitting involves fitting an empirical

distribution to G(X). The failure probability is then inferred from the fitted

distribution. There are two ways by which the PDF can be fitted. The first

approach is to calculate the the statistical moments of the performance function

G(X). An empirical distribut ion is then fitted to G(X) by the Method of Moments.

Another approach involves fitting an empirical distribution using the knowledge

of the statistical moments as well as the bounds of G(X). These two approaches

will hereafter be called the approach A and approach B respectively.

2.6.2.1 Approach A

Knowing the joint PDF of X, the moments of G(X) can be calculated using

Gaussian quadratures or simulation. It should be mentioned that the simulation

technique is used herein as a numerical integration technique for calculating the


moments of G(X) whereas it is used directly for estimating the failure probability

in a rigorous Level III analysis. The moments of G(X) can usually be calculated

with a reasonable accuracy using a much smaller number of simulations than

that required for the estimation of the failure probability. The method of PDF

fitting avoids the trouble of having to find the limits of integration required for

Eqn.2.16. Furthermore, in performing the numerical integration, only the values

of G(X) at the quadrature points or the simulation points are required and the

exact functional form of G()Q does not need to be known. Therefore, the method

of PDF fitting is also applicable to implicit performance functions.

Grigoriu and Lind (1980) used the so-called optimal estimator, which is a

combination of two suitably chosen distributions, for the empirical distribution.

The parameters of the empirical distribution are obtained by matching the first

two moments of the optimal estimator and G(X). The difficulty of this method

lies in the choice of the component distributions in the formation of the optimal

estimator. The choice is very often guided by hindsight rather than foresight. The

use of this method is therefore somewhat limited. Parkinson (1978b&1983) and

Grigoriu (1983a) used the Johnson’s translation system of curves as the empirical

distribution. Grigoriu (1983b) used the lambda distributions while Li and Lumb

(1985) adopted the Pearson’s curves. Johnson’s curves, lambda distributions and

Pearson’s curves all have four parameters which can be obtained by matching the

first four moments of the theoretical and empirical distributions. The calculation

of moments of G(X) also necessitates the evaluation of n-dimensional integrals

for a set of n random variables. If n is large, the use of numerical integration is

also impracticable. For large problems, the approximate techniques developed by

Evans (1967&T972) and Cox (1979) can be used for the calculation of the moments.

2.6.2.2 Approach B

All physical quantities have bounds which may be estimated from test results


or based on the subjective judgement of an experienced engineer.

Knowing the bounds, mean value and variance of a physical quantity, it is

convenient to model the quantity by a beta distribution. The PDF of a beta

distribution has the form

ai and a2 are the lower and upper bounds of X. £q and t/2 govern the shape of

the distribution.

Fig.2.8 shows a wide variety of shapes covered by a beta distribution. Because

of this, a beta distribution would usually model the distribution of a bounded phys

ical quantity with a reasonable accuracy. For example, an excellent fit by a beta

distribution has been reported by Lumb (1970) for the PDF of soil strength and

by Mirza and MacGregor (1979) for the strength of steel reinforcing bars. The

estimation of the parameters of the distribution involves the first four sampling

moments (Elderton and Johnson, 1969). If the sample size is small, the sampling

variance of the third and fourth sample moments is large. A more realistic ap

proach is to establish the shape parameters *q and ia> on a larger sampling basis

for the physical quantity (for example, as reported in the literature). Assuming

that the shape parameters remain constant, the scale parameters ai and a2 can

be estimated from the sample mean value x and variance s2 using the Method of

Moments:

fx(x) <x (x - di)"1 • (a2 - x)U2 (2.19)

(2.20)

a<2 — CL\ +(iq 1^2 T 2)“ • (iq -f t'o + 3) o

[v\ 4- 1) • [v<2 -f 1)(2.21)

Alternatively, the bounds may be known for physical or engineering reasons, and

the shape parameters calculated as discussed below.


ii - a -ii = —3■1 a

/ k - oo l—00

normal distribution

Figure 2.8 Shapes of Beta Distributions (after Oboni and Bourdeau, 1985)


In this approach of PDF fitting, the bounds of the performance function are

firstly established from the knowledge of the bounds of the input parameters. This

is a problem of constrained optimization. The mean value and variance of G(Xj

can be evaluated using the FOSM method or by means of numerical integration

discussed above. Once the bounds, mean value and variance of G()Q are known,

a beta distribution can be fitted to the PDF of G(X). The failure probability

of the slope can then be inferred from the fitted distribution. The calculation of

probability for a beta variate is discussed in Harr (1977) and Kennedy and Gentle

(1980). The bounds of G(2Q would define the scale parameters a! and a2 of the

fitted distribution and the shape parameters can be obtained using

V\kl2 -21 - k

~k+lk2l -2k-l

HI

(2.22)

(2.23)

where / = (pG - «i)/crG and k = (a2 - Vg)/vg-

Obviously, the viability of the method of PDF fitting depends on the comput

ing effort required for the estimation of the bounds of G(X). The method will be

used in Chapter 6 for estimating the failure probability of slopes.

If the lower bound of G(X) is known, a Pearson’s curve can also be fitted to

G(X) by matching the lower bound and the first three statistical moments of both

distributions. The procedure is discussed in Li and Lumb (1985).

The method of PDF fitting has the advantage of simplicity and generality.

The method is reasonably accurate for well behaved distributions. The accuracy

may drop for very skewed distributions, but this is perhaps true for all approximate

methods.

2.6.2.3 Examples

Let us consider two examples of PDF fitting (Approach A) using Pearson’s


curve as the empirical distributions. For the first example, the function is G(20 =

X\ X2 — X3. The variables are taken to be independent and distributed as follows.

X\ : beta distribution

fxAx 1) a (*i “ 15)3(25 - £1)1-5

X2 : normal distribution

/i2 = 1.0 cr2 = 0.1

X3 : gamma distribution

fxAx3) « (x3 - 8) exp{—0.5(2:3 - 8)}

A Monte Carlo simulation is used to generate the empirical distribution for

(7(2Q using a total of 10,000 sets of (Xi,X2, A3). A Pearson curve is also fitted

to the PDF of G(X) by matching the first four moments. Fig.2.9 shows the

probability plot of the fitted distribution against the simulated distribution. The

fitted distribution gives excellent agreement over the range of probability level

from 0.001 to 0.999.

The second example considers the product Z of two independent standard

normal variates X\ and X2. The exact probability density function of Z is given

by (Kendall and Stuart, 1969),

fz(z) = -/STo(kl) (2.24)7T

where Ko(-) is the modified Bessel function of the second kind of zero order. The

probability integral of fz(z) is calculated from the tabulated values of the integral

of Ko(z) given in Abramowitz and Stegun (1970). Fig.2.10 shows the probability

plot for the event Pr{^ > c}. The fitted Pearson distribution gives satisfactory

approximation to the theoretical values.

TED

DISTR

IBU

TIO

N

FITT

ED


G P(g)

-10.0

SIMULATED DISTRIBUTION

(“)

G P(g)

SIMULATED D1STRIBUTIONd>)

Figure 2.9Probability Plot for Fitted and Simulated Distribution

(a) Full Range (b) Lower Portion

NO

IinaiHJLSIC

I X

3VX

J


P(z)

,~2 '

FITTED DISTRIBUTION

P(/)

Figure 2.10Probability Plot for Exact and Fitted Distribution for Z


2.6.3 Method of probability bound

The third approach is the method of probability bound. Knowing the mo

ments of G(X), it is possible to establish the upper bound of the failure proba

bility (Veneziano, 1979). The bounds are absolute, i.e. they are applicable for all

distributions having the same moments based on which the bound is established.

Unfortunately, this also means that the bound must necessarily be wide. The use

of the method is therefore limited.

CHAPTER 3

PERFORMANCE FUNCTION OF SLOPES

3.1 INTRODUCTION

There are several methods currently available for performing a slope stabil

ity analysis, viz, limit equilibrium method, limit analysis and the finite element

method.

Limit equilibrium methods consider the static equilibrium of the slip surface

in a state of incipient instability. It is perhaps the oldest numerical model for

the analysis of slope stability. The basic assumptions are (a) the failure criterion

is satisfied along the assumed slip surface and (b) the soil behaves as a perfectly

plastic material. Most of the existing limit equilibrium approaches are based on

the method of slices which was originated by Petterson in the 1910s (Petterson,

1955). The method was later extended by Janbu (1954&1973), Bishop (1955) and

Nonveiller (1965) to give the so-called generalized procedure of slices (GPS).

Limit equilibrium methods do not take account of the stress-strain relation

ship of the soil. The problem is therefore statically indeterminate. Assumptions

have to be made regarding the stress distribution within the sliding soil mass or

along the slip surface to obtain a solution. Numerous approaches to the GPS have

been proposed. They all differ in the assumptions used for the stress distribution.

Although the assumptions by these approaches vary widely, the numerical differ-

3-1

3.1. INTRODUCTION 3-2

ences of the solutions seem to be minimal (Duncan and Wright, 1980; Fredlund

and Ivrahn, 1976; Li and White, 1987c). This perhaps explains why the limit

equilibrium method can still survive this ‘modern’ age when people endeavour to

develop the most sophisticated model using, for instance, non-linear finite element

analysis.

An alternative approach called limit analysis (Chen, 1975) has also been used.

This approach derives the equations from the balance of energy at failure. How

ever, results obtained from limit analysis are essentially the same as those from the

limit equilibrium method (Chen, 1975). The approach is simple to apply for dry

homogeneous slopes and in many cases provides a closed form solution. However,

for a non-homogeneous slope with pore-water pressure, limit analysis is much more

difficult to implement than the limit equilibrium method.

With the advent of high-speed computers, the finite element method is becom

ing more and more popular in geotechnical analysis. The finite element method

is a powerful numerical tool for analyzing the stability of slopes as it can take ac

count of the stress-strain relationship of soils, follow the stress path which the soil

would experience during construction, and accomodate the changes in material

properties for different soil strata.

Though versatile it may seem, there are certain philosophical questions to

be addressed and technical problems to be solved before stochastic finite element

methods can used for probabilistic design of slopes. They are:

1. The input parameters required for finite element analysis are usually difficult

to obtain. For instance, in predicting the failure of a slope, one important

parameter is the initial stress state of the soil in the field. Except for special

projects, such information is usually not available. In the end, assumptions

which may be quite arbitrary have to be made to furnish an analysis. This

subjective uncertainty is not necessarily smaller than the model uncertainty


associated with a simpler model such as the limit equilibrium method.

Use of the finite element method also poses a problem in the definition of the

performance function. In some analyses (e.g. Kraft and Mukhopadhyay, 1977),

the performance function is defined as a function of displacement; the problem

is then to decide which displacement (at the toe, the crown or elsewhere?) and

what displacement should constitute ‘failure’ (i.e. for the value of G(X) = 0).

The choices here are as arbitrary as choosing allowable factors of safety. In

other cases (e.g. Ishii and Suzuki, 1986), the finite element analysis may be

used to predict stresses. The performance is then defined for instance as the

safety margin of the strength values minus the stresses predicted by the finite

element method. The problem is then to establish what should be the relation

between the spatial variability of the constitutive relationship and that of the

soil strength. Clearly, there is a relationship between the two. For instance,

stiffer soils tend to have higher strength values and hence the deformation

characteristics and the strength of the soil should possess a positive cross-

correlation. In some constitutive models, such as the linear elastic model,

there is no explicit relationship between the stress-strain relationship and the

strength of the soil, thus giving us no guidelines on how the spatial variability

of these two soil properties should be modelled in statistical terms. Other

constitutive models may predict the strength values to be used. Therefore,

the statistical properties of the soil strength are established once the statistical

properties regarding the spatial variability of the parameters of the constitutive

model are specified. Of course, it is easier said than done.

A recent study by Wong (1984) shows that the total uncertainty associated with

the definition of failure, the discretization of the continuum and the choice of a

suitable constitutive model in a finite element analysis can amount to 40% to

60% of the predicted answer. In this case, one would question the credibility


of using such a sophisticated model when it does not necessarily produce more

reliable results than simple classical methods.

Of course, there are problems such as settlement prediction which cannot be

handled by classical methods and the finite element method remains a powerful

tool for getting an answer.

2. In Chapter 4, a random field model will be introduced for modelling the stochas

tic nature of soil properties in the field. Test results do suggest that the spatial

variability of the soil parameters required in a limit equilibrium analysis (such

as strength and density) can be adequately modelled by the random field model.

To implement a stochastic finite element analysis, one has to know the spatial

variability of the constitutive relationship of the soil. Although a substantial

amount of work has been done on this subject of soil plasticity over the past

two decades or so, there are no relevant published test results regarding the

spatial variation of the parameters of constitutive models that warrant a proper

statistical analysis. Whether the random field model is suitable for modelling

the spatial variability stress-strain relationships is still unknown at this stage.

3. In a deterministic analysis, calculations only need to be performed once. In a

probabilistic analysis, the use of finite element models would usually require re

peated calculations of the performance function. For instance, using the FOSM

method, it is necessary to calculate the derivatives of the performance func

tion with respect to individual random variables. For a linear elastic analysis,

explicit expressions can be derived for the calculation of the derivatives (Ishii

and Suzuki, 1987). However, it is questionable whether a linear elastic finite

element model would produce a more accurate prediction than a limit equilib

rium analysis. For non-linear finite element analysis, the performance function

is invariably implicit. The derivatives would have to be estimated numerically,

say by finite difference approximation. As the soil properties for each element


should be regarded as random variables, there would be at least a total ofn

1 + J2 rt calculations, where n is the number of elements and rt is the numbert=i

of random variables for each element. For n = 50 and rt = 3, which is not

atypical, a large computing effort of 151 repetitions would be required. Unless

a more efficient procedure is available, the stochastic finite element method is

likely to remain a tool of academic research.

In current approaches, the performance functions of slopes are formulated

using simple stability models such as the friction circle method (Stoyan et al, 1979;

Forster and Weber, 1981), the ordinary method of slices (e.g. Yucemen et a1, 1973;

Harr, 1977; Vanmarcke, 1980; Lee et a1, 1983; Bao and Yu, 1985; Ramachandran

and Hosking, 1985), simplified Bishop’s method (e.g. Alonso, 1976; Tobutt and

Richards, 1979; Tobutt, 1982; Anderson et al , 1982; Felio et al , 1984; Moon,

1984; Bergado and Anderson, 1985), simplified Janbu’s method (e.g. McPhail

and Fourie, 1980; Prist and Brown, 1983; Ramachandran and Hosking, 1985).

These simplified models tend to give a larger model uncertainty especially for

the ordinary method of slices and simplified Janbu’s method (e.g. Li and White,

1987c). Rigorous models have only been used very recently by Luckman (1987)

who used Spencer’s method and Li and Lumb (1987) and Li and White (1987b&e)

who adopted Morgenstern and Price’s method. A formulation based on limit

analysis was also proposed recently by Gussman (1985).

In this work, the performance function is formulated using the GPS. Different

stability models will be discussed. However, only Morgenstern and Price’s (M&P)

(1965) method will be used in subsequent analyses. M&P’s method is chosen

because it is generally regarded as one of the accurate models and is considered

to be numerically more stable than many of the existing models (Li and White,

1987c). Furthermore, some existing models can also be regarded as special cases

of M&P’s method, for instance, simplified Janbu’s method, simplified Bishop’s

3.2. BASIC EQUATIONS 3-6

method and Spencer’s method.

Conventional solution schemes for the GPS involve two levels of iteration -

one for the calculation of the interslice forces and the other for updating the FOS

(see e.g. Janbu, 1973; Fredlund and Krahn, 1976). When used in the formulation

of the performance function of slopes, such a procedure cannot give an explicit

function. As a result, the derivatives of the performance function required for a

FOSM analysis have to be calculated numerically. A new solution scheme for the

GPS is proposed herein. The new scheme has the advantage that it provides an

explicit definition of performance function without recourse to iteration for the

calculation of the interslice forces. As a result, the derivatives of the performance

function can be explicitly defined and evaluated analytically. Furthermore, the

performance function is formulated in terms of the safety margin. The advantage

of this will become clear in Chapter 6. The following presentation follows closely

to that of Li and White (1987c).

3.2 BASIC EQUATIONS

Some of the symbols and notations used in the derivation of G(X) are defined

in Fig.3.1. The subscript * denotes properties pertaining to the tth slice and the

superscript ' represents effective stress properties. The symbol ~ represents the

spatial average of a slice and the slices are numbered from 1 to n in the postive

x direction. In the derivation of the following equations, it is assumed that Ait-

is sufficiently small so that the lines of action of the vertical force and the

normal force <rt • A/t can be assumed to pass through the mid-point of the base

of the slice. Suppose that the slip surface is in a state of limiting equilibrium, the

following equations hold.


AQ: -J-

'T;Al;

AREA = A:

Figure 3.1Definitions and Notations Used in the Generalized Procedure of Slices


3.2.1 Limit equilibrium criterion

According to the Mohr-Coulomb failure criterion, the average shear stress over

a slice tx is given by

where t is the coefficient of internal friction tan</>'. To simplify the calculation, it

can be assumed that the variation of the effective stress (a — u) or the coefficient

of internal friction t is uniform along the base of the slice. Assuming a uniform

variation is equivalent to neglecting the variance reduction due to spatial averaging

over the slice. In a slope stability analysis, the overall uncertainty is usually

dominated by the variability associated with the cohesion intercept c! and the pore-

water pressure u (Alonso, 1976). Therefore, neglecting the variance reduction for

t has a smaller effect on the predicted value of the failure probability, Pj. Because

of this, t is assumed to be uniform for each slice and represented by the point

property tt at the mid-point of the slice. Thus

where <rt = average total normal stress acting at the base of slice i

\L{ = average pore-water pressure at the base of slice i

c[ = average cohesion of soil at the base of slice i

3.2.2 Vertical equilibrium of a slice

By considering the vertical equilibrium of a slice, we obtain

(3.1)

f, = c' + (a, - «,) • t, (3.2)

bi Ax{ = AW{ + AT{ — T{ Ax{ tan a, i = 1, n (3.3)

where AW{ — Aiii + qiAxi + A Pi = PiAxi

'ji = average density of soil over slice i


3.2.3 Horizontal equilibrium of a slice

By equating the horizontal forces acting on a slice, we have

AEt = (-at tan at + fl)Axl - AQt i = 1, n (3.4)

By combining Equations 3.2, 3.3 and 3.4 to eliminate and ft, an alternative

expression for AE{ is obtained:

A Ex = clAxl + (AWi + ATi - UiAxi) ■ tt

- ^A(?t + (AWi + ATt) • tan at i=l,n (3.5)

where m, = ■

3.2.4 Moment equilibrium of a slice

By taking the moments about the centre of the base of each slice, we obtain

AOATt = 2+ Ei-i tan at + AE{ • nt + ——- • zQi) i = 1, n (3.6)

where nt =

3.2.5 Boundary conditions

By considering the boundary conditions at x = xo and x = xn, the following

are obtained.

T0=Ta

Tn=Tb

Eq = Ea

En — Eb

E)qn.3.8 can be re-written in the following form

Ta + ytATi = Tbl=\

(3.7)

(3.8)

(3.9)

(3.10)

(3.8a)


and similarly

E a + ^2 A Et = Ebi=\

(3.10a)

By using Eqn.3.5, Eqn.3.10a can be re-written as

7a + ^ j [ c'tAxt + (AWl + ATi - utAxt) • txi— i

- |AQ, + (AWi -f A Ti) • tan a, | = Eb(3.106)

Equations 3.8 and 3.10 are commonly referred to as the conditions of overall

vertical and horizontal equilibrium respectively. With n slices, there can only

by 3 x n equilibrium equations based on Newton’s Law of forces. These 3 x n

conditions have been utilized in deriving Equations 3.3, 3.4 and 3.6. Equations

3.8 and 3.10 are in fact based on the consideration of the forces at the boundaries

of the slope and should best be interpreted as such. However, in view of the fact

that the term overall equilibrium is so commonly used in the literature, the usual

terminology will be retained here.

3.2.6 Overall moment equilibrium

By taking moments about point O and equating the overall resisting moment

Mr to the overall disturbing moment Ms, we obtain

n^|c'A:rt 4- (A Wi + A Ti - utAxt) • ttt=i

n

• mt • ym.

• VQi + (AW,- + ATt) • ym. ■ tana, - AT, • x,f=( *■

(3.11)

+ (Ebyb - Eaya + Tbxb + Taxa)

in which the expression on the left of Eqn.3.11 represents Mr and that on the

right represents Ms- The resisting moment is due to the shear stress developed

on the slip surface; all other forces are considered to contribute to the disturbing

moment.


As mentioned earlier, all the 3 x n equilibrium equation have already been

utilized. Eqn.3.11 is in fact a redundant equation. To obtain a consistent solution,

one of the equilibrium equations derived earlier has to be omitted. It is convenient

to omit Eqn.3.6 for the nth slice and ATn can then be obtained by invoking the

condition of overall vertical equilibrium (Eqn.3.8) so that

n— 1ATn =rt-T„_, =Ta - Y AT, (3.12)

Given a particular set of values of random parameters, the slope is not nec

essarily in a state of limiting equilibrium. However, if the values of all but one

of the parameters are specified, the value of the remaining parameter required to

achieve a state of limiting equilibrium can be calculated. This would constitute

an extra unknown to the problem. Any of the parameters can be chosen to be the

unknown parameter. Here, cn is chosen arbitrarily as the unknown parameter.

Unknowns Number

ox n

Ti n

Ex ii -f 1

Ti n + 1

hi n — 1

Cn 1

Total 5n + 2

Table 3.1 Number of Unknowns for the Generalized Procedure of Slices

3.3. LIMIT EQUILIBRIUM MODELS 3-12

Equations Number

Failure criterion (Eqn.3.2) n

Vertical equilibrium (Eqn.3.3) n

Horizontal equilibrium (Eqn.3.4) n

Moment equilibrium (Eqn.3.6) n — 1

Overall Moment eqm. (Eqn.3.11) 1

Boundary conditions (Eqn.3.7 to 3.10) 4

Total An 4- 4

Table 3.2 Number of Equations for the Generalized Procedure of Slices

Table 3.1 summarizes all the unknowns in the GPS procedure and Table 3.2

summarizes all the equations based on limit equilibrium and boundary conditions.

The total number of unknows is 5n-f 2 while the total number of equations available

is 4n + 4. The number of unknowns therefore exceeds the number of equations by

n — 2. The analysis is therefore statically indeterminate, and n — 2 assumptions

regarding the stress acting on the slices have to made to obtain a complete solution.

3.3 LIMIT EQUILIBRIUM MODELS

As the constitutive relation of the soil is not considered in the analysis, the

problem is statically indeterminate. Assumptions have to be made to obtain an

answer. The assumptions are different for different stability models. In the fol

lowing, the equations are derived on the assumption that the slope is in a state of

limiting equilibrium. Some terminology which will be used later is defined here.


1. Approximate solution - As will be discussed later, the performance function of

slopes can be formulated based on the condition of overall force equilibrium or

overall moment equilibrium. In many existing models, the failure probability

inferred from the condition of overall force equilibrium is different to that from

the condition of overall moment equilibrium. These approaches are said to

yield an approximate solution.

2. Rigorous solution - If a stability model gives the same value of failure probabil

ity based on either the condition of overall force equilibrium or overall moment

equilibrium, it is said to have a rigorous solution.

3.3.1 Taylor’s method

The method of friction circle developed by Taylor (1948) is well-known. Tay

lor has also proposed a stability analysis based on the method of slices which is

reported in Sherard et al (1960). The salient features of Taylor’s method are the

assumptions that the inclination of the resultant effective interslice forces is equal

to the average angle of the slope face and that the slip surface is circular in shape.

Using Taylor’s assumption, we have

l7 = g - Vs = tan 0 i=l’n~1 (313)

where 0 is the average angle of the slope face and U* is the total thrust due to

pore-water pressure acting on the ith interface of the slices. Eqn.3.13 gives

Tt_! + AT{ = (£t_! + AEi - Uts) tan 0 i = l,n - 1 (3.14)

Combining Equations 3.5 and 3.14, we obtain

^ _ -Tt_! + [jEj-i 4- (c[ + (pt - ut)tj)miAxl - (AQt -f AVTt tanat) - £//] tan 61 — (mi ■ t{ — tan at) tan 9

i = 1, n — 1 (3.15)


where pt = (n — 1) assumptions have been made regarding the interslice

forces and therefore Taylor’s method overspecifies the system by yielding an extra

equation than required, hence inconsistency will arise. Therefore, Taylor’s method

will only give an approximate solution. In his original proposal, the factor of safety

of the slope is computed based on the condition of overall force equilibrium and

the condition of overall moment equilibrium is ignored in the analysis.

There is no reason why the analysis cannot be based on the condition of overall

moment equilibrium. The assumption of a circular slip surface is also unnecessarily

restrictive. The method should also be applicable to a non-circular slip surface.

Taylor’s method can be extended as follows. Eqn.3.I3 can be modified as

' T' = X tan# • E' (3.16)

where A is a variable to be determined. This would introduce an additional un

known A to the system and the number of unknowns and equations are now equal.

It is then possible to obtain a rigorous solution.

3.3.2 Janbu’s method

Janbu (1954) was the first to develop a equation similar to Eqn.3.10b for the

condition of overall force equilibrium. The method is applicable to a general slip

surface. Janbu’s method assumes that the locations at which the interslice forces

act are known. That is to say, the n — 1 values of /it are prescribed in the analysis.

For convenience, we can write ht = ijj(xi) • zt where rp(x) is a known function

describing the variation of the moment arm ht across the slip surface and Z{ is the

dimension of the ith interslice boundary. Again, Janbu’s method overspecifies the

analysis by yielding an extra equation.

Combining Equations 3.5 and 3.6, an explicit expression for ATt for the first


n — 1 slices can be obtained.

A Tt = 2 • | - Xt_ i + Ei-1 tan at{

+ [ (c- + (pi ~ Ui)ti) ■ mt • Axt

1 — 2rii • (£, • — tan at)

— (AQt + Atan at) ] • nt

t = 1, n — 1 (3.17)

Janbu (1954&1973) also proposed a simplified method by taking AT; = 0.

Janbu’s method can be generalized by defining the moment arm h{ using

in which A is a multiplier. In this way, the parameter A can be adjusted to give a

rigorous solution.

3.3.3 Bishop’s method

Bishop (1955) assumed that the slip surfaces can be approximated by circles

and derived an equation similar to Eqn.3.11 for the condition of overall moment

equilibrium. Bishop (1955) did not suggest any guidelines for the calculation of the

interslice forces. However, he observed that the factor of safety based on overall

moment equilibrium is insensitive to the assumptions used for the interslice forces

and he therefore suggested the use of ATj = 0 to simplify the calculation. Such a

method is usually called the simplified Bishop method. The assumption of circular

slip surface is too restrictive and it is possible to generalize the simplified Bishop

method to non-circular slip surfaces by Eqn.3.11.

ht = Aip(xi) • (3.18)


3.3.4 Lowe and Karafiath’s method

Lowe and Karafiath’s method (1960) is similar to Taylor’s method except

that the inclination of the effective interslice forces at the interslice boundary is

assumed to be the average of the inclination of the ground surface at the top of

the interslice boundary and the inclination of the slip surface at the bottom of the

interslice boundary. Denoting the average angle by <pi, the expression of ATt for

Lowe and Karafiath’s method is given, similar to Taylor’s method, by

A Ti =-Ti-i -f [Ej-j + (c[ + (pi - Ui)ti)mjAxi - (A Qt + A Wj tan c^) - Uf] tan (pi

1 — (mi • ti — tan at) tan <pi

f = 1, n — 1 (3.19)

The same extensions as Taylor’s method can be applied to Lowe and Karafiath’s

method.

3.3.5 Morgenstern and Price’s method

M&P’s (1965) method is generally accepted as an accurate method of slope

stability because it gives a rigorous solution. The original formulation by M&P

(1965&1966) is very complex and involves the solution of two simultaneous differ

ential equations. In the following, a much simpler solution procedure is proposed.

The salient feature of M&P’s method is the assumption that the interslice

forces T{ and E{ can be related by

T{ ~ A f(xi) • Ei i = 1, n — 1 (3.20)

where A = a constant to be evaluated ;

f(x) = interslice force function.


Eqn.3.20 can be re-written as

Ti_! + ATt = Afi • (Ei-i + AEi) i = 1, n - 1 (3.21)

in which /, = /(xt). Combining Equations 3.5 and 3.21, we obtain

A Tx =— -f tan a;

t

i = 1, n — 1 (3.22)

If A is fixed, n — 1 assumptions are made and the problem will be overspecified

by yielding one redundant equation. However, if A is treated as a variable to be

determined, a rigorous solution can be obtained.

3.3.6 Spencer’s method

Spencer (1967) assumed that the inclination of resultant force of Tt and Ei is

constant, that is

where d is a value to be adjusted to give a rigorous solution. By choosing f(x) =

1 and A = tan $, Spencer’s method becomes a special case of M&P’s method.

Spencer (1973) later extended his method and suggested that T{ and Ei may be

related by

Ti—• = tan d i = 1, n — 1 (3.23)

't

Tt—f = k{ tan 0 Ei

(3.24)

where ki is a prescribed set of coefficients. Eqn.3.24 is fact identical to Eqn.3.20

if we write f(xi) = ki and A = tantf. Hence Spencer’s method is no different to

M&P’s method.


3.3.7 Bell’s method

Bell (1968) suggested that the distribution of the normal stress crt along the

slip surface can be approximated by

x — X— A • pt cos2 at + <; sin 2tt — c,

xCi - XQi = 1, n (3.25)

where A and f are coefficients to be determined and Xo, xn and xCi are defined in

Fig.3.1. n assumptions have been used in Eqn.3.25, two more than required. This

explains why two additional coefficients have to be introduced to obtain a rigorous

solution. Combining Equations 3.3 and 3.25, we obtain

A Ti = Xpi cos2 at + fsin(27r— ------—)xCi *^o .

• ft — P;+(c' — utU) • tan ai | • AX{

* = 1, n — 1 (3.26)

where = 1 -f fitanat. The value of f can be obtained by invoking the overall

vertical equilibrium condition (Eqn.3.8a) giving

nTt, — Ta + ^2 I^Pt cos2 ' ft — Pi + (c[ - Uiti) ■ tan aj] • Ax,

( = ------------- —-----n--------- —-------------------------------------- (3-27)Esm 2*(§=^).fc. Ax,

t = l '

The remaining coefficient A can be adjusted to give the rigorous solution.

3.3.8 Method of transmission of unbalanced thrust (TUT)

This is a method commonly used in China (Guo, 1979). In essence, the

method assumes that the resultant of T; and E{ is inclined at an angle equal to

the inclination of base of the slice, that is

T' f— = tan al+l i = 1, n — 1 (3.28)


The analysis is based on the condition of overall force equilibrium. In fact, the

TUT method can be regarded as an approximate solution of M&P’s method. The

expression for AT, is the same as Eqn.3.22 except that A has a value of one and

/, is replaced by tana,+ 1. By writing

T,■rf = A tan a,+ 1 i = 1, n — 1 (35)

it is possible to obtain a rigorous solution for the TUT method using exactly

the same procedure as M&P’s method. The expression of AT, for this extended

method is the same as Eqn.3.22 except that /, is now replaced by tancq+i.

3.3.9 Pan’s method

Pan (1980) suggested that AT, can be expressed as

AT, = Xrji t = l,n—1 (3.29)

where 77, is a set of prescribed values describing the distribution of AT, across

the slip surface and A is a multiplier to be adjusted to obtain a rigorous solution.

Based on the linear elastic solutions for homogeneous slopes, Pan (1980) suggested

the following value for /c,, viz,

rji = x ’ Al• (tan 6 — tan a,) t = 1, n — 1 (3.30)

where x is a constant depending on the angle of the slope and the elastic constants

of the soil. Here, the multiplier A in Eqn.3.29 and the constant x in Eqn.3.30 are

combined together to form one single variable, also denoted by A for convenience.

Thus

AT, = A • AIT, • (tan# — tan a,) i = 1, n — 1 (3.31)

3.4. FORMULATION OF PERFORMANCE FUNCTION 3-20

It can be seen that all the expression of ATx discussed above involve only

the interslice forces on the left of each slice. With the known conditions at the

left x — x0, all the interslice forces can be calculated explicitly and successively

without the iteration required in conventional procedures.

3.4 FORMULATION OF PERFORMANCE FUNCTION

So far it has been assumed that the slope is in a state of limiting equilibrium.

To formulate the performance function, it can be argued as follows.

When the realization of the random parameters is such that the slope is in a

state of stability, the full strength would not be mobilized. Therefore, the overall

resisting moment Mr (left hand side of Eqn.3.11) based on full mobilization of

soil strength must be greater than the maximum overall disturbing moment Ms

(right hand side of Eqn.3.11) the slope can offer. On the other hand, if the slope

is in a state of instability, the maximum overall resisting moment Mr offered by

full mobilization of soil strength would not be sufficient to balance the disturbing

moment Ms acting on the slope. Therefore, the slope would be in a state of failure

or safety according as Mr < Ms or Mr > Ms. A performance function for the

slope can then be formulated as Mr — Ms. By referring to Eqn.3.11, the following

is obtained.

The subscript m signifies that the performance function is based on the condition

of overall moment equilibrium condition.

Gm(X) = ^ c[Axi 4- (AfE, + ATi - utAx,) • tx • mx • ym.t:

[AQ; • yQ{ -p (AIEt -F AT,-) • ?/m,- * AT, • xmi

- (Ebyb - Eaya +Tbxb +Taxa)(3.32)

Another performance function based on the condition overall horizontal force


equilibrium can be formulated as follows. The force En can be interpreted as the

total horizontal resistance that can be offered by the slope. In a state of limiting

equilibrium, En is equal to the disturbing horizontal force Eb as indicated by

Eqn.3.10. If the slope is in fact in a stable state, the total resistance En that is

available should be greater than Eb and vise versa. Therefore, by subtracting Eb

from En and utilizing Eqn.3.10b, the following performance function is obtained.

Gf{X) =^T j [ c'iAxi + (AIT, + AT, - ^Ax,) • t-

- j^AQ* + (AWi + AT,) • tancv, | - (Eb - Ea)(3.33)

The subscript / signifies the condition of overall force equilibrium. It should be

realized that Gm(X’) and Gf()Q possess their physical meanings (Mohr-Coulomb

failure criterion and equilibrium conditions satisfied) only at the limiting condition

of G(X) = 0. At any other conditions, G(2Q only serves as an indicator function

depicting the safety-failure state of the slope.

To be correct, the moment arms xm{ and ym. should be regarded as random

variables as a result of the random forces acting on the slices. However, the

variability of xm. and ym, should be small unless AX{ is large. For practical

purpose, xmi and ymi can be treated as deterministic quantities and measured

from the centres of the bases of the slices.

An advantage of the present scheme is the explicit definition of the perfor

mance functions through the solution procedure given in previous sections. Be

cause of this, the derivatives of the performance functions Gm(}Q and Gf(2Q can

be evaluated analytically. A comprehensive list of formulae for the derivatives of

Gm(K) and G/(X) based on M&P’s method is given in Appendix A.

As will be seen later in Chapter 6, the failure probabilities inferred from

Gm(K) and Gj(X) are in general different. However, the value of A can be adjusted

so that the value of Pf derived from both performance functions are equal. The


procedure of adjustment will be discussed in Chapter 6.

M&P’s method is chosen herein as the stability model for reasons given earlier.

However, other stability models based on the generalized procedure of slices can

be used in lieu of M&P’s method without much difficulty. By using the above

unified solution scheme, it is only necessary to provide one separable subroutine

for the calculation of AT; and another for evaluating the derivatives of ATt with

respect to the basic input parameters. The solution procedure for other models is

the same as that of M&P’s described in this work.

Studies indicate that all rigorous stability models give essentially the same

value of factor of safety (e.g. Duncan and Wright, 1980; Li and White, 1987d).

Although a comparison for the failure probability of slopes has not yet been done

for different stability models, it is believed that different rigorous methods will

also yield similar values of Pf.

CHAPTER 4

PROBABILISTIC MODELLING OF SOIL PROFILES

4.1 INTRODUCTION

As is commonly known, soil properties exhibit variations from point to point

even within a seemingly homogeneous soil profile. The soil properties will in

evitably fluctuate spatially and perhaps temporally in response to the changes

in the processes governing its formation such as material source, environmental

conditions and others.

Except at the sample points, the soil properties at a particular location in

the field are generally not known and therefore have to be regarded as random

variables. To properly model the stochastic nature of soil properties, infinitely

many random variables within the field have to be considered. The random field

model is now commonly used to characterize the stochastic nature of soil properties

(e.g. Vanmarcke, 1977a).

Soils generally exhibit plastic behavior although to a differing degree. As a

result, the stability of a soil slope tends to be controlled by the average soil strength

rather than the soil strength at a particular location along the slip surface. Also

the disturbing force acting on the slope is related to the average density of the soil.

The study of the statistical properties of spatial averages is therefore important

4-1

4.2. HISTORICAL DEVELOPMENT 4-2

in analyzing the stability of slopes.

This Chapter deals with the inherent variability of soils. The basics of the

random field model will be discussed and extented to cover more general types of

soil profiles. In particular, emphasis will be given to the study of the statistical

properties of spatial averages of soil properties. The relevance of sampling uncer

tainties is also discussed. This Chapter concentrates on the theory of probabilistic

modelling of soil profiles. The procedure for estimating the statistical parameters

of the random field model will be discussed later in Chapter 5. The results devel

oped here will also be used in the probabilistic analysis of soil slopes in Chapter

6.

4.2 HISTORICAL DEVELOPMENT

Unlike time series analyses, the study of spatial random processes has not been

one of the main streams of orthodox statistics. Although a spatial random process,

or random field as is called in some literature (e.g. Haining, 1977; Vanmarcke,

1984), bears some resemblance to a time series, there are importance differences

between the two. By its very nature, a time series has a time scale. There is a

natural distinction of past and future. Furthermore, the realization at a particular

instance of time depends only upon past events. Current approaches of time series

such as the autoregressive model, the moving average model and the autoregressive

moving average model are based on this fact.

A spatial random process deals with the variation of random quantities within

a spatial domain (or field). There is no such distinction of past and future and

the dependence of soil properties extends in all three directions. In consequence,

some of the classical theories of time series are not applicable to spatial processes.

Whittle (1954&1962) has done some useful work on extending the classical


Ki.H

K. . .1-lj

K .L J

K . -

K • -bJ-1

Figure 4.1Second Order Symmetric Autoregressive Model

for a Two-dimensional Random Field

time series theory to random fields. A general discussion on the subject is also

given by Raining (1977). In describing a random field, statisticians tend to specify

the model and then generate the mean and covariance. For example, Whittle

(1954) described a discrete two dimensional field by the following second order

symmetric autoregressive model (Fig.4.1)

Kij — a(/C{+ij + Ki-ij + Ki,j+l + Ki'j— i) -f stJ (4.1)

where a is a constant and slJ is a uncorrelated random process. Generalizing

Eqn.4.1 to the continuous case yields the autocorrelation function (ACF)

p(r) = ar • Ki(ar) (4.2)

where a is a constant, r is the radial lag distance in two dimension and K\{-) is

the modified Bessel function of the second kind and order one. The calculation of


the ACF from a given model is easier (although it is still very involved and the

procedure is discussed in Raining (1977)) than the inverse problem of finding the

model from a given ACF. Furthermore, the inverse problem does not necessarily

have an interpretable solution (Whittle, 1954; Raining, 1977).

Mining engineers and geotechnical engineers tend to specify directly the ACF

(or the variogram) without going through the procedure of model building as in

the case of Eqn.4.1. The choice of the ACF is based on convenience and also guided

by experimental results. The underlying model for the ACF is seldom of interest

to soil engineers. Provided that the assumed function is an admissible ACF and

fits well to the experimental ACF, it would be a good ACF from an engineer’s

point of view. Christakos (1984) gave some useful discussions on the criteria for a

function to be an admissible ACF.

The use of the random field theory was first introduced by Vanmarcke (1977a)

for modelling the probabilistic nature of a homogeneous soil profile. Vanmarcke

(1984) also presented the random field theory in a way that is most convenient for

application in a geotechnical reliability analysis.

In the 1960’s, a new subject called geostatistics began to take shape. The the

ory stemed from the need in the mining industry to characterize the stochastic na

ture of mineral ores. D.G. Krig is generally regarded as the creator of geostatistics,

but the mathematical formality was developed by G. Matheron and his associates

in Centre de Morphologie Mathematique, France. The standard references on the

subject are Matheron (1971), David (1977) and Journel and Huijbregts (1978).

There are many similarities between the theory of geostatistics and random

field. The major difference is that in the random field model, the correlation

structure of the spatial process is described by the autocovariance function (ACVF)

while the variogram is used in geostatistics. The ACVF and the variogram are

virtually the same, but the existence of the ACVF requires a stronger assumption

4.3. RANDOM FIELD MODEL 4-5

of second order stationarity which is not always satisfied for mineral ores. The use

of ACVF is more favoured by geotechnical engineers (e.g. Lumb, 1974<Vl975a)

partly because the concept of the ACVF is more in line with the theory of time

series and more importantly the ACVF normally exists for soil properties. For

this reason, the random field model will be used in this work, although results

established in geostatistics will also be quoted from time to time.

4.3 RANDOM FIELD MODEL

Denote the value of a soil property at a point f = (x, y, z) by /c((). In general,

K(t) can be decomposed into a trend component g(t) and a random component

e(t) with zero mean value, viz,

The trend component can be expressed as a polynomial which can be estimated

from test results at various locations tt within the field. For example,

where Pj are the terms PG = 1, Pi = x, P2 = y, P4 = z etc. For ease of reference,

the right hand side of Eqn.4.5 will hereafter be called a generalized polynomial.

While the soil properties can be measured continuously in a site investigation using,

for instance, CPT tests, there are usually physical and/or financial constraints on

the number of boreholes that can be sunk or the number of soundings that can be

k(L) = sit) + £(t) (4.3)

g(L) — ao + atx -F a2y + a3z + aAx2 + a5y2 -F aGz2 4- a7xy + etc (4.4)

For convenience, g(t) is written as

(4.5)


made within the field. Therefore, the trend with depth can normally be established

with a greater precision than the lateral trend component.

Except for the sample points, the realization (i.e. the actual value) of a soil

property at location < is not known and must therefore be regarded as a random

variable. The realization of a soil property at location l is in general different from

that at location V even within a so-called homogeneous soil profile. To model

the soil property correctly, one has to consider infinitely many random variables

at all locations t. This important probabilistic nature of the soil property has

not been properly recognized in much of the current literature on probabilistic

geotechnical analysis. Very often, the soil property is represented as a single

random parameter. The list for this is in slope stability analysis is overwhelmingly

long (e.g. Biernatowski, 1969,1976,1979V1987; Matsuo and Kuroda, 1974; Harr,

1977; Grivas et al, 1979; Grivas and Harr, 1979; Grivas and Nadeau, 1979; He and

Wei, 1979; Tobutt and Richards, 1979; McPhail and Fourie, 1980; Pentz, 1981;

Chowdhury, 1981; Forster and Weber, 1981; Grivas, 1981; Edil and Shultz, 1982;

McGuffey et al , 1982; Sivandran and Balasubramaniam, 1982; Tobutt, 1982;

Cheong and Subrahmanyam, 1983; Prist and Brown, 1983; Felio et al , 1984;

Moon, 1984; Nguyen and Chowdhury, 1984; Bao and Yu, 1985; Gussman, 1985;

Nguyen, 1985c; Ramachandran and Hosking, 1985; Young, 1985V1986; Wolff and

Harr, 1987). Examples in other fields of geotechnical engineering are also readily

available such as Hoeg and Murarka (1974), Ivovas and Yao (1975), Grivas (1979),

Smith (1985) and Goni and Ilaji-Hamou (1987) in retaining wall designs; Grivas

and Ilarr (1977), Madhav and Arumugam (1979), Webb (1980), Nguyen (1985a)

and Gao (1985) in foundation designs; and Krizek et al (1977), Chang and Soong

(1979), Gao (1985) and Koppula (1987) in settlement analyses; and Kuwahara and

Yamamoto (1987) in a braced excavation design.

This has the implicit implication that the soil property is perfectly correlated


over the soil profile which means that the realization of the property is the same

at all locations. For example, if the cohesive strength at point A is 10 units, the

strength at all other locations is also 10 units. It this is the true statistical rep

resentation of the soil profile, one sample will be adequate to establish the in-situ

property of the soil and there will be no uncertainty involved in the estimation of

the soil property. Obviously, this is not the case for a real soil profile. The assump

tion of perfect correlation will usually lead to gross over-estimation of variance of

the performance function.

In considering the variability of soil properties, three main types of patterns

can usually be identified for the soil profiles, as indicated in Fig.4.2 (Lumb, 1966;

Matsuo, 1976; Matsuo and Asaoka, 1977; Asaoka and Grivas, 1982).

4.3.1 Type I soil profile

For Type I profiles, K,(t) is composed of a constant mean trend and a random

term with constant statistical properties. This type of soil profile is best modelled

as a homogeneous random field in which the variation of k([) is described by means

of the first and second order statistical moments (Vanmarcke, 1977aV 1984).

E{k(Q} = g(t) = a0 = m = constant (4.6a)

var{/c(/)} = var{e(t)j — o1 — constant (4.66)

cov{K(t), k(|/)} = cov{e(t_), s(i')} = C(v) = a2 • p(v) (4.6c)

E{-}, var{-} and cov{-} are the expected value, variance and covariance respec

tively. C(v) and p(-) are respectively the autocovariance function (ACVF) and the

autocorrelation function (ACF), and v = (vx,vy,vz) = \t' — 11 is the lag distance

between the points t and

Table 4.1 gives some examples of one and two dimensional ACFs. The param

eter <5 is called the scale of fluctuation (Vanmarcke, 1977a& 1984) and is a measure

of the spatial extent within which the soil property shows a strong correlation. A


oCl


O<3

cdrd(X<vmItc

.2'53CO£

o£

GcdGJGo

-225


large value of <5 implies that the soil property is highly correlated over a large

spatial extent, resulting in a smooth variation within the soil profile. On the other

hand, a small value of 8 will indicate that the fluctuation of the soil property is

large.

Some researchers (e.g. Baecher et a/, 1980) use the ‘correlation distance’ as

an alternative measure to the scale of fluctuation. The correlation distance vQ is

defined as the value of v such that p(va) = e~l where e is the natural number.

The scale of fluctuation and the correlation distance are related to each other. For

example, for an one-dimensional Type 1 ACF, vQ = 8/2. Table 4.2 summarizes

the value of vQ reported in or derived from the literature.

Equations 4.6a and 4.6b concern only the statistical property at a particular

point, called the point property of the soil. On the other hand, Eqn.4.6c describes

the cross moment at two particular locations, called the cross point property of

the soil. The point properties, such as the coefficient of variation (COV) and

the distribution are now well documented (Lumb, 1966,1970&;1974; Hooper and

Butler, 1968; Schultze, 1971&1975; Krizek et a/, 1977; Baecher et a/, 1980; Lee

et a/ , 1983; Chowdhury,1984). However, information regarding the cross point

properties is relatively sparse.

In geostatistics, the cross property of a random field is described by the the

semi-variogram 7(4;), defined by

27(v) = E{k(Q - K(t')}2 (4.7)

where v = \t — t'\. For ease of reference, i(v) will simply be called the variogram

in this work. If the ACVF exists, 7(2;) is given by

7(v) = £7(0) - C(v) =o2- C(v) (4.8)


Material Property direction* Autocorrelation function

P{v)Correlation distance

v0 (m)Source

clay shear strength V exp{-0.234u} 4.27 Wu (1974)

unweathered cl ly shale liquid limit H 0.445vA'i (1.31v)** « 0 Lumb (1974 & 1975a)liquidity index H uncorrelated 0

weathered clay shale liquid limit H uncorrelated 0liquidity index H uncorrel ated 0

sand fill compressibility H 3.81u/\ i(6.25u) 0.16marine clay index properties V exp{-1.23u} 0.81

undrained shear strength V exp{-0.33u} 3.03undrained shear strength V exp{-3.75u} cos(7.73u) 0.12

quick clay undrained shear strength V exp{— 0.35u} cos(6.28v) 0.19

clean sand CPT V exp{-0.9u} 1.11 Alonso & Krizek (1975)

CPT V exp{-1.6v} 0.63

CPT V exp{ —1.91u} cos(2.62u) 0.32

silty loam index properties V exp{— an} cos fiv a=0.158-3.10 2.14-3.10p =0.23-0.41

plastic clay dry density V ? 1.3 Vanmarcke & Fuleihan (1975)

clay undrained shear strength V exp{-au} a = 0.75 — 1.6 0.63-1.33 Matsuo (1976)

seabed deposits CPT H exp{-v2/900} 30 Tang (1979)

bay mud water content V exp{-au} 9 Vanmarcke (1977c)

void ratio V exp{-au} 10sand SPT V exp{—au} 2.4sand 1/SPT H exp{-au} 55

* V: Vertical H: Horizontal** K\: modified Bessel’s function of second kind and first order

Table 4.2.Autocorrelation Function and Correlation Distance of Soil Properties

4.3. RANDOM FIELD MODEL4-12

Material Property direction* Autocorrelation function

p{v)

Correlation distance

v0 (m)

Source

coastal sand CPT ? ? 5 Baecher et a/ (1980)compacted clay dry density V ? 1.2compacted clay dry density II ? 5compacted clay dry density V ? 5dune sand SPT ? ? 20

sand hydralic conductivity H&V ? < lm Smith (1981)sand porosity H&V ? < lmsand Do o H&V ? < lm

sand In Dr H exp{-v/34} 34 Fardis &: Veneziano (1981)sand In Dr H exp{-u/1.8} 1.8

soft clay undrained shear strength V exp{—vj\.2\] 1.21 Asaoka Grivas (1982)soft clay undrained shear strength V exp{-u/3.11} 3.11

alluival deposit l/CPT V exp{-au} a = 0.78 - 23.13 0.4 -1.3 Ximenez de Embun & Romana (1983)

soft glacial clay undrained shear strength V ? ~lm Wu & El-Jandali (1985)

soft glacial clay undrained shear strength V ? ~lm

silty seabed sand CPT V ? ~lm

silty clay CPT V exp{-u/0.1} 0.1m Author’s data

Table 4.2. (cont.)

4.4. STATISTICS OF SPATIAL AVERAGES 4-13

Since C(v) normally approaches zero for a large distance, 7(2;) will approach the

point variance a2 of the soil property as v increases.

4.3.2 Type II soil profile

In a Type II soil profile, ic(t_) can be decomposed into a non-constant trend

component and a random component with constant statistical properties. The

random component can be modelled in the same way as a Type I soil profile.

II

'-w (4.9a)

var{K,(t)} — var{e(i)} — o2 — constant (4.96)

/c(f), /c(f')} = cov{e(l),£(i')} = a2 • p(v) (4.9c)

Typically, g(t) will be estimated as a linear function.

4.3.3 Type III soil profile

In this type of soil profile, the random component of the soil property possesses

a constant coefficient of variation. However, e(t) can be transformed into another

random component r/(f) with zero mean value and constant variance, a2, via,

e(L) = g(L) ■ v(L) (4.10)

The transformed random component 77 (^) can also be modelled as a homogeneous

random field (Asaoka and Grivas, 1982). Thus

E{k(Q} = g(t) (4.11a)

var{K,(t)} = var{e(t)} = g2(t) • a2 (4.116)

cov{K(t),K(t_')} = cov{e(t),e(t')} = g(t) • g{t_') • a2 • p(y) (4.11c)


4.4 STATISTICS OF SPATIAL AVERAGES

The performance of geotechnical structures is usually governed by average

soil properties. In slope stability analysis, the average soil properties such as the

spatially averaged cohesion or soil density are of interest rather the point properties

of the soil. The spatial average of a soil property *c(£) is defined as

where V can be the length L, area A or volume V of the spatial domain depend

ing on the case and gy and ey are respectively the spatial average of the trend

component and the random component. In a slope stability analysis, the domain

V would typically be the base or the area of a slice. The mean value, variance and

covariance of the spatial averages are given by

(4.12)

(4.13)

var{/c\/} — Eygy + iy —

= E{gy + Sy — gy}2(4.14)

— E{iy}2

= var{ey}

— E{iy • £yi}

= COv{iy,6y>}

(4.15)


4.4.1 Type I and II soil profiles

For Type I and II soil profiles, Eqn.4.14 can be written as

var{ky} = var{ev)

= ^e{ [ 6(t)dt j2

= ^ ett'm' 1

= T / / £{£(<)£(<')}<M'

= hjvjv p(-)dLdt-= <t2- r2(K)

(4.16)

where r2(E) = ^ fv fv p{v)dtdi' is called herein the variance reduction factor.

For the covariance, we have

where B(V, V') = ^77 Jy// JV p{v)dtdt'. B(-) is called herein the covariance factor.

To implement the two dimensional stability analysis using Morgenstern and

Price’s method in Chapter 6, it is necessary to know the variance reduction and

covariance factors for spatial averages over the bases of the slices and the area of

the slices. The variance reduction and covariance factors can always be evaluated

using numerical integration. However, owing to the fact that the lag distance does

not vary smoothly with the location variables t, a quadrature with a relatively

COv{kv,ky>} = COv{iy, Sy)

(4.17)

= <t2 ■ B(V,V')


large number of integration points is required to achieve a reasonable accuracy.

This will incur a heavy time penalty in generating the covariance matrix of the

spatially averaged soil properties. Here, a procedure will be outlined by which the

variance reduction and covariance factors can be evaluated semi-analytically and

more efficiently.

4.4.1.1 Variance reduction factor for line averages

Referring to Fig.4.3, the variance reduction factor for a line average is given

by

r2(L) = T fL [L P(Vl,vy)didi' (4.18)

where vx = \l' — /|cos# and vy = |/' — /|sin0. By introducing the change of

variables (tx = /, 72 = — /), the domain of integration will be changed as indicated

in Fig.4.3. By carrying out the integration with respect to Ti, Eqn.4.18 becomes

p(|r2|cos0, |72|sin0)d7i dr2+

p(\r2\cos0, |72| sin 9)dTldT2

Z2 (L- 72)/?(|72|cos0, |r2|sin0)dr2+

fo/ ^(L + t2)p(\t2\cos 6, \t2\sm 0)dr2 (4.19)

Note that the first integral is evaluated over the domain Si in Fig.4.3 where

72 is always postive and hence the absolute sign can be removed from the first

integral. On the other hand, the second integral is evaluated over the domain S2

where 72 is always negative. However, by introducing a further transformation

r2 = —72, the domain of integration for the second integral can now be converted

to B2 within which r2 is always positive and therefore the absolute sign can also


x2-

(b)

Figure 4.3Calculation of the Variance Reduction Factor for a Line Average

(a) Configuration (b) Transformation of Integration Domain


be removed. After simplification, Eqn.4.19 becomes

(4.20)

Once the ACF is given, Eqn.4.20 can be integrated directly to give the variance

reduction factor. Alternatively, numerical integration can be used. As Eqn.4.16

involves only a single integral and the arguments of p(-) now vary smoothly with

the integration parameter, a low order numerical quadrature will be sufficient to

give a reasonable accuracy. The variance reduction factors for the separable two

dimensional ACFs listed in Table 4.1 are given in Table C.l in Appendix C.

r2( l)1.0

0.5

0.00 5

l6

10 15 20

Figure 4.4

Variance Reduction Factor for Line Average - Type I ACF

The variance reduction factor is bounded by 0 and 1. Therefore, the variance

of the spatial average is smaller than that of the point property. Fig.4.4 shows the


general trend for T2(-) for a one-dimensional Type I ACF. The pattern is similar

for two or three dimensional cases. The figure indicates that T2(•) diminishes as

the ratio L/8 increases. For line averages, it can be proved for Svell-behaved’

ACFs (i.e. it decays sufficiently fast as v increases) that the variance reduction

factor can be approximated for a large averaging length by (Vanmarcke, 1984)

The approximation is good when L > 28. Similar expressions can also be estab

lished for two and three dimensional spatial averages (Vanmarcke, 1984).

For some natural soils, the correlation distance of soil properties is small, of the

order of l-2m (see Table 4.2). The reduction of variance due to spatial averaging

can therefore be appreciable without the averaging dimension being very large.

This has a very important implication. Although it may be very discomforting to

realize that a coefficient of variation of greater than 40% (point property) is not

uncommon for the undrained shear strength of soil (Alonso, 1976), the variability

of the average shear strength, which governs the performance of slopes, is usually

much less than the point variability.

4.4.1.2 Covariance factor for line averages

Referring to Fig.4.5 and using Eqn.4.17, the covariance factor of two line

averages is given by

where vx = \vxo + l' cos 02 — l cos 9\| and vy = \vyo + l' sin 02 — / sin 9\|. Two cases

have to be considered.

(4.21)

(4.22)


(xo,yo)

Figure 4.5Calculation of the Covariance Factor for a Line Average

(a) Configuration (b) Transformation of the Integration Domain


1. Case 1 — 61 7^ 02

By making the transformation

74 = yxo + l' cos Qq, ~ l cos 61

(4.23)r2 = vyo + l' sin 02 — / sin 9i

Eqn.4.22 will be transformed into

J JsP(\n\,\T2\)dTidT2 (4.24)

where t] is the Jacobian determinant and S is the domain of integration in the

7i~t2 space. An example of the transformed domain is given in Fig.4.5. To

perform the integration, the region S can be divided into suitable subregions.

If Ti and r2 are positive within the subregion, the absolute sign can be removed.

If tj or r2 is negative, an additional transformation of 74 = —T\ or r2 = —r2

or both is necessary to change the subregion into one in which T\ and r2 are

positive so that the absolute sign can be removed. An example in given in

Fig.4.5. To is negative in subregion S4. By transformation r2 = —r2, the domain

of integration is now transformed to B4 in which r2 is positive. Following the

above procedure and integrating first with respect to 74, Eqn.4.24 can be re

written as

all subregions. The Jacobian determinant r/ and the constants of the integral I

are given in Table C.2 for some basic configurations of lines L and L' that will

be encountered in the analysis of a general non-circular slip surface. Formulae

for calculating he integral It for the two dimensional ACFs given in Table 4.1

(4.25)

where /, = j£‘ P(T 1 > T2)dTt dr2 and the summation in Eqn.4.25 is over


are also given in Table C.4.

2. Case 2 — 6\ — 02 — 0

In this case, the lag distances in Eqn.4.22 are given by

vx = vxo + (/' - /) cos 6

vy = vyo + (/' - /) sin0(4.26)

Using the transformation Ti = /'—/ and r2 = l and employing similar procedures

as in Case 1, the covariance factor becomes

over all subregions. The Jacobian determinant r] is equal to 1 for this case. The

constants of the integral J are listed in Table C.3 in Appendix C for some basic

configurations of lines L and L'. Formulae for the calculation of the integral Jt

are given in Table C.5 in Appendix C for the two dimensional separable ACFs

listed in Table 4.1.

The covariance factor for other non-basic configurations can be deduced from

the basic configurations. The procedure is outlined in the illustrative example

later.

4.4.1.3 Variance reduction factor for areal averages

The second spatial average of interest in a slope stability analysis is the areal

average of soil density over a slice. When expanded in full, the variance reduction

and covariance factors for areal averages become quadruple integrals. It is not dif

ficult to imagine the complexity involved in obtaining an exact analytical solution

for the integral. Even if numerical integration is used, it is troublesome to derive

the limits of integrations.

(4.27)

where J{ = f£'{ Ja^+b-t^ /9(Pi+<7i7’i, rl+str1)dr1dr2 and again the summation is


Slice

l)X

H

K<

Uy

h

(xo,yo)-1"

r

\equivalent

rectangle

Figure 4.6Equivalent Rectangle for the Calculation of the Variance Reduction Factor for an Areal Average

To simplify the calculation, a slice can be transformed into an equivalent

rectangle of dimension H x L having the same area as the slice, as indicated in

Fig.4.6. In consequence, the variance reduction factor can be approximated by

r2(^4) = 22772 // J" jQL J0L /»(*>*. vy)dldl'dhdh' (4.28)

where vx = \l' — /| and vy = \h' — h |. By applying the same transformations used

for the derivation Eqn.4.20 twice, first to / and l' and then to h and /i', Eqn.4.28


can be reduced to the following double integral.

r2M) = JYJp J0 L -T2)p(TuT2)dTldT2 (4.29)

If the ACF is separable, it can be written as /9(r1,r2) — Pi Eqn.4.29

then becomes

r2(A = r2(L) -r2(H) (4.30)

Eqn.4.30 is simply the product of the variance reduction factors of the line averages

over the sides of the rectangle. The reduction in computing effort for this case is

very large.

4.4.1.4 Covariance factor of areal averages

The equivalent rectangle approximation can also be used for the covariance

factor. In this case, the covariance factor is given by (Fig.4.7)

1 r H/ r H r L* r LB(A, A') = 77777777 / / / / p(vx,vy)dldl'dhdti (4.31)

LL'nrT Jo Jo Jo Jo

where vx = |/' — /| and vy = \h' — h|.

Two basic configurations as shown in Fig.4.8 are of interest. For the configu

ration in Fig.4.8a, the covariance factor is given as (Vanmarcke, 1984)

b(a’ a']=£(_i)t ■ <l* x h]2 ■ r2(-4,) (4-32)

where r2(At) is the variance reduction factor for a rectangle of dimension H x L*.

The covariance factor for the second basic configuration (Fig.4.8b) is (Vanmarcke,

1984)

B(A, = D->)‘+y • W x A*)2 • r2(^>) (4-33)t=0j—0


A= L H

Figure 4.7

Equivalent Rectangles for the Calculation of the

Covariance Factor for an Areal Average

where T2(AlJ) is the variance reduction factor for a rectangle with dimension

L* x H* which can be obtained using Eqn.4.29 or Eqn.4.30 if the ACF is separable.

Therefore, instead of evaluating a quadruple integral of Eqn.4.31, it is now only

necessary to calculate a sum of double integrals. Note that in Fig.4.8, all the

dimensions L* and H* are non-negative. Other non-basic configurations can be

assembled using the basic configurations. The procedure will be depicted in the

illustrative example later.

4.4.2 Type III soil profiles

For Type III soil profiles, the evaluation of the variance and covariance of


A - L' H"A =LH '

a=l'h;

L L'

(b)

Figure 4.8

Terminology for the Calculation of the of the Covariance Factor

for Areal Averages (a) Eqn.4.32 (b) Eqn.4.33


spatial averages is more complicated. They are described by

2var{ev} = ^ g(t)g(t')p(v)dtdt! = o2 • E2(V) (4.34)

cov{ev,Ev} = g(L)g{t_')p(v)dt_dt_' = a2 • D(V, V') (4.35)

where E2(-) and D(-) are also called the variance reduction and covariance fac

tors for convenience despite the fact that they are defined differently to T2(K)

and B(V,V). Denoting the two dimensional trend component g(t;) by g(x,y) and

following a similar procedure as in Equations 4.19 and 4.20 for the calculation of

T2(L), the variance variance factor of a line average for Type III soil profiles is

given by (Fig.4.3)

1 rL rLE 2(L) = — I g(x0 + l cos0,yo + ls\n9) ■ g(x0 + l'cos9,y0 + l's\n9)-1 fL CL

L* „

p(\l' — /| cos#, \l' — l\ sin 6)dldl'■ rL rL—r21

LrL, rL-r2

I g[x0 4- Ti cos9, yQ + Ti sm 9)-

g(x0 + (n + T2) cos0, y0 + (ti + r2) sin0)-

p(\r2\cos9, \t2\s\n9)dTldT2 +

J lJ 9(xo + Ticos0,y° + TisinO)-

g(x0 + {Ti + T2)cos6,y0 + (ri + 72)sin0)-

p(\t2\cos 9, \t2\ sin 9)dTidr2

i rLl= j2 L ^(r2 c°s0, t2 sin 9)

\lL—t2

g(xQ + ri cos 9, yQ + Ti sin 9)-

g(xQ -f (rx + t2) cos9, yQ + (ti + t2) sin 0)dTx +

g(xc + Ti cos9, yQ + Ti sin ^)*

g(xQ + (t - 72) cos 9, y0 + (ti - T2)s\n9)dTi dr2


LK(tz)p(t2 cosO, ro sin 0)(It2 (4.36)

K(t2) represents the integral inside the square brackets in the second last expres

sion. Knowing the ACF and the trend component, Eqn.4.36 can be evaluated

analytically or by numerical integration. Again, the formulation has the advan

tage of removing the absolute sign in the arguments of /?(•), thus a lower order

numerical quadrature can be used in performing the integration.

Similar expressions as Equations 4.25 and 4.27 can be derived for the covari

ance factor of line averages for Type III soil profiles except that the ACF in the

integrand is now multiplied by the function g(t)g(t') which has to be transformed

into the new integration domain in the same way the argument of the ACF is

transformed from the (x, y) space to (ri, To) space.

Unlike the variance reduction factor r2(Vr) and the covariance factor B(V, V'),

the values of E2() and D(-) depend on the absolute location of lines. Therefore,

the covariance factor D(V, V') cannot be assembled from the basic configurations

as for the covariance factor B(V, V'). As a result, all possible configurations of

lines have to be considered, and consequently numerical integration is mostly used.

For the areal average over a slice, the variance reduction factor using an

equivalent rectangle is given by (Fig.4.6)

~ IJfP L Jo Jo Jo + /, I/o -+- /i) - -f- A')-

p(\l‘ - /|, |A' - h\)dldl'dhdti (4.37)

By repeating the procedure in Eqn.4.36 twice, first to l and /' and then to h and

h', the absolute signs in the argument of p(v) can be removed.

Equations 4.32 and 4.33 are also valid for the calculation of covariance factors

of areal averages for Type III soil profiles except that the variance reduction factors

r2(Aty) are replaced by the appropriate factors E2(Aty) which can be obtained

4.5. WHITE NOISE PROCESS 4-29

using Eqn.4.37.

4.5 WHITE NOISE PROCESS

As the scale of fluctuation 6 approaches zero, the random field will degenerate

into an uncorrelated random process and T2(-) and £2(-) will also approach zero.

It might be thought that the variance of the spatial average would also approach

zero as b tends to zero, but this is not the case. The associated point variance of a

white noise process must go to infinity (see e.g. Priestley, 1981), but the products

cr2 - r2(Vr) and <r2 • £2(P) remain finite.

The ACVF of a white noise process can be represented as a Dirac ^-function.

C(v) =u = 0

(4.38)

Given a continuous function f(t), for a point € V and v = L0~L we have

Jvf(L)C(v)dL = f(t0)-w (4.39)

where w is called the white noise intensity with the dimension of variance x V.

For Type I and II soil profiles, the variance of a spatial average is given by

var{ky} = ^ J j C(v)didt'

= ±Jv wdt_' (4.40)

w = V

The covariance of two spatial averages in two non-overlapping domains is given by

cov{£v, k'V'} W> JvfvCMW (4.41)

4.6. COMPOSITE RANDOM PROCESS 4-30

where t G V, t' G V and v_ — \i — L'\- As v is always greater than zero, C(v)

will become zero. Consequently, cov{ky, ky>} — 0 for non-overlapping spatial

averages.

For Type III soil profiles,

Similarly, it can be proved that cov{ky, k y} = 0 for two non-overlapping regions.

4.6 COMPOSITE RANDOM PROCESS

The ACFs listed in Table 4.1 are all ‘simple’ functions, in the sense that

they decay from 1 at zero lag to zero at large lag distances in a smooth and

uninterrupted manner. Furthermore, the ACF is governed by a single parameter

which can be directly related to the scale of fluctuation.

For more complex random processes, the random component e(<) can be rep

resented as a sum of statistically independent random components

(4.42)

e(L) — e\ {Q + £2(0 + • • • + £k{L) (4.43)

each with a different scale of fluctuation. For example, S\ (t) may represent the

nugget effect (see later) and €2(1) a strongly correlated random component etc. A

number of simple results follow from this model. For example

4.7. NUGGET EFFECT 4-31

kC(t>) =

tTTi(4.45)

kvar{iy} = var{iy^}

t=i(4.46)

kcov{ev, ev>) = y cov{Ey\ey}} (4.47)

i = l

where the subscript and superscript i represent quantities related to £,(/). The

above composite random process (or nested structure as known in some literature)

is widely used in geostatistics (e.g. David, 1977; Journel and Iluijbregts, 1978) to

approximate a random field with complex correlation structure.

4.7 NUGGET EFFECT

Sometimes, a discontinuity is found to occur in the sample autocovariance

function and autocorrelation function at zero lag.

An example of this is shown in Fig.4.9. Other examples can be found in Lumb

(1975a), Baecher (1984) and VVu and El-Jandali (1985) and are also commonly

found in books on mining geostatistics such as Agterberg (1974), David (1977),

Journel and Huijbregts (1978) and Clark (1979). In mining geostatistics, such a

phenomenon is called a nugget effect. There are three possibilities which give rise

to the occurrence of the nugget effect.

1. The sampling interval is large compared with the scale of fluctuation so that

the details of the ACVF at small lag distances can not be reviewed.

2. The soil property possesses a random component which has a very small value

of 6 and behaves like a white noise process. For ease of reference, this kind of

process is called a small-<5 process.

3. The presence of measurement errors also gives rise to the nugget effect. Using

Aut

ocor

rela

tio


Distance of Separation (m)

Figure 4.9 Illustration of Nugget Effect (after Wu and El-Jandali, 1985)

an additive model, the measured soil property Km(t) can be written as

Km(t) = K(t) + et_ (4.48)

where e*_ is the measurement error. Assuming that measurement errors are

uncorrelated and that is independent of ac(/), it follows that

C(v) =CK(0) + &e

CK(V)

v — 0

v>0(4.49)

where C'K(-) is the ACVF of the ‘true’ soil properties and is the variance of

the measurement error. As CK (v) < CK(0), a discontinuity at zero lag will

occur.


If the nugget effect is due to the first cause, the question arises of how small

should be the sampling interval so that sufficient details of the ACF at small lag

distances can be reviewed. The following concept is useful. In a slope stability

design, preliminary investigations can be carried out to establish the variation of

the failure probability with the scale of fluctuation based on some prior knowledge

of the variability of soil properties. Suppose it is established for a particular design

that a scale of fluctuation less than 3m will correspond to a sufficiently low value

of failure probability.

With this in mind, a sampling interval equal to, for instance, 1/10 of this

value, i.e. 0.3m, can be used. If the scale of fluctuation is indeed greater than

3m, the design will not be acceptable. However, a sampling interval of 0.3m will

be sufficient to give enough details of the actual ACF of the soil property. This

information can be utilized for the re-design of the slope.

If the results indicate a scale of fluctuation of less than 3m, the design will

become acceptably safe. Further details of the ACF at lag distance smaller than

0.3m will only provide a slightly more precise estimate of the failure probability,

but it will not affect the acceptance of the design.

In mining geostatistics, the nugget effect is commonly attributed to the pres

ence of a small-<5 random component, although a combination with the testing error

has also been suggested. In geotechnical data analysis, there is a tendency to at

tribute the nugget effect entirely to testing errors (e.g. Baecher, 1984; Tang, 1984;

Wu and El-Jandali, 1985). Wu and El-Jandali (1985) also used the magnitude

of the nugget effect as a basis for comparing the variability of the measurement

errors due to different testing procedures.

However, it is considered that the possibility of a small-<5 random component

cannot be ignored in geotechnical soil properties. In fact, the spatial variability

of soil properties can be explained to a certain extent by the randomness of the


process governing the formation of the microscopic structural units (Yong, 1984).

Therefore, a sample ACF with a large nugget effect does not provide a confirmatory

evidence that the variance of the testing error is high. However, a sample ACF with

a small nugget effect does suggest that the variance of the testing error is small.

Thus it is necessary to establish the testing error variance by independent means

(e.g. Lumb, 1974) in order to separate the two effects. Discounting the nugget

effect in the analysis will lead to unconservative results if part of the nugget effect

is indeed due to a small-<5 random component, because the point variance and the

variance of the spatial average will be underestimated. Therefore, in the absence

of specific information on the testing error, it should be assumed that the nugget

effect is due to the small-<5 component.

An ACVF with a nugget effect can conveniently be modelled as a composite

random process consisting of a white noise random component and a transition

random component (i.e. S ^ 0). Thus

C(v) = Cn(v) +cr;ps(v) (4.50)

where Cn(v) is the ACVF of the white noise random component (see Eqn.4.38).

cr“ and ps(v) are the point variance and ACF of the transition random process.

Suppose the dimension of the sample is d where d can be the length, area or

volume depending on the case. The ACVF of the soil property averaged over the

dimension d, Cd(v) is given by

Cd(v) = <

5 + ci(o)*5 + *i v = 0

(4.51)

. C'd(v) ^ a~ps(v) v^O

where C'd(v) is the ACVF of transition component of the soil property averaged

over the dimension d. In practice, the scale of fluctuation of the transition random

4.8. SAMPLE SPATIAL AVERAGES 4-35

component is usually greater than the dimension d of the sample. For practical

purposes, C'd(v) can be approximated by the ACVF of transition component of

the point property as indicated in the above equation. Denote Cd{0) by ad and

let cr2 = c • o2, we have

°d ~ 7 + C '

=> w = (1 — c) • d • ad (4.52)

The constant c can be obtained by extrapolating the sample ACF to zero lag.

As an example, consider the results by Wu and El-Jandali (1985) as indicated in

Fig.4.9. The variance of the test results, was estimated to be 65.8(kN/m2)2.

From Fig.4.9, the value of c is given as 1.0 — 0.24 = 0.76. The dimension of the

sample is not given in the paper and a length of 0.1m is assumed for the sake of

illustration. Therefore, the white noise intensity becomes

w = (1 — 0.76) x 0.1 x 65.8 = 1.6 kN/m

Knowing the white noise intensity, the variance and covariance of the white noise

random component and the transition random component can be evaluated sep

arately using the procedures discussed previously and assembled together using

Equations 4.46 and 4.47.

4.8 SAMPLE SPATIAL AVERAGES

So far, it has been assumed that the trend component is known. In practice,

the mean trend and variances have to be estimated. Therefore, the overall un

certainty associated with the average soil properties consists of two parts - the

inherent variablity associated with the point to point variation of soil properties


in the field, and the sampling uncertainty associated with the estimation of the

trend component. In what follows, the variance associated with the estimation of

the trend component will be called the trend variance for ease of reference.

Denote the sample trend component by g(t) = ^ bjPj, where bj is the sample

estimate of the coefficients a3 of the ‘true’ trend component g(t) = J^ayPy, and

the covariance matrix of the coefficients b3 by Vf,. Both bj and can be estimated

for instance by the method of least squares from test results at N sample locations

in the field (Lumb,1974). It is further assumed that bj is an unbiased estimate of

ay i.e. E{bj) = ay. As the true trend of the mean soil properties is never known,

the true spatial average has to be estimated by the sample spatial average which

is defined by

The unbiasedness of the sample spatial average follows from the unbiasedness of

the sample trend coefficients bj.

The calculation of the variance and covariance sample spatial averages is more

complicated. The variance of a sample spatial average is expressed as

(4.53)

4.8.1 Type I and II soil profiles

The expected value of the sample spatial average is

E{kv} = y Jv E{g(Q + e(t)}dt = ^ [g(L)dt_ = gv (4.51)

var{Ky} = E{kv — gvs

(4.55)


where Pyj is the 7th term of the generalized polynomial averaged over the spatial

domain V. For simplicity, the correlation of the soil property at the sample points

and the soil property within the spatial domain V is neglected and Eqn.4.55

becomes

var{Ky} = var^Y^-aAPy^+varley) = {Py}TVi,{Py} + cr2-T2(V) (4.56)

{Py} represents the vector of Pyj and the superscript T denotes the transpose of

a matrix. The first term of Eqn.4.56 is the trend variance. Note that for Type II

soil profiles the trend variance is position dependent.

For Type I soil profiles, the trend is simply a constant which can be estimated

by the sample mean value 7c. The trend variance becomes the variance of the

sample mean value var{7c} and Eqn.4.56 therefore becomes

var{Ky } = var{7c} + a2 • r2(Vr) (4.57)

If the sample locations t_t are sufficiently far apart, the samples can be regarded

as independent. In this case, the point variance o2 can be estimated by the usual

unbiased estimator

N —y^v*. - k) (4.58)

where N is the total number of samples and the arrow means ‘estimated by’. The

trend variance can be estimated by

s2var{K} —► — (4.59)

However, if the sample location are close together, the point variance and

the variance of the sample mean value have to be estimated using the procedure


discussed in Chapter 5.

The covariance of the sample spatial averages is expressed as

COv{tCy, /Cy'}

By neglecting the correlation of the point property between the samples and the

points within the spatial domains V and V\ we obtain

Again, for Type I soil profiles, the first term in Eqn.4.61 reduces to the var{/c}.

4.8.2 Type III soil profiles

As would be expected, the evaluation of the variance and covariance of sample

spatial averages for Type III soil profiles is more involved. The complication lies in

the fact that the variability of the random component depends on the mean trend

component. Therefore, the uncertainty associated with the estimation of the mean

trend component is also reflected in the random component. The sample spatial

average for Type III soil profiles is

Taking expectation and neglecting the correlation between the soil property at the

sample points and the points within the domain V, we obtain

cov{kv,kv,} = {Pv}TVb{Pv,} +a2 -B(V,V') (4.61)

(4.62)

E{kv} = £ f[E{g(t)} + E{g(t)}• £{>?(<)M(4.63)


Denote Pt(L) to be the value of zth variable of the generalized polynomial at point

t. Again neglecting the correlation of the soil property at the sample points and

the spatial domain V, the variance of the sample spatial average Ky is given by

var{Ky} = E{tzv — gy}4

(4.64)

= {Pv}TVb{Pv} + Q

The last term Q can be written as

q = E{ h Jv jv fiWAU'bUtou'm'}

Jv JvPi(L)PAL’)E{t,(LHt!)}dtdt'

= ^7; y^y~',[cotj{6i,6y} + a<aj Pt(t)Pj(t') ■ a2 ■ p(v)dt_dt’

2 2

= ^2 YJ cov{bt,bj} Jv Pi(t)Pj(t')p(v)dtdt' + £2 Jv Jv 9(t)g(t!)p(n)dLdt!

2= VtEECOv{bi’bj} V vPAi)PAL')P(n)dtdL' + <T2 -^(V) (4.65)

t J J J

In summary,

var{KV} = {Pv}TVb{Pv}+a2 • E~(V) + a2 £ £ covib^ bj} • Kij (‘1-66)* J

where Kij = fy fy Pi(L) Pj[L') p(v)didt'. The last term in Eqn.4.66 accounts

for the coupling effect between the mean trend component and the random com-

4.9. NON-HOMOGENEOUS SOIL PROFILES 4-40

ponent. Similarly, the covariance for sample spatial averages is given by

cov{KVKV,} = {Pv}TVb{Pv.} + o*D(V,V') +a‘2YYdCOv{bi,bJ} ■ MtJ (-4.67)l J

where MXJ = fv, fv Pi(t)Pj(t')p(y)dtdt'.

4.9 NON-HOMOGENEOUS SOIL PROFILES

A soil profile is often not a single-layered soil. However, it can usually be

delineated into a series of soil strata each of which can be modelled as a Type I,

Type II or Type III soil profile. Unless proven otherwise, it is reasonable to assume

that the soil properties between two different soil strata are independent.

Layer 1

Layer 2

Figure 4.10 Linear Averages in Non-homogeneous Soils

To illustrate the procedure for calculating the variance and covariance of spa

tial averages over a multi-layered soil profile, consider the case of a line average as

shown in Fig.4.10. The cases for two-dimensional and three-dimensional spatial

averages are similar.

4.10. ILLUSTRATIVE EXAMPLE 4-41

The variance of the line average over line L in Fig.4.10 is

f ~ i r L\ ^h\ + L^^l-2 lvar{KL} — var j-------——---------

L\ • var{kLl} + 2LiL2 • cov{kLl, kL2) + L% • var{kL2}

L2

By the assumption of independence, Eqn.4.68 reduces to

var{ki,}L\ • var{kLl} + L% • var{/cL2}

L2

(4.68)

(4.69)

varf/c^i} and var{k^2} can be calculated using the procedure above for a single

layered soil.

The covariance of line averages over lines L and L' is given as

~ i f L\ _ L/2 - L'^ _ L2 _cov{kl, kl>) = couJ —/cLl + ~JTkl\ + ~jjkl2

= ^jjy-cov{kL,, kl, } + ^j^-cov{kLi, }+ (4.70)

LL'L Z/

cou{/cL2,^L/} + -jjrcov{kL2,kL'2}

Again, by the assumption of independence of the soil properties between two

different soil strata, we have

cov{kL,kL/} = U L[ LL'

cov{kLl,kLli} + L2L'2LL'

cov{kL2,kL.2} (4.71)

4.10 ILLUSTRATIVE EXAMPLE

Consider the cohesive slope as shown in Fig.4.11. The variance and covariance

of the sample spatial average are to be estimated on the basis of the following

information:

1. Eight soil samples taken at widely scattered locations have been tested for


undrained shear strength and soil density giving the following sample mean

value 7c and sample standard deviation s.

k s

cu 30 kN/m2 10 kN/m2

7 18 kN/m3 1.5 kN/m3

2. Local experience indicates that the soil profile can be modelled as a Type I soil

profile and the scales of fluctuation in the horizontal direction 6X and vertical

direction 6y can be taken as 3m and lm respectively.

In the following, we will illustrate the procedure for calculating:

a) the variance of the sample spatial average cohesive strength over the base of

slice 2 (Case 1);

b) the variance of the sample spatial average soil density over the area of slice 2

(Case 2);

c) the covariance of sample spatial averages of cohesive strength over the bases of

slices 2 and 3 (Case 3);

d) the covariance of sample spatial averages of soil density over the slices 2 and 3

(Case 4).

A separable two-dimensional simple exponential ACF is used. The spatial

average of the cohesive strength over a domain V is denoted by cy and the sample

spatial average by cy. Similar notations are used for soil density,

a) Case 1

Details of slice 2 are given in Fig.4.12. According to Eqn.4.20, the variance

reduction factor is given as

r 2(Z,) = -p / (L-T2)-e~2{ «. + Ut2 (4.72)

Eqn.4.72 can be evaluated using the formula in Table C.l in Appendix C for a


y

D

1 i ne Pt co-ordinate (m) x y

length(m)

anglee

A 2.465 -5.877AB

B 11.977 -6.8809.56 6.02*

BCC 15.029 -5.877

3.21 18.20°

CDD 21.489 -3.753

6.80 18.20°

_ Gj=

y9.51m

01=18.20

Figure 4.12Illustrative Example - Geometry of Slices for Calculation of the

Variance and Covariance of Line Averages


Type I ACF. From Fig.4.12, the parameter values for the formula are given as

Therefore,

L = 9.56m

0 = \a2\ = 6.02°

Sr =f cos 6.02° sin 6.02° I 3 + I

-1

= 2.292

r2(L) = T + A(e^-,)_ 2.292 2.2922~ 9.56 + 2 X 9.562 ^

2X9.562.292 -1)

= 0.21

Hence from Equations 4.57 and 4.59,

var{cL) = s2 j-L +r2(I)J + 0.211

= 33.5(kN/m2)2

b) Case 2

Fig.4.13 shows the details of the equivalent rectangle for slice 2 from which

the dimensions of the equivalent rectangle are obtained as follows.

L = 9.51m H = 9.01m

Since a separable ACF is used, the variance reduction factor for the soil density

T2{A) is given as (Eqn.4.30)

r2(A) = r2(L)-r2(#)


-------- 1

3.36 m

Slice 2 11.31m7.95m7.95m9.01 m

Slice 3Total area = A

1.06 m

9.51m9.51m

1.06m~]~

A' 11.31m

l A,-

9.51m 9.51m

a2 Ai 7.95 m

9 51 m 9.51 m

A'2 3.36m

7.95 m A2

9.51 m 9.51 m

Figure 4.13Illustrative Example - Geometry of Slices for Calculation of the Variance Reduction and Covariance Factor for Areal Averages


Using the formula in Table C.l in Appendix C, T2(L) is as follows

. f cosO0 sinO0 'l 1^ = Hr+ —1 =3

r2{L) = +9.51 2 x 9.512

= 0.27

(«' -1)

Similarly

_ f cos 90° sin 90° 'j 1 ~ | 3 * 1 } = 1

r 2(H) = — +v 1 9.01

= 0.10

1-

2 x 9.012 (C -1)

Therefore, r2(A) = 0.27 xO.l = 0.03. The variance of the sample spatial average

density over slice 2 becomes

var{iA} = s2 + r2M) J = i-52 jg + 0.031 = 0.34(kN/m3)2

c) Case 3

The configuration of slices 2 and 3 can be decomposed into two basic config

urations as shown in Fig.4.12. Let us consider the first basic configuration - lines

AB and BC. This configuration corresponds to the configuration of case 8 in Table

C.2 in Appendix C. From Fig.4.12, the following parameter values for the formula

can be derived, viz,

v%o — 9.51


Vy0 = 1.00

0 \ = 6.02°

02 = 18.20°

Lx = 9.56

L2 =3.21

1sin(6.02° + 18.20°)

= 2.44

The constants of the integral in Eqn.4.25 are given as (Table C.2)

Ai = vxo — L\ cos61 = 9.51 — 9.56 x cos6.02° = 0.00

B{ — cot Ox = cot 6.02° = 9.48

Cx = vxo + L2 cos 02 — 9-31 + 3.21 x cos 18.20° = 12.56

Dx = - cot 02 = - cot 18.20° = -3.04

Ex = 0.0

F\ = Lx sin Ox = 9.51 x sin 6.02° = 1.00

Using the formula for I in Table C.4 in Appendix C for Type I ACF, we obtain

lx = 0.18. Similarly, I2 = 0.40. Therefore,

B(L, L\) = jUh + h)

2.449.56 x 3.21

• (0.18 + 0.40)

= 0.046

The second basic configuration in Fig.4.12 corresponds to the mirror image of case

7 in Table C.2. To use the formula, the local axes have to be changed as shown in


Fig.4.12. The parameter values of the formula are as follows.

vxo = 21.49 - 11.97 = 9.51

vy0 = -3.75 - (-6.88) = 3.13

6{ = 18.20°

02 = 6.02°

h = 6.8

L2 = 9.56

1m — ________________ — 2 441 sin( 18.20° + 6.02°)

The constants for the integral /t can be calculated from the formulae of case 7 in

Table C.2 and using the expression for / in Table C.4, we obtain ^ It = 0.013.

Therefore

2.449.56 x 6.81

x 0.013 = 0.0005

The covariance factor B(L, L') can be obtained from the following procedure.

{ L' Lo )cov{cL,cL/} = covl cL, -jjcL/ + -JjtL'z >

_ ^ l r ~ , . ^2 r= —cov{cL, cL/ } -f -y7Cou{cL, cL^}U

= ^ ■ a- ■ B(L, L\) + ^ • a2 • B(L, L'2)

But cov{cl, cl>) = a2 • B(L, L'). Therefore

B(L, V) = B . B{L, L[) + ^- B(L, L'2)

3.21 x 0.046 + 6.81 x 0.000510.01

= 0.015


The covariance of the sample spatial average cohesive strength becomes

cov{cL,cL>} = s2 £')}

|= 14.0(kN/m2)2

d) Case 4

Using equivalent rectangles, slices 2 and 3 can be decomposed into 3 basic

configurations as shown in Fig.4.13. Consider the first configuration - rectangles

Ai and A'. The covariance factor for this configuration can be evaluated using

Eqn.4.33. From Figures 4.8 and 4.13, the parameter values for the formula are

L*0=0

L\ = 9.51

L\ = 19.02

L% =9.51

H*0= 0

H* = 11.31

HZ = 12.37

H* = 1.06

= 102 / - +0.015l8

The calculations of Eqn.4.33 are shown in the following Table.

As a separable ACF is used, the variance reduction factor r2(AtJ) is calculated

using Eqn.4.30. For example, r2(Au) = r2(Lj) • T2)//*). The covariance factor

B(Ai, A') is given by

B(AUA')1

4AXA'


i J 4* r2M.,) (-ly+ULtHf)2 -r2(Aij)

0 0 0.0 0.0 - 0.00

0 1 0.0 11.31 - 0.00

0 2 0.0 12.37 - 0.00

0 3 0.0 1.06 - 0.00

1 0 9.51 0.0 - 0.00

1 1 9.51 11.31 0.0225 259.91

1 2 9.51 12.37 0.0206 -285.45

1 3 9.51 1.06 0.15 14.94

2 0 19.02 0.0 - 0.00

2 1 19.02 11.31 0.0123 -568.30

2 2 19.02 12.37 0.0113 624.12

2 3 19.02 1.06 0.1465 -32.69

3 0 9.51 0.0 - 0.00

3 1 9.51 11.31 0.0225 259.91

3 2 9.51 12.37 0.0206 -285.45

3 3 9.51 1.06 0.15 14.94

sum=1.97

Table 4.3. Some Results for Calculating the Covariance Factor.


1“ 4 x (9.51 x 1.06) x (9.51 x 11.31)

= 0.0005

x 1.97

The covariance factor for the second basic configuration can be obtained us

ing Eqn.4.32 and the third by repeating the procedure above. The values are

B(A2, A[) = 0.0117 and B(A2, A'2) = 0.0009. The covariance factor B(A, A') can

be obtained using the following procedure.

cov{7A,7A/} = cou{7Ai+A2,7A/}

r A\ _ A2 ~ ~ x= cov{ — iAl + —7a2,7a'}

= ^-cov{iAl,7A'} + ^-cov{7A2,7A/}

At A2A\= ^cov{ 7a157a'} + ^-^{7^,7^-} +

= -± • a2 ■ B(Ai,A/) 4- • cr2 • B(A2, AA'x) +

cov{7a2,7a;}

A2A2 2AA' a ■ B(A2, Aa> )

But cov{7A,7A'} = • B(A, A'). Therefore,

B(A, = B(AUA') + AA> • B(A2, Aa.) + ^ • B(A2, Aa.JAA'

= 0.002

and

cov{'ia,'1a-} = s2 + B(A, A1)

I = 0.3

For line averages, the uncertainty associated with the inherent variability,

measured in terms of the contribution to the variance of the sample spatial average,

is comparable to the sampling uncertainty associated with the estimation of the

= 1.52 x /- +0.002l8


mean trend for this example. Therefore, none of the components can be neglected

in the analysis in the calculation of variance of the average cohesive strength.

However, because the soil density is averaged over a larger spatial domain, the

variance reduction from spatial averaging is significant. The uncertainty of the

sample spatial average soil density is dominated by the sample uncertainty as

shown in the example where the variance and covariance of the spatial averages

are only a small fraction of the sample point variance s2. If the soil properties are

modelled as perfectly correlated variates, the variance reduction and covariance

factor of the sample spatial average will be equal to 1 and hence the variance

and covariance of the sample spatial averages will be grossly over-estimated. This

explains why many of the analyses in the current literature give an unrealistically

high value of failure probability of slopes.

CHAPTER 5

STRUCTURAL ANALYSIS OF SOIL DATA

5.1 INTRODUCTION

In the previous chapter, the principles of probabilistic characterization of soil

profiles were discussed. In particular, the importance of the correlation structure

of soil properties and the role it plays in spatial averaging were emphasized.

This chapter addresses the estimation of the statistical parameters of soil prop

erties. In geostatistics, the statistical analysis of mining data is called a ‘structural

analysis’ which means extracting the statistical structure of the governing random

process from the data. This term will also be used herein. The structural analysis

of soil data is a vast subject in its own right and its development lags far behind the

theoretical development of geostatistics or random field. It is fair to say that the

autocovariance function (ACVF), autocorrelation function (ACF) or variogram of

soil properties is still commonly estimated using some semi-empirical procedures

such as fitting by ‘eye’. Active research in the field of structural analysis of soil

data only started in the 1980s and is still in a stage of development. Ripley (1981)

also commented in his book that very little work has been done on fitting the

parameter of the ACVF. The statement is still true. This chapter is an overview

of the recent theoretical developments in the subject. The implications of some

of the theories to the practice of soil sampling in the field are highlighted. Some

5-1

5.2. TYPE I SOIL PROFILES 5-2

of the geostatistical approaches relevant to the structural analysis of soil data will

also be mentioned.

Not surprisingly, the volume of literature on structural analysis of soil data

decreases in the order of Type I, Type II and Type III soil profiles, in agreement

with the complexity of the problem.

Before commencing a structural analysis of soil data, it is useful to carry out

an exploratory data analysis (EDA) to detect any anomaly that may be present

in the data. Cressie (1983) gave a good discussion of such a procedure.

The statistical analysis and interpretation of geotechnical data requires skills

and experience which can only be built up through practice. Although in many

cases the analysis can be automated using computer programs, the results must

not be accepted without cross validation with the observed data. The experience

of the the mining engineers in the statistical analysis of mining data such as David

(1977) and Journel and Huijbregts (1978) is very useful.

5.2 TYPE I SOIL PROFILES

To characterize a Type I soil profile, the mean value, the point variance, the

autocorrelation function and the trend variance (which is also the variance of the

sample mean value for a Type I soil profile) of the soil property are required. Three

sets of data will be used to illustrate the procedure of structural analysis of soil

data of a Type I profile. Details of the data are summarized in Appendix F. In

the following, the first set of data will be represented by the symbol the second

by o and the third by •.

5.2.1 Estimation of mean value

Given N samples of K(t{) at location £t, i = 1, N in the field, the mean value


of the soil property can be estimated by

(5.1)

Applying Eqn.5.1 to data set 1, the mean CPT value is given as 1.95 MPa

which is also shown on Fig.Fl of Appendix F.

Eqn.5.1 is an unbiased estimator of the mean soil property and the sampling

variance associated with the estimation of the mean value (which is also the trend

variance for a Type I soil profile) is given by

where a2 is the point variance, a is a factor and vt] is the lag distance between

sample points lt and tj. a has a lower bound value of l/N which corresponds to

the case of zero correlation between samples.

In examining the property of Eqn.5.2, there are two cases to be considered.

The first is to sample within a fixed domain V. An increase in the total number

of samples will mean a greater sampling density (i.e. number of samples per unit

volume) and eventually the entire domain V will be covered as N tends to infinity.

In this case, Eqn.5.2 will approach the following limiting value.

This implies that var{K} would remain finite as N —► oo. In this case, k would

not be a consistent estimator for the population mean and sampling uncertainty

still exists even when a huge number of samples are tested. This is a consequence

of the autocorrelation of soil property in the field. The test result at a particular

location ta not only gives an indication of the soil property at that point, it also

carries additional information about the likely value of the soil property in its

a2 N N(5.2)

var{:c} = a2 ■ r2(K) (5.3)


vicinity. Hence, an extra test carried out very near to ta will only give very limited

additional information about the population mean value of the soil property over

the entire field.

The second way of soil sampling is to increase the sampling domain V as N

increases. For soil sampling at a gridded network with fixed grid size, this would

mean an expansion of the network in space. In this case, Eqn.5.2 will have an

asymptotic value of order 1 /N for well behaved ACF (i.e. ACF which diminishes

sufficiently fast with an increase in lag distance) and 7c would become a consistent

estimate of the population mean.

Of course, the concept of an infinitely large domain exists only in theory.

In practice, the spatial domain would be finite, being the dimension of the site.

But the implication of the above discussion is that in a site investigation, an

extensive sampling of soil specimens over a small area within the site is not a

cost effective way of soil sampling as far as the estimation of the mean value is

concerned. Given a fixed number of soil specimens to be taken from the field, it

is preferable to maximize the separation distances between the samples, taking

specimens from different parts of the entire site rather than concentrating on a

particular small area within the site. If the site dimensions are of the order of

the scale of fluctuation, it has to be recognized that the ‘site mean’ may be the

different to the true population mean.

Figures 5.2 to 5.5 show the variation of the factor a in Eqn.5.2 for equally-

spaced data as shown in Fig.5.1. For a fixed value of A1/6, an increase in N would

mean an increase in the sampling length. By overlapping the different graphs, it

can be observed that the difference in a between different ACFs is small.


Figure 5.1 Equally-spaced Sampling Along a Straight Line

5.2.2 Estimation of point variance

The point property is commonly estimated by

1s2 = (5.4)

1 = 1

where k is given by Eqn.5.1. The expected value of s2 is given by

E{s2} — a2 — uar{7c}

= (1 — a) ■ a2(5.5)

where a is defined by Eqn.5.2. As soil properties are usually positively correlated,

a would tend to be positive and therefore E{s2} < a2. s2 is therefore a biased


Type I ACF

30 40 80 90 100sample

Figure 5.2 a Factor for Type I ACF


Type II ACF

/ 0.6

. • t 1 < r i i | < i • . . k | , . . , , .. , 1, , , , , , , , , i | , , . , |, , . . , I

10 20 30 40 50 60 70 80 90 100sample size

Figure 5.3 a Factor for Type II ACF

5.2. TYPE I SOIL PROFILES

1 10 20 30 40 50 60 70 80sample size

Figure 5.4 a Factor for Type III ACF


Type IV ACF

80 90 10060 70sample size

Figure 5.5 a Factor for Type IV ACF


estimator, consistently underestimating the point variance a2. The bias term,

which is equal to the theoretical variance of 7c, is of order l/N when N is large.

Therefore, s2 is an asymptotically unbiased estimator of a2.

It can be seen from Figures 5.2 to 5.5 that the bias is significant if the sample

size is small and increases with a decrease in the value of A1/6. Knowing the

ACF of the soil property, the bias can be easily accounted for by multiplying s2

by a factor of 1/(1 — a). The corrected estimator of point variance, denoted by s2

hereafter, will be given as

(l-a)(5.6)

In particular, if the sample locations are far apart, the samples can be regarded

as statistically independent. In this case, a = l/N and s2 becomes the usual

unbiased estimator given by Eqn.4.58.

A question often of interest is what strategy should be used to minimize the

value of a and the variance of s2 in a site investigation program. For a fixed A1/6

ratio, the longer the sampling length (i.e. more samples) is, the smaller will be the

value of a as indicated by Figures 5.2 to 5.5. The same is also true for the variance

of s2. A more common requirement is to determine the number of soil samples to

be taken for a fixed sampling length. A typical example is a site investigation for

a pile design in which soil samples would normally be taken over a depth roughly

equal to the anticipated design length of the pile.

Fig.5.6 shows the variation of the a against the number of samples taken

within a fixed sampling length of 4 x 8. The a factor drops initially as the sampling

size increases (i.e. a smaller sampling interval). However, when the sampling

interval decreases to about half of the scale of fluctuation, a further increase in

the sample size gives almost no change to the value of a. Also shown in the figure

is the variance of s2, var{s2}, which is calculated under the assumption that the


sampling length = 46

var{ s2}

10 20 30 40 50 60 70 80 90 100number of samples

Figure 5.6 a Factor and var{s2} for a Fixed Sampling Length

soil property follows a joint Gaussian distribution and the point variance is unity.

Again, the threshold vale of a is about half of the scale of fluctuation. Therefore,

as far as the precision of the estimation of the mean value and the point variance

is concerned, there is no advantage in reducing the sampling interval to less than

half of the scale of fluctuation. The only way to reduce the alpha factor is to

sample over a greater length. However, in order to estimate the ACF of the soil

properties, it is necessary to have samples taken at sampling intervals less than

this value.

Based on the above observations, a useful strategy in soil sampling is as follows:

1. An inexpensive test is used to measure a property of the soil (e.g. cone resistance


from a CPT or index properties) at close spacings. This would enables the ACF

of the property to be estimated with sufficient precision. The ACF so obtained

can be taken as indicative of the ACF of other properties of the soil.

2. Samples can then be taken at a larger spacings from the field so that tests

can be performed to determine the soil parameters (e.g. strength, density etc)

which are required as input parameters in the analysis. The spacing should

preferably be larger than the scale of fluctuation to minimize the correlation

of samples so that the value of a would become near to its minimum value of

l/N.

For data set 1, the value of s2 is obtained as 0.139 (MPa)2. As will be discussed

later, the autocorrelation of cone resistance for data set 1 is well described by a

Type I ACF with a scale of fluctuation of 0.2m which also implies from Fig.5.2 a

value of 0.12 for the factor a. Using Eqn.5.6, the corrected variance estimate is

given as

*2 = (r^=0157<MPa>2

The corrected standard deviation s is shown on Fig.Fl of Appendix F. As the

sampling length is large compared to the scale of fluctuation, the bias of s2 is not

significant for this case.

5.2.3 Estimation of trend variance

If the point variance of the soil properties is known, the trend variance can

be estimated using Eqn.5.2. As discussed in the previous section, the a factor and

point variance for data set 1 are respectively 0.12 and 0.157 (MPa)2. The trend

variance is therefore given as

var{lc} = 0.12 x 0.157 = 0.019(MPa)2

Some difficulties arise in the simultaneous estimation of the cohesion and


coefficient of frictional resistance of the soil at a point. One way to obtain tfre

strength components of the soil from one sample is to perform a multi-staged test.

In this case, the procedure of structural analysis for the strength parameters would

be the same as that discussed above for the CPT data. However, it is not always

practical to carry out a multi-staged test for soil samples.

An alternative way is to carry out shear tests for samples close to each other

in the field, e.g. neighbouring soil specimens trimmed from the core of a borehole.

If the samples are close together, the strengths can be regarded as perfectly cor

related and the strength parameters obtained from these samples will constitute

a single observation in the statistical sense. However, there are seldom sufficient

soil specimens close enough to each other to justify the above assumption. This is

particular true when the scale of fluctuation is small. For instance, if the scale of

fluctuation of the soil property is of the order of a few decimeters (10cm), which

is not uncommon in real situations (for example, the CPT data above), soil speci

mens 0.5m apart may be regarded as independent samples even though they may

be very close physically. In this case, the test results have to be aggregated and

the mean cohesion and tan (f> obtained from the regression analysis of the p-q plot

of the data.

As a side product of regression analysis, the variances of the y-intercept and

of the gradient of the straight line in the p-q plot can also be obtained. These

variances are in turn related to that of the cohesion and tan </>. However, it must

be pointed out that the variances of cohesion and tan (f) so obtained are not the

point variances of the soil properties. They represent only the trend variance

var{7c} associated with the estimation of the mean cohesion (which is related to

the intercept) and mean tan</> (the gradient). Thus lumping all the test results

in a single p-q plot does not give the estimate of the point variance of the soil


property. Accordingly, the variance of the sample spatial average will be given as

var{KV} = a2-T2(V) + a2eg (5.7)

where a2eg is the variance of the sample mean property obtained from regression

analysis.

As lumping the test results does not enable the point variance to be obtained,

a judgemental vale based on local experience of the soil properties will have to be

used for the point variance a2. This is not as discomforting as it may seem because

any uncertainty in the estimation of a2 is scaled down by a factor of r2(K). For

a soil with a small scale of fluctuation, the dimensions of the slope may be such

that the uncertainty is dominated by the sampling uncertainty making an accurate

estimation of point variance a2 unimportant.

5.2.4 Estimation of correlation structure

To estimate a truly three dimensional ACF, a prohibitively large number of

samples is required. For one dimensional ACF, it has been suggested by Lumb

(1974) that a minimum of 20-50 samples are required to give a reliable estimate

of the ACF. Similar figures have also been quoted for mining data in Journel and

Huijbregts (1978). Therefore, for a truly three dimensional ACF, this will mean a

sample size of the order of 104 to 105 which can hardly be possible in practice.

A more practical approach is to assume a separable correlation structure for

the soil properties. In consequence,

Pin) =Pi(Ui) ' P2(v2) -P3(vi) (5.8)

where Vj, v2 and are the lag distance in the principal directions. The as

sumption of a separable correlation model leads to considerable simplifications.

Furthermore, the ACF can be estimated separately using a smaller number of soil

measurements taken along the principal directions.


At present, published results on the ACF of soil properties are usually limited

the principal directions (see Table 4.2). This assumption may be reasonable for

fill embankments or clay deposits because the compaction or deposition of soils

usually follow a vertical sequence. If the ACF in different horizontal directions are

not dissimilar, a further assumption of circular symmetry may be made so that

Eqn.5.8 becomes

where the subscripts y and r denote vertical and radial lag distances.

For residual soil profiles on hillslopes, the weathering progress may process in

a direction normal to the slope face. It is likely that that principal directions are

normal and parallel to the slope face. However, little has been published on the

ACF of residual soils and hence no conclusive remark can be made at this stage.

Other anisotropic autocorrelation models have been considered by David (1977)

and Journel and Huijbregts (1978). The process of constructing an autocorrelation

model is also well illustrated in these two references.

5.2.4.1 Sample ACVF

To get a ‘feel’ for the correlation structure of the soil properties, it is common

to plot the variation of the sample estimate of the ACVF with lag distance. The

resulting graph is called a sample ACVF.

The ACVF is commonly estimated by one of the following two estimators.

to the vertical and the horizontal directions which are commonly regarded as

p(v) = py(vy) ■ Pr(vr) (5.9)

(5.10)

(5.11)

where N is the total number of samples used for the estimation of k (Eqn.5.1)


and Ny_ is the number of sample pairs having a lag distance of v = |/t — L\. For

equally spaced data on a straight line (Fig.5.1), Equations 5.10 and 5.11 become

j Nh

Ct(h) = — - ?c) • («<+* - K) (5.12)1= 1

1C2(h) - — ^(/c* - 7c) • (Ki+h ~ k) (5.13)

where Nh = N — h is the number of sample pairs with a lag distance h • A/,

Ki — K(ii) and C'f/i • A/) is written simply as C(h).

In many circumstances, the sampling interval is not uniform. In such case, a

practical procedure is to divide the range of separation distance into suitable equal

discrete intervals. The lag distance for those sample pairs within a particular class

is represented by the mid-point of the class. Similar procedure can be used for

two and three dimensional cases. Such a procedure would incur additional bias

for Equations 5.10 and 5.11, but reduces the large sampling variance that would

otherwise arise as a result of the small number of samples that have exactly the

same separation distance.

a. Biasedness of C\ (t>) and Co(v)

The biasedness of the estimators Ci(v) and C2{u) depends on the configura

tion of the sample points. For the special case of equally spaced data on a straight

line (Fig.5.1), it can be proved (see Priestley, 1981) that the biasedness of C\(y)

and C2 (^) is of order 1 /N for large N and hence the two estimators are asymp

totically unbiased. The result should also be true in general provided that the

sampling domain enlarges with an increase in N. However, for small values of N,

the bias of the estimators can be quite appreciable. Fig.5.7 shows the expected

values of C\(h) and C2(h) assuming that (a) the soil properties follow a joint

Gaussian distribution (b) the ACVF is of simple exponential type with a2 equal

to unity and (c) the ratio of sample interval to the scale of fluctuation (i.e. A//<5)


lag distance

6

lag distance 6

Figure 5.7 Expected values of (a) C\(h) and (b) Co(h)


is 0.1. L/b in the figure denotes the ratio of the total sampling length to the scale

of fluctuation. Since Al/b = 0.1, a value of L/6 — 1 will imply a total sample size

of 11 and so on. It can be seen that sample ACVF based on a small sample size

is highly biased.

To reduce the bias of the sample ACVF, Cressie and Glonek (1984) and Cressie

and Hawkins (1984) have recently proposed an alternative estimator

1 N*

C3(h) = — y^(/c» - med(/Cj)) • (/cl+/l - med(/c;)) (5.14)

where med(-) is the median of the test results. Eqn.5.14 has the implicit assump

tion that the distribution of the point property is symmetrical,

b. Variance and covariance of Ci (t>) and Cr>(v)

The derivation of the variance and covariance of the estimators Ci(v) and

C2{v) requires the additional assumption that /c(£) is stationary up to the fourth

order, i.e.

C{kUi)k((2)k(<3)k(<4)}

is a function of the lag distances — /2|, \Li — etc. Here discussion will only be

confined to the special case of equally spaced data along a straight line (Fig.5.1).

The covariance of the estimator C\ (h) for a Gaussian process is given as

(Priestley, 1981)

I —n iy — n — s

eW{c,(A),Cl(A + «)} = ;p g V [C(j - i)C(j + s-i)

+ C(j + h + a - i)C(j - i - h)\

(5.15)

Setting s = 0 and assuming a Gaussian process, the following expression for

the variance of C\(h) is obtained.

N-h-l

E ,r= — (N — h — l)

var {6,(/i)} = ^ E* (Af — A — |r|) • |C2(r) + C(/j + r)C(A — r)| (5.16)


Similar results can be derived for C2(h). Since the only difference between C$h)

and C2(h) is the divisor, all that is necessary is to replace N2 in Eqn.5.15 by

(N — h)(N — h — s). In particular,

var{62(h)} = * £ (N-h-\r$.{C*(r) + C(h + r)C(h-r)}

(5.17)

For ‘well-behaved’ ACVF, we have for large values of N (Priestley, 1981),

var{Ci(h)} = O(-L) (5.18)

var{C2(h)} = 0(^-) (5.19)

Therefore, for a fixed lag distance h, both C\(h) and C2(h) are consistent estima

tors of C(h). Fig.5.8 shows the variance of C\(h) and C2(h) respectively for the

same problem considered previously in Fig.5.7. The sampling variance of C\(h) is

consistently lower than C2(h) (note the change in scale in Figures 5.8a and 5.8b.

The variance of C\(h) diminishes with h while the variance of C2(h) blows up as

h approaches N.

A more valid comparison between the two estimators C\(h) and C2(h) is by

means of the mean squared error (MSE) defined by

MSE = E{C(h) - C(h)}2

= eI [6(h) - £{C(/i)}] + [E{C(h)} - C(h)

= var{C(h)} + {bias C(h)}2

(5.20)

Fig.5.9 shows the MSE of C\[h) and C2(h) for the same problem considered

previously. The figure indicates that C\(h) has a smaller MSE than C2(h) and

therefore the former is generally a better estimator than the latter. Ci(h) also

possesses an additional desirable property of being a semi-positive function.


lag distance

6

-f- = 1

lag distance

6

Figure 5.8 Sampling Variance of (a) C\(h) and (b)


Figure 5.9 Mean Square Error of C\(h) and Co(h)


As the sample estimates of C(h) at different lag distances are obtained from

the same set of data, the estimates are correlated as indicated by Eqn.5.15. For

‘well-behaved’ ACVF, Eqn.5.15 would also approach zero as N —► oo. However,

for small value of N, the correlation will be fairly high. In consequence, ‘ripples’

may appear in the sample ACVF and the curve may not decay as quickly as the

parent ACVF. This is a further complication in the interpretation of the sample

ACVF.

5.2.4.2 Sample ACF

The graph of sample estimates of the ACF with lag distance is called a sample

ACF or correlogram. Since the ACF p(v) is related to the ACVF by

p(u) =

natural choices of the estimator will be

Pi(h) =

M10 =

Cfe)0(0)

c'Ait)

C,(0)

Ci(h)<3-2(0)

(5.21)

(5.22)

(5.23)

The statistical properties of pi(h) and p2(/i) are even more complicated than

Ci(h) and C2(/i). No exact results have yet been derived for the two estimators.

However, for large N, approximate expressions for the covariance of p\(h) and

p2(/i) are given in Bartlett (1946) and Priestley (1981). The behaviour of p\(h)

and p2(/i) is similar to that of C\{h) and C2(/i).

Another estimator of the ACF which has been used in the past is the sample

correlation coefficient

N-h _ _(*1 “ *1 )(*•+* - «2>

P = i = l

N-h N-h' J2 {k% ~ *i)2 • (ac1+/i - 7c2);

i=1 t=l

(5.24)


N-h _ N-hwhere Tci = JZ aRd = TjZk JZ Ki+h- The estimator has become ob-

solete for reasons given in Jenkins and Watts (1968) and Priestley (1981), namely

that it does not (a) fully utilize the stationary properties of n(t) and (b) yield a

positive semi-definite autocorrelation matrix.

5.2.4.3 Sample variogram

An alternative way to display the correlation structure of the soil properties

is by means of the sample variogram. Since

7(0 = 0(0) - C(v) (5.25)

the information on the correlation structure carried by 7(1;) and C[v) is essentially

the same. The autocorrelation of soil properties usually diminishes with lag dis

tance. Consequently, 7(u) will approached the point variance C(0) as v increase.

The point variance which is C(0) can therefore be estimated by the asymptotic

limit, also called the sill, of the sample variogram at large lag distance.

As the variogram is defined by

2l(v) = E{k{L) ~ *(*')}" (5-26)

where \t — l_'\ = v, a natural choice of the estimation for the semi-variogram 71 (h)

will be

N- 27i(li) = ^3{K(-*) — k(->)}2 (5-27)

where |ft — tj\ = v and Ny. is the number of sample pairs with lag distance v.

7i(n) due to Matheron (1971) is the earliest estimator proposed for 7(1;) and is

still widely used nowadays. For equally spaced data on a straight line (Fig.5.1),

7(t>) will be denoted simply as q(/i) and similarly for its estimators.

71 (u) is an unbiased estimator for 7(u). However, it is not a resistant estimator

in the sense that the value of (t>) is badly affected by outliers due to the square


term in Eqn.5.27. Over the past few years, mining geostatisticans have proposed

many alternative resistant estimators for 7(1’). The following are some examples.

Armstrong and Delfiner (as reported in Dowd, 1983) have proposed two al

ternative estimators. The first one is

where Qq is a suitable quantile of the experimental cumulative distribution func

tion of the squared difference for lag h. For example, Qq can be chosen to be me

dian of the squared difference. The second estimator by Armstrong and Delfiner is

based on the M-estimator by Huber and iterations are required for the calculation

of the estimate.

Cressie and Hawkins (1984) proposed a series of estimators based on the trans

formed difference data Y{ — |/ct- — Act+/j|2. The estimators have the form

where T is an estimator for the central location of Tr. A number of estimators have

been used for this purpose including the mean, median, trimmed mean and .the

M-estimators. It is observed (Cressie and Hawkins, 1984) that the mean, median

and the M-estimators yield satisfactory results. However, the M-estimators are

more complicated and an iterative procedure is required for the calculation of T.

Dowd (1983) proposed four other resistant estimators for 7(h). The first two

estimators are

12(h) = (5.28)

1 T*(5.29)2 ' 0.457 + 0.494/N + 0.043/JV-

(5.30)

(5.31)


where yt(h) = /ct — Kl+h and y(h) ^median of yt(h). The third estimator is

where wt = (/ct- - kl+h)/(K • Ml) and Ml = median |/cx —/ct+1|. The sum is over

all sample pairs (totalling Nh) with lag distance h and K is a constant depending

on the data, usually from 6 to 9, the same as for the fourth estimator 77(t>):

where = (yi(h) — y(h))/(K ■ M2) and M2 = median \yt(h) — y(h)|. yt(h) is

defined as in Eqn.5.31.

Omre (1983) has also developed another resistant estimator based on the

estimation of the bivariate probability density function of Kt and Kv+h-

Each proponent of the resistant estimators has demonstrated the superiority

of his own estimator(s), normally by means of simulated data. Perhaps it is fair to

say that each of the resistant estimators discussed above would be more resistant

at large to contamination by outliers than the classical estimator 71 (t>), but none

of the estimators will be resistant enough to be the most resistant estimator for

every possible form of contamination.

Cressie (1979) proposed a somewhat different approach based on straight-line

fitting for the estimation of the variogram. Define Dt^ — «,•+/, — Kt. Assuming a

second order stationarity and that Dl+k,i-k and A,/i+A: follow a bivariate Gaussian

distribution, it can be shown that

%{h) = (5.32)

. _ Nhj2(y'(h) - y(h))2 ■ U - w.2)4l7{n) — r -,22[E(l-»,2)'(l-5»r)

(5.33)

(5.34)

where E{X\Y} is the expectation of X given Y. Thus if a linear regression of

Di^+k against Dl+k,h-k is performed, a straight line through the origin with

gradient b — (7(h) — ~i{k))l^(h — k) would be expected.


In particular, if k = h— 1, then b = (7(h) — 7(h— 1))/7(1) or 7(h) = 7(h— 1) +

6-7(1). By successively plotting the data points (Dl+h~ 1,1, Di^h-i), h = 1,2,-*

and calling 6^ the gradient of the best fit straight line through the origin, we have

7(2) = 6! • 7(1)

7(3) = 7(2) +62 • 7(1)

7(/i) =i(h- 1) + 62 • 7(1)

or

l(h) — (1 + 61 -f • • • + bh-i) • 7(1) (5.35)

The estimate of 7(h) can therefore be estimated by the recurrent relation of

Eqn.5.35. 7(1) can be estimated using any of the estimators mentioned above.

Similarly, by putting k = 1, we have

7{h) — (1 + Ch-i + Ch-i Ch-i + * * • + c/jC/i—i • • • Ci) -7(1) (5.36)

where Ch-1 is the gradient of the regression line of A,/i+i against Dt+\,h-\ etc.

The graphical method by Cressie (1979) is simple, but viable only when a

substantial amount of data is available. Taking h = 2 and k = h — 1 (Eqn.5.35),

it can be proved that the coefficient of correlation p between A+1,1 and D, 3 is

p(h) - p(2h)

1 - P(3A))(1 - p(h))(5.37)

Using the simple exponential model and writing f = exp{—2/iA//<5}, Eqn.5.37

becomes

sj 1 + f + f2f * 1 (5.38)

As ^ tends to 1} p approaches the theoretical asymptotic maximum value of 0.578.


* * * — r

sample correlation coefficient

Figure 5.1095% Confidence Interval for Coefficient of Correlation p

Values of Curves are Number of Samples


Fig.5.10 shows the 95% confidence interval for the experimental correlation

coefficient. Unless the sample size is large, the confidence interval is wide. For

example, for N = 20, the 95% confidence interval is approximately (0.2,0.8) for

the theoretical maximum value of p. This would imply a high sampling variance

for bh and the estimates of t(/i) will quickly become very unreliable as 7(h) is

built up successively using Eqn.5.35.

5.2.5 Parameter estimation of autocorrelation models

Both the sample ACVF (or ACF) and sample variogram are very useful in

choosing of a suitable autocorrelation model. Once a model is chosen, the next

step is the estimation of the parameters for the chosen model. Various methods

will be discussed, in the order of their complexity.

5.2.5.1 Fitting by ‘eye’

The procedure of fitting by ‘eye’ is widely used in geotechnical data analysis

and geostatistics (e.g. David, 1977; Journel and Huijbregts, 1978; Ripley, 1981)

because of its simplicity.

The procedure of fitting by ‘eye’ involves the fitting of a theoretical model to

the sample ACVF, correlogram or variogram. The selection of the parameter is

often based on the visual harmony between the fitted model and the experimental

curve. There are various ways by which the autocorrelation model can be fitted.

The following are some examples.

Fig.5.11 shows a plotting paper for the fitting of a simple exponential ACF.

To use the graph, the sample ACF (Eqn.5.22) is plotted against the lag number.

The value of A//<5 which gives the best fit to the sample ACF is read from the

graph. Knowing the sampling interval A/, the scale of fluctuation can be evaluated

accordingly.

The sample ACFs of data sets 1 and 2 are also plotted in Fig.5.11 using the

estimator p\(h) (Eqn.5.22). The curve with A//<5 = 0.1 gives a satisfactory fit


Figure 5.11 Plotting Paper for Type I ACF


to both data sets. As A/ is 0.02m for data set 1 and 2m for set 2, the scales of

fluctuation are therefore given respectively as 0.2m and 20m.

Another commonly used technique of fitting is by matching the gradient of the

experimental and theoretical curves at zero lag. The gradient of the experimental

ACF can be estimated by fitting a straight line to the first few points of the curve.

For example, fitting a straight line to the first five data points of Fig.5.11 for data

set 2 yields an estimate of —0.18 for the gradient of the experimental ACF at zero

lag. Since p(h) = exp{—2hAl/8}, we have

dp(h) _ 2A / dh &=o 6 (5.39)

Therefore,

-0.18

=> 8

2A /T

2x20.18 = 22m

which is similar to the estimate obtained previously for data set 2.

As fitting a straight line is always easier than fitting a curve, it is preferable,

if possible, to perform a suitable transformation to produce a linear relationship.

For instance, the simple exponential ACF will appear as a straight line on a semi

logarithm plot as illustrated in Fig.5.12 for the data set 1 and Fig.5.13 for data set

2. The gradient of the fitted straight linesfby eye) are —9.4 and —0.21 respectively

which correspond to a <5 of 0.21m for data set 1 and 19m for data set 2.

For more complex models involving more than one parameter, further manip

ulation of the data may be necessary. Examples of this can be found in David

(1977) and Journel and Huijbregts (1978).

5.2.5.2 Variance plot

The method of variance plot to be discussed in this section is also based on a

procedure of fitting by ‘eye’. But the method involves a more in-depth theoretical


loge ACF

-1.0 -

-2.0 -

10 11 12 13 lag number

Figure 5.12 Sample ACF for Data Set 1 - Semi-log Plot


1 o ge ACF

-0.5-

-1.0 -

-1.5 -

9 lag number

Figure 5.13 Sample ACF for Data Set 2 - Semi-log Plot


consideration which deserves a separate discussion.

Agterberg (1967) appears to be the first to apply the technique of variance

plot to the structural analysis of mining data. The method is also discussed in

his book Agterberg (1974). A similar procedure was also proposed by Vanmarcke

(1977a&c) for the estimation of the scale of fluctuation of soil properties. Define

K1 n —ln

n

^ ^ Ki+k— 1 (5.40)

7cm is simply the average of n adjacent observations starting from Act. Consider the

theoretical variance <r2 of 7cm of a equally-spaced spatial series (Fig.5.1). According

to the assumption of stationarity, uar{7c;n } is independent of the location. Without

loss of generality, take t = 1. Thus

K = var{lcln}j n n i—1

— {^%qr{/Ct} + y]co^{/Ci,/ct+J}|

2 9 n *—1(5.41)

= a2 -rz(n)

n—1where T2(n) = £ jl + 2 (1 — ^-)/?(r). In fact, the a factor discussed earlier

in Section 5.2.1 is T2(N) where N is the total number of samples. Note T2(n) is

analogous to the variance reduction factor for the continuous case and hence the

same symbol is used here. r2(n) has similar characteristics as T2(L), namely

1 if n = 1

^ if n is larger2(n) = (5.42)

where 6n is a constant related to the scale of fluctuation 6 and it can be treated

as the scale of fluctuation for the discrete case. For the simple exponential ACF,


we have

<5n -2A(

1 + e *

1 — e'2AI

66 =

2A /, ft” T 1 1 *„-l

(5.43)

Similarly for the square exponential ACF, we have

Sn =„( 42

1 + e ( s )

1 - e~6 =

\7tA/2

(5.44)

If the sample estimate of <r2, say s2, is plotted against n, s2 would tend to follow

a similar variation with n as T2(n). Therefore, the scale of fluctuation can be

obtained by matching the sample curve of s2 with the theoretical curve of T2(n).

Agterberg (1967&1974) suggested a different procedure of plotting n • s2 which is

to be matched with the theoretical curve of n • T2(n).

A simple estimator for <r2 is

= A yV,„ - <c„)27=i

(5.45)

Nnwhere /c„ = ^ Kin- Nn is the number of /ctn that can be formed for a record

i=iof N samples, i.e. Nn = N — n + 1. The expectation of s2 is

E{sl) =°n- var{Kn} (5.46)

The bias of s2 is therefore given as

bias s2 = var{icn}

—2 Nn-1 n — 1

W E E (lVB-|r|)(n-M)p(|r + .|)(5.47)


Figure 5.14 Bias of - Type I ACF (A//<5 = 0.1, o2 = 1)


Fig.5.14 shows the bias of the estimator s2 for a Type I ACF for the case

of A1/8 = 0.1 and a2 = 1. Also shown on the figure is the theoretical value of

r2(n). It can be seen that the bias is very substantial for a small sample size.

When the lag distance is larger than about half of the sampling length, the bias

is comparable to the theoretical value of T2(n) and therefore data points with lag

distances greater than this value should not be used for the estimation ACF. To

show the precision of the estimator s2, a more logical indicator would be the COV

defined in terms of the MSE, denoted as RCOV.

RCOVMSE of si

^/t>ar{s%} + {6ias sg}2(5.48)

The calculation of var{s„} is discussed in Appendix D. Fig.5.15 shows the plot

of RCOV for the same problem considered previously in Fig.5.14. The RCOV in

creases with lag distance indicating that the estimation s2 become more imprecise

as n increases.

Fig.5.16 shows the graph of T2(n) for the simple exponential ACF assuming

cr2 = 1. To use the graph, s2 has to be normalized by s2. In essence, r2(n) is

estimated by s2/s2. The value of A1/8 is read from the best fit curve and the

scale of fluctuation can be calculated accordingly.

Vanmarcke (1977c) suggested a simpler plotting procedure using the asymp

totic results of Eqn.5.42 for T2(n). Curves of 8n/n can be constructed for various

values of Sn. The value of Sn is determined from the curve which gives the best

fit to the data for large values of n. The scale of fluctuation can then be eval

uated accordingly using Equations 5.43 or 5.44 for examples. The procedure by

Vanmarcke (1977c) is simpler than the complete model, but suffers from the fol

lowing disadvantages. The value of s2 at large lag is more unreliable as indicated

in Fig.5.15. The information carried by the more reliable observations at smaller


Figure 5.15 Plot of RCOV for s“ - Type I ACF

5.2. TYPE 1 SOIL PROFILES 5-38

r2(n)

i Al

number

Figure 5.16 Plotting Graph for T2(n) - Type I ACF


lag distances is wasted. Furthermore, the curvature of r2(n) is small at large lag.

Consequently, the data at large lag distances can normally be fitted equally with a

range of values of Sn and the determination of a suitable value for Sn becomes less

certain. All well-behaved ACFs have the same asymptotic equation as Eqn.5.42.

Unless an ACF can be chosen based on other considerations, the knowledge of 6n

does not enable the scale of fluctuation to be calculated.

With the availability of computer plotting facility, graphs can be easily con

structed for different ACFs with different parameter values. By this method, the

procedure using the complete model is as simple as the simplified procedure by

Vanmarcke (1977c), but more accurate.

The variance plot for data sets 1 and 3 are shown in Fig.5.16. It can be

seen that the autocorrelation structure of the cone resistance (data set 1) is well

described by a Type I ACF with A1/6 = 0.1, which in turn implies a value of

5 equal to 0.2m. This is in good agreement with the values obtained from other

methods discussed above. A value of Al/S = 0.4 gives the best fit to the third set

of data (although it is apparent that a Type I ACF does not give an exact fit).

This yields a scale of fluctuation of 12.5ft as compared to a value of 10ft obtained

from the simplified procedure by Vanmarcke (1977c) who also assumed a simple

exponential ACF for the soil property.

5.2.5.3 Curve fitting by least squares

Instead of fitting by ‘eye’, a more objective way for fitting a model to the

sample ACVF, ACF or variogram is by means of least squares. The least squares

method is often used purely as a numerical criterion for obtaining a good overall

fit to the experimental curve.

Suppose G(h\9), where 9 is the collection of parameters, is the curve to be

fitted to the experimental curve G(h\9). For instance, G(h\9) may be the ACVF

C(h) = p(y\c*) where a is the parameters of the ACF and 9 = (<r2,a). G(h\9}


may be the estimator C\(h) for C(h). The parameters are obtained by minimizing

the sum of squares of the residuals, i.e.

min£yC7(Ai|£) -G(/i,|0)}2 (5.49)~ t

The use of the least squares method is to avoid the subjectivity inherent in the

procedure of curve fitting by ‘eye’. Sometimes, weights W{ are assigned to the

residuals to obtain a more reasonable criterion of curve fitting. The weighted least

squares procedure is described by

min Wi{G(hi\9_) — G(hl\9)}2 (5.50)— i

A larger weight is assigned to more reliable estimates and vice versa. Here the

choice of wt is also empirical. The program developed by Tough and Leyshon

(1985) is an example of this.

The least squares method can also be used as a statistical procedure for es

timating the parameters of the model, also called regression analysis. Neglecting

the biases of G(h\0). the following model may be used.

G(hi\0) =G(ht\6) + ei (5.51)

where et are the random errors with zero mean. The error term is related to the

random component (.(tj) of the soil property at the sample locations. As G(ht\0)

and G(hj\9) are estimated from the same set of observations, the error terms

et and e3 would also be correlated. The ordinary least squares method, which

assumes independence of error terms, is not appropriate for this case. Instead, the

generalized least squares (GLS) method should be used and the following problem

will have to be solved.

min|{G(/lt|£) - G(ht\0)}TVe~l {G(h,\0) -G(A,-|0)}} (5.52)


where G(h{\9) = [G(hi |0), G(/i2|$), • • -]r and similarly for G(ht\9). Vg_ is the co

parameters. The solution for Eqn.5.52 is quite involved. To reduce the computing

effort, a weighted least squares method could be used in which the off-diagonal

Cressie (1985) seems to be the first to develop this approach to fitting a theoretical

model to the sample variogram and derived a series of formulae for the calculation

of Ve_ for the estimators 7i(^) and l3{h) (Equations 5.27 and 5.29). For model

fitting of ACVF, the covariance matrix of the error terms can be obtained using

Eqn.5.15.

Although the use of Equations 5.52 and 5.53 is more justifiable than Equations

5.49 and 5.50, the latter are often used for expediency.

5.2.6 Effects of regularization

The properties of a soil at a point are purely a mathematical abstraction.

In practice, soil properties can only be measured from a specimen with finite

dimensions. In geostatistics, the average property kv measured from a sample with

dimension v(£) centred at a point t is called the regularized property over a support

v(t). The same term is used herein for soil properties. Only one dimensional

regularized soil properties will be considered here, but the conclusions drawn from

the following discussion apply equally well to the case of two or three dimensional

regularized properties.

Consider a length of dimension / centred at a point t (Fig.5.17), the regularized

variance matrix of the error terms. Note that the matrix V0 1 also depends on the

terms of Ve 1 are neglected. As a result, Eqn.5.52 becomes

(5.53)

soil property is defined as


K(t)

Figure 5.17 Regularization of Soil Property over a Length /

The ACVF Ci(v) of ki(t) can be defined as

Q(v) = cov{ki(t),ki(t + t>)} = a2 • B(l, /; v) (5.55)

where B(IJ; v) is the covariance factor of two line average each having an averaging

length of / and separated by a lag distance of v. As var{ki(t)} = a2 • r2(/), the

ACF pi(v) of ki(t) can be obtained as follows.

Pl{v) =-----TTBxvar{Ki(t)}

B(l,hv)F 2(/)

_ Jo(/ ~ t)(p(v + r) + p(v - T))dr

2 fo(l~ r)p(r)dr

(5.56)

Note Ci(v) and pi{v) are defined only for v > l.

Fig.5.18 shows the ACF of ki(t) for different supports v for a Type I ACF.

The ACF of ki{t) deviates from the point ACF as the ratio 1/6 increases. Note

that in Fig.5.18, b is assumed to be known. It is therefore not suitable as plotting

graph of the ACF to be used for the estimation of the ACF.

Define c to be the ratio of the length of the sample to the sampling interval,

i.e. c = l/Al. The correct plotting graph is shown in Fig.5.19 for the case of


0.2 0.3 0.4 05 0.6 0.7 0.8 0.9 1.00.0 0.1

Figure 5.18 ACF of Regularized Soil Properties


A C F

c = 0.0

c = 0.5

\\N

\\\\ v\\\\V

number

Figure 5.19 Plotting graph for Regularized ACF


c = 0.5 and a Type I ACF. The plotting graph for the point property (c = 0) is

also shown in the figure for comparison. Consider the extreme case of A1/6 = 0.5

which means that the support l is equal to one quarter of the scale of fluctuation.

As indicated in Fig.5.19, the difference between the two sets of curves is small

even though the length of the sample is relatively large as compared to the scale

of fluctuation in this case. Therefore, regularization of soil properties has little

influence on the estimation of the point ACF.

The effect of regularization on the a factor is examined next. For regularized

soil properties, the a factor should be calculated using the following equation (for

the one-dimensional case).

(5-57)t=ij—\

where cq is the a factor for regularized soil properties and vl3 is the lag distance

between sample points.

Fig.5.20 compares the a factors for point properties and regularized properties

for the case of c — 0.5. The a factors are so close for the cases of A1/6 = 0.05 and

0.2 that they cannot be distinguished in the figure. Even for the somewhat extreme

case of Al/8 = 1 in which the soil properties are regularized over a length equal

to half of the scale of fluctuation, the difference is still less than 10%. Therefore,

the effect of regularization on the ot factor can be ignored in practice.

However, regularization does have some influence on the estimation of the

point variance. The estimator s2 defined by Eqn.5.6 only gives the estimate of

the variance of the regularized property. To calculate the point variance, a further

correction for the variance reduction due to spatial averaging over the domain of

the support is required. Referring the Equations 5.4 to 5.6, the corrected point

variance estimator based on regularized properties should be


a , a(

number of sample

Figure 5.20 a Factor for Regularized Soil Properties


where sf is the variance estimate obtained from regularized samples using Eqn.5.4,

viz

= 0~«i)2 (5-59)i=i

If the sample size is relatively large in comparison with the scale of fluctuation,

the variance reduction due to regularization is significant. For instance, if 1/8 =

0.5, r2(/) becomes 0.74 for the Type I ACF. Therefore, the point variance will be

underestimated by 25% if the effect of regularization is not accounted for.

For a Type I soil profile, the mean value of regularized properties is the same as

that of the point properties. Therefore, the sample mean value of the regularized

properties 7q can be used directly as an estimator of the point mean value. In this

case, the trend variance becomes var{ki} instead of var{Jz} derived in Chapter 4

based on the sample point mean value 7c. Therefore

trend variance = t>ar{/q} = cq • var{ki(t)} (5.60)

where cq is given by Eqn.5.57 and var{ki(t)} can be estimated using a equation

similar to Eqn.5.6. Therefore,

vqt[ki} (5.61)

where sf is given by Eqn.5.59. Note there is no need to correct for the variance

reduction due to regularization for the trend variance. In fact, the larger the

support is, the smaller will be var{ki(t)} due to variance reduction and hence the

trend variance. This is one of the reasons why the testing of large block samples

in the field is usually considered to give a more accurate estimate of the strength

parameters than that obtained from small samples in the laboratory.

In summary, regularization has the following consequences on the structural

analysis of soil data.

5.3. TYPE II SOIL PROFILES 5-48

1. Regularization has little effect on the estimation of ACF. However, the sample

ACF at lag distances smaller than the dimensions of soil samples cannot be

obtained from regularized samples. If the scale of fluctuation is comparable to

the dimensions of the soil samples, details of the ACF at small lag distances

cannot be revealed. The estimate of the ACF will invariably be crude in this

case. However, it is not considered to be a handicap. In practice, the dimen

sions of soil samples are usually small, of the order of 10cm or less. If the

scales of fluctuation are comparable to the dimensions of the samples, it would

necessarily mean that the scales of fluctuation are also small. In this case, the

variance reduction due to spatial averaging in a slope would be so significant

that total uncertainty will be dominated by other factors such as the trend

variance rather than the spatial variability of soil properties, and an accurate

estimation of the scale of fluctuation becomes unimportant.

2. The influence of regularization on the a factor is small. In practice, the a

factor based on point properties can be used even for regularized properties.

3. The variance of regularized properties is smaller than that of the point prop

erties. The estimating the point variance by the variance of the regularized

properties will lead to unconservative results. However, this effect can be ac

counted for easily using Eqn.5.57.

4. Regularization has the effect of reducing the trend variance. Therefore, estima

tion of mean soil properties based on larger specimens is more accurate than

that on small specimens.

5.3 TYPE II SOIL PROFILES

5.3.1 Introduction

For a Type II soil profile, the trend component as well as the correlation


structure of the random component have to be estimated. The structural analysis

of a Type II soil profile is complicated by the fact that:

• If the correlation structure is known, the trend component can be estimated

efficiently (in the statistical sense) using the generalized least squares (GLS)

method. However, if the correlation structure is not known a-priori, the proce

dure of GLS cannot be carried out. Although the ordinary least squares (OLS)

method can be used and produces the same asymptotic efficiency as the GLS

method (Priestley, 1981), the method will be less efficient for small samples

and always underestimates the variance of the trend component.

• If the trend component is known a-priori, the random component of the sample

can be obtained simply by subtracting the trend value from the observed value

and the procedure for estimation of the correlation structure becomes identical

to that of Type I soil profiles. If the trend is not known a-priori, the correlation

structure of the random component may be estimated from the residuals of the

fitted trend, via £; = K(t{) — g(tt). However, such a procedure will produce

highly biased results and the resulting correlogram or variogram may bear

little resemblance to the true situation (Matheron, 1971; Johnston, 1972).

The difficulty now becomes clear; the trend component cannot be estimated

efficiently without the knowledge of the correlation structure of the random compo

nent and the correlation structure cannot be established without some knowledge

of the trend component.

In the following, a simplified procedure of structural analysis of soil data

for Type II soil profiles is introduced, followed by the discussion of an iterative

least squares method and the maximum likelihood method for the simultaneous

estimation of the trend component and the random component. The techniques

by Matheron (1973) and Delfiner (1976) for filtering out the trend component will

also be mentioned.


5.3.2 A simplified procedure

If the soil property has no horizontal trend, the following procedure may be

used.

Figure 5.21 Sample Locations in a Soil Profile

The principles can be illustrated using the simple example in Fig.5.21. Sup

pose soil measurements are made on samples taken from boreholes A and B at

the locations indicated in the figure. The arrows shows the scales of fluctuations

of soil properties based on the prior knowledge. One way to minimize the corre

lation between the samples is to use only the test results for borehole A having

a separation distance greater than 6y for the estimation of the trend component

for instance the test results at Ai, A^, Ay etc. The test results so obtained will

be largely independent and the trend component can therefore be estimated using


the ordinary method of least squares.

Residuals are then calculated by subtracting the estimated trend component

from the test results. The structural analysis of residuals would then be the same

as a Type I soil profile. Data from other boleholes can also be aggregated to obtain

a more accurate estimate of the trend component. For instance, test results with

separation distance greater than 6y are chosen from borehole B (e.g. at locations

Bi, B4, B7 etc). As boreholes A and B are separated by a distance greater than 6X,

the test results obtained from the two boreholes can be regarded as independent

of each other. Consequently, all the test results from Ai, A4, A7,...and Bi, B4,

B7,.. .can be lumped together from which the trend component can be estimated

using the ordinary method of least squares.

The above procedure breaks the circularity discussed above, but at the expense

of wasting some of the information carried by the test results which have not been

chosed for the estimation of the trend component.

5.3.3 Iterative least squares method

As mentioned earlier, the difficulty in analyzing a Type II soil profile arises

from the fact that the estimation of the trend component requires the knowledge

of the correlation structure and vice versa. This circularity may be broken by an

iterative least squares method as suggested among others by Johnston (1972), Ord

(1975), Sabourin (1976), Ripley (1979) and Bennet (1979).

Firstly, the trend component can be assessed using the generalized least

squares method and a reasonable ACF for the random component. The ACF is

then estimated from the residuals obtained from the fitted trend. With the newly

estimated ACF, the trend component is then re-calculated using the GLS method.

The process is repeated until (hopefully) the changes in the trend component and

the ACF are small.

There is no common consensus as to how the iterative least squares method


should be implemented. The following criterion based on the GLS method is

proposed herein. The criterion can be stated as

min J(0) = min eT A le e_ e_

(5.62)

where e = [fi, €2, • • •, £n]T denotes the realized random components of the soil

samples, given by

(, = «(<,) - g(ii) = KiU) - ^2 aj PjiU) (5.63)3

and A is the theoretical autocorrelation matrix (AM) of e. Under the assumption

of stationarity, the AM has the form

r 1 P12

1

A(a) = Sym

P13 Pin

P23 ••• P2N

1 * • • P3N >

1

(5.64)

where pij is the autocorrelation between sample i and j described by the ACF

p(i;|a) with parameters a. 0 is the parameter vector comprised of the trend coef

ficients a and the parameter a of the AM. Note that t{ does not have to follow a

regular pattern in space.

If the AM is given a-priori, the estimate of the coefficient of the trend compo

nent given by the solution of Eqn.5.62 is BLUE (best linear unbiased estimator,

see e.g. Kendall and Stuart, 1961). Furthermore, the least squares estimate of the

trend coefficients are given by (Draper and Smith, 1981)

2= (PtA-1P)-1PtA~1k (5.65)

1s = (a - P a)T A l(tz -Pa)N-k (5.66)


where N and k are respectively the number of samples and the number of terms in

the generalized polynomial (Eqn.4.5). a is a k x 1 column vector representing the

coefficients of the generalized polynomial, k is the vector of measured soil property

at the N sample locations. P_ = (Fj, jP2,... ,P.^)T is a N x k matrix where P_t is

a A: x 1 vector representing the values of the variable of the generalized polynomial

at the ith sample location and the arrow means ‘estimated by’. The covariance

matrix V± is estimated by

Va -* s2(XTA~lX)-' (5.67)

If A(o) is not known a-priori, the parameter 6 has to be optimized to give the

minimum value of J{9). It is easily proved that

min J(9) = min J(9) t 2.-a

mina

mina •J(fi)]

min[/F(a)]a

(5.68)

Eqn.5.68 can be interpreted as an iterative least squares procedure. Given the

value of a (i.e. the ACF is prescribed), the trend component is estimated using

the GLS procedure. The parameter a is then adjusted until the change of H(a)

and hence the trend component is small. Note that when a is specified, say Oy,

the function Ff(ay) is simply the solution of the GLS given A(a) — A(ft;) and

the solution for a and cr2 is given by Equations 5.65 to 5.67. In fact, Ff(oy) can

be solved readily by available statistical computing package, such as the IMSL

packages (IMSL, 1984). Therefore, the optimization problem of Eqn.5.68 is now

reduced to a much simpler problem of optimization H(a) with respect to a. In

particular, if a contains only one parameter, Eqn.5.68 can be solved efficiently

using the technique of rational approximation (see Appendix E). If a has more

than one variable, the problem may be solved by the optimization technique for


multivariate functions outlined in Appendix F. Other optimization techniques can

be found in Jacoby et a1 (1972).

Different ACFs or generalized polynomials may be used. The model which

gives the smallest value of J(6) can be regarded as the most suitable model.

Although the legitimacy of the iterative least squares procedure does not seem

to have been proven in any general manner, the method is commonly used for

expediency and as a compromise between theory and practice. A recent example

of the application of the iterative least squares procedure is given by Sadeghipour

and Yeh (1984) for the estimation of the transmisivity of a homogeneous aquifier.

It must be noted that the iterative least squares method does not guarantee

a feasible estimate for the parameters. To tackle this problem, Eqn.5.68 has to be

converted into a constrained optimization problem in which the parameters are

restricted to within their feasible ranges as is done in Sadeghipour and Yeh (1984).

5.3.4 Maximum likelihood estimation

If the joint distribution of the soil property is known, the parameters of the

model can be estimated using the method of maximum likelihoods (ML). In prac

tice, the joint distribution of soil properties is not known. A joint Gaussian distri

bution is usually assumed partly because the marginal distribution (which is also

the point distribution) can usually be approximated by a Gaussian distribution

with reasonable accuracy (e.g. Lumb, 1966; Matsuo, 1976), but mostly because of

convenience. The use of a joint Gaussian distribution is analogous to the use of

linear elasticity theory in settlement analysis. The assumption is not strictly cor

rect but often gives reasonable results in practice. The method of ML is discussed

in most references on statistics such as Cremer (1946); Arnold (1981) and Mood

et al (1974). If a Gaussian distribution is assumed for the random component of


the soil property, the likelihood function is given by

m(27TCT2) T

exp(-^iT^ li) (5.69)

The same notation of Eqn.5.62 is used in Eqn.5.69 and \A\ is the determinant of

the autocorrelation matrix.

Maximizing the likelihood function with respect to the parameters 0 is equiv

alent to maximizing the log-likelihood function M — In L(0).

M — — — In 27r — — In a2 — - In \A\-------eT A~ls (5.70)2 2 2 2 (j

The ML estimate of the point variance g2 is given by

a2 eTA~lsN (5.71)

Substituting Eqn.5.71 into Eqn.5.69, we have

M = -y (1 + In 2w — Id N) — j 7(5)1 (5.72)

where J(6) is defined in Eqn.5.62. Thus, maximizing the log-likelihood function

is equivalent to the minimization of the following function

Q(i) = min|yl| *./(£) (5.73)0_

Note if A = I, the identity matrix (which also means that the random component

is an uncorrelated process), the GLS procedure and the ML estimation are iden

tical. can be treated as a weighting function and the ML estimation can be

interpreted as a ‘weighted’ GLS procedure. Compare the weighting function \A\k

for a time series and a spatial series. For a first order autoregressive model AR(1),


the AM is given by (Vinocl and Ullah, 1981)

'(1 -p2)* 0 0 O'

~P 1 0 ••• 0

A = 0 -p 1 0 (5.74)

0

0 0 ••• -p 1

and the determinant \A\ is (1 — p2)?. The nullity of the upper triangle of the

matrix A is indicative of the fact time events are only related to past events. The

the GLS procedure and the ML estimation are asymptotically equivalent.

Fig.5.22 shows the weighting function for an AR(1) series. The figure indicates

that the weighing function is very close to unity even for modestly large values of N.

In consequence, the GLS and ML estimators are essentially the same in practice for

a time series. Now, consider a one-dimensional spatial series with equal sampling

intervals (Fig.5.1). Suppose that the ACF is of the simple exponential type (Type

I), the AM will be given by

weighting function of an AR(1) series is therefore (1 — p2)™ . Since

lim (1 — p2) 2^ = 1 (5.75)

1

P

A= P2 P (5.76)

P

P 1

where p = exp{—2A//<5}. The off-diagonal elements of the autocorrelation matrix

A are non-zero. This shows one of the fundamental difference between a time series

and a spatial series. The determinant of the matrix A is given as (1— p2)N~l. Thus,


time series

spatialseries

0.0 0.1 0.2 03 0.4 0.5 0.6 07 0.8 09 1.0

Figure 5.22 Weighting Function for ML Estimation

the weighting function becomes (1 — p2)~Fr~. As n oo, the weighting function

approaches the limiting value of (1 — p2). Fig.5.22 also shows the variation of the

weighting function for the spatial series. The variation of the weighing function is

for the spatial series much more pronounced than that of the AR(1) time series. As

a result, a larger difference between the GLS and ML estimates of the parameters

for the spatial series would be expected. The GLS procedure would yield a larger

value of p and hence a larger scale of fluctuation than the ML procedure.

Which method is better, the GLS procedure or the ML estimation, is debat

able. The influence of the choice of the GLS or ML estimates on the final predicted

value of failure probability of slopes remains a question of further inquiry. For ex-


ample, although the GLS procedure gives a larger value of the scale of fluctuation

6 which also means that the predicted value of the variance reduction due to spa

tial averaging is smaller, a larger value of 6 will also imply a smaller predicted

variance for the coefficient of the trend component and hence a smaller predicted

trend variance. Further investigation needs to be done to establish whether the

failure probability of slopes based on the GLS procedure is greater or smaller than

that based on the ML estimation.

Given the value of a, say the ML solution under the assumption of a joint

Gaussian distribution is the same as that obtained using the GLS procedure and

hence the conditional estimates of the parameters are also given by Equations 5.65

to 5.67. Using the same argument as in Eqn.5.68, the MLE can be re-formulated

as

min Q(([) = min6_ a

I A\*H(a) (5.77)

where H(a) is defined by Eqn.5.68. Eqn.5.77 can be solved using the same tech

nique as discussed previously for the GLS procedure.

Again different ACFs and generalized polynomials may be used. The most

suitable model can be identified using the Minimum Akaike’s Information Criteria

as suggested in Matsuo and Asaoka (1977).

It is well known that the maximum likelihood estimation yield asymptotically

efficient estimates for independent identically distributed samples (see Cremer,

1946, pp500-504). For spatial data, the observations are autocorrelated among

themselves. Intuitively, the same property should also hold for spatial data as N

becomes large provided that the an increase in N is accompanied by a simultaneous

increase in the sampling domain. The problem has been recognized by statistician

(e.g. Arnold, 1981), but a formal proof is not yet available to the knowledge of the

Author.


Once the optimum value of a is ascertained using Eqn.5.77, the estimates of

the coefficients of the generalized polynomial and its covariance matrix can be ob

tained using Eqn.5.65 and Eqn.5.67 except that the parameter a is replaced by the

ML estimate and the s2 is given by Eqn.5.71. Alternatively, the covariance ma

trix of the parameters can be approximated by the inverse of Fisher’s information

matrix M, i.e. Vg = M~l and the entry of the matrix M is given by

r _ ^ In L(0)~ aq an. (5.78|

where 9t is the element of 6 and the derivative is evaluated at the ML estimates.

The exact evaluation of Eqn.5.78 would be very involved. Alternatively, the deriva

tives can be calculated numerically using the technique of rational extrapolation

(Li and White, 1987d). One advantage of using ML estimation is that estimates

of the sampling variance of a can also be obtained using Eqn.5.78. Note that the

ML estimation normally gives biased estimates of the parameters. However, the

bias should be small if N is large.

5.3.5 Filtering out of the trend component

The approach of filtering out the trend component is first proposed by Math-

eron (1973) and later elaborated by Delfiner (1976). However, active research on

this approach only started in the 1980s and a series of paper on the subject has

appeared over the past few years. The approach is still in a state of develop

ment. The drawbacks of the original proposal by Delfiner (1976) are pointed out

by Starks and Fang (1982) and also discussed in Kitanidis (1983). The approach

is later modified by Kitanidis (1983&;1985), Marshall and Mardia (1985) and Stein

(1986).

The technique is discussed in detail in Delfiner (1976), Kitanidis (1983) and

Starks and Fang (1982). The following only briefly outlines the basic principles of

the method. Define a new random variable zi, called the increment, as a combi-


nation of a suitable subset of the samples, viz

zi = ]Tp/yK%)(5.79)

= +yt ^h€(kj)j j

The subscript l denotes the increment formed from the /th subset of the samples.

The trend component can be removed by suitably choosing the coefficients Ay

such that = 0- Consider the simplest case of a soil profile with a

constant trend (Type I). The increment can be formed from any two samples and

the trend can be removed by simply subtracting the two values i.e. /ct — ACy. The

coefficients will then be Xu = 1 and A/2 = —1. For a more general case of a

non-constant trend, the coefficients A; depends on the configuration of the sample

points. Several methods for determining of Ay are given in Starks and Fang (1982).

The increment zi is a random variable in its own right. It has the following

statistical properties.

E{zl} = J^XljE{elj}=0 (5.80)

var{zi} = a2Y^^\ij\lkp(vjk) (5.81)

cov{zi,zi>} = a2 V]y" kjh'kp(vjk) (5.82)o k

Both the variance and covariance of zi depend on the theoretical ACF of the

random component e(t). Here, it is assumed that the ACVF for e(Q exists. In fact,

Equations 5.81 and 5.82 are well defined under a weaker assumption of intrinsic

stationarity (Delfiner, 1976).

5.4. TYPE III SOIL PROFILES 5-61

Delfiner (1976) suggested that the parameter 0 can be estimated using the

weighted least squares method. The criterion of estimation is

where wi is the weight assigned to the residuals and E{zf\9} is given by Eqn.5.81.

Note that the method of least squares is used only as a numerical criterion for

obtaining the estimate of the parameters. Other statistical procedure such as the

maximum likelihood estimation, the minimum unbiased quadratic estimation, the

minimum norm estimation and the modified norm quadratic estimation are also

used in the literature (Kitanidis, 1983&T985; Stein, 1986).

Once the parameters of the ACF is obtained, the coefficients of the trend

component and the point variance can be estimated using the GLS procedure as

if the ACF is known.

5.4 TYPE III SOIL PROFILES

The structural analysis of a Type III soil profile is the most complicated among

the three types of profiles. Literature on this subject is scarce. Asaoka and Grivas

(1982) presented a structural analysis of the undrained cohesive strength of a Type

III soil profile. However, it was assumed that the COV of cohesive strength Su

is directly proportional to the depth y (i.e. the trend is assumed to be known) so

that the normalized strength Su/y could be modelled as a homogeneous random

field. Therefore, the analysis is more like a structural analysis of a Type I profile

than that of a Type III profile.

The difficulty of analyzing a Type III profile lies in the fact that the random

component is also related to the trend component. Even if the trend component

can be filtered out using the procedure in Section 5.3.3, the dependence of the

mine_ (5.83)

5.4. TYPE III SOIL PROFILES 5-62

random component on the trend component still exists and no suitable procedure is

yet available for filtering out such a dependence. Consequently, the GLS procedure

and the ML estimation discussed previously for Type II soil profiles seems to be

the only two options that can be extended to the structural analysis of Type III

soil profiles.

For Type III soil profiles, the GLS procedure will lead to an equation similar

to Eqn.5.62, but according to Eqn.4.11c, the element atJ of the AM would become

= g(Li) • g(Lj) • Pij (5-84)

Eqn.5.73 has to be modified in a similar manner for Type III soil profiles. Since the

covariance matrix also contains the coefficient of the trend component, Equations

5.68 and 5.77 are not valid for Type III soil profiles. The functions J(9) and Q(0)

have to be optimized using multivariate optimization techniques.

Equations 5.65 to 5.67 are not valid for the calculation of the trend component.

No suitable procedure is yet available for estimating the covariance matrix of

the trend coefficients for the GLS procedure. However, Eqn.5.78 remains a valid

approximation for the covariance matrix using the ML estimation.

A review of the current literature indicates that a Type III soil profile is

usually associated with the undrained cohesive strength of clays. Following a

similar procedure as in Lambe and Whitman (1969), the undrained shear strength

can be related to the effective strength parameters as follows:

\K0 + (1 — K0)Af \ sin (f)'a'y + c‘ cos (f)' u [1 + sin <f>'(2Af — 1)]

where

K0 = coefficient of earth pressure at rest

Af — pore pressure coefficient at failure

5.5. PLANNING OF A SITE INVESTIGATION 5-63

c' — effective cohesion

(f)' — effective angle of shearing resistance

Gy — overburden pressure

The overburden pressure is given by 7y • y where 7y is the average soil density over

a depth y. For normally consolidated clays, c' is small so that Eqn.5.85 becomes

Su = ? * 7y • V ' (5.86)

where f = [K0 + (1 — /60) Ay] sin 0'<ry/[l + shuf),(2Af — 1)]. If K0, A/ and </>'

have constant statistical properties within the soil profile, Su will appear like a

Type III soil profile with a linear trend and a constant COY. Therefore, a Type

III soil profile for the undrained shear strength may very well be a Type I soil

profile in terms of the effective soil parameters. If it is the case, the question arises

of whether it would be more cost effective to determine the statistical properties

of the undrained shear strength from direct measurements or indirectly through

the measurements of the effective soil properties. The cost effectiveness may be

measured in terms of the number of soil samples required and the total cost of soil

testing involved to achieve the same predicted value of failure probability. This

question remains unresolved at this stage.

5.5 PLANNING OF A SITE INVESTIGATION

The basic procedure of structural analysis of soil data has been discussed

above. This section summarizes some of the guidelines for planning a site investi

gation.

To design a slope to a specified failure probability, it is necessary to know

the mean value, the point variance and the ACF of soil properties. As discussed


earlier in Chapter 4, the magnitude of variance reduction due to spatial averaging

depends largely on the ratio of the scale of fluctuation to the dimension of the

slope.

In a preliminary site investigation, soil properties can be measured at close

spacings using some inexpensive tests such as the CPT or index tests. Techniques

for obtaining the scale of fluctuation were discussed in Sections 5.2.4 and 5.3.3.

If the results indicate a small scale of fluctuation in comparison to the di

mension of the slope, the variance reduction would be appreciable. The total

uncertainty will then be dominated by the uncertainty in the estimation of the

trend component. In this case, further tests to obtain a more accurate estimate

of the ACF are of little value. Instead, emphasis should be put on the estima

tion of the trend component so as to reduce the variance of the sample spatial

average, which is roughly proportional to l/N for Type I soil profiles when the

variance reduction factor is small. To maximize the effectiveness of soil sampling,

samples should be taken at spacings greater than at least one scale of fluctuation

so as to minimize the correlation. The trend component can then be estimated

using the procedure discussed above in this Chapter (e.g. Section 5.2 for Type

I soil profiles). As the samples are sufficiently far apart, the structural analysis

would be very simple. For instance, for a Type I soil profile, the point variance

and trend variance can be estimated using the usual estimators for independent

samples (Equations 4.58 and 4.59).

However, if the scale of fluctuation is large in comparison to the dimension

of the slope, the total uncertainty will tend to be significantly influenced by the

spatial variability of the soil properties. An accurate estimate of the ACF will

therefore be necessary so as to give a more precise calculation of the variance and

covariance of the spatial averages. As the scale of fluctuation is large, it may

not be feasible to take samples at wide enough spacings. Therefore, the effect of


correlation between the samples has to be accounted for in the calculation of the

a factor and the point variance (see e.g. Sections 5.2.2 and 5.2.3)

The strategy for a site investigation is different for analyzing the stability of

an existing slope for which some prior knowledge of the statistical parameters is

available. Soil samples may be taken at wide spacings to determine the mean soil

properties required for the analysis. The failure probability can then be estimated

using the measured mean soil properties and some tentative values of COV or 8

based on the prior knowledge. If the failure probability is sufficiently small, there

will be no need for further data collection. However, if the preliminary analysis

indicates a high value of failure probability, a more precise analysis or a re-design

of the slope may have to be considered. The planning of the site investigation

would then be the same as that discussed above for the design of slopes.

CHAPTER 6

PROBABILISTIC DESIGN OF SLOPES

6.1 INTRODUCTION

Over the past decade or so, there is a trend towards use of a probabilistic

approach for assessing the safety of slopes. Some elementary analyses based on

FOSM approach have already appeared in some textbooks in geotechnical engi

neering (e.g. Harr, 1977; He and Wei, 1979; Lee et a/, 1983). There is little doubt

that the more rational probabilistic approach will become more popular in the

future.

The literature on probabilistic analysis of slopes is now very extensive. How

ever, many of the these papers have fallacies in one way or the other and therefore

care must be taken not to be confused or misled when reading references on this

subject.

This Chapter therefore attempts to outline the historical development of prob

abilistic slope design and more importantly gives a critical discussion on some of

the shortcomings of existing approaches. The discussion is then followed by the

presentation of three probabilistic approaches of slope design. Examples are also

given to illustrate the implementation of these approaches.

6-1


6.2 HISTORICAL DEVELOPMENT

The application of statistics to soil mechanics was pioneered by Lumb (1966,

1967,1968,1970) in the sixties whose work concentrated on the statistical descrip

tions and modelling of the stochastic nature of soil properties.

Wu and Kraft (1970) appears to be one of the earliest attempts at probabilis

tic analysis of slopes, but only a <f> = 0 analysis was considered. The performance

function of a slope was formulated perhaps for the first time as a safety margin.

They also recognized the importance of spatial variability and the sampling un

certainty of soil properties, and considered the model uncertainty in the analysis.

Although the analysis presented by Wu and Kraft (1970) is somewhat rudimentary

in the light of present-day development, it has all the ingredients of a ‘modern’

analysis.

Cornell (1971) also presented a probabilistic analysis for 0 — 0 slopes in

which he made two major contributions. Cornell was the first to point out the

relevance of autocorrelation of soil properties in a slope design and proposed a

stationary spatial process for its modelling. He also elucidated the concept of

system reliability in slope design and pointed out that a slope should be treated

as a system in series with the failure probability for the most critical slip surface

serving as the lower bound of the system failure probability. The notion of spatial

autocorrelation was later treated in greater detail by Lumb (1974& 1975a).

The work of Wu and Kraft (1970) and Cornell (1971) was later extended by

Yucemen et a/ (1973) to cover c-0 slopes using the ordinary method of slices as

the stability model.

Following up the work by Cornell (1971), his student Morla Catalan (1974)

performed a few analyses on the system reliability of slopes for 0 = 0 slopes

and came up with the tentative conclusions that the system failure probability is

substantially greater than the lower bound value given by the failure probability

6.3. REVIEW ON EXISTING APPROACHES 6-3

for the most critical slip surface.

Another milestones in probabilistic design of slopes is the work by Alonso

(1976) which forms the basis of many of the later studies. In his work, a more ac

curate simplified Bishop’s method was used for the formulation of the performance

function and more rigorous treatments of the autocorrelation of soil properties wTere

introduced.

However, the most comprehensive treatment of spatial autocorrelation of soil

properties was given by Vanmarcke (1977a) who developed what is now commonly

known as the random field model. The random field model is now more or less a

standard tool for probabilistic modelling of soil profiles. Vanmarcke (1977b) also

presented the first three dimensional analysis of (f> — 0 slopes using a level crossing

approach.

After about a decade of cultivation, probabilistic geotechnical design has be

come a popular research topic since the mid-seventies and numerous papers has

appeared in various international journals and conference proceedings. However,

many of these later analyses have not progressed beyond the level of Alonso (1976)

and some of them may even lag behind those of Wu and Kraft (1970), Cornell

(1971) and Yucemen et a1 (1973).

However, a recent study by Luckman (1987) needs special mention here. In ad

dition to the conventional /^-approach commonly used in the literature, Luckman

(1987) has also used two other techniques for analysis, namely the /?///,-approach

and the first-order-marginal-distribution (FOMD) approach. He also used a rig

orous stability model by Spencer (1967) for the formulation of the performance

function. Further comments on Luckman’s work will be given later in Section 6.3.


6.3 REVIEW ON EXISTING APPROACHES

The following is a review of some of the aspects of the current probabilistic

approaches to slope design. Comments on some other aspects will also be men

tioned where appropriate in later discussions. Most of the current approaches have

one or more than one of the following characteristics:

(1) The performance function G(2Q is typically formulated as

researchers (e.g. Wu and Kraft, 1970; Li and Lumb, 1987; Li and White,

1987b&e) used the safety margin

The popularity of Eqn.6.1 is understandable as it is more in line with the con

ventional concept of factor of safety. However, Eqn.6.2 is preferred to Eqn.6.1

for reasons to be discussed later in Section 6.4.

(2) The performance function is currently based on simplified stability models such

as the friction-circle method, the ordinary method of slices, simplified Janbu’s

method or simplified Bishop’s method. The use of simplified models reduces

the computational effort and more importantly enable the derivatives of the

performance function required in a FOSM analysis to be evaluated analyti

cally. However, it is at the expense of having a less accurate stability model.

In the past, the use of a rigorous stability model was hampered by the fact

that conventional solution schemes cannot provide an explicit definition for the

performance function. The interslice forces have to be obtained by iteration. In

consequence, the derivatives of the performance function had to be evaluated

- 1 = F-1 (6.1)

where F is the ratio of total resisting force to total disturbing force. Some

G(X) = R(X) - 5(X) (6.2)


numerically by finite difference approximation as is done in Luckman (1987)

when using Spencer’s (1967) method. This problem is now' tackled by means

of the unified solution scheme outlined in Chapter 3.

(3) Although the importance of autocorrelation of soil property was pointed out in

the early 1970’s by Cornell (1971), explained more fully in Lumb (1974&1975a)

and emphasised time and again in more recent papers by Alonso (1976), Van-

marcke (1977a&;1980) and Castillo and Alonso (1985), papers which assume

perfect correlation of soil properties, either knowingly or not recognizing the

relevance of autocorrelation, still appear in various international journals or

conferences. As remarked by Baecher et a1 (1984), these analyses w'ould give

engineers the impression that probabilistic slope analysis have not progressed

beyond the level of those in early seventies.

As indicated in Fig.6.1 the failure probability under the assumption of perfectly

correlation is exceedingly high, of the order of 0.1-0.2, for the typical range of

design factor of safety of 1.2-1.4.

In the present w'ork, the autocorrelation of soil properties is properly considered

in the analysis by using the random field model discussed in Chapter 4.

(4) Current probabilistic analysis are usually based on the FOSM approach and

the reliability index (3 is used to measure the safety of slopes. As discussed

earlier in Chapter 2, the reliability index (3 is a ‘variant’ risk measure. Because

of this, some researchers suggested the use of Hasofer and Lind’s reliability

index (3hl in lieu of the conventional Cornell’s reliability index (3 (e.g. Nguyen,

1985c; Li and Lumb, 1987; Li and White, 1987bAe; Luckman, 1987).

Both the /^-approach and the (3hl-approach are used in the present work and

comparisons will also be made between the two reliability indices.

Since the joint probability distribution of soil properties is generally not known,

a Level III analysis is seldom performed in slope stability problems. Even if


S R _

0.2 — 0.2 -

0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.4 0.6 Q8 1.0 1.2 1.4 16

F F

0.4 0.6 0.8 1.0 1.2 14 1.6

F

Figure 6.1

Variation of Pj with factor of safety (after Matsuo and Kuroda, 1974)


the assumption of joint Gaussian distribution is used, the large dimension of

the problem would render such an analysis impracticable.

Nevertheless, a number of Level III analyses did appear in the literature. Mat

suo and Suzuki (1983) presented a Level III analysis for 0 = 0 slopes. In their

analysis, the soil density is treated as deterministic and they are therefore left

with only two random variables, namely the average cohesive strength over the

length of the slip surface which can be regarded as a Gaussian variate and the

model error which is considered to be uniformly distributed. The simplicity of

the performance function also enables the integrations to be performed without

much difficulty.

A number of Level III analyses for c-(j> slopes can also be found in the liter

ature. In these analyses, the failure probability is calculated either by direct

integration (e.g. Tobutt and Richards, 1979) or Monte Carlo simulation tech

niques (e.g. Tobutt, 1982; McPhail and Fourie, 1980; Prist and Brown, 1983;

Ramachandran and Hosking, 1985; Nguyen and Chowdhury, 1984). However,

these analyses are made possible by the incorrect assumptions that soil prop

erties are perfectly correlated so that they can be modelled as simple random

variables. Under this assumption, the cohesion for example for the entire slope

is treated as one simple random variable instead of n random variables (the n

values of average cohesion for all the n slices) for the random field model. Thus,

the total number of random variables can be limited to a few and consequently

a Level III analysis, although time consuming, may still be tractable. These

analyses, however complicated or rigorous they may seem, grossly overestimate

the failure probability giving no real indication of the safety of slopes.

Some approximate Level III analyses based on the advanced-first-order-second-

moment (AFOSM) method have also been proposed. Ramachandran and Hosk-

ing (1985) used the technique of Normal tail approximation developed by Rack-


witz and Fiessler (1978). However, the assumption of perfect correlation was

used in their analysis.

Luckman (1987) used the so-called first-order-marginal-distribution (FOMD)

method developed by Der Kiureghian and Liu (1986). This may be regarded as

the first approximate Level III analysis for c-(p slopes to date (other approaches

in the literature are not exactly a Level III analysis for reasons discussed above).

The basic principle is as follows. Given the marginal distributions and co-

variances (or coefficients of correlation) of the random variables, whether it

be simple random variables or spatial variables, a joint Gaussian distribution

can be fitted to the random variables using the procedure developed by Der

Kiureghian and Liu (1986). Having obtained the fitted joint Gaussian distri

bution, the procedure for determining the reliability index is similarly to that

of a Level II analysis. Of course, the two central questions to be answered are

how to determine (a) the marginal distributions and (b) the covariances of the

random variables, especially the spatial random variables.

To answer the first question, Luckman (1987) assumed that the marginal distri

butions of the spatial random variables were the same as their respective point

distributions, the implication being that spatial random variables (e.g. c[) were

perfectly correlated within the domain under consideration, typically the di

mension of a slice. This assumption has two shortcomings. Firstly, for this

assumption to be justified, the averaging dimension must be small compared

to the scale of fluctuation. For a typical slope stability analysis, it would often

imply a large number of slices of the order of 102. Secondly, even if the assump

tion of perfect correlation is justified for very small slice width, the marginal

distribution of the spatial random variables to be used should not be the point

distribution. Instead, it should be the sampling distribution of the point prop

erties. For instance, if the point distribution is Gaussian and the mean value is


estimated from N independent samples, the marginal distribution to be used

should be the t distribution with N — 1 degrees of freedom. Otherwise, the

spread of the distribution or total uncertainty will be underestimated giving

unconservative results. Unfortunately, the sampling distribution is often diffi

cult to obtain even if the parent point distribution is given, and not to mention

the fact that the latter is generally not known in geotechnical analysis.

To answer the second question, Luckman (1987) modelled the spatial random

variables using a one-dimensional random field. The correlation of spatial ran

dom variables is described by a one-dimensional ACF in the horizontal direc

tion, the implicit assumptions being that spatial random variables are perfectly

correlated or the scale of fluctuation is infinitely large in the vertical direction.

Thus the covariance of spatial averages depends only on the horizontal lag dis

tance between them. As indicated in Table 4.2, the scale of fluctuation of soil

properties in the vertical direction is usually small and tends to be smaller

than the scale of fluctuation in the horizontal direction. Therefore, the as

sumption of perfect correlation in the vertical direction does not seem to be

justified in the light of field observations. A more realistic representation is the

two-dimensional random field described in Chapter 4.

Although the FOMD method has yet to be improved before it gives realistic

modelling of the spatial random variables, it has the potential to be used as

an approximate Level III approach by which information other than the mean

and variance can be incorporated into the analysis.

In the present work, the method of PDF fitting is used as an approximate Level

III method. The implementation of this approach is considered to be simpler

than the FOMD approach adopted by Luckman (1987).

(5) Save for a few exceptions (e.g. Wu and Kraft, 1970; Cornell, 1971; Yucemen et

a 1 , 1973; Alonso, 1976; Yong et a/ , 1977; Li and Lumb, 1987; Li and White,

6.4. /^-APPROACH 6-10

1987c«A.f), current analyses seldom take account of the sampling uncertainty.

As illustrated in the example in Section 4.6, the sampling uncertainty may

be comparable to the uncertainty arising from the innate variability of soil

properties. Unlike the spatial correlation of soil properties which diminishes

with lag distance, the sampling uncertainty tends to be perfectly correlated

and there would be no variance reduction due to spatial averaging. Thus,

ignoring the contribution of sampling uncertainty will underestimate the failure

probability. This can be very significant when the averaging dimension is large

compared to the scale of fluctuation so that the variance reduction due to

spatial averaging is large or when the trend component is estimated using only

few samples.

In the present work, the sampling uncertainty associated with the estimation

of the trend component is considered using the procedure outlined in Chapter

4.

In the following, three different probabilistic approaches to slope design are

discussed. The first one is the /^-approach, the second one is the ^//^-approach

and the third one is the method of PDF fitting. These three approaches have

been discussed in some detail in Chapter 2. Here, emphasis will be given on the

implementation of these approaches in relation to slope stability analyses. For

ease of reference, the following terms are defined.

1. Mean centroid - A mean centroid is the point /i defined by the mean values j.il

of the input random variables A^, i = 1,/, namely // = (fz1? " * , /r), where

/ is the total number of random variables.

2. Sample centroid - A sample centroid is the point x defined by the sample mean

values xl of the input random variables Xt, i = 1, /, viz x = (xy, xo, • • •, xi).

6.4. /^-APPROACH 6-11

6.4 /3-APPROACH

The reliability index f3 is most commonly used for characterizing the safety

of a slope. Recalling Eqn.2.11, the calculation of /3 requires the knowledge of the

mean and standard deviation of the performance function G'(X).

In current literature (e.g. Hahn and Shapiro, 1967; Benjamin and Cornell,

1970), the formulae for calculating the mean and variance of the performance

function G(X) are usually derived using a first-order Taylor’s series approximation

and the function is linearized at the mean centroid, which implies that the mean

values of the random variables are assumed known. In practice, the mean centroid

is never known and the performance function will have to be evaluated at the

sample centroid. Failure to realize this fact will often result in the neglect of the

uncertainty associated with the estimation of the trend component.

By linearizing the performance function at the sample centroid and following

the same procedure as in Benjamin and Cornell (1970), for instance, the mean and

variance of G(2Q can be estimated by

E{G(X)}

i>ar{G(X)}

G(X)

i i

EEi= i j

dGidXi

dGOX4

cov{xi, Xj)

(6.3)

(6.4)

w'here the arrow —► means “estimated by”. The expressions for the partial deriva

tives are given in Appendix A.

In slope stability problems, xt includes the sample spatial averages of soil

properties. The calculation of cov{•} for sample spatial averages of soil properties

(e.g. cx) has been discussed in detail in Chapter 4. However, if a large number

of slices is used in the analysis, the variance reduction due to spatial averaging

for each slice is small. The variances and covariances of the spatial averages can

therefore be approximated by the point properties. The question of how many

6.4. ^-APPROACH 6-12

slices would constitute a “large number’ is judged in terms of the ratio of the slice

width to the scale of fluctuation of the properties. Take a one-dimensional Type I

ACF as an example. To limit the error to within 10%, the width of the slice has

to be less than one tenth of the scale of fluctuation. For a typical range of scale

of fluctuation of 0.5m to 10m, this will imply a slice width of 0.05 to lm or 10 to

200 slices for a 10m wide slip surface. Thus there is always a trade-off between

a saving in computing time in the calculation of the variances and covariances of

the spatial averages and an increase in effort in the evaluation of the derivatives of

the performance function and other quantities arising from the increase of random

variables of the system.

In the derivation of Equations 6.3 and 6.4, G(X) is linearized at the sample

centroid using a first order Taylor’s series approximation. It becomes immediately

apparent that the smaller the non-linearity of G(X) is the better. This is precisely

the reason why the performance functions Gm(X) and G/(X) are formulated in

terms of the safety margin. It can be observed that Gm(X) and G/(X) are linear

with respect to ct, ^ and ut etc; and are only non-linear with respect to tt. If

the performance function is formulated as (see below), it will then be non

linear with respect to all the parameters mentioned above and will therefore incur

a greater error to the mean value and variance using Equations 6.3 and 6.4.

Knowing the first two moments of G(X) is not sufficient to define the PDF of

Z or G’(X). However, an approximate PDF for G(X) can be inferred from other

considerations. Consider the following two formats of G(X)

i. G,(X) = /?(2C) -S{X)

ii. G2(X) = - 1

The first performance function is the safety margin used in this work. The

second format is the one commonly adopted in the literature. In slope stability

problems, /r’(X) and S(X) are sums of a large number of random components (see

6.4. (3-APPROACH 6-13

Equations 3.10b and 3.11). By virtue of the well-known central limit theorem,

(S#i (AT) can be well approximated by a Gaussian distribution. Consequently, the

failure probability can be related to the reliability index by

Pf = Pr(Gi (X) < 0) « $(-/?i) where px — —— — —- . (6.5)Ja2R - 2poRos + cr2

$(•) is the CDF of a standard Gaussian distribution and p the correlation coeffi

cient of R and S.

For the second format, a log-normal distribution is often assumed for the ratio

i) — The commonly held argument for this is that by taking the logarithm of

i/, we obtain

In r] = In R — In S (6.6)

Since the right hand side is a sum of random variables, it is argued that In t) can

be approximated by a Gaussian distribution by virtue of the central limit theorem

and hence t] is log-normally distributed. As a result, the failure probability is

given by

where

Fy = Pr(G2(X) < 0) <h( In /.iq2 — \ In(1 + P2 ~)

^/ln(l + Po ~)(6.7)

The use of Eqn.6.7 is not recommended because of the following reasons. Firstly,

Cef^K) has a higher degree of non-linearity than GhfX). Therefore, a larger error

is expected for the calculation of using the FOSM approach. Secondly, the

argument for inferring a log-normal distribution for rj is not justified. The use of

6.5. ^-APPROACH 6-14

the central limit theorem is valid only for a sum composed of many terms and this

is obviously not the case for Eqn.6.6.

Other distributions such as a Gaussian or a beta distribution have also been

assumed for Go(20 (Grivas, 1979; McGuffey et al 1982; Anderson et al , 1982;

Moon, 1984; Nguyen, 1985c; Chowdhury and Deroy, 1985). However, there does

not seem to be any justification for these assumptions.

In fact a better procedure for estimating the value of Pj for Go (20 is available.

As argued previously, a Gaussian distribution can reasonably be assumed for R(X)

and 5(X). The disturbing forces are mainly due to the weight of the soil mass and

as the variability of soil density is usually small, the assumption that the GOV for

5(X) is small can be made. Geary (1930) showed for this situation that the PDF

of ?i is related to the PDF of the following function

___________R - 5 • r;2_______

• r - 2^(75 • t] + o%

which follows a standard Gaussian distribution. Now,

(6.8)

Pf = ?r(G2(X)<0)

= Pt(j] < 1)

= Prk <R-S ■ 1

• 1 - Spends • 1 +<?s

= Pr(f <-A)

= *(-A)

(6.9)

Although Eqn.6.9 is approximate, it is very accurate if the COV of 5(20 ^ess than 20%. Therefore, by making the consistent assumption that the distribution

of R(X_) and 5(20 are Gaussian variates, the same value of Pj is obtained, as it

should be, for the two formats of G(20, even though the values of (5 are different

in the two cases.

6.5. f3HL-APPROACH 6-15

6.5 /^-APPROACH

The use of Pul as a risk measure has gained popularity in structural reliability

analyses, but it has not been used in slope stability analyses until very recently

(e.g. Ramachandran and Hosking, 1985; Gussman, 1985; Li and Lumb, 1987; Li

and White, 1987b&e, and Luckman, 1987). The reliability index Pul has the

advantage of being an invariant index of risk measure.

Iteration is required for calculating Pul and this can be done conveniently

using the following algorithm by Parkinson (1978a).

X0 + 1) = X+ Vx ■ vG- (Xlj) - X)T ■ yG )VGT ■ ■ VG J

(X0) - X)T • vG' • {vGT ■ Vx • vG'} *

(6.10a)

(6.106)

where yG = which is evaluated at the jth trialax! ’ ax2 ’ • • • ’ ax, > • • • > axt point X^i V\ is the covariance matrix for X and the superscript T means the

transpose of a matrix. The iteration stops on convergence of Pul• + has

to be adjusted to satisfy the limit state equation GfA) = 0 before it can be used

for the next iteration. This can be conveniently done by choosing all but one

of the parameter values to be the same as X_G + l) and the remaining parameter

value obtained using G(X) — 0- Here, cn is arbitrarily chosen to be the parameter

for adjustment. As the performance functions G^fX) and G/(X) are non-linear

with respect to tt, a correct choice of the initial estimate of tt is essential to the

success of the iterative algorithm. The following procedure is useful. Initially, t{

is assigned a value equal to its mean value and it is assumed to be deterministic

(i.e. variance of t{ is assigned a zero value) for the first iteration. When tt is

assumed to be fixed, the performance function becomes linear (with respect to ct,

7i and ul etc). Therefore the algorithm will converge to the same design point after

the first iteration independent of the values of the linear parameters used. This

6.6. METHOD OF PDF FITTING 6-16

‘conditioned’ design point forms a robust starting point for the general iteration.

For a linear performance function with jointly Gaussian random variables,

the reliability index Phl is related to the failure probability via Eqn.2.22. The

equation is often used for other cases of non-linear performance functions and/or

non-Gaussian variables to give a rough estimate of the likely magnitude of the

failure probability.

If the performance function is linear, it can be proved that p and Phl are

equal. The proof is outlined in Appendix B.

6.6 METHOD OF PDF FITTING

In the above two approaches, only the information on the first two statistical

moments of the input parameters are utilized in the analysis. Very often, soil

engineers would have some knowledge on the bounds of the soil properties, either

through subjective judgement or inferred from available soil data. This additional

information cannot be incorporated into the analysis using the above two methods.

In this section, a new approach based on the method of PDF fitting is introduced

whereby the knowledge on the bounds of the soil properties can be utilized to

produce a sharper estimate of failure probability. The method of PDF fitting has

already be outlined in Chapter 2.

To implement the method, the mean value, variance, lower and upper bounds

of the performance function are required. The first two quantities can be obtained

using Equations 6.3 and 6.4. The following discussion addresses the general solu

tion procedure for estimating the bounds of G(X), followed by the presentation of

a simplified solution procedure. A special case of Morgenstern and Price’s method

for which the bounds of G(X) can be calculated analytically will also be discussed.


6.6.1 Bounds of performance function - Rigorous method

It suffices to discuss the calculation of the lower bound of G'(X); the evaluation

of the upper bound is similar. In performing the optimization of G'(X), it has to

be realized that G(X) is linear with respect to ct, 7* and ul etc and is only

non-linear with respect to tt. Consequently, the optimum value of G(X) must

occur at the lower or upper bound of the linear parameters. Denote Iz to be the

interval [zL, zl ] bracketed by the lower bound and upper bound of the variable

Z. Let i — (G,...,fn) and denote X! = (X[, X2, ■ ■ ■) to be the collection of

linear parameters such as ct, 7f/t, AQt etc. The minimization of G'(X) can be

formulated as

The function H(t) is the conditional optimum value of G(X) given the value of f

H(t) can be obtained as follows:

where = (xlci, xlc2, • • •) is the conditional minimum point for the linear parame

ters which is given by

subject to t{ G It,

= min H(t) subject to tx G It (6.11)

H(l) = G(x‘cJ_) (6.12)

/

lower bound of x[ if 77' ^ 0

(6.13)


Note that --^7- is a function of £ only. The constrained problem in Eqn.6.11 can *

be converted to a unconstrained optimization problem by replacing tt by <7 with

the transformation £t = tf + (£[' — tf) sin2 Thus

nnnt H(t)

= min Q(^) f

subject to tt E hx(6.14)

where <3(<r) is the transformed function of II(X_). Either the optimization of Equa

tions 6.11 or 6.14 can be solved readily by available computer packages such as

IMSL (1984). Appendix E also outlines a method for solving the optimization

problem of Eqn.6.14. Experience shows that the minimum value of <S'(AT) occurs

at :

1. the lower bounds of ct, £t;

2. the upper bounds of

3. lower bound of 7; at the lower portion of the slip surface and the upper bound

at the upper portion.

The opposite is true for the maximum value of G'(X). These results are to be

expected from the physics of the problem.

6.6.2 Bounds of performance function — Simplified method

Because of the above observations, the following simplified optimization pro

cedure is suggested.

1. Give an initial estimate of the optimum point. The above observations

can be used as guidelines.

2. Calculate the derivatives of G'(X) at the estimated optimum point

3. The (* + 1)th estimate of the optimum point is estimated as follows.

For minimum value, assign Xt = xf if

For maximum value, assign Xt — if

dGdXtdGJx,

>0,

>0,

otherwise = xf

otherwise Xt = x[


4. Calculate the value of G(X) at the (« + 1)th estimate.

5. If G(A^! + 1)) = G(X^), the procedure terminates.

Otherwise, repeat step 2. If the procedure does not converge in three itera

tions, this is an indication that the optimum value occurs inside the feasible region

of C, not on the boundaries. In this case, the general procedure has to be adopted

for searching the optimum. This situation has not been encountered so far.

6.6.3 Bounds of performance function for Spencer’s method

For the special case of the interslice force function f(x) being a unit function,

which is also Spencer’s (1967) method, an analytical procedure can be derived for

the calculation of the lower and upper bounds of G(X)- The expression for ATt

(Eqn.3.22) now becomes

Eg —-f [cj Aii + ( AU i —U] Axi)fi|-7»i — (AQi -f AH i tan ct\)

-F — 11 ni] + tan a ii = 1

A Tt =

[c( Ax,- 4- (AIV, — u, Ax,-) t, ] m,—(AQ. + AIV, tan o,)

4—t, m,-f tan a,>, A1 < i < n

(6.15)

Firstly consider the performance function G'm(A). Denote Gm, to be the ith term

of the summation in Eqn.3.32. For example, Gmn is

G, Axn + (AlFn + ATn - unAxn) ■ tn • mn Vm n

[^Qn ■ UQn + (All n + A Tn) • ymn ' tan ctn A Tn • xmn(6.16)

It can be observed from Eqn.6.15 for this special case that ATt is related only to the

input parameters for the ith slice. Therefore, Gmi and Gmj are also independent

of each other for i < n, j < n and i ^ j. However, Gmi i — 1, n — 1 is coupled

with Gmn due to the fact that the value of ATn is connected to ATt via

ATn =Tb- rn_jn — 1

Tb-J2 AT,1=1

(6.17)


Denote Y_t = (c', 7t, t{, ul, APt, qu AQt) to be the random input parameters for

the fth slice. The minimization problem for G^fX) can be formulated as follows

(the procedure for maximization is similar). It is understood that the optimization

is subject to c[ £ Ic> etc.1 x

min Gm(X) = x mm mm 777,-Vln.! I

[-^62/6 Eaya -T T Ta.ra]

(6.18)

Now, examine the individual components in Eqn.6.18. Eqn.6.16 can be written in

the form of

(n) (n),fll T d‘2 tn

Gm. =b[n) + b{"}tn

(6.19)

where a\”'\ aX\ 6, and bin' are quantities independent of tn. It is easy to

recognize that . Since no stationary point can occur for the

type of function given in Eqn.6.19, the optimum point must occur at the lower or

upper bound of tn. As all the other parameters in En and also Ta, 1\, Ea and

Eb are linear parameters, the optimum point must occur at the bounds of all the

parameters in the first minimization operator in Eqn.6.18. As mentioned earlier,

GJrin is related to Gm. via Eqn.6.17. By combining the term Gmi and the term

ATt in Gmn to form one single quantity, designated as Gmi+n we have

Ga(*) +

mt+n = -------(Tj-2 - [Aiy* • ymi tana; + AQt ■ yQx\T t>2 11

(6.20)

where

a['} =c[Axt ■ b[:) ■ ym. ■ sec2 at

— oj1 ■ (c[Axl sec2 at + \i ~ A\\\ tan (6.21a)


(i) _/ &Wia'2 (a

— ulAxl • 6 (0 + ‘ 2/m, sec2 a.

— ojt • (AH^ + \i tan at — iilAxt sec2 a;) (6.216)

i (0 1b\ =— + tan otiA

b <•> -1A

(6.21c)

(6.21c/)

and

^t —2/m, tan xmi + (tn • Tnn tan c*n) • ymn “I- %mn

Ea — ^ — AQi i — 1

-AQi 1 < i < ii

It can also be recognized that dG^E) _ G^»+n . Again, no stationary point

for tt can exist for the function Gm.+n given in Eqn.6.20. Since all the other

parameters in Tt are linear, the optimum point must occurs at the bounds of

all the parameters in Y_l. Consequently, the minimization problem in Eqn.6.18

becomes

Cn — 1 -min Gm(AQ = min J min Gm (X)

X Yn,Ta,Tb,Eb,Ea ^ [ Y.,

-f- Axn T (AH n -F If) nn Axn)bi Vm n ’ Wln

AQnVq„ (AHn 1 b) Vmn tan ctn -F 1 b • xmn

+ {TbXb + Taxa) -F [Lb yb — Eaya) 1 (6.22)

An algorithm for solving Eqn.6.22 is outlined in Fig.6.2. The procedure for

finding the lower and upper bounds of G/(A) is similar. The minimization for


Given kt = dimension of Y_t gmin=LARGE gmax=SMALL do in = 1,2hn +4

set Y_n, Ta, Tb, Ea, Eb bounds combination tempmin = tempmax = last 3 lines of Eqn.6.22 do i = 1, n — 1

termmin = LARGE termmax = SMALL do it = l,2k>

set Et bounds combination term = Gmi+ntermmin = min(termmin,term) termmax = max(termmax,term)

enddotempmin = tempmin+termmin tempmax = tempmax+termmax

enddogmin = min(gmin,tempmin) gmax = maxfgmax, tempmax)

enddo

Figure 6.2Algorithm for Solving Eqn.6.22. (after Li and White, 1987b)

6.7. SYSTEM RELIABILITY OF SLOPES 6-23

Gj(X) can be formulated as

n —1minG/(X)= min J * ~ Yn,Ta,Tb,Ea,Eb \

minG/i+n(X)

+ \c„ Ai„ + (AWn + Tb-

AQn + (AWn + Tb) • tan c*n

-(Eb-Ea)

where Gfi+n is given by

a*'1 + ai'kj

/,+n 6<*> + 6<*)<,- (A(Jt + AW{ tan or,-)

The coefficients for Gfi+n are given by

and

=AW, tan2 a, + :i-s—tt* - x, tana,a

+ cjn • (clAxl sec2 ctl -f Xi ~ AWt tan a,-)-

a^ =— • (AW{ — iii • Axt) • sec2 at — AWt tan a; -f \iA

+ cjn • (AWi + Xi tan at-----ut Axt sec2 at)

b[l) =- + tan atA

^(t) _tan cxt 1——-1

cdn — tan cxn tn • Tnn

Ea-^f- AQ, 1 = 1

Xi = <

• mn

(6.23)

(6.24)

(6.25a)

(6.256)

(6.25c)

(6.25rf)

l-A Qt 1 < i < n

6.7. SYSTEM RELIABILITY OF SLOPES 6-24

6.7 SYSTEM RELIABILITY OF SLOPES

So far discussion has concerned the failure probability for a particular slip

surface. In fact, there are infinitely many admissible slip surfaces although the

failure probability of each of them may differ. The slope should be considered

as a system in series. Each component represents a feasible slip surface. Failure

of any slip surface (component) will imply the failure of the slope (the system).

The system failure probability of the slope should be evaluated using the system

reliability theory, although it is not an easy task and no suitable procedure is yet

available for its calculation.

The system failure probability Pfs of slopes is bounded by (Cornell, 1967)

(Pf) max < ^/s < 1 (6.26)

where (P/)mdLX is the failure probability for the most critical slip surface. If high

correlation exists between different slip surfaces, the system failure probability

will be close to the lower bound. A study by Morla Catalan (1974) on cohesive

slopes indicated that the system failure probability wa5 significantly higher than

(f/)max for the normal range of correlation existing for real slopes. However, the

sampling uncertainty (which is perfectly correlated) had not been considered in the

analysis and hence the correlation between slip surfaces would be underestimated.

Therefore, the conclusion may not be true in general. More research needs to be

done before any definite conclusions can be drawn.

Although a complete analysis of system failure probability is not available,

the value of (P/)max can serve as a convenient and valuable index in assisting the

engineer to exercise judgement in the design of slopes. The procedure for locating

the critical slip surface is discussed in Chapter 7.

It must be pointed out that some misconceptions concerning this system reli-

ablity of slopes exist in the literature. Grivas et a/ (1979) and McPhail and Fourie

6.8. PROBABILISTIC MODELLING OF PORE-WATER PRESSURE 6-25

(1985) treated the location of the slip surface as another random variable in the an

alysis and assigned a certain probability distribution for it. Grivas et al (1979) also

evaluated the ‘mean’ location of the slip surface. This is a misconception because

although the position of the slip surface is an important factor to be considered

in the analysis, it is by no means a random variable. To explain this, consider a

simple analogy of a portal frame having two plastic collapse mechanisms, Al and

A2. Each failure mechanism can be likened as the failure of a slip surface in a

slope. As the system is in series, the system failure probability of the portal frame

can be calculated as

Pfs = PrfAj U An) = Pr(Al) + Pr(A2) - Pr(Al D A2) (6.27)

As individual slip surfaces are treated as failure events in the analysis, the mean

location of the slip surface, which can be likened as the mean collapse mode of a

portal frame, bears no meaning in system reliability theory.

6.8 PROBABILISTIC MODELLING OF PORE-WATER PRESSURE

Pore-water pressure fluctuation within a slope is always difficult to predict.

It is influenced among other things by the spatial variability of hydrological prop

erties of soil, location and undulation of bed rock stratum, local distribution of

rainfall, variation of vegetation cover and presence of joints in the soil profile*.

The situation is aggravated by the fact that the variability of hydrological prop

erties of soil is often large. A commonplace COV of permeability of soils is larger

* The presence of joints will greatly affect the local variation of water flow. It

may also give rise to the formation of a soil pipe (Pierson, 1983). Some of the

slope failures in Hong Kong can be attributed to local built-up of water pressure

in the soil pipes (Brand et a1 , 1986).


than lOO^c. The sampling uncertainty is therefore high.

Although the importance of pore-water pressure has long been recognized in

slope stability designs, theoretical studies on stochastic modelling of water pres

sure fluctuation started only within the last decade or so (e.g. Smith and Freeze,

1979aAb; Chirlin and Dagan, 1980; Andersson and Shapiro, 1983; Kitanidis and

Vomvoris, 1983; Bergado and Anderson, 1985; Anderson and Howes, 1985; Lee

and W u. 1987). In these analyses, the hydrological properties of soils are modelled

as random fields and the response (which may be the rise of water level or others)

is predicted using the relevant differential equation governing the flow of water.

The mean value and variance of the predicted response are calculated using, typ

ically, a stochastic finite element method or simulation. At present, literature on

statistical analysis of measured water pressure fluctuation is extremely scarce. The

validity of the above theoretical models has yet to be verified.

A detailed study of pore-water pressure fluctuation is beyond the scope of the

present study. In this section, two simple models are proposed for characterizing

the uncertainty of pore-water pressure. In the first model, the pore-water pressure

is described by means of the pore-water pressure ratio rx which is defined as

= rx ■ ArnAij

(6.28)

where Ar,, Ax and r)l are respectively the width, the area and average soil density

for slice i. rt is treated as a random variable in the analysis.

In the second model, the variation of pore-water pressure is characterized by

three phreatic surfaces as shown in Fig.6.3. yu(x) represents the upper bound of

the water level, yP(x) the most probable location and yi(x) the lower bound. The

average water pressure ut acting on the slice base can be approximated by


Figure 6.3 Modelling of Pore-water Pressure

lw(wi ~ Vi) = lw ’ hi h> 0

0 h< 0

= lw ■ {Wi - Vi) • H(v)i - Vi)

— lw ' hi ' H

(6.29)

where iw is the unit weight of water and H( ) is the unit step function. The

location n\ of the phreatic surface becomes a random variable.

To evaluate the mean and variance of wt, it is required to know the PDF of


2

Vu ~ yl

Figure 6.4 PDF of w(x)

w(x). Here, a triangle distribution as shown in Fig.6.4 is assumed. The location

of yp(x) coincides with the maximum point of the PDF. The PDF of w(x), fw(w),

is given as below. For simplicity, w(:r) is denoted simply as w, w(xt) as wt and

the like.

fw (u>) =

fr

fr

w—yiyp-yi

Vu—rjuy«-yP

yi <w < yp

yP < w < yu

(6.30)

where fm = 2/(yu - yi).

The mean and variance of wt are given as

E{wt)

var{w{}

yi + yP + Vu3

fm~12 (yl + Vp)(yu + yP) - (yl + y?){yP + yi)

(yu + yP + yi)2

(6.31)

9(6.32)

6.9. ILLUSTRATIVE EXAMPLES 6-29

In theory, the variation of water pressure can be modelled as a random field as is

done in Luckman (1987). At present, little is known about the correlation struc

ture of water pressure. It is speculated that it may consists of two components,

one governed by the changes in regional water system with a scale of fluctuation

perhaps comparable to the scale of the slope and the other governed by location

variation of soil properties having a scale of fluctuation comparable to that of soil

properties. Pending more information on water pressure fluctuation, it is assumed

tentatively herein that the water pressure is perfectly correlated across the slope.

This would imply that the entries of the autocorrelation matrix for rt or wt are

all equal to 1.

6.9 ILLUSTRATIVE EXAMPLES

In this section, the implementation of the probabilistic approach will be de

picted by means of illustrative examples. The assumptions used in the following

discussion are discussed first.

1. In the analysis, only c[, tt and ul are taken as random variables. Other

loads (Ea, qx, APt, AQt) are taken as zero. The effect of a tension crack is also

neglected.

2. Little has been published in current literature on the joint PDF of soil prop

erties. Under controlled conditions such as constant soil density and moisture

content, Matsuo and Kuroda (1974) observed a strong negative correlation

between the strength components c and t. However, for natural soils, evi

dence (Lumb, 1970; Schultze, 1975) shows almost zero correlation between the

strength parameters while significant negative correlation has been reported by

Forster and Weber (1981) and Grivas (1981). However, the strength compo

nents c' and t are treated herein as independent for simplicity. The assumption


of mutual independence will simplify the calculation and also err on the con

servative side (Forster and Weber, 1981).

The influence of variability of soil density on Pj of slopes is usually small

(Alonso, 1976). This is due to two reasons. Firstly, can have positive and

negative values depending on the location of the slice. Thus it has a cancelling

effect on the contribution to the variance of the performance function. Secondly,

the variability of soil density is usually small. Furthermore, the averaging

dimension for soil density is large thus reducing the variance of the spatial

average density even further. In consequence, the cross-correlation of 7 with

c' and t, which is of secondary importance, can be neglected without incurring

significant errors.

3. The soil properties are modelled as random fields. The variance and covariance

for the sample spatial averages for c[ and 7i are evaluated using the formulae

given in Chapter 4. To be consistent with the assumption used in deriving

the performance function Gm(T) and G/(X), tt is represented by the point

property at the centres of the bases of the slices.

4. In the following examples, the pore-water pressure is represented by the pore-

water pressure ratio r. It is further assumed for simplicity that all rt have the

same mean value r and variance Furthermore, the cross-correlation of rl

with soil properties is neglected.

Because of the above assumptions, the covariance matrix of X has the form

(V?

Vx =

0 1

0 Vt 0 0

0(6.33)

The autocorrelation of soil properties is represented by the two-dimensional sepa

rable ACFs listed in Table 4.1. Although different values of scales of fluctuation


can be used for different soil properties, they are assumed to be equal.

T = /v f (x) * E

f(x)r constant

f(x) = half - sine

Figure 6.5 Types of Interslice Force Function

Two interslice force functions as shown in Fig.6.5 are used. Unless stated

otherwise, results presented below are based on the simple exponential ACF (Type

I), equal values of 8X and <5y, i.e. 8X = 8y = 6 and the constant interslice function.

The reliability index P and Phl are related to the failure probability using Pj —

$(—/?) and Pj — <&(—(3hl) respectively. Fig.6.6 shows the variation of Pj with

the reliability index (3 or Phl-

failu

re

prob

abili

ty


reliability index

Figure 6.6 Variation of Pj with Reliability Index


6.9.1 Example 6.1

centre of momentc

Figure 6.7 Geometry of Slope for Example 6.1

The geometry of the slope is shown in Fig.6.7. 10 slices are used through

out and the sample size Ar is taken arbitrarily to be 8. The following are input

parameters of the soil.

mean cov 1. bounds u. bounds

c' 18 kN/m2 20% 3.6 kN/m2 34.2 kN/m2

18 kN/m3 5% 15.3 kN/m3 20.7 kN/m3

t tan 30° 10% tan 25° tan 35°

r 0.2 10% 0.15 0.25

6.9.1.1 Adjustment of X

Fig.6.8 shows the typical variation of /?, (3hl and Pjj with A. The intersection

point of the curves gives the so-called ‘rigorous’ solution and the corresponding

value of A is denoted as Amf- At this point, the probability predicted from the

performance functions Gm(X) and G/(X) are equal. It can be observed that the


P

Phl

pff

Figure 6.8 Variation of (a) f3 (b) (3hl and (c) Pff with A


variation of j3, Phl and Pj based on Gm(X_) is much smaller than those of G f(X).

The same is true for the factor of safety of slopes (Li and White, 1987c).

Define the following function

9(A) = r,m - ,/ (6.34)

where r]m can be either p, 3hl or Pff based on the condition of overall moment

equilibrium depending on the case and so on. Fig.6.9 show's the typical monotonic

variation of (7(A), It is easily observed that the rigorous solution is given by the

root Amf of the equation g(A) = 0, which can be easily solved using the technique

of inverse rational approximation (Appendix E).

Table 6.1 shows the typical rapid convergence obtained using the technique of

inverse rational approximation. In the following, all the results presented refer to

the rigorous solution.

6.9.1.2 Accuracy of linear approximation for G(X)

There is ample evidence to suggest that the functions Gm(X) and Gf(X_)

are w'ell approximated by linear functions. Firstly, within the framework of the

FOSM approach, it is possible to use a more accurate second order expression for

calculating the mean value of the performance function (Hahn and Shapiro, 1967).

viz,

E r,,_, d-G{X)(£) + 2 E E dX.d.X, cov{xt, ary} (6.35)

This has been done for the illustrative example and it was found that the mean

value of the performance function so obtained differ from that calculated from

Eqn.6.3 by less than typically. Secondly, the rapid convergence for (3hl also

indicates that Gni(Ay) and G/(X) are reasonable linear. For example, for all the

data points shown in Fig.6.8, a maximum of six iterations is found to be sufficient

to achieve a tolerance of of 10~s for Phl- The convergence is extremely fast.


fW

Leqend

° f(x)= constant

a f(x) = half - sin e

simplified Bishop's method

6 = 10 m

simplified Jan bus method

-0.2 -

-0.4 -

-0.6 -

Figure 6.9 Variation of q(A)


A ,(A)§

0.54* 3.50962 3.44468 0.06494

0.66* 3.47745 3.55210 -0.07465

0.59583 3.49590 3.49541 0.0002

0.59598 3.49555 3.49555 0.000005

A 4 j. *ttPhl ,(A)§

0.54* 3.51329 3.44885 0.06444

0.66* 3.49823 3.57579 -0.07756

0.59445 3.50713 3,50652 0.0006

0.59497 3.50707 3.50707 -0.00000001

(b)

A ptrJI

pttrff «(A)S

0.54* 1.1458 x 10-4 1.5523 x 10"4 -0.4 x 10“4

0.66* 1.2692 x 10-4 0.9035 x 10~4 0.3 x 10~4

0.60316 1.2043 x 10"4 1.1687 x 10"4 0.04 x 10~4

0.59757 1.1986 x 10~4 1.1985 x 10~4 0.0001 x 10“4

(c)

* initial value of Avalues based on Gm(X_)

tt values based on Gj[X_)

Table 6.1.Adjustment of A Using Inverse Rational Approximation for

(a) (3 (b) (3hl and (c) Pff


Pf <Phl>

5= 1 m

8=8 = 88= 5m

8=10m

Figure 6.10 Failure Probability Plot inferred from (3 and Phl


As mentioned earlier, the values of p and Phl are equal if the performance

function is linear. Fig.6.10 shows the failure probability plot inferred from for [3

and Phl- The close agreement between the two values provides the third evidence

that a linear approximation is very accurate for Gm(X) and G/(X). As the values

of Pf inferred from (3 and Phl are similar, only results for Phl will be presented

below.

6.9.1.3 Comparison of different approaches

As mentioned earlier, values of Pj inferred from p and Phl are essentially

the same. This can be attributed to the minimal non-linearity of the performance

function formulated in terms of the safety margin. A larger difference between p

and Phl has been reported by Luckman (1987) for a slope in Hong Kong. This

may be due to their use of Go(X) (see Section 6.4) for the formulation of the

performance function giving rise to a higher degree of non-linearity and hence a

larger error in the calculation of mean and variance of the performance function

which is then reflected in the larger difference in two reliability indices.

Fig.6.11 shows the probability plot of the failure probabilities inferred from

Phl and the method of PDF fitting. It can be seen that the value of Pjf is

smaller than the values inferred from Phl (and also p). This is to be expected as

the approaches using p and Phl assume a unbounded tail for the distribution of

G(X)• In consequence, the probability inferred from a PDF with infinite tail will

be greater than that from a cut-off tail. At high probability level, the difference in

values is small. However at lower probability level, the values of Pf inferred from

P and Phl are conservative.

6.9.1.4 Influence of interslice force function on Pj

Fig.6.12 shows a comparison of Phl obtained for two different interslice force

functions. Although the constant and half-sine interslice force functions are vastly

different, the values of Phl are almost identical. Similar results are also obtained


Pf (PhiJ

o Type IType IIType IIIType IV

6 = 1 m

8 = 5m

8= 1 0m

Pff

Figure 6.11 Failure Probability Plot inferred from /3hl and Pff


Phl

A C FType IType IIType III

• Type IVu) 4.0 -

35 -

6 =10 m

f(x)= constant

Figure 6.12Comparison of Phl for Different Interslice Force Functions

for P and Pjf indicating that the failure probability of slopes is not sensitive to

the interslice function provided of course it is not very unreasonable.

6.9.1.5 Influence of the form of ACF on Pj

In probabilistic design of slopes, the order of magnitude of Pj is of more con

cern than its absolute value. It can be concluded from Figures 6.10, 6.11 and 6.12

along with other studies by the Author that the value of Pj is not very sensitive

to the type of the ACF used. All the four types of ACF used in this work give val-


ues of Pf with the same order of magnitude. The scale of fluctuation has already

captured the essential correlation structure of soil properties. A similar conclusion

has also been drawn by Luckman (1987). This 1ms significant practical implica

tions as the exact form of ACF is difficult to estimate in practice without a large

number of samples. Because of the simplicity of Type I ACF, it is recommended

for general use.

6.9.1.6 Influence of scale of fluctuation

Fig.6.13 shows the variation of Pf inferred respectively from 3hl and PDF

fitting with the scale of fluctuation. A Type I ACF (simple exponential) was

used in the calculation. It can be seen that Pj is very sensitive to the value of

<5, although it does not depend significantly on the function form of the ACF as

mentioned earlier. Therefore, more attention must be paid to the estimation of

this important parameter in soil investigation. Fig.6.13 reinforces the discussion

in Section 6.3, that ignoring the variance reduction due to spatial averaging results

in an astonishingly large value of Pf.

6.9.1.7 Location of Critical Slip Surface

Circular slip surfaces are considered first. Table 6.2 summarizes the positions

of the critical slip circles for different cases (<5 =5m). The locations of the critical

slip circle with minimum /?, (3hl and maximum Pjj are different, but they are

close to the critical slip circle with minimum factor of safety which is centred at

(-1,14).

Fig.6.14 compares the positions of critical non-circular slip surface with the

critical slip circles with minimum factor of safety or reliability index (3. It can be

seen that these surfaces are very close to each other. As the evaluation of failure

probability or reliability index requires more effort than the factor of safety, it is

preferable to search for the critical slip surface with the minimum factor of safety


Pf ( Phl>

20 m

20 m

Figure 6.13Variation of Pj inferred from (a) (b) Pjf

With Scale of Fluctuation


Centres of critical slip circle for

ACF P Phl Pff

Type I (-1.8.14.6) (-1.8,14.6) (-1.8,14.6)

Type II (-1.7,14.3) (-1.8,14.4) (-1.9,14.7)

Type III ( — 1.8,14.6) (—1.8,14.5) (-1.9,14.7)

Type IV (-1.8,14.7) ( — 1.7,14.4) (-1.8.14.6)

Table 6.2 Location of Centers of Critical Slip Surface - Example 1

first. The surface is then used as an initial estimate for the general search for the

critical slip surface with minimum reliability index or maximum failure probability.

6.9.2 Example 6.2 (Selset landslide)

In this example, the Selset landslide reported in Skempton and Brown (1961)

will be considered. The slip was within a deposit of non-fissured overconsolidated

boulder clay. No significant variation of mean soil properties was observed within a

depth of 60ft(18m). The soil profde could therefore be modelled as a homogeneous

random field. The slope was 42ft (13m) high with an inclination of 28°.

Eight samples were taken at different locations of the slope. For each sample,

at least three specimens were prepared for drained triaxial tests. Because of the

proximity in the field, the soil properties of the test specimens from each sample

will be highly correlated. The mean soil property determined from test specimens

of a sample would therefore constitute effectively one single sample in the statistical

sense. Hence, a value of 8 is used for the sample size N in this case. Since the

sample locations were far apart in the field, the soil properties determined from


Legendslip circle with min. FOSslip circle with min. |3HL

non-circular si ip surface with min. R̂

H L

5 = 10m

Figure 6.14 Location of Critical Surfaces - Example 1

each sample can be regarded as independent. The variance and covariance of the

sample spatial average are evaluated using equations given in Chapter 4 for a Type

I soil profile.

A summary of the test results is given in Skempton and Brown (1961) from

which the follow ing input parameters are derived.

mean COY

180 lb/ft2 (8.6KPa) 30%

130 lb/ft3 (21.8kN/m3) 0.7%

7%

7

0 32


Using a first order Taylor’s series approximation, the mean value and COV of

t (i.e. tan<^') are given as tan 32° and 9% respectively. A mean value of 0.45 was

suggested by Skempton and Brown (1961) as a suitable value for the pore-water

pressure ratio r of the slope. A judgemental value of 10% is assumed herein for

the COV of r.

no. of samples

6= 5ft

6 = 15ft

Height of slope ,H (ft)

Ll_

Oto

«*—

_ooa

Ljl

Figure 6.15

Variation of Failure Probability With Height

of Slope (Selset Landslide)

Fig.6.15 show's the variation of the failure probability with the height of the

slope. The reliability index Pul is based on Gm(2Q and a value of 0.6 for A. The


results presented in the figure correspond to the minimum value of Phl associated

with the critical slip surface. A toe failure is assumed throughout. It is also

assumed that the mean and variance of r are unaffected by the change in the

height of the slope. Since the scale of fluctuation of the soil properties are not

known, two values of 6 are used - 5ft (1.5m) and 15ft (4.6m). The results are

plotted in solid lines in Fig.6.15. For the actual slope height of 42ft (12.8m), the

failure probability of the slope is high (>50%). This is to be expected as the

failure of the slope had indeed occurred. As the slope height decreases, the factor

of safety increases and the difference in Pj given the two values of 8 becomes more

pronounced.

Assuming that the COY of the soil properties remains unchanged, the slope

was re-analyzed using a value of 30 for the sample size N. Since the sample size

is now larger, the sampling uncertainty is reduced resulting in a smaller value of

Pj. Note that the increase in reliability of the slope due to an increase in sample

size is greater for the case of 8 =5ft (1.5m). In fact, it is generally true that

increasing the sample size for soils with a smaller value of 8 is more effective in

reducing the failure probability than soils with a larger value of 8. It is because

when the scales of fluctuation is large compared with the dimensions of the slope,

the reduction in variance due to spatial averaging would be smaller. The variance

of the sample spatial average would then be dominated by the spatial variability

of the soil properties. An large increase in the sample size N can only reduce the

total variance by a small amount.

In the design of soil slopes, results like Fig.6.15 can be obtained using the prior

knowledge of the soil properties. This kind of information would be very useful

in the design stage for identifying the critical parameters to which more attention

should be paid and for determining a suitable sample size for soil testing.

CHAPTER 7

LOCATION OF CRITICAL SURFACE

7.1 INTRODUCTION

The reliability analysis of the stability of a slope involves two steps; one for

the calculation of the failure probability of a particular slip surface and the other

for locating the most critical slip surface which has the maximum risk. The former

has been discussed in Chapter 6. This chapter addresses the latter topic.

A number of approaches have been developed over the past decade for locating

the critical slip surface with the minimum factor of safety. The following briefly

reviews the existing approaches and the possibility of extending some of these

methods to the searching of critical slip surface in the context of probabilistic

analysis will also be discussed.

Garber (1973) and Baker and Garber (1977a, 1977b& 1978) presented a series

of papers on the application of calculus of variations to locating the critical slip

surface with the minimum factor of safety. In their approach, the functionals,

deduced from the consideration of the overall vertical, horizontal and moment

equilibrium, were minimized analytically with respect to the location y(x) of the

slip surface and the normal stress distribution a{x) along it. The general solution,

even for the simplest case of a homogeneous soil profile, requires the solving of the

Euler’s differential equation in addition to the determination of seven unknown

7-1


quantities (Baker and Garber, 1977b). Although the theoretical analysis based on

the calculus of variations gives greater insight to the method of limit equilibrium

in slope stability analysis and gives the smallest possible factor of safety with

no ‘arbitrary’ assumption about the internal force is required as emphasized by

Baker and Garber, it is difficult to use in practice. It is perhaps this that led Baker

(1980) to give up this analytical approach and revert to an approximate numerical

approach based on dynamic programming and the adoption of Spencer's method

of slope analysis.

Castillo and Revilla (1977) and Revilla and Castillo (1977) also presented a

theoretical study based on the calculus of variations. The approach is similar to

that by Baker and Garber except that the functionals to be optimized were now

based on the simplified Janbu method. That is to say, the distribution of normal

stress cr(x) is inferred by the assumption about the interslice forces rather than

obtained by means of calculus of variations.

The calculus-of-variation approach is entirely an deterministic approach. The

use of the method is very limited in the context of probabilistic design. The

difficulty arises from the fact that the functional of the failure probability cannot

be expressed as an explicit function of the location of the slip surface.

A different approach based on the random generation of slip surfaces was

developed by Boutrup and Lovell (1980). A series of admissible slip surfaces

are generated randomly and the critical slip surface with the smallest value of

the factor of safety is selected from those generated surfaces. Theoretically, the

method can also be used to locate the critical slip surface with the maximum

failure probability. However, it has the disadvantage that a lot of effort is wasted

in evaluating the failure probability of the surfaces which are far from the critical

surface and there is always a statistical uncertainty as regards to whether the

critical surface so obtained is the most critical slip surface or not.


Baker (1980) defined the slip surface by a number of nodal points joined

together by straight lines and used dynamic programming to determine the most

critical slip surface. However, Baker’s (1980) method has the disadvantage that

it is only applicable to Spencer’s (1967) method as explained in Li and White

(1987a). Furthermore, the return function cannot be formulated in a probabilistic

analysis and hence the dynamic programming approach cannot be used.

Celestino and Duncan (1981), Nguyen (1985b) and Li and White (1987a)

also described the slip surface by means of nodal points and the factor of safety is

treated as a multivariate function defined implicitly with respect to the coordinates

of the nodal points. The critical slip surface associated with the minimum FOS is

then obtained by optimizing the factor-of-safety function with respect to the nodal

coordinates. By defining the reliability-index function or the failure-probability

function in term of the nodal coordinates, the same approach can also be adopted

for locating the critical slip surface with the minimum reliability index or maximum

failure probability. The method is quite general and is valid for any stability

models. Of course, the viability of the method depends largely on the amount

of effort required for the optimization. Celestino and Duncan (1981) and Li and

White (1987a) used the alternating-variable approach in searching for the optimum

while Nguyen (1985b) adopted the simplex approach.

The success of the optimization approach has been demonstrated by Li and

White (1987a) for the calculation of the minimum factor of safety of a slope. How

ever, when applied to the calculation of the reliability index or the failure prob

ability, it has to be remembered that the calculation of these quantities requires

much more computing time than the calculation of the factor of safety. Although

the simple technique of alternating variables is adequate for finding the minimum

factor of safety, a more efficient optimization technique is desirable for locating

the critical surface with the minimum reliability index or maximum probability.

7.2. DEFINITION OF PROBLEM 7-4

In the following, the technique of steepest decent is used for the optimization of

the objective function.

7.2 DEFINITION OF PROBLEM

The problems of locating the most critical position for circular and non-

circular slip surfaces are different. They will be discussed separately in this section.

7.2.1 Non-circular slip surface

The following is a unified treatment of non-circular surfaces. Fig.7.1 shows

the position of a slip surface defined by a series of straight lines. The coordinates

of the 7th nodal point are denoted by (:rt, yt). As the coordinates (x^y,) vary,

they trace out infinitely many slip surfaces which are kinematically admissible.

The nodal points can be divided into two categories:

(i) Unconstrained nodal points (U)

For unconstrained nodal points, the abscissa xt and ordinate yt can vary

independently. Point B and E in Fig.7.1 are examples of unconstrained nodal

points.

(ii) Nodal points on prescribed curve (C)

Very often, the development of a slip surface is influenced by the presence

of a weak soil seam or strong rock layer. As in the case of Fig.7.1, points

C and D should only move in the direction parallel to the weak soil seam.

Other examples of of class C nodal points are points A and F in Fig.7.1. If

xd represents the initial abscissa of point D, the ordinate of the point will be

given by

vd = v(id +$d) (7.1)

where y(-) is the function describing the trajectory' of the nodal point along


F

y

Figure 7.1 Definition of Slip Surface by Nodal Points


the prescribed curve and sB is the horizontal distance of the point measured

from xB. The variable sB becomes the location parameter of the nodal point.

Define the risk function H(X_) to be either the factor of safety, reliability index

or the failure probability of a slip surface with location parameters X_. Suppose

that there are k class U nodal points and m class C nodal points. Without loss

of generality, the k nodal points of class U are represented as (sq, yt),i — 1, k and

location parameters of the class C points are denoted by Si,i = l,m. The risk

function can therefore be written as a multivariate function as follows;

H(X) = H(xl,x2,--- ,xk;yi,y2,---,yk\si,s2,---,sm)(7.2)

and 2 k + m = n

To locate the most critical slip surface is therefore equivalent to finding the 72-

dimensional minimum point and the minimum value for the function H. Some

constraints exist in an implicit form for the variables of the risk function. It can

be discerned by referring to Fig.7.1. The nodal points of the slip surface must be

in the order A-B-C-D-E-F. Therefore, the abscissa of the nodal points must satisfy

xA < xB < xc

xB <xc < xD (7.3)

xc <xD < xB etc.

Furthermore, the ordinates of the points must lie within the slope. Bell (1969)

also made further restrictions regarding the convexity of the slip surface.

7.2.2 Circular slip surface

The treatment for circular slip surfaces is simpler than for non-circular sur

faces. For circular slip surfaces, there are only three location parameters; namely

the abscissa xQ and the ordinate y0 of the centre of the slip circle as well as the

7.3. SEARCHING PROCEDURE 7-7

radius R of the circle. The risk function can be written as a function of these three

variables.

H(X) = H(x0,y0,R) (7.4)

Optimizing the function H(X_) will give the minimum value of F, the minimum

reliability index or the maximum failure probability depending on the case and

also the centre and radius of the most critical slip circle. Very often, the geology of

the slope dictates the radius of the slip circles. In these cases, the function H[X)

reduces to a function of the coordinates of the centre of the slip circle.

For both circular and non-circular slip surfaces, the minimum point and the

minimum value for the function H(X) can be obtained using the optimization

techniques for a multivariate function. The procedure is described in the following

section.

7.3 SEARCHING PROCEDURE

In an earlier work (Li and White, 1987a), the Author used the alternating-

variable approach for optimizing the factor-of-safety function of a slope. In this

work, the method of steepest descent will also be used. The method is generally

more efficient than the alternating-variable approach especially for non-circular

slip surfaces which have more location parameters than circular slip surfaces. The

alternating-variable technique and the method of steepest descent are well doc

umented in the literature (e.g. Beveridge and Schechter, 1970; Jacoby et a 1 ,

1972). A brief outline of the two methods is given in Appendix E. Here, some

useful procedures which have been incorporated into the computer program will

be discussed.

The derivatives of the risk function required in the implementation of the


method of steepest descent are obtained by means of a finite difference approx

imation. The following procedure is found to be useful. Both the forward and

backward finite difference approximations are used to estimate the derivatives of

the risk function H()Q- If the forward and backward finite difference approxima

tions of a location parameter have difference signs, this would indicate that the

global optimum is not likely to be in the direction of the location parameter. An

arbitrarily small value is then assigned to the partial derivative with respect to

the parameter. If both the forward and backward finite difference approximations

have the same sign, an average value will be used.

In theory, both the technique of alternating variables and the method of steep

est descent are only applicable to unconstrained problems. Therefore, the method

are strictly speaking not valid for optimizing the risk function H(X). One way

to tackle this is to introduce the so called barrier function or penalty function to

the risk function (Jacoby et al , 1972) so as to transform the problem into one of

unconstrained optimization. However, the following procedure is simpler.

It is observed that the physical constraints mentioned in Eqn.7.3 are violated

during the search usually when a poor approximation to the critical slip surface

associated with a relatively large number of nodal points is used as an initial guess.

To avoid such a problem, it is preferable to define the slip surface initially with a

minimum number of nodal points. The searching algorithm associated with a trial

surface having a small number of nodes is very robust. The physical constraints

are seldom invoked even for a crude initial trial surface. The use of a smaller

number of nodes in the initial stage also possesses other advantages which will be

mentioned later. Therefore, the following procedure is recommended.

(a) Initially, the slip surface is defined by a few nodal points and the surface is

then searched until the difference in //(X) between consecutive searches is

small. This will give an approximate location of the critical slip surface. The


tolerance can be less stringent at the begining.

(b) New nodal points are then introduced automatically at the mid-point of the

straight line joining adjacent nodes of this approximate critical slip surface.

The slip surface, which is now refined by more nodal points, is searched until

the required accuracy is attained.

(c) Step (b) is repeated until the addition of more nodal points does not result in

any appreciable change in the value of H(X_).

To avoid premature termination of the search in step (a) or (b), the following

termination criterion is suggested. If the difference in the value of H(X_) for three

successive searches, is smaller than the tolerance, this step can be terminated.

Provided that a reasonable initial estimate of the critical slip surface is present,

as is obtained by searching a slip surface having a small number of nodes in this

case, the physical constraints of the location parameters are seldom invoked during

subsequent refinements of the slip surface.

There is no guarantee that the above procedure will not violate the constraints

during the search. Consider more closely the constraints in Eqn.7.3. When the

constraints are violated, it simply means that two adjacent nodal points are ap

proaching each other, trying to cross one another. It indicates that the slip surface

is ‘over-specified’ by having too many nodal points. A simple solution to solve this

problem is to remove one of the two points which is causing the trouble. The

algorithm is restarted with the slip surface now defined with fewer nodal points.

Such a procedure is found to be successful.

For circular slip surfaces, the treatment is much simpler. Normally the only

likely constraint will be on the radius of the slip circle. In such cases, the radius

can usually be expressed in terms of the center of the slip circle. As a result,

the risk function will become a function of two variables and there is usually

no further constraints for this transformed function except perhaps that the slip


circle must cut the slope. But this constraint can usually be satisfied. Therefore,

the risk function can be regarded basically as an unconstrained function and the

alternating-variable technique or the method of steepest descent can be applied

without modification.

There is no guarantee that the critical surface given by above procedure is the

absolute minimum. The same is true for other optimization techniques. However,

by starting trial slip surfaces w:ith a small number of nodes at different locations

and observing whether they are approaching the same approximate final location

will give a quick and good indication of whether other local minimums are present

within the slope

7.4 ILLUSTRATIVE EXAMPLES

7.4.1 Example 7.1

Re-consider the homogeneous soil slope of Example 6.1 in Chapter 6. The

input parameters are the same as those used previously. Again the pore-water

pressure is represented by the pore-water pressure ratio with all rx assumed to

have the same mean value of 0.2 and COV of 10%. Toe failure is also assumed

herein. 15 slices and Type 1 ACF with b — 5m are used throughout. The slice

function f(x) is taken as a unit function. The analysis is based on (3 with a value

of 0.6 for A.

Circular slip surfaces are considered first. Fig.7.2 shows the contour plot of (3

against the location of the centre of the slip circle. Because of the homogeneous

nature of the slope, a well defined minimum exists. Both the technique of alter

nating variables and the method of steepest descent are used. Point O in Fig.7.2

is the initial trial location for both searching algorithms. The complete searching

sequence is summarized in Table 7.1 for the technique of alternating variables and


cP Y<r

non- feasibleregion

x-axis (m)

Figure 7.2 Contour Plot of /3 - Example 7.1

- axi

s (m

)


Point Centre p accs accg

O (0.00,15.00) 3.96 - -

A (-2.14, 15.00) 3.47 0.01 0.1

B (-2.14, 14.86) 3.47 0.01 0.1

C (-2.10,14.86) 3.47 0.01 0.1

D (-2.10, 14.75) 3.47 0.01 0.1

E (-2.10, 14.75) 3.47 0.0001 0.01

F (-2.10, 14.68) 3.46 0.0001 0.01

G (-2.03, 14.68) 3.46 0.0001 0.01

N (-2.04, 14.65) 3.46 0.0001 0.01

Table 7.1.

Searching Sequences for Circular Slip Surface - Example 7.1

(Alternating-variable technique)


Point Centre 0 accs accg

0 (0.00,15.00) 3.95 - -

A' (-1.65, 14.23) 3.46 0.01 0.1

B' (-1.69,14.19) 3.46 0.01 0.1

C (-1.65,14.24) 3.45 0.01 0.1

D' (-1.70, 14.20) 3.46 0.01 0.1

E' (-1.70, 14.21) 3.45 0.0001 0.01

F' (-1.70, 14.21) 3.45 0.0001 0.01

G' (-1.70, 14.21) 3.45 0.0001 0.01

H' (-1.70, 14.21) 3.45 0.0001 0.01

N' (-1.70,14.21) 3.45 0.0001 0.01

Table 7.2.

Searching Sequences for Circular Slip Surface - Example 7.1

(Method of Steepest Descent)


in Table 7.2 for the method of steepest descent. The quantity accs is the stopping

criterion for each search. When the differences of p for three consecutive searches

are all less than accg, the program is either terminated or re-started by stipulat

ing more stringent values for accs and accg. It can be seen from Tables 7.1 and

7.2 that the greatest reduction in p occurs in the first search for both methods.

Subsequent searches only results in minute changes in 3. As shown in Fig.7.2,

the search approaches quickly the bottom of the valley of the contour plot after

the first search (point A for the alternating-variable technique and and A' for the

method of steepest descent) and the final locations N and N7 are very close to A

and A'.

Figures 7.3 and 7.4 show respectively the searching sequences for the non-

circular critical slip surface using the technique of alternating variables and the

method of steepest descent. The slip surface is at first defined by a 3-point surface

which is then refined successively by introducing nodal points midway between

two adjacent nodal points until the slip surface is finally defined by a total of 9

points. Although it is not shown here, the non-circular surfaces obtained using

both methods are almost coincident. The non-circular slip surfaces are slightly

more critical than the circular slip surface. The alternating-variable approach is

found to be more time consuming than the method of steepest descent as expected.

Both methods give a similar value of /?.

7.4.2 Example 7.2

Fig.7.5 shows the geometry of the slope in this example. The input soil prop

erties are as follows:

mean COV sample size k

c' 5 kN/m2 20% 10 2 m lm

Layer 1 7 18 kN/m3 5% 5 2 m lm

t tan 30° 5% 10 2m lm

rem

arks


o o o

p Oo o

• --- 4-< '.tr CL CL Ql

rn cn

“D

£cna>

l—

a>&

Sear

ch fo

r Non

-circ

ular

Slip

S

emar

k


h-OG*

t-<Dt-3bp

U-t

Sear

ch fo

r Non

-circ

ular

Slip

Surf

ace -

Exa


mean cov sample size 6*

c' 10 kN/m2 20% 8 2 m 2 m

Layer 2 7 18 kN/m3 5% 5 2m 2 m

t tan 30° 5% 8 2 m 2m

The pore-water pressure is descrbed using the second model described in Sec

tion 6.6. The curves yu(x), Vp(j) and iji(x) represent respectively the upper bound,

most probable level and lower bound of the phreatic surface. Other details of an

alysis are the same as Example 7.1.

Circular slip surfaces are considered first. Fig.7.6 shows the contour plot

of (3 against the location of the centre of the slip circle. It can be seen that

multiple local minimums exist as would be expected for non-homogeneous slopes.

To locate the absolute minimum, the search process has to be started at different

initial positions. Fig.7.6 shows two search sequences started at two different, initial

locations (points O and O' in the figure). The method of steepest descent is used.

Points A and A' in the figure represent the positions after the first descent with

accs set at 0.01. The search is then continued until the difference in /3 between

three consecutive searches are all less than 0.1. The search is then re-started with

accs set at 0.001 and accg set at 0.01. The final position of the search is marked

as points N (3 = 4.14) and N' (3 = 4.26). in the figure. It can be seen that

the greatest reduction of (3 occurs during the first descent. The search quickly

approaches the minimum point after the very first descent. Once the minimum

location is near, subsequent searches will only result in minute changes of (3 and

locations of the centres of the slip circles.

Fig.7.7 shows the results for non-circular slip surface. Both the technique of

alternating variables and the method of steepest descent are used.


17

- 16

15

14

12-5 -4 -3

x -axis (m)

Figure 7.6Contour Plot of {3 Against Center of Slip Circle - Example 7.2

- axi

s (m

)

lege

nd

rem

arks



At first, the slip surface is defined by two straight lines having three nodal

points as shown in the figure. The 3-point critical slip surface is located. The

slip surface is then refined by introducing nodal points mid-way between the two

straight lines. The slip surface, now defined by five nodal points, is again searched.

The process is repeated until the slip surface is finally defined by 9 nodal points for

the alternating-variable technique and 8 nodal points for the method of steepest

descent (one point has been deleted as two of the points are too close together

during the search). The final values of accg is set at 0.01. The alternating-variable

technique gives a slightly more critical slip surface with a smaller value of fl.

In slope stability analysis, a critical band near the critical slip surface can

usually be found. All slip surfaces within this band would have factor of safety,

reliability index or failure probability very close to that of the critical slip surface.

This is demonstrated by the results shown in Fig.7.7. Although the slip surfaces

obtained using the two methods are some distance apart, the numerical difference

of (5 is less than 59o indicating that these two surfaces are within or close to the

critical band.

CHAPTER 8

LIMITATIONS AND SUGGESTIONS

This thesis has presented a general probabilistic approach to slope design.

However, the discussion will not be complete without mentioning the limitations

of the present approach and pointing out some of the areas where further research

would be necessary.

1. A two-dimensional stability model is used herein. This is equivalent to saying

that soil properties are perfectly correlated in the transverse direction. The

consequence of such an assumption remains a question of further inquiry. But

no doubt a three-dimensional analysis, especially for c-0 slopes, will be much

more complicated than a two-dimensional analysis both in terms of the formu

lation of the performance function and the generation of the covariance matrix

of the spatial average soil properties. No detailed probabilistic study on three-

dimensional soil slopes has yet been published in the literature for c-^> slopes,

although attempts have been made to analyze a three-dimensional 0 = 0 slope

using a level-crossing approach (Yanmarcke, 1977bYl980; Yeneziano and An-

toniano, 1979).

2. Model uncertainty and measurement errors of soil properties have not been

considered. The failure probability given by the present model will therefore

be the lower bound values. Given the statistical properties of the model un

certainty and the measurement erros, the procedure for incorporating these

uncertainties into the analysis is relatively straight forward (see e.g. Yuceman

8-1

8. LIMITATIONS AND SUGGESTIONS 8-2

et a1 ,1973; and Ang and Tang, 1984). Of course, the questions are how to

calibrate the accuracy of the limit equilibrium method and to determine the

testing errors of soil measurements. These questions cannot be answered solely

by the use of statistics. A better understanding of the physics of the problem

and the fundamental concepts of soil mechanics is necessary.

3. In a limit equilibrium analysis, soils are assumed to be perfectly plastic mate

rials. On this basis, the spatially averaged soil properties will be the pertinent

parameters to use in the analysis. However, for strain-softening soils, the ef

fect of ‘brittle’ failure cannot be overlooked. In a conventional deterministic

analysis in which the soil properties are assumed to be constant, the yield zone

always initiates at the location with the highest stress level. However, the pic

ture will be somewhat different when looked at from a probabilistic point of

view. Since soil properties van' from point to point within a slope, there may

be a chance that the soil strength is very low at a location where the stress

level is not the highest. Failure can well initiate from this point instead of the

most highly stressed region. On the other hand, if it so happens that the soil

strength is the lowest at the most highly stressed region, the yield zone may

propagate catastrophically to the adjoining area leading to a sudden failure of

slope.

The spatial variability of soil has therefore two opposing consequences. On

the one hand, the spatial variability reduces the variance of the average soil

properties and hence the failure probability of slopes. On the other hand,

spatial variability of soil will increase the likelihood of progressive failure as

failure can initiate at any location along the slip surface. Which effect will

dominate depends on the strain-softening behaviour of soil, at present study

on this topic is limited. Further discussion is given in Tang et al (1985).

4. The system reliability of failure probability of slopes remains a relatively ‘unde-


velopecP area of research. The study of system reliability of slopes would give

the answer of whether the low’er bound value given by the failure probability

of the most critical slip surface is close to the system failure probability or not.

5. The prediction of critical groundwater conditions is considered to be one of the

most important elements in assessing the safety of slopes. At present, there

is little information regarding the statistical characteristics of measured pore-

water pressure variation although a number of theoretical models have now

been proposed as mentioned in Section 6.8. These theoretical models have yet

to to be verified by field observations (of which few exist at present).

An empirical procedure has been developed by Lumb (1975b&1979) for predict

ing rain-induced slope failures in Hong Kong. Matsuo and Ueno (1978&T981)

have also developed a more elaborate procedure based on probabilistic calcu

lations for the prediction of rain-induced slope failures in Japan. There is still

ample scope for interesting and challenging research in this area both for the

theoretically-minded and practitioners with an aim of crystallizing the theoret

ical matters into some sound and preferably simple design rules. Although the

effort required may be tremendous, the reward of success is also great in terms

of a sharper estimate of the failure probability and a more reliable warning

system of slope failures induced by rain-storms.

6. The most logical way to assess the safety of slope is by means of a probabilistic

approach and the state-of-the-art of the first-order-second-moment approach

has advanced to a stage that has become practical for routine slope designs.

However, before a probabilistic code can be introduced, it is necessary to have

sufficient prior knowledge on the statistical properties of local soils such as the

coefficient of variation, the scale of fluctuation, upper and lower bound values

etc. A statistical analysis of soil data obtained from the existing data bank,

the current literature or a extended research program would be of great value.


Further studies on the cross-correlation of soil properties and the simplification

of the structural analysis of a Type III soil profile would also be useful.

CHAPTER 9

CONCLUSIONS

A general probabilistic approach to slope design using the random field model

and Morgenstern and Price's method is presented. This study utilizes a rigorous

model for the formulation of the performance function. By following the unified

solution scheme discussed in Chapter 3, this approach has the potential to be

extended to other rigorous stability models currently in use in the literature. An

other salient feature of the scheme is that the performance function can be defined

explicitly in terms of the basic random variables with no need of iteration for the

calculation of the interslice forces, thus enabling the derivatives of the performance

function to be evaluated analytically.

Formulating the performance function as a safety margin has the advantage of

minimizing the non-linearity of the function. This results in fast convergence for

the calculation of the reliability index Phl and simplifies greatly the calculation

of the lower and upper bounds of the performance function. The use of a safety

margin also makes the assumption of normal distribution for the performance

function more reasonable by virtue of the central limit theorem.

The random field model has been extended to cover non-homogeneous soil

profiles and take account of the sampling uncertainty. A series of useful formulae

is developed to facilitate the calculation of the variances and covariances of the

spatially averaged soil properties.

The state-of-the-art of the structural analysis of soil data is also critically

9-1

9. CONCLUSIONS 9-2

reviewed. It is considered that more research should be carried out to establish

some practical yet sound procedures for the estimation of the statistical properties

of soils.

Comparisons are made for different probabilistic approaches to slope design,

namely the approaches based on ;3, 3hl and PDF fitting. Studies indicate that

the reliability index ,3hl is essentially identical to the reliability index (3 when the

performance function is formulated as a safety margin. This can be attributed to

the fact that the performance function so formulated is ‘almost' linear.

Experience indicates that at a high level of failure probability (> 10~2 - 10-3),

the values of Pf inferred from all the above three methods are similar. However,

a larger difference exists between the value obtained from PDF fitting and those

from the reliability indices (3 or (3hl when the failure probability is small.

It is also observed that the failure probability based on Gm(X) is much less

sensitive to the parameter A than that based on Gj(X). Although the use of a

rigorous solution is always desirable, an approximate result based only on Gm(X)

and a reasonable value of A would very often give a sufficiently accurate answer.

Results also show that the failure probability of slopes is not sensitive to

the types of interslice force functions and autocorrelation functions. However,

it depends significantly on the magnitude of the scales of fluctuation which are

considered to be the most important parameter that characterizes the correlation

structure of soil properties. More attention should be paid to the estimation of

this parameter than has hitherto been the case.

An algorithm for locating the critical slip surface is developed. The critical

surface with minimum factor of safety very often gives a good indication of the

likely location of the critical surface with maximum failure probability.

Although there are still problems to be solved regarding the probabilistic mod

elling of soil behaviour, variation of pore-water pressure and the methodology for

9. CONCLUSIONS 9-3

reliability calculations, it is pedantic to delay the use of the available probabilis

tic methods, especially the FOSM approach, for want of a completely rigorous

analysis. The main advantage of using a probabilistic approach is to provide an

operational procedure by which the uncertainties of the design can be considered.

It also helps the engineer to quantify experience by building up knowledge on

the values of the statistical parameters such as the COV or scales of fluctuation

of local soils. These judgemental values can always be updated and uncertain

ties sharpened when more information becomes available. Moreover, experience

is more easily transmitted to an inexperienced engineer by conveying the likely

values of the statistical properties of the soil than just specifying a magic number

for the factor of safety.

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Vanmarcke,ETI.(1984) Random Fields: Analysis and Synthesis, M.I.T. Press, Cambridge, Mass., 382pp.

Vanmarcke,E.II. and Fuleihan,N.F.(1975) ‘Probabilistic Prediction of Levee Settlements’, Proc. 2nd International Conference on Applications of Statistics and Probability in Soil and Structural Engineering, Aacher, Germany, ppl75-190.

Veneziano,I).( 1974) Contributions to Second-moment Reliability Theory, Research Report R74-33, Department of Civil Engineering, M.I.T., Cambridge, Mass.

Veneziano,D.( 1979) ‘New Index of Reliability’, Journal of Engineering Mechanics Division, ASCE, Vol.105, No.EM2, pp277-296.

Veneziano,D. and Antoniano,J.( 1979) ‘Reliability Analysis of Slopes: Frequency- domain Method’, Journal of Geotechnical Engineering Division, ASCE, Vol. 105, No.GT2, ppl65-182.

REFERENCES R-17

Vinod,H.D. and Ullah,A.( 1981) Recent Advances in Regression Methods, Marcel Dekker Inc., New York, 361pp.

Webb,D.L.(1980) ‘Probability Analysis Applied to the Design of Piles at Richards Bay’, Proc. 7th Regional Conference for Africa on Soil Mechanics and Foundation Engineering, Accra, pp873-878.

Whit tie, P. (1954) ‘On Stationary Processes in the Plane’, Biometrika, Vol.41, pp434- 449.

Whittle,P. (1962) ‘Topographic Correlation, Power-low Covariance Function, and Diffusion’, Biometrika, Vol.49, pp305-314.

Wolff,T.F. and Harr,M.E.(1987) ‘Slope Design for Earth Dams’, Proc. 5th International Conference on Applications of Statistics and Probability in Soil and Structural Engineering, Vancouver, Vol.2, pp725-732.

Wong,F.S.(1984) ‘Uncertainties in FE Modeling of Slope Stability’, Computers and Structures, Vol.19, pp777-791.

Wu,T.H.(1974) ‘Uncertainty, Safety and Decision in Soil Engineering’, Journal Geotechnical Engineering Division, ASCE, Vol.100, No.GT3, pp329-348.

Wu,T.H. and El-Jandali,A.(1985) ‘Use of Time Series in Geotechnical Data Analysis’, Geotechnical Testing Journal, ASTM, Vol.8, No4, ppl51-158.

Wu,T.H. and Kraft,L.M.(1970) ‘Safety Analysis of Slopes’, Journal of Soil Mechanics and Foundation Engineering Division, ASCE, Vol.96, No.SM2, pp609-630.

Ximenez de Embun,J.R. and Romana,M.R.(1983) ‘The Formulation of a Soil Model from Penetrometer Probabilistic Information - Application to Differential Settlement Prediction’, Proc. 4th International Conference on Applications of Statistics and Probability in Soil and Structural Engineering, Pitagora Editrice, Italy, ppl601-1613.

Yang,Z.(1982) ‘The Preliminary Study of the Calculation Methods of the Broken Line Slip Surface of the Landslides (in Chines)’, Proceeding of Landslides, Vol.4, ppl 16-129.

Yong,R.N.(1984) ‘Probabilistic Nature of Soil Properties’, Probabilistic Characterization of Soil Properties: Bridge Between Theory and Practice, (Eds) D.S. Bowles and H.Y. Ko, ASCE ppl9-73.

Yong.R.N., Alonso,E., Tabba,M.M. and Fransham,P.B.(1977) ‘Application of Risk Analysis to the Prediction of Slope Stability’, Canadian Geotechnical Journal, Vol.14, pp540-553.

Yong,S.C.( 1967) ‘On the Stability of Clay Masses: How Safe are the “Factor of Safety”?’, Proc. 3rd Pan American Conference on Soil Mechanics and Foundation Engineering, Vol.2, pp255-270.

REFERENCES R-18

Young.D.S.( 1985) ‘A Generalized Probabilistic Approach for Slope Analysis’, Journal of Mining Engineering, Vol.3, pp215-228.

Young.D.S.( 1986) ‘A Generalized Probabilistic Approach for Slope Analysis: Practical Application to an Open Pit Iron Ore Mine’, International Journal of Mining and Geological Engineering, Vol.4, pp3-13.

Yucemen,M.S.; Tang,W.H. and Ang,A.H-S.(1973) A Probabilistic Study of Safety and Design of Earth Slopes, Technical Report, University Illinois, Urbana, Illinois.

Zhou,H.( 1982) New Methods of Calculation in Optimal Design (in Chinese), Xin- shidai Chubanshe, 250,pp.

APPENDIX A

PARTIAL DERIVATIVES OF

PERFORMANCE FUNCTIONS

The following abbreviations are used in subsequent expressions.

ft = | jj- ~ ■ mi + tan-l

(A. 1)

0 i = 1

dt

i > 1

(A.2)

A.l COHESION

d Gy

dc'

, , <9 ATi ,(Axi + • ti) ■ mi ■ ym,

dc'

dA Tidc'

(Urm ■ tan Q[ - xm .) * / n=

n dSTj r(A.3)

+ ^ dc'ij=i+1

{f j mj tan Qj) • y^j T xmj

Axn " mn ■ ymn z = n

A-l

A.2. AW A-2

dGj~dti

, » dATt , c)A7t (Axt H---- • ft) • mt------------ • tan at

+

3c'

ru <9A7y= < £ dc'j=i+i i

dc[

[tj • nij — tan ctj)i / n

andAxn ’ Win

Axt • 77? t -ft = i

dA T, dTj-i , . •" x-i ~ dc'. ‘ dj ' 0 i < J < n

dc'-

dTn_1dc'. J = 71

i = n

(A.4)

(A.5)

3T,dc' — <

a at, ac;.

ar,-! a at,ac' ^ dc'.

j = i

i < j < n

(A.6)

A.2 A 4V,

dAWi

l1 + ) • U • ’ !/m, -

r,, dAT,- <9ATt1(1 + : „, ) • 2/m, ' tail a, - • xL dAIFt m,

n dAT+ 2L, oE\v '(,J'n,J ~tan n>'' + s

/=* + ! 1

i 7^ w

(A.7)

{tn ‘ Hln tan CVn) ' Urn l = 71

A.3. PORE-WATER PRESSURE A-3

dGf d AWi

, dATt , , dATi ,(I + aAiU)'i'-’n‘-(1 + 5A^)-tanQ*

n dA T+ E dEw -Vi-™,-taD °A]=l+1 1

i ^ n

(A.8)

„ t„ ■ rn„ - tana,, i = n

andr (ti • ml - tana,-) • ft j — i

dATj~dM\

dTj-i d A dj • $j i < j < n (A.9)

arn_! , a aw, j = n

dTjdAWi

a at, a aw,

dTj-x a A Ty a a iv7, ^ a aw7,

j = i

i < j < n

(A. 10)

0G _” Al dAWl ’

5G A <9G 36' dG3pt 1 3 A I E; ’ dAPt dAWi (A.11)

A.3 PORE-WATER PRESSURE

If the first model is used, the pore-water pressure iil is expressed in terms of the

pore-water pressure ratio rt and the following equations apply. ulAxl = rz • Al •

A.4. PORE-WATER PRESSURE A-4

where 7 is the mean soil density.

dGmdrt

' SAT, _ dATl( 777 A{ 7t) ■ 11 • in l • ym i 77 (2/m, 1an oa xTn.

= <

drt

^ 3ATy+ E.^fj=i+i

drl

(tj • rrij - tan ay) •

, An ' In '^n ‘ ^Tln • Um ,

i ^ n

(A.12)

i — n

dGfdrt

f ,d&T{ dATt(-77------At) • A • mi----- ^--- - tan atdrt dr{

—

" 9ATj ,+ y —7—- ■ (tj • nij — tan ay)i 7^ n

j—t+1 drt (A. 13)

and

d AT j

y An ' Hn ' tn ’

-Ai • l{ • U • mt • j = i

1 = n

drt (A.14)ar;_

9r, • dj • <7 i < j < n

dTjdrt

f a at,dri

dTj-! a A Tj< dri dr<

j = i

i < j < n

(A. 15)

In the illustrative examples given in the text, rt is assumed to be perfectly corre

lated, i.e. the pore-water pressure ratio for the whole soil mass is represented by

a single variable, r. The derivative of the performance function with respect to r

is simply given by TjT using the formulas given above.

If the second model is used, the variable r, in the above equations has to be

replaced by u\ and the term Al^ji, i — 1, n replace by 7^ • H(wt — yt) or 7w ■ H(ht)

A.4. COEFFICIENT OF INTERNAL RESISTANCE A-5

where H(-) is the unit step function.

A.4 COEFFICIENT OF INTERNAL RESISTANCE

dU

(1 + t{ tan Qj) ■ t,dM\dtx

+ ( ATt T~ Ail i — ul Axx )

c[ Axj tan

dATi

■ m2 • ym. • cos2 Q;

= <

+

dt{

A, dATjdti

(ymx • tan Qj - xm.)

Xj=i+i(tj ‘ Wlj tan Oj) • Hm, "h Zyrij

(AWn+Tb-Tn-i

- iin Axn - cnAxn tan an) • m2 • ymn • cos2 at

i 7^ n

(A.16)

2 = n

d Gf dix —

o \ 'T'

{(1 + t, tan a) ■ U ■ + [ AT, + AIV, - i.Ax, )

A . 'i 2 o— clAxl tan at j • ??i“ • cos“ Qt-----~qJ~ ’ ^an Ql

+A, 5 A TvX (tj ■ mj — tan etj)

(A\Vn+Th-Tn-X

- un Axn - cn Axn tan q„) • m2 • cos2 at

i / n

(A.17)

i — /i

and

A.5. AQ{ A-6

dATjdtt

(AWt — utAx{ — c'lAxl tan at) • m2 • ft • cos2 al

+ < - ^7T+ J = lc'lAxl -f (AWi — ulAxl) ■ tt ■ mi

(A Qt + A Wi tan ctj) j • (f • m2 • cos2 at(A-18)

dTj-:dt{ ■ d. $3 i < j < n

dTjdt{

' dATi dti

dTj-! 9AT,dti dti

j = *

i < j < n

(A-19)

A.5 A Q{

dGmdA Qi

dA Tio A ^ -ti-mi- ym-O x

dATt ,yQl + a A ^ (ym,- • tan Qi - arm,)

+

dA Qi

^ dA7y dAQi£ (*y • mj - tan aj) ■ ym + x,

i 7^ n

(A.20)

< = /i

A.6. END FORCES A-7

dATtdAQt

(tt ■ ml — tan at) — 1

dGf dA Qi

n

+ Ej=t+i

dA Tj dA Qi

■ (tj • inj — tan aj)i ^ n

(A-21)

-1 i = n

and

d AT3 dAQt

ft j = i

dT,_ dA Qi, * dj ' 0 i < j < n

(A.22)

dTjdAQt

a at,dA Qi

ar,'_! a at,dAQi + dAQi

j = i

i < j < n

(A-23)

A.6 END FORCES

dGmdEa ya + J2

t=l

<9ATt<9£a

(0 • - tan a,) • ymi + xmi

dGf

~dE~a

dATi~dE^

(ti ■ m'i • — tan at)

(A-24)

(A.25)

A.6. END FORCES A-8

0 j = 1

dAr,d Ea ^kr ■ dJ • & 1 < i <"

arn_idEa j = n

(A. 26)

a at,dEa

<

dTj_, 9AT,k dEa dEa

j = 1

1 < j < n(A.27)

dTaDATi dTa

(tl ■ wt - tan at) • ym. + xmi (A.28)

dGf~dT\

n OAT,dTa

(ti • mi tan at)

j = 1

dATjdTa

1 < j < n

_dTn-1

, 3Ta j = n

dTjwa

/ 1 + 3 ATX 3Ta<

3T;_! d AT, 3Ta dTa

j = 1

(A.29)

(A.30)

(A .31)1 < j < n

A.6. END FORCES A-9

OGrn

dEb = -Vb (A. 32)

dGf~dE~b

(A.33)

dGmdTb (tn ■ rnn ~ tan an) ■ ymn 4- xmn - xb (A.34)

dGfr.rp — n tan QnoTb

It is worth pointing out that once the values of the parameters are given, the

derivatives can be calculated explicitly and successively using the equations given

in this Appendix. The above formulae are applicable to the case where A is not

zero. If A is zero, which corresponds to the case of simplified Bishop or Janbu

analysis, ATt is identically zero. Therefore, all the derivatives of ATt and Tt are

to be replaced by zero in the above expressions.

APPENDIX B

PROOF OF EQUALITY OF /? AND aHL

FOR LINEAR PERFORMANCE FUNCTIONS

This appendix outlines the proof that p and Phl are equal for a linear perfor

mance function. It is assumed that the sample centroid lies in the safety domain.

Suppose that G{X) is expressed as

lG(X) =ao + Ya,:r, (B.l)

1=1

Using Equations 6.3 and 6.4, the mean value and variance of G'(X) is given

by

_ jllG — ao + ^ ^ flt^t — ao + yG • A (B.2)

i=i

l i

°~G ~ ai(lj ' cov{*i'*3} = VGT • Vx ■ \/G (B.3)i= i j = i

Therefore, the reliability index p is given as

a0 + VG • X1 \JV GT • Ux • yG'

(B.4)

Referring to Eqn.6.10, as the iteration always converges after one iteration, it

suffices to show that P = Phl,- Eqn.6.10a can be rewritten as

X[l) -X = Vz-yG- (V0) -X)r v G~1VGT ■ \ V ■ vC J

(B.5)

B-l

B. PROOF OF EQUALITY OF P AND fiHL FOR LINEAR PERFORMANCE FUNCTIONSB-2

Note that v^(2Q is constant and independent of the trial value of X_. As the

initial trial point must satisfy the limit state equation G'(X) = 0, therefore

a0 + a‘x\°] = «o + {A0)}r • VC = 0t— 1

=> !I(0)}T-vG=-«o

(B.6)

Consequently,

X*1* -X = Vx- vG-VGr • Vx • VG |

(B.7)

Substituting Eqn.B.7 into Eqn.G.lOb, we obtain

Phl = \ivx'VQT • V<2- (~a0 ~XT V G)(\/Gt ■ Vx •

= |(V<2T • Vx • yG) • (-a0 ~XT V G){\/Gt ■ Vx •

= |(—a0 -XT V G')(vG;r • Vx •

APPENDIX C

FORMULAE FOR VARIANCE REDUCTION

AND COVARIANCE FACTORS

Table C.l summarizes the variance reduction factors T2(L) for the separable

two dimensional ACFs given in Table 4.1. The parameter 8l in the table represents

the scale of fluctuation in the direction of the line L. When 0 is equal to 0 or 7r/2,

8l reduces to the respective scale of fluctuation in the x or y direction.

Table C.2 lists out the constants of the integral It and Jacobian determinant

of Eqn.4.25 for some basic configurations of lines L and V for the case 0\ 7^ Oo

and Table C.3 lists out the Jacobian determinant and the constants of integral Jt

for the case 6X =60.

The formulae for calculating the integrals /t and Jt for the two dimensional

separable ACFs of Table 4.1 are given in Tables C.4 and C.5 respectively.

The function F(a,/3,p,q) in Table C.4 has the form

No close-form solution for Eqn.C.l is known to the Author, and numerical integra

tion is therefore used. The error function erf(-) can be approximated accurately

using the rational Chebyshev polynomial by Cody (1969) and the integral can be

evaluated using Gaussian quadrature (Davies and Rabinowitz, 1975). The nodal

points and coefficients of Gaussian quadrature can be obtained from Abramowitz

and Stegun (1970).

(C.l)

C-l

C. FORMULAE FOR VARIANCE REDUCTION AND COVARIANCE FACTORS C-2

fa)c*fa

■-o’+

faXCD

+

+

CV <0

— S'I

faX<u

+

-s’

0>

<^bCl sr

+

T| ST

+

+e

IIOCL

-I®l<N

+

+

«?

«?

<-o*+

«?

oJl)fa

< •—i i—i >fa o »—i W—Ho<

CX>>H

<Dfa>~>

E-

<Dfa>,o>cx>>

H

Var

ianc

e Red

uctio

n Fac

tor a

nd Sc

ale o

f Flu

ctua

tion o

f Tw

o-di

men

sion

al Se

para

ble A

CF


Case Configuration Criteria • -4, ct A Ei Fi

61 < 62

1 t’xo ^1 Cl (l’yo — LjSi )CT2

CT2 V XO Vy0CTl CTy Vyo LyS 1 Vy0

L1 Sy < L'2 ■?2 t'xo > LyCy

Vyo ^ LySy

2 vxo LyCy

(vy0 - LySy)CT2

ct2 Vxo Vy0CT2 ct2 Vyo Vyo ~ Li [ S1 + L2S2

3 Vxo + L2C2—

(v y0 + I^S^CTl

CTy Vxo Vy qCT'2 ct2 Vy0 — LySy + L2S2 1 \jO + L2S2

1 Vx0 LyCy

(v,jo — LySy )CT2

ct2 Vxo VygCT 1 Cl 1 Vyo — LySy Vyo -t- LySy + L2S2

61 <C d2

L\S\ > L2S2

Vxo > LXCX

2 t’xo + L2C2 —

(Vyo + LjSoJCTl

CTy Vxo Vy0d T\ CTy Vyo — LySy + A Sn Vyo

Vy0 > LySy

3 t'xo + L2C2 —

(Vyo + LlS^CTy

CTy Vxo Vy0C T2 CT2 Vyo tj’yo + L0S2

1 t’xo ct2 t'xo + L2C2 — LyCy Ty 0 LySy

Oy >62

LySy = L2S2

t'xo Zl L1 Cy2 t'xo -CTy t'xo + L2C2 — LyCy -T2 0 LySy

Vy0 -- 0

n

i| sini — >|

1| sin(0i — S3j|

1| sin(tfi—tf;}|

Table C.2.Jacobian Determinant and Constants of Integral / in Eqn.4.25

for the Basic Configurations of Line Averages [9j 7^ Oo)

C. FORMULAE FOR VARIANCE REDUCTION AND COVARIANCE FACTORS

Case Configuration Criteria

4

61 > S2 L\ S\ >

i'xo > A A Vyo > L2S2

61 > S2 L\S\ < L2S2t'xo ^ L1 C1

Vyo > A So

6

L,S, < L2S2 rJO > A A

I’yo > A Si + L2S2

•At Bi Ct A Ei Ft n

1 t-’xO “f" VyoCT'l -CTi Vxo — A A + -ct2 Vy0 -r L\S\ — L2S2 Vyo -fj L\Sl

[vy0 + AA)CT2

2 l-'xo VyoCTx —CTi vI0 + L2C2+ -CTi Vyo Vyo + A. ~>i — L2S2 i| sin(e1-V2 )|

(ryc) - L2S2)CTi

3 I'xo + VyoCTo -ct2 vT0 4- L2 C2 -f- -CTi Vyo ~ L2S2 JO

(Vyo - L2S2)C7\

1 Vxo -f Vy0CT\ -CTi i’xo ~ L i A -f

{Vyo + L\Sl)CT2—CTo Vyo

2 Vxo "f Vy0CT2 -ct2 Vxo — A A +

(vy0 + AS\)CT2-ct2 Vy0 + L\S i — L2S2

3 Vxo -f Vyo CT2 -ct2 vI0 + L2C2 +(Vyo - L2S2)CTl

-CTi Vyo L2S2

1 Vxo — A C\ -f -CTo Vxo + L2C2 — CTi Vyo — A Si — L2S2

U'yo — L\Si )CT2 (Vyo ~ L2S2)CTl

2 vI0 ~ A A +

(vyo- LiSi)CT2-CTo Vxo + Vy0CT2 -ct2 Vyo L 2*5*2

3 V XO Vy qC'T 1 CTi Vxo "f Vy0CT2 ct2 Vyo L1 5,

t'yo -f A ‘Si

lV‘

tyo -f i — L2S2

Vyo -ft L252

t?yo H A A

Cvo

1| s i n (01 — 0 ;) [

1| sin(6j +»;)|

Table C.2. (cont.)

C. FORMULAE FOR VARIANCE REDUCTION AND COVARIANCE FACTORSC-5

Case Configuration Criteria t BC D E* n

*

st t~ -----------r 1 LlCl+ CT- + L2C2 CTx vy0 - LxSx - L2S2 riyo - LxSx' LXSX>L2S2 [Vyo-LxSx)CT2 {Vyo - L2S2)CTx

12/.Q J v >°r~q * r 0 2 vxo - vyoCTx CTX vxo + L2C2- CTX Vyo-LxSx vyo - L.Sn —. 1_______

«Jxo ‘3 vxo - vyoCTx CTX vxo + vyoCT2 —CT2 Vyo — L2S2 vyo

,

1 vxo LXCX CTX vxo + L2C2 —CT2 0

K01 L2/ Uvn vX0>LxCxy^Q2 Vyo = LXSX = L2S2 2 vxo-LxCx CT2 vxo + L2C2 -CTX 0

uxo

LXSX

1 | sintSi+il2)|

----------------------------------------—------------------------------------------- -----

Table C.2. (cont.)

Cas

e


al

c*:

o

•X

o

4’

o

Cl

Co co

O o

o oH H3 S

<3 >-3

o ^

Tabl

e C.3

.)b

ian D

eter

min

ant a

nd C

onst

ants

of In

tegr

al I

in Eq

n.4.

27

br th

e Bas

ic C

onfig

urat

ions

of L

ine A

vera

ges (0

\ = 02

)


ACF 1 = Ie Ja+Bt[2 p( n. ^)dri dr2

Type I 12 ( p,] (exP{ 2(at + (3tF)} exp{ 2{al + (3tE)}i— i

a,=A Q2=c /J2 = A + g

Type II £ £(_iPi,qi)l— 1

cv = ^ ft — ^ Pi — A P2 — C qi = B q2 = D

Type IIIQ E £(-1)‘+*: exp{-[p. + (9.+ /5)6]}'

i— 1 k— 1f (1 +/?£fc)(2+p, +(], £k) | 0(- + Pi + Qx £k ) + <7i( l + | 2/3(7, 1\ q>+0 1 (<7,+/?)2 1 (<7,+/?)3J

<* = Ji ft~Tv Pi = 01A p2 = aC

ql = aB q2 = oD = E f2 = F

Type IV 2 2 21 y~> y^ y^ / |\t + A: + l exP( "“^7ifc)*[^i sin + cos $ijk\lj -r+"?

« = ^ ft = ^ Pl = ^ P2 — Cqi = B q2 = D = E &> - E

Vik = api + UiZk Vi = otqt + ft Vi = aqt + (3 v2 = aql - (3

$ijk — Vj kk + &Pi ~

Table CU.

Integral I for the Covariance Factor of Line Average (Eqn.4.25)


ACF J = Je Ia + r’t? PiP + (F\ + )dr{ dr2

Type I (i.) (3 = 0

e~2a(F-E)- (C-A) + (D-B)(F + E)/2

(ii.) /3 ^ 0; B — 0 and D ^ 0

|I^re-2 S(C+DF) _e-20(C+DE) + 2/s De~2l3A (F - E)

(iii.) /? ^ 0; B ^ 0 and D ^ 0

4 /?2,-2 0C (e-2(3D F _ e-2f3D Ej _ e~2/?~4 ^-20B F _ e-2pB Ej

K — JL + JL a ~ 6, +

Type II (i.) B ^ 0 and D ^ 0

2 2Le-»(7-VJ 2 E(-l)<+d(A)- + a±^).erf(Vir«fty + ^)+2 ^/a

i = l J= 1

1 +7Ts/ah,

(ii.) i5 = 0 and D^O

+ £±gC) . + >$■)+

- A, •erf(v/ia/l +3 — 1

i___c-(N/?raf2/+:^f)'7t sfaD

n - al j_ if. - E£ + II - El 4- if.a “ 62 + 62 p - b* ^ bl I — + 61

gx — A g2 = C hi = B ho — D A \ — E A 2 = F$ij — 9i + hi • Ay

Table C.5.

Integral J for the Covariance Factor of Line Average (Eqn.4.27)


ACF J=r /e Ia+Bt? p(p + (Fi T + Su )dn dr2

Type III (i.) (3^0, B ^ 0 and D ^ 0

i= i j — iI /*»£ j_ , f'JJ_ Jl___ I 3 qs

2 ’r /? ‘ ' (3 ' bxV

(ii.) (3 0, B = 0 and

‘-*a

,-APA

S i "2/ 3qrs^2; I'Zj i ^2; _*__,2 0 * by ^ 0 ’ bx ' 0^SX6~

+ AJ ' hf" + + WbTTy

e"4aii.(-i)i+y{< i+^,(i+$)^+(iii.) f3 = 0, B ^ 0 and D/0

2 2E Ii = 1 j = 1

u + £)-£ + (‘ + £)-£

(iv.) f3 = 0, B = 0 and D 0

1 ^ *y

C7 ■ 4 q 4 s <~»>6/i, 6* 5y I2hx

e~4a E(-ip({1 + g)(i + £)^+J = 1

—A

O + fHf + O + fr) 134p 4r

<5y

£y ' 6*4p \ 4s 6 J ‘ 6y

c3.

4r 6y> 6X

4 s *2/ ]*y 12D f

4q~bx

.41 + 2 '

a = t + i

li.ii. a!£y 3

^=*+tg{ — A g-2 — C hi = B h2 — D A \ = E X2 = F

Vij = 1 + If + If * fry *fry - i + fr + I; * fry fry - ft + ^ • h

Table C.5. (cont.)


ACF J = Se Ia+Bti P(P + + STijdTi dTo

Type IV (i.) v ^ 0, B ^ 0 and D/0o o

e - ( e.+ i + ir^£ E(-D,+J+Ii = 1 j = 1 hi(7 —1'~) cos(a+7C>y )+2~jiy sin(a + 7f.'> ) . sin(/?+i'f,-y)

(72 + *'2)a ‘ + 2 i/ *

(ii.) v i=- 0, B = 0 and D 0

2 f —ur~‘. r / .2 ,2

5"JS(_1)y+,{1=^ (7* — i/-) cos(g + 7C2j )+27t/ sin(o + 7<~2>) , sin(-?+i,f2;)(7“ -f- i/2 J 2 l o,, 2

+e-vD 7 sin (a+ 7/? ) — i' cosfa-f vB ) , sin(/3-fi'£?) — cos( f3+i.'B ) (72 + t/2) 21'

2i'~

(iii.) i/ = 0, B ^ 0 and D ^ 0

tEE(-')’w -cosfg1=1J=1 L

2 2 cos(a+7f,y) C,

(iv.) i/ = 0, Z? = 0 and D yt 0

!r||(-i))>1{c”y - cos/? ■ § + p-y+^? - cos/3 ■ y|] ■ |

a — JL _ JL /? — + JL ^ — JL _ JL -y = JL + 2La - 6, 5V P ~ Sx ^ 7 5, ^ ty

gi = A g2 — C hi — B ho — D A \ — E \o — F

$ij — 9i d~ hi * Ay

Table C.5. (cont.)

APPENDIX D

SAMPLING VARIANCE OF VARIANCE PLOT

This appendix shows the calculation of variance of in Section 5.2.3.2. With

out loss of generality, it is assumed that E{Kt} = 0. It is also assumed that

follows a joint Gaussian distribution. The soil property at t — (see Fig.5.1) is

denoted simply as Kt and p(tt) by p(i). We have

Nn

N,

Nr

D*"1 = 1

Nn

i = li Nn n

Ki+j-1 - Nn n E E *+>■n " 7~r 77Ti=ij=lN n n n

N„n2 EEE^-1n i=i j = i j'=i

i iVn ATn n n

S S ’ **'+>'-1Ar2;l2.................n i= 1 i' = 1 j — 1 j ' = 1

(D.l)

By definition

rar{s“} = £{(s“)2} - |E{s“}|“ (D.2)

The second term on the right hand side of Eqn.D.2 can be obtained using Equations

D-l

D. SAMPLING VARIANCE OF VARIANCE PLOT D-2

5.47 and 5.48. Therefore, it suffices to discuss the first term in the equation.

£{(4)2}Nn n n

1

Ki+j— 1 Ki+j' — 1

Nn Nn n nELEE Ki+y-,K,■<+;<-1\ ro o—,i=i t'=i j = i j'= i

ATn Nn n n n n_£{ A'2n4 J2YU212J2Y, Ki+j-lKi+j'-lKi'+p-lKi'+p'-ln 1= 1 j' = 1 J = 1 j' = 1 p= 1 p' = 1

cy N n N n Nn Tl Tl Tl Tl

~ JpZA zL ]C Ki+J-l^+j'-l^ + g-lKy + g'-ln i= 1 p= 1 p' = 1 _/ = 1 j ' = 1 g= 1 q'= 1

1 Nn Nn Nn Nn n n n n

+ A'l„4 V + , <-ln t—1 t' —1 p=l p'=l J = 1 j'=l gr=l “1

(D.3)

Denote the first, second and third term of the last expression by Ai, A2 and

A3 respectively. Here, a standard result is used for jointly standardized Gaussian

variates Xt (Isserlis, 1915&1918).

£{-Y, A2.Y3.vn = £T{.Y,A2} • E{AVY,} + E{.V,.Y,} • £{.Y2.Y4}

+ E{A’, A’-i} ■ E{ A'o A'3}(D,l)

Using the above result, the following expressions for Ai, An and A3 can be ob

tained.

^1 A

Arn Nn n n n n

p(\i - i' + j- p|) • p(\i - i' + j' - p'\) +

p(\i - % +j - p'\) • p(\i - i' + j' - ;;|)|

P(\j ~ j'\) ■ P(\P~P'\) +

D. SAMPLING VARIANCE OF VARIANCE PLOT D-3

a n4

4 n —1

- M) ■ p(r)

r4 Nn-1

2p(\i - i' + j - p|) • p(\i -i* + j ~p

2[ E i“Lr = ^-l)

O^j-4 A rV2-? 0v„-|rl) ]T («- |s|) •P(lr + «l) (D.5)i r=-(AC,-l) Ls = -(n-l) J

Similarly

A^> — y y y (Nn -M)(n-M)(n-i(i)-/j(s) •/>(»•+o' r=-(N^-l) s = —(n —1) t = -(n —1)

4(7 4 Nn _Nn n n

+ wDLEE'l|i+J-p-,l)t = 1 P=1 J = 1 <7— 1

(D.6)

A** — iVirc4 (An - M)(»- M) -p(V + *1)Lr= — (Nn — 1) s = —(n —1)

+9 4 A'n Nn n n

wEEEE'i"n n Li=l p = l j = l g=l+y-p-?i) (D.7)

APPENDIX E

TECHNIQUES OF

RATIONAL APPROXIMATION

The Technique of rational Approximation has wide applications in engineer

ing analysis which are discussed in some detail in Li and White (1987d). This

Appendix only outlines the relevant techniques which have been used in solving

some of the problems in this work.

E.l SOLVING NON-LINEAR EQUATIONS

Given a non-linear function g(x). The problem arises of solving the equation

g{*) = o (E.i)

It is assumed that the equation has a single root at x = a. In engineering analysis,

it is common that g(x) is only expressed implicitly in terms of x or that the function

is so complicated that it does not lend itself to solution using techniques which

require the knowledge of the derivatives such as the Newton-Raphson's method.

Initially, the values gt = g(xt). i — l.m. are calculated for in different values

of x. The inverse function of g{x), designated as i](g), can be approximated by

the following rational polynomial expressed in the form of a continued fraction.

E-l

E.l. SOLVING NON-LINEAR EQUATIONS E-2

x - T](g) & ai 4-

d 2 +

g- gi

g-g2(E-2)

a 3 +. g gm—i

’ • H--------------dm

The coefficient at in Eqn.E.2 can be computed using the procedure described in

Table E.l.

d\ dn d3 a4

d\ = XX

d'2 1 — Xo a — Jfc--------Q±_doX - ax

d3l = X3 n — 17.3 - 1713- a3l - ax

_ 173 - g-23 a32 ~ a2

a41 ~ X\ n — 174 — g\4“ a41 -

- _ 174 ~ 172043 a42 - «2

„ _ 174 — 17.34 a43 - a3

Table E.l Coefficients of Rational Polynomial

Since a = r;(0), an approximate solution of the root can be obtained by

substituting g — 0 into Eqn.E.2 giving

9ict ~ -r m + i — ai

a o —172

(E.3)

gm-ldm

d 3 -

E.2. OPTIMIZATION E-3

The value of g(x) corresponding to the approximate solution xm+i is then

computed. With this new data point, a better approximation to T](g) can be

obtained by lengthening the continued fraction to (m + 1) terms. The above

procedure is then repeated until the required tolerance for g(x) is achieved. It

should be noted that the addition of an extra data point to the continued fraction

does not require any re-ordering of the continued fraction or re-calculation of

the previous coefficients. Only the coefficient am+l needs to be evaluated. The

iteration procedure of Eqn.E.3 has an order close to 2 (Li and White, 1987d) and

hence it has efficiency comparable to Newton-Raphson’s method.

E.2 OPTIMIZATION

E.2.1 Univariate function

Only the case of minimization will be considered here. The procedure for

maximization of functions is entirely similar.

E.2.1.1 Theory

Given a unimodal function g(x) with a unique minimum value gp at the point

xp. The basic approach for searching for the position of xp may be divided into

two steps:

(1) locating the interval within which the optimum position xp lies;

(2) searching for the optimum position within the interval.

The first step can be done using the forward-backward method. Suppose that

Xo is the initial estimate of the optimum point xp and h (h is positive) is the initial

step size. It is required to find an interval [ xi,xu ] such that xp E [ xi,xu ]. The

procedure is as follows:

Forward calculation:- Firstly, the value of g(x0) and g(x0 -f h) are computed. If

(7(2:0) > y(-To T-/i), the optimum position should be on the right of x0. The step size


is then doubled and the value g(x0 +3h) is calculated. If g(x0 +h) < g(x0 +3/i), it

is obvious that xp is bounded by Xq and Xq +3h. The interval [ x*, xu ] is therefore

[ 2r0, xQ +3h ]. If <7(2:0) > g(x0 +3/i), the search continues in the forward direction,

doubling the step size of each iteration.

Backward calculation:- If <7(2:0) < g(x0 + /i), the value of <7(2:0 — h) is evaluated.

If g(x0 — h) > <7(2:0), the required interval becomes [x0 — h, x0 + h }. Otherwise,

the current step size is doubled and the search continues in the backward direction

until the interval is identified.

The procedure of the forward-backward method is summarized in the flow

chart of Fig.E.l.

initialise x0, hXC - Xqxu = xc + h xi — xc ~ hwhile g[xu) < g(xc) do

h = 2 hXL = Xc

xc — xuXu — xu -{- h

endwwhile g(xt) < g(xc) do

h = 2hXu — X c

X c X[

xi — X[ — hendw

Figure E.lComputer Pseudo-code for Forward-backward Method

(after Li and W hite, 1987c!)


The second step is to locate the optimum position xp within the the interval

[ xi, xu ]. It can be done using Zhou’s (1982) method. The procedure is as follows

(1) Approximation of given function

Given m sets of values of xt and gt = g{xi), xx E [ xi,xu ], i = 1 , m, the

function g(x) can be approximated by means of the following rational polynomial.

/ X , X- Xig(x) « ci +-----------------------------X — X‘2

Co H---------------------------

C3 + X ~ XJn_l•• + ------------

Cm

Eqn.E.4 can also be written in a more compact form as follows.

(EAa)

g(x) & 0!(x);

</>i(x) = ct +X - Xj

(EAb)

&771 (*C) — C7n

The coefficients in Eqn.E.4 can also be calculated using the procedure discussed

in Table E. 1.

(2) Calculation of derivatives

Using Eqn.E.4d, the derivative of the function g(x), denoted by g'(x), can be

approximated by

ff'(x) PS # (x) (E.5)


The derivatives of g(x) at the trial points xt, i — 0, m are given by

g[ = g'(xt) & <t>\(xi)

) ~ (art - xi) • [03(*i)]2

&] + 1 (^t) (•£» j'-(. i (*^t)[0j + l(*t)l2

j < i

( 0$(*») = < 1

if i — m;

otherwise.(E.6)

(3) Approximation of derivative function

As the function gf(ar) is assumed to be unimodal, <7'(a;) is therefore a monotonic

function and hence an inverse function for it exists. Given the m initial trial values

of xt and with the corresponding values of g[ calculated using Eqn.E.6, the inverse

function of g'(x), designated as w(g'), can be approximated by another rational

polynomial:

x = w(g') di + g' - g\

do +g' - g-2

(E.7)

do, T / /g - gm-1

*• + ------ :--------

Again the coefficients in Eqn.E.7 can be computed using a procedure similar to

that described in Table E.l.

(4) Iteration for solution

Since the optimum point xp is given by the solution of the equation g'(:r) = 0,

an approximate solution for xp can be obtained by substituting g' — 0 into Eqn.E.7.


Thus

Xp K, Xm -)_ i (E.8)

(h — g'm-1

d m

The corresponding value of <7m+1 is then evaluated. With these new values of

xm + i and gm +1, it is possible to obtain a better approximation to the function

by lengthening the continued fraction of Eqn.E.4 to ra + 1 terms. The procedure is

then repeated using Equations E.6, E.7 and E.8 until the change of x for successive

iterations is small or the value of g'(x) is sufficiently close to zero.

Four is usually a good starting value for the number m. The current value

of xi, xc and xu obtained from the first step (see Fig.E. 1) can be used as the

first three trial values xt, i — 1,3. The fourth trial value can be obtained using

the quadratic interpolation technique. Since the values of the function g(x) at

xi,xc and xu have already been calculated, a quadratic response curve may be

fitted through these three points. The location of the optimum point xp can be

estimated by the minimum point of the response curve, denoted by z4, and it is

given by

1 (xl - xpgx + (.rg - x\)g2 + (j? - x%)g32 (Xo - x3)gi + (x3 - xi)go + (xx - x2)g3

(E.9)

Since gc is always less than gi and gu, x4 must lie between xi and xu and

therefore it must be bounded by the interval [ xi,xu ]. These four trial values can

be input into Eqn.E.4 to iterate for the minimum point.


H * 4m-0.25

Figure E.2Geometry of Vertical Cut Slope for Example E.l

E.2.1.2 Illustrative Example

Example E.l The factor of safety with respect to height for a vertical cut slope

with straight-line failure surface is given as (Fig.E.2) (Jumikis, 1962)

F =2c't/72c't77

i

cos 9 (sin 9 — cos 9 • tan (})') x2 + H2

xH — x2 tan(E.10)

where c' is the effective cohesion, the effective angle of internal resistance and 7

the unit weight of the soil. Find the minimum factor of safety given that 0' = 30°

and c’ftH — 0.25.

Solution: Table E.2 show the calculations for finding the optimum location and

the minimum value of the factor of safety. Two methods were used, namely, the

method of quadratic interpolation and Zhou’s method. The method of quadratic

interpolation is discussed in Beveridge and Schehter (1970). Although in practice

it is meaningless to evaluate the factor of safety to such a high accuracy, it is for

the sake of illustrating the efficiency of Zhou’s method. It can be seen from Table

E.2 that the convergence for Zhou’s method is extremely fast. Only eight data


points are enough to achieve a tolerance of less than 10~9. If the method of ‘golden

section' (see Beveridge and Schehter, 1970) were used, at least 45 steps would have

been required to achieve the same accuracy. The method has also been applied

successfully to searching for the critical slip surfaces in slip stability problems (Li

and White, 1987a).

E.2.2 Multivariate functions

Numerous numerical techniques for optimizing multivariate functions are avail

able in the literature. Only two methods which have been used in the present study

will be mentioned here. The first method is the technique of alternating variables

and the other is the method of steepest descent. These two methods are discussed

in many books on optimization such as Beveridge and Schehter (1970) and Jacoby

et a/ (1972) to which readers may refer for details.

E.2.2.1 Technique of alternating \~ariables

The use of the technique of alternating variables involves the following steps.

Given a n-dimensional function H(X) and an initial estimate of the optimal point.

Each variable is chosen in turn and all the other n — 1 variables are held constant.

The optimal point with respect to the chosen variable is obtained. The variable is

then fixed at this ‘conditioned’ optimum point and the procedure is repeated for

the next variable. Once all the variables have been searched, the process will be

started again from the first variable. The procedure is repeated sequentially until

the required accuracy is achieved.

The procedure is best illustrated by means of a two-dimensional function as

shown in Fig.E3. Starting at the initial point A, the function is searched along

the AVaxis until the ‘conditioned’ optimum point is reached at point B. Keeping

the value of X\ at this optimum value, the function is then searched along the

Xo direction until again it reaches the second ‘conditioned’ optimum point C. By

repeating the above process, the search will gradually advance towards the global


Zhou’s method Quadratic interpolation

i •O F F

1 1.0* 2.48345600 1.0* 2.48345600

2 2.0* 1.75728427 2.5* 1.74057641

3 3.0* 1.83719577 4.0* 2.36602540

4 4.0* 2.36602540 2.564365348 1.74714334

5 2.342749098 1.73231977 2.398572062 1.73395225

6 2.309974309 1.73205089 2.367814767 1.73287179

7 2.309400883 1.73205081 2.341121033 1.73229423

8 2.309401077 1.73205081 2.329031402 1.73214427

2.320679156 1.73208171

29 2.309401430 1.73205081

Exact 2.309401077 1.73205081 2.309401077 1.73205081

* initial trial valuesTable E.2.

Comparison of Zhou’s Method and the Technique

of Quadratic Interpolation

E.2. OPTIMIZATION E-ll

X2

Xl

Figure E.3 The Technique of Alternating Variables

optimum point.

It can be seen that the technique of alternating variables consists of sequences

of one dimensional search. For each step, the ‘conditioned’ optimum point can

be located using the technique of rational approximation discussed above or other

simpler techniques such as the method of quadratic interpolation.

E.2.2.2 Method of steepest descent

Instead of searching along the variable axes as in the case of alternating-

variable technique, an alternative method is to search along the direction which

has the greatest rate of improvement. For a function of two variables, it would

imply searching along directions of steepest gradient in the contour plot of the

function and hence the name of the method. For n-dimensional functions, the

direction of greatest improvement would be along the gradient vector VH(X) of

the function. Normalizing the gradient vector syH(X) gives the unit vector e for


the direction of search, viz,

V //(V)I V H(X)I

(Ell)

where

VH(X) =dll dll dHdXi ’ dX-2 ’ " ‘ ’ dXn

dH\212

x2

Figure E.4 Method of Steepest Descent


Fig.E4 illustrates the method of steepest descent for a function of two vari

ables. The unit vector ex for the first descent is calculated at the initial point A.

The search is then started along this direction until it reaches the ‘conditioned’

optimum point B. The unit vector e2 of the steepest descent at B is then evaluated

and the search then proceeds along this direction. The procedure is then repeated

until the stopping criterion is satisfied. For explicit functions, the derivatives may

be calculated analytically as in the case of finding the lower and upper bounds of

the performance function in Section 6.4.1. For implicit functions, finite difference

approximation can be used.

For each descent, the locus of the movement X is described by

X — Xi + A • el+1 (E. 12)

where Xi is the location after the z’th descent, el+x is the unit vector of steepest

descent for the (i *f 1) search and A is a location parameter. The parameter

A is searched until a ‘conditioned’ minimum value along the direction el+i is

reached. The resultant location will then form the starting point for the next

search. Therefore, the method of steepest descent can be treated as a series of

one-dimensional search with the objective function treated as a one-dimensional

function of A for each step. Again to find the optimum value of A, the technique

of rational approximation or the method of quadratic interpolation can be used.

APPENDIX F

SOIL DATA

DATA SET 1

The first set of data is extracted from a cone resistance record obtained from a

cone penetrometer test carried out on the southern slope of Mount Cullarin, south

of Goulburn, New South Wales. The data correspond to measurements taken

between a depth of 3m to 4.5m. The soil is mainly a silty clay. No significant

trend is observed for this section of the profile and it can therefore be modelled

as a Type I profile. The measurements are taken at equal intervals of 0.02m and

the total number of data points is 76. The data are listed in Table F.l and also

plotted in Fig.F.l.

DATA SET 2

The second set of data is modified from Table 28 of Agterberg (1974), which

gives the sample ACF of the logarithmically transformed zinc values. The sample

ACF shows a distinct nugget effect which is common for mineral ore. The nugget

effect has been removed and the resulting sample ACF values are listed in Table

F.2. The sampling interval of the zinc values is 0.2m.

DATA SET 3

The third set of data is read from Fig.6 of Yanmarcke (1977c). This corre

sponds to the variance plot of initial void ratio of a soil. The values read from the

figure are given in Table F.3. The sampling interval is 5ft.

F-l

F. SOIL DATA F-2

1.0 30__ i

Qc (MPa)

Figure F.l Cone Resistance - Soil Data Set 1

F. SOIL DATA F-3

Depth Qc Depth Qc Depth Qc

3.00 2.26 3.52 1.70 4.04 2.483.02 2.33 3.54 1.05 4.06 2.283.04 2.41 3.56 1.26 4.08 2.053.06 2.50 3.58 2.05 4.10 2.023.08 2.39 3.60 1.93 4.12 1.983.10 2.50 3.62 1.81 4.14 1.983.12 2.21 3.64 1.77 4.16 1.933.14 2.40 3.66 1.71 4.18 1.943.16 2.02 . 3.68 1.69 4.20 1.923.18 1.88 3.70 1.63 4.22 1.893.20 1.83 3.72 1.60 4.24 1.963.22 1.91 3.74 1.60 4.26 1.993.24 2.28 3.76 1.53 4.28 1.983.26 2.19 3.78 1.46 4.30 1.973.28 2.68 3.80 1.40 4.32 1.993.30 2.62 3.82 1.42 4.34 1.953.32 2.45 3.84 1.39 4.36 1.843.34 2.25 3.86 1.44 4.38 1.843.36 2.34 3.88 1.55 4.40 1.843.38 2.61 3.90 1.55 4.42 1.853.40 2.89 3.92 1.46 4.44 1.863.42 2.84 3.94 1.39 4.46 1.893.44 2.26 3.96 1.58 4.48 1.923.46 2.14 3.98 1.81 4.50 1.913.48 1.91 4.00 1.703.50 1.72 4.02 1.81

Table F.l. Cone Resistance (qc) in MPa - Data Set 1.

F. SOIL DATA F-4

h P(h) In p(h)

1 0.916 -0.087

2 0.508 -0.677

3 0.465 -0.766

4 0.345 -1.064

5 0.447 -0.805

6 0.349 -1.052

7 0.173 -1.754

8 0.254 -1.370

9 0.179 -1.720

Table F.2. Modified Sample ACT of Zinc Values.

F. SOIL DATA F-5

h 0/0

»n/« T

1 1.0

2 7.6

3 6.1

4 5.0

5 4.2

6 3.6

7 2.7

8 2.7

Table F.3. Variance Plot of Initial Void Ratio (Vanmarcke, 1977c).

PROBABILISTIC APPROACHES TO SLOPE DESIGN

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