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AN EXAMINATION OF SEARCH ROUTINESUSED IN SLOPE STABILITY ANALYSES

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An examination of search routines used in slope stability analyses

Gillett, Susan Gille, M.S.

The University of Arizona, 1987

U-M-I 300 N. Zeeb RtL Ann Aibor, MI 48106

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AN EXAMINATION OF SEARCH ROUTINES

USED IN SLOPE STABILITY ANALYSES

by

Susan Gille Gillett

A Thesis Submitted to the Faculty of the

DEPARTMENT OF CIVIL ENGINEERING AND ENGINEERING HECHANICS

In Partial Fulfillment of the Requirements For the Degree of

MASTER OF SCIENCE WITH A MAJOR IN CIVIL ENGINEERING

In the Graduate College

THE UNIVERSITY OF ARIZONA

19 8 7

STATEMENT BY AUTHOR

This thesis has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.

Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

SIGNED:

APPROVAL BY THESIS DIRECTOR

This thesis has been approved on the date shown below:

Qm *7- \JzA ̂Dr. J. S. DeNatale

Assistant Professor of Civil Engineering and Engineering Mechanics

Date

ACKNOWLEDGMENT

The author would like to express her gratitude to

Dr. Jay S. DeNatale, who served as the advisor for this research

project. His guidance, encouragement, and friendship throughout the

preparation of this thesis are deeply appreciated. Dr. Edward A.

Nowatzki and Dr. Panos D. Kiousis also deserve thanks for their many

helpful comments and suggestions.

Very special thanks go to my husband, Paul, and our sons, Brian

and John. This work would not have been possible without their love,

patience and understanding. The support and encouragement of my

parents and brothers are also appreciated.

iii

TABLE OF CONTENTS

Page

LIST OF TABLES • v

LIST OF ILLUSTRATIONS vi

ABSTRACT viii

1. INTRODUCTION 1

2. LITERATURE RE7IEW 4

2.1 Existing Search Strategies ... 4 2.2 Available Optimization Techniques 18

3. MATERIALS AND METHODS 21

3.1 Modelling Slope Stability Problems for Optimization ..... ............ 21

3.2 Example Problems 24 3.3 The STABR Slope Stability Program 25 3.4 Criteria for Evaluating Relative Efficiency

of Search Strategies ..... 26

4. PRESENTATION AND DISCUSSION OF RESULTS 27

4.1 Steep Slope in Homogeneous Soil ............ 27 4.2 Mild Slope in Homogeneous Soil ............ 37 4.3 Slope in Stratified Soil 40

4.4 Birch Dam 52 4.5 Slope in Soil with Continuous

Variation of Strength 57 4.6 Stepped Slope in Homogeneous Soil 66 4.7 Case Histories 70 4.8 Comparison of Grid and Pattern Search Results ..... 70

5. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS 72

5.1 Summary ....... 72 5.2 Conclusions 72 5.3 Recommendations for Further Study ........... 73

APPENDIX A: SINE CASE HISTORIES 74

APPENDIX B: STABR DATA FILES 83

REFERENCES 100

iv

LIST OF TABLES

Table Page

1 A Comparison of Available Stability Programs 5

2 A Comparison of the Various Classes of Optimization Techniques (After DeNatale* 1983) .... 20

3 Pattern Search Results for Example #1 ........... 35

4A Effect of Step Length Logic on Efficiency for Example #1 . . 36

4B Effect of Expertise on Efficiency for Example #1 36

5 Grid Search Results for Example #1............. 38




8 Grid Search Results for Example #2............. 44

9 Pattern Search Results for Example #3 50



11 Pattern Search Results for Example #4 55


12B Effect of Expertise on Efficiency for Example #4...... 56

13 Grid Search Results for Example #5. 62




16 Comparison of Grid and Pattern Search Results ....... 71

v

LIST OF ILLUSTRATIONS

Figure Page

1 Typical Grids for Grid Searches ...... 9

2 A Typical Three-Gr?.d Analysis ............... 10

3 Example of a Slope With Two Local Minimum Safety Factors (After Duncan and Buchignani, 1975) 12

4 The Pattern Search Method Used in STABR (After Lefebvre, 1971) 13

5 A Typical STABR Search Path ................ 15

6 Alternating Variable Search Method

(After Swann, 1972) 17

7 Slope Stability Models .................. 22

8 Example #1 - Steep Slope 28

9 Safety Factor Contour Plot for Example #1......... 29

10 Grid Sizes and Sequences Used in the Grid Searches .... 31

11 Initial Grids and Starting Points for Example #1 33

12 Example #2 - Mild Slope ............. 39

13 Initial Grids and Starting Points for Example #2 ..... 41

14 Example #3 - Stratified Soil Deposit ........... 46

15 Critical Circles for Three Different Tangent Elevations . . 47

16 Variation in Safety Factor With Depth for Example #3 . . . 48


18 Example #4 - Birch Dam 53


vl

vii

LIST OF ILLUSTRATIONS—Continued

Figure Page

20 Example #5 - Shear Strength Which Increases With Depth . . 58

21 Critical Circles for Various Tangent Elevations 59

22 Variation in Safety Factor Hith Depth for Example #5 . . . 61


24 Example #6 - Benched Slope Face .............. 67

25 Safety Factor Contours for Circles Tangent to Firm Stratum for Example #6 68

26 Safety Factor Contours for Circles Passing Through Upper Toe for Example #6 ............... 69

27 Congress Street Open Cut 76

28 Brightlingsea Slide 77

29 Seven Sisters' Slide 78

30 Northolt Slide ...................... 79

31 Selset Landslide ..................... 80

32 Green Creek Slide ..................... 81

33 Bishop and Morgenstern*s Example 81

34 Spencer's Example Problem 82

35 Morgenstern and Price's Example Problem .......... 82

ABSTRACT

Slope stability analyses are commonly performed using limit

equilibrium solutions, in which the safety factor of an assumed failure

surface is calculated. Many computer programs are available which

perform safety factor calculations and search for the critical, or most

probable failure surface. The searches are always performed using

"direct search" techniques, which are the simplest but least efficient

optimization methods. A variety of more advanced optimization

strategies are available, including conjugate direction, conjugate

gradient, restricted-step, and quasi-Newton methods. In the future,

some of these more advanced algorithms will be incorporated into

existing slope stability programs, which will greatly increase the

speed with which the search converges to the critical slip surface.

The relative efficiency and reliability of these new search strategies

must be established by comparative testing on a wide variety of slope

problems.

This paper presents a set of problems that will serve as a

basis for future comparative testing of different optimization

procedures. These problems span the range of slope problems

encountered in geotechnical engineering practice. Baseline measures of

efficiency are obtained using an existing slope stability program with

grid search and pattern search capabilities.

viii

CHAPTER 1

INTRODUCTION

The analysis of earth slope stability is one of the primary

concerns of the geotechnical engineer. Slope stability must be

analyzed when designing a wide range of projects including earth dams,

highway and railway embankments, cut slopes, and excavations.

The stability of an earth slope is its safety against failure

or movement. A slope stability analysis seeks to determine the most

probable failure surface and its degree of stability, or safety factor.

These are referred to as the critical slip surface and minimum factor

of safety. There are five available classes of analytical procedures:

(1) Empirical methods based on field slope charts,

(2) Variational methods,

(3) Limit analysis methods,

(4) Finite element methods, and

(5) Limit equilibrium methods.

The empirical procedures predict stability using historical records of

slope failures in a particular geographical area. The last four

classes are analytical procedures which are used when a more precise

solution is desired. In practice, limit equilibrium methods are the

most commonly used.

1

2

In a limit equilibrium stability analysis, a potential failure

surface is assumed and its safety factor is calculated using one of

many available solutions. This calculation is repeated for a number of

potential failure surfaces until the minimum safety factor is found.

'Limit equilibrium solutions have been developed by Bishop, Fellinius,

Janbu, Horgenstern and Price, Lowe and Karafiath, Spencer, Taylor, the

U. S. Army Corps of Engineers, and others. Detailed examinations and

comparisons of the different methods have been made by Siegel (1975a),

Fredlund and Krahn (1977), Duncan and Wright (1980), and Fredlund,

Krahn, and Pufahl (1981).

Many computer programs have been developed to perform the

necessary safety factor calculations and to search for the critical

slip surface. The searches are always performed using "direct search"

techniques, which are the simplest but least efficient optimization

methods. In the future a variety of advanced optimization strategies

(including conjugate direction, conjugate gradient, restricted-step,

and quasi-Newton algorithms) will be incorporated into existing

programs. These algorithms will greatly increase the speed with which

the search converges to the critical slip surface. However, the

relative efficiency and reliability of these new search strategies must

be established by comparative testing on a wide variety of slope

problems. It should be noted that the theoretical accuracy of a

particular safety factor formulation (such as Bishop's or Spencer's) in

no way depends on the strategy that is implemented to guide the search

for the minimum safety factor and critical slip surface.

3

The purpose of this research is thus to develop a set of

problems to serve as a basis for the future comparative testing of

different optimization procedures. These "test functions" span the

range of slope problems encountered in geotechnical engineering

practice. Baseline measures of efficiency are obtained using an

existing slope stability program with both grid search and pattern

search capabilities.

CHAPTER 2

LITERATURE REVIEW

2.1 Existing Search Strategies. Since the advent of the

computer, geotechnical engineers have come to rely on computer codes to

analyze slope stability problems. Table 1 presents a chronological

list of many of the programs that have been developed based on limit

equilibrium solutions. Commercially available codes as well as limited

access and public domain codes written by researchers at universities,

engineering consulting firms, and government agencies are in use today.

As may be seen in Table 1, the primary difference between the

various slope stability programs lies in the method used to calculate

the factor of safety. Many of the programs use Bishop's Modified

Method, which is a highly regarded method for analyzing circular

failure surfaces. Several of the programs can be used for the analysis

of noncircular slip surfaces.

The vast majority of existing slope stability programs include

some type of routine which directs the search for the minimum safety

factor. The most commonly used search method in existing slope

stability programs is the grid search. In a grid search, a coarse

network of pointB corresponding to slip circle centers is set up over

an area large enough to include the centers of all probable failure

surfaces. It is common to initially use a relatively large and coarse

grid to get a general idea of the critical circle center location, and

4

5

TABLE 1. A COMPARISON OF AVAILABLE STABILITY PROGRAMS

Program Safety Factor Search

Name References Formulation Technique

Little & Price (1958)

Horn (1960)

ICES- Bailey ft Christian (1969)

LEASE-1 Newman (1985)

STABR Lefebvre (1971)

MALE Schiffman (1972)

Schlffnan & Jubenville (1975)

SSTAB1 Wright (1974)

SSTAB2 Chugh (1981)

SLOPE Fredlund (1974)

Fredlund & Krahn (1977)

CIVILSOFT (1976)

SLOPE-II Fredlund & Nelson (1985)

Geo-Slope (1987)

PC-SLOPE Fredlund & Nelson (1985)

Geo-Slope (1987)

STABL Siegel (1975a)

Siegel (1975b)

Siegel et al (1979)

STABL2 Boutrup (1977)

Boutrup et al (1979)

Bishop's Modified

Method

Swedish Circle

Method

Bishop's Modified

Method

Bishop's Modified

Method and Ordinary

Method of Slices

None

Pattern Search

Grid Search

Pattern Search

Morgenstern1s Method Grid Search

Spencer's Method

Spencer's Method

All State-of-the-Art

Methods

Swedish Circle Method


Methods


Methods

Bishop's Modified

Method and Janbu's

Simplified Method

Bishop's Modified

Method and Janbu's

Simplified Method

Grid Search

Grid Search

Grid Search

Grid Search

Grid Search

Randomly

Generated

Grid Search

Randomly

Generated

Grid Search

6

TABLE 1. A COMPARISON OF AVAILABLE STABILITY PROGRAMS (Continued)

Program

Name References

STABL3 Chen (1981)

STABL4 Lovell et al (1984)

PCSTABL4 Carpenter (1985)

SSDP Baker (1980)

Celestino & Duncan (1981)

Safety Factor

Formulation

Bishop's Modified

Method and Janbu's

Simplified Method

Bishop's Modified

Method and Janbu1s

Simplified Method

Bishop's Modified

Method and Janbu's

Simplified Method

Spencer's Method

Spencer's Method

Search

Technique

Randomly

Generated

Grid Search

Randomly

Generated

Grid Search

Randomly

Generated

Grid Search

Dynamic

Programming

Alternating

Variable

REAME

SWASE

Huang (1981)

Huang (1983)

Huang (1983)

Cross (1982)

Bishop's Modified

Method

Sliding Block

Bishop's Modified

Method

Grid and

Pattern Search

None

None

Nguyen (1985) Bishop's Modified

Method & Morgenstern-

Price Method

Simplex Method

of

Spendley et al

SB-SLOPE Von Gunten (1985) Bishop's Modified

Method

Grid Search

STABRG GEOSOFT (1986) Bishop's Modified

Method and Ordinary

Method of Slices

Pattern Search

7

TABLE 1. A COMPARISON OF AVAILABLE

Prograa

Naae References

SL0P8RG GEOSOPT (1986)

STABILITY PROGRAMS (Continued)

Safety Factor Search

Formulation Technique

Spencer18 Method

Handles Noncircular

Surfaces

GEOSLOPE GEOCOMP (1986) Based on STABL

8

then to construct smaller grids with finer spacings to pinpoint the

exact location of the failure surface and the precise value of the

minimum safety factor. The radii of the circles are determined from

the condition that all circles must pass tangent to a particular user-

specified elevation or through a particular user-specified point.

Figure 1 shows typical preliminary grids for slopes whose

critical slip surfaces are toe circles and base circles. Once the

computer program has evaluated the safety factors corresponding to each

point in the grid, a second smaller and finer grid is constructed

around the point with the smallest safety factor. Analyses are then

performed for circles centered at each of these new grid locations.

Sometimes, for accuracy, it may be necessary to repeat the analyses a

third time using a still smaller and finer grid. A typical three-grid

analysis is shown in Figure 2. For a slope in a layered system or in

soil whose shear strength increases with depth, the critical slip

circle will be tangent to an unknown depth. In these cases, it is

necessary to examine grids corresponding to various trial depths of

tangency.

In a grid search, the grid is either set up by the user or

generated by the computer. The STABL series of programs, developed at

Purdue University, usea. random, computer-generated grids. In this

technique, described by Boutrup et al. (1979), trial piecewise-linear

failure surfaces are generated from a series of starting points with

equal horizontal spacing along the ground surface at the base of the

slope. The failure surface is defined by a series of line segments

9

grid spacing

slip circle centers <

5a» potential failure surfaces

(a) Toe Circles

(b) Base Circles

Figure 1. Typical Grids for Grid Searches

10

Lowest value in grid

Second grid

(a) Initial (coarse) Grid

Lowest value in grid

Final grid

(b) Second (finer) Grid

True minimum

(c) Final Grid

Figure 2. A Typical Three-Grid Analysis

11

beginning at these starting points. The orientations of these line

segments are defined as functions of direction limits and a random

number function. The user specifies the number of trial surfaces

desired, and the computer plots the ten surfaces having the lowest

safety factors.

The grid search method is useful in determining safety factor

values for many specified circle centers. With this information, it

becomes possible to plot contours of equal safety factor values. This

information may be helpful when analyzing problems which have more than

one local minimum safety factor. Such a situation is depicted in

Figure 3. The major disadvantage of the grid search procedure is its

inherent slowness. The trial evaluation points are always preselected

by the analyst, and thus the efficiency of the technique depends solely

on the expertise and insight of the user. It may take several trials

to identify the true tangent depth of the critical circle. Once this

has been established, further grid refinements may be required to

ensure that the true minimum safety factor has been determined.

A second common search method is the pattern search, which is

included in the STABR (Lefebvre, 1971) and REAME (Huang, 1981 and 1983)

programs. It is similar to the grid search in that a fixed circle

center spacing or "step length" is predetermined by the user. In the

STABR program, the coordinates of the center of the first circle to be

analyzed are specified on input. The safety factor is calculated for

this center and for centers spaced symmetrically around it, as shown in

Figure 4a. The centers are generated in the order shown by rotating

12

2.0

1.8 F= 1.75

Figure 3. Example of a Slope With Two Local Minimum Safety Factors (after Duncan and Buchignani, 1975)

4

1

2

(a) First clockwise rotation around the given center, A. The rotation starts at point 1, with radius of rotation twice the final step length.

x -*

(b) The 45-degree clockwise rotation around the center, B. The rotation starts at point 1, with radius of rotation 1.414 times the final step length.

Figure 4. The Pattern Search Method Used in STABR (after Lefebvre, 1971)

14

around the specified center with a radius of rotation equal to twice

the specified final step length. If a safety factor less than that at

the center of rotation is found for any point, that point becomes the

new center of rotation. If a full rotation is completed and no safety

factor smaller than that at the center of rotation is found, the length

of the search circle radius is reduced to the final step length and a

second 4-point rotation is performed. If a smaller safety factor is

still not found, another rotation around the same center is initiated,

starting at a 45-degree angle with a search circle radius length of

1.414 times the final step length, as shown in Figure 4b. After a

tentative minimum safety factor is found, it may be necessary to repeat

the analysis using a smaller step length in order to further pinpoint

the center of the true critical circle. It is often necessary to

repeat this process for several different tangent elevations, since the

depth to which the critical circle is tangent is unknown for many

problems.

Figure 5 shows a typical path taken by the STABR pattern search

routine. The search path is superimposed on a safety factor contour

plot. Each point represents—a- safety factor evaluation for a

particular circle center. In this example, 25 trials were

automatically performed to reach the actual minimum, then seven more

evaluations to verify that the minimum had been found. As Figure 5

shows, several centers were evaluated twice during the search (for

example, Points 11 and 14, Points 17 and 20, etc.), which causes

unnecessary delay and computer expense.

15

safety factor contours

minimum safety factor = point 25

Figure 5. A Typical STABR Search Path

16

The pattern search method represents a major improvement over

the grid search, since the computer does much more of the work in

identifying the critical slip surface. However, the method has several

drawbacks. If the tangent depth for the critical slip surface is

unknown, several trials are needed to locate the actual critical

tangent elevation, just as in the grid search method. As shown in

Figure 5 and discussed above, the path followed by a pattern search can

often be quite inefficient. If the starting point of the search is far

from the actual critical center, many evaluations are needed to find

the minimum safety factor, since the step length does not vary. In

addition, the minimum may not always be identified, as the total number

of evaluations is generally limited by the particular program being

used. Even if a minimum is identified, it may be necessary to repeat

the analysis using a still smaller step length in order to further

pinpoint the center coordinates.

A limited variety of other direct search procedures have also

been used in conjunction with slope stability analyses. Celestino and

Duncan (1981) used the alternating variable method to search for the

critical piecewise linear slip surface. In the alternating variable

method, the search for a minimum safety factor is accomplished by

searching in directions parallel to each coordinate axis and then

changing directions each time a minimum is located. A diagram of a

typical alternating variable search is shown in Figure 6.

Nguyen (1985) recently applied an early version of the simplex

method to slope stability analysis. In this method, a regular simplex

(n + 1 mutually equidistant points, where n defines the number of

17

Figure 6. Alternating Variable Search Method (after Swann, 1972)

18

independent variables) is set up and the safety factor is evaluated at

each vertex. An iteration consists of replacing the vertex associated

with the highest safety factor value with its mirror image about the

centroid of the remaining vertices. More recently, Awad (1986) adapted

the more efficient simplex method of Nelder and Mead (1965) to slope

stability analysis. In the method of Nelder and Mead (1965), the

regularity of the simplex design is abandoned, and the simplex

automatically rescales itself by expanding or contracting according to

the local geometry of the function. The family of simplex methods is

generally regarded as the most efficient of direct search techniques

(Swann, 1972).

This literature review examined all slope stability codes

mentioned in the literature as well as many commercially available

codes. While several direct search optimization methods have

successfully been used, no advanced optimization techniques have ever

been incorporated into slope stability analysis programs. The merits,

limitations, and general requirements of these more advanced classes of

optimization techniques may now be briefly described.

2.2 Available Optimization Techniques. An optimization

routine is designed to search for the minimum value of the function

being examined. Many search algorithms have been developed, and they

can be classified based on the level of information that is required to

direct the search. They can be categorized as follows (DeNatale,

1983):

19

(1) Second derivative (or Newton) methods,

(2) Quasi-Newton methods,

(3) Discrete (or finite-difference) Newton methods,

(4) Restricted step (or trust-region) methods,

(5) Conjugate gradient and conjugate direction methods,

(6) Direct search (or ad hoc) methods, and

(7) Sum of squares methods.

The classes of algorithms have been arranged roughly in order of

decreasing efficiency, with second derivative methods being the most

efficient, and with direct search routines being the slowest and most

basic. The exception is the sum of squares methods, which are

advantageous in certain limited applications. A thorough discussion of

the merits and shortcomings of the various general classes of

optimization techniques is presented by DeNatale (1983), and a summary

of this discussion is given in Table 2.

The safety factor expression in a generalized limit equilibrium

procedure (such as in a Bishop's or Spencer's analysis) is

nondifferentiable. Therefore, gradient and curvature expressions are

not explicitly available. However, finite-difference procedures can be

used to calculate derivatives, and all classes of optimization

algorithms could potentially be used in conjunction with slope

stability analyses. In fact, it is very likely that the more advanced

techniques, such as the quasi-Newton methods, may be the most efficient

overall, despite the additional trials associated with derivative

calculations by finite-differences (DeNatale, 1983).

20

TABLE 2 , A COMPARISON OF THE VARIOUS CLASSES OF OPTIMIZATION TECHNIQUES

Class Requirements Advantages Disadvantages

Second

Derivative

Methods

•function

•gradient

•curvature

•superlinear

convergence

•self-corrective

•possible to

distinguish between

local minima and

saddle points

••ay not converge

from poor initial

guess

•requires second

derivatives

•requires solution

of n-linear equations

at each iteration

Discrete

Newton

Methods

•function

•gradient

•sane as for

Second Derivative

Methods

•inefficient for

large-dimension

problems

•optimal differencing

Intervals must be

determined

Quasi-

Newton

Methods

•function

•gradient

•requires first

derivatives only

•no equation solving

is required

•round-off errors can

have large effect on

performance

Restricted

Step

Methods

•function •excellent

convergence

•requires many arith

metic operations

Conjugate

Direction

and

Conjugate

Gradient

Methods

•function •requires little

core memory

•few arithmetic

operations per

iteration

•excellent for

large problems

•less efficient and

robust than Newton-

type methods

Direct

Search

Methods

•function •extremely general

and simple to code

•immune from

rounding errors and

ill-conditioning

•requires little

core memory

•rather slow

convergence

•function

evaluations increase

exponentially with

the dimension of the

problem

•a large number of user

specified constants

is required

CHAPTER 3

MATERIALS AND METHODS

3.1 Modelling Slope Stability Problems for Optimization.

Before optimization techniques can be used with slope stability

programs it is necessary to characterize slope problems as optimization

problems. This may be done by analyzing the general classes of

stability problems to determine the number of independent variables (or

dimensions) to be searched for each type of problem. General

characterizations are shown in Figure 7.

Figures 7a and 7b show planar slip surfaces. The general case

involves two independent variables, these being the angle of

orientation of the slip surface and the height above the toe at which

this surface intersects the slope face. For the special case of a

plane passing through a fixed point, such as the toe of the slope, the

critical angle of inclination is the only quantity that would need to

be identified.

Figures 7c, 7d, and 7e show circular failure surfaces. In

general, a problem involving circular slip surfaces is a three-

dimensional problem. The three degrees of freedom are the x- and y-

coordinates of the circle center and its radius. For certain cases

circular slip surfaces can be analyzed as two-dimensional problems. In

a purely cohesive (<{> = 0) soil with a slope angle greater than

53 degrees, the critical circle passes through the toe of the slope

21

22

one degree of freedom: inclination of slip surface

(a) Plane Through Toe

two degrees of freedom: slip surface inclination, height above toe of intersection with slope face

(b) Any Planar Slip Surface

two degrees of freedom: x- and y- coordinates of circle center

(c) Circle Through Toe

Figure 7. Slope Stability Models

23

two degrees of freedom: x- and y- coordinates of circle center

(d) Circle Tangent to Firm Stratum

three degrees of freedom: x- and y- coordinates of circle center, radius of

circle

(e) Any Circular Slip Surface

X.

2s degrees of freedom, where s = number of line segments defining slip surface

(f) Piecewise Linear Slip Surface

Figure 7. (continued)

24

(Terzaghi, 1943), as shown in Figure 7c. In a purely cohesive (<f> = 0)

soil with a slope angle less than 53 degrees, the critical circle

passes tangent to the top of the underlying firm stratum (Terzaghi,

1943), as shown in Figure 7d. In both cases the radius of the circle

is fixed once the x-and y-coordinates of the center are selected.

Another type of slope problem is one which has a piecewise-

linear slip surface. The general case is shown in Figure 7f. This

type of problem involves 2s degrees of freedom, where s is the number

of line segments used, and where the orientation and length of each

segment become the independent variables. Composite circular-linear

surfaces are also possibilities.

3.2 Example Problems. In optimization research, analytic

"test functions" are normally used to evaluate the relative merits and

limitations of competing search strategies. A large number of these

functions have been developed, and many are described by Cornwell

et al. (1980). A similarly wide range of problems is developed herein,

with the following purposes in mind: (1) to find a set of problems

which represent typical slope stability problems encountered in

geotechnical engineering practice, and (2) to pinpoint those problems

in which direct search methods present severe limitations or give

inaccurate results. The following types of problems are therefore

considered:

(1) Steep slopes in homogeneous soils,

(2) Mild slopes in homogeneous soils,

(3) Slopes in soils whose shear strength increases with depth,

25

(4) Problems with a stepped slope face,

(5) Slopes in stratified soil deposits, and

(6) Published case histories that have been examined by others.

One aspect of special interest in choosing these example problems was

to identify those conditions that result in multimodality, or the

existence of more than one local minimum safety factor.

3.3 The STABR Slope Stability Program. The STABR slope

stability program developed by Lefebvre (1971) is used in the present

study. It analyzes circular failure surfaces using both Bishop's

Modified Method and the Ordinary Method of Slices. The program can be

used for both total and effective stress analyses. (A total stress

analysis uses the undrained strength parameters c and $ to

establish the short-term stability of the slope. An effective stress

analysis uses the drained strength parameters c* and to estimate

the long-term stability of the slope.) It is capable of handling

irregular slope profiles, tension cracks, soil layers with different

properties and nonuniform thicknesses, complicated pore pressure

patterns, variation of undrained strength with depth, and seismic

forces.

As mentioned previously, a pattern search routine is included

in the STABR program. To employ the search routine the user must

specify either a horizontal line to which all slip circles are tangent

or a specific point through which all circles pass. Additional

information needed by the program includes the coordinates of the

26

center of the initial circle and the final step length desired. To

minimize the number of evaluations, the User'B Manual recommends that a

fairly large step length be used initially, so that the approximate

center of the critical circle can be found with relatively few circle

analyses (Lefebvre, 1971). The program can then be rerun using a finer

step length to pinpoint the critical circle center with more accuracy.

A modified version of the program has also been developed by DeNatale.

In this version the pattern search strategy is replaced with the basic

grid search procedure.

3.4 Criteria for Evaluating Relative Merits of Different

Search Strategies. The test problems may be used to compare two

different search strategies by analyzing the same problem using each of

the two different search techniques and then comparing the results for

accuracy and efficiency. Items which should be included in a

comparison of two methods are (1) the effect of starting point (a

reflection or indication of user expertise), and (2) the effect of grid

spacing (in the grid search methods) or step size (in the pattern

search methods). The effects of these factors should be examined

through a parametric study. Since the number of arithmetic operations

used by any optimization scheme to select a trial evaluation point (or

circle center) is always small relative to the number of operations

needed to compute the safety factor, the number of trial evaluation

points required to locate the minimum safety factor is used as the

basis for evaluating efficiency. It is desirable to obtain a minimum

value using the least number of evaluation points in order to reduce

the total computer time.

CHAPTER 4

PRESENTATION OF RESULTS

4.X Steep Slope in Homogeneous Soil. The first problem

studied is a slope in homogeneous, cohesive soil with a slope angle of

60 degrees, as shown in Figure 8. It is known that the critical circle

will pass through the toe of the slopet since a <t* = 0 shear strength

characterization is being used and the slope is steep (Terzaghi, 1943).

These features cause the problem to reduce to two-dimensional search

for the critical safety factor.

The location of the critical circle center was first identified

by running the STABR program for a one-foot step length. A grid of

circle centers at five-foot intervals was then set up around the

critical center in order to identify the safety factor contours. The

contour plots in Figure 9 show a regular contour pattern with a single

minimum safety factor.

The other factors studied for this problem are the effects of

starting point and grid size (or step length) on the total number of

function evaluations required in both grid and pattern searches. To

define errors in the initial grid size and spacing for grid searches,

four levels of expertise are defined using the following four grid

logics:

27

28

(100,100)

60 (111.55,120)

120 pcf

500 psf

40'

Figure 8. Example #1 - Steep Slope

29

! - 2 1 . 2

20*

Minimum Safety

Factor - 1.093

60

Figure 9. Safety Factor Contour Plot for Example #1

30

Initial Grid Logic

Side Length of Grid Perimeter

Initial

Grid Size Intermediate

Grid Size Final

Grid Size

Expert 30% 6% Rcc 1 ft

Good 60S: Rcc 9% Rcc 1 ft

Fair 130% Rcc 13% Rcc 6% Rgg 1 ft

Poor 200% Rcc 20% Rcc 6% Rcc 1 ft

The grid perimeters and grid sizes are shown in Figure 10. As may be

seen, the initial grid is always centered about the true critical

center. All initial grids extended vertically downwards to the level

of the top of the slope. Rcc is defined as the radius of the critical

circle.

The choices of expert, good, fair, and poor grid logics are

subjective. They are intended to model the various abilities and

degrees of experience of slope stability analysts. Someone with little

or no previous experience in the area of slope stability would be less

likely to choose a small initial grid close to the actual critical

center than a person with a great deal of experience in analyzing

slopes. The better grid logics (expert and good) do not require an

intermediate grid because they are fairly close to the critical center

initially.

To define errors in starting points for pattern searches, four

levels of expertise are defined, based on the distance of the starting

point from the actual critical center:

31

Initial Grid Perimeter Side Length1-1''

Lowest value in grid-

1—*—f

J I

I Initial Grid Size

Intermediate Grid Perimeter Side Length ( = 2 times initial grid size)

' ̂ Intermediate Grid Size

•Lowest value in grid

Final Grid Perimeter Side Length ( = 2 times intermediate grid size)

Final Grid Size (1 ft.)

True minimum

Figure 10. Sizes and Sequences Used in the Grid Searches

32

Type of Estimate

Expert

Good

Initial Distance Error

Upper corners of expert initial grid

Poor

Fair

Upper corners of good initial grid

Upper corners of fair initial grid

Upper corners of poor initial grid

The initial grids and starting points for this problem are shown in

Figure 11. In order to obtain representative measures of efficiency,

two different starting points were examined for each of the four levels

of expertise. The following two step length refinement logics are used

with each level of expertise:

Step Length Refinement Logic A

Trial 1 - Start the search at the predetermined starting point.

Use a 5 ft final step length.

Trial 2 - Start the search at the critical center determined

in first trial. Use a 2 ft final step length.


in second trial. Use a 0.5 ft final step length.

Step Length Refinement Logic B

Trial 1 - Start the search at the predetermined starting point.

Use a 5 ft final step length.


in first trial. Use a 0.5 ft final step length.

These two step length refinement logics are intended to define two

strategies of different effectiveness. The relative superiority of

33

poor initial grid 39,62)

(129,72) fair initial grid-^ (91,72)

(119,82) ,82)

(114,87) (96,87)

(110,91)

(100,100)

(111.55,120)

Figure 11. Initial Grids and Starting Points for Example #1

34

each refinement logic may depend on the closeness of the initial

starting point.

The total number of safety factor evaluations required in any

given pattern search analysis is obtained by adding together the number

of function evaluations performed in each trial of a given step length

logic. The total number of function evaluations required in any

particular grid search analysis is obtained by adding together the

number of function evaluations performed for each grid size (i.e.,

initial, intermediate, final) for each type of estimate.

Table 3 gives the results of the various pattern search

analyses. The average number of function evaluations for each level

of expertise and step length logic is summarized in Table 4A. Table 4A

shows that the number of function evaluations required to identify the

critical center decreases as the starting error decreases. An

examination of Table 4A also shows that the expert and good starting

points require about the same number of function evaluations. However,

once the starting error increases past the good level of expertise,

efficiency decreases rapidly, especially for Step Length Refinement

Logic B.

The relative efficiency of the two step length refinement

logics depends on the distance of the starting point from the actual

critical center. This is clearly shown in Table 4A. When the starting

guess is fairly close (expert and good levels of expertise), Step

Length Refinement Logic B is the more efficient logic. When the

starting guess is far from the critical center (fair and poor levels of

35

TABLE 3

PATTESM SEARCH RESULTS FOR EXAMPLE #1

Trial Starting

Point (ft)

Step Length Logic

Total No.

of Function Evaluations

Minimum F.S.

1 (139,62) A 59 1.093

2 (139,62) B 74 1.093

3 (81,62) A 63 1.093

4 (81,62) B 77 1.093

5 (129,72) A 47 1.093

6 (129,72) B 54 1.093

7 (91,72) A 50 1.093

8 (91,72) B 54 1.093

9 (119,82) A 47 1.093

10 (119,82) B 34 1.093

11 (101,82) A 39 1.093

12 (101,82) B 36 1.093

13 (114,87) A 44 1.093

14 (114,87) B 34 1.093

15 (96,87) A 41 1.093

16 (96,87) B 32 1.093

(x,y) Radius of Minimum (ft)

(111,91),29

(111,91),29

(111,91),29

(111,91),29

(111,91),29

(111,91),29

(111,91),29

(111,91),29

(111,91),29

(111,91),29

(111,91),29

(111,91),29

(111,91),29

(111,91),29

(111,91),29

(111,91),29

TABLE 4A

EFFECT OF STEP LEHGTH LOGIC ON EFFICIENCY FOR EXAMPLE fl

Starting Step Length Average Number of Guess Logic Function Evaluations

Poor A 61 B 76

Fair A 49 B 54

Good A 43

B 35

Expert A 43

B 33

TABLE 4B

EFFECT OF EXPERTISE ON EFFICIENCY FOR EXAMPLE #1

Starting Guess

Step Length Logic

Number of Function Evaluations

Poor B only 76

Fair A-B average 52

Good A-B average 39

Expert B only 33

37

*

expertise), Step Length Refinement Logic A is the more efficient logic.

Logic B requires more function evaluations for the poorer starting

points since the intermediate step length refinement is omitted. The

expert, good, fair, and poor analyses could be further quantified by

assuming that .a person who uses an expert starting point would also use

the most efficient step length refinement logic (Logic B), a person who

uses a poor starting point would also use the least efficient step

length refinement logic (Logic B), and that persons who use

intermediate starting points would use intermediate logics (the average

of Logics A and B). With such assumptions, it becomes possible to

develop Table 4B.

Table 5 presents the results of the grid search analyses. Once

again the size of the initial and subsequent grids has a large effect

on the total number of function evaluations. About twice as many

evaluations are required as the level of expertise decreases from

expert to poor. A comparison of Tables 4B and 5 reveal that many more

function evaluations are required by the grid search than by the

pattern search for any given level of expertise.

4.2 Mild Slope in Homogeneous Soil. The second problem

studied is a slope in homogeneous cohesive soil with a slope angle of

30 degrees, underlain by a firm stratum, as shown in Figure 12. It is

known that the critical circle is tangent to the top of the firm layer

(Terzaghi, 1943). Therefore, this problem again reduces to a two-

dimensional search for the minimum safety factor.

38

TABLE 5

GRID SEARCH RESOLTS FOR EXAMPLE #1

Grid Total Number of Mininum (x,y) Radius of Logic Function Evaluations F.S. Critical Circle (ft)

Expert 75 1.093 (111,90),30

Good 98 1.093 (111,90),30

Fair 130 1.093 (111,90),30

Poor 144 1.093 (111,90),30

39

(100,100)

30'

(152,130)

"Jf = 120 pcf s!= o

c = 800 psf 30'

Figure 12. Example #2 - Mild Slope

40

Grid and pattern searches are once again used to identify the

critical circle, and the effect of varying the initial grid size and

starting point is again studied. The previously defined initial gridB,

starting errors, and refinement logics are used. Specific initial

grids and starting points are shown in Figure 13.

The results of the various pattern searches are given in

Table 6, and the average number of function evaluations for each level

of expertise and step length refinement logic is summarized in

Table 7A. The results presented in Tables 7A and 7B reveal that many

more function evaluations are required to find the minimum safety

factor as the starting point is moved farther from the actual critical

center. The magnitude of the starting error has a more pronounced

effect on efficiency in this problem than in the toe circle problem,

Example #1. This is because the larger critical circle radius in this

problem caused the starting points to be farther from the actual

critical center than in Example #1.

Table 8 gives the results of the grid searches. As with

Example #1, more function evaluations are once again needed for a grid

search than for a pattern search. In this problem, the good grid logic

required more safety factor evaluations than any of the other levels of

expertise. This is because an intermediate grid refinement is not

used, and the one-foot final grid required a large number of

evaluations due to the magnitude of the critical circle radius, Rcc.

4.3 Slope in Stratified Soil. Example #3 consists of a slope

in cohesive soil with an intermediate sand layer. The soil profile is

(210.5,-9) poor initial grid

(41.5,-9)

fair initial gridN

(181,20.5) 1 (71,20.5)

goor initial grid, (151,50.5)

grill/

yaoi.50.5)

(138.5,63)(113.5,63)

7^ \ (126,75.5) ̂ -expert initial grid (100,100)

(152,13C


42

TABLE 6

PATTERN SEARCH RESULTS FOR EXAMPLE #2

Trial

Starting Point (ft)

Step Length Logic

Total No. of Function Evaluations

Minimum

F.S.

(x,y) Radius of Critical Circle

(ft)

1 (210.5,-9) A 96 1.289 (126,75.5),84.5

2 (210.5,-9) B 178 1.289 (126,75.5),84.5

3 (41.5,-9) A 80 1.289 (126,75.5),84.5

4 (41.5,-9) B 140 1.289 (126,75.5),84.5

5 (181,20.5) A 73 1.289 (126,75.5),84.5

6 (181,20.5) B 132 1.289 (126,75.5),84.5

7 (71,20.5) A 63 1.289 (126,75.5),84.5

8 (71,20.5) B 108 1.289 (126,75.5),84.5

9 (151,50.5) A 51 1.289 (126,75.5),84.5

10 (151,50.5 B 75 1.289 (126,75.5),84.5

11 (101,50.5) A 47 1.289 (126,75.5),84.5

12 (101,50.5) B 65 1.289 (126,75.5),84.5

13 (138.5,63) A 45 1.289 (126,75.5),84.5

14 (138.5,63) B 47 1.289 (126,75.5),84.5

15 (113.5,63) A 41 1.289 (126,75.5),84.5

16 (113.5,63) B 39 1.289 (126,75.5),84.5

TABLE 7A

EFFECT OF STEP LENGTH LOGIC ON EFFICIENCY FOR EXAMPLE #2


Poor A 61

B 76

Fair A 49 B 54

Good A 43

B 35

Expert A 43

B 33

TABLE 7B


Starting Step Length Number of Guess Logic Function Evaluations

Poor B only 159

Fair A-B average 94

Good A-B average 60

Expert B only 43

44

TABLE 8

GRID SEARCH RESULTS FOR EXAMPLE #2

Grid Logic

Total Number of Function Evaluations

Minimum F.S.

(xfy) Radius of Critical Circle (ft)

Expert 157 1.289 (125.5,74),86

Good 338 1.289 (126,75.5),84.5

Fair 234 1.289 (126,75.55,84.5

Poor 240 1.289 (125.5,74),86

45

shown in Figure 14. In this type of problem, the depth to which the

critical circle is tangent is unknown. However, slope stability theory

predicts that the critical circle will pass tangent to a boundary

between two layers rather than at some point within a layer (Taylor,

1948). The STABR program was run specifying three different depths of

tangency. The results of the three searches are shown in Figure 15.

It can be seen that the minimum safety factor occurred for the slip

circle tangent to the top of the sand layer.

The variation in safety factor was studied by evaluating the

safety factors for a given circle center while varying the depth of

tangency. The variation in safety factor with depth is shown in

Figure 16 for three circle centers along the critical vertical section.

As may be seen, the safety factor decreases as the depth of the circle

is lowered until the minimum value is reached at the top of the sand

layer. The safety factor increases for circles passing through the

sand layer, then decreases again as the tangent depth passes through

the lower cohesive layer. This shows a multimodal type of behavior or

the existence of more than one local minimum safety factor.

The effects of initial grid size, starting point, and refinement

logic are examined by the techniques previously described in

Section 4.1. Figure 17 shows the specific initial grids and starting

points, and Table 9 presents the results of the various pattern

searches. The average number of function evaluations for each level of

starting guess and refinement logic are summarized in Table 10A. As

may be seen from Tables 10A and 10B, more than twice the number of

(100,100)

20'

(135,120)

IS = 120 pcf / - 0 10* c = 500 psf

= 120 pcf, ff = 30°, c = 0 5'

y = 120 pcf / = 0 15' c = 500 psf

Figure 14. Example #3 - Stratified Soil Deposit

min

rain 2.224

= 1.417 raxn

Figure 15. Critical Circles for Three Different Tangent Elevations

*-sJ

48

Safety

Factor

4.0

3.0

2 .0

Sand layer

1 . 0 110

3.0

2 . 0 '

1 . 0

3.0

2.0

1 . 0

Circle Center (115,85)


110 120 130 140 150


110 130 150

Depth (ft.)

Figure 16. Variation in Safety Factor with Depth for Example #3

(161,43) poor initial grid

1 (74,73)

fair initial grid-* (145.5,58.5) \ (89.5,58.5)

good initial grid* (130.5,73.5) ^104.5,73.5)

124,80)(111,80)

—a—».

(117.5,86.5) —expert initial grid

(100,100)

(134.6,120)


50

TABLE 9


Trial Starting

Point (ft)

Step

Length Logic


Minimum

F.S.

(xfy) Radius of Minimum (ft)

1 (161,43) A 69 1.271 (117.5,86.5),43.5

2 (161,43) B 98 1.271 (117.5,86.5),43.5

3 (74,73) A 49 1.271 (117.5,86.5),43.5

4 (74,73) B 50 1.271 (117.5,86.5),43.5

5 (145.5,58.5) A 54 1.271 (117.5,86.5),43.5

6 (145.5,58.5) B 70 1.271 (117.5,86.5),43.5

7 (89.5,58.5) A 51 1.271 (117.5,86.5),43.5

8 (89.5,58.5) B 58 1.271 (117.5,86.5),43.5

9 (130.5,73.5) A 44 1.271 (117.5,86.5),43.5

10 (130.5,73.5) B 49 1.271 (117.5,86.5),43.5

11 (104.5,73.5) A 42 1.271 (117.5,86.5),43.5

12 (104.5,73.5) B 42 1.271 (117.5,86.5),43.5

13 (124,80) A 46 1.271 (117.5,86.5),43.5

14 (124,80) B 35 1.271 (117.5,86.5),43.5

15 (111,80) A 44 1.271 (117.5,86.5),43.5

16 (111,80) B 30 1.271 (117.5,86.5),43.5

TABLE 10A


. Starting Step Length Average Number of Guess Logic Function Evaluations

Poor A 59 B 74

Fair A 53 B 64

Good A 43 B 46

Expert A 45 B 33

TABLE IOB


Starting Guess

Step Length Logic


Poor B only 74

Fair A-B average 59

Good A-B average 44

Expert B only 33

52

function evaluations are required as the level of expertise decreases

from expert to poor. Grid search analyses were not performed.

It should be noted that the particular study described above

assumed that the actual critical tangent elevation was known. Finding

the critical depth is actually the first step in solving for the

minimum safety factor. One could get some idea of the total number of

function evaluations needed in any analysis by multiplying the values

shown in Tables 10A and 10B by the number of tangent elevations

searched (in this case, three).

4.4 Birch Dam. The fourth example studied was Birch Dam, a

case history which has also been examined by Gelestino and Duncan

(1981) and Nguyen (1985). The soil configuration is shown in

Figure 18. Celestino and Duncan found the critical slip surface to be

a base circle tangent to the top of the firm stratum. This was

verified by running STABR searches using a variety of different trial

tangent elevations.

Figure 19 shows the initial grids and starting points for the

various levels of expertise. The pattern searches are presented in

Table 11 and the average number of function evaluations are summarized

in Table 12A. Table 12B presents the effect of expertise on the

efficiency of the pattern searches. In studying these results, it can

be concluded that both starting point and step length have a pronounced

effect on the total number of function evaluations required to find the

minimum safety factor. In all but the expert starting guess in

Table 12A, step length refinement logic B required up to twice as many

53

(100,100)

c = 1023 psf X = 3 1 ° ,

Y = 127 pcf/

c = 1629 psf / = 18° Y = 127 pcf

(245,157) c = 1023 pcf 37'

Y = 127 pcf

Figure 18. Example #4 - Birch Dam

(297,-49) poor initial grid'

(54,-49)

(254.5,-6.5) fair initial grid

) (96.5,-6.5)

good initial grid (212,36) ^ (139,36)

(193.5.54.5)(157.5.! 4.5)

(175.5,72.5) •expert initial grid (100,100)

(245,157)


in

55

TABLE 11


Trial

Starting Point (ft)

Step

Length Logic


Minimum

F.S.

(x,y) Radius of

Critical Circle (ft)

1 (297,-49) A 127 1.096 (175.0 73.5) 120.5

2 (297,-49) B 259 1.096 (175.0 73.5) 120.5

3 (54,-49) A 118 1.096 (174.0 75.5) 118.5

4 (54,-49) B 237 1.097 (175.5 72.5) 121.5

5 (254.5,-6.5) A 93 1.096 (174.5 74.5) 119.5

6 (254.5,-6.5) B 179 1.097 (175.5 72.5) 121.5

7 (96.5,-6.5) A 94 1.097 (175.5 72.5) 121.5

8 (96.5,-6.5) B 172 1.097 (175.5 72.5) 121.5

9 (212,36) A 69 1.096 (175.0 73.5) 120.5

10 (212,36) B 95 1.097 (175.5 72.5) 121.5

11 (139,36) A 66 1.096 (174.0 75.5) 118.5

12 (139,36) B 100 1.096 (175.0 73.5) 120.5

13 (193.5,54.5) A 51 1.097 (175.5 72.5) 121.5

14 (193.5,54.5) B 56 1.097 (175.5 72.5) 121.5

15 (157.5,54.5) A 62 1.096 (174.5 74.5) 119.5

16 (157.5,54.5) B 60 1.097 (175.5 72.5) 121.5

TABLE 12A

EFFECT OF STEP LENGTH LOGIC ON EFFIClEHCT FOR EXAMPLE #4

Starting Guess

Poor

Fair

Good

Expert

Step Length Logic

A

B

A

B

A

B

A

B

Average Number of

Function Evaluations

123 248

94 176

68 98

57 58

TABLE 12B

EFFECT OF EXPERTISE OH EFFICIENCY FOR EXAMPLE #4

Starting Guess

Poor

Fair

Good

Expert

Step Length Logic

B only

A-B average

A-B average

A only


248

135

83

57

57

function evaluations as step length refinement logic A. This suggests

that a larger initial step length would increase the efficiency of the

search even though two further grid refinements are needed instead of

one. This is due to the fairly large scale of this problem and the

resulting relatively large value for the critical circle radius Rcc*

A comparison of the various levels of expertise of starting guess shows

that a poor initial guess requires from two to four times more

function evaluations than an expert guess, depending on which step

length logic is used.

It may be noted in Table 11 that the coordinates and radius of

the critical circle and their corresponding safety factors vary

slightly. This is because the minimum safety factor occurs for a range

of points rather than for a single point. The fixed step length in the

pattern search causes the search to end at slightly different points.

The difference in safety factors is due to the number of significant

digits used in the STABR analysis.

4.5 Slope in Soil with Continuous Variation of Strength. The

fifth problem studied was a slope in soil whose strength increases

linearly with depth, shown in Figure 20. In this type of problem, the

depth to which the critical circle is tangent is unknown. To locate

the critical depth, searches were performed at several widely spaced

tangents. The resulting critical circles are shown in Figure 21. This

showed the lowest safety factor occurring for a circle tangent to a

depth 20 feet below the top of the slope. Further searches were then

c = 200 psf

(100,100)

20'

(134.64,120)

V = 120 pcf

100'

_ L _

^ c = 800 psf

Figure 20, Example #5 - Shear Strength Which Increases With Depth

Elev

—120 F.S.=

F.S. = 0.859

150

180

F.S. = 1.328

210

Figure 21. Critical Circles for Various Tangent Elevations

In vo

60

made using tangent depths 10, 15, 2 5 , and 30 feet below the top of the

slope. This showed the critical depth to be somewhere between 25 and

30 feet below the top of the slope. Final searches were made using

tangent depths spaced every 0.5 feet in this range, which determined

the critical depth to be 25.5 feet below the top of the slope.

The variation in safety factor with depth was also studied for

this problem. Figure 22 shows plots of this variation for three

different slip circle centers. The effects of the starting guess on

grid search efficiency was studied using the technique described

previously. The results are given in Table 13. It took nearly three

times as many function evaluations to locate the minimum safety factor

for a poor starting guess as for an expert guess. Xt may be noted that

the minimum safety factor is less than one, indicating unstable

conditions. This condition was ignored for efficiency comparison

purposes.

The effects of starting point and step length in pattern

searches were studied. Starting points for the searches are shown in

Figure 23. The results are given in Table 14 and the average number of

function evaluations required are summarized in Table 15A. Table 15B

shows the effect of expertise on the efficiency of the pattern

searches. It can be seen from these results that the choice of

starting point has little effect on the efficiency of the pattern

search for step length refinement logic A. The same trend of

decreasing efficiency as seen in earlier problems is seen in Table 15B.

However, it must be remembered that these results were obtained after

61

2.0

1.5

1 . 0


110 120 130 140

Safety

Factor

0.5 110 120

1.5 '

1.0

0.5


130 140


110 120 130

Depth (ft.)

140

Figure 22. Variation in Safety Factor With Depth for Example #5

62

TABLE 13

GRID SEARCH RESULTS FOR EXAMPLE #5

Grid Logic


Mininum F.S.

(xfy) Radius of Critical Circle (ft)

Expert 61 0.781 (117,86.5),38.5

Good 113 0.781 (117,86.5),38.5

Fair 135 0.781 (117,86.5),38.5

Poor 169 0.781 (117.5,87),38

(79.5,49) (150.5,49)

,62.5) [142,62.5)

(129,75.5) Xl06,75.5)

(12

(117.5,87) ^•expert initial grid

(100,100)

(134.6,120)

Figure 23. Initial Grids and Starting Points for Example #5 Ot co

64

TABLE 14


Trial Starting

Point (ft)

Step Length Logic

Total No.. of Function Evaluations

Minimum F.S.

(x,y) Radius of Minimum (ft)

1 (150.5,49) A 69 0.781 (117.5,87),38

2 (150.5,49) B 86 0.781 (117.5,87),38

3 (79.5,49) A 53 0.781 (117.5,87),38

4 (79.5,49) B 70 0.781 (117.5,87),38

5 (142,62.5) A 50 0.781 (117.5,87),38

6 (142,62.5) B 63 0.781 (117.5,87),38

7 (93,62.5) A 47 0.781 (117.5,87),38

8 (93,62.5) B 54 0.781 (117.5,87),38

9 (129,75.5) A 45 0.781 (117.5,87),38

10 (129,75.5) B 44 0.781 (117.5,87),38

11 (106,75.5) A 42 0.781 (117.5,87),38

12 (106,75.5) B 38 0.781 (117.5,87),38

13 (123,81.5) A 41 0.781 (117.5,87),38

14 (123,81.5) B 34 0.781 (117.5,87),38

15 (112,81.5) A 42 0.781 (117.5,87),38

16 (112,81.5) B 31 0.781 (117.5,87),38

TABLE 15A



Poor A 55 B 78

Fair A 49 B 59

Good A 44

B 41

Expert A 42

B 33

TABLE 15B


Starting Guess

Poor

Fair

Good

Expert

Step Length Logic

B only

A-B average

A-B average

B only


78

54

43

33

66

the critical tangent depth had been determined. Locating the critical

tangent depth is by far the most difficult and time-consuming part of

analyzing this type of problem.

4.6 Stepped Slope in Homogeneous Soil. The next example

studied was a stepped slope in homogeneous, cohesive soil. The slope

geometry is shown in Figure 24. For this type of slope profile, it is

not known initially whether the critical circle will be a toe circle or

a base circle. Safety factor contours were studied for both cases.

Figure 25 shows safety factor contours for slip circles which

are tangent to the firm stratum. The plot shows a fairly regular

contour pattern with a minimum value of 1.236.

Figure 26 shows safety factor contour for slip circles passing

through the upper toe of the slope. An examination of this figure

shows a regular contour pattern on the right side of the grid (above

the upper slope) having a minimum value of 1.718. However, the factor

of safety values gradually decrease as the corresponding circle centers

are chosen farther to the left side of the grid. This is an example of

a problem which does not have a single or unimodal minimum safety

factor value. Such a condition presents a problem in evaluating the

minimum safety factor using a pattern search procedure. If the

starting point for the search is chosen in the vicinity of the local

minimum (x,y) = (110,94), the search will converge to this point.

However, if the starting point is chosen farther to the left, in the

region of gradually decreasing safety factor values, the search will

not converge at all.

(140,115)

(120,115)

t = 120 pcf ff= 0 c = 500 psf

(160,130)

Figure 24. Example #6 - Benched Slope Face

68

min

Figure 25. Safety Factor Contours for Circles Tangent to Firm Stratum for Example #6

2.0 4.0 4.0 2.0

0.5

F.S. . = 1.718 mm

(local minimum)

upper toe of slope

Figure 26. Safety Factor Contours for Circles Passing Through Upper Toe for Example #6

CT> VO

70

4.7 Case Histories. A series of case histories was presented

by Krahn and Fredlund (1977) for the purpose of verifying the

University of Saskatchewan SLOPE program for slope stability (Fredlund,

1974). Krahn and Fredlund suggested that these problems could be used

as standard examples in future program verification studies. These

case histories are also suitable for use in verifying and comparing

search routines. However, it is very difficult to duplicate these

problems as they have been presented in the aforementioned paper due to

the extremely small scales used and incomplete or incorrect strength

parameter data. Therefore, the original references were consulted in

order to present accurate and complete information about each case

history. For convenience, complete information regarding each of these

nine problems is presented in Appendix A, together with sketches

showing the location of the critical circle.

4.8 Comparison of Grid and Pattern Search Results. The

results of the grid and pattern searches can be compared for the

various examples since they were found using identical procedures. i

These results are summarized in Table 16. An examination of this table

shows a general trend of decreasing efficiency with decreasing levels

of expertise. It can also be seen that a grid search requires many

more function evaluations than a pattern search for the same level of

expertise.

TABLE 16

COMPARISON OF GRID AND PATTERN SEARCH RESULTS


Starting

Guess

Example #1

Grid Pattern

Example #2

Grid Pattern

Example #3 Pattern

Example #4

Pattern

Example #5

Grid Pattern

Expert 75 33 157 43 33 57 61 33

Good 98 39 338 60 44 83 113 43

Fair 130 52 234 94 59 135 135 54

Poor 144 76 240 159 74 248 169 78

CHAPTER 5

SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

5.1 Summary. Slope stability analyses are commonly performed

using limit equilibrium methods. Many computer codes are available

that employ various limit equilibrium methods, as well as some type of

optimization strategy to locate the critical slip surface and

corresponding safety factor. A thorough search of literature and

commercially available computer codes revealed that all optimization

strategies currently in use in slope stability programs are direct

search methods, which are the most inefficient optimization techniques.

A wide range of slope stability problems, including several

published case histories, was presented. The problems are realistic

slope problems that can be easily duplicated. These problems will

serve as a basis for future comparative testing of optimization

algorithms as they are incorporated into slope stability analysis

programs. In this study, the efficiencies of two direct search

methods, the grid search and the pattern search, were evaluated using

the formulated set of problems.

5.2 Conclusions. Both the grid and the pattern search methods

are somewhat inefficient in locating the critical slip surface. This

is especially true for large-scale problems, such as Examples #2 and

#4, which require many function evaluations in the search for the

minimum safety factor. The grid and the pattern searches are largely

72

73

trial-and-error procedures for problems where the critical tangent

depth is unknown. In addition, the pattern search may produce

inaccurate results for problems such as stepped slopes, which may have

more than one local minimum safety factor. A grid search is the

preferred technique for these types of problems.

A comparison of the grid and the pattern search results reveals

that a grid search always requires more function evaluations than a

pattern search for the same level of expertise of the analyst.

Therefore, it can be concluded that the pattern search is generally

more efficient than the grid search.

The level of expertise of the analyst has a pronounced effect

on the efficiency of both the grid and the pattern search methods. A

poor choice of starting point combined with an inefficient step length

logic results in two to three times more function evaluations than an

expert starting point and logic for pattern searches. In grid

searches, using poor grid logic requires 1-1/2 to 3 times as many

function evaluations as an expert grid logic. The use of more advanced

optimization techniques in slope stability analysis would largely

eliminate the effect of user expertise.

5.3 Recommendations for Further Study. It is recommended that

a more efficient optimization algorithm be incorporated into one of the

existing slope stability analysis programs. This would greatly

increase the efficiency of slope stability analysis. The set of sample

problems presented here should be used as a basis of verification and

comparison of the resulting computer code.

APPENDIX A

NINE CASE HISTORIES

The first case history studied was the Congress Street Open Cut

in Chicago (Ireland, 1954). In consulting the original reference, it

was found that the single clay layer shown in the paper by Krahn and

Fredlund actually consisted of three separate layers having different

values of density and cohesion. A sketch of this example is shown in

Figure 27.

The second case history studied was the Brightlingsea slide,

which occurred in 1941 after a row of concrete blocks was placed on top

of a sea wall (Skempton and Golder, 1948). The configuration of this

problem is shown in Figure 28.

Another case history examined was the Seven Sisters' slide in

Western Canada (Peterson et al, 1957). A sketch of this example is

given in Figure 29.

A slip occurred in a railway cutting at Northolt in England in

1955 (Henkel, 1957). An effective stress analysis was performed. The

pore pressures were defined by the water level measured by piezometers

installed after the slip occurred. Figure 30 is a sketch of this

example.

Another case history presented by Krahn and Fredlund was the

Selset landslide (Skempton and Brown, 1961). This slide occurred in

74

75

APPENDIX A (Continued)

heavily overconsolidated clay. The slope . profile is shown in

Figure 31.

The Green Creek slide (Crawford and Eden, 1967) occurred in

Leda clay in Eastern Canada. An effective stress analysis was

performed using the pore pressure parameter ru to define the pore

pressures. Figure 32 shows the slope profile.

Three previously published example problems were also examined

by Krahn and Fredlund. These included an example from Bishop and

Morgenstern1s stability charts, Spencer's example problem, and

Morgenstern and Price's example problem. These examples are shown in

Figures 33 through 35. The Bishop-Morgenstern example was analyzed for

two pore pressure conditions, one with ru - 0 and the other with

ru = 0.45.

Figures 27 through 35 are intended to serve as references for

any future researchers who may wish to analyze these problems using

other, more advanced optimization techniques.

(X,Y) = (138,73)

(100,100) Elev. (ft.) J 100

y = no pcf

c = 0 psf = 30°

111 2f = 132 pcf c = 1420 psf

(130,117.6)

(124,117.6)

125

135 Y - 128 pcf c = 1060 psf = 0 (169.5,146.75)

•155 Stiff clay

Figure 27. Congress Street Open Cut

(X,Y) = (104,89)

Radius = 23' concrete block

/ .>(85,94.75) (87.8,94.

(74.5,97.75)

(92,99

(100,100)

(69,101.5) (101,101)

T = J V 5 pcf c 240 psf

1: ~)f = 120 pcf 110,110)

concrete: Y = 150 pcf c = 0

fS = 0

Figure 28. Brightlingsea Slide

•v!

(X,Y) = (121.7, 85.5)

(86,100) (100,100)

Elev. (ft.)

106 (37,114)

117.5 (58,114) (66,117.5 (126,117.5

133

c = 0. ff — 0

ff = 0

(2) r = 121 pcf c = 518 psf /! = 13°

Figure 29. Seven Sisters' Slide

03

79

F.S. . = 1.35 mm

(X,Y) = (153.7,62.3) Radius = 78.4'

(175,130), (177,131)

concrete wall

(177,136.5)'

piezometric line

107.5'

(175,137.5)

concrete: "jf = 150 pcf c = 2000 psf js = 35°

soil: Y = 125 pcf c'= 250 psf ff'= 20°

Figure 30. Northolt Slide

80

(X,Y) = 178.8,38.2)

Radius = 106.8'

(100,100)

piezometric line

y = 139 pcf

c' = 180 psf

Elev. 170.51

Figure 31. Selset Landslide

81

*min~ 9^?? F.S (X,YT'= (200.5,32.3)

Radius = 146.8'

(100,100)

t= 100 pcf c = 482 psf

20° (255,163)

Figure 32. Green Creek Slide

Case 1: r =0 u

F.S. = 1.87 (X,Y) = (350.4,-194.5) Radius = 397.7' Case 2:

(400,200)-^

r =0.45 u

(X.Y) = (315,-67.3) Radius - 280.8'

F.S. = 1.12

(100,100)

Y = 120 pcf

c = 300 psf fS = 25°

Figure 33. Bishop and Morgenstern's Example

82

(X,Y) = (304.4,-82.9) Radius = 282.91

(100,100)

Y = 120 pcf c = 240 psf / = 40e (300,200)

Figure 34. Spencer's Example Problem

F.S. = 1.80 (X,Y) = (246.6,1.6) Radius = 205.4'

(100,100)

Y= 120 pcf c = 1200 psf / = 20° (300,200)

Figure 35. Morgenstern and Price's Example Problem

APPENDIX B

STABR DATA FILES

83

PROBLEM #1: TOE CIRCLE 0 0 4 3 0 0 0 0 111.55 90.00 1 111.55 120. ,00

-90 100 100 100 160 200 100 100 100 100 160 200

111.55 120 120 120 160 200 300 120 120 120 160 200 1 500 0 120 2 4000 30 120

PROBLEM #2: BASE CIRCLE 0 1 4 3 0 0 0 0 126.00 75.00 1 160.00 -90 100 100 100 160 200 100 100 100 100 160 200 152.00 130 130 130 160 200 350 130 130 130 160 200

1 800 0 120 2 4000 30 120

PROBLEM #3: STRATIFIED SOIL 0 1 4 5 0 0 0 0 117.32 86.70 1 135.00 -200 100 100 100 130 135 150 200 100 100 100 100 130 135 150 200 134.64 120 120 120 130 135 150 200 550 120 120 120 130 135 150 200

1 500 0 120 2 0 30 120 3 500 0 120 4 4000 30 120

PROBLEM #4: BIRCH DAM 0 1 7 4 0 0 0 0 ISO 75 1 194 -200 100 100 100 100 194 194

44 100 100 100 100 194 194 72 100 100 100 157 169 194 86 100 100 100 157 157 194 100 100 100 100 157 157 194 500 157 157 157 157 157 194

1 1023 31 127 2 1629 18 127

3 1023 0 127

PROBLEM #5: CONTINUOUS COHESION VARIATION 0 1 4 2 0 2 0 0 117.30 86.00 1 124.00 -200 100 100 100 400 100 100 100 100 400

134.64 120 120 120 400 550 120 120 120 400 1-10 120 100 200 400 1700

PROBLEM #6: BENCHED SLOPE FACE 0 1 6 2 0 0 0 0 110 90 1 120 115 -100 100 100 100 130 100 100 100 100 130 120 115 115 115 130 140 115 115 115 130 160 130 130 130 130 300 130 130 130 130

1 500 0 120

CONGRESS STREET OPEN CUT (IRELAND 1954) 0 1 9 5 0 0 0 0 140 80 1 155 500 100 100 100 111 125 135 155

100 100 100 100 111 125 135 155

115 111 111 111. 111 125 135 155 124 117.6 117.6 117.6 117.6 125 135 155

130 117.6 117.6 117.6 117.6 125 135 155

140 125 125 125 125 125 135 155

160 135 135 135 135 135 135 155 169 146.75 146.75 146.75 146.75 146.75 146.75 155

600 146.75 146.75 146.75 146.75 146.75 146.75 155

1 . 0 30 110 2 1420 0 132 3 820 0 128 4 1060 0 128

BRIGHTLINGSEA SLIDE {SKEMPTON & GOLDER, 1946) 0 1 1 6 4 0 0 0 0

100 90 1 112 69 101.5 101.5 101.5 101.5 101.5 140

74.5 97.75 97.75 97.75 97.75 101.5 140

84.9 97.75 97.75 97.75 97.75 101.5 140 85 94.75 94.75 94.75 98.1 101.5 140

87.8 94.75 94.75 94.75 98.1 101.5 140 87.9 97.75 97.75 97.75 98.1 101.5 140

88.5 97.75 97.75 97.75 98.75 101.5 140 92 99.5 99.5 99.5 100.2 101.5 140 99.9 100 100 100 100.2 101.5 140 100 100 100 100 101.5 101.5 140 101 101 101 101 101.5 101.5 140 109.9 108 108 108 108.5 108.5 140 110 108 108 108 110 110 140 110.7 108.7 108.7 108.7 110 110 140 110.8 108.7 108.7 108.7 108.7 108.7 140 150 114.4 114.4 114.4 114.4 114.4 140

1 0 0 150 2 0 15 120 3 240 0 105

SEVEN SISTERS SLIDE (PETERSON ET AL, 1957)

0 1 13 5 ] L 0 0 0

125 75 1 133 -25 117.5 117.5 117.5 117.5 117.5 117.5 133.

19 117.5 117.5 117.5 117.5 117.5 117.5 133.

27 114.0 114.0 114.0 114.0 114.0 117.5 133. 37 114.0 114.0 114.0 114.0 114.0 117.5 133.

58 108.0 108.0 108.0 108.0 114.0 117.5 133. 66 105.7 105.7 105.7 110.0 117.5 117.5 133.

74 103.4 103.4 103.4 106.0 117.5 117.5 133. 86 100.0 100.0 100.0 100.0 117.5 117.5 133. 100 100.0 100.0 100.0 100.0 117-5 117.5 133.

115 106.0 106.0 106.0 110.1 117.5 117.5 133.

126 110.3 110.3 110.3 117.5 117.5 117.5 133.

144 117.5 117.5 117.5 117.5 117.5 117.5 133.

250 117.5 117.5 117.5 117.5 117.5 117.5 133.

1 0 0 115 2 518 13 121 3 432 0 105 4 432 0 105 -25 106 19 106 27 106 37 106 58 106

66 106

74 107 86 109 100 111 115 114 126 117.5

144 117.5 250 117.5 0

0 0 0 0 0 0

0 0

0 0 0 0

0

NORTHOLT SLIDE (HENKEL, 1957) 0 1 8 3 1 0 0 0 130 40 1

141 -100 100 100 100 100 250 100 100 100 100 100 250

175 130 130 130 130 250 175.1 130 130 130 137. 5 250

177 131 131 131 136. 5 250

177.1 131 131 131 136. 5 250

177.2 132 132 132 132 250 300 132 132 132 132 250

1 2000 35 150 2 250 20 125 -100 107.5 100 107.5 175 134 175.1 137.5

177 136.5 177.1 132 177.2 132 300 132

SELSET LANDSLIDE (SKEMPTON & BROWN, 1961) 0 1 6 2 1 0 0 0 150 50 1 145 -50 100 100 100 147 67 100 100 100 147 100 100 100 100 164.2

112 106.4 106.4 106.4 170.5

184 145 145 145 170.5 400 145 145 145 170.5 1 180 32 139 -50 100 67 100 100 104 112 106.4 184 145 400 145

95

GREEN CREEK SLIDE (CRAWFORD & EDEN, 1967) 0 1 5 2 - 1 0 0 0 150 0 1 179 200 100 100 100 400 100 100 100 100 400

255 163 163 163 400 331 168 168 168 400 450 175.8 175.8 175.8 400

1 482 20 100 0.62

BISHOP AND MORGENSTERN EXAMPLE FOR RU 0 1 4 2 - 1 0 0 0 250 -100 1 203 -200 100 100 100 600 100 100 100 100 600 400 200 200 200 600 700 200 200 200 600

1 300 25 120 0

97

BISHOP AND MORGENSTERN EXAMPLE FOR RU = 0.45 0 1 4 2 - 1 0 0 0 250 -100 1 213.5 -200 100 100 100 600 100 100 100 100 600 400 200 200 200 600 700 200 200 200 600 1 300 25 120 0.45

SPENCER'S EXAMPLE 0 1 4 2 - 1 0 0 0 250 0 1 200 -300 100 100 100 500 100 100-100 100 500 300 200 200 200 500 600 200 200 200 500 1 240 40 120 0.5

MORGENSTERN & PRICE EKAMPLE 0 1 4 2 0 0 0 0 200 50 1 203.6 -300 100 100 100 500 100 100 100 100 500 300 200 200 200 500 600 200 200 200 500 1 1200 20 120

100

LIST OF REFERENCES

Awad, B. M. (1986, "Application of the Simplex Method to Slope Stability Analysis," M.S. Thesis, The University of Arizona, Department of Civil Engineering and Engineering Mechanics,

Tucson, Arizona.

Bailey, W. A. and J. T. Christian (1969), "ICES-LEASE-l; A Problem Oriented Language for Slope Stability Analysis," MIT Soil Mechanics Publication No. 235. Massachusetts Institute of Technology.

Baker, R. (1980), "Determination of the Critical Slip Surface in Slope Stability Computations," International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 4, No. 4, pp. 333-359.

Boutrup, E. (1977), "Computerized Slope Stability Analysis for Indiana Highways, Vols. I and II," Joint Highway Research Project Report

Nos. JHRP-77-25 and JHRP-77-26, Engineering Experiment Station, Purdue University.

Boutrup, E. , Love 11, C. W., and R. A. Siegel (1979), "STABL2 - A Computer Program for General Slope Stability Analysis," Proceedings of the Third International Conference on Numerical

Methods in Geomechanics, pp. 747-757.

Carpenter, J. R. (1985), "PCSTABL4 User Manual," Joint Highway Research Project No. JHRP-85-7, Engineering Experiment Station, Purdue University.

Celestino, T. B. and J. M. Duncan (1981), "Simplified Search for Noncircular Slip Surfaces," Proceedings of the Tenth International Conference on Soil Mechanics and Foundation

Engineering, Vol. 3, pp. 391-394.

Chen, R. H. (1981), "Three-Dimensional Slope Stability Analysis," Joint Highway Research Project Report No. JHRP-81-17, Engineering Experiment Station, Purdue University.

Chugh, A. K. (1981), "User Information Manual, Slope Stability Analysis Program SSTAB2," Bureau of Reclamation, U. S. Department of the Interior.

Civilsoft (1976), Informational Brochure on SLOPE.

101

LIST OF REFERENCES (Continued)

Cornwell, L. W., Kocman, M. G., and M. F. Prosser (1980), "Computational Experience with Davidon's Least-Square Algorithm," Journal of Optimization Theory and Applications, Vol. 31, No. 1, pp. 27-39.

Crawford, C. B. and W. J. Eden (1967), "Stability of Natural Slopes in Sensitive Clay," Journal of Soil Mechanics Division, ASCE, Vol. 93, pp. 419-436.

Cross, J. P. (1982), "Slope Stability Program," Civil Engineering,

ASCE, Vol. 52, No. 10, pp. 71-74.

DeNatale, J. S. (1983), "On the Calibration of Constitutive Models by Multivariate Optimization, A Case Study: The Bounding Surface

Plasticity Model," Ph.D. Thesis, University of California at Davis, Department of Civil Engineering, 435 pp.

Duncan, J. M. and A. L. Buchignani (1975), "An Engineering Manual for Slope Stability Studies," University of California at Berkeley, Department of Civil Engineering.

Duncan, J. M. and S. G. Wright (1980), "The Accuracy of Equilibrium Methods of Slope Stability Analysis," Proceedings of the Third International Symposium on Landslides, Vol. 1, pp. 274-254.

Fredlund, D. G. (1974), "Slope Stability Analysis, User's Manual CD-4," Department of Civil Engineering, University of Saskatchewan.

Fredlund D. G. and J. Krahn (1977), "Comparison of Slope Stability Methods of Analysis," Canadian Geotechnical Journal, Vol. 14, pp. 429-439.

Fredlund, D. G., Krahn, J., and D. E. Pufahl (1981), "The Relationship Between Limit Equilibrium Slope Stability Methods," Proceedings of the Tenth International Conference on Soil Mechanics and Foundation Engineering, Stockholm, Vol. 3, pp. 409-416.

Fredlund, D. G. and J. D. Nelson (1985), "Course Notes: Slope Stability Analysis," Colorado State University, Geotechnical Engineering Program.

Geocomp Corporation (1986), Informational Brochure on GEOSLOPE.

Geo-Slope Programming Ltd. (1987), Informational Brochures on SLOPE-II and PC-SLOPE.

Geosoft (1986), Informational Brochures on STABRG and SL0P8RG.

102

LIST OF REFERENCES (Continued)

Henkel, D. J. (1975), "Investigations of Two Long Term Failures in London Clay Slopes at Wood Green and Northolt," Proceedings. Fourth International Conference on Soil Mechanics and Foundation Engineering, Vol. 2, pp 315-320.

Horn, J. A. (1960), "Computer Analysis of Slope Stability," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 86, No. SM3, pp. 1-17.

Huang, Y. H. (1981), "User's Manual, REAME, A Computer Program for the Stability Analysis of Slopes," University of Kentucky, Institute for Mining and Minerals Research.

Huang, Y. H. (1983), Stability Analysis of Earth Slopes, Van Nostrand-Reinhold, New York.

Ireland, H. D. (1954), "Stability Analysis of the Congress Street Open Cut in Chicago," Geotechnique, Vol. 4, pp. 163-168.

Krahn, J. and D. G. Fredlund (1977), "Evaluation of the University of Saskatchewan Slope Stability Program," Road and Transportation Association of Canada Forum.

Lefebvre, G. (1971), "STABR User's Manual," University of California at Berkeley, Department of Civil Engineering.

Little, A. L. and V. E. Price (1958), "The Use of An Electronic Computer for Slope Stability Analysis," Geotechnique, Vol. 8, pp. 113-120.

Lovell, C. W., Sharma, S. S. and J. R. Carpenter (1984), "Slope Stability Analysis with STABL4," Joint Highway Research Project No. JHRP-84-19, Engineering Experiment Station, Purdue University.

Nelder, J. A. and R. Mead (1965), "A Simplex Method for Function Minimization, Computer Journal. Vol. 7, pp. 308-3131

Newman, E. E. (1985), personal communication.

Nguyen, V. U, (1985), "Determination of Critical Slope Failure Surfaces, Journal of Geotechnical Engineering. ASCE, Vol. Ill, No. 2, pp. 348-352.

Peterson, R. Iverson, N. L. and P. J. Rivard (1957), "Studies of Several Dam Failures on Clay Foundation," Proceedings, Fourth International Conference on Soil Mechanics and Foundation Engineering, Vol. 2, pp. 348-352.

103

LIST OF BEFERENCES (Continued)

Schiffman, R. L. (1972), "A Computer Program to Analyze the Stability of Slopes by Morgenstern's Method," Report No. 72-18. Department of Civil Engineering, University of Colorado at Boulder.

Schiffman, R. L. and D. M. Jubenville (1975), "User's Manual for MALE-1, Version 2-A: A Computer Program to Analyze the Stability of Slopes by Morgenstern's Method," GESA Report D-75-16, University of Colorado at Boulder.

Siegel, R. A. (1975b), "STABL User's Manual," Joint Highway Research Project Report No. JHRP-75-9, Engineering Experiment Station, Purdue University.

Siegel, R. A. (1975b), "STABL Uder's Manual," Joint Highway Research Project Report No. JHRP-75-90, Engineering Experiment Station, Purdue University.

Siegel, R. A., Kovack, W. D. and C. W. Lovell (1979), "New Method of Shear Surface Generation for Stability Analysis," Proceedings of the 29th Annual Highway Geology Symposium, pp. 295-312.

Skempton, A. W. and J. D. Brown (1961), "A Landslide in Boulder Clay at Selset, Yorkshire," Geotechnique, Vol. 11, pp. 280-293.

Skempton, A. W. and H. Q. Colder (1948), "Practical Examples of the = 0 Analysis of the Stability of Clays," Proceedings, Second

International Conference on Soil Mechanics and Foundation Engineering, Vol. 2, pp 63-70.

Swann, W. H. (1972), "Direct Search Methods," in Numerical Methods for Unconstrained Optimization (W. Murray, Ed.), Academic Press, London, U.K., pp. 13-28.

Taylor, D. W. (1948), Fundamentals of Soil Mechanics, John Wiley and Sons, New York.

Terzaghi, K. (1943), Theoretical Soil Mechanics, John Wiley and Sons, New York.

Von Gunten Engineering Software, Inc. (1985), Informational Brochure on SB-SLOPE.

Wright, S. G. (1974), "SSTABl - A General Computer Program for Slope Stability Analysis," Research Report No. GE-74-1, Department of Civil Engineering, University of Texas at Austin.

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