EMSTO-PLASTIC INCREMENTAL APPROACH
IN SLOPE STABILITY ANALYSIS

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
VESHARN POOPATH
1971

 

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thesis entitled

ELA$T0~ PLASTlC INCQE 012me AFMOMH
IN sLoPE smeuLITY ANALYsus

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ABSTRACT

ELASTO-PLASTIC INCREMENTAL APPROACH
IN SLOPE STABILITY ANALYSIS

BY

Visharn Poopath

The formation of failure surfaces induced by incre-
mentally increasing loads on soil slope embankments was
studied through the use of the finite element method.

The soil was assumed to behave as an ideal elastic,
perfectly plastic and isotropic material. The effects of
strain hardening and nonhomogeniety were considered.

Two embankment building processes, single-lift
construction and incremental construction, were simulated
by the superposition method. Loading was considered to
result from an incrementally increasing soil unit weight
for the single-lift construction, and from the application
of surface forces to simulate a gradually increasing soil
thickness for the incremental construction. The investiga-
tions included slopes completely within the elastic range,
and slopes in the elasto-plastic range up to and including
limiting equilibrium or failure conditions. Plastic
deformations were assumed to be contained up until complete

failure,and no analysis was considered beyond this point.

Visharn Poopath

Very small values of modulus of elasticity were applied in
the contained plastic regions during increments of load.
Location and configuration of potential failure surfaces
were determined by the slip-line method and compared with
results of plasticity analyses by the method of Bishop
(1954). Stability was also determined by an average factor
of safety calculated from the factor of safety field in
the elasto-plastic stages. These results are compared with
the factor of safety calculated by the Bishop method.

f It was found that for homogeneous soil slopes,
the elasticity and elasto-plasticity analyses are in
agreement with the Bishop analysis in determining the
location and configuration of potential failure surfaces.
This indicates that geometrical changes of the slope body
due to applied loads, during the stages prior to failure
are not large enough to cause appreciable error in the
plasticity analysis at the final stage. The elasticity
analysis provides an indication of the maximum height of
slope at which first local yield occurs, the location of
first yield, the area where tensil stresses develop, and
the required strength characteristics of the soil to
prevent slope failure.

For nonhomogeneous soil slopes, the elasto-plasticity
analysis was found to be useful in studying failure behavior.
The potential failure surfaces predicted from elasto-plastic
stress fields differed considerably from potential failure
surfaces predicted by Bishop's method for nonhomogeneous

soil slopes.

Visharn Poopath

Factors of safety calculated from elastic stress
fields, elasto-plastic stress fields and by Bishop's method

all were in good agreement.

ELASTO-PLASTIC INCREMENTAL APPROACH

IN SLOPE STABILITY ANALYSIS

BY

Visharn Poopath

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Civil Engineering

1971

:33 f}
/ I
O

DEDICATION

To the memory of my father.

ii

ACKNOWLEDGMENTS

The writer would like to give his sincere gratitude
to his major professor, Dr. R. R. Goughnour, Associate
Professor of Civil Engineering, for his academic advice,
patience and encouragement throughout the completion of this
doCtoral thesis. Thanks also go to the other members of
the writer's committee: Dr. 0. B. Andersland, Professor
of Civil and Sanitary Engineering, Dr. W. A. Bradley,
Professor of Metallurgy Mechanics Materials Science, Civil
and Sanitary Engineering, Dr. H. F. Bennett, Assistant
Professor of Geology. The writer also appreciates the help-
ful c00peration given by Mr. L. Szafranski, Mechanical
Technician, in preparing laboratory equipment.

The writer is grateful to The Thai Government for

the financial assistance throughout his doctoral prOgram.

iii

TABLE OF CONTENTS

Page
ACKNOWLEDGMENTS . . . . . . . . . . . . . iii
LIST OF TABLES . . . . . . . . . . . . . vii

LIST OF FIGURES . . . . . . . . . . . . . viii

LIST OF NOTATIONS . . . . . . . . . . . . xiii

Chapter

I. INTRODUCTION . . . . . . . . . . . 1

II. REVIEW OF CURRENT METHODS OF STABILITY
ANALYSIS. . . . . . . . . . . . . 5
2.1 The Culmann Method. . . . . . 6
2.2 Point to Point Stress Analysis Method . 7
F__«2 3 Circular Arc Methods . . . . . . . 12
2.3.1 ¢-Circle Method . . . . . . 12
2.3.2 Jaky Method. . . . . . . . 17
~" {72.3.3 The Method of Slices. . . . . 31
2.3.4 Bishop's Method . . . . . . 38
2.4 Composite Surface of Sliding . . . . 46
2.4.1 Janbu's Method. . . . . . . 46
2.4.2 Morgenstern’s Method. . . . . 51
2.5 The Method of Characteristics . . . . 64
2.6 Numerical Method . . . . . . . . 72

III. CONCEPTS IN ELASTICITY AND PLASTICITY, AND

BEHAVIOR OF SOILS UNDER LOAD . . . . . . 79

79
80

3.1 Rheological PrOperties of Materials .
3.1.1 Elastic Solids. . . . . .

3.1.2 Viscous Fluids. . . . . . 83
3.1.3 Plastic Materials. . . . . 83
Idealized Stress-strain Curves. . . 85
Mohr's Theory of Failure. . . . . . 87

Slip Lines . . . . . . . . . 90
Behavior of Soils under Load . . 92
3. 5.1 Behavior of Cohesionless Soils 92
3. 5.2 Behavior of Cohesive Soils. . 97
3.6 Pore Pressure . . . . . . . . 113

WOOD-’00
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iv

Chapter
IV. ELASTO-PLASTIC INCREMENTAL APPROACH. . . .

4.1 Energy Approach in Elasto-plastic
Incremental Behavior . . . . .
4.1.1 Single-lift Construction .
4.1.2 Incremental Construction .
Strength and Failure Concepts . .
Stress Condition during Loading .
Simulation of Contained Plastic Zone
4.4.1 Different-E Method . . .
4.4.2 Truncated Plastic Zone . .
4.4.3 Propagation of a Slip Line.
4.5 Stability Analysis. . . . . .

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V. PROCEDURE . . . . . . . . . . . .

5.1 Basic Concepts of Finite Element Method.

5.2 Simulation of Embankment Stresses and
Strains by the Superposition Method . .
5.2.1 Single-lift Construction . . .
5.2.2 Incremental Construction . . .

5.3 Factor of Safety and Unused Shear
Strength . . . . . . . . . . .

VI. PRESENTATION OF RESULTS AND DISCUSSIONS . .

6.1 Homogeneous Soil Slopes . . . . . .
6.1.1 Elasticity Solutions. . . .
6.1.2 Elasto-plastic, Single-Lift

Construction . . . . . . .
6.1.3 Elasto-plastic, Incremental
Construction . . . . . . .

6.2 Nonhomogeneous Soil Slopes . . . . .
6.2.1 Elastic Range Analysis . . . .
6.2.2 Elasto-plastic, Incremental

Construction . . . . . . .

6.3 Stability Analysis. . . . . . . .

VII. SUMMARY AND CONCLUSIONS. . . . . . . .
7.1 Homogeneous Soil Slope . . . . . .
7.2 Nonhomogeneous Soil Slope . . . . .
7.3 Stability Analysis. . . . . . . .

BIBLIOGRAPHY O O O O O O O O O O O O O O

Page

116

119
120
126
127
128
132
132
133
134
136

140
141
145
145
148
150
157

157
157

163
165
168
168

169
172

210
210
213
213

215

Page
APPENDICES O O O O O O C O O O O O O O O 220

A. Computer Program "FEASTX" . . . . . . . 221
B. Computer Program "NEWST" . . . . . . . 249
C. An Example of Calculated Stress Fields. . . 250
D. Sample Calculation of Average Factor of

safety 0 O O O O O O O O O O O 254

vi

LIST OF TABLES

Page
Tabulation of Stresses for Undrained Loading . 103

Tabular Form of Stress Conditions at Different
Stages of Embankment Loading. . . . . . 130

Movement of Points along the SIOpe Boundary
During the Loading Process of two Layer
Embankment. . . . . . . . . . . . 208

Computed Average Factor of Safety Values. . . 209

vii

Figure

2-1.

2-14 0
2-15.

2-16 0

LIST OF FIGURES

Page

Orientation of Slip-lines for a SIOpe with

Lateral Restraint . . . . . . . . . 7
Orientation of Slip-lines for a Slope without

Lateral Restraint . . . . . . . . . 7
Excavated Slope . . . . . . . . . . . 10
Trajectories of Principal Stresses. . . . . ll
Sketches Showing Elements of the ¢-Circ1e

Method for Toe Circles. . . . . . . . 14
Sketch Showing Elements of the ¢-Circ1e Method

for Circles Passing Below the Toe of Slope . l6
Slope and Failure Configurations . . . . . 18
Component of Forces on an Element of Sliding

surface. 0 O O O O O O O O O O O 18
Stress Components on an Elementary Prism; and

Eliquibrium of an Elementary Prism-force

System at any Point on the Sliding Surface . 19
Boundary Stress-Conditions . . . . . . . 24
Stress Components on any Plane Surface other

than the Sliding Surface . . . . . . . 26
Sketches Showing Tension and Compression Zones. 26
Inclination of Tangent Lines at the Boundary

Points . . . . . . . . . . . . . 27
Stability Analysis by the Slices Method . . . 32
Stability Analysis by the Bishop Method . . . 40
Definition Sketch for Janbu's Method . . . . 47
Enlarged Slice . . . . . . . . . . . 48
Forces on a Slice Boundary . . . . . . . 48

viii

Figure Page

2-19. Potential Sliding Mass Used for Derivation of

Morgenstern's Equations . . . . . . . 53
2—20. An Element at an Interface between Two Slices . 55
2—21. Coordinates and Loading Systems. . . . . . 65
2—22. Stress States at Limiting Equilibrium. . . . 68

2-23. Finite Slope with General Potential Slip
surface. 0 O O O O O O O O O O O 74

2-24. Shape of Critical Potential Slip Surfaces with

Variable Pore-pressure (after Bell, 1969) . 78
3—1. Stress-deformation Relationships of Ideal

Materials . . . . . . . . . . . . 82
3-2. Idealized Stress-strain Curves . . . . . . 86
3-3. Failure Envelope of Mohr's Stress Circles . . 87

3-4. Common Forms of Stress EnvelOpes for Normally

Consolidated and Preconsolidated Soils . . 88'
3-5. Mohr's Circles for Plane-strain Conditions . . 90
3—6. Graphical Representation of Stress at a Point . 91

3-7. Mechanism of Deformation and Shear in a Mass

of Bulky Grains . . . . . . . . . . 94
3-8. Stress-strain in Cohesionless Soil. . . . . 95
3—9. Mohr's Envelope for Cohesionless Soil. . . . 96

3-10. Mohr's Envelope for Saturated-clay in Drained
Shear O O O O I O O O O O O O O 98

3-11. Mohr's EnvelOpe for Saturated Clay in Con-
solidated-undrained Shear. . . . . . . 100

3-12. Mohr's Envelope for Saturated Clay in Undrained
conditions. 0 O O O O O I O O O O 102

3-13. Stress-strain Curves of Clays in Undrained
Shear O O O O O O O O O O O O O 106

3-14. Peak and Residual Strengths of Sensitive Clays. 107

ix

Figure Page
3-15. Stress Path in the General Triaxial Test. . . 108

3—16. Stress Path for Plane of Maximum Shear Stress
in Trixial Compression. . . . . . . . 110

3-17. Stress-strain in Drained Shear for Different
Loading Systems . . . . . . . . . . 110

4-1. Elasto-plastic System . . . . . . . . . 117

4-2. Idealized Stress-strain Curves Applied in
Elasto—plastic Problems . . . . . . . 118

4—3. Elasto—plastic Incremental Behavior under
Gradually Increasing Body Force, Single-Lift
Embankment. . . . . . . . . . . . 121

4-4. Graphical Presentation of Internal Energy
Development . . . . . . . . . . . 123

4—5. External Work-done and Internal Energy Applied
for Case Shown on Figure 4-3. . . . . . 124

4-6. Relative Location of a Typical Point in an
Embankment during Construction . . . . . 129

4-7. Stress-strain Behavior of the Typical Point. . 129
4-8. Mohr's Circles Representing Total Stress

Conditions for a Typical Point in an

Embankment. . . . . . . . . . . . 131

4-9. Mohr’s Circles at Yield Condition for the
Typical Point. . . . . . . . . . . 131

4-10. Simulation by Different E-Method . . . . . 132
4—11. Simulation by Truncated Plastic-zone Method. . 134
4-12. Simulation by PrOpagation of a Slip Line. . . 135
4-13. A Potential Slip Surface in a Part of the Body. 139

5-1. Typical Slope Boundaries . . . . . . . . 144

5-2. Flow Diagram of Single Stage Construction
Simulation. . . . . . . . . . . . 147

5-3. Flow Diagram of Incremental Construction
SimUlatj-On. O I O O O O O O O 0 O 149

X

Figure Page

5—4. Definition Sketch for Factor of Safety and
Unused Shear Strength . . . . . . . . 150

5-5. Priority of Yield by Two Stability Concepts. . 152
5-6. Two Possible-failure Stress Conditions . . . 153
5-7. Typical Finite Element Configuration of a Slope 155

5-8. Comparison of Slip Surfaces for Different
Element Sizes. . . . . . . . . . . 156

6-1. Influence of v on Tensile Stress Development,
E = 342,000 psf, y = 110 pcf. . . . . . 176

6—2. Stress Enve10pes for Points along the Bottom
Layer of a Slope. . . . . . . . . . 177

6-3. Stresses in a Semi-infinite Body . . . . . 180

6-4. Effect of Strength Envelope on Failure Behavior
of Cohesionless Soil SlOpes . . . . . . 180

6-5. Effect of Strength Envelope on Failure Behavior
of Cohesive Soil SIOpes . . . . . . . 181

6—6. Change of Slip-surfaces for Different v-Values. 182

6—7. Comparison of Potential Sliding Surfaces
between BishOp's and Elastic Solutions, for
Different SIOpe Angles. . . . . . . . 183

6-8. Elasto-plastic Analysis of Single-step
Construction, H = 20 feet. . . . . . . 186

6-9. Elasto—plastic Analysis of Incremental Con—
struction, Elastic-perfectly Plastic
Assumption. . . . . . . . . . . . 187

6-10. Elasto-plastic Analysis of Incremental Con-
struction, with Elastic-strain Hardening
InClUded O I O O O O O O O O O O 188

6-11. Slip Surfaces Plotted According to the Stress
Field of the Final Stage of Incremental
Construction (Figure 6-9) for Different
Assumed ¢-Va1ues. . . . . . . . . . 189

6-12. Elasto—plastic Analysis under Drained Strength
Criterion and Incremental Construction,
Elastic Strain-hardening Assumption . . . 190

xi

Figure Page

6-13. Possible Stress Conditions during an Increment
Of Load. 0 O C O O O I O O O O O 193

6-14. Development of Tension Zone for Two—layer Case. 195

6—15. Elasto-plastic Analysis, Incremental Construc—
tion for Two-layer Slope . . . . . . . 197

6-16. Stress-path at Different L8cations in Slope
Body, 10 feet Lifts, 45 -Slope Angle . . . 198

6-17. Factor of Safety Contours for a Homogeneous
Soil Slope. . . . . . . . . . . . 200

6-18. Factor of Safety Contours for a Two-layer Slope 203

6-19. Potential Sliding Surfaces Used in Calculation
of Average Factor of Safety . . . . . . 206

xii

El

ii

NOTATIONS

pore-pressure parameters

a soil parameter

width of slice

fmnih order tensor of elastic constants
cohesion of soil; (prime(') denotes
effective stress, D=drained, U=undrained,
CU=consolidated-undrained)
compressibility of fluid

compressibility of soil skeleton

effective normal force acting on base
of slice

neutral normal force acting on base of
slice

void ratio or volumetric strain

modulus of elasticity, normal force at
interface of slices or internal energy

effective normal force at interface of
slices

summation of principal strains

slope of stress-strain curve in elastic
range

slope of stress-strain curve in plastic
range

factor of safety or body force
specific gravity of soil skeleton
total height of slope

depth or height of slice

xiii

{r}
{R}

W,dW,AW

X,T
{X}

XIYIZ

stiffness matrix

bulk modulus

length of slice base

porosity of soil

normal force acting on base of slice
hydrostatic stress or mean pressure

neutral normal force at interface of
slices

horizontal external force

pressure distribution along rupture
surface

vector of nodal point displacements
vector of nodal point forces

radius of circular rupture surface
pore-pressure ratio

distance along rupture surface or
length of slope surface

shear force acting on base of slice
stress transformation matrix

second order tensor

pore-pressure or neutral stress
volume or velocity

total weight of a slice or external
work-done

shear force at interface of slices
a column matrix of body forces

rectangular coordinates

xiv

6..
1]
{du}

{Ge}

6 s e
x' y' 2

angle of inclination, angle of slip-line
with major principal stress direction,
angle at base of slice or angle of
curvature

angle of line of thrust with horizon

angle of slip-line with minor principal
stress direction (8) or angle of slope

unit weight of soil

unit weight of soil submerged in water
unit weight of water

Kronecker delta

a column matrix of incremental of
deformations

a column matrix of increment of unit
deformation components

unit deformation in x, y and 2 directions
unit deformation in elastic range

unit deformation in plastic range
function of transformation

angle of major principal stress direction
with horizon

Lame's elastic constants
Poisson's ratio

mass density of soil (é)
normal stress

major principal stress

minor principal stress prime (')
denotes

normal stress in x-direction effective
stress

normal stress in y-direction

XV

If,S

¢.¢'.¢D.¢CU.¢CD

<1>(...)

A,d,6

over burden pressure
preconsolidation pressure

a column matrix of stress components
octahedral stress components

shear stress
shear stress in x-y coordinates

shear strength or shear stress at failure
angle of internal of friction (prime(')
denotes effective stress, D=drained,
CU=consolidated-undrained, CD=consolidated-
drained)

function

increment or decrement

Change;

summation

 

The Greek Alphabet

 

Alpha
Beta
Gamma
Delta
Epsilon
Xi

Eta
Theta
Lambda
Mu

tycpsmm0o<m9

Nu

Pi
Rho
Sigma
Tau
Phi
Chi

Psi

eexeaoosc

Omega

xvi

CHAPTER I

INTRODUCTION

Civil engineers commonly solve problems in continuum
mechanics by one of two assumptions concerning the material
properties of the medium: the material is elastic (normally
linearly elastic) or the material is rigid plastic. The
theory of elasticity, the first assumption, is normally
used to investigate the behavior of the material under
loads which cause stresses below the yield stress.
Elasticity solutions can provide a partial answer to the
question, how strong is the body? in the form of a factor
of safety or ratio of maximum safe load to the applied
load. Plasticity theory, the second assumption, can be
used to calculate the maximum safe load by limiting equili-
brium methods.

For simple problems, such as uniaxial tension,
elastic and plastic parts of deformation can be separated
by a critical yield point. More complicated problems such
that stress states at critical points do not reach the
yield condition at the same time, fall into the elasto-

plastic category. These problems can be reliably solved

by plasticity theory only if the material has very small
elastic deformations in comparison with the plastic parts.
Currently the stability of slopes is most often
considered in the realm of plasticity. A complete solution
of stresses and strains in a mass of soil under arbitrary
loading as the load is increased to failure is almost
always neglected. Plastic limit theorems (Drucker, Prager
and Greenburg, 1952) permit lower and upper bounds on
the failure load to be found conveniently when the soil
mass is idealized as perfectly plastic. Liam Finn (1967),
Chen (1969), Lysmer (1970), and others, have extended
the applications of the theory of plasticity to soil
mechanics problems by introducing different forms of
statically admissible stress fields for the lower bound
solution and some forms of velocity fields as required
for the application of the upper bound theorem. The maximum
safe load at limit equilibrium (which is actually the upper-
bound load) can also be predicted by the slip-line method
(Sokolovski, 1965). To obtain the stability of slopes,
equilibrium methods (Bishop, 1954; Morgenstern, 1965;
Janbu, 1954) under an assumption of simultaneous failure
along any complete surface are generally used to estimate
an average factor of safety. The factor of safety is the
ratio of resisting ability to the static stresses of the

points along the assumed sliding surface.

Plasticity analyses do not simulate the actual
behavior of slopes except at failure, when the factor of
safety is equal to unity. For slopes having large factors
of safety,equilibrium analyses may often provide a valid
indication of stability, but they do not provide informa-
tion concerning the deformations of slope, nor are they
capable of indicating which zones around the slope are most
critically stressed. These methods can be used to predict
the location of the critical failure surface in slopes
only when the soil is homogeneous throughout. In cases
of nonhomogeneous soil the methods of prediction are no
longer applicable. Some types of soils behave like
rigid plastic materials (very small elastic strains), but
most soils exhibit a certain range of elastic deformation
that may cause a considerable amount of error in the
plastic analyses. Elasticity theory is useful for design
purposes in finding required soil strength and the locations
of the most critical stresses. Elasto-plastic incremental
theory is useful in obtaining the true mode of failure,
and stress fields in the range between fully elastic,and
complete failure can be investigated.

It is the purpose of this thesis to present a
method of analysis for stress-strain behavior in the range
between fully elastic and complete failure, by considering
an incrementally increasing load. Location and configuration

of the critical failure surface is then predicted by the

slip-line field obtained from the stress field at full
failure. Behavior of elasto-plastic stress fields during
increasing load can be observed. These results are compared
with the classical equilibrium method of Bishop (1954).

Because of the difficulty in obtaining exact
solutions of elasto-plasticity problems, especially for
very complicated shapes and loading systems, numerical
methods (finite element) are used to calculate the stress
fields at different stages. The soil is assumed to have
a piece-wise linear stress-strain relationship. Calculation
in the elasto-plastic incremental range is done by
superposition.

This study is limited to the investigation of the
stability of slope embankments where loading is assumed to
result from gradually increasing body forces or from
gradually increasing surface loads simulating embankment
construction. Loading is assumed to be performed within a
short period of time so that the strength criterion can
reasonably be represented by the undrained strength envelope
(for saturated soils), but the failure surface is predicted
by a graphical method of slip-line construction under the

effective angle of internal friction concept.

CHAPTER II

REVIEW OF CURRENT METHODS

OF STABILITY ANALYSIS

Many methods of stability analysis have been pro-
posed to investigate the characteristics of failure
surfaces and the factor of safety against sliding along
these critical failure surfaces. Good agreement of results
between natural failures and calculated failures has been
obtained only for homOgeneous and isotropic soils. Various
types of trial failure surfaces have been investigated,
but the latest acceptable method for homogeneous soil
introduced by Bishop (1954) is analyzed by a cylindrical
failure surface. However, it is apparent that because
of the departures from homogeneity, the soil mass may slip
along non-circular surfaces. In this case the most critical
failure surface can be obtained later by trial and error
using as a guide the circular failure surface from Bishop's
method. For non-homogeneous soil the factor of safety can
be obtained for individual slip surfaces (Morgenstern,
1965; Janbu, 1954). An advanced method to investigate the
true failure surface in non-homogeneous soils has not been
accomplished. Each individual case must be investigated
separately, and very few general rules have been laid.

5

The same criterion of failure is applied in all of
the methods mentioned above. The shearing strength as
expressed by Coulomb's equation is used for the limit of
equilibrium, and two-dimensional, plane strain is assumed.
Most of the methods assume a constant cohesion and angle
of shear resistance along the failure surface. The shear
strength will vary only because of varying normal stress
along the surface under consideration. Review of these

methods is presented in this chapter.

2.1 The Culmann Method

 

Culmann (1866) assumed that rupture would occur
along a plane surface. This assumption was the first
rational one for the failure surface of a cut slope with
a lateral retaining wall. For an ideal frictionless
retaining wall, no shearing resistance develops along the
interface of the soil and the wall (Figure 2-1a). The
stress distribution within the soil mass will be the same
as that in a semi-infinite soil mass under the influence
of gravity (Figure 2-lb). The orientations of the principal
stresses are self-parallel, and this condition is also true
along the failure surfaces.

When the retaining wall is removed, the orientations
of the principal stresses change around the cut area.
Distortion of the failure surface will also occur. In

practical work it is observed that the failure surface

distorts as shown.on Figure 2-2. The radius of curvature
is minimum at the toe and increases along the failure

line to a maximum at the top surface.

 

 

 

\\ //§ //

 

 

 

Figure 2-1.--Orientation of Slip-lines for a Slope
with Lateral Restraint.

 

 

 

”F/IT/ff

Figure 2-2.--Orientation of Slip-lines for a Slope
Without Lateral Restraint.

2.2 Point to Point StressgAnalysis Method

 

This method was investigated by two investigators:

Brahtz (May and Brahtz, 1936), and Bennett (1951). The method

consists of investigating the state of stress at every point
of interest in the slope vicinity. Brahtz developed a stress
function in the form of Airy's stress function similar to
those used in plane stress problems of the theory of elas-
ticity. He introduced a comparative factor K = 50/y, where
the unit weight of saturated soil, y, is in pounds per cubic

foot. The condition of plasticity is given by
T = c + 0 tan ¢ (2-1)

where T = available shear strength, c = cohesion of the soil,
0 = normal stress at the shearing surface, and ¢ = angle
of shearing resistance.

The safety factor is defined by the ratio of the
available shearing strength to the shearing stress at each
point of the potential zone of failure. This point to
point stress analysis aims at finding the zone of failure,
where the limiting equilibrium of the soil is reached or
exceeded, characterized by lines of equal factor of safety
against flow or failure. The factors of safety in this
analysis are point functions and cannot be compared with
the mean value of safety against total rupture along a
continuous sliding surface.

In this analysis Brahtz assumed a horizontal soil
layer (Figure 2-3a), so that the stresses at any depth 2

are given by

02 = yz, 0x = O, o = 0 (2-2)

where 02 = vertical normal stress, Ox = horizontal normal
stress, and Oxz = shearing stress on vertical and horizontal
planes. Brahtz then removed the lateral soil to form a
trapizoidal shaped dam (Figure 2-3b). He then attempted
to correct the horizontal stress (0X) by assuming a stress
function, which does not affect the basic vertical and
shearing stresses, superimposed upon the basic stress
system of equation (2-2). The resulting corrected system
was thus obtained for the trapezoidal soil mass. His analysis
was intended primarily for the purpose of dam design, and
effects of pore pressure were also included for different
situations of reservoir water.

Finally an empirical slope formula was developed
as a guide in choosing the slope of a dam on both surfaces

as follows:

upstream slope, S1 = 275/(2c+hytan¢) (2-3)
downstream slope, S2 = (275h2-1.7B(c+Hytan¢))
/(H(2c+Hytan¢)) (2-4)

where h = depth of reservoir, H = total height of dam,
B = width of crown, c = cohesion of soil, ¢ = angle of
internal friction of soil, and y = unit weight of saturated

material.

10

 

 

Figure 2-3.--Excavated Slope.

Bennett (1951) advocated point-to-point stress
analysis without using stress functions but using the super-
position method of the elastic stress distribution laws up
to thelimit of elastic equilibrium. He stated that the
purpose of stability analysis is not to investigate the
elastic deformations, but to explore the possibilities of
a shear failure. The inelastic behavior of the soil will
not have much effect on this exploration, but it may be
unwise to ignore those features of stress analysis which
are so particularly well adapted to study the borderline
between elastic and plastic conditions. The study of
probable magnitude and orientation of principal stresses,
and direction of potential slip planes may be profitable.

He presented the result as a sketch of a system of
orthogonal curves approximating the trajectories of the
principal stresses (Figure 2-4). The potential surfaces

of rupture are also drawn, making a constant angle with

11

the directions of principal stresses. He stated that a
quantitative determination of the exact stresses within
the soil slope is very difficult, and the only accurate

current method was photoelasticity.

7‘ Potential
3‘ S lip -S urface
/
/

'l
Surface of Max. Shear
‘5

   

 

 

Direction of Major Principal Stress

__”__.Direction of Minor Principal Stress

Figure 2-4.--Trajectories of Principal Stresses.

To investigate the stability of a slope, the points
along a potential slip plane or within a zone of known
weakness such as an unconsolidated clay layer in the
foundation are examined. To account for the degree of
consolidation the pore pressure is estimated from consoli-
dation theory, and is subtracted from the total normal
stress acting on any plane through the point. The shearing
stress on the plane is assumed to be unaffected by the
neutral stresses.

Bennett concluded in his paper that this method

is a quick rough stress analysis at one or two points in

12

the critical zone, and it is therefore useful in preliminary
layout‘of slopes. Finally, he recommended that the point-
to-point stress analysis, as well as the sliding surface
procedure, may be used in the analysis of any important
embankment on a weak foundation and that the results be
given equal weight in a final judgement of the safety

factor.

2.3 Circular Arc Methods

 

The circular arc method was first proposed by
K. E. Petterson (1915). The justification that circular
arcs are close approximations of actual rupture surfaces
comes from field investigations of a large number of
actual slides. An infinite number of arcs may be drawn
for a given slope. The critical arc, the one used in
analysis, is defined as the one which is least stable.
The method of circular arc analysis has been widely used
and accepted as satisfactory. This method has undergone
many refinements by many authors. A few of the most

prominent methods will be presented.

2.3.1 ¢-Circle Method

 

This method was developed by Professor Glennon
Gilboy and Arthur Casagrande (Taylor, 1937). It was
initially proposed in terms of a complete graphical

solution of the slope problem. For any circle analyzed,

13

the method provides the required soil strength to obtain
equilibrium of the free body diagram. They found that
one graphical layout for any given slope and any given
angle of internal friction, could represent various
heights of slope merely by the use of different scales.
Finally Taylor (1937) introduced a dimensionless

parameter, c/(FwH), called the "stability number,"

(c = actual unit cohesion for the soil, F - factor of
safety, H = height of slope, and w = unit weight of the
soil), that is used to describe the stability of lepes.
Figure 2-5 shows an arbitrary circular sliding
surface, AB, through the toe of the slope. Possible
circular arcs are of three configurations: circles
passing through the toe of the slope, circles passing
below the toe, and circles passing above the toe. Taylor
developed only the first two cases. The problem is
considered astmadimensional with unit thickness
extended in the third dimension. Shearing resistance
on both faces of the third dimension is neglected. It
is assumed that the soil mass is in equilibrium and just

begins to slide along the circular arc.

l4

 

 

 

ARM:

 

 

 

 

a). b).

Figure 2—5.--Sketches Showing Elements of the
¢-Circle Method for Toe Circles.

Forces acting on the mass are the weight of the mass (W),
the resultant cohesion (C), and the resultant of force
transmitted from grain to grain of the soil across the

arc AB. The vector representing W will act vertically
through the center of gravity of the mass, and its magnitude
and line of action may be found easily by any one of

several methods. The magnitude of the resultant cohesion,
C, is given by clL, where cl is the unit cohesion required
for equilibrium and L is the length of the chord AB. The
line of action of C is parallel to the chord AB, and its

moment arm, a, is described by

L—‘I

(2-5)

t‘lw

clLa = clLR or a =

where L is the length of arc AB. Thus the line of action
of C may be found. P is the resultant force transmitted
from grain to grain of the soil across the arc AB.

The intersection of W and C is known. Since the
three forces, W, C and P, must be concurrent, P must also
pass through this intersection. With W known in magnitude
and direction and the direction of C also known, the force
polygon may be constructed if a second point can be
obtained on P to determine its direction. This is
accomplished by the basic assumption of the ¢-Circle Method.

P is made up of small elementary forces, such as
p (Figure 2-Sa) which, at failure, must act at an obliguity
of ¢ to the arc AB. If the line of action of p is
extended and a circle is drawn with center at 0 and tangent
to the extended line, it will be seen that the p force
for any other element of the arc AB will also be tangent
to this circle. This circle is called the ¢-circle. Any
two such p forces will intersect just outside of the
¢-circle so their resultant would pass just outside this
circle, and by the same reasoning P must pass outside. The
assumption introduced is that P is tangent to the ¢-circ1e.

Mathematical derivation of the stability number is
referenced to Taylor (1937), and the final results are

presented.

16

Case a (Figure 2-5), circles pass through the toe:

c = kcsc2x(y csczyfcoty) + cotx-coti (2-6)
FwH 2cotx cotv +‘2
(x,y,i and v are shown in Figure 2-5)

Case b (Figure 2-6), circles pass below the toe:

c = kcsc2x(y_csc2y-coty) + cotx-coti—Zn

FwH 2cotx cOtv + 2 (2-7)

(x,y,i and n are shown on Figure 2-6. v can be
obtained in the same fashion as in the first case)

 

DH

 

 

 

 

 

Figure 2-6.--Sketch Showing Elements of the ¢-Circle
Method for Circles Passing Below the Toe of Slope.

17

2.3.2 Jaky Method

 

Jéky (1936) proposed an analytical method of slope
stability analysis based on the assumption of circular
failure surfaces. Compatibility of strains is violated by
assuming simultaneous failure along a circular failure
surface, but equilibrium of streSses is still applied along
the assumed failure surface.

Slopes of all kinds remain stable if they are not
made steeper than a certain maximum angle, 8. If a slope
steeper than B is desired it must be supported by a retain-
ing wall. The forces that keep a slope stable are the
passive internal forces, friction and cohesion. Acting
against the direction of sliding, these forces prevent
movement on the potential surface of rupture. The relation-
ship between the normal (n) and the tangential (t) stresses
that act on the surface of rupture (Figure 2-7) may be

expressed by the well known Coulomb formula
t = n tan¢ + c (2'8)

where ¢ is the angle of internal friction of the soil, and
c is shearing resistance or cohesion.

Fr. Kotter (1903) was the first to derive a formula
of the sliding surface for cohesionless materials. Accord-
ing to Kotter the following relationship exists between the

pressure, q, and radius, ds/da, of the curved sliding surface

18

 

 

 

 

Figure 2-7.--Slope and Failure Configurations.

d d .
—§-- 2qtan¢ = d; $1n(a-¢) (2'9)

where a is the angle the sliding element makes with the

horizonta1,and ds is the length of the element (Figure 2-8).

Figure 2-8.--Component of Forces on an Element of
Sliding Surface.

If, in equation 2-9 we substitute

19

q = t/sin¢ and
%S,= 1 dt then

a s1nB dd
dt _ ds . . _ _
35 2t tan¢ — a y51n¢ s1n(a ¢) (2 10)

where t is the shearing stress distribution along the sliding
surface.

As derived, this formula is valid only for cohesion-
less soils. In the following (Jéky, 1936) it will be shown
that the K6tter formula is valid also for cohesive soils.

. Taking into consideration the most frequent practical case,
when the terrain is horizontal, the equilibrium of differ-
ential forces on a small prismatical element may be given

by the Cauchy equations (Figure 2-9).

30 ET
Tif— + Ii = y (2-11)

 

 

 

 

 

v /
”71?"! 96696

. Figure 2-9.--Stress Components on an Elementary
Prlsm; and Equilibrium of an Elementary Prism-force System
at any Point on the Sliding Surface.

20

Equilibrium of forces on the sides of a small
triangle that has one side on the sliding surface, to the
forces on the other two sides, may be expressed with the

formulae

0 -0
t = —X§—§-sin2a + T cosZa

(2-12)

 

x .
n = + c 2 - 2 .
-—7——- 2 os a T Sln a
These expressions, if solved for Ox and o produce

rcota+n-tcota

0
ll

(2-13)
0 = Ttana+n+ttana.

These may be substituted for Ox and oy in the Cauchy

equations

33--:-cot0L + 21 + §£__ 3E cota = 0
3x y 8x 3x

3T at an at _
5? tana + 5; + §§-+ gi-tand - Y

Because T, the shearing stress, is a function

merely of the x and y coordinates, the formula

31 cosa + 31 sina = 31
8x

"<
G)
(D

gives the differential of T by the ds element of arc.

Thus, the above equations may be changed to

21

at an . 8t _
5g + 3; Sing 3; 008d — 0

(2-14)
31 an at . _

These equations remain valid regardless of the laws governing
the condition at failure. The shear causing rupture in

soils is taken as

t = n tan¢ + c.

Therefore rupture will occur in such elements of the earth

in which (t-ntan¢) is maximum. This expression is a function
of x,y, and a. Therefore its maximum will develop where

the partial differential quotients by x,y, and a are zero.

Thus, on the surface of rupture

3t _ an

'53: " a; tam”

3t _ 8n _
5y - BY tan¢ (2 15)
3t _ an

’55 - so? tam-

Equations (2—15) and (2-12) may be combined to yield
(Cy-ox)c032a-2T51n2a = [-(Cy-ox)s1n2a-21cosZa]tan¢

or simply,

0-0'

I = 17$ cot(2d-¢) . (2-16)

22

And, using equation (2-12),

_ cos(2a-¢) _
- t cos¢ . (2 17)

This way we arrive at an expression for the shearing stress
(I) in terms of the sliding force (t). This expression

is valid only for the surface of rupture.

A further step will determine the gi-and g; partial

differential quotients. Let us put g; and gg-of equation

(2-15) into equation (2-14). Then simple computation will

result in the following:

 

at g; sin¢ dt ycosa sin¢- g; sin¢

a; = 51n(a-B) d?" cos(a-¢) ' (2-18)
From equation (2-12) we obtain

an _ _

53'_ 2t
and from equation (2-8)

at _ 3n

'5; — 3—(1- tané.
Therefore

3% = -2t tangS. (2-19)

The total differential of the sliding stress, t,
may be formed from the partial differential quotients,

as that of any function with multiple variables

23

dt _ 3t 3t . 8t dd
a-g - 53(- COSG + W 51nd + 507 Fig.
Substituting the respective values for the partial differ-

ential quotients, and arranging yields

dt dd _ . .
35 2t tan¢ a; — Y s1n¢ s1n(a-¢),
. . ds
and, multiplying by dd
dt _ ds . . _ _
55' 2t tano — y dd 51n¢ s1n(a ¢). (2 20)

The result is wholly Fr. Kotter's formula for
cohesionless material. In the above derivation, however,
not only friction, but also cohesion has been taken into
consideration. Thereby the validity of Kotter's formula
is ascertained to be entirely general, and applicable to
both cohesive and cohesionless materials.

There are, however, two unknowns in the formula,
the sliding stress, t, and the shape of the surface of
rupture. Of these two the curved surface of rupture, as
pointed out before, is known from observations to be, for
all practical purposes, a circular cylinder. This know-
ledge permits a simple integration of Kétter's differential
equation, and the resulting formula will yield the unknown

sliding stress, t.

24

The radius of the circular cylinder - 33': R (an

increase of 8 means decrease of a, therefore the minus

sign), if put into equation (2-20) gives

%5 - 2t tan¢ = - RY sin¢ sin(a-¢)a

The solution of this differential equation is

= Rysin¢
1+4tan ¢

 

[cos(a-¢)+2tan¢ sin(a-¢)]

(2-21)

+K eZtanda,

where K is an integration constant determinable from
boundary conditions. At the intercept of the surface of
rupture with the terrain, point C in Figure 2-10, the

stresses are zero. Thus

 

T = 0
(2—22)
0 = 0 .
Y
{t1=0
n1=0

 

 

Figure 2-10.--Boundary Stress-Conditions.

25

Stresses at point C are fully determined by these
conditions, because by combining equations (2-8), (2-12),
and (2-17) the following equations, expressing the stress

conditions in any point of the sliding surface, will be

 

 

 

obtained
0 = t 1+81n$ s1n(2a:p) _ c cot¢
y s1n¢ cos¢
= l-sin¢ sin(2a-¢) _ _
Ox t sin¢ cos¢ c cot¢ (2 23)
_ cos(2d-¢)
T _ o
cos¢

When I = 0, then 2a-¢ = 90°. Therefore inclination of the
sliding surface at point C is given by do = 45+¢/2. Because
also 0 = 0

y 7

t0 = c(1-sin¢). (2-24)

Incorporating these with equation (2-21),

C(l-sino) = RYSin%[cos(45-¢/2)

l+4tan ¢ (2-25)

 

+2tan¢sin(45-¢/2]+Ke2tan¢(45+¢/2).

Equation (2-25) permits determination of K. Its value,
however, is unecessary for our purposes. Another boundary

Stress-condition exists at point B

26

0 ./
Y ./

/’
/,/’ sliding plane

 

 

 

skxe

 

Figure 2-ll.--Stress Components on any Plane
Surface other than the Sliding Surface.

 

 

 

Figure 2-12.--Sketches Showing Tension and
Compression Zones.

27

\ /:>\

     

45°-¢/2
8-45°+¢/2

Figure 2-13.--Inclination of Tangent Lines at the
Boundary Points.

The t1 and n1 stresses that act on a plane inclined at 8
(Figure 2-11) may be calculated by substituting t1, hi,
and B in equation (2-12) for t, n, and a. Combining

equation (2-12) with equation (2-23) yields

t = t cos(ZB-2a+¢)

 

 

1 c056—
= l-sin¢sin(28-2a+¢) _ _
n1 t sin¢cos¢ c cot¢. (2 26)
At point B, t1 = 0. Therefore d1 = B-45+¢/2. Also n1 = 0,

and therefore t1 = c(l+sin¢). These values of a1 and t1

put into equation (2-21) give

28

c(l+sin¢) = Egigiggfibos(B-45-¢/2)+2tan¢sin(B-45-¢/2)]
+ tan

(2-27)

+Ke2tan¢(B-45+¢/2).

Now, by eliminating the constant K from equation
(2—25) and (2-27), the radius of the sliding cylinder, R,

may be computed

_ c l+sin¢ 2
R - —Y- W l+4tan d?)

1- -tan2 (45- ¢/2) e2tan¢(8- 90°)
[cosIB- 45- -¢/7)+2tan¢s1n(8- 45- -¢72XF[Cos(45- ¢/2)+2tan¢sin(15;¢/2)]

 

For simplifying this formula let us introduce the following

notations:

tan 1 2tan¢
(2-28)

m = 45°+¢/2+A.

Angles A and w are functions of ¢ only. Therefore, in a
particular case they are constants. Thus
_ o
c l+sin¢ 1-tan2(45-¢/2)etanMB 9° )

R = _ r _ o . (2'29)
0 y s1n¢cosl cos(B-w)-sinw etan>\(8 9O )

 

This RO is the radius of a cylinder that intercepts the
slope at point AO (Figure 2-12). At AO there are neither
normal nor tangential stresses, but there is the neutral

axis of the bent slope. The bending causes tension in the

29

upper half, AOA, and compression in the lower half, AOB.
When the algebraic sum of the normal forces equals zero,
the slope is in balance. Its length, AB, with a linear
distribution of stresses, is equal to 230 (Figure 2-12).
It is of interest to note that the plane of rupture
solution predicts a slope subjected to bending moment.
Inclination of the tangent to the failure surface
through point A0 is given by a1 = B - 45 + ¢/2 (Figure 2-13).

Inclination of the tangent through point C is given by

_ - - - B+¢

do — 45 + ¢/2. Thus AOC 1s inclined at an angle —§—. In
words, the angle of the chordplane of the cylinder, halves
the angle (8 - ¢) between the slope and the angle of repose.

The central angle of the arc with R0 radius is

given by (90° - 8). From Figure 2-13

AC = 2RO sin(45°-B/2), and

AC sin (Egg)

5 = . therefore
0 s1nB '

 

 

sin(45-B/2) sin(§%$)
s = 2R

0 o sinB ' (2_30)

The sliding cylinders intercept the slope invariably
at a constant angle a1 = B - 45 + ¢/2. The tangential

stress, t1, is everywhere zero, and from equation (2-26)

30

we obtain 28 - 2&1 - ¢ = 90°. Therefore, R is a linear
function of 5 (equation 2-30). In other words an increase
in length of the slope will be followed by a proportional
increase in the R radius of the sliding cylinder.

Linear relationships exist between t and R
(a = constant), between nl and t (equation 2-26), and
between R and 3 (equation 2-30). Therefore n1 is propor-
tion to the length 8 of the slope, giving a triangular
stress-distribution (Figure 2-12). The maximum length

of the slope is

4Rosin(45°-B/2)sin(§§$)

sinB

 

s = Zs =
0

which together with Ro's formula gives

. _ . B+¢
= 52 l+sin¢ $1n(45° 8/2)51n(—§—

y sin¢cosI sinB
(2.31)
tanA(B-90°)

l-tan2(45-g/2)e 1_
tanX(B-90v) ‘

cos(B-w)-sinw e
This is the polar equation of the limit curve for cohesive

soils.
Taylor (1937) commented that Jéky's method leads

to results which are somewhat on the unsafe side.

31

2.3.3 The Method of Slices

 

The method of slices was developed by W. Fellenius
(1936). It is based on the static analysis of the mass
above any trial failure arc, with this mass considered to
be made up of vertical slices.

Before the method and its assumption can be truely
understood a clear picture is required of all forces enter-
ing the analysis, and these forces will be explained in
some detail. Seepage forces are included in the following
explanation, although these were not included in the
original form of the slices method.

Let it be assumed that the circular failure arc
shown on the section of Figure 2-l4a is an arbitrarily
chosen trial arc. This section is assumed to be in
homogeneous soil through which a steady state of seepage
is occuring as represented by the equipotential lines,
which are shown by dashed curves. This section has been
arbitrarily divided into five vertical slices of equal
width.

From the equipotential lines the neutral pressure
may be determined at any point on the section. For example,
at point N, in Figure 2-1451, the pressure head is MN.
Point S is determined by setting the distance NS equal to
MN. A number of points obtained in the same manner as 8
give the curved line through S which is a pressure head
diagram. On the sides of the slices pressure head diagrams

are also shown.

32

Pressure Heads on Sides
of Slices \

 

Zh§/4='*;T_
Slice 1 j

 

L7R=//§

Slice 3

   
 

Slice 2

   
 
   

   

Equipotential

 

 

 

 

Triai
\ Failure
Arc

 

 

 

 

 

  

‘ Vector
Representing
Neutral Forces \

   
 

  

Pressure Heads on Arc.
(Measrued Radially as
Shown by SN=MNJ

(a) Cross Section Showing Slices and Neutral Forces.

   
  

 

 

 

 

 

 

 

In‘t13 1 Internal
intergranular Neutral
Vectors, DE Vectors, CD
7 3 1 2'
.lrxdiéf' ”“’“*
L “5 ‘TFFTIB
A1
ea
El
31
Slice 2 51.:e 3

Slice 5 Slice 4 Slice 3

(b) Vector Polygons.

Figure 2-14.--Stability Analysis by the Slices Method.

33

The actuating force may be taken as either the
combination of total weights and boundary neutral forces
or the combination of submerged weights and seepage
forces. The former will be used. Thus the forces acting
on any slice are its total weight, the neutral forces
acting on its sides and on its base, and the resultant
intergranular forces on its sides and on its base.

Granular forces on the base consist of the cohesion, the

normal component of intergranular force PN' and the
frictional resistance that is developed by PN and that is
equal to PNtan¢D.

In Figure 2-14a , vector polygons are given which
show all forces for all slices. For purposes of explana-
tion any slice may be used, since notations are the same
for each slice except for subscript. The total weight of
any slice is equal to the volume multiplied by the total
unit weight, and is represented by vector AB. The neutral
force across the base is represented by vector BC, which
acts normal to the arc. The combined neutral force on
the two sides of the slice is represented by vector CD,
acting horizontally. For the steady seepage case, with
the flow net known, vectors BC and CD are completely
defined, and from the pressure head diagrams, shown in

(a), these vectors may be determined.

34

Vector DE represents the combination of intergranular
force on the two sides of the slice. The magnitude of the
force represented by DE is dependent on strains and
stress-strain characteristics of the soil; it is therefore
statically indeterminate. The vectors DE in the figure
are merely arbitrarily chosen values that are reasonable;
to obtain a solution some assumption relative to these
forces must be made.

Reverting to considerations of the entire mass,
it may be noted that the total weight of the mass must be
transmitted across the rupture arc. The sum of all boundary
forces, including both neutral and intergranular components,
must equal the total weight, regardless of the distribution
among the several slices. The assumption mentioned at
the end of the previous paragraph is, therefore, an assump-
tion relative to distribution only and thus it is called
a distributional assumption. Fortunately the various
possible distributions usually lead only to minor differ-
ences in the stability conditions.

The remaining forces are the normal and tangential
components of the intergranular forces across the arc.

These forces are represented by EF and FA, and are commonly
designated by PN and PS. Once the assumption giving DE
is chosen and point B is thus known, EF and FA can be

determined. The requirement for equilibrium is that the

35

shearing force, Ps’ be supplied by friction and cohesion.
The figures show that PS is supplied by the sum of the

frictional force FG, which is equal to P tan ¢D' and the

N
cohesional force CD, represented in the figure by GA.

This may be written

P5 = PN tan ¢D + CD

With respect to the mass as a whole the requirement
for stability is expressed in terms of the equilibrium of
moments about the center of the circle. In the case under
consideration boundary neutral pressures exist only on
the circular arc and introduce no moment. Thus the actuating
moment is that of the total weight. If weight, AB, is
separated into shearing and normal components AJ and JB,

designated respectively by WS and W the normal component

N'

has no moment, and the actuating moment may be written

RZWS, in which R is the radius and the moment is clockwise.
For a correct consideration of the equilibrium of

individual slices, forces CD and DE, which act on the

sides of slices, must be included. However, relative to

the mass as a whole they form closed polygons, as shown

in the upper’ left on Figure 2-14b , and the summation

of their moments must be zero. Force EF also has no

moment about the center of the circle. Thus the only other

forces introducing moment are FG and GA. Forces FG give

36

a moment which may be expressed as R£(FG), or RtandaDZPN

Forces GA give the moment RZ(GA), or RcDLa, in which La

is the length of the arc. These moments are counter—

clockwise, and together they form the resisting moment.
Equating the actuating and resisting moments

gives the expression of moment equilibrium,

RZWs = RtancbDZPN + RcDLa (2-32)

All terms in this expression are moments, and therefore
they are vector quantities. If the equation is divided

by R it becomes

ZWS = tan¢DZPN + cDLa (2-33)

The terms of this equation have the dimension of forces,
but since they are tangential summations they are not
vector quantities in the ordinary sense.

In equation (2-33) the only term which depends on
the distributional assumption, and thus the only term

which offers any difficulty in the analysis is 2P If

N.
the resultants of lateral intergranular pressures DE are
correct, as shown on Figure 2-l4la,_ the vectors EF,

representing P are correct.

NI
Fellenius proposed the use of the assumption that
the lateral forces are equal on the two sides of each

slice. This is not simply an assumption relative to the

37

distribution, since in addition it involves other factors
which cannot be fully explained without an unduly long
discussion. A small amount of study shows that it gives
conditions in individual slices which are entirely
incorrect. For example, it indicates that slices 1 and

2 have much less resisting force than is required for
equilibrium, and that slices 4 and 5 have much more
resisting force than needed. However, these inconsistencies
counteract each other to a large degree, and for the mass
as a whole this assumption, which is the characteristic
assumption of the slices method, gives fairly reasonable
results. In analyses of the type shown<N1 Figure 2-14,
where the effects of seepage are included, there are two
possible interpretations of the Fellenius assumption.
These interpretations are considered separately in the
two following paragraphs.

In the first interpretation it is assumed that
the lateral intergranular forces are in balance on the
sides of each slice. According to this assumption forces
DE all become equal to zero, and XPN becomes 2(DH).

In the second interpretation it is assumed that
the lateral combined forces are in balance on the sides
of each slice. On this basis combinations of forces CD

and DE all become zero and ZPN becomes Z(CJ).

38

The use of the second interpretation does not
require the determination of the neutral pressures on
the sides of the slices, and thus it saves a considerable
amount of labor. Since it gives conservative results,
it is probably the more satisfactory procedure for general

1158.

2.3.4 Bishop's Method

 

In 1954 a modified method in the stability analysis
of slopes was proposed by Bishop. Consideration was given
to the effect of pore pressure distribution and variation
of shear parameters on the position and shape of the rupture
surface. Failure is assumed to occur simultaneously along
the surface. The surface is also assumed to be a circular
arc. The problem is assumed to be one of plane strain.

Under these conditions a quantitative estimate of
the factor of safety can be obtained by examining the
conditions of equilibrium when incipient failure is
postulated, and comparing the strength necessary to main—
tain limiting equilibrium with the available strength of
the soil. The factor of safety (F) is thus defined as the
ratio of the available shear strength of the soil to that
required to maintain equilibrium. The shear strength

mobilized is, therefore, equal to s, where

s = %{c' + (on-u)tan¢')} (2-34)

39

and c' denotes cohesion, ¢' denotes the angle of shearing
resistance, on denotes the total normal stress, and u
denotes pore-pressure.

In order to examine the equilibrium of the mass
of soil above the slip surface it follows from equation
(2-34) that it is necessary to know the value of the normal
stress at each point on this surface, as well as the
magnitude of the pore-pressure. It is possible to estimate
the value of the normal stress by the following method
developed by Fellenius (1927, 1936), in which the conditions
for the statical equilibrium of the slice of the soil lying
vertically above each element of the sliding surface are
fully satisfied.

Consider the equilibrium of the mass of soil (of
unit thickness) bounded by the circular arc ABCD, of radius
R and center at 0 (Figure 2-15). In the case where no
external forces act on the surface of the slope, equilibrium
must exist between the weight of the soil above ABCD and
the resultant of the total forces acting on ABCD.

The following definitions are given:
denote the resultants of the total

horizontal forces on the sections n
and n+1 respectively.

En' and Enfl

Xn’ and Xn denote the vertical shear forces.

+1

W denotes the total weight of the
slice of soil.

 

40
denotes the total normal force acting
on its base.

denotes the shear force acting on its
base.

denotes the height of the element.
denotes the breadth of the element.
denotes the length BC.

denotes the angle between BC and the
horizontal.

denotes the horizontal distance of
the slice from the center of rotation.

 

 

 

 

 

 

a) Force Components on a Sliced b) Force Polygon.

Element

Figure 2—15.--Stability Analysis by the Bishop Method.

41

The total normal stress is on where
_ P -
o — 2 (2 35)

Hence from equation (2-34) the magnitude of the
shear strength mobilized to satisfy the conditions of

limiting equilibrium is s,where
_ 1 . P .
s — f{c +(I -u)tan¢ }. (2-36)

The shear force, S, acting on the base of the slice
is equal to 5%, and thus, equating the moment about 0 of
the weight of soil within ABCD with the moment of the

external forces acting on the sliding surface, we obtain
2W X = ZSR = 282 R (2-37)

It follows, therefore, from equation (2-36) that

__B_

F = 2W x

£[c'£+(P-u£)tan¢'], (2—38)

From the equilibrium of the soil in the slice above
BC, we obtain P, by resolving forces in a direction normal

to the slip surface

P = (W+xn-Xn+l)cosa-(En-En+l)31nd. (2-39)
The expression of F thus becomes
_ R . u
F - m EEC £+tan¢ (WCOSCX-UIQ)
(2-40)

+tan¢' {(xn-x )cosa-(En-En+l)sina}].

n+1

42

Since there are no external forces on the face of

the slope, it follows that

I
O

£(Xn-X ) (2—4la.)

n+1

and

Z(E -E
n

l
O

) (2-41b.)

n+1

However, except in the case where ¢' is constant
along the slip surface and a is also constant (i.e., in a
plane slip surface), the terms in equation (2-40) contain-
ing Xn and En do not disappear. A simplified form of
analysis, suggested by Krey (1926), Terzaghi (1929) and
also presented by May (1936) as a graphical method, implies

that the sum of those terms,

| _ _ _ .
Ztan¢ {(Xn Xn+l)cosa (En E )51nd},

n+1

may be neglected without serious loss in accuracy.
Putting x = R sing, the simplified form may be

written as

_ l . u _ -
F — EVFEIBE'ZEC 1 + tan¢ (Wcosa u£)]. (2 42)

The pore-pressures are often expressed as a function of
the total weight of the column of soil above the point

considered, i.e.

43

u = E (11;) (2—43)

where B is a soil parameter based either on field data or
laboratory tests. In the case, puttingyg = b seca, the
expression for factor of safety can be further simplified

to

_ l I I _— _
F — ZWSina Z[c i + tan¢ W(cosq B secu]. (2 44)

 

This expression permits the rapid and direct compu-
tation of the value of F which is necessary if sufficient
trial circles are to be used to locate the most critical
surface. However, the values of F are, in general, found
to be conservative, and may lead to uneconomical design
(Bishop, 1954). This is especially marked where conditions
permit deep slip circleseuoumi which the variation in a
is large. To derive a method of analysis which largely
avoids this error, it is convenient to return to equation
(2-38). If we denote the effective normal force, (P - ui),
by P' (Figure 2-15b) and resolve the force on the slice
vertically, then we obtain, on rearranging

c .
P' _ W + Xn Xn+1 2(u cosa +F—-31nd)
— T I O
cosa + tan¢ 51nd

F

 

 

44

Substituting in equation (2-38) and putting 2 = b sine and

x = R sina.

 

 

_ 1 . . _- _ seca
F _ fW—EIHE Z[{C b+tan¢ (W(l B)+(Xn xn+1))} l+tan¢'tana]°
F
(2-45)

The values of (Xn - Xn+1) used in this expression
are found by successive approximation, and must satisfy
the conditions given in equation (2-41). In addition,
the position of the lines of thrust between the slices
should be reasonable, and no unbalanced moment should be
implied in any slice. The factor of safety against sliding
on the vertical sections should also be satisfactory,
though since the slip surface assumed is only an approxi-
mation to the actual one, overstress may be implied in the
adjacent soil.

The condition 2(Xn-X = 0 in equation (2-41a)

n+1)
can also be satisfied directly by selecting appropriate
values of Xn. The corresponding sum, 2(En-En+l) can be
readily computed in terms of the expression used in equation

Resolving the forces on a slice tangentially we

obtain the expression

(W+xn-Xn_1)51na + (En-E )cosa = S

n+1

or (2-46)

45

(En-E ) = Sseca-(W+Xn-Xn+l)tand.

n+1

Now, if equation (2—45) is written

_ 1 _
F - EWSIHEZEm] r (2 47)
then
s = IF“, (2-48)

and hence

m
2(En-En+l)- Z[§seca-(W+Xn-X )tana]. (2-49

n+1

The X-values must, therefore, also satisfy the condition

that

m
E [FeeCd-(W+Xn-Xn+l)tana] — 0. (2-50)

In the practical point of view it is of interest
to note that although there are a number of different
distributions of (Xn-Xn+l) which satisfy equation (2-50),
the corresponding variations in the value of F are found
to be insignificant (less than 1% in a typical case). It
should also be noted that since the error in the simplified
method given in equation (2-42) varies with the central
angle of the arc, it will lead to a different location
for the critical circle than that given by the more rigorous

method.

46

Bishop also developed a more general case for
partially submerged slopes.
Slip surfaces predicted by Bishop's method correlate

well with observed field data for homogeneous soil.

2.4 Composite Surface of Sliding

 

In many instances the geometrical or geological
conditions of the problem are such that the surface of
sliding may not be even approximately circular. The
following methods have been proposed in calculating the
factor of safety against sliding for noncircular rupture

surfaces.

2.4.1 Janbu's Method

 

The method was proposed by Janbu (1954). A sliding
mass with a noncircular surface of sliding is showncnu
Figure 2-16.

The soil mass above the slip surface is divided
into an appropriate number of slices. Equilibrium in each
slice must be maintained as follows:

Moment equilibrium about point M (Figure 2-17),

yields

(T+%—AT) Ax+(E+%AE) AYO = 0 (2-51)

For a vertical section (Figure 2-18) AE=AT=0.

Hence,

47

 

 

  

Line of
Thrust

 

 
   

 

 

 

 

 

 

 

 

 

 

WI, 19115;
2____._J___
Y I
. _ AW
Notations: p — K§'= yh when above water level, or
p = yWfifyhz when partially submerged
u = pore pressure, or excess pore pressure
above WL.
t = %§-derivative of T with respect to x.
Q = horizontal external force at top of slip
surface.

Figure 2-16.--Definition Sketch for Janbu's Method.

 

 

 

 

w—‘f—m
«jLine of Thrust
AW
T
AY0
-_1__’L/ ...___E
E+AE 1i<;_
r 7°12
T+AT ‘
134/i
n S
(W
AY N
Ax = tanat
_ Ax . =
L - cosa ’ S SeAL

Figure 2-17.--En1arged Slice.

 

 

Figure 2-18.—-Forces on a Slice Boundary.

49

T = -tandt(E) (2-52)

Equilibrium of each slice in y and x-directions

(Figure 2-16) yields
AW + AT = S sind + N cosa (2-53)
and
AB = -S cosa + N sina (2-54)

Equilibrium of the whole mass in y and x-directions

(Figure 2-16) yields

2(S sing + N cosa) = ZAW, i.e. EAT = 0 (2-55)
and

Z(—S cosa + N sing) = -Q, i.e. ZAE+Q=O (2-56)
Combining (2-53) sind— (2—54) cosa, yields

(AW + AT)sina - AE cosq = S. (2-57)
Introducing (2—57) into criterion (2-56), yields

Q + Z(AW+AT)tana = ZS/cosa = ZseAx/coszd.

(2-58)

Coulomb's equation for shear strength, is given by

N
s=c+

e e KAL - u)tan¢e (2-59)

50

in which, according to the definition of the safety factor,

F,

O

tan¢'
F

 

; tan ¢e = (2—60)

(D
“I

where c' and tan¢' denote the available shear strength
characteristics.

Introducing for N from (2-53) into (2-59) and
solving for Se'

ce + (p+t-u) tan¢e

Se = l + tana tan¢e (2_6l)

 

When combining (2—60), (2-61) and (2-58) and solving

for F, one finds

 

Z[c'+(p+t-u)tan¢']Ax
F = n“ = 5% (2-62)
Q + Z(p+t)tanan

 

where, na = cosza(l+tand tan¢'/F).

From the moment equilibrium condition (Figure 2-16) one

gets,
T = —tanat E. (2-63)
x
Since EX = 2 E, equation (2-62) and equation (2-57) yield,
0
Tx - —tandt 2(B-F) (2-6 )

O

51

From which T and AT are obtained for all slices. The
implicit equations (2-62) and (2—64) are solved by successive

approximations.

2.4.2 Morgenstern's Method

 

The method proposed by Morgenstern et_gl, (1965), is
used to calculate the factor of safety of non-circular slip
surfaces. The method can be used to consider a wide
variety of surfaces and any configuration composed of soils
with differing shear strength properties. It is also able
to make an allowance for complex pore-pressure distributions
and hence treat stability problems in terms of effective
stress. The main theoretical justification for the method
is that all equilibrium conditions are satisfied. (The
methods formerly presented in this paper do not satisfy all
the equilibrium conditions). It is also an application of
the method of slices. The solution of the equations is
currently obtained using a digital computer.

We investigate the equilibrium of the potential
sliding mass shown.on Figure (2-l9a). In this Figure: (l)
the equation of the assumed slip surface is y = y(x); (2)
the equation of the surface of the slope, which is taken
as known, is y = z(x); (3) the equation of the position
of action of the effective horizontal thrust between slices
is unknown and given by y = yé(x); and (4) the line of

thrust of the internal water pressure is y = h(x).

52

The forces acting on an infinitesimal slice of
width dx of the potential sliding mass are shown on Figure
(2-19b). In this Figure:

E' denotes the lateral thrust on the side of the
slice in terms of effective stresses.

X denotes the vertical shear force on the side
of the slice.

dW denotes the total weight of the slice.

P denotes the resultant water pressure acting
on the side of the slice.

dP denotes the water pressure on the base of
the slice.

dN' denotes the effective normal pressure on the
base of the slice.

dS denotes the shear force acting along the base
of the slice.

a denotes the inclination of the base of the
slice with respect to the horizontal.

The condition that there be no rotation of the
slice is satisfied if the sum of the moments about the
center of the base of the slice is equal to zero. By
taking moments about the mid—point of the base of the slice

we find that

E ‘[ (y—yp - (~31) J + PWE (y—h) - (4211) l- (E '+dE '> [y+dy—y;-dyg+ page) J

-x 521$ -(x+dX)%§-(pw+dpw)[(y+dy)-(h+dh) €11}de g = 0 (2—65)

53

  
  

a) Slope Geometry and
Force Location

 

 

 

 

 

 

 

l ('y+dy) - (yfiyp
_ -1Hdh)
(y h (y-yp (Y+°Y) _ (_ __1L

-__- | _____IL

I ‘ ’/ a?
.L/"

./~ (3%

b) Forces on an \r/
Infinitisimal dN'S

Slice

Figure 2—19.—-Potentia1 Sliding Mass Used for
Derivation of Morgenstern's Equations.

After simplifying and proceeding to the limit as dx+0, it
can be readily shown that

dP
_ d , . _ dE' d _ w _
X‘af‘E Yt’Yaf-+a;‘Pwh’Ya§’- (266’

For equilibrium in the N direction, we find

dN'+dPb = dWcosa - dX cosa-dE'sind -desina. (2—67)

54
From equilibrium in the S direction, we find

d8 = dE' cosa+de cosa-dXsind+dW sing. (2-68)

The Mohr-Coulomb failure criterion in terms of effective

stresses may be expressed as

as = %[c'dx secu+(dN')tan¢'] (2—69)

where c' is the cohesion intercept, ¢' is the angle of
shearing resistance, and F denotes the factor of safety.

It should be noted that equation (2-69) also
constitutes a definition of the factor of safety. The
factor of safety with respect to shear strength has been
adopted here. It is that value by which the shear strength
parameters must be reduced in order to bring the potential
sliding mass into a state of limiting equilibrium. It is
clear that the factor safety with respect to moment ratios
cannot be utilized in non—circular analyses where the
shape of the sliding surface is arbitrary.

Eliminating dS from equations (2-68) and (2—69)

we obtain

%[c'dx seca+(dN')tan¢'] = dE'cosa+dec05d-dXsind+dWsind,
(2-70)

55

 

 

 

 

 

 

 

 

a)
x o ,
=: y at
T XY
l xy r +————dx
__A XY 3X
b) 0' dw 80;
‘) ——>1x ‘ i ‘— O}'(+———dax X
y rxyl
‘rj_‘
8Txya 80'
xy+ 3y y o'+ y
Y
Y
Figure 2-20.--An Element at an Interface Between

Two Slices.

Eliminating dN' from equations (2-67) and (2—70),

and dividing by dx cosa yields

2'5e02a+£eei'_[911 - £25 - <11: tamfir tam—d}, seca]
F F dx dx dx dx dx
dP
_ dE' w _ dX dW _
— a;— + a;— dxtana + Etana. (2 71)
In the specified co—ordinate system, tana = -g%.

and equation (2-71) becomes

56

 

where
de = ru dW seca, (2-73)

and ru is the pore-pressure ratio defined by Bishop and
Morgenstern (1960).

Therefore, the two governing differential equations

are
dP
_g_. dilé. -__‘i
(l) X _ dx(E yt) y dx dx (Pw h) y dx
and

 

(2) ggitl- tani dYJ +§xtta§¢ + §§J= §L{1+(g§)23

(2-74)

dPwrtancb' dy tand' +dy

+deF 8‘52‘1]+a‘{ d_-r1.1(i[l+(x

 

 

—) zjtan¢ }

If y is specified as some function of x, we have,
in general, a statically indeterminate problem involving
the unknown functions E', X and Yé, and the two governing
differential equations. The indeterminacy arises from our
lack of knowledge of the stresses existing in the soil mass.

If the stresses could be determined, the displacements

could be predicted using a representative stress-strain

57

relationship. This would obviate the need of doing limit
equilibrium analyses. It is our inability to obtain an
adequate stress analysis that justifies the application

of limit equilibrium methods. At the same time, not knowing
the stresses makes it necessary to invoke an assumption in
order to render the problem statically determinate.

There are three classes of assumptions that can
be made:

1. The distribution of normal pressure along the
sliding surface can be assumed. The friction circle
analysis is an example of a method that adopts this type
of assumption. We have chosen to eliminate the normal
pressure from the equilibrium equations and thereby place
the burden of the indeterminancy on the internal forces.

2. An assumption may be made regarding the

position of the line of thrust. For example if

y-yt = a(y-Z) (2-75)

the moment equilibrium equation, neglecting the pore-pressure

terms, becomes

x = E g—i - a g§(E(y-z)) . (2-76)

Equations (2-76) and (2-74) now define a statically deter-
minate problem where the magnitude of a and F must be found

by satisfying the appropriate boundary conditions. An

58

attempt (Morgenstern et_§l., 1965) has been made to obtain
solutions using equations (2-74) and (2-76) but ill-
conditioned functions arise in the solution of the dif-
ferential equations and it was not found possible to
obtain a satisfactory numerical procedure.

3. Assumptions may be made regarding the relation
between the E' and X forces. If we isolate an element
at the interface between two slices, as shown on Figure
(2-20a), the effective stresses acting on this element will
be as given on Figure (2-20b). For a specified geometry

and slip surface the internal forces are determined by

El

fyo};(y) dy (2-77)
Z

and

Y
X = fzrxy(y) dy (2-78)

We may therefore assume that
X = Af(x) E'. (2-79)

If f(x) is specified the problem is statically determinate
and A and F may be found from a solution of the differ-
ential equations that satisfies the appropriate boundary
conditions. The function f(x) can take any prescribed form
in principle. However, the behavior of soil imposes certain
limitations so that only a certain range of functions will

be reasonable in practice. Estimates of the function can

59

be obtained from elasticity theory. More reliable field
observations of internal stresses in soil slopes may also
be useful in estimating the distribution of the internal
forces.

To simplify the equations it has been found con—
venient to define A f(x) by using the total horizontal

stress E instead of the effective stress E'. Thus we define

E = E'+Pw (2-80)
and the point of application yt of the total stress by

Eyt = E'yt, + Pwh (2-81)
Then instead of (2-79) we assume

X = Af(x)E. (2-82)

In order to investigate the stability of a soil
mass with any slope and properties it has been assumed
that the potential sliding body may be divided into a
number of finite slices by vertical lines with co-ordinates
x0,xl,...xn. This division is carried out so that within
each slice the portion of the slip surface is linear. Also
within each slice, the interface between different soil
types and pore pressure zones are linear and the function
f defined by equation (2-82) depends linearly on x. Hence

within each slice we have

60

y = Ax+B (2-83)

dW

a; = px+q (2-84)
and f = kx+m. (2-85)

Equations (2-83) and (2-84) allow a section with
any arbitary shape to be approximated in the analysis.
Equation (2-85) assumes that the ratio of the internal
forces changes linearly over a segment of the sliding body.
This assumption is not unduly restrictive because k and m
may be chosen to vary from segment to segment and any
continuous distribution of internal forces can be approxi-
mated in this way.

Using equations (2-80) to (2-85), equation (2-66)

becomes

_ d _ dE _
x — aglEyt) Ya; (2 86)

and equation (2-74) becomes

(Kx+L)%% + KE = Nx+P (2-87)

where

tan¢'

K = Ak( F

+ A) (2-87a)

 

61

 

 

 

L = m<tan¢ flA)+l-A tag¢' (2-87b)
N = p[—3§$—+A-r u(1+A 2)tan¢ J (2-87c)
and p = —-(l+A 2)+q[Ea_g¢_+A- ru(l+A2)ta2¢']. (2-87d)

Equation (2-87) can be integrated across each slice
in turn starting with the value E = O at the beginning of
the slip surface. If, for each slice, x is measured from
the beginning of that slice, then the solution which

satisfies
E = Ei when x = 0 (2-88)

is

2
Nx
“[E. iL+—§_'+ PX]. (2-89)

['11
I

_ L+Kx

The value of E at the end of the slice is determined
and this gives the starting value of E for the next slice
unless the end of the slip surface has been reached. The
boundary condition to be satisfied at the end of the slip

surface is

E = E when x = x (2-90)
n n

where En is usually zero.
Satisfying equation (2-89) and its boundary condi-

tions is alone insufficient to insure complete equilibrium

62

since equation (2-86) must also be satisfied. This can
be done by determining yt from the values of E found from
equations (2-89) and (2-82), provided they satisfy the
following necessary condition. By integrating equation
(2-86) we have
_ _ x dy
M - E(yt-Y) - f (X-Ea§)dx. (2-91)
x0

Since in general M = 0 when x = xn, then

X

_ n d _ _
Mn — fx0(x-Ea§)dx — o. (2 92)

If equation (2-92) is satisfied, values of yt can be found
from equation (2-91) thereby insuring that each slice is
in moment equilibrium.

Hence in order to find values of A and P such that
all the equations of equilibrium are satisfied, we start
with guessed values of A and F and then intergrate across
all the slices to obtain the values of En and Mn which in
general will not both be zero. Then, by a systematic
iterative method of modifying A and F, values are finally
obtained for which En and Mn are zero.

There is one complication, however, which will be
described here. In some cases there is more than one
pair of values of A and P which satisfy the equations for

a given surface and one must decide which is the physically

63

meaningful solution. The rule which has been adopted is
that the values of A and F should make the values of
(L + Kx) positive for the complete range of x of the
slip surface. Use of this rule appears to produce unique
values of A and F (Morgenstern _e_t__a_l., 1965). From equation
(2-89) it can be seen that if (L + Kx) is zero for any
value of x, E becomes infinite at that point. This is
not physically possible. Although the values of (L + Kx)
are discontinuous across any of the boundaries of the
slices xl,x2, . . . . , xn—l, one should not expect-
(L + Kx) to change sign across these boundaries. If we
consider a very thin slice near each xi, in which some
soil properties are very quickly varying but continuous,
then if (L + Kx) changes sign across a discontinuity, it
should be zero at some point in the corresponding thin
hypothetical region. This indicates that (L + Kx) should
be of the same sign for all x, and from experience it
appears that it should be positive due to the dominance of
the contribution of unity to L in equation (2-87b).

Some special assumptions relating the X and E

forces are of interest. If it is assumed that

_ Edy _
X — —dx (2 93)
it can be shown that
Y = Y (2—94)

64

and the factor of safety can be found from a modified form
of equation (2-90) alone. The assumption stated by equation
(2-93) is similar to that adopted in the conventional
circular analysis, Bishop (1954). Although this assumption
simplifies the numerical calculations it is not acceptable
because the implied internal forces are likely to be
physically inadmissible. There is also no reason for the
relationship between the internal forces to depend solely

upon the inclination of an arbitrarily chosen slip surface.

2.5 The Method of Characteristics

 

The method of characteristics was a development of
the ideas suggested by Rankine (1857). Kotter (1903)
considered differential equations of equilibrium, and the
condition of limiting equilibrium at each point from a
set of equations of limiting equilibrium and then trans-
formed them to curvilinear coordinates. Further develop-
ment of this theory was done by Prandtl (1920), Von Karman
(1927) and Caquot (1934). However, due to the absence
of a general method, all these investigations found only
a limited application in practice.

In 1939 a general method was derived by Sokolovski.
The general idea of the method is to derive an exact theory
of limiting equilibrium. To do this the material is assumed
to be perfectly plastic (the strain state is at or beyond

yield point) throughout the continuous region to which

65

thhstheory is applied. The limiting stress is assumed to
obey Coulomb's criterion, and no strain hardening occurs
beyond the limiting stresses. Within the region of interest,
the compatability of strain is violated, but equilibrium

conditions are still satisfied.

 

 

“‘ «90° T ‘.

Y
\ x
0x
Y
Y

Figure 2-21.--Coordinates and Loading Systems.

In the mathematical analysis, plain strain is
assumed. Referring to Figure 2-21, the z-direction is
perpendicular to the paper, and the stress components
Tyz = sz = 0. The remaining components ox, 0y, oz and

TXY are independent of the coordinate 2. For convenience

we take a compressive stress to be positive and a tensile

stress to be negative.

66

From the equilibrium condition (anywhere within

the soil mass)

 

 

80 BT
J XY: ' _
8x + By YSlna (2 95a)
and
3T 30
XY Y = _
8x + —§— ycosa . (2 95b)

With the Coulomb failure criterion (in terms of ox, 0y

and Txy)

. 2
1 _ 2 2 _ Sln ¢ 2 _
3(Ox oy) +TXY - 4 (ox+oy+2ccot¢) , (2 96)

 

and the given boundary conditions, the problem can be
considered as statically determinate.

Sokolovski (1960) developed a procedure called
"the method of characteristics and finite difference" to
solve the problem as follows.

The first consideration was to assume the existance
of slip lines when limiting equilibrium is reached. At
the limiting equilibrium the two-dimensional problem
xy’ as
indicated in equations (2-95). The three stress components

concerns three stress components, Ox' 0y and T

can be expressed in terms of two functions 0 and 6, shown

in Figure 2-22, where o is the distance from the point of

67

intersection of the Mohr-Coulomb envelope to the center of
the respective Mohr circles, and 6 is the angle between

the direction of the major principal stress and the x-axis.

That is
Ox = o(l+sin¢ c0526) - w (2-97a)
0y = 0(l-sin¢ c0526) - w (2-97b)
Txy= osin¢ sin26. (2-97c)

Equations (2—97) were obtained from the state of stress
at yield condition (Coulomb's circle). Substituting
equations (2-97) into the equilibrium condition (2-95) one

obtains the basic set of equations

a_o__

80 36 _ .
8y ——-cosZB )— YSina

. 80 . . . .
(l+Sin¢c0526)§§+Sin¢51n26 2051n¢(51n26ax '5?

(2—98a)

. . 80 . 30 . 38 . 36 _
Sin¢Sin26§§+(l Sin¢cosze)§§+2051n¢(c0526§§+51n26§§) — ycosa

(2-98b)

Multiplying the first of these expressions by sin(6tu) and

TT

the second by -cos(6iu), where u=4 %, after some elabora-

tion we get the two expressions

sin(a—¢)
cos¢

 

36_ycos(a-¢)]

30 86
|:ax 20tan¢8x Y 5; cos¢

Jcos(9-u)+[%%—20tan¢

sin(6-u) = O (2-99a)

68

.Edwunflaflzvm mGHuHEHA um wmumum mmmuumll.mmlm musmflm

.mmoam>cm nEoHsoonusozAnv

mmmu m>fluud

 

 

 

 

 

 

 

 

mmmu m>flmmmm u +
ommo
m>wuoms epoooua uh
mmmu m>flmmmm \ m
\ I
\
+H
.IH .
o k\ 6 mo 6 o e _
.0 Iii
: , e

I.|u nlnllul. Inlhnllu

+maom // mxv.»ov w maom _
z , _‘l W lb
., +.o GP

 

.mmcwamflam mo
coaumucwflqumv

 

 

 

69

and
[——+20tan¢gxy y§£%é%%gljcos(6+u)+[——+20tan¢ay Y39%é%%glj
sin(e+u) = O. (2-99b)
By introducing
x = % cot¢.2n % (2-100)

and recognizing that

Q)

30 = 224. 0 _ 3X

upon substitution in equations (2-99), we find

 

 

36 35 _ _ 1998(6+a-u) _

§§'+ tan(6+u)a _ - 20sin¢cos(6+uT (2 101a)
and,

£1 = = _ YCOS(9+a+p) _

3x + tan(6 “)3 3y b Zosin$cos(B-u)* (2 101b)
where g = x+6
and n = x-e .

Equations (2-101) are "Sokolovski's basic equations."

Noting the relationships

0 = c exp[(g+n) tan¢] (2-102a)

and

70

e = 9&1 (2—102b)

we see that the problem has been transformed to the deter—
? 9

mination of the functions E and n, (E é £(x,y), n én(x,y)).

By considering equations (2-101) and the expressions for

the total differentials,

85 dx + 35

 

 

 

d5 = 5;» gy'dY
(2-103)
-3n 3n
and using Cramer's rule, we have
35 = a dy-tan(6+p)d€ (a)
5;. dy-tan(6+u)dx
3Q = dE-adx (b)
By dy-tafiTB+fi§dx
(2-104)
§fl_= b-dy-tan(6-u)dn (c)
8x dy-tanTG-u)dx
d -b
§D.= '1 dx (d)

 

3y dy-tan(6-u)dx '

Equations (2-104) can be used to obtain solutions
of slip lines within the soil mass and solution of lines of
discontinuity.

To obtain the solution of slip lines within the

soil mass we set the numerators and the denominators of

71

Equation (2-104) equal to zero simultaneously and obtain
two families of characteristic curves (n and E).

For the n-characteristics

dx tan(6-u) . (a)

dy
and (2-105)

Cgs¢[sin(a-¢)dx+cos(d-¢)dy]. (b)

 

do-Zotan¢ d6

For the E-characteristics,

dy = dxtan(8+p) (a)

and (2-106)

do+20 tan¢ d6= E%§$[sin(a+¢) dx+cos(a+¢)dy]. (b)

To obtain the solution of the lines of discontinuity,
we set only the denominators of Equation (2-104) equal to
. . . 35 3E 33 31 -
zero (the four partial derivatives Saw 5;, 3x and By W111

become discontinuous and in turn the stresses are also

discontinuous) and obtain the locus of the lines of

discontinuity
dy _ - . dy — —
—§-— tan(6 u), and d§'- tan(6+p). (2 107)

The lines of discontinuity are actually envelopes of the
slip lines obtained in the first part (equations (2—105)

and (2—106)). Since along the line of discontinuity

72

equilibrium is violated, the restricted region in which
the equations of equilibrium and strength relation can
yield meaningful answers is the area within the envelope
only.

To obtain solutions of the differential equations,
equations (2-105) and (2-106), Sokolovski integrated the
expressions by a finite-difference procedure. Since it
is a long and tetious procedure, explanation is referred
to Harr (1966, pp. 252-324). Harr also presents solutions
by this method of some problems in bearing capacity and
slope stability. The final solution obtained from the
analytical method is in the form of the maximum load that

the structure can sustain.

2.6 Numerical Method

 

A numerical procedure has been developed (Bell,l969)
to approximate the location of the critical rupture
surface and to assess the importance of its shape in slope-
stability calculations. The analytical procedure is an
extension of the slip analysis developed by Bishop (neglect-
ing the unbalance of vertical shear forces on an individual
slice). A system of equations is set up with points on
any assumed sliding surface as unknown variables. By
the use of an optimization process, the depths can be
obtained which yield the smallest factor of safety (F).
The method can be used to analyze homogeneous soil slopes

only.

73

Bell (1969) has shown that the modified factor
of safety from the Bishop approximate method of slices
for a statically loaded finite homogeneous slope under
a homogeneous pore-pressure distribution, Figure 2-23,

may be expressed by

 

 

h.
*
N seca.+(l-r )—iseca.
l u hO i
[X tanai J (2—108)
l+——§T—
F = F*
h.
2 —£ sing
h .
where F* = —¥E—T (2—109)
tan¢
* C. 2
N - W ( "110)
_ u _
ru _ 7_ (2 111)

where u = the homogeneous pore-pressure acting on the
potential sliding surface, Y = the total unit weight of

the soil, and the strength parameters c' and ¢' are the
effective cohesion and effective friction angle respectively.
The geometric variables ai and hi are defined by Figure 2-23,
and hO is any convenient reference dimension (usually taken
to be the total height of the slope). For a particular

slope geometry, the dependent dimensionless variable, F*,

is a function of only two independent dimensionless variables,
the modified stability number N* and the homogeneous pore-

pressure ratio ru.

74

 

 

 

 

 

 

 

 

 

 

 

 

A Y.2
Tgo xn EJ
b '1
EEK—I-
y
' d
I nub
l
l Yi
I
/ d1-1I di
I .
I : ' I}11 /
I I I 5 I l 7'“ X
I I ' '~zi
d I, I ‘—”_f
hl 1 d2 d3 d4 0‘1
h h z
2 h 4

Figure 2-23.--Finite Slope with General Potential
Slip Surface.

For a particular slope geometry, N*, and ru, the
most critical potential slip surface might first be assumed
to be that for which F* is an absolute minimum. For the
case of a circular surface the problem practically involves
the minimization of the factor of safety with respect to
two coordinates of the circle center and the radius.

With less restrictions on shape, the factor of
safety with slope geometry, N*, and ru fixed, may be
considered to be a function of any number of primary space

variables. The following expression is proposed

75

F = ¢(d1, d d x , x0) (2-112)

2’ 3 ° ' ‘ n-l, n

in which n = number of slices of equal width, d1 = depth

of the right side of the 1th

slice, x0 = horizontal
coordinate of the toe of potential slip surface,and xn =
horizontal coordinate of the head of potential slip sur-
face. The relationship between the primary space variables

and those of equation (2-108) may be approximated for

relatively narrow slice widths by

 

 

(d. +d.)
_ 1-1 1 _
hi — 2 (2 113)
(xn-xo)
b = __ (2-114)
n
zi = yi-di (2-115)
-1
tan (2 -z._ )
a. = l 1 1 (2-116)
1 b

in which the slope geometry is specified by y = F(x) and
Figure 2-23.

The solution of the problem, in general terms,
now involves the minimization of the factor of safety with
respect to the primary space variables listed by equation
(2-112). The adoption of modified dimensionless para-
meters such as those of equation (2-108) through (2-lll)

simplifies both the calculations and the presentation of

76

results. The following expression is adopted effectively

in computation

* = ‘ ' ' - - _ -
F ¢(dl, d2, d3 . . . . dn-l’xn’xo) (2 117)

in which each dimensionless space variable listed is
defined as the ratio of the real distance to the reference
height, ho’ associated with the slope. In mathematical

terms, the solution is stated explicity by

 

 

3F* _ 0 . _
5T- (1 — 1, 2 o o o ., n‘l) (2-118)
1
8F* = 0 (2-119)
axn
*
and 35 = o , (2-120)
8x0

Three restrictions are ultimately placed on the form of the
potential slip surface. The first is already contained in
equations (2-112), (2-117), and (2-118), where vertical
cracks at the head or toe of the potential slip surface

are not permitted. The second restriction is that the
potential slip surface must be planar or concave upward.
The third restriction prevents the sliding surface from

penetrating a rigid stratum below.

77

The first restriction could well be invalid at
the head of the surface when a cohesive slope is near,
or at, yield and this is deserving of further investigation.
The second restriction is based on a lack of field evidence
to the contrary, and for constant slice width, it is

numerically required that

zi_l - zzi = zi+13 0 (2-121)

or the dimensionless equivalent statement that

2. - 22. + 2i > 0. (2-122)

An additional restraint was considered but not
invoked which involved the angle of inclination of the
potential slip surface at soil-air or soil-water interfaces.
Detailed laboratory or field studies on the shape of
failure surfaces would aid in clarification of this point,
at least for the yield condition.

Part of the results (Bell, 1969) obtained by the
numerical method is presented on.Figure 2-24 which shows
shapes of critical potential slip surfaces in comparison
with circular slip surfaces. Stability analysis by this
method was found to be in agreement with the method intro—

duced by Bishop and Morgenstern (1960).

78

 

 

 

 

 

 

 

Tan. Circle .___

Approximate General—+—

Slope = 421

Figure 2-24.--Shape of Critical Potential Slip
Surfaces with Variable Pore-Pressure, (After Bell, 1969).

CHAPTER III

CONCEPTS IN ELASTICITY AND PLASTICITY,

AND BEHAVIOR OF SOILS UNDER LOAD

3.1 Rheological Properties of Materials

 

The general behavior of materials under stress
according to a certain assumption can be used to approxi-
mate soil-deformation characteristics. Investigations of
material behavior follow two main lines of inquiry: the
study of the effect of the assumption of a property or
quality, or the examination of the effect of a quantita—
tive assumption. Two stress-deformation relationships
must be considered in two different categories: one that
connects the volume change with the hydrostatic stress,
and the other which relates shearing or distortional
deformation to the shearing stress. Furthermore, the
variation of the behavior in time and with temperature
may also enter in one or both relationships.

There are two kinds of ideal materials: elastic
solids (Figure 3-la) and viscous fluids (Figure 3-1b).
(If the behavior of the material under an applied stress
is dependent only on the applied stress and not on the
previous history of stress or deformation in time, the
substance is usually referred to as "ideal".)

79

80

3.1.1 Elastic Solids

 

A certain level of hydrostatic or shearing stress
applied to elastic solids causes an immediate deflection
or deformation of the body; later, when the stress is
removed, the body returns to its original shape and size,
and there is a one to one relationship between the state
of stress and the state of strain, for a given temperature.
Heat transfer is considered insignificant, and all the
input work is assumed converted to internal energy in the
form of recoverable stored elastic strain energy, which
can be recovered as work when the body is unloaded. In
general, however, the major part of the work input into
a deforming material is not recoverably stored, but
dissipated by the deformation process, causing an increase
in the body's temperature and eventually being conducted
away as heat. The relationship of the deformation of the
material to the applied stress can be of any manner, linear
or non-linear. If the relationship is linear or the
strain is directly proportional to the applied stress,
the material is referred to as a Hookean Elastic Solid.
The classical elasticity equations, often called the
Generalized Hooke's Law, are nine equations expressing
the stress components as linear homogeneous functions of

the nine strain components

= C

Tij ijkl Ekl (3‘1)

81

where Tij is the stress tensor, is the strain tensor,

Ekl

and Cijkl is the tensor of elastic constants.

Due to the symmetry of Tij and E for elastically

kl’
isotropic material, the eighty-one independent elastic
constants can be expressed in terms of just two elastic

moduli

T.. = A 6

1] Ekk + ZuEi. (3-2)

ij 3

where A and u are the Lame's elastic constants, and 6ij

is the Kronecker delta. Contraction yields

Tii = (3A+2p)Eii, or Ekk = Tkk/(3A+2p) (3-3)

Substitution of (3-3) into (3-2) yields

__ i' 1 _
Eij ‘ 277117? Tkk + ZITij (3 4)

The two Lame's elastic constants, A and p, are
related to the more familiar shear modulus, G, Young's

modulus, E, and Poisson‘s ratio, v, as follows:

_ _ E _ v E _

 

Equation (3-4) can also be written in the form

l+v

\)
Eij E Tkk

Equation (3-6) can be represented by two relation-

ships as follows:

and

where

stress

 

 

Tij 2GEij
P -Ke
l
' .—
Tij Tij §Tkk6ij
U l
Eij Eij_§Ekk6ij
P _ Tk_k.
3
e Ekk
E
K 3 - v
c-Linear
“'Non-
linear
strain

a) Ideally Elastic Materials.

82

(distortion part) (3-7)

(hydrostatic part)

= the deviatoric stress,

= the deviatoric strain,

= the mean pressure,

= the volumetric strain,

= the bulk modulus.

Nonlinear
U)
m i
m
H
4.)
U) /
H
m
m D
g Linear

 

 

Velocity Gradient, g;

b) Ideally Viscous Fluid.

Figure 3-l.--Stress-deformation Relationsips of
Ideal Materials.

83

3.1.2 Viscous Fluids

 

The qualitative postulate for a viscous fluid
is that the rate of shearing deformation of the material
depends on the applied stress. If a direct proportionality
between the deformation velocity gradient normal to the
shearing surfaces and the applied shear stress is assumed
as a quantitative description of the quality of the sub-
stance, the fluid is said to be Newtonean, with viscosity
being the proportionality constant. However, a further
relationship between hydrostatic stress and volumetric
strain must be specified for a fluid. Quantitatively the

material may or may not be compressible.

3.1.3 Plastic Materials

 

The two substances considered in sections 3.1.1
and 3.1.2 have the qualities of ideal elasticity and
fluidity for which various quantitative stress-deformation
relationships can be suggested. If, however, a material
experiences finite deformations up to a certain level of
stress only, after which it deforms continuously, it is
said to be "plastic" and to flow plastically above the
yield stress. If it is postulated that no deformation
takes place up to the limiting stress, and when the
stress is reached, the material deforms continuously with-
out stress increase, the material is referred to as

"ideally plastic."

84

The constitutive equations for plasticity are not
well established. Malvern (1969) states that several
different idealizations have been proposed, all failing
to account completely for the phenomena observed, but
some nevertheless are quite useful. Most of the applica-
tions made have used the perfectly plastic theories.

Most applications of plasticity theory fall into
one of two categories, as follows:

(a) Contained plastic deformations. When large
deformations are prevented by surrounding elastic material
or by elastic redundant members of a statically indeter-
minate structure, the plastic deformation is said to be
contained. Collapse will not occur even though some
parts of the body are loaded beyond the elastic limit.

For such contained deformation it is reasonable to neglect
work-hardening, and to use as the deformation variable

the small-strain tensor, whose increments for small

strain are negligibly different from the increments of
natural strain.

(b) Uncontained plastic flow. In metal-forming
processes (e.g., drawing, extrusion, rolling) the problem
is to produce a desired large deformation by cold-working.
The cold-working involves hardening, which may be desired
in order to have a stronger finished product, but which
interferes with the process itself not only by necessitat-

ing greater working forces, but also because nonuniform

85

hardening may produce residual stresses. It may be
reasonable to neglect the elastic-strain increments

in comparison to the plastic-strain increments at large
strains and to take as the deformation variables either
the rate-of—deformation tensor or the natural-strain
increment.

The two categories above represent two opposite
extremes of possible problems. One of these two extremes
is satisfactorily applied to most practically important
problems of metals. Problems intermediate between these
two extremes seldom occur in metals but may occur in non-
metals, for example, in soil mechanics or in polymers.
Stresses associated with deformations of such materials
are also much more rate-dependent than is the case with
metals, where the usual theories of plasticity neglect

rate-dependence altogether.

3.2 Idealized Stress-strain Curves

 

Idealized stress-strain curves frequently used for
applications in mathematical calculations are illustrated
in Figure 3-2. It has already been seen that cases (b) and
(d) can be used to characterize contained plastic deforma-
tions, while (b) and (c) can represent uncontained plastic
flow (where elastic strains are neglected). According to
Malvern (1969), perfectly plastic material has proved to

be quite useful in limit analysis of contained deformation.

86

 

stress
stress

 

 

 

 

strain strain

a) Perfectly Elastic, b) Rigid, Perfectly Plastic.
Brittle.

stress
stress

 

 

 

 

strain strain

c) Rigid, Linear Strain d) Elastic, Perfectly Plastic.
Hardening

Stress

 

 

strain
e) Elastic, Linear Strain Hardening.

Figure 3—2.--Idealized Stress-Strain Curves.

Limit analysis, without analyzing the detailed contained
deformation up to limit load, seeks to find the limit load
at which uncontained plastic flow would occur. If the
geometry change of structure below the limit load is
negligible, the same limit load is predicted by rigid,

plastic theory as by elastic-plastic theory.

87

3.3 Mohr's Theory of Failure

Mohr's theory of failure assumes that the critical
shear stress is not necessarily equal to the maximum shear
stress but depends also on the normal stress acting on
the shear plane. The limiting stress states are deter-
mined by the function If = f(o'), wherecf = effective normal

stress on the shear plane (see Figure 3-3).

 

 

I T =fO'
I) f ( )
Linear
Approximation
E’,/i
/
,/‘T /
. /
I '-
W‘Uniaxial Z:Pure Z:Uniaxial ZC-Combined 0'
Tension Shear Compression Stresses

Figure 3—3.--Failure Envelope of Mohr's Stress
Circles.

The curve representing this function in the system
of rectangular coordinates, (0',T), is the envelope of

all Mohr circles representing the limiting stress states.

88

The failure envelope of normally consolidated
cohesive soils is a good approximation to the straight

line

= t I , I _
If C0 + o tan¢ (3 8)

where If = shear strength, 0' = effective normal stress,
and the definitions of c8 and ¢' are shown in Figure 3-4.
This straight line approximation of Mohr's failure

envelope expresses Coulomb's failure criterion.

unload —7~7 [—————-Approximate for 0-C-

' Reload(Preconsolio.ted)

      
  

T

 

I I

(3.. I :L: ' \ 'UP 0’

“““Normally Consolidated

   
 

 

 

Figure 3-4.--Common Forms of Stress Enve10pes for
Normally Consolidated and Preconsolidated Soils.

The envelopes of the limiting stress states for
unloading or reloading of preconsolidated cohesive soils
are usually curved lines forming a hysteresis loop as
shown on Figure 3-4. The preconsolidation load is denoted
by a; in this figure. If, in the first approximation,
this hysteresis is replaced by a straight line, Coulomb's
failure criterion can also be applied to over consolidated

clays.

89

= I I _.
If c + 0 tan ¢r° (3 9)

For normally consolidated cohesive soils, cg is
small in value and can usually be neglected. From equation

(3-8), we have

If = 0'-tan ¢'. (3—10)
For plane-strain problems the failure condition

is assumed to be independent of the intermediate stress.

Using the notation in Figure 3-5, Coulomb's failure

criterion can be expressed as follows:

oi-oé oi+oé
—-T—- = (C ' C0t¢ ' + T) Sin¢ ' , (3‘11)
or oi-oé = ZC' cos¢' + (oi +o§)sin¢'. (3-12)

The influence of the mean principal stress is partly taken

into account when using the failure law

I = f(o'

oct ). (3-13)

Coulomb's failure theory has been experimentally
proven to represent the characteristics of soils, and will
be used as the failure criterion in this paper (even
though the effect of the intermediate principal stress

as assumed is not always true).

90

¢l

 

 

 

 

c'cot¢' 0'

 

 

 

by)

Figure 3-5.--Mohr's Circles for Plane-Strain
Conditions.

3.4 Slingines

 

The stress state at any point in a continuous body
can be represented graphically by Mohr's stress circle
on the 0"T plane (Figure 3-6a). According to Coulomb's
criterion, the strength envelope is represented by two
straight lines on the 0"T plane. The equations for this

envelope are

T = f(cz + O tan¢ ). (3‘14)

As the stress state reaches the yield point, Mohr’s
circle will touch the envelope at two points, P and P'.
The stress conditions at points P and P' will represent

stress states on the slip planes (or surfaces) at any

91

 

c-+otan¢

 

 

 

 

   

= -(c+(3tan¢)

 

a) Mohr's Circle Representing Stress Condition at Yield.

Slip Planes

lie/r:

 

      

b) Orientation of Slip Planes with Respect to the
Principal Stresses.

Figure 3-6.--Graphica1 Representation of Stress at
a Point.

92

point in the medium. The two surfaces are symmetrically
oriented relative to the directions of the principal
stresses (Figure 3-6b). The rupture surfaces are mutually
oriented at the acute angle of (g - ¢ ), or at the obtuse
angle (g- + q; ). With respect to the major principal stress,

the rupture surface makes an angle.

a = i <1} - ¢ /2). (3-15)

and relative to the minor principal stress, an angle
8 = :1 (% + 45/2). (3-16)

If a plastic region forms within the medium and the stress
field is known within the region, a field of slip surfaces

can be graphically plotted within the zone.

3.5 Behavior of Soils Under Load

 

Soil behavior under load depends heavily on the
type of soil. Two major categories, cohesionless and
cohesive, are conventionally classified for soils, with
different strength characteristics being obtained for each

of these soil types.

3.5.1 Behavior of Cohesionless Soils

 

Cohesionless soils are normally composed of bulky
grains ranging in shape from angular to well rounded.
The behavior of granular soils under normal and shear

stresses is presented briefly in this section.

93

When loads are applied, see Figure 3-7, the soil
particles deform more or less elastically in the first
stage. Secondly, there is local crushing at the most
highly stressed points of contacts. Thirdly, both the
elastic distortion and the crushing cause slight trans-
lation and rotation of the grains, increasing the size
of some of the voids and decreasing others.

Both the previous confining stress and the stress
level at the beginning of a stress increment influence
the deformation resulting from the stress increment.

The higher the degree of confinement, the greater the
previous crushing and local adjustments, and therefore
the less additional strain produced by additional incre-
ments of shear stress.

If the shear stress is increased further, two
additional responses are evident. First, the particles
tend to roll across one another (Figure 3-7). The
resistance depends on their angle of contact and is
proportional to the confining stress a. The total
resistance to rolling is the statistical sum of the
behavior of all the particles. The second mechanism is
sliding of one grain across the other. The resistance
to sliding is essentially friction, which is proportional
to the confining stress. A third mechanism involves the
interference and interlocking of the corners of the more

angular, irregular particles.

94

Rigid Rolling v// Sliding
Li /~’Plate up / Crushing

.' I”???
0"" "
(Ce A

Figure 3-7.--Mechanism of Deformation and Shear
in a Mass of Bulky Grains.

            

Interlocking
fracture

If the shear stress becomes sufficiently large,
the effect of the distortion, crushing, shifting, rolling
and sliding of the grains will be continuous movement
and distortion of the mass, or "shear failure."

A typical stress-strain curves for initially
loose and initially dense material subjected to an
increasing shear stress with a constant confining stress,
03, are shown on Figure 3-8. Both curves exhibit strains
that are approximately proportional to stress at low
stress levels, suggesting a relatively large component
of elastic distortion. If the stress is reduced, the
unloading stress-strain curve is nearly the same as the
loading curve. However, not all the strain is recovered
upon unloading, indicating some particle re-orientation

and point crushing, with consequent loss of energy. The

95

hysteresis loss, the area of the stress-strain loop,
represents this energy lost. Loose soils with larger

voids and fewer points of contact exhibit greater strains
and less recovery of strain upon unloading than dense soils.
At higher stresses the strains are proportionally greater,

indicating greater crushing and repositioning.

/fi/‘Peak —————— Loading
- — — — Unloading

   
  

stress

Confining Stress, G3,

Dense Sand constant

Energy Lost in Cycle =
Hysteresis

 

 

strain

Figure 3-8.--Stress-Strain in Cohesionless Soil.

The results of tests on dry cohesionless soils
show that the shear stress at failure, termed the shear
strength, S, is nearly pr0portiona1 to the normal effective
stress on the failure surface, 0'. The Mohr's envelope
is approximately a straight line through the origin which
makes an angle of ¢' with the d-axis (see Figure 3-9).

The equation for soil strength is given by
S = 0' tan¢'. (3-17)

The angle ¢' is termed the angle of internal friction.

96

Actual
Envelope /,
/

//

/
A
,I/ “'Straight Line Through Origin

/’/’ (Approximation)

 

 

0 Cl

Figure 3-9.--Mohr's Envelope for Cohesionless Soil.

In general the major factors in failure are rolling and
sliding. The sliding resistance of the grains is deter-
mined by the effective stress, the coefficient of friction
between the minerals, the surface roughness, and the angle
of contact between the grains. These in turn depend on
the grain shape and the soil structure as reflected in the
relative density. The resistance to rolling depends on
the particle shape, the gradation (particle sizes), and
the relative density. As a result the angle of internal
friction is greater than the angle of friction between the
minerals, and it varies with grain shape, gradation, and
relative density. The Mohr's envelope is not always per-
fectly straight nor does it always pass through the origin
because the resistance to rolling is present even with no
confinement. At very high stresses the envelope may be

curved concave downward due to fracture of some of the grains.

97

3.5.2 Behavior of Cohesive Soils

In cohesive soils, the shearing process is more
complex. Like coheSionless soil, clay is made up of
discrete particles which must slide or rotate for shear
to take place. However, there are a number of significant
differences:

(a) The soil is relatively compressible, there-
fore when a load is applied to saturated clay, it is
initially supported by neutral stresses and is not
immediately transmitted to the soil structure.

(b) The permeability of the clay is so low that
the neutral stresses produced by the load are dissipated
very slowly. Therefore it may be months or even decades
before the soil structure feels the full stress increases.

(c) There are significant forces developed between
the particles of clay by their mutual attraction and
repulsion.

Because of the slow changes in the neutral stress
and the corresponding slow changes in effective stress,
the strength of clays is defined in terms of neutral
astress dissipation. Three basic conditions are defined:

(1) Drained or slow shear (consolidated-drained).

(2) Consolidated-undrained.

(3) Undrained (or unconsolidated-undrained) or

quick shear.

98

In drained shear there is no neutral stress change
and any increase in total stress produces a corresponding
increase in effective stress. The soil consolidates,
reducing the void ratio and water content. As a result of
consolidation the particle spacings are reduced and inter-
particle bonds are increased in proportion to the confining
stress. The shear strength, therefore, increases in propor—
tion to the effective confining stress, and Mohr's envelope
is a straight line through the origin (Figure 3-10). The
angle, ¢D’ is called the angle of shear resistance of
apparent internal friction (typical values lie between

5° to 30°). When a clay has been preconsolidated to a

 

___f_¢D

__JL._.

’,a’ Over consolidated
,‘VflK—-Norma11y consolidated

 

 

Figure 3-10.--Mohr's Envelope for Saturated—Clay
jJ1 Drained Shear.

99

stress of 5c and then unloaded, the particles do not return
to their original spacing and previously higher void ratio.
As a result the interparticle attractive force is not
reduced and the strength at stresses less than the pre-
consolidation load is no longer proportional to the
effective confining pressure but is some what higher

(Figure 3-10). Strength under confining pressures above

the preconsolidation load is given by

S = 0' tan ¢D° (3-18)

Strength under confining pressures below the preconsolidation

load is given by
S=c' + O' tancb'

where c' = intercept on the T axis, and ¢' = is as shown
on Figure 3-10.
The drained shear condition represents the strength
of soil developed by a long-term stress change. However,
it can be used for any problem involving shear in saturated
clays if effective stresses are used. Many slope failures
are due to reduction of soil strength taking place over
a long time.
In consolidated-undrained shear, the soil consolidates
completely under some confining stress, with a correspond-

ing reduction in void ratio and water content.

100

Effective stress
Total stress

>750 ¢

stress/ CU
}$ on ‘t’bD
c
i:I% 7¢w I 4 f;§:-Effective

Total

 

 

 

 

./ 7 \ stress
0: 0° 0' ’ 3 a 0% o 0'
l 1 '
AI uIl 03IzAuI IAuI
a) Positive Au, (Standard Test) b) Negative Au

Figure 3-11.--Mohr's Envelope for Saturated Clay
in Consolidated-Undrained Shear.

A plot of the effective stresses will give the
Mohr's envelope for drained shear (Figure 3-11). A
different envelope will be produced, if total stresses
are plotted. The total stress circles are displaced
horizontally to the right or left by an amount equal
to the neutral stress (iu). The neutral stress developing
during the undrained shear process depends on two
factors: sign of stress changes and over-consolidation
ratio of the soil sample. Normally consolidated clays
develop positive pore pressure under increasing total
stress and vice versa, while heavily over-consolidated
clays give opposite results. Enve10pes of the total
stresses are presented in solid lines (Figure 3-11). The
straight line portion represents results of normally

consolidated clays. The curved portion represents

101

results of over-consolidated clays. The straight line
portion of the apparent Mohr's envelopes of total
stresses will pass through the origin and will have an
apparent angle of shear resistance, ¢CU’ which is about
half of ¢D for the standard triaxial test (Bishop gt_al.,
1957).

The effective stress envelope is always the same
under any stress condition, while the apparent Mohr's
envelOpe changes with changes of stress condition. The

apparent angle of shear resistance, ¢ is less than ¢D

CU'
under the condition which the total stresses are greater
than the effective stresses (Figure 3-lla), and vice
versa (Figure 3-llb).

This test is frequently employed in the analysis
of embankment foundations, where the clay soil is first
fully consolidated by its own weight and later subjected
to a sudden increase in stress by the embankment load.

In undrained shear, both the confining and shear
stresses are applied so rapidly that virtually no con-
solidation takes place. The soil void ratio and water
content remain essentially unchanged, and neutral stress
supports all of the added loads. The soil in its initial

state supported an overburden pressure 06 (or a pre-

consolidation load cé), under which it consolidated to

102

establish its void ratio, water content, and interparticle
spacing. The soil strength resulting from this initial
effective stress can be obtained by using the Mohr's
envelope of drained shear shown on Figure 3-12. An
increased confining pressure, A03, and axial load A01, is
carried almost entirely by neutral stress, and the void

ratio, interparticle spacing, and resulting undrained soil

strength remain essentially unchanged.

/_.Mohr Envelope in
,/ Drained Shear
I /A/ (Effective Stress)
~— ~ / ¢
7] / /(“'T D

f / /

/
fl’f Mohr Envelope in Undrained
j/ Shear (Total Stress)

j/mfl“

AGl I 0,0'
i
b T __

 

 

 

 

 

 

 

 

0' A0 A0

0 _ A0

3

Figure 3-12.--Mohr's Envelope for Saturated Clay
in Undrained Conditions.

The stress conditions during loading are tabulated

in Table 3-1.

103

.wabmu one ca confluommo muommmm mmmuum Hmuusm: one
umuam no: mwOp menu pan AH v M ..w.Hv mmmuum Hmmflocflnm momma one can» mmma mo

HHHB pmoa ampusnum>o wsu Eoum mmwuum Hmmflocflum Hosea may mmmmo acme :er

H ou Hmowm cmESmmm coon was .< .HmumEmumm .musmmmum whom mafia

 

 

He<I we u we me<l.ae< u s me< + me n me
we u we me<i.ae< u s He<+me< + we u He He< .emoa Hmexm mceeee
me n me me< u 9 mn& + we n me
me u we me< n 9 me< + we u He me< .ucmEmchcoo mcflpee
WC H m0 0 u 5 ¥¥WO " MD
oe I He I o co I He aoa a :00
e I e o I e I A .u.e HMfluflch cmcusnum>o
mmwuum o>euommmm mmmnum Heuuswz mmmuum Hmuoa mcwpmoq

 

¢.mcwpeoq omcwmucca How mommmuum mo coflumaonmall.HIm flames

104

The effective minor principal stress is independent
of the added confining stress, and therefore the effective
major principal stress at failure and the strength depend
only on the original overburden stress, 05, and the effective
(drained shear) envelope. A plot of the total stress,
the solid lines on Figure 3-12, shows a series of Mohr's
circles. All have the same diameter, and the resulting
envelope of total stresses is a horizontal straight line.
The intercept of the envelope on the I-axis is approxi-
mately equal to the shear strength (or cohesion) of the
soil in its original condition, consolidated by the over-
burden stress, 08. The apparent angle of friction, ¢U,
is zero. However, the angle of the failure plane a is
determined by ¢D' and is not 45° as might be assumed with
¢U = 0. Only soils for which the difference between ¢D
and the true angle of internal friction (Leonards,l962)
is negligible are considered.

The undrained strength depends on the original

overburden stress, 05(or a consolidation stress), and

g.
on the drained Mohr's envelope. In a compressible soil
such as an unsaturated clay, the overburden stress is
related to the void ratio by the stress-void ratio curve.
As a result the undrained strength of a saturated clay

increases with decreasing void ratio and also decreasing

water content.

105

Since most construction proceeds rapidly compared
with the rate of clay consolidation, undrained strength
is used in most problems of design. Even where the con-
struction is so slow that some strength increase will
develop, the undrained strength is frequently used because
it is the minimum strength and therefore conservative.
Caution must be exercised in using the undrained shear
strength in the analysis of the problems where the final
stress is less than the original overburden load (or
consolidation stress), such as the design of excavated
slopes or the study of land slides.

Undrained shear involves little or no volume
change, but only distortion of the mass (Poisson's ratio
is nearly 0.5). The shape of the stress-strain curve
(Figure 3-13) depends largely on the interparticle bonding
imposed by preconsolidation and by the structure. For
undisturbed clays, the initial portion of the curve is
straight, probably reflecting the distortion of the bonds.
The curve flattens as increasing numbers of bonds are
broken.

If a sample of undisturbed saturated clay is
completely remolded without changing in water content,
and then tested, it will be found that the undrained
strength has been reduced. The difference between the

undisturbed and remolded soil is greatest in sensitive

106

clays. Because of their flocculent structure and edge
to face_bonding, they are relatively rigid and have much
higher initial values of Young's Modulus for a given
water content than an insensitive clay with a more

oriented structure. The loss

 

 

 

 

'”undisturbed

__—undisturbed , insensitive
m m (undisturbed)
U) ’ 3 l”
f3) /’I i3 //
m ‘/ m

/,/K igE <

/ ‘—“Remold 5

// / Remold
/ /

0 strain 0 strain
a) Insensitive Clays. b) Sensitive Clays.

Figure 3-13.--Stress-strain Curves of Clays in
Undrained Shear.

of strength in remolded clays is caused by a break down
in the soil structure and a loss of the interparticle
attractive forces and bonds. In clays originally with
a dispersed structure, the loss is small, while in clays
with a highly flocculent structure or soils with a well-
developed skeletal structure, the loss in strength can
be large.

The sensitive clay reaches a peak strength similar

to that of dense sand, and then becomes weaker with

107

increasing strain (Figure 3-l4b). The strength remaining
after large strain (after failure) is the "residual strength"

and is approximately equal to the remolded strength (Figure

    

  

 

 

 

 

3-l4a).
Peak T

m ____________________

3 Sensitive Peak

5 Undisturbed

m __________ ---—. ___
Sensitive Residual I
Remolded Strength UltiIate

istrain 0'
a) Stress-strain Curve of b) Peak and Ultimate
Sensitive Clays. Failure Envelopes.

Figure 3-14.--Peak and Residual Strengths of
Sensitive Clays.

The reduction of strength can occur in slope
embankments on very sensitive clay foundations where the
change is not only in strength but also in elasticity
properties of the clay. The modulus of elasticity will
decrease with an increase in deviatoric stress (or shear
strain) for clay. For sand the effect of deviatoric
stress is very small in contrast with that of the mean

0 0 0
normal stress or the stress invariant ( 1 +3 2 + 3).

 

With an increase of the stress invariant, the modulus
of elasticity of sand will also increase. The effects

mentioned above can induce local failure in a slope

108

embankment, on sensitive clay, which changes the stress
pattern in the soil mass. The changes of stress state in
the soil mass in addition to the decrease in modulus of
elasticity in the failure zone (for sensitive clays) can
develop a progressive failure zone in the soil mass.
Lower elastic modulus in the area means larger deformation
of the zone which induces great change in the state of
stresses in the adjacent area and further complete failure
zones can be developed. Slope failure can occur by the
effect of local strength reduction.

In general, the stress-strain behavior of any soil
is obtained from the triaxial test. The stress path in
the conventional triaxial test can be any of those shown

on Figure 3-15.

Decreasing Chamber Pressure

q ‘ //*‘(Constant Axial Load)

1 —-Increasing Axial Load
1 (Constant Chamber
Pressure)‘
L P
Consolidation (Increasing
Chamber Pressure)

 
  

 

 

Figure 3-15.--Stress Path in the General Triaxial
Test.

109

Each stress path will produce a certain stress-
strain relationship. For example, in drained-shear,
consolidation of soil takes place in two stages: first
during the addition of the confining stress and second
during the addition of axial stress that produces shear.
In a conventional consolidation test where 62 = 83 = 0,
the lateral stress is a constant fraction of the vertical
stress. Although the mechanisms are similar, the stress-
void ratio curve is different because the stress fields
are different. The stress paths of the two processes
are shown on Figure 3-16. The stress-strain curves are
shown on Figure 3-17 for different stress paths in drained-
triaxial tests.

Consider the constitutive equation of the form

1:3 T!

ij E ij'

= _ X I _
Eij E T kké (3 19)

If the major principal axis coincides with the z-axis,

the equation can be written in engineering notation

= i. '_ I I
Ex EEOX V(0y + OZ) Jr
a = l[0'-v(0' + 0')] (3-20)
y E y x z
and e =[i0'-v(0' + 0')1
z E z x y '

110

 

 

 

 

q q
6
3
2
l 4
l l
p koé 0c p
a) Successive Stresses for b) Increasing Load with
Increasing Load with a Constant Ratio of Axial
Constant Ratio of Axial to Confining Stress (4-5)
to Confining Stress, Followed by Increasing
1-2-3 (as in Simple Axial Stress at Constant
Consolidation Test). Confining (5-6).

Figure 3—16.--Stress Path for Plane of Maximum
Shear Stress in Triaxial Compression.

N f(/////~—Isotropic (0X=0y=0z)
0 /

‘ /

,,—.0 =0 =k0

a / [f x y z

3 / / (simple condition)
13
2 To Failure
3
. = '
fl kOc

m

>

 

 

 

Vertical Strain, 52

Figure 3-17.--Stress-strain in Drained Shear for
Different Loading Systems.

111

In the triaxial test 0; = 0 , and the axial strain

I
Y
can be reduced to the form

_l._ . _
ez — E[0z 2v0y]. (3 21)
With the assumption of incompressible soil (v = 0.5),

we have

_l._. _
ez - -E-[0z 0y] (3 22)

0' -

O!
or E = (i—E—i) . (3-23)
Z

The relationship can be obtained from the test as
a typical result for elastic behavior of the soil during
the increment of axial stress. It can be seen from equation
(3-23) that the modulus of elasticity depends on two
parameters, (0; - 0;) and 62. The effect of chamber pressure
(0;) on E has been proposed for both clays and sands
(Lambe §E_31., 1969). In the real situation 0; and 0;
change simultaneously during the loading process, with the
ratio depending very much on the loading system. The
relationship can be of any type depending on the behavior
of the loading. In plane strain problems, the effect of
the intermediate stress (0;, 0; # 09) makes the relationship
rather complicated and unpredictable. The only reasonable
method is to investigate the stress path during the loading
process at each point, and apply to those points the E's

obtained experimentally according to the stress path.

112

For the conventional consolidation test 8x = 6y = 0,

and 0X = 0y. So equation (3—20) becomes

 

l_l_ u _
0X — 0y — (1—2) oz (3 24)
and
e - lII'V'ZVZIO' (3-25)
2 _ E -v 2'
For incompressible materials (v = 0.5), we have
0' = 0' = 0'
x y z
and
_ L 0 I _ _
ez - E[0.5] oz — 0, (3 26)

which cannot be identified. For compressible material
(v<0.5), and the relationship can be identified by equation
(3-25).

Experimental results show that failure cannot
occur (unless stresses are very high) under consolidation
stresses. For compressible material (a; = a; # 0;) a
certain amount of deviatoric stress (0; - 0;) may develOp,
but not large enough to produce failure.

It will be shown in the discussions that, develop-
ment of stress paths at different points in the slope body
during an increment of load are quite different. Only

two categories of stress path (conventional consolidation;

113

Figure 3-l6a; and triaxial-teSt; Figure 3-l6b) are described
here. The consequence of the variation in stress path

at different locations within the soil body is a variation
in stress-strain relationship or the modulus of elasticity.
The effect is quite complicated for calculation purposes,

and will not be considered in this paper.

3.6 Pore Pressure

 

DevelOpment of pore pressure during application of
load is considered only for clay. With time this fluid
stress will dissipate, transferring an increasing amount
of the applied stress to the soil skeleton, with resulting
increasing settlements with time. The process is called
"consolidation."

It is assumed in this derivation that the
structural skeleton of the soil is linearly elastic in its
behavior, and is homogeneous, nondilatant and isotropic.
The fluid in the pore space is also assumed to have a
linear relationship between volume change and stress.

At the instant of applying load (before drainage

can begin), the volume change of the soil skeleton is

l—2v

AV = V (——E—)[A0i + A0' + A05] (3—27)

2

where A0', A0', A05 are the increments of the effective
principal stresses due to the applied load, V = the initial
volume, E = modulus of elasticity, and v = Poisson's ratio

of the soil skeleton.

114

The decrease in volume of the soil skeleton is
almost entirely due to the decrease in the volume of the
voids. If n is the initial porosity, and Cw the compressi-
bility of the fluid in the pore space, this volume change
is related to the pore-pressure change, if no drainage

occurs, by the expression
-AV = nVCwAu. (3-28)
If follows, therefore, that

nCwAu = l-2v
E

 

[Aoi + A0'

2 + A03]. (3-29)

Equation (3-29) can be modified for the case of triaxial

 

tests, where A02 = A03 as follows:
Au = 1 [A0 + l(Ao - A0 )1 (3-30)
l+n(Cw/CCI 3 3 l 3 ’
_ 3(1-2v) _ . . . .
where Cc - ___E—__ — the compreSSibility of the 8011 skeleton,

or in terms of empirical parameters (A and B) by

Au = B[A03 + A(A0l - A03)]. (3-31)

For plane strain problems A02 # A03, so the modified

equation can be expressed as follows:

1 A01 + A02 + A03

Au: . I: ]I
(1+nCw/C;) 3

(3-32)

 

 

115

or in terms of the empirical parameter, B,

A0l + A02 + A03
3

 

Au = B[ ] = B°A0O (3-33)

ct'

CHAPTER IV

ELASTO-PLASTIC INCREMENTAL APPROACH

Very few investigations have been made into the
behavior of combined elastic and plastic zones (Figure 4-1)
within a body subjected to load. Exact solutions have
been obtained only in a few simple cases, for example
axisymmetric (Mendelson, 1968). A few numerical solutions
have been obtained (Hoeg, Christian, and Whitman, 1968;
DunlOp and Duncan, 1970; and Clough and Woodward, 1967).
Many uncontained plastic-flow problems (extrusion, drawing,
punch, etc.) have been solved by the slip line method, but
the solutions are not unique because of the ununiqueness
of the slip line field. Behavior of contained plastic
zones in connection with incrementally increasing load
has not evidently been investigated by any experimental
mean. H6eg, Christian and Whitman (1968) state that even
after a point yields, according to the Tresca yield
criterion, the stress condition continues to change at the
point. These stress changes are the consequence of
changes in stress at the surrounding elastic points. The
sum of the principal stresses can change, thus causing

elastic volume changes. The direction of the principal

116

117

stresses can change, thus altering the elastic portion
of the shear distortion. However, the maximum shear stress

at the point cannot increase.

P (Loading System)

    
 

Plastic region

”,7///Elastic region

Figure 4—l.--Elasto-plastic System.

The analytical procedure described here will
investigate the stresses and displacements within a soil
slope, beginning from the initial elastic condition,
continuing with the develOpment of local yield, and
finally spreading of the yield or plastic zone up to the
critical failure condition. The stress-strain relation-
ship of the soil is assumed to be ideal elastic-perfectly
plastic or ideal elastic with linear strain hardening in
the plastic portion (see Figure 4-2). The former is more

applicable for contained plastic behavior (Malvern, 1969).

118

 

 

 

U)

U)

8

u E (Elastic with Linear
m E =0 Strain Hardening)

Y.P.___- E’~{E)(Elastic-perfectly
Plastic)
Ee
strain

Figure 4-2.--Idealized Stress-strain Curves
Applied in Elasto-plastic Problems.

The contained plastic principle is applied to
represent the plastic zone (or zones) develOping within
the elastic regions of the soil slope. The actual stress
and deformation fields are calculated at different stages
of loading. Large deformations are allowed within the
plastic regions, but compatability of strains at the
boundary between the surrounding elastic regions and the
plastic zones is still applied. However, with one
technique to be described later the strain compatability
condition can be violated in some particular locations.
With these combined restrictions, the deformations will
be restricted by the surrounding elastic material, while
stress deviators within the plastic zone change in strain
hardening material assumption with drained strength
criterion and remain constant in perfectly plastic

material assumption with undrained strength criterion.

119

The plastic zone will propagate as the load increases,
while some local yields may develop in other locations.
The yield condition may result from stress states of pure
compression, pure tension (tensile strength is very low
for soil), or a combined state. The assumed contained
plastic flow condition is applied up to the limit of
equilibrium (equilibrium will be defined later in this
chapter). Beyond this limit, kinematic effects must be
considered in analyzing stress and deformation-rate
fields. At this stage, the problem becomes one of the
uncontained plastic flow category, and can be analyzed
by the slip line method (Mendelson, 1968). Since the
uncontained plastic flow problem is beyond the scope of
this paper, no further presentations will be made here.

4.1 Energy Approach in Elasto-
plastic Incremental Behavior

 

 

The energy approach is the most popular method
used in analyzing the behavior of a continuous medium
under load. Elastic perfectly-plastic or elastic-strain
hardening behaviors can be assumed for the material.
Gravitational forces and surface forces can be applied
to the body. For static conditions, the energy approach
yields equilibrium at any location in the body. The
equilibrium requirement is independent of material
behavior, i.e., whether the material is elastic or

plastic.

120

The material applied in this analysis is assumed
to be isotropic and ideal elastic-perfectly plastic. The
problem is considered only for conditions of increasing
load. Ideal elastic-perfectly plastic behavior can be
approximated as a piece-wise linear material. The
mechanical process will be explained according to
different processes of construction (i.e., different

loading systems) as follows:

4.1.1 Single-Lift Construction

 

Stress and deformation behavior of a single lift
embankment is shown in corresponding steps in Figure 4-3.
The height of embankment is assumed to be great enough to
cause collapse, and load increase is simulated by con-
sidering the soil unit weight to increase incrementally.
The soil mass will first deform elastically (Figure 4-3a)
storing internal energy within the soil mass due to the
external work done by a part of the soil weight. Graphical
presentation of elastic stored energy is shown on Figure
4-4a. During this range of deformation the stresses
everywhere within the soil body are below the limiting
failure condition. As the body force increases homo-
geneously, the external work done is increased, along with
the internal energy. At a certain stage, local yield (or
local yields) occurs (Figure 4-3b), and the internal energy

is partly dissipated in the yield or plastic zone. The states

 

 

 

 

  

 

 

 

 

 

 

1.2:1

Mathematical Presentation

 

 

 

T (u )Y
Work done by body force = fvizl-igl-dv - IV ; dv
Y
T
Internal energy stored = fv{0} {6:} dv
a.
Y1=Y+AY e 6u:AY
AW = Iv=ve éuzj'dv + fv=ve 2 dv
\— Local yield area
3 start of yield
U)
{P’.
T e
T e 60 6:
r = ___...
OE fv=veo 65 dv + fv=ve 2 dv
strain
b. Start of Local Failure.
Vp 6“:1A 1
= +
Awl [Ive Au Y1 dv + Ive 2 dv) [va éuzl Y1 dv]
m
3 T e
u 60 6:
u _ e l l T p
W --- [El — [Ive 0 651 dv + Ive 2 dv] + [va 01 6:1 dv]
strain
C) - Condition of new yield points
(3 - Condition of formerly yield points
c. Spreading of Plastic Zone.
GUEBAY
3w = [f 60 dv + f z” “ dv] + [I au" dv]
d n v9 z Yn ve 2 vP zn Yn
_- +.
Y :n- Lyn-l
- T
— T 'I‘ e Aon (56: T P
= d +
AEn [Ive 0n(6en) dv + Ive 2 v [fvp on(6en) dv]

Z

a continuous slip surface

d. Equilibrium State (Limit of
Static Condition).

 

 

Part II is assumed to be rigid,
no effect due to
traction-forces.

is also assumed to be a
rigid motion body.

Part I

k
W /unit time (F cos 8) V0

k
E /unit time 0.. .. ds
1] 1]

122

0, center of rotation

 

 

 

 

m

m

8

a @Q@

0/66/
4’_ y
Strain /
[5 Potential slip surface

<:>= drop of stress and strain states wk > Bk + Frictional dissipation

from plastic to elastic of points
in the plastic area, but not on
the critical slip surface.

(:>= stress and strain states of
points on the critical slip
surface.

e. Collapse of SlOpe.

f. Notations:

X = {X} = Io,o,y} = body force vector

d = {éu; = {flu ,fiu ,éu } = displacement vector

u x y z

3 = 13 = t5 ,3 ,7 ,r ,' ,T i = stress vector

xx yy 22 xy yz 2x
55 = {it} = 15: ,ée ,i5 ,69 ,5: ,oL } = strain vector
xx yy 22 xy yz zx
Eij = strain rate tensor
e . P -
V = total volume, V = volume of elastic part, V = volume of plastic part
. . , T . , .

Y = unit weight of SOil, T = total unit weight of SOil
aw = increment of external work done
AB = increment of internal energy

n = total steps of body force increment, i = number of step

e = elastic, p = plastic, K = kinetic, T = transposed, s = surface
V0 = average velocity of the total velocity field

F - body force of the soil mass I

8 = angle between direction of body force and direction of average velocity

Figure 4-3.--Elasto-plastic Incremental Behavior under Gradually Increasing
Body Force, Single-Lift Embankment.

123

 

 

 

 

 

 

 

 

m
0‘)
i..__ __
3 _4f— ' strain
a I hardening
l
I
3 ®l
g I
u 4//,//”///’ strain
U)
a Plastic flow
\ a f
stored energy u ‘v4
U) /|
/ l
I
a . on
strain l
L
strain
= Internal energy dissipated
C): Internal energy stored
a) Elastic. b) Elasto-plastic.

Figure 4-4.--Graphica1 Presentation of Internal
Energy Deve10pment.

of stress of points within the plastic zone has now
reached the yield condition (Mohr's envelope). At this
stage, slip lines develop within the plastic zone, but
the slope is still in a statically admissible stage, and
compatability and equilibrium conditons are not violated
anywhere in the body. Additional storage of the developing
internal energy is no longer admitted within the plastic
region (for perfectly plastic case). The additional part
of the energy absorbed by this zone will be transformed
into other forms of energy, e.g., heat (Figure 4-4b).

So the total incremental energy will be partly stored in

the elastic part and dissipated in the plastic zone. The

124

state of stresses in the plastic zone can be only
slightly changed. The deviatoric stresses in this zone
will increase slightly in work hardening material, but

must remain constant in perfectly plastic material.

 

 

P
J_ - ,___

P———

6!)

p y AW \\\\\\
_ F. L. u
Heal (Sue ‘ uE luPl l'—3up u

e
W=kpu ; AW=p6ue+k6p6ue W=We+wp=%pue+pup; Aw=p6up

Elastic Portion Plastic Portion

Aw

‘\\ \ \

 

 

 

 

 

 

a) External Work-done.

 

 

 

 

 

 

 

 

o o
flL-h-_-- /
60‘"—-‘ /
AB 0’ /AE
o /’
, , /’
e .
M” . y s
.e
E=8oe ; AE=06ee+860658 E=Ee+Ep=koee+oep; AE=06€p
Elastic Portion Plastic Portion

b) Internal Energy.

Figure 4—5.--Externa1 WOrk-done and Internal
Energy Applied for Case Shown on Figure 4-3.

125

As more external work is done by further increases
in the body forces, the plastic zone (or zones) will
enlarge (Figure 4-3c). The soil slope is still in a
static state as long as the elastic areas of the soil
slope are not completely separated by a plastic zone as
shown on Figure 4-3d. When this stage is reached the
soil slope is said to be in the limiting equilibrium
condition. The slope analysis methods discussed in
Chapter II assume the existence of this stage but neglect
geometrical changes due to elasto-plastic deformation up
to this stage. It is appropriate that an analysis at this
stage should be called the "limit-equilibrium-analysis."

A contradiction can be observed between the
analysis presented here and the method of characteristics
by Sokolovski (1960). The later method determines the
limit load that causes failure throughout the soil body,
designated as Part I shown on Figure 4-3e, and collapse
under this condition. Actually, the true plastic zone
does not deve10p in such fashion and the true limit load
will also be different. Any loading system obtained from
the characteristic method should be on the upper bound
side, and the computed collapse load will be greater than
that of the actual collapse load. A collapse mechanism
will exist at this stage if there is a continuous slip
surface between any two boundaries of the body (Figure

4-3d). If this slip surface has not developed, the

126

plastic zone will spread further with an additional
increase in soil unit weight until a complete slip
surface occurs. The slope is now in a kinematically
admissible state. It is the optimum possible state that
the slope can remain in static condition. After that,
movement of the soil body (Part I, Figure 4-3e) can occur
any time if only a small additional amount of external
work is applied to the body. The soil body (Part I,
Figure 4-3e) will move as a rigid body along the slip
surface. It is possible that the state of stress of

any point in the plastic zone, but not on the slip surface,
may drop below the yield enve10pe during the movement
(Figure 4-3e). Further analysis can be done to obtain an
upper bound solution. This analysis is concerned with
kinetic effects and the equations of motion are used
instead of equilibrium equations. A certain amount of
power is developed as this soil body I moves in the
direction of the average velocity of the velocity field.
In this unstable stage the load power must be equal to or

greater than the power dissipated along the slip surface.

4.1.2 Incremental Construction

 

For the case of incremental construction a similar
explanation applies, only now the applied external work is
due to the additional surface force from additional thin

layers of soil.

127

4.2 Strength and Failure Concepts

 

In the elasto-plastic incremental process, the
development and propagation of the plastic zone (or zones)
depends heavily on the strength envelope of the material.
As described in Chapter III, the strength characteristics
of clay are very dependent on history and on behavior of
loading. For saturated clays two strength enve10pes,
drained and undrained, are normally obtained. In the
embankment the application of load is fast enough so that
drainage of clays during the construction period can be
ignored. The concept of undrained strength (¢ = 0) will
be applied as the criterion for strength.

However, the orientation of slip lines in the
plastic zone cannot be determined from a value of ¢ = O,
which in general gives the angle a equal to 45°. The true
slip angle, a, must be obtained from the drained strength
concept (ignoring differences between ¢' and the true
angle of internal friction), and a = 45 - ¢'/2. The
failure surface will be picked from among those constructed
according to the true slip angle. Different types of
clays may give similar configurations of plastic zones
because of the similar strength concept, but their failure
surfaces may be different. In analyzing the stability of
a slope, a relationship between the soil strength and the

modulus of elasticity is required. Such a relationship

128

was presented by Skempton and Henkel (1957), who investi-
gated various samples of London clay. They found a linear

relationship between the two parameters

E = 140 CU (4-1)

This relationship will be used in the stability analysis.

3;} Stress Conditions during Loading

Stress conditions at a typical point in the slope
during construction by lifts up to failure is discussed in
this section. It is assumed that remolded soil is com-
pacted in place to full saturation, and that drainage is
not allowed during construction. Compaction is assumed
to be equal for each layer and the applied work done is
assumed to influence only the depth of the increasing layer.
Thus, the initial effective stress (06) is assumed to be
homogeneous throughout the soil body, and this establishes
the shear strength of the soil.

Figure 4-6 shows the relative location of a typical
point, A, for initial, intermediate, and yield conditions
during construction. Stress-strain conditions for this
point during construction are illustrated on Figure 4-7.
The letters a, b and c represent conditions at progressive
stages of construction, with yield occurring at c.

Figure 4-8 shows Mohr's circles of total stress at these
stages of construction. Figure 4-9 shows Mohr's circles

for total and for effective stress at failure for point A.

129

 

 

 

 

 

 

 

 

 

 

KN \ \\ K
[Y X \ X \:
Aquinm /{ A?
A A A, at failure
a) Initial b) Intermediate c) Yield
Condition Condition Condition

Figure 4-6.--Relative Location of a Typical Point
in an Embankment During Construction.

stress
deviator

 

 

strain

Figure 4-7.--Stress-strain Behavior of the
Typical Point.

Stress conditions for the various stages of construction
are summarized in Table 4-1.

It is obvious that in the present case the state
of effective stresses at failure is independent of the
deve10pment of neutral stresses. Since the total stress
state can be used to specify failure, it is not necessary
to recalculate the effective stresses. The effect of

neutral stresses will not be considered in this paper.

1130

.cofiu«ccoo came» mmuocmc =0:
.mcoHuHccoo mumacmfiumHCfl mmuocmc =n=
cofiuwccoo Hmfluwcfi mmuocop gm:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

m um as u a once + on u no
0 U
omoa I oHo< I moam moa + ca + oa
m M N 5 UNOQ + CO I.l.. ND
0 O OH .
omoa I oaoq I omoam moa + moa + ca
0 o H cone wusmflm
m m UH .I. 9 Ho< + .o .I. o .mmomum flame.»
U U
omo< I o~o< I oHoam moa + moa + oa
m m ha u s bmoc + wo H MD
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a
5.2 I flo< I nmo<m cc + 2 + cc Dole musmah
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nmo< I nmoq I QHOQN nmoq + nmo< + o<
m m u a «moa + as n mo
0 m. m m. .
mmoa I mHo< I moqm mo< + moo + Hoq
m m n : £3 + 00 u mo
m m .
mmo< I maoc I m~o<m moo + mmo< + Ho<
omIv muomflm
m m n : mao<.+oo "Ho .mmmum awauacH
am am m4 cm mm 3.. . . .
o< I o< I o<m o< + o< + o<
mommmuum m>Huomwmm Aaumv mmmuum Hmuusoz mommouum kuoe weapmoq

 

.mcHUMOA

ucwfixcmnEm wo mmmmum ucmumwwflo um mcofluflccou mmmuum mo Euom HmHSQMBII.HIv mqmée

 

131

//~— strength envelope

 

 

  
  
 

total
stress path

 

0' . . . p,o
0 Initial effective stress

Figure 4-8.--Mohr's Circles Representing Total

Stress Conditions for a Typical Point in an Embankment.

 

//
é effective tEI‘tal stresses
/ stresses
I ,,

AA A-
In -IJU

(00+ 3c 361c 3j 0,0
06+A03C

 

 

.ig1+2A‘Hc"Ache-Ache7.J ,_

0 3

 

 

 

‘._ 0'+Ao -T

0 1c

Figure 4-9.--Mohr's Circles at Yield Condition

for the Typical Point.

132

4.4 Simulation of Contained
Plastic Zones

 

Three methods of simulation will be explained

according to different failure assumptions as follows:

4.4.1 Different-E Method

 

In this simulation both equilibrium and continuity
are enforced on the boundary between elastic and plastic
zones. Equilibrium and continuity are also applied within
the confined plastic zone. After a point yields, a very
small value of modulus of elasticity, the plastic modulus,

will be applied (see Figure 4-10).

/////// El a) Elastic.

 

 

 

 

 

 

éC/C/’//////)://'El
r’
/////f b) Elastic and Plastic,
m
L32
E1 = Elastic modulus
E2 = Plastic modulus

Figure 4-lO.--Simulation by Different—E Method.

133

Changes in principal stresses (both direction and
magnitude) occur slightly during an increment of load. The
maximum shear stress at a point in the plastic zone is
constant throughout the loading increments if perfectly
plastic material is assumed, while for linear strain
hardening material, the maximum shear stress may increase
due to the change in yield surface. For this simulation,
an undrained strength envelOpe is assumed, where the strength
is independent of increasing normal stress, and this
assumed condition applies until complete discontinuity of
elastic zones occurs. For a strength envelope such that the
shear strength increases linearly with an increase of normal
stress, a change in magnitude of the principal stresses can

change the point from plastic to elastic.

4.4.2 Truncated Plastic Zone

 

In this method the points within the plastic zone
(or zones) can experience stress changes up to yield only.
After the points yield, the stress states are kept constant.
At the conclusion of any load increment the plastic part
(or parts) is truncated and the existing stress is applied
to the elastic boundary to calculate the stress field in
the body for the next load increment (Figure 4-11).

In this case the stress-strain behavior can only
be assumed as elastic-perfectly plastic. The strength

must also be that of the undrained case. If the strength

134

 

 

 

 

 

 

Contained plastic zone Y'Applied boundary-
\ pressure
\
a) Real condition. b) Simulation,
Figure 4-ll.--Simulation by Truncated Plastic-zone

Method.

is such that the shear strength increase linearly with

normal stress, this method is inapplicable.

4.4.3 Propagation of a Slip Line

 

In the third simulation violation of compatability
is allowed along a continuous surface. The surface is
indicated by a selected slip surface according to the slip
angle within the plastic region. The slip surface is not
unique, but the most practical one will be selected. The
stress states along the assumed surface are always at the
limiting equilibrium, which means constant boundary pressure
will be applied along the surface.

The simulation procedure is illustrated on Figure
4-12. When yield occurs over a small area, slip lines can
be drawn through the area according to the known stress
state. The slip surface will be simulated by applying a
new external boundary surface along the fictious slip

surface with boundary pressure according to the stress

135

 

Applied boundary
pressure

 

 

a) Real condition.

 

 

 

 

 

b) Simulation.

Figure 4-12.--Simulation by Propagation of a
Slip Line.

state at the location. When a new load increment is
applied, a larger area will yield. If the new yielded
area is attached to the initial one, the next slip line
can be drawn connecting from the original one. If the new
yielded area is apart from the proceeding one, simulation
of the new yield area is performed in the same fashion as
the former one. In a situation in which there is more
than one fictious slip surface, the potential failure
surface can be taken from the one that forms the first
complete cut surface (the surface that connects any two
lepe boundaries).

For lepe embankments the first yield area is quite
predictable, and normally continuous spreading of this
plastic zone occurs in embanking processes. Failure in
tension may occur at some other points away from the

initial compressive yield point. The tensile failures are

136

likely to be observed at the top surface, which produce
tension cracks.

The disadvantage of this method is that when a
discontinuity is applied along the slip surface, it is
impossible to control the overlaps of the two fictious
boundary surfaces as deformation proceeds in further stages
of loading.

Strain hardening material can be accommodated
only in the first simulative method. In the second and
third methods, the soil must be assumed to behave as an
elastic-perfectly plastic material. Only the first method
will be used to simulate contained plastic zones in this

paper.

4.5 Stability Analysis

 

To facilitate discussion of the elasto-plastic
incremental approach in the analysis of slope stability,
two general theorems will be briefly stated:

(I) Lower Bound Theorem

The lower bound theorem states that the collapse
load calculated from a statically admissible stress field
is a lower bound to the actual collapse load. A statically
admissible stress field is defined as a stress field which
satisfies all stress boundary conditions, is in

equilibrium, and nowhere violates the yield criterion.

137

(II) Upper Bound Theorem

The upper bound theorem states that the collapse
load calculated from a kinematically admissible failure
mechanism is an upper bound to the actual collapse load.
A kinematically admissible failure mechanism is defined
under the condition that collapse will occur if there is
any compatible pattern of plastic deformation for which
the external forces do work at a rate equaling or
exceeding the rate of internal energy dissipation.

Most existing methods of analysis for slope sta—
bility require an assumed failure mechanism, and there-
fore, are all upper bound methods. The lower bound
theorem has been applied to soil mechanics less fre-
quently than the upper bound theorem since it is con-
siderably easier to construct a good kinematically
admissible failure mechanism than it is to construct a
good statically admissible stress field. An upper bound
solution is often a good estimate of the collapse load,
but a lower bound solution is more valuable in engineering
practice as it results in a safe design.

A lower bound solution combined with an upper
bound solution can serve to bracket the actual collapse
load. If one could guess the actual stress field that
Inould be represented at collapse, then the predicted

loounds would be equal.

138

In this paper the analysis of slope stability is
done by developing actual stress fields at different
stages of loading up to the collapse load. The stress
fields are developed by a numerical technique (finite
element method). The solution satisfies most of the
requirements of the statically admissible stress field.
The stress fields do not provide equilibrium of stresses
along the element boundaries, but the nodal force
resultants are in equilibrium. One additional property
of the stress fields is the continuity of strains every
where in the body. This property is not required for the
statically admissible stress field. However, adding the
property does not violate any restriction of the stress
field.

The analysis yields actual stress fields at dif-
ferent stages of loading, and the true mode of failure
is obtained. Consequently the true collapse load for the
assumed material properties can be predicted. Lower
bound loads are obtained at different stages prior to the
limiting equilibrium conditon. Since it is not the
purpose of this paper in extending the investigation
beyond the limit load, the kinematically admissible
velocity field is not considered.

The factor of safety against failure can be

obtained from the average factor of safety calculated by

139

X T AL
average F.S. =-——————— (4.2)
X CU AL
where T = shearing stress along any finite length, AL,

along the potential sliding surface.

At any stage of loading, the factor of safety can
be calculated. The minimum factor of safety (at the
limiting condition) equals unity for this analytical
method.

It should be noted that the shearing stress on the
potential sliding surface (Figure 4-13) may not be the
maximum shearing stress. The plane tangent to the surface
at the point does not always orient in the plane of the
maximum shearing stress. However, for the case of
saturated clay (v = .45) the two planes will be proved to

be close to each other.

Potential slip-surface

 

Figure 4-l3.--A Potential Slip Surface in a Part
Of the Body.

CHAPTER V

PROCEDURE

The analysis of stresses and deformations in
embankments is performed by a plane strain formulation
of the finite element method. The calculation of stresses
and strains at each stage of loading is accomplished by a
computer program, "FEAST l," by E. L. Wilson and J. T.
Christian (1967). This computer program can be used to
determine the deformations and stresses within certain
types of stressed bodies. It will analyze problems of the
following categories: axially symmetric, plane stress,
and plane strain. Elastic, non-linear material prOperties
are considered by a successive approximation technique.
The effects of displacement or stress boundary conditions,
concentrated loads, gravity forces and temperature changes
can be included. Because of limited capacity of the
computing machine, another computer program (NEWST) was
constructed to calculate the stresses-addition (algebraic
adding) process between the original stress field and the
additional stress field resulting from the next load
increment. The soil is assumed to behave as an ideal
elastic-perfectly plastic material. This is a reasonable

assumption for embankments where the soil is in a remolded

140

141

stage (Scott, 1963). It is also assumed that the soil is
isotropic. Both homogeneous and layered cases are investi-
gated. The analysis applies mainly to cohesive saturated
soils, especially in embankment processes. The effect of
pore pressure development is not considered.

Plane strain conditions are assumed in the calcula-
tion of stress fields developed according to the gravita-
tional body force of the soil mass. The stress-strain
relationship or modulus of elasticity is assumed to be

homogeneous throughout the body. The ratio of

E
E2 (Ep = plastic modulus, Ee = elastic modulus) is about
e

l . . . l
T656 for elastic-perfectly plastic assumption and T55 for

elastic-strain hardening assumption. Poisson's ratio

for saturated soil is about 0.45 (v = 0.5 gives perfect
incompressibility). For the elastic problems, the material
is assumed to have a linear stress-strain relationship,
while a bi-linear relationship is applied to the elasto-
plastic problems. The stress fields in the elastic range
are directly obtained from a single step of calculation,
while the super-position method is performed for the
elasto-plastic stress fields.

5.1 Basic Concepts of Finite
Element Method

 

 

The basic concept of the finite element method

(Zienkiewicz et al., 1967; and.Connor, 1967) is an

142

idealization of an actual elastic continuum as an
assemblage of discrete elements interconnected at thier
nodal points. For the analysis of two-dimensional stress
fields, it has been found convenient to use triangular and
rectangular plate elements in the idealization. To
maintain compatibility between the edges of adjacent
elements, it is assumed that the deformations within each
element vary linearly in the x and z-directions. On the
basis of this assumption, it is possible to calculate the
stiffness of the complete structural assemblage. This is
obtained by merely superposing the appropriate stiffness
coefficients of the individual elements connecting to
each nodal point.

If the vector of all nodal point displacements in
the complete assemblage is designated as {r}, and the
vector of the corresponding nodal forces is {R}, the
structure stiffness matrix, [K], (which is obtained by
superposing the finite element stiffnesses) expresses the

relationship between these quantities as

{R} = [K] {r} (5-1)

The order of these matrices is 2N if there are N nodes.
Nodal displacement includes both x and 2 components.

Where boundary conditions impose displacement constraints
on any of the nodal points, the matrices may be reduced by

eliminating the corresponding rows and columns.

143

In a standard elastic finite element analysis,
these linear equations of equilibrium are solved for the
nodal displacements resulting from the given nodal
forces. Then the stresses in all of the elements, denoted
by the matrix {0}, are obtained from the nodal displace-

ments by the matrix transformation
{a} = [S] {r} (5-2)

in which the stress transformation matrix, [S], takes
account of the assumed linear displacement patterns in the
elements, as well as their given material prOperties.
Concerning the finite element plane strain analysis
procedure, in general, it may be noted that (l) compati-
bility is satisfied everywhere in the system; (2)
equilibrium is satisfied within each element; and (3) equi-
librium of stresses is not satisfied along the element
boundaries, in general, but the nodal force resultants
are in equilibrium. This local discrepancy in stress
equilibrium represents the type and extent of the approxi-
mation involved in the finite element method of analysis.
For analysis by the finite-element method, the slope must
be divided into a number of elements, as shown on Figure
5-1. For each nodal point in the system, displacements
are obtained in two directions (x and 2). After the dis-
placements of all of the nodal points have been evaluated,

the stresses within each element are evaluated and

144

represented by three stress parameters, 0 , o ,

O O
x z xz
The principal stresses are then calculated both in magni-
tude and direction. In plane strain analyses, the body is
assumed to behave in the same manner with respect to the

direction perpendicular to the area under consideration.

The thickness considered in this direction is one unit.

 

      
 
 

   

l
2 (Top Surface) '

   

l
(Slope Surface)

    
     
 

3

(Surface of
Symmetry)

Firm Base
(Bottom Surface)

Figure 5-l.--Typical Slope Boundaries.

To simulate a slope embankment, boundary conditions
must be comparable to those in the real conditions. In
simple lepes (see Figure 5-1) the boundary conditions can
be categorized as follows:

1 . Boundaries @ and ® are free surfaces . No
loads are applied along these boundaries, and points along

these boundaries are also free to move in any direction.

145

2. Boundary(:) the center line of the slope, is
simulated by requiring zero horizontal displacement and
zero vertical load along the boundary. This simulation
can also be used to represent any vertical plane that is
far enough from face(:)so that shear stresses along the
plane can be assumed equal to zero.

3. Boundary(:>is the interface between the firm
base and the bottom of the soil body under consideration.
The simulation is performed by applying zero displacements
in both directions (x and z).

5.2 Simulation of Embankment Stresses

and Strains by the Superposition
Methbd

 

 

The validity of the superposition method is dis-
cussed by Timoshenko and Goodier (1970). In applying the
principle of superposition it is assumed that very small
elastic deformations occur after each stage of loading.

The simulations are discussed under two categories
according to the construction processes, single-lift

construction, and incremental construction.

5.2.1 Single-Lift Construction

 

Although the assumption of a single lift is not
strictly valid for a constructed embankment, the method
is useful in studying cases where failure may be due to
soil strength deterioration. The stresses and deformations
in the soil body are obtained by direct application of the

Igravitational body forces on the completed structure.

fi'fl L43. .

146

If the embankment height is such that no plastic
failures occur, the actual unit weight of the soil is
applied and the stresses and deformations are calculated
within a single step. If, however, the height of the
single—lift embankment is great enough so that local
yield (or yields) occurs, the calculations are performed
by applying increments of unit weight. The total unit

weight of the soil is divided into parts, as follows:

 

— Y1 = (applied within elastic range up to
the first local yield).
Ytotal n
L Y2 = Z Ayl (divided into "n" increments).
i=1

The first part, Y1: is obtained such that first

yield just occurs at some point within the soil body. The

second part, YZ’ is divided into n fragments. The
fragments do not necessarily have to be equal. The steps
of calculation are illustrated by the simplified flow
chart, Figure 5-2. In the elastic range of the first
step, El and Y1 are applied throughout the body. In the
second step, an increment of unit weight, A73, is applied
throughout the body, while El and B2 are applied in the
elastic and plastic regions respectively. In the third
step, the calculated stress field must be checked so

that the stress condition in the generating failure zone

147

 

 

 

 

 

 

 

 

 

 

 

 

Step Calculations performed to obtain H 1 E
<::) Y' that causes first local yield —L_ Yl' 1
1
Calculations performed by applying —
® arbitrary Ayg- with very small E2 and AYI
large El applied to plastic and H 41 2
' ' I
elastic regions respectively E2 \‘E1

 

 

 

iierror > 10
anywhere

   

  
   
 

yield stresses are
checked with the allowable
strength. The allowable
error is 10°fi

 

     
 
 

error < 10 yCeverywhere

Calculations Jre continued
in the 5 me manner

 

 

 

 

 

 

 

    
     

 

error > 10-/.

 

 

 

v

@' Plastic Zone
<::>- Elastic Zone

error < 10 '/.

Figure 5-2.--Flow Diagram of Single Stage
Construction Simulation.

148

will not deviate from the yield stress more than 10 per
cent. A similar procedure is repeated until limiting
equilibrium is reached.

The potential slip surface is drawn on the slope
under the stress field at the limit equilibrium state.
The start of the slip surface is selected as the location
of the first-yield element and drawn continuously from
this location to touch the tension zone. It was found
that the slip surface normally touches the tension zone
around the area where the maximum tensile stress occurs.
Tension cracks were assumed to occur throughout the depth

of the tension zone.

5.2.2 Incremental Construction

 

Incremental construction is intended to simulate
stress and strain conditions in an embankment placed and
compacted in lifts. The load increments result from the
added weight of each lift as it is placed, and the soil
unit weight is kept constant. The resulting stresses are
again added algebraically. Complete presentation of the
process is shown in the flow diagram (Figure 5-3). The

initial height, H is obtained within the elastic range.

1!

At this stage YT and E are applied throughout the body

1

with height H In the second step the unit weight of the

1.
soil, YT' is applied only in the added layer, AHl, while

the original body is considered to be weightless (Y = 0).

149

 

Calculations performed to obtain a
proper height (H1) that yields first

<::) local yield, unit wt.=ytotal throughout H ' ”(77;Z7§;7|

the body. 1 T~"

Step

 

 

 

 

 

Calculations performed by imposing an
arbitrary thickness (AHI) of soil layer
of weight (YT) on the top of the body
(::> obtained in the previous step. The body

is assumed to be weightless and containa
plastic zone considered in the last
step.

 

 

 

 

 

 

     
  
  
  

A Error > 10 '/.
anywhere

@+@

Stresses states in
the generating plastic zone
are checked with the allowable
strength, (allowable error -
10 "/o) .

 
   
   
 

 

  

   
   

I Error < leL everywhere

Calculations are continued
in the sade manner

 

 

 

 

 

 

   
 

A Error > 10 °/.

 

 

 

E")

 

   

< .0
rror 10 / @- Plastic Region, y=0

<:)_.Elastic Region, y=0
QGCD- Elastic Region, Y=YT

Figure 5-3.--Flow Diagram of Incremental
Construction-Simulation.

150

E2 is applied in the plastic region and E1 is still used
in all other parts. In the third step the calculated
stress field must be checked so that the stress condition
in the generating failure zone will not deviate from the
yield stress more than 10 per cent. A similar procedure
is repeated until limiting equilibrium is reached. The
potential slip surface is drawn in the same manner as
described in section 5.2.1.
5.3 Factor gfisafety and

Unused Shear Strength

Some comments regarding factor of safety and

unused shear strength follow:

F = factor of safety

5 = c-+o tan¢

      
 

stress condi-
tion at a

 

 

 

 

 

 

 

 

 

Tf_ certain
,—” point
/ / /
, SI
0 "1 F _‘
% o 03 I 01 o
[i
c Ol+g3 ’1
(4......m + I) )
Lenny I.

Figure 5-4.--Definition Sketch for Factor of
Safety and Unused Shear Strength.

 

151

Referring to Figure 5-4, the Mohr's circle repre-
sents stress conditiOns below failure for some point, and
the solid line represents the limiting strength. The

factor of safety is defined as

Tf
FS -— T (5-3)
and unused shear strength is defined as
US = Tf - T (5-4)

where T = shearing stress on the potential slip plane
(angle = a), and If = shearing strength on the potential
slip plane (angle = a).

The concept of factor of safety in this presenta-
tion is slightly different from the one used in the con-
ventional stability analyses (Bishop's, Morgenstern, etc.).
According to the conventional methods, the stress state on
the slip plane will be at point A' instead of point A
(Figure 5-4), while the shearing strength is the same.
Point A is obtained from a point on the Mohr's circle
where a line is drawn parallel to the strength envelope
and tangent to the circle at A. Point A' is obtained from
a point on the Mohr's circle where a line (dotted line on
Figure 5-4) is drawn from point Q and tangent to the
circle at A'. The two points deviate from each other more

when the value of ¢ is larger. For small values of ¢

152

the two points are close together, and the two concepts
give about the same results.

In order to choose points at which failure is
most eminent, or the yield priority, it is convenient to
use the concept of unused shear strength. This can be
explained with reference to Figure 5-5. The stress
condition of(:)gives higher factor of safety than that
of(:» while the two conditions have equal amounts of
unused shear strength. With the principle of factor of
safety, the stress state of @will indicate higher
priority of yield which is not quite correct. Actually,
the stress states have the same yield priority. For

example, if yields are induced by increasing major

Unused shear—strength
(b)

  
 
     

  

Unused shear—
strength (a)

 

 

 

Cl‘r

Figure 5-5.--Priority of Yield by Two Stability
Concepts.

153

principal stress and decreasing minor principal stress
as indicated in condition I (Figure 5-6), or increasing
major principal stress while minor principal stress is
kept constant as indicated in condition II (Figure 5-6),

the amount of stress changes of@and@are equal in both

. I __ I I __ I II =
cases. That is, Ac1a - Ac1b and Ac3a - Ac3b, or Ac1a
Aoié. The previous example shows that the two stress

states,(:)and(:} have the same priority of yield while

the factors of safety are different.

TA

 

 

 

 

 

 

 

AclaII AolbII

Figure 5-6.-—Two Possible-failure Stress Conditions.

The above condition can affect the computed

location of the critical failure surface. If, for some

154

stress field, contour lines are plotted for the factor

of safety field and the unused shear strength field, a
difference in configuration of the contour fields will

be observed. Consequently, if an analytical method is
performed by selecting a continuous surface that gives the
least average factor of safety or least average unused
shear strength on each of the two contour fields, the
surface with least average factor of safety and the one
with least average unused shear strength will not locate
at the same position. However, the difficulty occurs
only when the drained strength envelope is used (¢ £ 0).
If the undrained strength envelope (¢ = 0) is used as the
failure criterion, the difficulty does not occur.

In the elasticity analysis, the potential slip
surface is drawn by the graphical method of slip-line.
The start of the slip surface is selected from the loca-
tion or element that gives least unused shear strength
and drawn continuously to touch the tension zone.

Figure 5-7 is the typical finite element configura-
tions of slope used in elasticity and elasto-plasticity
investigations. Figure 5-8 is the result of an accuracy
test to obtain appropriate element sizes applied in the

investigations.

155

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a

. . , a _ _
a) Elastic Investigations,(fi x H _ 8

 

sum
x
mur
u
one
x
may

b) Elasto-Plastic Investigations,(

Figure 5-7.--Typical Finite Element Configuration
of a Slope.

 

II II
0
.Is
U1

 

|:]a

 

 

 

a = size of element

Nonmflized

a a
Element Size (H x H)

 

X

X

X

X

[—l
NIH WIH (NH oolI—I NIH
X
NII-I wlI—I axlI-a <::>|I--I sly

 

Figure 5-8.--Comparison of Slip Surfaces for
Different Element Sizes.

CHAPTER VI

PRESENTATION OF RESULTS AND DISCUSSIONS

The analytical results and general discussions

will be presented in Chapter VI.

6.1 Homogeneous Soil Slopes

 

Results of analyses of homogeneous soil slopes
will be presented under three headings: Elasticity
Solutions; Elasto-plastic, Single-stage Construction; and

Elasto-plastic Incremental Construction.

6.1.1 Elasticity Solutions

 

The results of analyses of slope embankments
within the elastic range are presented for single-stage
construction. The results indicate formation of a tensile
stress zone (Figure 6-1). It was found that the behavior
of this tensile stress zone depends only on Posisson's
ratio, 0, and not on modulus of elasticity, E. As Poisson's
ratio decreases, the depth intensity of the tensile stresses
decreases. The size of the zone is less when Poisson's
ratio is smaller. At Poisson's ratio of 0.3 no tensile
stress zone is observed (slope angle = 45°, y = 110 pcf,

E = 342000 psi).

157

158

In general, for a slope with stresses within the
elastic range, development of stresses due to gravity
depends on the two elastic parameters, E and v, on the
unit weight of the soil, and on slope geometry. To
explain the above phenomenon, one can observe that the
modulus of elasticity and the unit weight can influence
only the magnitude of the stresses, and that only Poisson's
ratio can influence the principal stress directions at
the various locations within the slope. For a simple soil
slope, such as that shown on Figure 6-1, horizontal
deformation is not restrained laterally at the lepe
surface (leftward, on Figure 6-1). Since the boundary is
considered to be fixed at the bottom of the slope, the
horizontal displacement of points along the slope face
are greatest at about midheight. For incompressible
material (v approaches 0.5), the magnitude of these
horizontal displacements is greatest, and decreases for
smaller values of 0. Thus, the tensile stresses developing
in the top part of the slope are of greater intensity as v
approaches 0.5.

Normally, clays possess values of Poisson's ratio
varying from 0.4 to 0.5 for saturated clays and 0.1 to 0.3
for unsaturated clays, and sands have values varying from
0.2 to 0.4 (Bowles, 1968). Tensile cracks will occur in
the locations where tensile stress states reach or exceed

yield conditions. This will be observed mostly in saturated

159

clay rather than sand. Vertical cracks always precede
failure in clay slopes. For sand, tensile failure will
develop along slip surfaces.

Figure 6-2 shows several series of Mohr's stress
circles drawn in corresponding order of the locations from
left to right along the bottom horizontal layer in the
slope bodies. The results shown are only the envelopes
for points along the bottom layers, but the results are
similar for any other horizontal layers. For small values
of v, the envelopes to these stress circles start as a
straight line passing slightly above the origin and curve
downward to approach a horizontal asymtote. If the value
of v is greater than about 0.35 the stress envelopes
curve downward and form a peak. Also, for values of v
greater than about 0.35, the initial part of the stress
envelope is curved. An explanation of these two
characteristics of the strength envelopes is furnished in
the following.

Consider the case of a semi-infinite medium with
the surface parallel to the horizontal plane (x-y plane),
and the principal stress directions parallel to the x, y
and 2 directions (see Figure 6-3). With the gravity load,
strain in the horizontal directions is assumed to be zero,

and the horizontal stresses are equal,

ey=-}]§-[o -v(oi+ox)]=0 . (6-1)

Y

v o (6-2)

With Ox = Cy we have Ox = (I:V) 2

For a value of v = 0.5, the magnitude of Ox equals that of
02, which in turn gives the magnitude of deviatoric stress
(01 - 03) equal to zero. (In the plane strain condition

the magnitude is not exactly equal to zero.) As v decreases,
the value of (TE?) decreases while the magnitude of

(al - 03) increases.

If we consider the soil in an area that is far from
the slope surface so that the principal stress axes are
parallel to the x, y and 2 directions, the magnitude of the
deviatoric stresses will depend on magnitude of v. In the
cases of 0 less than 0.5, the deviatoric stress equals to
(%E%¥) oz, which gives a definite value of the deviatoric
stress for each v. For v less than the threshold values,

l-Zv

around 0.35, the magnitude of (TEST) 0 is greater in

2
comparison to the deviatoric stresses of the points on the
horizontal layer near the slope. The deviatoric stresses
increase in magnitude for points at locations farther from
the lepe surface, and approach the maximum value of

(%fi%¥) 0 . For v greater than the threshold value, the

z
magnitude of (%fi%¥) 02 is not the maximum deviatoric stress
for points along the horizontal layer. Now the magnitude

of the deviatoric stresses increase to a maximum value at

a certain location which is the peak of the envelope, and

161

then decrease approaching (1f2v

 

-v) oz for points far from
lepe surface.

The results (Figure 6-2) indicate the requirements
of the strength characteristics for the slope to maintain
a stable condition. Soils with low values of v require
low values of cohesion, or intercept of the T-axis, and
high values of 0, or slope of the strength envelope in the
initial part to maintain a stable condition. As the soil
property, v, is larger, required values of the cohesion are
greater, while the ¢-va1ues required are smaller. Maximum
shear stress, Tmax (Figure 6-2), developed is greatest
at lowest v, and decreases as v increases.

The stress envelopes shown on Figure 6-2 can be
used to indicate the type failure phenomena, and the
location of first local failure. For dry soils (sand) with
a drained shear strength envelope (Figure 6-4a), failures
occur first at the points along the slope surface. The
failure phenomena is known as skin failure. Experimental
results show that development of skin failures for sand
occur for slope angles greater than about the angle of
internal friction (¢D) of the sand (Lambe g£_§l., 1969).
In the case of wet soils (wet sand or clay), which
possess a certain amount of cohesion, strength and stress

envelOpes are shown on Figure 6-5a. The location of first

local yield is indicated by point A. The location of the

162

first local yield depends on the characteristics of the
strength envelope, c and 0. A lower value of 0 moves
the location of the first local yield in the positive
x-direction, and vice versa.

Figure 6-6 shows the effect of changes in Poisson's
ratio on the position of slip surfaces. The slip surfaces
are drawn according to the angle of internal friction
( 0) equals to 30°, and for a 45° slope angle. It can be
seen that the position of the slip surfaces moves toward
the slope surface as v decreases.

The position of slip surfaces for a given soil
type changes also according to different 0 angles. In
Figure 6-7, the slip surfaces are drawn for 0 = 30° and 0°.
Comparison between the slip surfaces obtained from an
elastic stress field (a lower bound condition), and the
potential rupture surfaces obtained from Bishop's analysis
is established for saturated clay (v = 0.45). Slope
angles are 45°, 38.6°, and 30°. Soil properties are,

u = 0.45, E = 56000 psf, ¢ = 0° and 30°. Three interesting
points should be mentioned for the comparison.

1. The similarity in shape of the surfaces, i.e.,
the slip surface from the elastic solutions yields a shape
very close to circular.

2. Correspondence in position of the surfaces,

i.e., both cases give the position of the surfaces in such

163

a fashion that the position moves toward the slope surface
as the value of 0 increases.

3. Similarity in type of failure, i.e., both cases
give toe failure for 0 = 301 and base failure for 0 = 61

It is noticable that the rupture surface obtained
by Bishop's analysis does not give any indication of
tensile cracks, while in the elastic stress field, tensile
cracks are evident (Figure 6-7a).

It appears that the position and shape of a
potential slip surface for homogeneous saturated clays
with homogeneous strength can be predicted by the slip
surface obtained from elastic stress fields. (Since
the potential rupture surface analyzed by Bishop's method
has been shown able to predict the location of the true
rupture surface in homogeneous soils [Sevaldson, 1956]).

6.1.2 Elasto:plastic, Single-
lift Construction

 

 

Figure 6-8 shows the results of analysis of a single-
lift embankment. The soil properties are assumed as
E = 56000 psf, v = 0.45, c = 400 psf, 0 = 0°. Height of the
lift is 20 feet, and the gravitational body force of the soil
was gradually increased from 96.5 pcf up to 128.75 pcf at
the limiting equilibrium condition. With undrained
strength criterion, the first local plastic yield occurred
at a point on the firm base (stage 1), and not at the toe

of the slope. Stages 1 through 4 show how the plastic zone

164

spread as the body force was increased (geometry and
boundary conditions were assumed unchanged). At the final
stage, when limiting equilibrium was reached, the plastic
zone touched the tensile stress zone along the top surface.
It should be observed that the behavior of the tensile
stress zone changes as the plastic zone spreads. Maximum
tensile-stress occurs at point(:>(on Figure 6-8a), in the
elastic condition (stage 1). As the plastic zone spreads,
occurrance of the maximum tensile stress condition shifts
from point(:)toward others and closer to slope apex

(point a+b+c). With this body shape and loading system,
it is found that the plastic zone spreads continuously
without development of plastic yield in other locations
(disregarding the tensile stress zone). Spreading rate

is faster in the later stages when a large plastic area
has developed.

Comparison of potential slip surfaces (0 = 6) is
shown on Figure 6-8, stage number 4. Slip surfaces
obtained from the elasto-plastic stress field, and from
the elastic stress field are essentially the same as that
predicted by Bishop's method. Locations and overall
appearance are about the same for all analytical methods.
The tensile stress zone near the potential slip surfaces
is deeper for the elasto-plastic analysis than for the

elasticity analysis.

165

6.1.3 Elasto—plastic, Incremental
Construction

 

 

In practice the final shape of an embankment is
attained by a building sequence, usually involving the
placing of material in layers, one on top of another. In
order to arrive at a practicable simulation process, an
incremental approach is required, in contrast to the con-
ventional solution that presumes the construction to be
completed in a single stage and the gravity load to be
withheld until the final shape is attained.

Results of elasto-plastic, incremental behavior by
gradually increasing the height of slope embankment are
shown on Figure 6-9. With the same material prOperties
as those used in the single stage method, height at
failure is found to be the same for both cases. Develop-
ment of tensile stresses is less in this process of
construction. Rate of spreading of the plastic zone is
about the same at each stage. The general behavior of
plastic zone development is similar for both simulations
even though the configurations are not exactly alike.

Figure 6-10 shows results of including elastic-
strain hardening soil behavior. Configurations of plastic
region development are not much different from those
obtained by the elastic-perfectly plastic assumption. The
main difference is the creation of tensile stresses, which

are less intense for this assumption. The slip surfaces

166

in the plastic region generated by the stress field at the
final stage appear to be of the same form as those in the
former cases.

The resulting location and configurations of slip
surfaces presented in the previous sections depended on a
¢ = 0°failure envelope. As discussed formerly in Chapter
V, the slip angle should be obtained from the drained-
strength envelope (0 # 03, which means a = 45 - ¢D/2.
Construction of slip-lines for different assumed ¢D
(20° and 30°) are presented on Figure 6-11. The stress
field used for this construction was the final stage of
the incremental construction with elastic-perfectly plastic
material assumption. It should be noted that in this
presentation the strength criterion follows the undrained
strength envelope while the failure angle depends on the
drained strength envelOpe. The results show that base
failure is still obtained for any value of ¢D' but the
location of the potential sliding surface does change.
Steeper slip surfaces are found with higher 00, which
agrees very well with elasticity solutions and solutions
according to Bishop's method in the sense of their
orientation. The shape of the slip surfaces are also
close to circular. It is obvious that application of
Bishop's analysis in predicting the potential sliding
surface is impossible in this case, because of the dif-

ference in strength and failure criteria mentioned above.

167

Figure 6-12 presents the results of elasto-plastic,
incremental analysis for unsaturated soil, using the
drained shear strength envelope for both strength and slip
surface orientation. Elastic-strain hardening is assumed.

The results show a great effect of the strength
characteristics in the generation of the plastic region.
Local yield starts at the toe of the slope instead of at
the base and continuously spreads out as the load increases.
However, during an increment of load the stress condition
of any point in the plastic region can shift from the
yield condition (circle 1, Figure 6-13) to an elastic
state of stress (circle 2, Figure 6-13). Location of the
tensile stress zone also differs from the undrained analysis.
The plastic region in the drained analysis develops closer
to the slope surface so that the potential rupture surface
can pass through the toe of the slope (shown in the final
step of Figure 6-12), and is closer to the slope surface.
Comparison of loaction and configuration of potential
rupture surfaces analyzed by three procedures, elastic
stress field, BishOp's Method, and elasto-plastic stress
field, are shown in stages 5 and 6 of Figure 6-12. Good
agreement is obtained among these solutions. The strength
parameters are c = 200 psf, and 0 = 30°. Other soil

properties are shown on Figure 6-12.

168

A number of additional factors which could be
considered are:

1. As formerly pointed out, Chapter III, soil
strength is often reduced after soil is subjected to
pressure sufficiently large to cause rearrangement of the
soil skeleton.

2. Progressive formation of the failure surface
can cause strength reduction in sensitive clays along part
of the surface.

3. Other soil properties may also change during

deformation, for example the modulus of elasticity.

6.2 Nonhomogeneous Soil Slopes

 

The results of analyses of nonhomogeneous soil
slope embankments are presented in two parts, Elastic

Range Analysis and Elasto-plastic Incremental Construction.

6.2.1 Elastic Range Analysis

 

Investigations were performed for embankments con-
sisting of two layers of different soils. Only the case of
hard soil overlaying soft material was considered. The
materials were chosen to simulate saturated clays, and the

following properties were assumed:

Hard clay; E 360000 psf, Y = 140 pcf.

Medium clay; E = 170000 psf, y = 128 pcf.

Soft clay; E 86500 psf, Y = 110 pcf.

 

169

Results of tension zone development are presented

%, is defined as the height

of the bottom layer divided by the total height of the

on Figure 6-14. Height ratio,

slope. Poisson's ratio values of 0.4, 0.45, and 0.49 are
considered.

The results indicate that the height ratio has some
effect on the configuration of the tension zone, but not as
much effect as variation in Poisson's ratio. Also, softer
soil in the bottom layer induces a longer magnitude of
tensile stresses in the top layer.

Envelopes of required strength for different
values of v are essentially the same as those for homo-
geneous lepes, and are not shown.

6.2.2 Elastojplastic, Incremental
Construction

 

 

Only analyses of firm soil overlaying softer

material are considered as before. The soil properties

are:
top layer soil; ' E1 = 210,000 psf,
cl = 1500 psf,
Vl = 0.45,
and bottom layer soil; E2 = 56000 psf,
c2 = 400 psf,
v2 = 0.45.

A4 ;._

«___-

 

170

The results (Figure 6-15) show similar behavior in the
location of the first yield point, and the configuration
and rate of plastic zone propagation as those obtained in
the homogeneous soil cases. Characteristics and occurrance
of tensile stress zone are also similar, except that the
zone propagatesdeeper in the two-layer case. Configuration
and location of the potential slip surfaces are also alike.
Notice that the soil properties of bottom layer are the
same as those of the homogeneous soil lepe. Since higher
quality soil is applied in the top layer, the height

(Hf = 22.5 feet) at limit equilibrium in the two-layer

case is greater than that of the homogeneous soil condi-
tion (Hf = 20 feet). Behavior and configuration of the
failure surface tends to be in the same neighborhood as
that of the homogeneous case. Nonhomogeneity does

not appear to affect failure behavior greatly.

Notice the discrepancy between the failure surfaces pre-
dicted for the two-layer case by Bishop's method and that
predicted by the elasto-plasticity analysis. It appears
that Bishop's method is not applicable in locating the
actual failure surface, in nonhomogeneous soil slopes of
this type. Many other factors, such as construction
processes (excavation or embankment) and failure behavior
(long term failure or sudden failure), could be considered.
Each case must be investigated individually. An elasto-

plastic incremental analysis with consideration of the

A” A‘.-n...- “‘ _

171

cause of failure, soil strength at failure, and development
of pore pressures at failure could be of great value.

Some comments should be made about the difference
in stress paths generated at different locations of the
slope body during load increments. Results of investiga-
tion are presented on Figure 6-16. The results indicate
that the stress paths for points closer to the slope
surface more nearly approximate those of triaxial shear
tests. The triaxial test (Chapter III) produces a shearing
process which guarantees the occurrance of failure. The
stress paths of locations (point D) quite far from the
lepe surface, give results similar to the consolidation
test (Chapter III), which does not guarantee failure. It
is obvious that failure can occur only in the slope
vicinity and will not occur in the areas where stress
states are similar to those of the semi-infinite body under
gravitational body forces. As explained in Chapter III,
the stress-strain relationships are different under dif-
ferent applied stress paths. Laboratory tests obtaining
the stress-strain relationships (or E-values) must be per-
formed to simulate stress conditions similar to those
really occurring in the field. Different E-values should
be applied to those corresponding locations in the compu-
tation of stresses and deformations. A similar situation
is explained and performed by Huang (1968) in his work

concerning the nonlinearity of soil media.

 

hi.-

was"

172

6.3 Stability Analysis

 

The results are presented in terms of a factor
of safety field at different stages of construction. Only
the incremental construction case is considered.

Figure 6-17 shows the factor of safety contour
fields for homogeneous soil at various stages of construc-
tion. Conditions are the same as those represented on
Figure 6-12. The magnitude of the factor of safety values
in the fields decreases rapidly as the embankment height
increases, especially in the potential failure zone.
Spreading of the factor of safety equal one contour line
is continuously toward the top surface. Since the factor
of safety cannot go below one, the average gradient of the
factor of safety field in the potential failure area
decreases rapidly as construction progresses. In this
case using drained strength criterion, the low factor of
safety contours concentrate in the area close to the
slope surface and the lowest factor of safety line starts
at the toe of the slope.

In Figure 6-18 contour lines of the factor of
safety for a two layer slope are shown. Since the undrained
strength is used for the failure criterion, different con-
figurations are found in this case comparing to that shown
on Figure 6-17. Concentration of the low factor of
safety contour lines is at the base area instead of the

toe area. The factor of safety equal to one contour line

 

Fuse 1

173

is within the slope body. The magnitude of the factors of
safety and the gradient of the fields are lower in this case
which indicates a more unstable condition. Built up slope
embankments under undrained shear strength (sudden failure)
will have higher potential to fail than those with drained
shear strength (long term failure).
To illustrate the deformed shape of the slope at '“

different stages, displacements of points along the

sen—n—‘

boundaries of the lepe are given in Table 6-1. It is
quite obvious that the magnitudes of the displacements are
much greater in the final stage than those in the first
stages. The large displacements in the final stage are
obtained because of development of the plastic zone in the
soil body.

Average factor of safety values were calculated
along the potential sliding surface predicted from the
critical failure surface according to Bishop's method at
different stages to indicate the stability of the slope.
Undrained strength criterion is used in the analysis.
These are presented on Table 6-2. Height of the slope is
20 feet for both single-stage and incremental constructions.
The average factor of safety values were calculated, using

the relationships for E and c given in Chapter III. These

U
are compared with those given by BishOp's analysis. The
results show good agreement between the two methods, and

the two values converge more closely in the final stage.

174

Since comparison of the stability values as per—
formed in this section requires knowledge of the relation-

ship between the strength parameters (c for undrained,

U

c' and 0' for drained) and the elastic parameters (E and v),

 

the author would like to see the development of a more

complete definition of these relationships. Only a few

L

investigations have been performed so far.

Figure 6-19 shows the potential sliding surfaces

fiv -— .

used in the calculation of the average factors of safety.

I . HOMOGENEOUS--SOIL SLOPE

175

176

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5') V = 0'49 d) v = 0.35
[7.1
I
i N
b) v = 0.45 e) v = 0.325
c) v = 0.4 f) v = 0.3

(:3) Tension Zone
E = 342,000 psf

c = 110 pcf

Figure 6.l.--Inf1uence of v on Tensile Stress
DevelOpment.

177

 

   

 

 

 

 

 

 

 

P
T
max
¢ . v = 0.2
required.
c required
_L
T 0
F
-__ 0=0.3
' 0
P
_________ "_ v = 0.35
0
v = 0.4
0
P
v = 0.49
0

a) Slope Angle 45°.

CD-indicates location of points along
bottom layer (see Figure 6.2c)

Figure 6.2.—-Stress Envelopes for Points Along the
Bottom Layer of a Slope.

 

 

 

rkTmax v=0.35

L- Peak 0
T J _

 

 

 

 

b) Slope Angle 27% 0
Figure 6.2.—-Continued.

179

234567891011231 5161718

 

     

7 8 9101112131415

0) Location of Points Along Bottom Layer of Slope.

Figure 6.2.-—Continued.

 

180

 

0' 2=0Y=O 3

 

Figure 6—3.--Stresses in a Semi—Infinite Body.

 

 

e
T qéyo
“ $9 ,
e I
60‘?» \16102 ’
E“ ,
9" ties: , ’ ’
w ’ ”
\C.Failure‘ [ Z Stable ’6
Condition Condition

a)Stress Circles, and Strength Envelopes.

Approximate Original Surface
inal Surface

 

Failure
Surface

 

 

 

b)Failure Behavior.

Figure 6-4.--Effect of Strength Envelope on Failure

Behavior of Cohesionless Soil Slopes.

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181

 

 

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182

 

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183

 

 

 

 

 

184

 

 

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185

ll
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oom

 

(3 Stage No.

20 feet [:::]

=96.5 pcf

 

Yi

 

 

 

I
__L_L_LJ

20 feet E::]

y2=109.4 pcf

 

 

 

 

 

 

20 feet

 

 

 

 

 

 

20 feet [::]

y4=128.75 pcf

i

Ee=56,000 psf, Ep=560 psf, v=0.45, c=400 psf, ¢=0°

 

 

 

 

OElastic Zone @Plastic Zone® Tension Zone

 

Elasto-Plasticity,(¢=0) -------- Bishop's,(¢=0)

Figure 6-8.--Elasto-plasticity Analysis of Single-
Lift Construction, H=20 feet.

 

187

 

15 eet

 

 

 

 

17.5 fe t

 

 

 

  

 

As\\\\

20 feet

 

I

 

Stage No.

ED

ED

y=128 pcf, Ee=56,000 psf, EP=56 psf, v=0.45, c=400 psf, ¢=0°

( )Elastic Zone ®Plastic Zone ®Tension Zone

 

Elasto-Plasticity, (0:0) ------ Bishop's,(¢=0)

Figure 6-9.-—Elasto-Plasticity Analsyis of Incre-
mental Construction, Elastic Perfectly-Plastic Assumption.

188 0 Stage No.

15 felet EL)

 

 

 

     

17.5 feet 6)

 

\ A \{N

/— Max. Tensile Stress

 

V‘ 20 f et
:\‘\ Elasto Plasti ©
_ Slip Surface
X .\\-.
A \\

E e=56, 000 psf, Ep= 560 psf, v= 0. 45, y= —128 pcf,
c=400 psf, ¢=0°
\‘ >Elastic Zone @Plastic Zone @Tension Zone

‘\

 

Figure 6-10.--Elasto-plasticity Analysis of Incre-
mental Construction, with Elastic Strain Hardening Included.

 

189

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190

Stage No.

1. I Q)

 

 

 

 

 

25 feet 6)

Ee=56,000 psf, Ep=560 psf, v=0.45, y=128 pcf,

 

 

 

c=200 psf, ¢= 30 degree

0 Elastic Zone ® Plastic Zone

Figure 6-12.--Elasto-plasticity Analysis Under
IDrained Strength Criterion and Incremental Construction,
lElastic Strain-Hardening Assumption.

191

Stage No.

 

4.

Q 30 feet (3)

 

 

 

 

 

 

35 feet
G

\\\\‘

 

 

 

Elastic Zone Plastic Zone Tension Zone

Figure 6.12.--Continued.

 

 

 

WElasto-olasticity

 

 

 

 

 

Elasto-plasticity

 

 

 

 

 

 

 

SSSSS

193

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II. NON-HOMOGENEOUS SOIL SLOPE

194

195

 

 

 

 

 

 

 

 

 

:3"

a) v = 0.4, — = 0.375

m

 

 

 

 

 

 

 

 

b) 0 = 0.4, %= 0.375

(ED Tension Zone

Hard Soil

Soft Soil

Hard Soil

Medium Soil

Hard Soil

Soft Soil

Hard Soil

Medium-
Soft Soil

Figure 6.14.--Development of Tension Zone for

Two—layer Case.

196

 

 

 

 

 

 

 

 

 

= 0.375

O

C

II

C

.5

L0
mlD‘

 

 

 

 

I“

T

 

 

A

 

 

 

d) v = 0.49, %= 0.5

(ED Tension Zone

Figure 6.14.--Continued.

 

Hard Soil

Soft Soil

Hard Soil

Medium Soil

Hard Soil

Soft Soil

Hard Soil

Very Soft Soil

197 Stage No.

 

 

 

 

 

 

 

 

 

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15 feet [::]
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15 ieet @
5 feet

 

 

 

 

 

 

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¢=O ' \ if
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\ / I" I
‘ . ‘x‘ I

 

IT

 

 

   

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Top-layer Bottom-layer (:) Elastic Zone
IEe = 210:000 PSf E = 55:000 PSf 6:9 Plastic Zone
0 = 0.45 v = 0.45
c = 1500 psf c = 400 psf Tension Zone
¢=O° ¢=00 ®
7 = 135 pcf y = 135 pcf

Figure 6-15.--Elasto-p1asticity Analysis, Incre-
mental Construction for Two-Layer Slope.

198

 

 

 

 

g
P
g
P
g
(«D
rfi' .
P
q @

 

 

Figure 6.16.--Stress-path at Different Locations in
Slope Body, 10 feet lifts, 45°-slope Angle.

III . STABILITY ANALYSIS

199

200

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206

 

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b) Incremental Construction.

Figure 6.19.--Potential Sliding Surfaces Used in
Calculation of Average FS.

 

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209

TABLE 6-2.--Computed Average Factor of Safety Values

 

a. Single-stage Construction

 

 

 

 

 

H = 20 feet, E = 56000 psf, v = 0.45, cU = 400 psf.
Elastic-perfectly plastic material
Stage y. F S Av. F S
(pcf) (Bishop's Anal.)
1 96.5 1.25 1.622
2 109.4 1.102 1.292
3 122.3 .994 1.052
4 128.75 .947 1.0
b. Incremental Construction
y = 128 pcf., E = 26000 psf, v = 0.45, cU = 400 pcf.
Elastic-perfectly plastic material
Stage H F S
(ft) (Biship's Anal.) AV' F S
l 15 1.26 1.635
2 17.5 1.081 1.5

3 20 .947 1.11

 

CHAPTER VII

SUMMARY AND CONCLUSIONS

Presentation of the results and discussions have
been made in the preceeding chapter. These results will
be summarized under the headings: Homogeneous Soil Slope,
Nonhomogeneous Soil Slope, and Stability Analysis. Items
covered under each heading are intended to reflect the
findings of this investigation and are limited to the

methods used, under the assumptions described previously.

7.1 Homogeneous Soil Slope

 

Elasticity and elasto-plasticity approaches have
been used in the investigations of homogeneous soil slopes.
Two construction processes, single-step construction and
incremental construction, were assumed to simulate loading
behaviors.

Elasticity analyses were found to be useful for
design purposes. Results obtained from the elasticity
investigations are:

1. development of a tension zone in the top
part of the slope.

2. required soil strength characteristics to
prevent slope failure.

3. location and configuration of the potential
failure surface.

210

211

The behavior of tension zone development was found
to depend on Poisson's ratio (v). Values of v closer to
0.5 (incompressible, v = 0.5) give a larger tension zone
area. The tensile stresses are normally observed along the
top surface and slightly appear along the slope surface.
Magnitude of the tensile stresses developed in this zone
increase with an increasing value of modulus of elasticity
or unit weight of the material.

The required strength characteristics, as investi-
gated by elasticity analysis, are also found to be
dependent on the value of the soil parameter, v. The
required angle of shearing resistance to prevent lepe
failure increases as the v-value decreases, while the
required cohesion increases with an increase in v. The
required soil strength, or the stress envelope developed
in the elastic stress field can be used to indicate the
location of the first yield point on a sloPe, if the
actual strength envelope of the soil is known. Under the
elastic stress field, a slip surface can be drawn according
to the orientations computed from the soil strength
envelope, which predicts the location of the critical
failure surface. For saturated or dense soils and simple
slope geometry, the predicted failure surface was found to
agree well, in location and configuration, with the
potential sliding surface predicted by the classical

method of Bishop.

212

The results obtained from elasto-plasticity
analyses show the development of a plastic zone and a
tension zone during the various stages of loading.
Different plastic behavior assumptions, strain hardening
and perfectly plastic, were applied in the investigation.
It was found that the behavior of the plastic zone develop-
ment is only slightly influenced by the construction
sequence and by the material behavior assumption. The
plastic, or failure zone develops closer to the slope
surface if the soil has a higher angle of internal
friction (dry soil). The general appearance of the
potential sliding surface obtained from a slip surface
was found to be the same for all cases using the same
assumed soil strength. The potential sliding surface
was also found to have good agreement with the one given
by BishOp's method (and elasticity analysis).

The tension zone development appears to be
greatly influenced by two factors, construction sequence
and material behavior. A smaller area and of less
intensity was observed for incremental construction. The
location of the maximum tensile stresses, develOping during
the collapse stage, gave good continuity with the possible
sliding surface predicted by the slip line field in plastic

zone.

u- .4.“

213

7.2 Nonhomogeneous Soil Slope

 

Only the case of a two layer soil slope with a
stiff upper layer was considered. It was found from the
elasticity analysis that development of a tensile stress
zone depends on the following:

1. 0 values (more intense and larger area appears
as v approaches 0.5).

2. Modulus of elasticity of the soil in the
bottom layer (higher values of E in the bottom layer
produce less tensile stress development).

3. The relative thickness between the bottom
layer and the top layer.

From the elasto-plasticity solutions, it was found
that the potential slip surface of the two layer soil slope
does not deviate much from that of the homogeneous soil
slope. The potential sliding surface located by Bishop's
method does not agree with that by the elasto-plasticity
results. In this analysis, where the soil in the top
layer is stiffer than the bottom one, it was found that
tensile stresses developed have a marked influence on the
cause of slope failure. Maximum tensile stresses with
deep propagation develop near the area where possible

sliding surfaces in the plastic region are found.

7.3 Stability Analysis

 

Slope stability was presented in terms of the

average factor of safety along the potential sliding

 

214

surface given by Bishop's analysis (or elasticity analysis
for homogeneous slopes). Contours of factor of safety
were plotted to show the potential failure region and the
decreasing stability as the load increases.

It was found that under undrained strength analysis
the calculated average factor of safety and the factor of
safety given by BishOp's method at each stage were in fair
agreement. At the failure stage for both the undrained and
the drained strength analysis, the factor of safety given
by Bishop's analysis was in close agreement with the one
from the elasto-plasticity analysis.

With the undrained strength analysis, the Bishop's
factor of safety is about 1.25 when first local failure
occurred. Bishop (1952) concluded that local overstress
occurs where the factor of safety lies below a value of

about 1.8.

 

LI

BIBLIOGRAPHY

215

BIBLIOGRAPHY

Bell, J. M. "Noncircular Sliding Surfaces." J. of Soil
Mechanics and Foundation Div., ASCE, Vol. 95, SM3,
1969, pp. 829-844.

 

 

Bennett. P. T. "Notes on Embankment Design." 4th Cong.
on Large Dams, Newdelhi, Vol. 1, 1951, pp. 223-245.

 

Bishop, A. W. "The Stability of Earth Dams." Thesis
presented to the University of London, London,
England, 1952.

Bishop, A. W. "The Use of the Slip Circle in the Stability
Analysis of Slope." Proc. EurOpean Conf. on
Stability of Earth Slopes, Stockholm, V01. 1,

1954, pp. 1-13.

 

 

Bishop, A. W., and Henkel, D. J. ‘The Measurement of Soil
Properties in the Triaxial Test, Edward Arnold
(Publishers), Ltd}, London, 1957.

 

 

Bishop, A. W., and Morgenstern, N. "Stability Coefficients
for Earth Slopes," Goetechnique, Vol. 10, 1960,
pp. 129-150.

 

Bowles, J. E. Foundation Analysis and Design. McGraw-
Hill Book Co., Inc., New York, 1968.

 

Chen, W. F. "Soil Mechanics and Theorems of Limit Analysis.‘
J. of Soil Mechanics and Foundation Div., ASCE,
Vol. 95, SM2, 1969, pp. 493-517.

 

Clough, R. W., and Woodward, R. J. "Analysis of Embankment
Stresses and Deformations." J. of Soil Mechanics
and Foundation Div., ASCE, Vol. 93, SM4, 1967,
pp. 529-549.

 

 

Connor,£L "Introduction to the Finite Element Displacement
Method." M.I.T., 1967.

Culmann, K. Die Graphische Statik. Zfirich, 1866.

 

216

217

Drucker, D. C., Prager, W., and Greenberg, H. J. "Extended
Limit Design Theorems for Continuous Media."
Quarterly_of Applied Mathematics, Vol. 9, 1952,
pp. 381-389.

 

Dunlop, P., and Duncan, J. M. "Slopes in Stiff-Fissured
Clays and Shales." J. of Soil Mechanics and
Foundation Div., ASCE, Vol. 95, SM2, 19H), pp. 467-
489.

 

Fellenius, W. "Calculation of the Stability of Earth
Dams." Second Cong: on Large Dams, Washington,
D.C., 1936, pp. 445-459.

 

Fellenius, W. Erdstatische Berechnungen mit Reibung und
Kohaesion. Ernst, Berlin, 1927.

 

 

Hoeg, K., Christian, J. T., and Whitman, R. V. "Settlement
of Strip Load on Elastic-Plastic Soil." J. of Soil

 

Mechanics and Foundation Div., ASCE, Vol.494,

 

Huang, Y. H. "Stresses and Displacements in Nonlinear
Soil Media." J. of Soil Mechanics and Foundation
Div., ASCE, Vol. 94, SMl, 1968, pp. 1-15.

Jaky, J. "Stability of Earth Slopes." Proceeding of the
First Internat. Conf. on Soil Mechanics and
Foundation Engr. Harvard School of Engr.,
Cambridge, Mass., Vol. 2, 1936, pp. G-9.

 

Janbu, N. "Application of Composite Slip Surfaces for
Stability Analysis." Proc.‘European'Conf.‘on
Stability of Slopes, Vol. 3, 1954, pp. 43-49.

 

Karman, T. "Uber elastische Grenzzustande." Proc.
Second Cong. Appl. Mech., Zurich, 1927.

 

Kotter, F. Die Bestimmung des Druckes an gekrfimmten
Gleitflachen, eine Aufgabe aus der Lehre vom
Erddruck, Berlin, 1903.

 

Krey, H. Erddruck, Erdwiderstand und Tragfaehigkeit des
Burgrundes. Ernst, BerIIn, 1926.

 

Lambe, T. W., and Whitman, R. V. Soil Mechanics. John
Wiley and Sons, Inc., New York, 1969.

Leonards,G. A. Foundation Engineering. McGraw-Hill Book
Co., New York, 1962.

 

| 3

218

Liam Finn, W. D. "Application of Limit Plasticity in Soil
Mechanics." J. of Soil Mechanics and Foundation
Div., ASCE, Vol. 93, SMS, 1967, pp. 101-119.

 

Lysmer, J. "Limit Analysis of Plane Problems in Soil
Mechanics." J. of Soil Mechanics and Foundation
Div., ASCE, VoI. 96, SM4,’I970.

 

Malvern, L. E. Introduction to the Mechanics of a
Continuous Medium. Prentice-Hall, Inc., New
Jersey, 1969.

 

 

May, D. R., and Brahtz, J. H. A. "Proposed Methods of
Calculating the Stability by Earth Dams." 2nd
Cong. on Large Dams, Vol. 4, 1936, p. 539.

 

Mendelson, A. Plasticity Theory and Application. The
Macmillan Co., New York, 1968.

 

Morgenstern, N. R., and Price, V. E. "The Analysis of the
Stability of General Slip Surfaces." Geotechnique,
Vol. 15, 1965, pp. 79-93.

 

Promdtl, L. "Uber die Harte plastischer Korper." Nachr.
figl. Ges. Wiss. Gottingen, Math. Phys. Kl., 1920.

 

Rankine, W. J. M. "On the Stability of Loose Earth."
Phil. Trans. Royal Soc., London, 1857.

Scott, R. F. Principles of Soil Mechanics. Addison-Wesley
Publishing Co., Inc., Massachusetts, 1963.

 

Sevaldson, R. A. "The Slide in Lodalen, October 6th, 1954."
Geotechnigue, Vol. 6, 1956, pp. 167-184.

 

Skempton, A. W., and Henkel, D. J. "Tests on London Clay
from Deep Boring at Paddinton, Victoria and the
South Bank." Proc. of the 4th Internat. Conf.
Soil Mechanics and Foundation Engr., 1957, pp.
100-106.

 

 

Sokolovski, V. V. "Limiting Stress State of Granular and
Fine-Grained Soil Media." Dokl. Akad. Nauk. SSSR,
Vol. 24, No. 8, 1939.

 

Sokolovski, V. V. Statics of Granular Media. Perfamon
Press, Oxford, 1965.

 

Sokolovski, V. V. Statics of Soil Media. Buttercorths
Scientific Publications, London, 1960.

 

Taylor, D. W. "Stability of Earth Slopes." Boston Soc. of
Civil Engineers, Vol. 24, 1937, pp. 337-385.

 

 

219

Terzaghi, K. "The Mechanics of Shear Failures on Clay
Slopes and Creep of Retaining Walls." Pub. Rds.,
Vol. 10, 1929, pp. 177.

Timoshenko, S. P., and Goodier, J. N. Theory of Elasticity.
McGraw-Hill Book Co., Inc., New York, 1970.

 

Wilson, E. L. (Univ. of Calif., 1966), Christian, J. T.
(MIT, 1967). "Program FEAST 1-65," MIT.

Zienkiewicz, O. C., and Cheung, Y. K. The Finite Element
Method in Structural and Continuum Mechanics.
McGraw-Hill Publishing Co., Ltd}, London, 1967.

 

APPENDICES

220

 

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APPENDIX B

PROGRAM NEWSTP(INPUTcOUTPUT-PUNCH)
DIMENSION 51(25093)oSN(25093)¢S(250o3)
RFADIOO¢C9PHIQN9M
[.W "HRFADTOT7T(SITIEUTTJETT3TTTETTNTW'
! READIOI.((SN(I¢J)9J=193)9[=19N)
PRINT 102
100 FORMATT2F10:3¢ZISTHMT"
101 FORMAT(23X92E18.4112X9E12.4)
102 FORMAT(*ELtM-NO. X-STRESS Z-STPFSS XZ'STRESS MAX-STRFSS
l ”IMTN-STRESS"“W_"ANGEE—"-TMWFTST"M”"”UNUSEU:SHEAR“FAILANGI"*"”FATLA"
i 2N62*I
I 103 FORMAT(IS¢10F12 3) y,
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P[=(3.147*PHI)/180.
DOll=loN
T 2002lei3‘ M...,. "_11- 11“.... 11-)--- .A1.H. “A“ i__m.t .U_ L
E 2 S(IoJ)=SI(IoJ)+SN(IoJ)
i TX=S(191)
VT7=S(]92)
TXZ=-S(Io3)
T1=(TK+TZ)/2.+SQPT((TX+TZ)*“?-4.*(TXBTZ-TXZ#*8))/2.
l’—MM WTP§TTX7T7)7?::SORT(TTYTTZT¥¥Z:E.aTIXEIZ-TYZEEE)572.
TFTA=(ATAN(2.*(-TXZ)/(TX-TZ)))/2.
TFTA=(180.*TETA)/3.143
‘Kfifil=1£7A-45.4PH1/P.
ANGZzTETA+45.+PHl/2o
TA1=-T2
[___ “"112: 4- A.-.“
R=(TAl-TA2)/2.
RM=((TA1+TA2)/20*(C/IAN(PI)))*SIN(pI)
Rfis=pM:P
RE5T=HES*COS(PI)
F5: RM/H
f -_____ NF: Ntff _
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' l CONTINUE
-"wh’ HIF(M)4;A:3"" w
3 DOSI=IQN
S PUNCH1010(S(19J)9J=193)
[”UFW4'CONTINUE
5 STOP
I END

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APPENDIX C

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

EL.N0. SIGHA1 SIGHAJ FS FAILANbl FalLAN02 UNUSED SHEAR STJENGTH
1 1767.690 1653.00t 2.390 «0.10( -00.030 232.970
2 17Q719‘9J l“521“VU 20355 ‘02-‘06? '9’0‘333 632.1(70
3 1766-970 lkbO-CU.’ ZOJOZ “00705 -990295 8300939
0 1791.563 1000.795 2.320 38.789 -51.210 627.000
5 1790.950 1991.009 £.(~0 30.590 -53.~$9 ((2.0(0
0 1000.117 14(9.071 (.120 33.757 -50.(~3 (11.0777
7 1010.703 1&00.39( 1.990 30.200 ~59.000 193.790
0 1030.000 1307.309 1.090 25.073 ~0h.127 103.650
9 16290‘09.’ IZUJOCIJ 10"le 81-099 'OUOVOA 1150600

In 1169.911 11050301 1010'”, 160351 '7Job‘99 370565
11 1079.(77 070.970 .990 12.000 -77.992 -1.909
12 1972.77( 800.002 .900 7.900 -8(.0~0 -~1.900
13 1107.2~( 200.000 .090 «.70: -0S.(90 -«9.(01
14 001.390, 13.630 .9(( 2.907 —07.093 -3~.050
TS 1001.0(0 1292.0(0 2.393 «9.1(7 -~0.073 («0.700
10 1001.511 1290.931 (.570 «(.300 -~I.032 899.710
17 1053.000 1237.305 (.030 90.007 -99.~33 (9(.(75
10 1000.011 1(31.73« (.903 30.090 -51.302 (37.011
19 1003.200 12(1.7(S (.30( 30.733 -03.(07 2(9.((0
20 1079-013_ 1(23-049 4-101-4, 39-001 -00.309 {19.003
21 1007.201 1171.401 1.9(3 3z.30~ -57.090 191.900
22 1090.990 1113.093 1.000 29.7u3 -00.(07 109.099
23 1301-711 .1010-Véb 1.“£( fib.5b“ ‘0J1*”° 110.007
20 1017.000 070,770 1.2a? 22.309 -07.0«1 79.300
25 1377.017 007.9(3 1.100 17.109 —7(.0~0 00.103
20 1100.001 «90.003 1.1(3 10.009 ~7~.190 93.750
27 037.733 106.00! 1.100 9.191 -00.009 0(.~00
20 1319.101 1031.990 (.030 04.109 -«0.091 (00.9((
29 1319.709 1009.003 (.000 92.3(0 -~7.07~ (07.00(
30 1310.«*( 10(0.379 (.790 90.007 —~9.~~3 (09.009
31 13(0.(00 1017.000 (.0~~ 30.003 -01.1~7 («0.716
3a 1547.290 1000.031 (.903 37.300 -0(.09a (3h.09(
33 1337.0(0 903.737 (.(09 30.00( -03.996 de.909
3« 1309.900 9.7.700 ..909 3~.070 -00.1d9 190.910
35 13ba-L53 firm-37H 1.713 31.0?”7 -bo.931 100.09%
30 1330.300 797,903 1.907 31.9(7 -00.073 131.00(
37 1(00.«90 070.(~9 1.300 (7.00( —0(.910 10«.070
Ad 1099-3”: 033-972 1-339 41.310 '00."69 100.391
39 031.09( (03.900 1.90c 16.770 -7/.:d~ 1dh.~9d
«0 090.0(0 09.793 1.003 3.003 —00.~17 197.300
01 1010-099 0((-(09 3-19: 90.119 -~;.0n0 (12.069
«2 1077.313 019.047 3.10( «(.301 -97.039 271.057
«3 1079.100 019.1(0 3.01A 90.000 -09.31( (07.000
«0 100(-0(9 ~00--(( (-0: 39-((J -00.700 (01.(9o
Ab 1000.7(c 791.196 (.009 J".193 -D1.h37 (31.633
«0 1097.980 709.090 (.«3/ 37.0(0 —0(.37. (30.030
97 1100-111 739-001 (.19: 37.003 -0g.330 (13.<0d
«a 111(.079 079.009 1.0.7 37.90d -0(.090 103.390
99 109(.1vc 090.917 1.019 30.099 -03.900 10:.113
so 1013-090. 000-13. 1-0(0 3(.(00 -0l.739 130.093
51 03(.000 3.0.19. 1.001 (0.00: ~0~.990 107.798
52 030.Z9o 1.0.30. (.00 10.701 -70.(39 (00.100
53 039-13; 013.000 3-097 9“.1b( -&2.”~fi KUILQCV
59 039.0(3 010.993 5.091 «(.993 -~/.007 (00.910
55 000.901 00~.103 3.377 90.977 -e7.0(3 (01.001
00 093-013 099-101 3.(01 39-7H0 -00-4fi0 (lb-£7“
57 DQOoI‘iU SDUQUJJ (.8511 JV.1‘3J '5005‘07 805.961
50 009.010 000.003 (.713 39.3(9 o0u.071 (0(.000
o 0 um 01 ou9.7|7 (33,013

60 000.1(( 005,003 (.001 «1,510 -«n_7~g 507,700
01 004.20: *1h.(30 1.0co 90.070 -«9.3(( 100.909
02 709.770 3:1.300 1.79( 30.00( -03.~30 170.701
63 395.7393 107.1710 (.515 (0.7093 -OJ.JII‘) (IV-166
6‘0 390.61! 109.095 (.755 1‘7.th -/0.h‘_)9 850.991
05 DUU-fifij «um-L00 9-113 9~.d3/ '951’01 JU(.’””
00 000.991 ~0(.015 9.033 9(.703 -~7.(3/ 300.017
07 001.030 390.991 3.070 91.971 -90.0(9 (90.002
00 003-979 309-117 3.000 90.001, -~9.939 (90.031
09 000.119 171.703 3.399 90.300 -~9.09~ (0(.317
70 010.230 303.091 3.117 90.991 -~9.009 (71.001
71 010-770 331-030 (.010 “(.707 ~«I.(0J (07.930
72 029.240 301.339 (.977 93.009 -99.990 (30.069
73 019.973 (90.000 (.103 90.(09 -~~.711 (10.793
79 030-703 199-337 (.«1~ 90.930 -~9.070 cJ~.tl?
7S 3‘09.JJ~ 77.501; (.993 JI).71D -‘:~9.cd9 (00.935
70 301.320 (00.009 «.970 04.901 -~0.099 319.069
77 301.900 190.9(( ~-0~I 93.407 ~90.7~J 117.906
70 301.03: 100.377 ~.01c ~2.(9b -~/.700 313.403
79 30(.~71 170.700 «.300 91.099 —00.300 307.1“Q
31. 303.019 10fi.o3( 3.909 “1.007 -~0.313 (99.000
81 3070.750 197.539 3.0133 92.351 -~I.~~9 (VI.‘¢““
02 307.009 13~.033 3.930 ~~.00~ -~a.~lo (03.002
H3 J/jléfj 151‘.n7 3,190 07,970 -«c.0d2 (70.911
09 100.709 10~.1ZL (.070 00.000 -39.919 (01.010
05 (01.190 01.19: 9.616 39.090 -00.13~ 300.0:0
an, JJ~.010 9K.703 3.310 30.030 ~01.10( (79.191
87 1(1.n~( -3.c(9 0.930 «9.710 -~0.(0( 337.007
80 141.009 -7.n01 0.:00 «0.107 -u>.n|J 340.305
89 1(1-099 -1h.973 0.790 «3.701 -«0.(39 330.979
90 1(1.0(0 -30.017 0.(90 «3.0(9 -~0.~71 3(u.~77
gl ‘dUodJu -Vfio”) hoH‘J “JODUU -‘000“UU JIOQVUO
92 1;0.2L* -3~.990 9.993 99.109 ‘93.091 310-015
93 119.709 -~0.1~( 9.303 90.30! -h~.093 307.099
99 1((.09( -00.o30 9.003 «7.900 -~(.0~0 311.101
95 141.419 -c3.1~0 0.019 07.173 -3(.0(7 3(7.0(0
Q0 110.003 10.171 0.-~c 30.003 -01.~«7 30(.00~
M1NIM0H faCIUN OF baFLIV 10 .0903 AT LLthNT wuH0En 13

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FdA LUADER 050771 FaA 1500E5 067001
-PNUGHAH----AUUNESS- --LAOELED---CUHHON—-
NEu5TK 000100
0070A 013036
5103 013053
515TEM. 016622 EXECHse 016622
CUNLClU 016622
1NPUTC5 015600
KUOEHs 015570
AHAKLNQ 017202
UUTPTCt 020723
AIANE 021017
SINCOS: 021100
50H]? 021155
TANE 021177
----u«5A115F100 EXTLKNALS"--‘ REFERENCES
LEM-NO. x-5IRESS {-518055 12-516255 HAA-5IHE55 H1N‘SlHt55 ANGLt F.5. UNuseo-SHEAw FAILANGI FAILANGZ
1 -lQJ‘¢.§111 'dulb.“°u -I.93( -1036.66b '291‘).025 '1.192 2.200 261.216 -66.b9d “6.308
2 -1033.160 -2016.000 ~23.960 -1031.061 -2015.579 -3.500 2.269 239.815 -69.080 61.920
3 “10300560 ”(U1106“0 "900‘037 '10dboJlo -2015056‘0 ’50V118 6.21, 23b0956 '51.938 39.512
‘0 -10‘70J69 -CUQQIZTU ‘3".69" -101607‘13 -60170655 ‘60‘030 201°“ 23(0102 -530930 370070
5 -1023.370 -200/.500 -77.66o -1000.365 -2022.505 -10.976 2.006 226.523 -50.676 36.520
0 -1021.600 -2011.330 ~100.no9 -1590.063 -2035.007 ~13.076 1.907 212.139 -59.176 31.820
7 -1011.903 -2u11.330 -132.n19 -1571.015 ~2051.195 -10.010 1.001 191.809 -02.310 20.002
8 “13(00130 -(1-1DoQUJ '1770300 -13500655 'd1020J55 '19-‘ifio 10582 156365 -650“90 250510
9 -156U0‘0JU 'lHIC-DIU -aJfioJVJ “1"51-JD-I ‘dualole ’2‘..be 10367 1150691 ‘70006', 200931
10 -1006.020 -2603.100 -315.376 -1no9.201 -2530.039 -23.260 1.005 2.062 -00.760 22.256
11 -1110.363 -166o.3ca ~307.037 -ddo.360 -1oo6.303 ~32.960 1.051 20:330 -70.660 12.556
12 ‘Vlfl-hjd 'l 19(059’ 'QCQQHQU '3950050 '16dd.970 “JOIVOJ 0996 -260025 -820463 80537
13 -0/0.06l ~027.n61 -663.0I9 -JUJ.579 -120£.309 -60.105 .919 -3b.257 -85.085 5.315
16 -61n.302 -511.251 -631.731 -30.551 -099.002 —61.909 .960 ~20.157’ -07.609 3.591
15 «1309.010 1769.510 -7.513 -130n.053 -1766.007 -1.193 2.309 260.929 -60.093 66.307
10 -1307.000 1760.300 -22.IJI -1305.035 -1769.605 -3.505 2.367 265.226 -69.085 61.915
17 -1353.360 .7.n.630 -30.5II —1JIV.¢6, -1750.603 -5.660 2.302 261.002 -51.690 39.506
18 ~137I.700 1766.560 -55.511 -1309.563 -1732.757 -0.617 2.230 235.500 -S3.917 37.003
19 -1370.910 1765.030 -76.3nn -|350.003 -1759.Hbl -10.022 2.119 225.500 -50.322 36.078
20 -135n.500 1760.610 -9n.6°9 -1335.995 ~1710.915 -13.100 1.906 209.575 -50.000 32.360
21 -1365.330 1705.000 -123.llu -1310.507 -1000.303 -15.666 1.766 182.207 -00.966 30.056
22 -1303.900 1750.370 -103.500 -1251.600 -1012.966 ~17.009 1.520 165.933 -03.309 27.091
23 -1309.220 1050.250 -26n.6nl -1216.562 -1950.920 -21.003 1.101 59.607 -00.503 26.697
26 -1095.910 1703.010 '251.069 -1u13.001 51000.039 -10.007 .997 -1.000 -03.507 27.613
25 ”1117750de 1‘99A.“OU “17730.3'91) -10d’.60‘* '1D‘~2.191 -1HOC91 1.0“8 1650168 '630791 27.209
(b -/£J.:>'33 '87'0.0“3 -JdK’ID—’, "OO1-U31 -1157019‘3 '36.béd 102‘“ 75.616 '8“.025 6.972
27 -6o2.6ud -531.21H -326.660 -170.560 -023.000 -61.955 1.252 02.365 -87.655 3.565
20 -1166.700 1603.210 ~n.965 -1166.010 -1603.352 '1.176 2.697 253.506 -60.076 66.320
29 -1162.110 1602.200 -21.003 -1160.010 ~1603.500 -3.529 2.600 251.690 -69.029 61.971
30 -1130.d20 1610.030 -35.030 -1133.126 ~1606.320 -5.005 2.600 267.100 -51.305 39.015
31 ~1129.130 1679.900 -51.hJc -1121.000 -1607.362 -0.190 2.312 239.001 -53.096 37.302
32 ~111I.200 1600.913 -0n.o92 ~1106.717 -1693.653 -10.363 2.176 220.267 -SS.063 35.157
33 '110101d1} 1‘05‘70‘QUU '0/051’3 ‘10”20J7J “15050.7“, -120170 1095‘. 2090506 -570070 330330
36 —1009.020 1699.070 -110.009 -1063.200 -1j20.606 -13.56d 1.769 100.016 -59.060 31.952
3: -1036.300 1537.070 -165.I03 -995.097 -1570.273 —15.n67 1.656 131.800 -00.567 30.653
30 -926.951 .500.230 -103.597 -n75.710 -1009.6l3 ~15.uu7 1.169 56.772 500.507 30.693
17 -Ou1.hn5 .p2n.991 -1I6.36+ -752.029 -1500.026 -12.no1 1.062 17.012 550.301 32.099
36 ‘0‘)U."3, 113301“U '1U005‘011 -hd‘*.2'15 -11I'30L’1" -llo““d 10515 1‘1085“ ‘56.?62 3‘0058
39 -609.01, -06h.761 -196.129 -350.000 -77l.022 -33.973 1.957 200.631 -79.673 11-527
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62 -n9n.5lu 1215.690 -10./9/ -097.607 -1210.001 -3.301 2.022 250.012 -60.081 62.119
‘03 'HVI.O/U 181“.“‘3J -JC0UDI -055031I -181700173 -500715 6.5“8 2530767 -510115 390865
66 -ddu.9ll 1213.100 -60.109 -ulé.060 ~1220.029 -7.760 2.622 265.520 -53.260 37.700
‘05 '100-001 1(15-17‘0 -0110016 '"D‘JOHUU 'ldfbo‘Q51 -g0§7“ 80256 2320779 -5500?“ 35.926
66 -n63.139 1222.750 -75.2nl -~2n.l53 -123/.130 -10.013 2.06! 213.773 -50.313 36.007
67 -010.720 1262.733 -90.590 -792.37I ~1201.nl3 -11.605 1.!03 103.512 ~50.905 36.095
60 -I60.1d2 1210.000 -100.l9l -/25.207 -1291.515 -11.070 1.675 136.626 -50.576 36.626
«9 -n63.703 1296-700 -119.on1 -nga.6ou -1J1n.UH5 -10.009 1.202 70.071 -55.589 35.611
50 -500.005 1220.301 -9n.705 ~695.900 -1239.120 -7.560 1.117 63.520 -53.066 37.956
51 -1d3.0/1 -~13.3/0 -ou.220 -377.995 -097.660 -0.700 1.503 151.330 -52.200 38.000
52 -t{/.6/3 -302.035 -on.659 -10/.295 -622.2JJ -d6.~10 3.650 207.756 -69.916 21.086
53 -5o1.33/ -9~.0.3c\u v5.209 4161.262 “460.655 -1.UJ‘4 (.083 270.362 '56.D39 “5.561
56 -05o.95l -96/.unu -.5.001 -050.095 -960.722 -3.110 2.029 207.012 -60.010 62.390
55 -O‘OU.\IHI‘ -‘4‘o/Qu‘flu -(/.LUN -n“5.0““ -99V.3(, -b.130 807‘“ 201.901 '50.b36 “003°“
50 -036.033 -96o.930 -3n.9o/ -029.065 -951.726 -7.000 2.571 252.780 -52.506 38.696
57 -015.0/5 -960.510 -50.I90 -000.097 -95n.005 -H.6H2 2.377 239.500 -53.982 37.018
515 -‘39u,~ona -9')3.I-a.1 -b1,1U6 _DPJU.D1Z -VDD.O“U “Q.t“5 6.1“7 220.531 ’5“.755 36.255
5“} —DDC.UJD '”/1.V"U '07.UC’7 “3‘710397 '96d.“d9 “50'5“5 1057s 1920516 -5“o~’“a 360652
00 -690.003 -9~~./Iu -05.690 -602.509 -1000.006 -7.216 1.572 150.190 -52.716 38.280
01 -391./1o 101/.010 -5o.020 -310.739 -1021.9n9 -5.077 1.290 96.013 -50.577 60.623
02 ~209.911 -91~.900 -J0.0ud -207.n53 -921.030 -3.210 1.250 83.720 -6B.710 62.286
03 -103.165 -b90.1o1 -1u.775 -102.073 -590.633 -1.666 1.902 192.715 ~60.966 66.050
06 -5~.211 -19/.075 —12.020 -53.210 -19d.o70 -6.756 5.529 329.300 -50.256 60.760
05 -622.956 -o/v.625 -3.99o -622.092 -0l9.6bl -.oVJ 3.192 201.236 -662393 “6.607
00 -617.007 -olw.120 -12.201 ~517.039 -079.D96 -2.006 3.110 270.150 -hu.106 62.836
07 —600.002 -0/n.705 -2u.936 -605.200 -000.310 -6.375 2.970 271.831 -69.d7b “1.125
00 -390.259 -b/a.526 -30.160 -30/.1~1 -001.062 -5.903 2.779 ’201{§§1 ~51.603 39.597
09 -3ol.910 -nlv.737 -39.001 -303.100 -0o6.550 -7.029 2.565 260.331 -52.529 38.671
70 —339.010 -0n6.209 -65.365 ~333.153 -090.000 ~7.357 2.291 230.391 ~52.057 38.163
71 -302.101 -095.023 -66.¢79 -297.202 -I00.762 —0.337 2.020 200.091 651.037 39.163
72 -251.070 -71!.230 -29.603 -250.010 -719.090 -3.009 1.761 173.062 -69.109 61.091
7.7 ‘17,,55‘0 “’JC11“U 'CaUL’U '1’7057’ “7.78.1“, .0(07 1.“?! 130.56b -“50107 “5.293
76 -93.560 -03..310 17.999 -92.930 -031.912 1.916 1.500 130.707 -63.530 67.616
75 6.339 -330.356 30.100 0.023 ~3J6.030 0.611 2.367 231.123 '39.089 51.911
70 '16,0291 '“U’olb'fl ‘6053‘9 -167OCOU ‘“UV01’V -0059 30651 296.152 '“bnlsg ““06“1
77 -1r31.1161 -4'.u-4..,6:1 -l.523 -10(1.77.3 -61)9.Juu '1.'962 3.595 290.755 -‘07.662 “3.5—3'8
78 -100.200 -600.371 -13.690 -107.525 -609.020 -3.199 3.360 203.090 v68.099 62.301
79 -16o.732 -600.79n -19.55l -167.270 -610.200 ~6.275 3.019 273.275 -69.775 61.225
00 -122.236 -609.192 -25.320 -120.000 -611.610 ~5.006 2.777 250.073 -50.505 “0.690
81 -d9.91l -610.n39 -2n.761 -0I.303 -613.393 -5.075 2.600 261.205 -50.575 60.625
82 -56.009 -615.057 ~26.993 -52.907 -617.379 -3.960 2.210 221.010 -69.660 “1.560
03 -21.320 -62a.090 -o.032 -21.230 -620.700 -.052 1.992 201.067 -60.352 66.668
86 7.200 -66J.305 31.002 9.625 -662.510 3.969 1.707 177.730 -61.551 69.669
05 36-331 -336-650 61.032 30.035 ~336.603 0.202 2.120 213.305 -39.230 51.702
80 03.932 -261.257 59.030 75.171 -252.690 10.000 2.651 237.019 -36.032 50.100
00 52-27! -137-315 -2.7II 52.316 -1Jl.350 -.h39 6.225 305.799 -60.339 66.001
09 00.779 -13/.351 -6.061 00.096 -137.600 -1.357 3.920 290.330 -60.057 66.163
90 09.307 -13/.372 —I.105 09.009 -137.596 ~1.7°2 3.526 200.713 ~67.292 63.700
91 120.302 -1J/-;1; —6.406 120.020 -13/.021 -2.005 3.090 270.922 -67.505 63.635
92 157.753 -13I.020 -10.562 15n.159 -137.390 -2.066 2.705 251.961 -67.566 63.656
93 195.337 -135.956 -n.021 195.501 -130.170 -1.609 2.600 233.516 -60.969 66.011
96 (13.6071 -1JDA‘4JJ 1.7".) (lbIQUU -130.9‘02 .390 (02°C 223.007 -“Sodlo “5.790
95 180.696 -125.399 31.306 191.597 -12n.502 5.068 2.695 239.301 -39.052 51.168
90 100.565 -127.705 27.371 111.076 -130.916 0.510 3.299 270.771 ~30.982 52.018

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1 -1000.130 -2232.020 -12.900 -1003.702 -2233.000 -1.722 2.020 220.550 -07.222 03.770
2 -1799.900 -2227.090 -30.300 -1790.500 -2231.300 -5.003 2.002 217.733 -50.503 00.017
3 -1790.790 -2219.100 -02.595 -1705.709 -2220.101 -0.215 1.907 213.777 -53.715 37.205
0 -1700.100 -2207.030 -03.070 -1772.009 -2223.201 -10.900 1.920 209.230 -50.000 30.592
- I ‘ u '99.6;'0 -176Y.~73 -2226-077 -‘2- 67'. 10696 2030;1~ -5803,“ 3—20626
0 ~1005.200 -2213.270 -100.090 -1770.093 -2200.017 -10.017 1.002 203.073 -59.517 31.003
7 -1007.050 ~2290.900 -111.957 —1000.053 -2310.777 -13.937 1.023 190.900 -59.037 31.503
0 ~1900.070 -2000.520 -107.092 -1901:053“‘:2099.537' -10.791 1.000 139.300 -00.291 30.709
9 -1975.910 -2755.520 -327.520 -1050.579 -2070.051 -20.010 .007 —07.029 -05.510 25.090
10 -1017.200 ~2010.052 -310.020 -1002.000 -2551.070 -23.100 1.000 2.039 -00.000 22.310
- . -10 . -305.000 -902.337 -1705.922 ~32.700 1.052 20.903 -70.200 12.732
12 -905.777 -1190.220 -023.700 -029.179 -1512.020 -30.751 .900 -23.157 -02.251 0.709
13 -713.502 —071.231 -001.005 -303.530 -1201.195 -39.922 .922 -35.g32 —05.022 5.570
t“ '559008'0 ‘56101‘07 -‘300762 -7603“: '9“30589 ‘510503 09‘03 -2“0913 '513103 3.6;?
15 ~1510.77o -1939.990 -12.002 -1510.020 -1900.330 -1.035 2.039 219.105 -07.135 03.005
10 -1515.050 -1933.950 ~30.100 -1512.500 -1937.050 -0.903 2.020 217.005 -50.003 00.597
{7 -1510.700 —1923.020 -00.213 -1502.099 -1932.221 -0.120 1.999 210.005 -53.020 37.370
10 -1509.390 -1912.070 -03.000 -1092.720 -1929.330 -11.202 1.909 211.513 -50.702 30.230
19 -1510.290 -1903.700 —105.512 -1007.500 -1930.520 ~10.210 1.901 200.290 -59.710 31.202
20 ~1539.100 -1921.000 0120.227 -1501.227 -1950.993 -10.727 1.079 201.271 —02.227 20.773
21 -1573.100 -1903.300 -107.900 -1525.300 02031.112 ~17.099 1.705 170.120 -03.399 27.001
22 -1030.310 -2220.000 -151.703 ~1001.150 -2250.010 -13.753 1.320 105.225 -59.253 31.707
2! -1010.970 -2501.000 -175.709 ~1392.009 -2500.501 -0.090 .739 -153.003 -50.190 30.010
20 -1102.007 -1709.059 o251.070 -1020.709 -1071.010 -10.070 .999 -.331 -03.570 27.030
25 -1093.321 -1070.339 -155.009 -1037.205 -1520.395 -19.779 1.727 177.750 -05.279 25.721
20 -737.000 ~001.190 —319.293 -002.159 -1130.079 -30.000 1.205 00.010 «00.100 0.052
27 -031.231 ~550.020 —323.730 -101.052 -020.005 -39.707 1.201 79.331 —05.207 5.733
20 -1237.207 -1005.170 -11.177 —1230.901 ~1005.070 -1.500 2.001 220.033 -07.o00 03.932
29 -1233.329 ~1039.990 -30.000 -1230.505 -1002.010 -0.705 2.002 210.000 ~50.205 00.755
30 -1220.007 -1030.900 -50.159 -1220.232 ~1039.215 ~0.055 2.020 215.010 -53.555 37.005
31 ~1220.000 -1010.320 ~001030 -1207.290 -1035.502 -11.550 1.900 210.050 -57.050 33.950
52 -1220.000 -1012.290 -111.509 -1190.300 -1002.302 -15.070 1.913 202.702 -00.570 30.022
33 -1239.500 —1020.300 0139.050 -1190.202 ~1009.710 -17.970 1.707 107.107 -03.070 27.522
30 -1200.070 -1700.990 -103.970 -1210.095 01759.305 -10.337 1.552 151.001 -03.037 27.103
35 o1201.000 -1023.000 -150.203 -1202.702 -1001.790 -13.950 1.295 97.100 -59.050 31.550
30 -1200.050 -2300.030 30.000 -1203.109 -2305.311 1.002 .000 -79.753 -03.010 07.102
37 0005.350 -1532.200 -173.027 -705.922_ -1571.715 -12.770 .1.003 17.073 -50.270 32.720
30 ~090.090 -1095.979 ~113.521 -000.070 -1125.997 -10.005 1.010 105.920 -00.305 30.095
39 ~000.003 ~505.131 -101.159 -300.192 -721.302 -30.923 2.170 220.000 ~02.023 0.577
00 0320.029 -390.330 -210.300 -100.505 -570.202 -00.011 1.073 109.339 -00.111 0.009
01 -950.000 ~1350.100 -10.107 -950.199 -1350.001 -1.070 2.102 223.900 -00.970 00.020
02 -953.725 -1305.500 -31.212 -951.250 -1300.031 -0.520 2.117 221.015 -50.020 00.970
03 -905.500 -1337.070 -50.055 -930.000 -1300.010_ :7.023 2.005 210.557 -53.323 37.077
00 -937.002 -1327.500 -02.200 -920.032 -1300.190 -11.021 1.901 207.023 -50.921 30.079
05 -930.505 -1320.000 -113.202 -900.025 -1350.900 -15.003 1.001 190.120 -00.503 30.037
00 ~929.911 -1337.900 -100.390 -003.979 -1303.032 -17.030 1.000 109.010 -03.130 27.002
07 -917.000 —1390.050 -101.030 -007.050 -1000.000 ~17.100 1.007 133.000 -02.000 20.350
00 -907.210 -1520.020 -107.352 ~070.000 -1501.790 -12.001 1.225 77.300 -50.101 32.019
09 -057.090 -1000.020 -00.000 -052.500 11073:100 -0.530 1.029 11.720 -50.030 00.970
50 -972.290 -1500.070 120.520 -905.017 -1591.503 11.719 1.307 99.139 -33.701 57.219
51 -030.271 -730.310 -25.000 -032.020 —732.553 -0.950 2.729 259.017 -50.050 00.500
52 -227.501 -320.000 -07.500 -193.030 -301.990 -20.003 0.000 320.503 -72.103 10.037
53 —003.525 -1050.290 -0.057 -003.323 -1050.092 -1.330 2.237 229.510 -00.030 00.100
55 -000.300 -1o00.091 -00.970 -050.103 -1050.090 -7.210 2.113 210.000 -52.710 30.202
50 -005.090 -1030.055 -75.075 -032.030 -1050.711 -10.037 1.900 205.500 -50.137 30.003
57 -027.300 -1032.537 —107.521 -000.599 -1059.302 -13.972 1.007 105.009 -59.072 31.520
50 ~003.200 -1001.502 -137.030 -503.755 -1001.111 -10.039 1.002 155.015 —01.539 29.001
59 -577.035 -1009.010 -150.915 -530.201 -1130.500 -15.250 1.395 117.330 -00.750 30.200
00 -520.702 -1100.220 ~122.530 -500.073 -1210.209 -1o.107 1.100 03.109 -55.007 35.313
01 -059.729 -1330.330 -20.109 -050.027 -1335.232 -1.030 .900 -22.570 -07.330 03.002
02 -325.270 -1005.330 171.930_ 473200.191V _11122.013_ -152-X95 .900 -0.001 -33.330 _57.000 ___
03 -220I930“ :5091071 99.010 -109.007" -500.510 17.300 2.310 230.973 -20.190 02.000
00 -57.001 -107.395 37.533 —03.000 -100.952 19.051 0.000 302.939 -25.009 05.351
05 -012.905 -750.050 -0.710 -012.030 -750.507 -1.110 2.300 230.200 -00.010 00.302
00 -000.000 -750.500 -21.303 -002.773 -755.001 -3.071 2.323 233.510 -00.971 02.029
07 -305.900 -750.590 -39.530 -301.702 -750.030 -0.115 2.197 223.290 -51.015 39.305
59- —350.003 -705.7507 ___003.109fi_ 73300.0107 _;755.003_ _Hg9.032 2.011 205.005 150.532_ 30.000
"‘09 -320.057 -7021100 " -91.031 " -301.325 _' -701.230 -11.703 1.700 179.203 -57.203 33.737
70 -270.027 -700.101 -119.000 -200.032 -770.530 -13.371 1.501 103.000 -50.071 32.129
71 ~220.033 -775.902 -120.090 -191.709 -000.200 -12.000 1.330 102.010 -57.900 33.100
72 -101.275 —050.190 ~09.920 -109.730 -001.735 -7.312 1.100 52.770 -52.012 30.100
73 -70.330 -930.000 39.003 -70.555 -930.235 2.000 .909 -22.002 -02.900 00.100
70 -3.513 -702.500 190.900 _ _00.053 “__0792.700______10.100 .900 -13_237 -31.350 59.000
"75 59.201 ':259.517“—_””175.000' _“ 137.039 -337.275 23.070 1.090 100.507 -21.020 09.370
70 -107.107 -050.007 -0.270 -107.000 -050.500 -.790 2.019 250.000 000.290 00.710
77 —130.229 -055.720 -13.703 -135.030 -050.313 -2.005 2.527 200.737 -07.905 03.035
70 —113.002 —050.300 -20.235 -ITi.057 -0507305 -0.300 2.305 232.200 -09.000 01.132
79 -70.009 -052.299 -03.227 -09.507 -057.101 -0.001 2.007 210.710 —51.901 39.059
00 -17.910 -059.793 -00.003 -0.390 __ 1000.309 _101300' 1.700 170.051 -53.000 37.100
01 57.730 -051.025 -00.070 "722070 -005.971 -9.390 1.099 130.330 -50.090 30.100
02 103.130 -003.002 -93.907 157.350 -077.710 -0.005 1.200 05.192 -50.105 30.095
03 221.900 -095.007 -55.039 220.130 -099.005 -0.300 1.109 39.013 -09.000 01.100
00 250.150 -527.305 05.050 203.032 -532.007 0.757 1.010 0.002 -00.703 50.257
05 209.120 -000.755 197.250 200.720 -000.355 10.271 1.090 35.101 -29.229 01.771
00 210.100 -120.000 203.920_ 309.129 3323.029 20.900» 1.099 132.792 -20.530 70.000
07 113.271 -153.711 -1.000 113.279 -153.719 -.313 2.990 200.753 -05.013 05.107
09 152.320 -150.200 —9.051 152.011 -150.091 -1.700 2.005 200.300 -07.200 03.730
90 199.030 -150.509 -10.231 200.572 -155.331 -2.015 2.205 221.550 -00.115 02.005
91 275.170 -150.750 -25.000 270.070 -150.250 -3.373 1.003 102.390 -00.073 02.127
92 303.010 -150.079 -35.090 355;§7° -150.§39 -3.701 1.000 120.017 -09.201 01.719
93 515.030 -151.937 -00.003 510.295 -150.002 -3.009 1.100 00.001 -00.909 02.031
95 019.730 ~119.509 07.001 022.010 —122.050 3.007 1.001 22.020 -01.013 09.107
90 350.030 -150.100 121.202 301.033 ~105.371 12.005 1.000 110.002 -32.055 50.105

 
 
 

  

  

  

  

  

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1 -1666.167 -2336.160 -15.691 -1665.665 -2336.663 -1.909 1.929 210.333 -57.509 53.591
3 ~1676.775 -2316.626 -70.012 ~1665.951 -2329.659 -6.765 1.663 205.733 -55.265 36.716
5 -1673.655 -2306.765 -66.627 -1656.326 -2325.301 -11.150 1.665 202.565 -56.650 35.350
6 6 O 5 O 6 O O O O
6 ~1927.560 -2359.100 -99.566 ~1905.661 -2360.919 -12.367 1.651 199.759 o57.667 33.133
7 02015.500 ~2596.050 -111.557 -1991.005 ~2522.536 -12.391 1.653 173.600 -57.691 33.109
9 -1992.696 -2775.079 -326.229 -1675.696 -2693.279 -19.906 .667 -67.665 -65.506 25.592
10 -1655.229 -2550.295 -313.205 ~1711.557 -2563.066 -22.963 1.005 1.673 -66.563 22.537
I O 6 O O I O O O O
12 ~961.366 -1239.522 -522.353 -666.607 -1552.061 ~36.567 .950 -22.261 -61.967 9.013
13 -753.273 -919.660 -550.563 -366.265 -1265.666 -39.626 .925 -33.722 -65.126 5.672
W o o o I o o I
15 -1562.155 -2031.013 -15.232 -1561.693 -2031.565 -1.613 1.919 206.605 -57.313 53.667
16 -1579.920 -2022.260 -52.557 -1575.665 -2026.316 -5.529 1.916 206.179 -50.929 50.071
I O I O O C O O O O
16 -1565.596 -1995.911 -93.626 o1565.161 -2015.356 -12.263 1.912 205.603 -57.763 33.217
19 -1605.936 ~1996.560 -112.996 -1575.605 -2026.692 -15.019 1.912 205.753 -60.519 30.561
6 6 5 110MB 6 o o a ' c o
’1 -1691.360 -2202.700 -150.159 -1650.526 -2253.535 -15.206 1.565 137.552 -60.706 30.295
22 -1662.156 -2515.310 -159.756 -1656.056 -2551.510 -9.662 .966 -6.051 -55.362 35.616
25 —1066.330 -1597.356 -153.553 -1051.157 -1552.519 -17.560 1.666 171.616 -62.960 26.050
. 9097531——‘—=3107366————=5667093———‘11367076”‘—“*‘flfln . . . .
27 -377.671 -596.320 -329.760 -150.270 ~635.721 -35.735 1.175 60.735 -61.235 9.765
26 ~1261.715 ~1722.150 -13.579 -1261.296 -1722.566 ~1.765 1.932 205.529 -57.265 53.736
25 31(73.657 -1715.305 -51.323 ~1275.970 -1716.192 -5.371 1.923 205.561 o50.671 50.129
30 -1277.699 -1700.696 -70.556 -1266.251 -1712.156 -9.220 1.911 202.965 -55.720 36.260
31 -1262.703 ~1665.290 -100.967 -1256.736 -1706.255 -13.355 1.695 201.065 -56.655 32.156
"532‘—'=T3037755“—‘51665:656—""71267636““‘71255.575_——717257126“‘_“=167969“_—-_'f?6 . - . .
33 -1332.756 -1735.120 -157.657 ~1265.377 -1763.591 -16.130 1.710 177.172 -63.630 27.370
35 -1355.572 -1692.360 -156.259 -1306.927 -1929.925 —15.205 1.375 116.716 '59.705 31.296
'36‘ -1275.770 ~2110.110 ‘IIV.DDB -1267.992 ‘(1lb-UUU -7.665 {969 5’455.920 -53.565 37.516
36 -1263.666 ~2302.565 31.916 51262.669 -2303.565 1.792 .655 -76.996 -53.706 57.292
37 -603.732 -1529.097 -173.972 -765.165 -1566.665 -12.607 1.055 16.076 -56.307 32.693
36 -750.533 -11077276“‘—”—597.392"—‘=716.160*‘—=1131:531“‘—‘=13.977 2.005 206.363 -59.577’ 31.523’
39 -552.269 -561.625 -175.536 -323.173 -690.950 -36.320 2.223 225.666 -61.620 9.160
50 -379.517 -556.160 -229.066 -222.953 -715.725 -35.335 1.660 162.225 ~79.635 11.166
51 5965.175 -151 3. 656 ‘12. 713 fi65.796 51515. 025 -1.66‘7‘ 1.961 206. 263 -57.197 53. 603
52 —961.053 -1507.171 -39.560 -977.530 -1510.795 -5.255 1.952 205.101 -50.755 50.256
53 -975.151 -1393.759 -69.927 -963.769 51505.121 -9.233 1.906 199.939 -55.733 36.267
‘“55‘——_=9737269—"=T3617227"“—=105.935“‘_—=9571660“‘_=1506:636‘———‘113.605—“————T:633-—"‘I91T103‘-——=597105—‘———’317695_“”—‘
55 o976.667 -1375.003 -150.576 -933.977 -1516.913 -17.709 1.735 176.011 ~63.209 27.791
56 -1002.236 -1521.269 -157.255 -959.767 ~1573.716 -16.537 1.606 159.133 -63.937 27.063
57' o666.361 ~1555.590' 5132.027 -956.710 -16?5.z51 -1z.652 1.367 113.260 -56.162 32.656
56 -936.711 -1610.120 -65.699 -930.517 -1616.315 -5.511 .955 -19.929 ~51.611 39.969
59 -662.769 -1672.330 -63.151 -657.672 ~1677.227 -5.532 1.030 12.519 -59.932 51.066
.‘96—‘—=15233356 52.272 -576.602 ’551926.363 6.556 1.629 159.667 ~50.65z 50.956
51 -572.367 -705.530 66.962 -553.529 —723.366 15.360 3.039 275.205 -30.150 60.660
52 -205.912 -375.956 -70.562 -179.556 —501.320 -19.756 3.653 295.065 -65.256 25.752
53 -692.669 -1105.501 -11.256 -692.361 31105.709 ‘1.)09 2.016 209.552 o57.065 53.536
55 -665.596 -1099.537 ~35.565 -662.562 -1102.575 -5.676 1.979 205.505 -56.376 56.625
55 o672.592 -1C90.250 -65.062 -662.566 -1100.155 -6.659 1.696 196.535 -55.159 36.651
"56““=6552511“——:10767976————T1017656’—‘*:6327132—_‘=1100T205_-"’=127661“‘————“1:775 161.053 -56.367 32.633
57 -652.199 -1073.372 -151.916 -599.661 -1115.690 -16.671 1.606 156.607 -62.171 26.629
56 -632.366 -1o79.637 -166.367 -577.291 -1135.912 -16.309 1.566 136.075 -63.609 27.191
59 -666.051 -1211.520 -116.555’ -653.311 -1236.160 -11.762 1.505 119.925 —57.262 33.716
60 -556.571 -1571.510 -29.069 -557.556 -1572.335 -1.621 .913 -39.706 -57.321 53.679
61 ~565.795 -1339.130 -26.976 -563.963 -1339.961 -1.765 .959 -22.266 ~57.265 53.735
62 -327.660 -1067.756 171.250 -291.055 -1125.562 12.125 '963 o5.550 ~33.375 57.625
“63““‘*11?T530'_"-335.660 116.361"_"-62.651———”6365.569”—”“— 23.151“— ‘"2.510 —"252.696 ‘—“”522.359“—"‘—66.651
65 -166.910 -553.330 53.519 -179.961 -560.259 9.052 2.695 265.356 -36.556 55.552
65 -505.595 -793.522 -6.963 -505.267 -793.730 -1.325 2.113 216.165 -56.625 55.175
"66“——‘3395:553—‘——?T90:503‘————726:676 -332.566 ‘73(.5VU -5.156’ 2.051“—"2103255”——'-*59.656 51.352
67 -372.077 -765.229 -55.306 -365.051 ~791.265 -7.379 1.925 196.905 -52.679 36.121
66 o336.607 -777.351 -67.600 -321.669 -795.269 -10.902 1.735 173.377 -56.502 35.596
‘ 69 -292.525 -767.753’ "'5127.605*"‘3260.333‘0 5799.939M—" -15.131‘ —"‘“ 1.517‘—“”'139.363““‘—‘559.631"“‘“‘*31.369555‘
70 -253.657 -770.603 -163.205 -206.635 -616.016 -16.127 1.336 103.169 -61.627 29.373
11 -236.016 —776.920 -153.512 -200.561 -315.575 -13.916 1.332 101.766 -59.516 31.562
. - .010—————*657360——-—*3707flfir-—112037016—_—‘"”17556—“——" 7995 ‘—’——'27659————1W17055——'———59T956-“—’
73 -69.513 -925.670 50.261 -67.625 -927.756 2.665 .950 ~21.525 -52.615 56.165
75 3.613 -733.355 199.066 53.957 -763.697 15.165 .970 -12.507 -31.315 59.665
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76 -125.272 -579.556 -5.650 -125.176 -579.652 -.953 2.260 227.555 -56.553 55.557
77 o109.752 -576.376 -19.102 -106.755 -579.365 -2.957 2.166 219.751 -56.557 52.553
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62 230.290 ~555.621 -157.553 265.790 -569.121 -12.357 1.066 25.950 -57.657 33.153
63 199.657 -593.369 -69.026 209.519 -603.261 -6.325 .993 -3.006 -51.625 39.176
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66 -26.020 ~605.530 151.556 5.617 -636.267 13.055 1.261 63.921 -32.556 56.555
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66 166.371 -162.233 -6.606 166.512 ~162.375 v1.166 2.532 235.525 -56.666 55.315
69 205.569 -162.725 ~13.620 205.993 ~163.229 -2.120 2.170 215.530 -57.620 53.360
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92 552.360 -162.707 -56.097 557.115 ~167.563 -5.677 1.110 39.325 -50.177 50.623
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95 1012.090 -157.915 -63.961 1016.065 -163.909 -5.062 .665 ~196.595 ~59.562 51.516
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