cr s-69-8 'finite element analyses of stresses and

[ ', .· i 1 .. ~ ') ,.

u t£ r'.J ;) .. /' ,.t, .,i

v

CONTRACT REPORT S-69-8

FINITE ELEMENT ANALYSES OF STRESSES AND MOVEMENTS IN EMBANKMENTS DURING CONSTRUCTION

A Report of an Investigation

by

F. H. Kulhawy

J. M. Duncan

and

H. Bolton Seed

under

Contract No. DACW39-68-C-0078

with

u-. S. Army e-ngineers waterways Experiment Station

CORPS OF ENGINEERS

Vicksburg, Mississippi

November 1969

College of Engineering

Office of Research Services

University of California

Berkeley, California

Report No. 'TE-69-4

.UtMY·MRC VICKBI!IUAQ, MISS.

T~IS DOCUMENT ~AS BEEN APPROVED FOR PUBLIC RELEASE AND SALE; ITS DISTRIBUTION IS UNLIMITED

FOREWORD

The work described in this report was performed under Contract No. DACW39-68-C-0078 "Behavior of Zoned Embankments and Embankments on Soft Foundations" between the U. S. Army Engineer Waterways Experiment Station and the University of California. This is the first report on investigations performed under this contract. The research was sponsored by the Office, Chief of Engineers, under the Civil Works Investigations Engineering Studies 525, "Shear Characteristics of Undisturbed Weak Clays."

The general objective of this research, which was begun in June, 1968, is to develop methods for analysis of stresses and movements in embankments. Work on this project is conducted under the supervision of Professor J. M. Duncan, Associate Professor of Civil Engineering and Professor H. Bolton Seed, Professor of Civil Engineering. The project is administered by the Office of Research Services of the College of Engineering. The phase of the investigation described in this report was performed by F. H. Kulhawy, and the report was prepared by F. H. Kulhawy, J. M. Duncan, and H. Bolton Seed.

The contract was monitored by Mr. D. C. Banks, Chief, Rock Mechanics Section, Soil and Rock Mechanics Branch, under the general supervision of Mr. J. P. Sale, Chief, Soils Division. Contracting Officer was COL Levi A. Brown, Director or tin: U. S. Army- En-gi1re-er Waterways- Experimenr- Statiun~

3

SUMMARY

The objective of this investigation was to develop procedures for conducting finite element analyses of stresses and movements in embankments during construction. The procedures developed involve incremental analyses, simulating successive stages during construction of the embankment, and employ nonlinear stress-strain parameters determined from the results of laboratory tests.

Previous studies of the nonlinear, stress-dependent stress-strain behavior of soils were extended during this investigation to include variations of Poisson's ratio values as well as modulus values for use in incremental analyses" In order to examine the suitability of these procedures for representing the stress-strain characteristics of a wide variety of soils under both drained and undrained test conditions, the procedures were applied to 46 different soils, ranging from cobble sizes to highly plastic-clays, for which stress-strain information had been published or was available from other sources. In each case it was found that the simple procedures developed for representing nonlinear, stressdependent soil stress-strain behavior were convenient and provided reasonably accurate representations of the actual soil behavior"

A finite element computer program was developed for incremental analyses of embankment stresses and deformations, incorporating these nonlinear stress-strain characteris-ticsr and this- computer program v,tas- used to conduct a series 0f analyses of the deformations in Otter Brook Dam during construction These analyses showed that the vertical diGplacements (settlements) with1n an embankment during construction are affected very strongly by the value of soil modulus, and the horizontal displacements are affected very strongly by the value of Poisson's ratio. The vertical and hor1zontal displacements calculated using nonlinear stressstrain characteristics were in close agreement with those measured during construction of the dam.

Studies of embankment stability showed that the values of stress calculated by the finite element method may be used to define a factor of safety with respect to either local overstress or overall stability, Provided that the factor of safety with regard to overall stability is defined in a manner consistent with that employed in limit equilibrium analysis procedures, the value of the factor of safety calculated using finite element stresses is nearly identical to that calculated using the best limit equilibr1um procedures of slope stability analysis.

Studies were also conducted to determine the effectiveness of these finite element analysis procedures for calculating stresses and displacements in zoned dams. Analyses were performed for two hypothetical zoned dams which had the same cross-section, but which differed with regard to the stiffness of the core materiaL These analyses showed that the settlements of embankments are influenced considerably by the stiffness of the core material, and that the stress conditions are strongly affected by the relative stiffnesses of the core and shelL

5

Foreword

Summary

List of Figures

List of Tables

List of Symbols

English Letters

Greek Letters

CHAPTER 1 INTRODUCTION

TABLE OF CONTENTS

CHAPTER 2 FINITE ELEMENT ANALYSIS PROCEDURES

Characteristics of the Finite Element Method

Nonlinear Stress-Strain Behavior

Incremental Analyses

Required Number of Layers

CHAPTER 3 NONLINEAR STRESS-DEPENDENT MODULUS RELATIONSHIP

Nonlinearity

Stress-Dependency

Tangent Modulus

Modulus Parameters for Drained Conditions

Modulus Parameters for Unconsolidated-Undrained Conditions

CHAPTER 4 NONLINEAR STRESS-DEPENDENT POISSON'S RATIO RELATIONSHIP

Nonlinearity

Stress-Dependency

Tangent Poisson's Ratio

Poisson's Ratio Parameters for Drained Conditions

Poisson's Ratio Parameters for Unconsolidated-Undrained Conditions

7

3

5

11

15

17

17

19

21

24

24

25

25

25

34

34

41

43

44

49

55

57

60

60

62

65

CHAPTER 5 ANALYSIS OF OTTER BROOK DAM USING NONLINEAR STRESSDEPENDENT STRESS-STRAIN PROPERTIES

Finite Element Mesh

Properties of Otter Brook Dam Fill

Comparison of Calculated and Measured Displacements

Additional Results of Finite Element Analysis

CHAPTER 6 COMPARISON OF VARIOUS FINITE ELEMENT ANALYSIS PROCEDURES FOR OTTER BROOK DAM

Types of Analyses

Basis o~ Comparison

Comparisons of Displacements

Comparison of Stresses

Usefulness of Various Types of Analyses

CHAPTER 7 EVALUATION OF EMBANKMENT STABILITY USING FINITE ELEMENT STRESSES

Local Failure

Overall Stability

CHAPTER 8 FINITE ELEMENT ANALYSIS OF ZONED EMBANKMENTS

Finite Element Mesh

Material Properties

Modification of Analysis Procedure

Results of Analyses

CHAPTER 9 SUMMARY AND CONCLUSIONS

Simulation of Construction Sequence

Stress-Strain Behavior of Embankment Materials

Analyses of Otter Brook Dam

Evaluation of Embankment Stability

Analysis of Zoned Embankments

Conclusions

LITERATURE CITED

8

70

70

70

73

76

82

82

82

83

88

94

95

95

100

111

111

111

116

118

126

126

126

126

127

127

128

129

APPENDIX A LABORATORY TESTING PROCEDURES AND RESULTS 137

Soil Classification 137

Specimen Preparation 137

Specimen Compaction 138

Specimen Storage 138

Unconsolidated-Undrained (UU) Triaxial Shear Test Procedure 138

Equipment Calibration 138

Unconsolidated-Undrained (UU) Triaxial Shear Test Results 141

APPENDIX B DERIVATION OF RELATIONSHIP BETWEEN STRESS LEVEL AND SHEAR STRESS FACTORS OF SAFETY

APPENDIX C COMPUTER PROGRAM FOR THE FINITE ELEMENT ANALYSIS OF EMBANKMENTS

APPENDIX D COMPUTER PROGRAM FOR THE INTERPOLATION OF FINITE ELEMENT STRESSES AND STRAINS FROM KNOWN LOCATIONS TO DESIRED LOCATIONS

APPENDIX E COMPUTER PROGRAM FOR THE STABILITY ANALYSIS OF EMBANKMENTS OR SLOPES BASED UPON FINITE ELEMENT STRESSES

9

145

148

162

165

Fig. No.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

LIST OF FIGURES

Maximum Heights of Earth and Rock Dams from 1850 to Dateo (Modified after Glossop, 1967)

Solutions for Displacements in an Incrementally Loaded, Linear Elastic Columno

Solutions for Displacements of an Infinitesimally Layered Elastic Column with Stress-Dependent Modulus.

Selected Displacements in Infinitesimal and Finite Layer Column Models with Stress Dependent Modulus.

Correlation of Displacements in Infinitesimal and Finite Layer Column Models with Stress Dependent Modulus.

Hyperbolic Representation of Stress-Strain Curve.

Transformed Hyperbolic Representation of Stress-Strain Curve.

Deviations from Ideal Behavior on Transformed Plots.

Experimental and Hyperbolic Stress-Strain Curves for a Poorly-Graded Sand. (Data from Hirschfeld and Poulos, 1963)

Experimental and Hyperbolic Stress-Strain Curves for a Well-Graded Gneiss Rockfill. (Data from Casagrande, 1965)

Variations of Initial Tangent Modulus with Confining Pressure.

Variation of Nonlinear Modulus Parameters with Relative Density for Sacramento River Sand.

Variation of Nonlinear Modulus Parameters with Maximum Particle Size for Soils with Parallel Grain Size Curves. (Data from Marachi, 1969)

Strength Parameters for Compacted Pittsburg Sandy Clay Under Unconsolidated-Undrained Test Conditions.

Nonlinear Modulus Parameters for Compacted Pittsburg Sandy Clay Under Unconsolidated-Undrained Test Conditions.

11

22

27

29

30

32

35

36

38

39

40

42

46

48

52

53

Fig. No.

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

Typical Strain Patterns in Soils During Shear.

Experimental and Hyperbolic Axial Strain-Radial Strain Curves for a Dense Poorly-Graded Sand. (Data from Lee, 1965)

Experimental and Hyperbolic Axial Strain-Radial Strain Curves for a Dense Poorly-Graded Basalt Rockfill. (Data from Casagrande, 1965)

Variations of Initial Tangent Poisson Ratio with Confining Pressure.

Variation of Nonlinear Poisson Ratio Parameters with Relative Density for Sacramento River Sand.

Variation of Nonlinear Poisson Ratio Parameters with Maximum Particle Size for Soils with Parallel Grain Size Curves. (Data from Marachi, 1969)

Nonlinear Poisson Ratio Parameters for Compacted Pittsburg Sandy Clay under Unconsolidated-Undrained Test Conditions.

Otter Brook Dam. (After Linell and Shea, 1960)

Finite Element Mesh for Otter Brook Dam.

Displacements in Otter Brook Dam Using a Nonlinear Modulus and Poisson Ratio.

Displacements within Otter Brook Dam Using a Nonlinear Modulus and Poisson Ratio.

Principal Stress Contours in Otter Brook Dam Using a Nonlinear Modulus and Poisson Ratio.

Contours of Mobilized Strength in Otter Brook Dam Using a Nonlinear Modulus and Poisson Ratio.

Elastic Parameters in Otter Brook Dam.

Measured and Adjusted Displacements of Bridge Pier in Upstream Face of Otter Brook Dam.

Combinations of Elastic Constants Required for Various Analyses to Obtain Correct Bridge Pier Displacements in Otter Brook Dam.

12

56

58

59

61

64

66

69

71

72

75

77

78

79

81

84

85

Fig. No.

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

Displacements of Bridge Pier in Upstream Face of Otter Brook Dam as Determined by Various Methods of Analysis.

Displacements of Upstream Face of Otter Brook Dam as Determined by Various Methods of Analysis.

Comparisons of Horizontal Displacements in Otter Brook Dam as Determined by Various Methods of Analysis.

Comparisons of Vertical Displacements in Otter Brook Dam as Determined by Various Methods of Analysis.

Comparisons of Major Principal Stresses in Otter Brook Dam as Determined by Various Methods of Analysis.

Comparisons of Minor Principal Stresses in Otter Brook Dam as Determined by Various Methods of Analysis.

Comparisons of Maximum Shear Stresses in Otter Brook Dam as Determined by Various Methods of Analysis.

Contours of Major Principal Stress in Otter Brook Dam Using a Nonlinear Modulus and Different Poisson Ratios.

Contours of Major Principal Stress in Otter Brook Dam Using a Nonlinear Modulus and Different Poisson Ratios.

Contours of Mobilized Strength in Otter Brook Dam Using a Nonlinear Modulus and Constant Poisson Ratio.

Development of Failure Zones in Otter Brook Dam When Using a Nonlinear Modulus and Constant Poisson Ratio.

Contours of Mobilized Strength in Otter Brook Dam Using a Nonlinear Modulus and Poisson Ratio, and Most Critical Circular Arc,

Variation of Stresses Along Critical Arc in Otter Brook Dam.

Comparison of Different Factors of Safety,

Comparison of Factors of Safety for Otter Brook Dam Cross-Section.

Cross-Section and Finite Element Mesh for Example Zoned Embankment,

13

86

87

89

90

91

92

93

96

97

99

101

103

106

109

llO

ll2

Fig. No.

48

49

50

51

52

53

54

55

56

57

58

59

60

61

Contours of Tangent Modulus in Example Zoned Embankment.

Contours of Tangent Poisson Ratio in Example Zoned Embankment.

Contours of Settlement in Example Zoned Embankment.

Contours of Horizontal Displacement in Example Zoned Embankment.

Contours of Maximum Principal Stress in Example Zoned Embankment.

Contours of Vertical Soil Pressure Measured in Gepatsch Dam (After Schrober, 1967).

Contours of Minimum Principal Stress in Example Zoned Embankment.

Contours of Maximum Shear Stress in Example Zoned Embankment .

Contours of Mobilized Strength in Example Zoned Embankment.

-Ho±sture-Density -Relationships for Compacted Pittsburg Sandy Clay.

UU Triaxial Shear Test Results for Compacted Pittsburg Sandy Clay, Low Compactive Effort.

UU Triaxial Shear Test Results for Compacted Pittsburg Sandy Clay, Medium Compactive Effort.

UU Triaxial Shear Test Results for Compacted Pittsburg Sandy Clay, High Compactive Effort.

Comparison of Mobilized and Failure States of Stress.

14

114

115

117

119

120

121

122

124

125

139

142

143

144

146

Table No.

1

2

3

4

5

6

7

8

9

10

LIST OF TABLES

Classification Data and Stress-Strain Parameters for Soils Tested Under Drained Conditions.

Typical Values of Stress-Strain Parameters for Clean Sands and Gravels.

Classification Data and Stress-Strain Parameters for Soils Tested Under Unconsolidated-Undrained Conditions.

Classification Data and Stress-Strain Parameters for Soils Tested Under Drained Conditions.

Average Values of Poisson's Ratio Parameters for Clean Sands and Gravels Under Drained Conditions.

Poisson's Ratio Parameters for Soils Tested Under Unconsolidated-Undrained Conditions.

Soil Parameter Values for Nonlinear Analyses of Stresses and Deformations of Otte~ Brook Dam.

v-alues of Factor of- Sa-rety- Bas-e-d- orr &tress Level for Otter Brook Dam Cross-Section.

Values of Factor of Safety Based on Shear Stress for Otter Brook Dam Cross-Section.

Unit Weights and Stress-Strain Parameters Employed in Analyses of Zoned Embankments.

15

45

50

51

63

67

67

74

104

107

113

a

A

b

B, B

c

c'

d

D

D r

ei

f

E

Ei,Et

F

{F}

FSL

F 'T

G

G s

h

H

K

(K]

LIST OF SYMBOLS

English Letters

reciprocal of initial tangent modulus

pore water pressure coefficient

reciprocal of hyperbolic deviator stress at failure

pore water pressure coefficients

total stress cohesion intercept

effective stress cohesion

rate of change of initial tangent Poisson ratio with strain

grain diameter

relative density

initial void ratio

initial tangent Poisson's ratio

elastic modulus

initial tangent modulus, tangent modulus

rate of change of initial tangent Poisson ratio with confining pressure

nodal point force matrix

stress level factor of safety

shear stress factor of safety

initial tangent Poisson ratio at one atmosphere or shear modulus

specific gravity

layer height in one-dimensional column

column height in one-dimensional column

modulus factor or bulk modulus

stiffness matrix

17

K coefficient of lateral earth pressure at rest 0

L length

n exponent for stress-dependent modulus

Pa atmospheric pressure

p ratio of major principal stress to overburden pressure

PI plasticity index

Rf failure ratio

S degree of saturation r

u pore water pressure

{u} nodal point displacement matrix

U uniformity coefficient

w1

liquid limit

wi,w ,w t initial, natural, optimum water content n ~

Y layer elevation in one-dimensional column

18

LIST OF SYMBOLS

Greek Letters

y,yd,yd total, dry, optimum unit weight opt

ot,o.,o total, initial, subsequent displacement 1 s

6 change

E ,Ef,E vertical, failure, axial, radial, volumetric strain y a

E ,E r v

v Poisson ratio

v.,v initial tangent Poisson ratio, tangent Poisson ratio 1 t

a ,a horizontal, vertical stress X y

a1

,a3

major, minor principal stress

(a1-a3)m mobilized deviator stress

(a1-a

3)f deviator stress at failure

~,~' total stress, effective stress angle of internal friction

T shear stress on X-Y plane xy

T maximum shear stress max

19

CHAPTER 1

INTRODUCTION

Embankments are among the earliest major structures constructed by man. Hathaway (1958) has noted that homogeneous earth dams were constructed for flood control and irrigation purposes in Egypt as long ago as 2300 B.C., and that by 1100 A.D. dams were built which incorporated many of the features of modern dams. For example, the Bhojpur Lake dams in India were constructed with a central earth core and upstream and downstream faces of rubble masonry. One of these dams, with a base width of 300 ft and a height of 87 ft, is still in use today. As well as the advanced concepts of zoning they incorporate, these dams are believed to be the first with spillways.

Although some early dams had features similar to those of modern dams, many early dams failed because there was no rational basis for dam design. Up to about 1850, the design of dams was based primarily upon empirical procedures and "rule-of-thumb" (Wilson and Squier, 1969). During the period from 1850 to 1940, the development of new construction procedures and equipment made possible the construction of higher dams and dams constructed of new materials such as rockfill and hydraulic fill, but failures were still quite frequent. In the period since 1940, very efficient earthmoving and compaction equipment has been developed, and construction control has been greatly improved. In addition, governmental agencies now supervise the design and construction of many dams. These advances in the art of dam design, coupled with advances in procedures for analysis of seepage and slope stability, have made possible the construction of dams of increasing size and complexity, as indicated by the characteristics of dams recently built or under construction:

(1) The height of dams is increasing rapidly, as shown in Fig. 1. Nurek Dam, currently under construction, will reach a height of 1040 ft (Fox, 1968).

(2) The volume of earthwork for dams is also increasing very rapidly. The earthwork for Tarbela Dam will involve an estimated 160 million yd 3 (Engineering News-Record, 1968c).

(3) Because of the large number of dams already built (Gruner, 1967, estimates the world-wide total is 150,000), marginal damsites are being used which would not have been used previously. For example, Muddy Run Dam was constructed on a foundation of weathered mica schist (Wilson and Morano, 1968).

(4) The quality of available construction materials for dams is also declining due to the necessity for using less desirable sites. For example, the shell of Brianne Dam was constructed of cleaved

21

FIG. I

HEIGHT, (~

'"' 1000

NAME jHEIGHT,f .. t

TORSIDE, ENGLAND 123 SAN LEANDRO, u.s.A. 155

900 PONTHOOI<, U.S.A. 276 CRANE VALLEY, U.S.A. 145 NECAXA, MEXICO 184 SAN PABLO, U.S.A. 220

800 BULL CORRAL,U.S.A. 240 TIETON, U.S.A. 235 BILDON, AUSTRALIA 260 WINDSOR, (QUABBIN I U.S.A. 295 S. GABRIEL, U.S.A. 377

700 BOUHANIFIA, ALGERIA 325 ANDERSON RANCH, U.S.A. 456 AMBUKLAO, PHILIPPINES 424 TRINITY, U.S.A. 537 GEPATSCH, AUSTRIA :100

600 OROVILLE, U.S.A. 770 NUREK, U.S.S.R. 1040

KEY ROCK a ROCK EARTH DAMS • ·o

500 EARTH DAMS 0

(~

400

300

(

200

100 (~

~ 8 0 0 0 ~ 0 C7l z 0 N ,., .,. cD cD 2

!! !!! !! !!! !!! !!! !!! !!! C7l cro--~g

I I I I I I I I I zcr

~ cD 0 N ;;; ;: in - :J~ .... cD z ~ ~ !!! C7l !!! !!! !!! !!! C7l 0 - u

DATE

MAXIMUM HEIGHTS OF EARTH AND ROCK DAMS FROM 1850 TO DATE (MODIFIED AFTER GLOSSOP ,1967)

22

mudstone (Engineering News-Record, 1968a) and the shell of Muddy Run Dam is comprised of mica schist (Wilson and Morano, 1968).

Because of the increasing height of dams and the declining quality of embankment and foundation materials at many sites, instrumentation has been used with increasing frequency in recent years to monitor the behavior of dams during and after construction. At the present time, almost all major dams under construction are instrumented to measure surface and internal movements and pore water pressures. By means of such instrumentation, the behavior of a dam may be monitored closely during construction, reservoir filling, steady-state seepage and rapid drawdown. If the instrument readings indicate unforeseen problems, remedial measures may be undertaken at an early stage. The cost of instrumentation, which Wilson (1968) estimates at one-half to one percent of the cost of a dam, is generally considered to be a worthwhile investment to evaluate performance. Studies of the results obtained from instrumented embankments have already revealed a great deal about the behavior of embankments (Casagrande, 1965; Squier, 1967; Wilson and Squier, 1969).

The results of instrumentation studies would be of even greater value if procedures were available for calculating embankment stresses and deformations: Analyses of th1s type would provide information which would be very helpful in planning instrumentation studies, and would help to insure that important aspects of the behavior would not go undetected. The analyses would also be useful for interpreting instrumentation studies. If the results of the analyses and the measurements were in agreement, the analyses could be used to derive information for locations where there were no instruments, and informat1on concerning aspects of the behavior which were not instrumented. Thus instrumentation studies and analyses of displacements and stresses in embankments together provide a very effective combination of techniques for study of embankment behavior. Just as analytical results are useful for planning and interpreting instrumentation studies, so instrumentation results are useful for judging the accuracy of the analyses. The procedures developed for analysis of embankments during construction, and the use of these procedures for examination of the behavior of an instrumented embankment, are described in subsequent chapters of this report.

23

CHAPTER 2

FINITE ELEMENT ANALYSIS PROCEDURES

A number of procedures have been used for analyses of stresses in elastic wedges and embankments. These include infinite elastic wedge analyses (Terzaghi, 1943; Goodman and Brown, 1963; Richards and Schmid, 1968), photoelastic analyses of gelatin models (Brickell, 1962; Richards and Schmid, 1968), finite difference numerical analyses (Bishop, 1952), and finite element analyses (Brown and King, 1966; Clough and Woodward, 1967; Finn, 1967). While some interesting and useful results have been obtained using each of these procedures, the finite element method is the most generally useful. It may be used for analyses of stresses and displacements in nonhomogeneous as well as homogeneous embankments, and with suitable techniques, may be used to obtain approximate solutions for problems involving nonlinear material properties. The important characteristics of the finite element method as applied to analyses of embankments and procedures for its use are described in subsequent sections of this chapter.

Characteristics of the Finite Element Method

Since its introduction by Turner et al. (1956) the finite element method has been shown to be a very powerful procedure for stress analyses and has been used for many different purposes. A number of excellent papers have been published on this method (notably Clough, 1960, 1965, and Wilson, 1963) as well as a recent textbook (Zienkiewicz and Cheung, 196 7) 0

For analysis by the finite element technique, the continuous body is represented by a set of elements which are connected at their joints or

--no-dal -points. -On -the -bas-i-s -of -an -assumed -va-ri-a-tion -o-f -s-t-rai-n-s -wit-hin elements together with the stress-strain characteristics of the element material, the stiffness of each nodal point of each element is computed. For each nodal point in the system, two equilibrium equations may be written expressing the nodal point forces in terms of the nodal point displacements and stiffnesses. These equations are solved to determine the unknown displacements. With the displacements of all nodal points known, strains and stresses within each element may be computed. Analyses of realistic systems commonly requires formulation and solution of several hundred simultaneous equations, and the technique is only practicable when formulated for high-speed digital computers.

Various types of elements have been developed; these elements differ in shape, number of nodal points, and assumed mode of strain variation within elements. The element used in this study is a quadrilateral consisting of two linear strain triangles (Felippa, 1966), Within this element strains are assumed to vary linearly, but to insure compatibility between elements, the strains on the outside boundaries of the quadrilateral are assumed to be constant. Studies by Felippa (1966) have shown that this element provides a good combination of efficiency and accuracy.

24

The analyses described in this report are plane strain analyses of transverse embankment sections" It is expected that plane strain analyses represent a close approximation of the actual strain conditions within embankments which are long in comparison to their height" Even in the case of shorter embankments it seems likely that plane strain analyses will often provide a useful and reasonably accurate approximation,

Nonlinear Stress-Strain Behavior

Nonlinear stress-strain behavior was approximated in the analyses described in this report by assigning modulus values to each element consistent with the values of stress in .that element. The analyses are performed using a step-by-step or incremental analysis procedure in which various successive stages in the construction of the embankment are simulated in the analysis. During each step or increment the relationship between stress and strain for each element is assumed to be linear; nonlinear stress-strain behavior is approximated in the analyses by appropriate changes in the values of modulus and Poisson's ratio during succes~ive stages of the analyses. The procedures developed for determining the stress-strain parameters required for use in these analyses are described in subsequent chapters of this report.

Incremental Analyses

Brown and Goodman (1963) have shown on a theoretical basis, that for precisely accurate analyses of embankments, it 1s necessary to simulate the placement of successive layers of embankment materiaL Clough and Woodward (1966) have examined the usefulness of both incremental finite element analyses (in which the placement of successive layers was simulated) and simpler gravity turn-on finite element analyses (in which the gravity body fo:rces were applied to the entire structure at one time). Their studies indicate that while gravity turn-on analyses may provide reasonable stress distributions for homogeneous embankments, they predict displacement patterns which err~ bas-±c-crlly- d±ff~rem:- from- thuse- ea-iculat~-d

by means of incremental analyses and measured in real embankments, The vertical displacements calculated by gravity turn-on analyses are largest at the top, decreasing to zero at the bottom in the case where the foundation displacement is zero, For the same case, the displacements calculated by incremental analyses are largest near midheight and smaller at both the top and the bottom, This latter type of variation corresponds to the results of measurements made on real embankments. It is thus readily apparent that if finite element analyses are to be employed to calculate embankment displacements, the analyses should be performed using incremental analysis procedures which closely simulate the actual sequence of construction operations.

Required Number of Layers

Construction of an embankment may frequently involve placement of a large number of layers, each of which is only a foot or so in thickness. Although it is desirable from ,:he point of view of accuracy to simulate

25

the actual construction sequence as closely as possible, and, therefore, to use a large number of increments in the analyses, considerations of computer storage and computer costs impose practical limitations on the number of layers which may be used in analyses" It is therefore necessary to investigate the relationship between accuracy of results and the number of increments employed in the analyses. As evidenced by comparisons of the results of gravity turn-on and incremental analyses, the values of stress calculated for embankments are not strongly affected by the number of increments employed in the analyses. The calculated displacements, however, are affected appreciably by the number of increments and may therefore be used to establish criteria for the number of increments required for accurate results.

Studies by Clough and Woodward (1966) showed that for a homogeneous embankment of linear elastic material, the vertical displacements calculated using 14 layers of equal thickness were essentially the same as those calculated using 7 layers. In these analyses, the modulus value of each layer was assumed to be very small when the layer was placed, The procedure employed for assigning small modulus values to newly placed elements (multiplying the modulus values of these elements by a "modulus reduction factor" with a value much less than unity and then multiplying the calculated displacements at the top of the new layer by the same small number) had the effect of eliminating that portion of the displacement at the top of the new layer which resulted from compression of previously placed layers. Although the modulus reduction factor was employed to reduce the magnitude of shear stresses between the new layer and previously placed material, its use and the resulting elimination of part of the initial displacements has the effect of improving the accuracy of calculated displacements as compared to values calculated in analyses employing a large number of thin layers.

Calculated values of vertical displacement in a progressively builtup column of linear elastic material constrained to deform in the vertical

-dtrec-tion only are shown in -Fig. 2. The di-spl-ac-ement£ in this figure are plotted in dimensionless form, in which 8 is the vertical displacement, E is Young's modulus, y is unit weight, H is the total height, and F is a function of Poisson's ratio, v, given by the expression v

(l+V)(l-2V) (1-V) (1)

The displacements on the left include the initial displacements at the top of each new layer due to its own weight. It may be noted that the displacements vary considerably depending on the number of layers employed in the calculation, and that as many as 40 layers would be required for displacements closely approximating those corresponding to an infinite number of layers of infinitesimal thickness" The displacements on the right were calculated using a modulus reduction factor, and the portions of the initial displacement due to compression of previously placed material have thus been eliminated. It may be noted that the displacements are not so strongly influenced by the number of layers

26

1.0

~ 0.8

..... ~ :z: Cl

w 0.6 :z:

(/) (/)

w ..J z 0.4 0 (/)

N z ....., w :E 0 0.2

0

FIG. 2

0

...... 1.0 ......, '

' 3

0.8 \ I I I

I 0.6 Infinitesimal I Infinitesimal I

0.1

Layers I

0.4

0.2

0 0.2 0.3 0.4 0 0.1 0.2

DIMENS,IONLESS VERTICAL DISPLACEMENT ~ 8 Fv yH

Layers

0.3

a) TOTAL DISPLACEMENTS b) DISPLACEMENTS MODI FlED BY

MODULUS REDUCTION FACTOR

0.4

SOLUTIONS FOR DISPLACEMENTS IN AN INCREMENTALLY LDADED, LINEAR ELASTIC COLUMN

employed, and that those calculated using 8 layers are very close to those corresponding to layers of infinitesimal thicknesso These conclusions are the same as those derived from the finite element study of embankment deformations conducted by Clough and Woodward (1966), indicating that the one-dimensional displacements within a column may provide a means of studying the number of layers required for accurate values of displacement. Clough and Duncan (1969) have performed a number of analyses of this type to establish criteria for the number of layers required to simulate excavation as well as fill placement.

These studies have shown that the choice of the reference position for displacements is an important consideration in the required number of layers. If the reference position of the top of each layer is taken as that immediately after placement, the displacements calculated using any finite number of layers is the same as that calculated using infinitesimal layers. This choice of reference position is equally logical for analytical studies and instrumentation studies, and has therefore been adopted for the analyses conducted during the course of this investigation.

Studies were also conducted to calculate displacements in progressively built-up columns in which the modulus value of the material varied with confining pressure according to the equation

E Kp a

(2)

in which E is Young's modulus, o3 is the minor principal stress, K is a modulus number and n an exponent representing the rate of variation of E with o

3• The term p , denoting atmospheric pressure, is introduced so

that K is a pure numSer. For a material with modulus related to confining pressure as indicated by equation (2), vertical displacements have been calculated for various values of the exponent no The results of these ca] culations _are _shown -in -Ei&· _3_, where -the -displacements -ar-e plott~u in a dimensionless form in which 8 is the vertical displacement, K is the modulus number, n is the exponent, and K0 is the coefficient of earth pressure at rest. It may be noted that the magnitude of the dimensionless displacement increases with increasing values of the exponent n, but that the maximum value always occurs at midheight regardless of the value of n.

The displacements shown in Fig. 3 were all computed assuming that the material was placed in layers of infinitesimal thickness. If the material was placed in layers of finite thickness, there are several possibilities with regard to the value of o 3 to be used for evaluation of the soil modulus E: One possibility is to use the value of o 3 for each layer before the new layer is added. A second possibility is to use the values of o3 after the new layer is added, and a third possibility is to use the average of these two values. These procedures have been termed "past stresses," "present stresses," and "average stresses" solutions. Each of these three procedures leads to different results, as shown in Fig. 4,

28

~ ,..

1-l: (!)

w l:

en en w ...J z 2 en z w :E 0

FIG. 3

1.0

E • Kp0

( _!!! )" Po

0.8

0.6

0.4

2

0 0 0.2 0.4 0.6 0.8 1.0

8KK n •-n DIMENSIONLESS VERT! CAL DISPLACEMENT 0 Po

F 1-n Hz-n vY'

SOLUTIONS FOR DISPLACEMENTS OF AN INFINITESIMALLY LAYERED

ELASTIC COLUMN WITH STRESS DEPENDENT MODULUS

29

1.0 1.0 1.0

0.8 n •0.5 0.8 0.8 n • 0.5

0.6 0.6 0.6

~ >- 0.4 0.4 0.4

~ Infinitesimal :::r: 0.2 0.2 0.2

(!)

&&I ~ 0 0 0

en 0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0 en

1.0 1.0 &&I 1.0 _, z

w 0 0.8 OB 0.8 0 en

z &&I :::E 0.6 0.6 0.6 0

0.4 0.4 5 0!4

02 02 0.2

0 0 0 0 0.2 0.4 0.6 0.8 1.0 1.2 0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0

8KK n •-n DIMENSIONLESS VERTICAL DISPLACEMENT F y•~n ~ 2 .n ..

•PAST STREss• SOLUTION • AVERAGE STREss• SOLUTION •PRESENT STREss• SOLUTION

FIG. 4 SELECTED DISPLACEMENTS IN INFINITESIMAL AND FINITE LAYER COLUMN MODELS WITH STRESS DEPENDENT MODULUS

where dimensionless displacements are shown for two values of the exponent n. Values for n = 0.5 are shown on the top row and values for n = 1.1 are shown on the bottom. The displacements calculated using "past stresses" with any finite number of layers are larger than those corresponding to infinitesimal layers, and those calculated using "present stresses" are smaller, for both values of n. The values of displacement corresponding to "average stresses" are also smaller than the infinitesimal layer solution, but are more accurate than the "present stresses" soluti::>n,

To perform finite element analyses of fill placement, it is necessary to select a calculation procedure and to determine the number of layers required for accurate evaluation of displacements, Solutions using either "average stresses" or "present stresses" require twice as much time for analysis of a single increment: as a solution using "past stresses": One analysis must be performed, using estimated modulus values, for the purpose of calculating the stresses at the end of the increment. Then the increment must be analyzed again using the appropriate modulus values. Because "average stress" and "present stress" solutions require the same amount of time per increment, and the "average stress" solution is always more accurate, there is no reason for using the "present stress" solution.

The remaining two procedures differ in two ways, First, the "average stress" procedure requires t:w1.ce as much t1me per increment as the "past stress" procedure, as expla1.ned previously. Second, the "average stress" procedure is some\vhat more ac~urate than the "past stress" procedure, Thus it is conceivable that either .'Jf these procedures might provide the best: combination of accuracy and efficiency. Data are presented in Fig .. 5 which may be used to de termi_ne which of these two procedures is better for a given value of n and required degree of accuracy. For example, with a value of n = 0,5, Fig. 5 shows that d1.splacements within 5% of the correct values could be obtained using the "average stress" analysis procedure and 5 1.ncrements, whereas the same accuracy could only be achieved using a "past stress" solution if 18 increments were used. Even though each increment requires twice as long for analysis using "average stresses," this procedure is still more efficient because fewer than half as many layers are required. Inspection of Fig, 5 shows that the "average stress" procedure is a more efficient method of simulating fill placement for almost all conditions.

Studies of the nonlinear stress-strain behavior of soils described in the following chapter show that modulus values for soil is affected by the stress level, or percentage of strength mobilized, as well as by confining pressure. Whereas the tangent modulus increases with increasing confining pressure, it decreases with increasing stress level. Because both confining pressure and stress level generally increase during embankment construction, it: would be expected that the tangent modulus values would be more nearly constant than assumed in the previously described studies of one-dimensional deformations, which neglected the effects of stress level, Therefore it seems likely that the actual numerical accuracies for embankment displacements will be at least as great as indicated by Fig. 5. Even though the curves of Fig. 5 are

31

1.4

J n • 1.1 <(

!I! 1.3 w ;n

.... laJ 0.9 .... z ~ Str e11 • Solution

1&. 1&.

.a z 0.7

10 1.2 . 0

~ 0.5 0:: ------.... z 1.1 w 0.3 ---------::!: w -----0 ct _.

(Infinitesimal) Q. 0 en 1.1 0 0.3

----------- ------------

_. 0.5 --------ct 0.7 0 -.... 0.9 0:: 0.9 w >

• Average Streas" Solution

!.!

0.8 L-.---L----"-----.L.----.1..----..L..------1 0

FIG. 5

5 10 15 20 25 30

NUMBER OF LAYERS

CORRELATION OF DISPLACEMENTS IN INFINITESIMAL AND FINITE LAYER COLUMN MODELS WITH STRESS DEPENDENT MODULUS

32

based on a somewhat simplified representation of the variables governing the accuracy of calculated vertical displacements in fills, these curves provide a simple and useful means for determining the number of layers required for embankment analyses.

33

CHAPTER 3

NONLINEAR STRESS-DEPENDENT MODULUS RELATIONSHIP

The stress-strain behavior of any type of soil depends on a number of factors including density, water content, structure, drainage conditions, duration of loading, stress history, confining pressure, and shear SLress. In many cases it may be possible to take account of these factors by selecting soil specimens and testing conditions which simulate the corresponding field conditions. Even when the soil specimens and test conditions are carefully selected to duplicate field conditions, it is commonly found that soil behavior over a wide range of stresses is nonlinear and dependent upon the magnitude of the confining pressure employed in the tests.

The studies described in this chapter were conducted to examine the range of applicability of a simple method of representing the complex stress-strain characteristics of soils developed by Duncan and Chang (1970), and to determine the values of the parameters employed in this relationship for various types of soils and drainage conditions.

Nonlinearity

A simple method for representing nonlinear stress-strain curves for soil has been proposed by Kondner and his co-workers (Kondner, 1963; Kondner and Zelasko, 1963a, 1963b; and Kondner and Horner, 1965), In their method, a nonlinear stress-strain curve is represented by a hyperbola of the form

t. a

(crl - 0 -3) = a-+ -bt:: (3) a

in which (o1

- o3

) is the principal stress difference, E is axial str~~' and a and b are parameters whose values are determined empirically. As shown in Fig. 6, these parameters are the reciprocals of the initial slope (initial tangent modulus) and the asymptote to the stress-strain curve.

For purposes of determining the values of the parameters a and b it is convenient to transform equation (3) into the following linear form

(4)

As shown in Fig. 7, when the relationship is represented in this transformed manner, the parameters a and b are respectively the intercept and the slope of the straight line"

34

w VI

-.: I

b---C/) C/)

w 0:: 1-C/)

a: 0 ..,:... <t

> w 0

Asymptote ------~------------------

D'3 = Constant

AXIAL STRAIN, E 0

( Oj - o-3 ) = 1.. u It b

FIG. 6 HYPBERBOLIC REPRESENTATION OF STRESS-STRAIN CURVE

-0

b.., ~ I

b--· -(/) z (/)

w <t a: a: ..... ..... (/)

(/) a: w

_J 0 "' <t .....

<t X -<t >

a= _L w 0 E· I

AXIAL STRAIN, c0

FIG. 7 TRANSFORMED HYPERBOLIC REPRESENTATION OF STRESS- STRAIN CURVE

The somewhat failure,

value of the asymptotic stress difference, (o1 - 03) 1 , is always larger than the compressive strength or stress diffe¥e~ce at (o

1- o

3)f. These two values may be related as follows

(5)

in which Rf is a correlation factor called the "failure ratio," which always has a value less than unity. The value of Rf, which is determined empirically by comparing the values of (o1 - o 3)f and (o 1 - o3)ult' is a measure of how nearly the shape of the stress-strain curve may be approximated by a hyperbola. Values of Rf equal to unity correspond to stressstrain curves of precisely hyperbolic shape, and smaller values to stressstrain curves of other shapes. Values of Rf for a variety of different soils have been found to range from 0.5 to 1,0 and to be essentially independent of confining pressure.

The stress-strain curves for most soils are not precisely hyperbolic in shape, and when stress-strain data are plotted in the transformed manner shown in Fig. 7, the data do not describe a straight line. Two types of deviations from ideal behavior are shown in Fig. 8, If the initial portion of the stress-strain curve is linear, the data will describe a nonlinear variation of the type shown on the left, Alternatively, if the initial portion of the stress-strain curve is more sharply curved than a hyperbola, the transformed data will deviate from a straight line as shown on the right in Fig. 8. Because the data do not describe a linear variation in either case, it would be possible to approximate the actual variations with many different straight lines, To reduce the degree of subjectivity involved in this aspect of the test interpretations, a study has been made to evaluate various procedures for fitting a straight line to transformed data. This study has shown that a consistently good match with the actual stress-strain curve may be achieved if the hyperbola is chosen so that it coincides with the stress-strain curve at three points: the origin, and the points where 70% and 95% of the strength are mobilized. This may be accomplished by choosing the straight line in the transformed representation so that it coincides with the actual data at the 70% and 95% points as shown in Fig. 8.

The curves shown in Figs. 9 and 10 demonstrate the usefulness of this simple hyperbolic representation for two soils. Stress-strain curves for a series of drained triaxial tests on poorly graded glacial outwash sand with a relative density of 80% (Hirschfeld and Poulos, 1963) are shown in Fig. 9, together with hyperbolic representations of these same curves. The average value of the failure ratio for this soil is very low (0.55) indicating that the actual stress-strain curves are not close to hyperbolic in shape; the hyperbolic curves, which are shown as dotted lines in Fig. 9, would coPtinue to much greater values of stress difference than the actual compressive strength, (o 1 - OJ)f. As shown in Fig. 9, the hyperbolic representation is not employed for values of stress difference exceeding the compressive strength. At larger strains the curves are represented by nearly horizontal straight lines. (Because of numerical difficulties it is not possible to simulate a reduction in stress difference beyond the

37

-wa b"'

I

b-.. -en en z

c( I&J a: a: ... ... en

(I) a:

.J 0 ... c( c(

)( c( >

I&J Q

'ii '> .. .J

• • .. ... -Cl)

~ 0 ,.._

A .. t~

'> .. .J

• / • .. / ... -/ Cl)

/ ~ It)

en

--------------'-AXIAL STRAIN. £

0

-~ b"

w ~-Cl)

z (I)

1&.1 c( a: a: ... ... Cl)

en a: .J 0 c( ...

c( )( > <I(

1&.1 a

~ 0 ,.._

AXIAL STRAIN. £0

a) LINEAR INITIAL PORTION OF CURVE b) VERY CURVED INITIAL PORTION OF CURVE

FIG. 8 DEVIATIONS FROM IDEAL BEHAVIOR ON TRANSFORMED PLOTS

125

100

.: b-

en· ...... 15 en• t&J:E a:u ........ enc:»

:ill: a:- 50 0

w ... \0 c[

> l&J Q

25

0 0

FIG. 9

HYPERBOLIC EXPERIMENTAL

5 10 15

AXIAL STRAIN, €0

(%)

20

en enw' ..... a:• ... :E

0.0125

en ~0.0075 a:c:» olll: ... -c[

> l&J Q 0.0050 ..... z ~ a: ... en ..J 0.0025 c[

X c[

0 0 5 10 15

AXIAL STRAIN, € (%) a

20

EXPERIMENTAL AND HYPERBOLIC STRESS-STRAIN CURVES FOR A POORLY GRADED SAND (Data From Hirschfeld And Poulos,l963)

60 r-------r-----~~----~-------, 0.012

so

-en_ U)N 1&12; a:u .... '30 U)C)

::.c a:-0 .... ct 20 > LLI 0

10

0

FIG. 10

0

--- HYPE BOLIC -b"' I

- EXPERIMENTAL b-' 0.010

0 w rn· U) l&J a: 0.008 .... _ en• -a:"' o2: .,_u ct '0.006 -0 >::.C w-0 ....... z 0.004 ct a: .... U)

...J 0.002. ct X ct

0 5 10 15 20 0 5 10 15 20

AXIAL STRAIN, €0

(%) AXIAL STRAIN, £0

(%)

EX PERl MENTAL AND HYPERBOLIC STRESS -STRAIN CURVES FOR WELL- GRADED GNEISS ROCK FILL (Data From Casaqrande. 1965)

peak in incremental finite element analyses of the type described in this report.) It may be noted that the hyperbolae and straight lines provide a reasonable representation of the stress-strain curves for this sand even though the failure ratio is very low,

Similar comparisons for drained triaxial tests on the shell of Mica Dam, a well-graded gneiss rockfill, are shown in Fig. 10. The four stressstrain curves shown are for tests on specimens compacted to 95% relative density. The value of Rf for this material is 0.74, and it may be noted that the stress-strain curves are represented very closely by hyperbolae and straight lines.

These examples show that hyperbolae of the form suggested by Kondner and his associates provide a simple and accurate means of representing stress-strain curves for soils. Studies of the stress-strain characteristics of 47 soils described in subsequent sections have demonstrated the suitability of this representation for a wide variety of soils.

Stress-Dependency

The stress-strain characteristics of soils commonly depend on confining pressure. As shown in Figs, 9 and 10, the steepness of the initial portion of the stress-strain curves and the strength values both increase with increasing magnitude of the confining pressure employed in the tests. The influence of confining pressure on the stress-strain characteristics may be incorporated in the stress-strain relationship by relating the values of the initial tangent modulus and soil strength with confining pressure,

The variation of initial tangent modulus with confining pressure may be expressed very conveniently in the following form, which was suggested by Janbu (1963}:

E = Kp i a

a n (~) Pa

(6)

in which Ei is the initial tangent modulus, cr 3 is the minor principal stress, Pa is atmospheric pressure expressed in the same units as Ei and cr3, K is a modulus number, and n is the exponent determining the rate of variation of Ei with cr 3 ; both K and n are pure numbers. Values of the parameters K and n may be determined readily from the results of a series of tests by plotting the values of Ei against cr 3 on log-log scales and fitting a straight line to the data, as shown in Fig. 11. The data shown in Fig. 11 represent tests on clay, sand, and gravel which were conducted under unconsolidated-undrained, consolidated-undrained, and drained test conditions. In each case the variation of initial tangent modulus with confining pressure may be represented to a reasonable degree of accuracy by a straight line on the log-log plot.

The relationship between compressive strength and confining pressure may be expressed in terms of the Mohr-Coulomb failure criterion as follows:

41

G:' (/)

A- OTTAWA SAND (LEE, 1965) B- CLAYEY SANOY GRAVEL- OROVILLE DAM CORE (OWR, 1969)

C • DRAMMEN CLAY ( BJERRUM AND SIMONS, 1960) D - GLACIAL OUTWASH SAND (HIRSCHFELD AND POULOS, 1963) E - CLAYEY SAND - OTTER BROOK DAM ( LINELL AND SHEA, 1960)

+

~ 10,000 ~-----------------------4~~--------------------~

-W·

K • 270 n•0.50

_(CD T _EST SJ

n • 0.76 ( UU TESTS)

100 ~----~~--L-~--L-~~~~----~----L--L~~~-L~ I !0 100

' CONFINING PRESSURE, cr3

OR a-3

( TSF)

FIG. II VARIATIONS OF INITIAL TANGENT MODULUS WITH CONFINING PRESSURE

42

2c cos~+ 2o 3 sin~

(ol- 0 3)f = 1- sin~ (7)

in which c and ~ are the Mohr-Coulomb strength parameters,

Equations (6) and (7), in combination with the previously described hyperbolic relationship, provide a means for relating stress to strain by means of the 5 parameters, K, n, c, ~' and Rf. Use of this relationship in nonlinear finite element stress analyses is discussed in the following section.

Tangent Modulus

The nonlinear, stress-dependent stress-strain relationship discussed previously may be used very conveniently in incremental stress analyses, because it is possible to determine from this relationship the value of tangent modulus corresponding to any point on the stress-strain curve, If the value of o 3 is assumed to be constant, the tangent modulus may be expressed in the form:

E = t

(8)

Performing the indicated differentiation on equation (3) and substituting the parameters discussed previously, the tangent modulus may be expressed as

1 E. E = ___________ 1 ________ __

t 2 r _!_ + Ri~ l [ Ei (01 - OJ)r j.

(9)

Although this expression for the tangent modulus value could be employed in incremental stress analyses, it has one significant shortcoming: The value of tangent modulus, Et, is related to the strain, which has a completely arbitrary reference state. Because the reference state for strain is arbitrary, and because stresses may be calculated more accurately than strains in many soil mechanics problems, it seems logical to eliminate strain and express the tangent modulus in terms of stress difference. The resulting equation for tangent modulus is

(10)

This equation may be employed in either effective or total stress finite element stress analyses. For effective stress analyses drained test conditions, with 03' constant throughout, are used to determine values of the required parameters. For total stress analyses unconsolidated-undrained

43

tests, with o 3 constant throughout, are used to determine the parameter values.

The usefulness of equation (10) results from its simplicity with regard to two factors:

(1) Because the tangent modulus is expressed in terms of stresses only, and not strains, it may be employed for analyses of problems involving arbitrary initial stress conditions without any complications.

(2) The parameters involved in this relationship may be determined readily from the results of laboratory tests. The amount of effort required to determine values of the parameters K, n, and Rf is not much greater than that required to determine values of c and ¢.

In order to study the applicability of this stress-strain relationship to various types of soils and rockfills, and to determine values of the required parameters for these materials, a review of published stressstrain information has been made. The results of this review are summarized in the following sections.

Modulus Parameters for Drained Conditions

Classification data for the 36 soils included in the study of drained stress-strain characteristics are listed in Table 1. Except for the last soil listed (Cannonsville silt) which was undisturbed, all of these soils were compacted before testing by either vibratory or impact methods; the initial void ratios, dry unit weights, and relative densities before testing are given in the table. The values of <f>' for these granular soils were determined for each test by assuming that the Mohr envelope passed through the origin (c' = 0). It may be noted that the values of <f>' determined in this way decrease with increasing confining pressure. The range of confining pressures and the corresponding range in friction angles are shown in the table together with the values of K, n, and Rf determined from the test results. The values of K, n, and Rf shown in Table 1 are the average values of these parameters over the range of pressures employed in the tests.

Examination of these data shows some general relationships among the classification characteristics, relative densities, and stress-strain parameters:

(1) The value of the modulus number K increases roughly in proportion to relative density. Variations of Ei with 03 for Sacramento River Sand (SP-4) at four relative densities are shown in Fig. 12. The linear interpretations of these data shown in the figure correspond to the same value of the exponent (n = 0.54) for all relative densities, and values of K which are roughly proportional to relative density. It may be noted that these linear interpretations match the plotted points closely except for

44

Table 1. Classification Data and Stre$s-Strain Parameters for Soils Tested Under Drained Conditions.

Cobble

Cobb !a

Cobble

Soil N.-er

c-u C-lh

C-2

Gll-1

SoU Deacriptioa

Sante Fe Andesite R.ockf111

Sante Fe Andedte R.oc.k£111

Granitic: Cneiu l.ockfill {Mica o- Shell)

Coaalo.erate R.ockfill (Netzahualcoyote D• Shell)

Craaitic Gneiss lockfill (Mica Da Shell)

Quartzite Rocltfill (Furnu Da• Shell)

Quartzite llockfill (Furnas Da• Transition)

Quutdte Rocltf111 (Furnas o- Tranaitioo.)

Pinzandaran Gravel

Diorite Rockflll (El Infiernillo Da Shell)

Harsal (1963)

Harsal (1963)

Reference

Casagrap.de (1965); Marui. et al (1965)

H.arul. et al (1965); GUiboa and Senau1ni (1967)

Casagralflde (1965); H.araal. et al (1965)

Casagra:pde (1965)

Caugrapde (1965)

Casagrapde (1965)

Marsal ,et al (1965)

Hars&l ,et al (1965)

GW

GW

GW

GW

GW

GW

GW

GW

GW

GW

GW

GW

GW

GW

CP

CP

CP

CP

CP

CP

CP

CP

CP

cc S\1

sv SP

SP

SP

SP

SP

SP

SP

SP

SP

SP

SP

Gll-2

Gil-)

Gll-4

Gll-5

CW-6

Gll-7

Gll-8

GW-9

CW-10

CW-11 Gll-12

Q.l-13

GW-14

CP-1

GP-2

GP-3

Silicified Conglo.!!rate Rod:fill (El lnfiernillo Du Shell) Marsal tH a1 (1965)

Silicified Conglo.rate Roc.kfill (El lnfiernillo Da• Shell) Ma.rsal ,et al (1965)

CP-4a

CP-40

CP-5

GP-6

CP-7

CP-8

GC-1

SW-1

S\1-2

SP-1a

SP-lh

SP-2

SP-J

SP-4&

SP-QI

SP-4c

SP-4d

SP-ja

SP-5b

SP-6

Ar&illite locltfill (Pyr&ldd D-. Shell)

Arailllte Rockfill (PyraiRid Dn Shell)

Crushed Olivine B&~~&lt

Crushed Olivine Basalt

Gravel (Mev Doa Pedro Daa Shell)

Quartzite Rockfill (f'urn .. Da• Shell)

Sandy Gravel (Mica o .. Shell)

Budt Rockfill

Coatreras Andeaite Gravel

Coatreru Andeaite Gravel

~bibolite Gravel (Oroville Da. Shell)

Silty Sandy Gravel (Oroville Daa Transition)

.A.phibolite Gravel (Oroville Daa Shell)

~hiboll te Gravel (Oroville Daa Shell)

Clayey Crave 1 (New Hogan Daa Core)

Araillite lock£111 (Pyraaid Daa Shell)

Crushed Olivine Basalt

texcoco Sand

Tezcoco Sand

OttMta Sand

Glacial Outllaah Sand

Sacra~~~~ento River Sand

SacraD!'Dto liver Sand

Sacra.nto River Sand

Sacra.ento R.iver Sand

Ha• i.iver Sand

B•• River Sand

-Uiphibolite Sand (Oroville D&• Shell)

SM-SC SM-SC-1& Silty Clayey Sand (Mica Da• Core. Dry)

SM-SC SM-SC-lh Silty Clayey Sand (Mica Daa Core. Std. AASHO Opt.)

SM-SC SM-SC-1c Silty Clayey Sand (Hica o .. Core. Wet)

CL CL-1 Silty Clay (Arkabutla Daa. Std. AASHO Opt.)

ML ML-1 CannOMville Silt (Undisturbed)

Karachi ( 1969)

Karachi (1969)

Harachi (1969)

Karachi (1969)

Bechtel (1969)

Casagra1

nde (1965)

Caaa.gra1

nde (1965)

Casagra1

nde (1965); Karsal (1967)

Marsal (1963)

Marsa1 (1963)

Hall an1

d Gordon (1963)

Hall an,d Gordon (1963)

Harachi (1969)

Karachi (1969)

Bird 0,961)

Marachi (1969)

!iarachi (1969)

Haru.1 (1%3)

Haraal (1963)

Lee (191

&5)

Hirschf.eld and Poulos (1963)

Lee (191&5)

Lee (1965) I

Lee (19155) I

Lee (191&5) I

Bishop {1966)

Bishop (1966)

Harachi (1969)

Casagra11r1de (19&5); Insley and !::Iillis (1965)

CasagraJr1de (1965); Insley and Hillis (19&5)

' Casagra11r1de (1965); Insley and Hillis (1965)

Cas.agramde. et al (1963) I

Hirschfdd and Poulos (1963)

Grain Size <-> Dry Unit Weight

(1b/ft') 060 °30 °10

130

130

133

47.0

84

<10

<25

<10

21.0

93

64

64

17.8

53

17.8

53

19.0

19.0

22.0

120

120

124

7.5

26.0

2. 7

42.0

20.0

20.0

7.4

23.1

7.4

23.1

1.6

16.0

1.2

19.0 3.6

75 65

75 65

25.0 13.0

18.0 4.8

13.2 4.6

39.6 14.2

12.0 0.6

4.1 1.8

4.1 1.8

2.4 1.8

2.4 1.8

0. 73 0.68

0.83 o. 40

0.22 0.17

0.22 0.17

0.22 0.17

0.22 0.17

0.25 0.17

0.25

3.1

0.17

1.1

o. 34 0.03

0.34 0.03

o. 34 0.03

0.023 0.005

0.033 0.018

110

110

53

0.9

6.0

0.25

17.0

4.5

4.5

2.7

8.0

2. 7

7.6

0.13

12.0

0.23

1.0

44.0

44.0

5.1

0.4

0. 36

1.1

0.6

0.6

1.2

1.2

0.64

0.14

0.15

0.15

0.15

0.15

0.10

0.10

0.09

0.002

0.002

0.002

70.2

77,0

101.0

118.9

123.7

132.1

105.7

106.9

114.1

1U.2

113.0

125.1

125.0

U3.l

133.8

88.1

96.1

144.0

148.0

152.0

149.3

113.0

111.6

U5.4

90.3

99.4

111.0

112.3

89.5

94.0

97.8

103.9

146.5

0.0008 110.0

0.005 108.0

Initial Void Ratio

1.06

0.88

0.62

0.39

0.32

0.34

0.56

0.55

0.45

0.46

0.45

0.43

0.43

0.39

0.30

0.68

0.54

0.21

0.16

0.20

0.22

0.46

0.43

0. 74

0.58

0.49

0.50

0.87

0. 78

1. 71

0.61

0.82

0.64

0.23

0.49

0.57

Relative Density

Loose

Dense

79%

70%

95%

65%

50%

~toot

~toot

~toot

-100%

~tOO%

50%

95%

Loose

Dense

100%

100%

~100%

-100%

~100%

~1oot

Loose

Dense

100%

80%

38%

60%

78%

100%

Loose

Dense

~100%

Stress ..... (T/ft 1

)

<1

<1

5-26

1-26

5-26

4-37

4-37

.C.-37

1-26

1-26

1-26

2-26

2-47

2-47

2-47

2-47

9-47

4-37

7-33

5-26

<1

<1

9-40

9~40

2-47

2-47

1-4

2-47

2-47

<1

<1

1-41

1-41

1-41

1-41

1-41

1-41

7-71

7-71

2-47

4-35

4-35

4-35

2-8

1-40

Muaber of

Teats

10

10

• (degrees)

40

47

34-29

49-37

37-32

45-39

50-42

45-39

53-39

46-34

46-37

46-36

47-36

47-36

48-37

48-36

40-35

42-34

39-37

46-39

42

47

43-37

45-39

49-40

47-38

18

50-30

52-39

37

45

39-28

44-37

34-27

37-28

39-27

41-26

34-30

38-31

51-41

33-35

33-35

3).-35

36

44-30

340 0.21

400 0.20

65 0.61

440 0.45

., 0.90

0.84

0.52

0.54

372 0.35 0.74

755 0. 35 0.80-0.95

1210 0.39

875 0.50

715 0.50

290 0.30

320 0.38

335 0.41

&50 0.25

650 0.25

1115 0.12

1115 o.u 665 0.28

950 0.11

520 0.37

0. 73

0.58

0.61

0.70

0.64

0.62

0.68

0.68

0. 70

o. 70

0.77

0.88

0. 76

640 0.26 0.65

730 0.53 0.91

975 0.50 0.88

1730 0.33 0.89

1850 0.29 0.69

3780 0.19 o. 76

3780 0.19 o. 76

95 0.98 0.75-1.0

650 0.25 0.68

lllS 0. U 0. 70

375 0.67 0.98

1075 0.56 0.87

2490 0.58 0.91

270 0.50 0.55

345 0.54 0.85

545 0.54 0.86

780 0.54 0.85

1210 0.54 0.87

370 0.46 O.Bl

1440 0.!15

3780 0.19

1195 0.18

525 0.50

150 0.84

240 0.54

350 0.57

0.88

0. 76

0.81

0. 71

0.62

0. 78

0.60

10,000

II.. t/) ... ....,

w 6

t/)

:::> ..J :::> 0 0 :::l: 1000 ... z w (.!)

z c{ ... ..J c{

... z

100 I

FIG. 12

0 0

Sacramento River Sand (S P-4)

Curve ei Dr<"4) K n Rf a 0.87 38 345 0.54 0.85 b 0.78 60 545 0.54 0.86 c 0.71 78 780 0.54 0.85

--d -o~ai t00

1

I i2iO -o.-54 -<la7

(Data From Lee, 1965) I I I

10 50

, EFFECTIVE CONFINING PRESSURE, u

3 (TSF)

VARIATION OF NONLINEAR MODULUS PARAMETERS WITH

RELATIVE DENSITY FOR SACRAMENTO RIVER SAND

46

tests on loose specimens at confining pressures higher than about 10 tons per sq ft.

(2) Soils which consist of particles of similar shape and mineral composition, and which have parallel grain size curves, also have similar stress-strain characteristics, Marachi (1969) showed that if the grain size distribution of a soil is "modelled"--that is, if the soil is sieved and the various sizes are recombined to form a finer soil with a parallel gradation curve--the value of ¢' is virtually independent of particle size, The classification data and stress-strain parameters for the soils Marachi tested are shown in Table 1, and the variations of initial tangent modulus with confining pressure for his tests are shown in Fig. 13, In each of the three cases illustrated in Fig. 13, the variations of Ei with 03 for tests on specimens with various maximum particle sizes have been represented by a single straight line, corresponding to a single value of K and a single value of n. Although somewhat different values of K and n might be more representative of tests on one particular size, the lines shown represent reasonable interpretations of the data. Therefore it may be concluded that by modelling grain size distributions it is possible to form soils which have similar stress-strain characteristics as well as similar strength characteristics. The type of modelling investigated by Marachi may thus be employed to determine stress-strain and strength characteristics for soils containing gravel or cobble sizes, wi~hout testing large-size specimens.

(3) The values of the exponent n vary over a fairly wide range even for soils of the same classification, but characteristic values may be established for sands and gravels. Inspection of the values of n listed in Table 1 shows that the values of n for GW and GP soils range from 0.11 to 0.53, and the values for SW and SP soils range from 0.12 to 0,67, The average value of n for the gravelly soils is slightly less than one-third (0.32) and the average value for the sandy soils is slightly less than one-half (0.46).

(4) The average value of the failure ratio, Rf, is smaller for wellgraded soils than for poorly graded soils. The average value of Rf for GW and SW soils is 0.69, while the average value of Rf for GP and SP soils is 0.82.

(5) Soils with high values of modulus number (K) tend to have low values of the exponent (n), and vice versa. This fact, and the fact that the modulus number K increases with increasing relative density, are both reflected in the fact that the ratio Kn/Dr is nearly constant for similar soils. Examination of the data in Table 1 shows that most of the wellgraded soils are characterized by values of Kn/Dr ranging from 120 to 180, while most of the poorly graded soils are characterized by values of Kn/Dr ranging from 500 to 750.

(6) Based on these observations, it is possible to infer typical values of the stress-strain parameters for clean sands and gravels. For these materials at 100% relative density, representative values are given

47

-LL. en ..... .... .

en :> _J

~ 0 0 :E

..... z LLI C)

z ~ ..J cr ... z -

FIG. 13

10,000

1000

10,000

1000

15000

1000

!500

0 6" MAX. SIZE (GP-8) 0 2" MAX.SIZE(GP-71 A 0.47• MAX. SIZE (SP-6)

0 6• MAX. SIZE (GW-131 0 2" MAX. SIZE (GW-121 A 0.47" MAX. SIZE (SW-21

K • 1115 n • 0.12

6" MAX. SIZE (GW·I3) 0 2• MAX. SIZE (GW-10) A 0.47• MAX. SIZE (SW-1)

10

OROVILLE DAM SHELL (Amphibolite Gravel)

AVG. e.• 0.22 I

o, • 100 "·

Crushed Basalt AVG. e1•0.43 D,•IOO%

A

100

I EFFECTIVE CONFINING PRESSURE, o-

3 (TSF)

VARIATION OF NONLINEAR MODULUS PARAMETERS WITH MAXIMUM

PARTICLE SIZE FOR SOILS WITH PARALLEL GRAIN SIZE CURVES (Data From Marachi ,1969)

48

in Table 2. It may be noted that most of the SP soils listed in Table 1 are uniform, whereas most of the GP soils are not uniform, but are poorly graded because the curvature requirement is not satisfied. This difference in grain size distribution, which is not reflected in the Unified classifications of these soils, is probably responsible for the difference in the typical values of ¢' for GP and SP soils; the poorly graded sands, being in general uniformly graded soils, are characterized by smaller values of ¢' than the poorly graded gravels,

These general relationships among the classifications of these soils, their relative densities, and the values of their stress-strain parameters may provide a useful context for interpreting the results of tests on other soils, and the typical values of the parameters may be useful for studies of a preliminary nature, However, in view of the wide variation in the values of the stress-strain parameters for soils of the same classification and having the same relative density, it may be concluded that values of these parameters for use in accurate analyses should be determined by conduct~ng tests on suitably selected and prepared soil specimens.

Modulus Parameters for Unconsolidated-Undrained Conditions

Classification data for the eleven soils included in the study of stress-strain behavior under unconsolidated-undrained test conditions are listed in Table 3. All of these soils contain some silt or clay, and all were compacted before testing by impact or kneading compaction procedures. The permeabilities of these soils are sufficiently low so that they would be suitable for use as core material in zoned dams or for homogeneous dams, and it would be expected that only a limited amount of drainage of these soils would occur during construction, It would therefore be expected that the behavior of these soils during construction would be most suitably studied using unconsolidated-undrained test conditions,

Previous studies of the strength and stress-strain characteristics of compacted cohesive soils [Seed, Mitchell, and Chan (1960); Seed and Chan (1961)] have shown that their behavior depends on compacted dry density, compaction water content, and method of compaction. However, the soils for which information was available in the literature (the first ten soils in Table 3) were each tested at only one condition of water content and density. In order to examine the relationship between compaction density and water content and the values of the stress-strain parameters c, ¢, K, n, and Rf, a number of unconsolidated-undrained triaxial tests were conducted on specimens of Pittsburg sandy clay which were prepared by kneading compaction to a range of water content and density conditions, The compaction and triaxial test procedures employed are described in Appendix A. The results of these tests are shown in the form of contours of c, ¢, K, and n in Figs. 14 and 15. The values of Rf for these tests ranged from 0.91 to 0.96.

On the basis of the results shown in Figs. 14 and 15, it is possible to make a number of conclusions regarding variations of the parameter

49

Table 2. Typical Values of Stress-Strain Parameters

for Clean Sands and Gravels.

4>' (degrees) Soil Group K n

Low 0 3 High 03

GW 47 35 500 0.3

GP 46 38 1800 0.3

sw 50 35 300 0.5

SP 40 30 1200 0.5

50

R f

0.7

0.8

0.7

0.8

Table 3. Classification Data and Stress-Strain Parameters for Soils Tested Under Unconsolidated-Undrained Conditions.

Un1U<r:d Grain Size C=l Cof!Pactioa Dry t:nit Water Stress NUIIIber

Syste• Soil

Soil Description Reference Uqo.ad Plasticity w 'Weight Content R~ge of c •

Group Nuat.er 060 030 010

Limit Index Energy yd .. x op< (lb/ft') (%) (Tfft1 ) Tests (T/ft 2 ) (~greu) .,

(ft-lb/h 1) (lb/ft J) (%)

- ------GC GC-2 Clayey Sandy Gravel (Oroville Daa Core) Department of \olater Resources (1969) 9.0 0.12 0.005 30 16 20,000 138.6 8.1 ~139 8-1 )-4) 1.32 25.1 341 0. 76 0.88

SK SH-1 Gravelly Silty Sand (Ball Hountain DUI) L1ndl and Shu (1960) 0.85 0.074 0.05 N.P. N.P. Std. MSHO 122.9 10.0 -124 9.4 1-4 0.20 39.) l8S 0.69 0.44

Silr:y Clayey Sand (l:lopkinton oa .. ) Linell and Sh.,a (1960) 0.22 0.014 0.001 21 Std. AA.SHO 129.2 ,_, -ut 8.8 l-b 1.15 38.0 270 0.59 0.86

Vl sc SC-1 Clayey Sand (Ot.te-rbrook D.i.IJI) Linell and Shea (1960) 0. 30 0.017 0.001 27 11 Std. AASHO 126.0 11.3 -126 12.1 1-4 1.08 14.0 40 0.48 0.68 1-'

sc SC-2 Clayey Sand (Thoaaston Da•} Linell and Shea (1900) O.loO 0.028 0.003 29 lZ Std. AASHO 123.3 12.0 -122 12.0 1-4 0.90 17.0 30 0.9lo 0.61

sc SC-3 Clayey Sand (New Don Pedro D.a111 Core) Bechtel (1969) O.Slo 0.020 0.005 27 11 20,000 125.8 9.8 -123 9.5 5-lo3 ,_80 20.3 4520 -0.12 0.82

sc SC-lo Clayey Sand (U lnfiernillo Dati! Core) Marui and de Arellano (1965) 0.04 0.003 -40 -20 Std. AASH.O -106 -20 ~106 20.3 1-10 0.40 0.5 85 0.35 0.93

CL CL-1 Silty Clay (Arkabutla Di!llll) Casag<ande et al (1963) 0.023 0.010 0.0008 30 Std. AASHO IIO.O 18.0 -108 19.0 1-llo 1-80 17.0 85 0.21 0. 73

CL CL-4& Silty Clay (Canyon Dam) Casagrande et al {1963) 0.019 0,002 " 25 Std. AASHO 108.0 19.0 -110 18.7 l-14 2.10 3.0 205 0.42 0.92

CL CL-4b Silty Clay (Canyon D.lm) Cas.111gnnde et a1 (1963) 0.019 0.002 40 25 Std. AASHO 108.0 19.0 -uo 18.7 l-14 2.10 3.0 175 0.41 0.90

CL CL-5 Pittsburg Sandy Clay Thh study 0.040 0.003 35 16 Mod. AASHO ll8.9 u.s Sever•l Seven.! 1-6 27 Values vary with density and

0.91-0.96 water content; See fias. 14,1.5.

-- -·----- --~- ·-- -- . -----

FIG. 14

125

120 C (TSF)

115

110

105

-u.. 100 u

Q. ..... 6 12 18 24 >-1-en z l&.l 0

,... 125 a:.

c

120 ; (OEGEES)

115

110

105 15

100 6 12 ial 24

WATER CONTENT (%)

STRENGTH PARAMETERS FOR COMPACTED PITTSBURG SANOY CLAY UNDER UNCONSOLIDATED- UNDRAINED TEST CONDITIONS

52

125

120

115

110

105

LA.. 100 u

Cl. - 6 12 18 24 >-1-C/)

z LIJ 0

>- 125 0: 0

120

115

110

105 I

•0.5 100

6 12 18 24

WATER CONTENT (%)

FIG. 15 NONLINEAR MODULUS PAR A METERS FOR COMPACTED PITTSBURG SANDY CLAY

UNDER UNCONSOLIDATED· UNDRAINED TEST CONDITIONS

53

values with variations in compaction conditions for Pittsburg sandy clay:

(1) The values of cohesion intercept increase with increasing dry density, and are largest for water contents near optimum.

(2) The values of ~ increase with decreasing water content, and are largest for specimens compacted at very low water contents with high compactive effort.

(3) The values of K increase with decreasing water content, and are largest for specimens compacted at very low water contents with high compactive effort.

(4) The values of n increase with increasing water content, from negative values at very low water contents, through zero at water cont~nts close to optimum, to values greater than zero at water contents wet of optimum. Ultimately, as shown in Fig. 15, it would be expected that n would decrease back towards zero at very high water contents approaching complete saturation. The negative values of n are indicative of the fact that specimens compacted at very low water contents are quite brittle at low pressures and less brittle at high pressures; the initial portions of the stress-strain curves are actually steeper at low pressures than at high pressures. It may be noted that the New Don Pedro Dam core material (SC-3 in Table 3) is also characterized by negative values of n.

The results of these tests on Pittsburg sandy clay show that the shear strength and stress-strain behavior of compacted cohesive soils under unconsolidated-undrained test conditions may vary widely depending on compaction density and water content. It would seem to be important therefore that the density and water content of specimens tested to determine values of the strength and stress-strain parameters for embankment materials should duplicate as nearly as possible the conaitions of aensity and water content used in the actual embankment. Previous studies have shown that the method of compaction may also have an important influence on the behavior of compacted soils (Seed, Mitchell and Chan, 1960), and the influence of this factor should also be considered.

54

CHAPTER 4

NONLINEAR STRESS-DEPENDENT POISSON'S RATIO RELATIONSHIP

More than a single stress-strain coefficient is required to relate stress and strain under two-dimensional and three-dimensional loading conditions. In the finite element analyses conducted during this investigation, strain increments were related to stress increments by means of the generalized Hooke's law for isotropic materials, which contains two independent parameters. These parameters may be either Young's modulus and Poisson's ratio, or deformation modulus and bulk modulus (Clough and Woodward, 1967); either of these sets of parameters may be defined in terms of the other. For purposes of representing the nonlinear, stressdependent behavior of soils determined in laboratory tests, it has been found to be convenient to express the results in terms of Young's modulus , and Poisson's ratio. For purposes of analyzing stresses and displacements in embankments by the finite element method, particularly after failure, it has been found to be desirable to express the relationship between stress increments and strain increments in terms of deformation modulus and bulk modulus.

In a manner consistent with the definition of tangent modulus discussed in the previous chapter, the tangent Poisson's ratio may be defined as the rate of variation of radial strain with axial strain under conditions of axial compression without radial restraint:

= -dE

r d£

a (11)

in which vt is the tangent Poisson's ratio, Er is radial strain and Ea is axial strain. Although radial deformations of triaxial test specimens are not frequently measured, the average value of radial strain may be expressed in terms of the volumetric strain as follows

E r

(E - E ) v a

2 (12)

in which Ev is volumetric strain. Typical variations of volumetric and radial strains with axial strain are shown in Fig. 16. It may be noted that the value of tangent Poisson's ratio (slope of the curves shown in the lower part of Fig. 16) depends on both the confining pressure and stress difference or axial strain. An empirical relationship which incorporates these aspects of the behavior of soils is described in subsequent sections of this chapter.

55

~

"' .. z ct It: 1-en

0 0:: 1-L&J :E ;::) _. g

.. "' .. z ct It: len ...1 ct 0 ct It:

FIG. 16

z Q

~ _. 0 z Q en en L&J It: 0. :E 0 0

Increasing Confining

\Pressure

Axial Strain, £0

a) VOLUMETRIC STRAIN VS. AXIAL STRAIN

I

AXIAL STRAIN, f:a

Increasing Confining

. \Preuure

b) RADIAL STRAIN VS. AXIAL STRAIN

TYPICAL STRAIN PATTERNS IN SOILS DURING SHEAR

56

Nonlinearity

Nonlinear relationships between axial and radial strains, like those shown in the lower part of Fig. 16, may be approximated by an empirical hyperbolic equation of the form

E: r

E: = ---a f + dE

r (13)

in which f and d are parameters whose values are determined empirically. If equation (13) is rewritten as

E:

......!:. = f + dE E r

(14) a

it may be noted that the parameter f is the value of the ratio Er/Ea at zero strain. Thus the parameter f is equal to the value of tangent Poisson's ratio at zero strain, which herein is called the initial Poisson's ratio, v .• The parameter dis the slope of the line represented by equation (14).

1

A study of the stress-strain behavior of a variety of types of soil conducted during the course of this investigation has shown that the volume change characteristics of most soils may be represented to a reasonable degree of accuracy by the empirical relationship expressed by equations (13) and (14). Data derived from tests on a very dense, poorly graded sand (SP-4d) are shown in Fig. 17. It may be noted that the same data plotted in the transformed manner on the right in Fig. 17 do not describe linear relationships,- and similar deviations from linearity- are typical of many other soils as well. Therefore, as in the case of the stress-strain curves, it has been found to be desirable to reduce the degree of subjectivity involved in fitting a straight line to such data by selecting the two points on the curve corresponding to 70% and 95% strength mobilized for purposes of defining a representative straight line. The relationship between the hyperbolic variations corresponding to the linear relationships shown on the right in Fig. 17 and the actual data are shown on the left in Fig. 17, where it may be seen that the empirical hyperbolic representations (the dashed lines) correspond quite closely with the actual variations (the solid lines). A similar comparison, for a dense poorlygraded basalt rockfill, is shown in Fig. 18: Although the transformed data deviate appreciably from linearity at small strains, matching the data with straight lines at the 70% and 95% points results in a close correspondence between the hyperbolic representations and the test data, as shown on the left in Fig. 18.

The values of tangent Poisson's ratio, represented by the co-tangents of the curves shown on the left in Figs. 17 and 18, increase with increasing strain and decreasing confining pressure. The dense sand, for which data are shown in Fig. 17, is dilatant at low confining pressures and the values of initial Poisson's ratio for this material range fTom

57

20 1.00

----HYPERBOLIC -:o

"' ~ EXPERIMENTAL

..,

tTj • 40.1 KG/Ctl z - 15 <f 0.75 ...: " ex: • 1-

fuo (/)

.. ..J z <f <f 10 X 0.50 ex: 1-

<f

(/) ...... z

..J <f <f

1.11 X a: co <f 5 1- 0.25 (/)

• ..J <f 0 <f ex:

0 0

0 2 3 4 5 0 2 "3 4 5

RADIAL STRAIN, £, ,.,., RADIAL STRAIN, €r (%)

FIG. 17 EXPERIMENTAL AND HYPERBOLIC AXIAL STRAIN- RADIAL STRAIN CURVES FOR A DENSE POORLY-GRADED SAND (Data From Lee, 1965)

20

~ 15

0 'u

z 10 ct

a:: 1-(/)

~ ct

""' \0 X 5 <

0 0

FIG. 18

0.5 ...,o

HYPERBOLIC ...... ... EXPERIMENTAL

.., 0.4

z Ci a::

0.3 1-(/)

~ ct X ct 0.2 ....... z Ci a:: 1-(/) 0.1 ~ < 5 < a::

0 2 3 4 5 0 2 3 4 5

RADIAL STRAIN, Er ( %) RADIAL STRAIN, e, (,.J

EXPERIMENTAL AND HYPBERBOLIC AXIAL STRAIN-RADIAL STRAIN CURVES FOR A DENSE POORLY-GRADED BASALT ROCKFILL

~DATA FROM CASAGRANDE .1965)

0.7 at a confining pressure of 1 kg/cm2 to about 0.2 at a confining pressure of 40 kg/cm2

• The values for the dense basalt rockfill range from about 0.27 at 5 kg/cm2 to 0.15 at 25 kg/cm2

• Although the empirical hyperbolic relationship may be used for any values of Poisson's ratio, finite element stress analyses of the type described in this report may only be performed for materials having values of Poisson's ratio less than one-half. Therefore, if the value of Poisson's ratio determined from laboratory test results is greater than or equal to one-half, it is necessary to assign a value slightly less than one-half for purposes of analysis.

Stress-Dependency

As shown by Fig. 16, the variations of radial strain with axial strain for soils depend on the value of confining pressure as well as the value of strain. For many soils the value of vi (Poisson's ratio at zero strain) has been found to decrease with increasing pressure, the variation of vi being approximately linear with logarithm of confining pressure, as shown in Fig. 19 for six cohesionless soils, The variations shown in Fig. 19 may be represented by equations of the form

v. 1

a = G - F log (_l)

Pa (15)

in which G is the value of vi at a confining pressure of one atmosphere, a3 is the minor principal stress or confining pressure, Pa is atmospheric pressure expressed in the same units as 03, and F is a parameter whose value is determined empirically and which represents the rate of decrease of v. with increasing confining pressure.

·1

Tangent Poisson's Ratio

The relationships expressing nonlinearity and stress-dependency may be used to define a value of tangent Poisson's for any state of stress. According to equation (11), which defines the tangent Poisson's ratio

1 -= v

t

dE a

dE r

(16)

By performing the indicated differentiation on equation (13), the tangent Poisson's ratio may be expressed as

f V t = (1 - dE ) 2

a (17)

As shown previously, the parameter f is equal to v., the value of tangent Poisson's ratio at zero strain, which is related t6 confining pressure• as shown by equation (15). Substituting equation (15) into equation (17) results in the following expression.

60

>-.. 0 ~ ~ a: z 0 (/) (/)

0 Q..

~ z LIJ C)

z ~

~ ~ ..... _, ~

~

z

0.6

'} CT3 i • G - F I og { p )

a

0.6

0.4

0.2

A- OTTAWA SAND (LEE, 1965)

B- GLACIAL OUTWASH SAND (HIRSCHFELD AND POULOS, 1963 C -GRANITIC GNEISS ROCKFILL (CASAGRANDE,1965) 0- QUARTZITE ROCK FILL (CASAGRANDE, 1965)

E- SANDY GRAVEL (CASAGRANDE,I965) F- GRANITIC GNEISS ROCKFILL ( CASAGRANDE,I965)

0.0 ...__ __ .._____, _ _.__......_. ........... ......._._.__ __ _..__ ...... _ __.___.__.__L..-L..._,

I 10 100 10 100

I. EFFECTIV:E CONFINING PRESSURE, 0'3 (TSF)

FIG. 19 VARIATIONS OF INITIAL TANGENT POISSON RATIO WITH CONFINING PRESSURE

\) t

G- F log (o 3/pa)

( 1 - dE ) 2

a

(18)

The axial strain may be eliminated from equation (18) by expressing this strain in terms of the stresses and stress-strain parameters, using equation (3), as follows:

(19)

When this equation is substituted into equation (18), the tangent Poisson's ratio may be expressed as

G - F log (O/pa) \) = (20) t

l- d(o1 - o 3) ~2 03 n [ _ Rf(o1 - o 3)(1- sin¢1

Kp (-) 1 a Pa 2c cos¢+ 2o 3 sin¢

This expression contains eight parameters: The five modulus parameters K, n, c, ¢, and Rf; and three additional parameters G, F, and d. The values of all of these parameters may be determined from the results of a series of triaxial or plane strain compression tests with volume change measurements. Studies conducted to determine values of these parameters for

-various soils under drained test conditions and unconsolidated-undrained test conditions are described in the following sections.

Poisson's Ratio Parameters for Drained Conditions

Classification data and stress-strain parameters for 35 soils tested under drained conditions are summarized in Table 4; this table contains data for the same soils as Table 1 in the previous chapter. For all of these soils except CL-1, for which volume change data were not available, values of the Poisson's ratio parameters G, F, and d have been calculated and are listed in the last three columns of the table.

Examination of these data shows some general relationships among the classification characteristics, relative densities, and Poisson's ratio parameters:

(1) The value of the Poisson's ratio parameter G increases with increasing relative density. Variations of initial tangent Poisson's ratio with 03 for Sacramento River Sand (SP-4) at four relative densities are shown in Fig. 20. The values of G corresponding to the linear interpretations shown in this figure increase from 0,47 at a relative density of 38%

62

Table 4. Classification Data and Stress-Strain Parameters for Soils Tested Under Drained Conditions.

Unifhd Syate• Cro~

Cobble

Cobblt!

Cobb a

Soil ~~~u

Soil Ducriptioa

S&nte Fe Andeaiu iodtfill

Sante Fe Aade:aite Rockfill

Granitic Codn J.ockfill (Mica n.. Shell)

Conalo.:rate I.DCk.fUl (lletzllh~lcoyote o .. Sbell)

Cunit.ic Coeiaa l.ockfill (Kica o- Shell)

Quartdte J.ockfill (Fum.u Daa Shell)

Qual'tdte l.oclr.Ull (Fumu Da. Tran.it.ioa.)

Quartdte lockf111 (Furnas Da. Transitioa)

Pinz.&D.diltAD Crawl

Diorite. Rockfill (El lnfiernillo o- Shell)

Marui (1963)

Hand (1%1)

Referll'nce

C..u.arande (1965); K.rn~. et al (1965)

Maual, et al (1965); C~oa and lena .. ini (1967)

Ca .. u&r&n~ (1965); Maua~, et al (1965)

Cauararuie {1965)

c. .. grande (1965)

c-a&rande (1965)

Manal et al (1965)

Marsal et al (1965)

cw cw cw cw cw cw cw cw cw cw cw Cll

C-Ia

C-lb

C-2

CW-1

CW-2

cw-> GW-4

cw-5

CW-6

CW-7

CW-8

GW-9

Ql-10

GW-ll

cw-u

Silicified Coaal~ute loc:kfill (El Infiernillo 0.. Shdl) Kanal et al (1965)

SUieified Con&lc.erate Rocltf111 (El hlfiemillo 0.. Sbdl) X.nal et al (1965)

Araillite iocltfill (Pynaid D.o Shell) M.nchi (1%9)

A.raill1tc l.odtfill (Pyraaid ea. Shell)

Cru.hed Olivi~W l&.salt

Ql GV-13 Cna~hecl Olivine !&salt

Ql GW-14 Cuvd (New DoD Pedro n.. Shell)

CP GP-1 Qu.artdte l.ockfill (Furnas o- Shell)

CP CP-2 Sandy Crawl (Mica 0.. Shell)

GP GP-3 la&&lt ioc:kfill

f:' CP-4& Coatrer .. Andesite Crawl

CP CP-~ Coatrer- Andesite Cr&W!1

CP CP-S Uphibolite Crawl (Oroville 0.. Shell)

CP CP-6 Silty Sandy Crave! (Oroville 0.. tu.nsition)

GP CP-7 .. ibolite Gravel (Oroville D- Shell)

CP CP-8 Allph1bol1te Cn'Wl (Oroville Daa Sb.tll)

CC cc-1 Clayey Crawl (Nev Boaan D- Core)

sw sw-1 Araillitc lockf111 (Pyraa1d o- Shell)

SV Sw-2 Cru~Md OUvioe !ualt

SP SP-1.& TeKCoco Sand

SP SP-lb Texcoco Sand

SP SP-2 Ot~a Sand

SP SP-3 Cladal Outwash Saad

SP SP-4& Sacra-ato R.i'Wr Sand

SP SP-4b SacraaHtto l.iver Sand

SP SP-4c. Sacraaento J.iv.r Sand

SP SP-4d Sacr .. ftto li"'r Saad

SP SP-5a IIA• l.iwr Saud

SP SP-5b H• l.her Sand

SP SP-6 Upbibolita Sand (Oroville Da• Shell)

51'1-SC SM-SC-la SUty Clayey Sand (Mica Da• Con, Dry)

SM-SC SH-SC-lb Silty Clayey Sand (Mica D- tore, Std. AASMO Opt.)

SK-SC SK-SC-lc Silty Clayey Sand (M1ca o- Core, Vet)

CL CL-1 Silty Clay (Arkabutla o-. Std . .V.SHO Ope.)

ML ML-1 Carmcmsvilll! Silt (Undhtu.rbed)

Marachi (1969)

Marachi (1969)

Karachi (1969)

Bechtel (1969)

Cuaarande (1965)

Cuaarande (1965)

c-aarande (1965); Manal (1%7)

Hanal (1963)

Hand (1963)

Hall &Dd Cordon (1963)

Ball and Cord01:1 (1963)

Karachi (1969)

Karachi (1969)

Bird (1961)

Karachi (1969)

Karachi (1969)

Marsd (1%3)

Mana! (196 3)

Lee {1965)

Hirschfeld and Poulos (11

1~3) Lee (1965)

Lee (1965)

LH (1965)

Lee (1965)

lbhop (1966)

lhhop (1966)

Karachi (1969)

c .. aar-ande (1965) i Inllle~r and Hillh (1965)

C..aarande (1965); Insle~1r •ad Hillis {1965)

Ca•aarande (1965); Insle~1r and Hillis {1%5)

C..a&rande, et al (1963)1

Hirschfeld and Poul05 (11

1J6J)

Craia She <-> Dry lla1t ldtial

D60 D)O 010 VeiiJht Void

(lb/ft '> Ratio

llO

llO

1ll

47.0 .. <10

'" <10

21.0 ., .. 64

11.8

" 17.8

uo uo U4

7.5

26.0

2.7

42.0

20.0

20.0

7.4

23.1

7.4

1l0

1l0

" 0.9

6.0

0.25

17.0

4.5

4.5

2.7

8.0

2.7

70.2

77.0

101.0

118.9

123.7

132.1

105.7

106.9

114.1

112.2

113.0

125.1

53 23.1 7.6 125.0

19.0 1.6 O.ll U3.1

19.0 16.0 12.0

22.0 1.2 0.23

19.0 3.6 1.0 133.8

1S 65 44.0 88.1

75 65 44.0 96.1

n.o 13.0 5.1 1u.o 18.0 4.8 0.4 141.0

1).2 4.6 0.36 1.52.0

39.6 14.2 1.1 149.)

12.0 0.6 113.0

4.1 1.8 0.6 111.6

4.1 1.8 0.6 125.4

2.4 1.8 1.2 90.)

2.4 1.8 1.2 99.4

0. 73 0.68 0.64 111.0

o.n o.4o 0.14 112.1

0.22 0.17 O.lS 89.5

0.22 0.17 O.lS 94.0

0.22 0.17 0.15 97.8

0.22 0.17 0.15 103.9

0.25 0.17 0.10

0.25 0.17 0.10

).1 1.1 0.09 146.5

0.34 O.OJ 0.002

0.34 0.03 0.002

0.34 0.03 0.002

0.02) 0.005 0.0008 110.0

0.033 0.018 0.005 108.0

1.06

0.88

0.62

0.)9

0.)2

0.)4

0.>6

0.55

0.45

0.46

0.45

0.43

0.43

o."

o. )0

0.68

0.54

0.2.1

0.16

0.20

0.22

0.46

0.4)

0.74

0.58

0.49

0.50

0.87

o. 78

1.71

0.61

0.82

0.64

0.2)

0.49

0.57

ldativoe Density

Loose

19%

rot

95t

65t

50t

-lOOt

-lOOt

-lOOt

-lOOt

-100%

50t

95t

Lo~·

-·· lOOt

100t

-100%

-lOOt

lOOt

80t

l8%

60%

78%

100%

....... -100%

" <1

5-26

1-26

S-26 ... , 4-37

4-37

1-26

1-26

1-26

2-26

2-47

2-47

2-47

2-41

9-47 ... , 7-ll

S-26

<1

<1

9-40

9-40

2-47

2-47

1-4

2-47

2-41

<1

<1

1-.U

1-41

1-41

1-41

1-41

1-41

7-71

7-71

2-47

4-15

4-)5

4-35

2-8

1-40

10

10

$ (dearees)

40

" 14-29

49-31

37-32

45-39

S0-42

45-39 ,,..,. 46-34

46-17

46-)6

47-)6

47-36

43-37

41-36

40-lS

42-34

39-37

46-39

" " 4)-37

45-39

49-40

47-38

18

50-lO

52-39

" 4S

)9-28

44-37

34-27

H-28

39-27

41-16

l4-l0

lt-ll

51-41

3l-35

3l-35

33-35

l6

44-)()

., 340 0.21 0.90 0.2S 0.25 6.4

400 0.20 0.84 0.2) 0.27 7.3

65 0.61 0.52 0.12 0.04 3.4

440 0.45 0.54 0.37 0.16 4.0

372 0.35 0.74 0.28 0.10 3.8

755 o. 35 0,80-0.95 o. 35 0.09 t..8

1210 0.39

875 0.50

715 0.50

290 0. )0

320 0. 38

335 o • .u 650 0.25

650 O.H

1115 0.12

0.73 0.23 0.11 16.8

c. 58 0.27 o. u 16.)

0.61 0.50 0.21 7.2

0.10 O.ll 0.23 4,,

0.64 0.3) 0.18 4.2

0.62 0.25 0.12 5.1

0.68 0.29 0.12 4.6

0.68 0.29 0.12 4.6

0.70 0.30 0.1) 5.5

1115 0.12 0.70 0.30 0.13 5.5

665 0.28 0.77 0.38 0.14 2.6

950 o.u 0.88 0.30 0.14 3.1

520 0.17 o. 76 0.30 0.08 2.5

640 0.26 0.65 0.32 O.ll 6.0

7.]0 0.51 0.91 0.28 0.20 12.2

975 o.so 0.18 0.18 0.18 15.5

17JO 0.3] 0.89 0.47 0.09 1.9

11SO 0.29 0.69 0.54 0.16 2.6

)7&0 0.19 0.76 0.63 0.19 16.8

3710 0.19 0.76 0.43 0.19 14.8

95 0.98 0.75-1.0 0.29 0.22 4. 7

650 0.25 0.68 0.29 0.12 6.6

1115 0.12 0.10 0. 30 O.ll ,. 5

)75 0.67 0.98 0.28 0.25 7.1

1015 0.56 0.17 0.60 0.32 6.)

2490 0.58 0.91 0.11 0.15 4.3

270 O.SO 0.55 O.H 0.19 2.5

145 o.56 o.l5 o.n 0.22 2.0

545 o.56 o.86 o.~s 0.26 2.1

710 0.54 0.85 0.68 0.)0 1.9

1210 0.54 0.87 0.76 0.10 1.6

)70 0.46 0.81 o.u 0.15 1.4

1440 0.45 0.88 0.58 0.20 3.4

)780 0.19 0.76 0.43 0.19 14.8

ll95 0.111 0.81 0.)7 0.12 5.3

525 0.50 0.71 0.38 0.12 5.)

ISO 0.14 0.62 0.39 0.12 5.3

240 0.54 0.78

350 0.57 0.60 0.49 0.17 ~-1

? SACRAMENTO .

0 RIVER SAND 1- 0.6 < a::

z 0 (I) (I)

0 Q,

0.4 1-z w (!)

z < 1-

..J 0.2 < -1-

z

(Data 0

10 100

I EFFECTIVE CONFINING PRESSURE, 0"3 ( TSF)

Curve ei Dr

j_%} G F d

a 0.87 38 0.47 0.22 2.00 b 0.78 GO 0.55 0.26 2.07 c 0.71 78 0.68 0.30 1.92 d 0.61 100 0.74 0.30 2.50

FIG. 20 VARIATION OF NONLINEAR POISSON RATIO PARAMETERS WITH RELATIVE DENSITY FOR SACRAMENTO RIVER SAND

64

to 0.74 at a relative density of 100%. The values of the parameters F and d listed in Fig. 20 also increase slightly with relative density.

(2) Soils which consist of particles of similar shape and mineral composition, and which have parallel grain size curves, also have similar volume change characteristics as indicated by the values of the parameters G, F, and d. The variations of vi with o3 for several soils tested by Marachi (1969) are shown in Fig. 2lo The results for tests on specimens with various maximum particle sizes have been represented by a single straight line in each of the three cases shown in Fig, 21. Although somewhat different values of G, F, and d might be more representative of tests on one particular size, the lines shown represent reasonable interpretations of the data. Therefore it may be concluded that by modelling grain size curves it is possible to form soils which have similar volume change characteristics.

(3) Average values of the parameters G, F, and d for the sands and gravels are given in Table 5. It may be noted that the values shown in Table 5 for well-graded sands are based on data for only two soils. It may also be noted that most of the SP soils listed in Table 5 are uniform, whereas most of the GP soils are nor uniform.

In view of the wide variations in the values of the stress-strain parameters for soils of the same classification and having the same relative density, ic may be concluded that values of these parameters for use in accurate analyses should desirably be determined by conducting tests on suitably selected and prepared soil specimens.

Poisson's Ratio Parameters for Unconsolidated-Undrained Conditions

A review of published strength test results did not provide any information concerning volume changes during unconsolidated-undrained tests on partly saturated soil specimens. Volume changes are seldom measured in this type of test because of experimental difficulties, requiring measurement of changes in the volume of fluid contained in the pressure chamber, and because volume change data are not often used for purposes of design or analysis. The only data available at the time of this study were those from tests on the New Don Pedro Dam core material (Bechtel Corporation, 1969) and the Oroville Dam core material (Department of Water Resources, 1969). The values of the Poisson's ratio parameters determined in these tests are given in Table 6, and supplementary information concerning these soils is given in Table 3. As indicated in Table 3, both materials were compacted at water contents quite close to optimum, but the values of the Poisson's ratio parameters are quite different for these two materials, The value of G is 0,60 for the New Don Pedro Dam core, indicating that the material dilates at low confining pressures. The value of F for the Oroville Dam core material is negative, indicating that the value of Poisson's ratio for this material increases with increasing confining pressure in the range of pressures encompassed by the tests, presumably as a result of increased saturation at higher confining pressures.

65

·-? ~

0 ;:::: <t a:

z 0 f/) f/)

0 0...

1-z w (!)

z <t 1-

..J <t 1--z

FIG. 21

0.5

0.3

0.1

0.4

0.2

0

0.4

0.2

0

06" MAX. SIZE (GW-13) 0 2" MAX. SIZE (GW-12) t:::r. 0.47"MAX. SIZE (SW-2)

10

OROVILLE DAM SHELL (AMPHIBOLITE GRAVEL)

AVG. ei • 0.22 Dr • 100%

CRUSHED BASALT

AVG. e i • 0.43 D, •100%

-PV-RAMW {lAM Sfi£LL (ARGILLITE ROCKFILL)

AVG. ei •0.46 Dr. 100%

I EFFECTIVE CONFINING PRESSURE, o-3 (TSF)

100

VARIATION OF NONLINEAR POISSON RATIO PARAMETERS WITH MAXIMUM PARTICLE SIZE FOR SOILS WITH PARALLEL GRAIN SIZE CURVES

(Do to From Morochi, 1969)

66

Table 5. Average Values of Poisson's Ratio Parameters for

Clean Sands and Gravels under Drained Conditions.

Soil Group G F d

GW 0. 32 0.14 6.4

GP 0. 38 0.15 8.0

sw 0.30 0.13 5.0

SP 0.54 0.23 4.3

Table 6. Poisson's Ratio Parameters for Soils Tested

under Unconsolidated-Undrained Conditions.

Soil Soil

G F d Number

New Don Pedro Dam Core SC-3 0.60 0.27 2.4

Oroville Dam Core GC-2 0.30 -0.05 3.8

67

Because so few data were available concerning the volume change behavior of partly saturated soils under unconsolidated-undrained test conditions, a comprehensive series of U-U triaxial tests were conducted on compacted specimens of Pittsburg sandy clay (CL-5). Tests were conducted on 1.4 in dia, specimens prepared using kneading compaction procedures to a variety of different density and water content conditions; the details of this test program are described in Appendix A. The results of the tests were used to calculate values of the Poisson's ratio parameters G, F, and d as indicated previously, and contours showing the variations of the values of these parameters with compaction density and water content are shown in Fig. 22.

The values of all three parameters may be seen to vary quite widely with initial density and water content: The value of G decreases from about 0.55 at low water content and high compactive effort to about 0.35 at higher water content and lower compactive effort. For a saturated condition, G would be expected to be equal to one-half, as shown in Fig. 22. The value of the parameter F, which reflects the rate of decrease of Poisson's ratio with increasing confining pressure, decreases from about 0,3 at low water contents to negative values at high water content. It seems reasonable that the value of F would be equal to zero for completely saturated specimens, because the value of Poisson's ratio would be equal to one-half for all values of confining pressure, The value of the parameter d decreases from about 2,0 at low values of degree of satu~ation to zero for complete saturation,

As in the case of the strength and modulus parameters discussed in the previous chapter, these tests on Pittsburg sandy clay show that the values of Poisson's ratio for compacted cohesive soils may vary widely

___d_epending _on _Ccompaction density and water content. Accordingly, it is desirable that the density and water content of specimens tested to determine volume change characteristics of embankment materials for analytical studies of behavior should duplicate as nearly as possible the density and water content used in the actual embankment.

68

..

FIG. 22

125

120

115

110

105

100 6 12 18 24

125

lL. 0 120 n. ..... >- 115 ._ en z 110 LLI 0

>- 105 a: 0

100 6 12 18 24

125 1 I

120 F -115

110

\ 105 \

\o.3 100

\

6 12 18 24

WATER CONTENT (%)

NONLINEAR POISSON RATIO PARAMETERS FOR COMPACTED PITTSBURG SANDY

CLAY UNDER UNCONSOLIDATED· UNDRAINED TEST CONDITIONS

69

CHAPTER 5

ANALYSIS OF OTTER BROOK DAM USING NONLINEAR

STRESS-DEPENDENT STRESS-STRAIN PROPERTIES

Otter Brook Dam, a rolled earth dam constructed in New Hampshire during the summer of 1957, is about 130 ft high and has a crest length of about 1300 feet. As shown in Fig. 23, the embankment section is symmetrical, with 2.5 on 1 slopes upstream and downstream. Except for thin blankets of gravel and rock fill at the surface of the slopes and a chimney drain within the downstream portion of the dam, the embankment is homogeneous. The main portion of the embankment consists of a glacial till which is a very well-graded clayey sand with between 10% and 20% gravel and larger size particles. The foundation consists of a thin layer of very dense basal till overlying bedrock.

When the dam had been constructed to about 55% of its final height, it was noticed that the footing of the bridge pier shown in Fig. 23 had tilted and moved outward. Reference stakes were placed to measure additional movements, and construction was halted when the movement rate was found to be increasing. After the stability of the embankment was re-evaluated and found to be satisfactory, construction was resumed and the dam completedo By the end of construction the upstream face of the dam had bulged outward more than three feet, Thus, although the dam was stable as designed, it deformed sufficiently to cause concerno Although these aeforrnations could not have been anticipated at the time the dam was designed, they can be calculated at the present time using the finite element analysis procedures and stress-strain relationships described in previous chapters.

Finite Element Mesh

The finite element configuration employed in analyses of Otter Brook Dam, which is shown in Fig. 24, consists of 44 elements and 62 nodal pointso Because the embankment was symmetrical (except for the chimney drain) only a half-section was used for the analyseso The dense till and bedrock foundation were assumed to be completely rigid and were not represented in the mesh. In accordance with the results of the studies described in Chapter 2, the analysis was performed using 8 layers and one cycle of iteration per layer to improve the degree of correspondence between the calculated values of stress and the values of tangent modulus and tangent Poisson's ratio for each element

Properties of Otter Brook Dam Fill

The properties of the Otter Brook Dam fill (a sandy clay, SC-1) are listed in Table 3. Because the permeability of the soil is low, it would

70

3'

Upstream

BRIDGE PIER

l i

"TILL• .nm=-u-----"?'h- -=-----

2' ROCK FILL 3' GRAVEL FILL

---- -u/'_--q:::= ,~/.:. ~-=- - ~ ~~- - -- ~'.=-;<=- - -

~- -7~ BEDROCK '

~~

FIG. 23 OTTER BROOK DAM (AFTER LIN ELL AND SHEA, 1960)

44 ELEMENTS 62 NODAL POINTS

POINT A IS THE LOCATIOf~

OF THE BASE OF THE

BRIDGE PIER.

327.5

HORIZONTAL DISTANCE (FT.)

FIG. 24 FINITE ELEMENT MESH FOR OTTER BROOK DAM

8o2

788

773 ....: lL

756

z 739 0

.... 722 ct

> l&J

705 ..J l&J

688

671

.jt2.~

be expected that virtually no drainage would take place during construction, which required only one season. Therefore it seems logical that unconsolidated-undrained tests would be most appropriate for determination of strength and stress-strain parameters applicable duririg construction and at the end of construction.

Linell and Shea (1960) presented stress-strain curves for the results of unconsolidated-undrained tests on the Otter Brook fill which showed that the values of stress difference continued to increase even at very large values of strain. The stresses corresponding to axial strains of 20% were selected by Linell and Shea for determining values of c and ¢ for use in stability analyses, and these values were also employed in the finite element analyses described in this chapter. Volume changes were not measured during the tests reported by Linell and Shea, and the values of the Poisson's ratio parameters employed in these analyses were estimated on the basis of the series of unconsolidated-undrained tests described in the previous chapter. The soil on which these tests were performed is a sandy clay (Pittsburg sandy clay) which is similar to the Otter Brook fill material. The values of the Poisson's ratio parameters were selected in accordance with the compaction conditions for Otter Brook Dam, with dry density approximately equal to the Standard Proctor maximum and water content slightly wet of optimum. The values of the soil parameters employed in the nonlinear analyses of Otter Brook Dam are shown in Table 7.

Comparison of Calculated and Measured Displacements

Reference stakes were installed in Otter Brook Dam when the fill reached elevation 7 44, five feet above the base of the bridge pier, which is about 55% of the full height of the dam. Besides four reference ~takes, two ahove and two helm-.r- elev-ation- 744,- measurements_ were- alS-o_ made_ to_ ascertain the horizontal and vertical movements of the bridge pier. Measurements made ac locations below elevacion 744 do not reflect all of the displacement at these locations, but only that part induced by placement of fill above elevation 744. In order that the calculated and measured values could be compared on an equal basis, the calculated displacements at these points were corrected by subtracting those portions of the displacements resulting from placement of fill below elevation 744.

The calculated displacements are compared to the measured values described by Linell and Shea (1960) in Fig. 25. The variations of the horizontal displacement with height above the base of the dam are shown on the left, and the measured and calculated variations of the horizontal and vertical bridge pier displacements with fill height are shown on the right. It may be noted that in bach cases the agreement is very good, indicating that the values of the stress-strain parameters determined from triaxial test results, together with the incremental analysis procedure described previously, provide a rational basis for estimating embankment deformations.

The close agreement between the measured and calculated displacements provides evidence chat the results of the analysis closely reflect the actual behavior of the embankment. For this reason it would be expected

73

Table 7. Soil Parameter Values for Nonlinear Analyses of

Stresses and Deformations of Otter Brook Dam.

Soil Parameter Symbol Value Employed in Analyses

Unit Weight y 140 lb/ft 3

Cohesion c 1. 08 T /ft 2

Friction Angle ¢ 14 degrees

Modulus Number K 40

Modulus Exponent n 0.48

Failure Ratio Rf 0.68

' Poisson's G 0.43

Ratio ) F -0.05

Parameters d 0.60

74

830~----~------~------~---~--~

.,.: "--

790

~ 750 1-

~ LIJ ..J 1&1

710

670

0

Crest of Dam

"-... 'a--~

~ 'j

~_.,~ /,/'

jJ 0 MEASURED POifiTS --CALCULATED

Base of Dam

2 3

HORIZONTAL DISPLACEMENl"

OF UPSTREAM FACE (FT.)

4

815~------r------.r------.-------,

....,: u. 795

..J

..J u.

~ 775 a.. 0 1-u. 0

z 755 0

1-<t > LIJ ..J 735 LIJ

670 0

Crest of Dam

HORIZONTAL DISPLACEMENT

MEASURED --- CALCULATED

Base of Bridge Pier

2 3 4

DISPLACEMENT OF BRIDGE PIER (FT.)

FIG. 25 DISPLACEMENTS IN OTTER BROOK DAM USING A NONLINEAR MODULUS AND POISSON RATIO

that the calculated values of displacement for other points within the embankment, and other calculated results, such as embankment stresses, would also provide useful information concerning the behavior of the embankment.

Additional Results of Finite Element Analysis

Calculated values of deformation within the embankment are shown in Fig. 26; the values shown are total displacements resulting from construction of the entire embankmento The horizontal displacements along several vertical lines are shown in the upper portion of the figure. The magnitude of these displacements which are everywhere directed away from the embankment centerline, vary from zero at the base of the embankment and at the embankment centerline to a maximum value of about three and onequarter feet about midway beneath the toe and the crest, slightly beneath the surface of the slopeo The calculated vertical displacements are largest at the embankment centerline about two-thirds of the height above the base, where the settlement is about three and one-half feet. Along the lower portion of Lhe slope there is an upward movement of the surface which results from the large outward bulging deformation; the deformed shape of the embankment is shown in the lower part of Fig. 26,

It should be noted that compared to the embankment height (about 130 ft) the displacements are very largeo It was this fact which caused some concern with regard to the SLability of the embankment. The maximum vertical displacement at the embankment centerline amounted to more than 2.5% of the embankment height" These large deformations undoubtedly resulted from the fact that the material of which the dam was constructed, at the conditions of water content and density employed in the dam, was

-characterized by a srr.all value of modulus -as comp-ar-ed to shear strength. Thus although the embankment was stable (the factor of safety with regard to slope failure is about 2) the deformations were large enough to cause difficulties with the bridge pier and even to alter the appearance of the dam.

Contours of the calculated stresses within the dam are shown in Fig. 27. The values of major principal stress, shown in the upper part of the figure, are roughly equal to the overburden pressure (yh) at any point varying from about 90% of the overburden pressure beneath the centerline, as shown by the scale at the right, to values exceeding 100% beneath the outer portions of the slopes. The variations of the minor principal stress (cr 3), which are shown in the center part of the figure, are quite similar to those of the major principal stresses, being highest at the base beneath the crest and decreasing to zero on the slopeso The values of maximum shear stress, shown in the lower part of the figure, range from low values near the face of the slope to values of approximately one and one-quarter tons/ft 2 at two locations, in the lower half of the dam at the centerline and at the base of the dam midway between the toe and the crest.

The contours of stress level or percentage of strength mobilized are shown in Fig. 28. It may be noted that the maxima, which are slightly

76

HORIZONTAL DISPLACEMENTS ALONG SELECTED SECTIONS (FT.)

..±-

VERTICAL DISPLACEMENTS ALONG SELECTED SECTIONS (FT.)

r

VECTORS OF

FACE DISPLACEMENT ( FT.)

DISPLACEMENT SCALE 0 I

2 I

4 FT. I

FIG. 26 DISPLACEMENTS WITHIN OTTER BROOK DAM

USING A NONLINEAR MODULUS AND POISSON RATIO

77

+

+

MAJOR PRINCIPLE STRESS Dj (TSF)

MINOR PRINCIPAL STRESS o-

1 (TSF)

MAXIMUM SHEAR STRESS

t' MAX (TSF)

5

7

FIG. 27 PRINCIPAL STRESS CONTOURS IN OTTER BROOK DAM

USING A NONLINEAR MODULUS AND POISSON RATIO

78

yh

(TSF)

0

I

2

3

4

5

6

7

8

9

FIG. 28 CONTOURS OF MOBILIZED STRENGTH IN OTTER BROOK DAM USING A NONLINEAR MODULUS AND POISSON RATIO.

79

t

I

greater than 50%, are located near the places where the largest values of maximum shear stress occur. For a material with ¢ equal to zero these maxima would coincide exactly, and it is interesting to note that their variations are very similar for the Otter Brook Dam fill material which has an angle of internal friction of 14 degrees under the undrained conditions reflected in these analyses.

The values of tangent modulus and tangent Poisson's ratio corresponding to the stress conditions at the end of construction are shown in Fig. 29. It may be noted that the values of tangent modulus vary by a factor of three, from about 20 tons/ft 2 in regions of low stress to about 60 tons/ft 2 in regions of high stress, and the values of tangent Poisson's ratio vary from about 0.4 near the slopes to about 0.48 near the base of the embankment. While average values of these elastic parameters might be found which would result in values of displacement or stress in substantial agreement with those calculated using nonlinear, stress-dependent properties, it might be very difficult to select such a value rationally on the basis of laboratory test results. Nonlinear analyses of the type illustrated in this chapter, however, may be performed in a straightforward manner using the results of laboratory tests. Based on the results of this analysis of Otter Brook Dam, it appears that the calculated results are in good agreement with the actual behavior.

80

TANGENT MODULUS (TSF)

TANGENT POl SSON RATIO

FIG. 29 ELASTIC PARAMETER IN OTTER BROOK OAM

81

CHAPTER 6

COMPARISON OF VARIOUS FINITE ELEMENT ANALYSIS

PROCEDURES FOR OTTER BROOK DAM

Types of Analyses

In addition to the nonlinear, stress-dependent finite element analysis of Otter Brook Darn described in the previous chapter, a number of analyses of the darn were performed using analysis procedures in which:

(1) The placement of fill in successive layers was not simulated, but gravity forces were applied throughout the entire embankment simultaneously (termed "gravity turn-on" analysis procedure). The embankment material was assumed to be linear and to be homogeneous with respect to the values of Young's modulus and Poisson's ratio.

(2) The placement of fill in successive layers was simulated, using incremental analysis procedures. The embankment material was assumed to be linear and to be homogeneous with respect to the value of Young's modulus and Poisson's ratio.

(3) The placement of fill in successive layers was simulated, using the same incremental analysis procedures as in the previous method. The value of Young's modulus was varied in accordance with the calculated values· of -stress, as in the analyses desc-r±}}ed in the previnus chapter, but the embankment material was assumed to be homogeneous with respect to the value of Poisson's ratio.

These analyses were performed for the purpose of comparing the values of displacements and stresses calculated by means of the various procedures. Previously, Clough and Woodward (1967) performed gravity turn-on analyses and incremental analyses using constant values of Young's modulus and Poisson's ratio. Finn (1967), and Finn and Troitskii (1968) presented results of incremental analyses performed using nonlinear modulus values and constant Poisson's ratio; and Clough and Woodward (1967) presented results of an incremental analysis performed using nonlinear shear modulus values and constant values of bulk modulus. Each of these types of analyses resulted in good correspondence between measured and calculated displacements, and it is therefore of some interest to examine the results of these types of analyses in detail, to determine in what respects they are similar and in what respects they differ.

Basis of Comparison

In the case of each of the three types of analysis, the values of the elastic parameters employed in the analyses were adjusted so that the

82

calculated values of horizontal and vertical displacement at the location of the bridge pier were equal to the estimated values of total displacement at Lhis location. It was necessary to estimate the magnitudes of the total displacements because measurements were begun at the time when the fill height had reached elevation 744, seven feet above the base of the bridge pier. The variations of the measured values and the adjusted (total) displacements with increasing height of fill are shown in Fig. 30; the estimated total horizontal displacement at the bridge pier is 3.2 ft and the vertical is 1.0 ft.

A number of analyses of each type were performed using various values of Young's modulus and Poisson's ratio, for the purpose of finding values of these parameters which would result in calculated displacements in agreement with those observed. The results of these calculations are shown in Fig. 31. The upper part of the figure represents results of gravity turn-on analyses, the center represents results of incremental analyses with constant values of E and v, and the lower part represents results of incremental analyses using nonlinear modulus values given by equation (10) together with constant values of Poisson's ratio, The dotted lines in the figure represent combinations of values of E (or K) and v which result in horizontal displacements equal to those observed, and the solid lines represent combinations of values of E (or K) and \! which result in vertical displacements equal LO those observed" (In conjunction with the incremental analyses performed using nonlinear modulus values, the values of the parameters c, ¢, n, and Rf were kept c,onstant, and were in all cases equal to the values determined from the laboratory test results, as summarized in Table 7 .) Where the solid and dotted lines cross, the combination of parameter values results in values of both horizontal and vertical displacement in agreement with those observed. Thus by Judic1ous selection of the values o£ the required paramet.eJ::s~- it is possible to arrange- tha-t thecalculaLed and observed bridge pier displacements are the same,

The three solutions represented by the points on Fig. 31 where the solid and dotted lines cross are thus all equivalent with regard to final bridge pier displacements, and it is of some interest to compare other aspects of these analyses. The variations of bridge pier displacements with increasing fill height for all of the incremental analyses are shown in Fig. 32, and it may be noted that these variations are all very similar and are in good agreement with those observed. Only the final value is shown for the gravity turn-on analysis, because intermediate displacements are not calculated when this analysis procedure is employed.

Comparisons of Displacements

Calculated and measured final displacements for four points on the face of the embankment are compared in Fig. 33. On the right in this figure, the final calculated finite element values are shown together with the observed displacements; the agreement between the observed values for the upper two points (above elevation 744) and the values calculated by any of the three incremental analysis procedures is quite good, The agreement is not so good in the case of the gravity turn-on arl.alysis. As

83

815 815 r------r--------,

-.,_; D~'!!_ __ _

"'-- 795

..J _J

"'-"'- 775 0 775 a. 0 ..... "'-0

co z ~ 755

0 ..... ct > LLI _, LLI 735

6 7 0 ~:o..=-=.:...;.=~-----------___.

0 2

DISPLACEMENt (FT.) DISPLACEMENT (FT.)

a) OUTWARD HORIZONTAL DISPLACEMENT b) DOWNWARD VERTICAL DISPLACEMENT

FIG. 30 MEASURED AND ADJUSTED DISPLACEMENTS OF BRIDGE PIER IN UPSTREAM FACE OF OTTER BROOK DAM

FIG. 31

~ 60 ...l :::>_ OIJ.. Ow :::Et-

CONSTANT MODULUS CONSTANT POISSON RATIO

(GRAVITY TURN-ON ANALYSIS)

u-j::: 40 U) c(

~ 36.1 ------

U)

3 =>OIJ.. QU)

:::Et-0

~ c( ...l 1.1..1

rr.· 0 .... u c( IJ..

--0.465 ---

20~----L-~------~------~------~----~

0.50 0.46 0.46 0.44 0.42 0.40

30

POISSON RATIO, '\)

CONSTANT MODULUS CONSTANT POISSON RATIO

(BUILD-UP ANALYSIS)

30.0 ------10~---------~-------~---L--~~-------~--------~

0.50 0.4B 0.46 0.44 0.42 0.40

POl SSON RATIO, V

NONLINEAR MODULUS 50 CONSTANT POISSON RATIO

U) 30 :::> ...l :::> 0 0 :e

10 ~------L-------~--~--~-------L------~ 0.50 0.46 0.46 0.44 0.42 0.40

POISSON RATIO, '\)

--- COMBINATIONS OF E (OR K) AND v TO OBTAIN CORRECT HORIZONTAL DISP.

- COMBINATIONS OF E (OR K) AND v TO OBTAIN CORRECT VERTICAL DISP.

COMBINATIONS OF ELASTIC CONSTANTS REQUIRED FOR VARIOUS ANALYSES TO OBTAIN CORRECT BRIDGE PIER DISPLACEMENTS IN OTTER BROOK DAM

85

~ II..

..J

..J

II..

II.. 0

a.. 0 ~

II.. 0

z 0 ~

0> ct > (7\ liJ ..J liJ

FIG. 32

815 ...--------.----.

CREST OF DAM CREST OF DAM

795 795

JUS TED

775 775

755 755

735 735

670 ~~~~~~~~£~~~~~---~ 670 ~~~~~~~~ 0 3 4 0 2

DISPLACEMENT ( FT.) DISPLACEMENT (FT.)

O) OUTWARD HORIZONTAL DISPLACEMENT b} DOWNWARD VERTICAL DISPLACEMENT

------NONLINEAR MODULUS, NONLINEAR POISSON RATIO

---- NONLINEAR MODULUS, CONSTANT POISSON RATIO -·-CONSTANT MODULUS, CONSTANT POISSON RATIO (BUILD-UP ANALYSIS)

0 CONSTANT MODULUS, CONSTANT POISSON RATIO (GRAVITY TURN-ON ANALYSIS)

DISPLACEMENTS OF BRIDGE PIER IN UPSTREAM FACE OF OTTER BROOK DAM AS DETERMINED

BY VARIOUS METHODS OF ANALYSIS

00 _,

830r-------r-------.-~---..------, 830r-------.------..------,--------,

CREST OF DAM CREST OF DAM ------------------~

790 790

-.,..: 1.&..

z 750. 750 0 ~ ct > LIJ _, LIJ 710 710

0 MEASURED POINTS 0 MEASURED POINTS

670 -------------------- 670 --------------------BASE OF DAM BASE OF DAM

0 2 3 4 0 2 3 4

HORIZONTAL DISPLACEMENT OF UPSTREAM FACE (FT.)

a) FEM VALUES ADJUSTED FOR LATE MEASUREMENTS b) FEM VALUES NOT ADJUSTED

------NONLINEAR MODULUS, NONLIN~AR POISSON RATIO

---- NONLINEAR MODULUS, CONSTA,NT POISSON RATIO I

--CONSTANT MODULUS, CONSTANT POISSON RATIO (BUILD-UP ANALYSIS)

---CONSTANT MODULUS, CONSTAr,H POISSON RATIO (GRAVITY TURN-QN ANALYSIS)

FIG. 33 DISPLACEMENTS OF UPSTREAM FACE C>F OTTER BROOK DAM AS DETERMINE£? BY VARIOUS METHODS OF ANALYSIS

mentioned previously, agreement could not be expected in the case of the lower two points, because measurements were not begun until the embankment fill reached elevation 744o In the case of the incremental analyses this may be accounted for by correcting the calculated values for displacements induced by filling below this elevation, as shown on the left in Fig, 33. It may be noted that when this is done the agreement is quite good in all cases. Adjustments of this type are not possible in the case of the gravity turn-on analyses, because only a single value of displacement is calculated for each point.

The final values of horizontal displacements are compared further in Fig. 34, where contours of horizontal movement are shown for the three analysis procedures discussed previously in this chapter, as well as for the procedure using nonlinear modulus and nonlinear Poisson's ratio discussed in the previous chapter. The contours for all three of the incremental procedures are virtually identical to each other, and the contours for the gravity turn-on procedure is also very similar, In each case the maximum horizontal displacement is slightly greater than three feet, and occurs at a point beneath the center of the slope, A similar comparison for vertical displacements is shown in F1g,. 35. The results of all three incremental analysis procedures are again very nearly the same, with the maximum settlement occurring at the centerline beneath the surface and a small amount of bulging on the lower part of the slope in all cases, The results of the gravity turn-on analysis pro~edure, however, are markedly different, being characterized by sectlemem:s which increase to a maximum at the top of the embankment. Th1s difference in settlement patte1:ns represents a basic difference between the results of i;:1cremem:al and gravity turn-on analyses. The pattern of measured settlements in real embankmen:cs corresponds to that calculated by means of the incremental analjsis procedures, being largest in the center of the embankment where the combination of added weight above and thickness of compressible material beneath is largest. The d1fferent pattern of settlements in the gravity turn-on analysis corresponds to the assumption of simultaneously applied gravity forces throughout the embankment, which does not correspond to reality.

Comparison of Stresses

The stresses calculated by means of these four analysis procedures are shown in Figs. 36, 37, and 38. Contours of major principal stress (o1), shown in Fig. 36, are very nearly the same for all methods of analysis with gravity turn-on or incremental analysis procedures using linear or nonlinear material properties. As discussed previously, these stresses are slightly less than the overburden pressure (yh) at the centerline, and they slightly exceed the overburden pressure beneath the outer portion of the slope. Contours of the minor principal stress (03) shown in Fig. 37 are also very nearly the same for all methods except for the zone of tension near the crest of the slope which was calculated in the gravity turn-on analysis but not the incremental analyses, The contours of maximum shear stress (Tmax) are also similar for all procedures with the exception that the largest value calculated by the gravity turn-on procedure is about 20% smaller than the largest values calculated by means of the incremental analysis procedures.

88

ALL DISLACEMENTS ARE IN FEET (~)

CONSTANT MODULUS

CONSTANT POISSON RATIO


CONSTANT MODULUS

CONSTANT POl SSON RATIO

(BUILD· U.P ANALYSIS)

NONLINEAR MODULUS


NONLINEAR MODULUS NONLINEAR POISSON

FIG. 34 COMPARISONS OF HORIZONTAL DISPLACEMENTS IN OTTER BROOK DAM

AS DETERMINED BY VARIOUS METHODS OF ANALYSIS

89

ALL DISPLACEMENTS ARE IN FEET (' + )

CONSTANT MODULUS



CONSTANT MODULUS


(BUILD-UP ANALYSIS)

NONLINEAR MODULUS


NONLINEAR MODULUS

NONLINEAR POISSON RATIO

FIG. 35 COMPARISONS OF VERTICAL DISPLACEMENTS IN OTTER BROOK DAM


90

ALL STRESSES ARE IN TSF

CONSTANT MODULUS



CONSTANT MODULUS


(BUILD- UP ANALYSIS)

NONLINEAR MODULUS


NONLINEAR MODULUS


FIG. 36 COMPARISONS OF MAJOR PRINCIPAL STRESSES IN OTTER BROOK DAM


91


CONST~NT MODULUS

CONSTANT POISSON RATIO (GRAVITY TURN-ON ANALYSIS)

CONSTANT MODULUS


(BUILD-UP ANALYSIS)

NONLINEAR MODULUS


NONLINEAR MODULUS


FIG. 37 COMPARISONS OF MINOR PRINCIPAL STRESSES IN OTTER BROOK DAM


92


CONSTANT MODULUS



CONSTANT MODULUS


(BUILD-UP ANALYSIS)

NONLINEAR MODULUS CONSTANT POlS SON RATIO

NONLINEAR MODULUS

FIG. 38 COMPARISONS OF MAXIMUM SHEAR STRESSES IN OTTER BROOK DAM


93

Usefulness of Various Types of Analyses

Although the vertical displacements calculated using gravity turn-on and incremental analysis procedures are widely different the calculated values of stress are nearly the same, suggesting that even the simpler gravity turn-on analysis procedure may be useful for calculating approximate stress distributions in homogeneous embankments. The advantage of the simple gravity turn-on procedure is that such analyses may be performed using standard options in many readily available finite element computer programs.

More accurate values of stress may be calculated using incremental analysis procedures, even if the soil is assumed to be linear elastic and to be represented by single, constant values of Young's modulus and Poisson's ratio. Because the calculated values of stress are independent of the value of modulus employed in the analysis of homogeneous embankments, incremental analyses therefore provide a means of calculating accurate stress values, even without an accurate value of Young's modulus. The calculated values of displacement are inversely proportional to the value of Young's modulus employed in the analysis, and are also influenced by the value of Poisson's ratio, as shown in Fig. 31. Therefore, in order to calculate accurate values of displacemen~ it is necessary that appropriate values of Young's modulus and Poisson's ratio be used in the analysis. The nonlinear, stress-dependent stress-strain relationship described in previous chapters provides a means of determining appropriate parameter values from laboratory test results in a straightforward manner, and this procedure may therefore be used to calculate accurate values of both stress and displacement. In the case of the other procedures, in which constant parameter values are employed, some judgment is inevitably involved in selecting appropriate parameter values and it is therefore more difficult to calculate accurate displacements using these procedures.

The difficulties of using constant parameter analysis procedures are compounded in the case of non-homogeneous embankments for which the stress distribution depends on the relative displacements between adjacent zones. In such cases nonlinear analysis procedures would be expected to provide not only more accurate values of displacement but also more accurate values of stress. It may therefore be concluded that the incremental analyses using nonlinear, stress-dependent properties are the most generally applicable of the procedures studied, and would be expected to give the best results for the widest variety of conditions.

94

CHAPTER 7

EVALUATION OF EMBANKMENT STABILITY USING FINITE ELEMENT STRESSES

Procedures for calculating factors of safety against local and overall failure using values of stress obtained from finite element analyses are described in this chapter. The value of the factor of safety against overall failure determined from finite element stresses is very nearly equal to the value determined using the accurate methods of limit equilibrium analysis. For the most critical point within an embankment the factor of safety against local failure may be considerably smaller than the factor of safety against overall failure.

Local Failure

One method of assessing the stability of an embankment, using the results of nonlinear stress analyses of the type described in previous chapters is to examine the values of stress level or fraction of strength mobilized, (o1 - o 3)/(o1 - o 3)f. This procedure is very convenient because the mobilized strength value is calculated for each element as an intermediate step in the calculation of the value of tangent modulus.

Effect of the Value of Poisson's Ratio. To examine the effect of the value of Poisson's ratio on the development local failure within an emgankment, analyses of Otter Brook Dam were performed using a wide range of values of Poisson's ratio. The analyses were performed using the strength parameters and nonlinear modulus parameters determined from the results of the laboratory tests reported by 1inell and Shea (1960), together with values of Poisson's ratio which were assumed to be constant throughout the embankment.

Contours of the principal stress (o1 and o 3) within the embankment calculated using v = 0.475, v = 0.30, and the nonlinear Poisson's ratio values as described previously are shown in Figs. 39 and 40. The values of o 1 shown in Fig. 39 are very nearly the same for all three analyses, but the values of 03 shown in Fig. 40 vary significantly with the value of Poisson's ratio. The values calculated using v = 0.475'are slightly higher than those calculated using the nonlinear variation, and the values calculated using V = 0.30 are considerably lower. As shown previously (in Fig. 29) the values of Poisson's ratio for the nonlinear Poisson's ratio analysis ranged from 0.40 to 0.48, with an average of about 0.45. Therefore it may be seen that the values of o 3 within the embankment decrease as the value of Poisson's ratio decreases. In fact, if an analysis was performed using v = 0 the values of o 3 would be essentially equal to zero throughout the embankment. This is because a material with v = 0 does not tend to bulge horizontally when compressed vertically, and therefore no horizontal stresses would be generated by internal restraint resulting from interference between neighboring elements. A mater~al with a high value of Poisson's ratio, on the other hand, tends to bulge a large

95

FIG. 39

CONTOURS ARE IN TSF



\) •0.475

CONSTANT POISSON RATIO ~ • 0.30

3

7 __ _,

5

5

7

CONTOURS OF MAJOR PRINCIPAL STRESS IN OTTER BROOK DAM

USING A NONLINEAR MODULUS AND DIFFERENT POISSON RAllOS

96

FIG. 40

CONTOURS ARE IN TS F .



'0 •0.475

CONSTANT POISSON RATIO '0 • 0.30

2

3

CONTOURS OF MAJOR PRINCIPAL STRESS IN OTTER BROOK DAM '·

USING A NONLINEAR MODULUS AND DIFFERENT POISSON RATIOS

97

amount horizontally when compressed vertically, and therefore relatively large horizontal stresses develop within the embankment, Thus the magnitude of the horizon~al stress in an embankment as well as the amount of bulging deformation is determined primarily by the value of Poisson's ratio.

The value of Poisson's ratio also has an impor~ant effect on the values of stress level within embankments, as shown by the contours of stress level in Fig, 41, which were calculated using the values of c and ¢ determined from the laboratory tes~s on the Otter Brook fill material. Even though all of these values were calculated using the same s~rength parameters, the maximum values vary from slightly more than 50% for v = 0,475 to virtually 100% in the case of v = 0.30, The variations in these values may be attributed to the variations in the values of o 3 shown in Fig. 40: Because the values of o1 are nearly the same for all cases, the values of stress difference, (o 1 - o 3), increase as the values of o 3 decrease, Furthermore, the values of stress difference at failure, (o 1 - o 3)f, decrease as the values of o 3 decrease, as indicated by equation (7), As a result, when the value of o 1 is constant, the values of stress level, (o1 - o 3)/(o

1 - o 3)f, increase very rapidly as the value of o 3 decreases.

The reciprocals of the calculated values of stress levelJ which are (ol- o3)f/(ol- 03), may be interpreted as being values of factor of safety against local failure. The numerator of th1s factor of safe·cy, the stress difference at failure, is calculated assuming that a 3 is the same at failure as for the mobilized stress stat:e; as discussed subsequently, this is a different definition from that commonly employed in equilibrium slope stability analysis procedures. The minimum values of factor of safety against local failure for the conditions illustrated in Fig, 41

-vary -from -a -value -sli-ghtly l-ess -timn two -for v = 0. 475 to a value very close to unity for v = 0. 30, It may thus be concluded that the likelihood of local failure is very strongly influenced by the value of Poisson's ratio.

Effect of the Value of Bulk Modulus after Failure, In standard plane strain computer programs which employ a generalized form of Hooke's law, the relationship between stresses and strains is expressed as

(1-V) \) 0

{d (21) {o} E 0 (1-V) 0 (l+v)(l-2v)

0 0 (l-2v)/2

In simulating the occurrence of local failure, the value of E is set equal to a very small number with the result that the values of normal stress {o} do not change appreciably subsequently, no matter how great rnay be the changes in the values of strain {t.:} Thua, during placement of subsequent layers these elements cannot carry any appreciable increase in loadJ so the amount of load carried by their neighbors is increased, sometimes to an unreasonable degree. The progressive development of failure zones in

98

FIG. 41

t

v. 0.475

v•0.4

v. 0.3

CONTOURS OF MOBILIZED STRENGTH IN OTTER BROOK DAM USING A NONLINEAR MODULUS AND CONSTANT POISSON RATIO

99

Otter Brook Dam which were calculated using this approach with v = 0.2 are shown on the left in Fig. 42. It may be noted that nearly all the elements have failed by the time layer 6 has been placed.

The mode of post-failure behavior resulting when equation (21) is employed and the value of E is reduced differs from the behavior of real soils; even after failure real soils retain the ability to carry additional normal stress. If the normal stress does increase subsequently, the capacity of the soil to carry shear stress is also increased. Thus, even after failure real soils may carry additional shear stresses provided the bulk stress increases.

Clough and Woodward (1967) have suggested an alternate stress-strain formulation which is obtained by rewriting equation (21) in the form

(K+G) (K-G) 0

{a} = (K-G) (K+G) 0 {d (22)

0 0 G

in which K =bulk modulus= E/[2(l+v)(l-2v)] and G =shear modulus= E/[2(l+v)]. Studies conducted during the present investigation have shown that the values of K and G prior to failure may be calculated using values of Et and vt determined by means of equations (10) and (20) together with the nonlinear stress-strain parameters discussed previously. When this is done, the results are precisely the same as those obtained using the same values of Et and vt in equation (21). Post-failure behavior corresponding more closely to the behavior of real soils may be achieved by reducing the value of G _to 2aro aLter _failure, _b_ut _maintaining t:he vruue o£ X at the same value it had for the increment prior to failure. Using this procedure, failed elements are able to sustain additional bulk stress after failure.

Failure zones within Otter Brook Dam calculated using this modified procedure with v = 0.2, which are shown in the right in Fig. 42, are much smaller at the end of construction than those on the left which were calculated using the same value of Poisson's ratio, but reducing the value of Young's modulus after failure. After layer 5 had been placed, the failure zone as calculated by means of either procedure was the same, encompassing about half of the elements in the bottom row. Upon placement of the next layer, nearly all the elements failed when the value of Young's modulus in the failed elements was reduced, but only a few additional elements failed when the value of G was set equal to zero while the value of the bulk modulus was kept constant. It is believed that the latter procedure more nearly represents the behavior of real soils because there is no reason to believe that the bulk modulus of soil decreases markedly after failure. This procedure was therefore adopted for use in subsequent analyses of embankment behavior performed during this investigation.

Overall Stability

Finite element stresses may be used to calculate factors of safety against overall failure using the concepts of stress level discussed

100

~r--T,--.,r:----f LAYER 6

LAYER 7

LAYER 8

~a) ELASTIC MODULUS (E) SET

EQUAL TO 0.001 AFTER FAILURE """ .,,_ SHEAR FAILURE T -TENSION FAILURE

LAYER 6

LAYER 7

LAYER 8

b) SHEAR MODULUS (G) SET EQUAL TO ZERO AFTER FAILURE

FIG. 42 DEVELOPMENT OF FAILURE ZONES IN pTTERBROOKDAMWHENUSINGANONLINEARMODULUSANDCONSTANTPOISSON RATIO

previously or using the same definition of factor of safety employed in equilibrium analysis procedures.

Factor of Safety Based on Stress Level. A factor of safety with regard to overall stability based on stress level may be defined using the values of stress level for points on a continuous shear surface like the circular arc shown in Fig. 43. The contours of stress level shown in Fig. 43 were calculated using the nonlinear modulus and nonlinear Poisson's ratio described in Chapter 5. A weighted average value of stress level calculated for this circular arc was found to be 47.2%; the corresponding value of factor of safety based on stress level (FSL), defined as the reciprocal of this average value, is 2.12.

A computer program was written to calculate the value of FsL for any circular arc using the values of stress calculated from finite element analyses and punched on cards as input; a listing and user's guide for this program are given in Appendix E. Using this computer program it was possible to calculate values of FSL for many different arcs and to determine the most critical by repeated trial. For the stress level values shuwn in Fig. 43, the circular arc shown on the figure is the most critical one found, and the corresponding value of FsL is the lowest. As might be anticipated, the circular arc corresponding to the lowest value of FSL passes through the locations where the stress level is the highest,

Similar studies were also performed using the values of stress level calculated using various constant values of Poisson's ratio, together with nonlinear modulus values, which are shown in Fig. 41. The results of these calculations, which are summarized in Table 8, show that the value of FsL decreases rapidly with the value of Poisson's ratio employed in the

_atreas anal)[ses. As _di_scussed _pre'\donsly_, _this increase in stress level or reduction in the value of FsL may be attributed to the fact that the values of cr 3 throughout the embankment decrease markedly as the value of Poisson's ratio decreases.

Factor of Safety Based on Shear Stress. In equilibrium procedures of slope stability analysis, the factor of safety is defined with respect to the value of shear stress for a potential failure surface. This factor of safety may be expressed as

= l:(c+otancp) tiL l:TliL

(23)

in which the summations (l:) indicate that both the shear strength and the shear stress are summed over a number of increments of length (liL) along the shear surface. The value of a, the normal stress on the shear surface, is assumed to be the same in the equilibrium and failure states.

Values of normal stress and shear stress for a number of circular shear surfaces were calculated from the results of the finite element stress ana1yses discussed previously, using the computer program described in Appendix E. The variations of normal stress and shear stress for one

102

I I

I I

I I

I

' I

I I

I

,, ,..., I '

I ' I '

I ', I ',

I ' I ',

I ' I '

I ', I '

I ' I ', l

I ' i I ' I '

I ' I '

I ' I ~

AVERAGE STRESS LEVEL ALONG CIRCULAR ARC • 47.2 %

FIL • 2.12

FIG. 43 CONTOURS OF MOBILIZ EO STRENGTH IN OTTER BROOK DAM

USING A NONLINEAR MODULUS AND POISSON RATIO, AND MOST CRITICAL CIRCULAR ARC.

103

Table 8. Values of Factor of Safety Based on Stress Level

for Otter Brook Dam Cross-Section.

Value of Poisson's Ratio Value of FSL

0. 475 2.22

0.40 1.90

0.30 1. 36

104

circular shear surface are shown in Fig. 44. The surface for which the stresses are shown is the one shown in Fig. 43 and is for all practical purposes the critical surface for all analyses. It may be noted that the normal stress distributions in the upper part of Fig. 44 are nearly identical for all values of Poisson's ratio. Although the shear stress distributions differ appreciably, their average values are nearly exactly the same, as dictated by requirements of moment equilibrium of the mass bounded by the circular surface.

The values of FT corresponding to various values of Poisson's ratio are shown in Table 9, together with values of FT for the same critical circular arc which were calculated using the Ordinary Method of Slices and Bishop's Modified Procedure. It may be noted that the values of FT calculated by finite element analyses are influenced only a small amount by the value of Poisson's ratio employed in the analysis, as would be expected from the high degree of similarity in the normal stress distributions shown in Fig. 44. It may also be noted that the values of FT calculated using finite element stresses are only slightly greater than those calculated using Bishop's Modified Method, the difference varying from 2% to 8% depending on the value of Poisson's ratio used in the finite element analyses. Wright (1969) has shown that equilibrium analysis procedures which satisfy all conditions of equilibrium* give values of FT which are very nearly the same as those calculated by Bishop's Modified Method for homogeneous slopes, and Bishop's Modified Method may thus be considered representative of the best equilibrium procedures for Otter Brook Dam, It may be concluded, therefore, that values of factor of safety calculated using finite element stresses are for practical purposes the same as those calculated using the best equilibrium analysis procedures, provided the factor of safety is defined in the same way in both cases,

Comparison of Factors of Safety based on Stress Level and Shear Stress. The definition of factor of safety based on stress level differs from the definition of factor of safety based on shear stress in two ways:

(1) The shear stress compared to the shear strength in the shear stress factor of safety is referred to a single continuous surface--the shear surface. The shear stresses considered in the factor of safety based on stress level are the maximum shear stresses at each point along a continuous surface, disregarding the orientation of the surface, and are always greater than or equal to the shear stresses on the shear surface. For this reason the stress level factor of safety tends to be smaller than the shear stress factor of safety.

(2) The shear strength employed in the shear stress factor of safety is calculated assuming that the normal stress on the failure plane (0) is the same in the mobilized stress state and the failure stress state. On the other hand, the shear strength employed in the stress level factor of safety is calculated assuming that the minor principal stress (0 3) is the

*Bishop's Modified Method does not satisfy horizontal equilibrium, 'and the Ordinary Method of Slices satisfies neither horizontal nor vertical equilibrium.

105

en en w .... 0:1.1. ._en en._ -~cr a 2: b 0: 0 z

e~--------~--------~----------T---------~

0: .... "2r---------~--------~----------T---------~ C(I.L. wen % ... en-

E 0,., w .. Nen -en ~w ma:: 0 ... :ecn

_J w > w _J

en (/)

1.&1 a: ... (/)

0.5

0 0

TOE

I ____ ! ... ------~ ... -o# ...

,,." " "

100 200 300 400 CREST

DISTANCE ALONG CIRCULAR ARC(FT.)

---NONLINEAR MODULUS, NONLINEAR POISSON RATIO ·---NONLINEAR MODULUS, CONSTANT POISSON RATIO N•0.475) ------NONLINEAR MODULUS p CONSTANT POISSON RATIO (\) • 0.30)

FIG. 44 VARIATION OF STRESSES ALONG CRITICAL ARC IN OTTER BROOtc DAM·

106

Table 9. Values of Factor of Safety Based on Shear

Stress for Otter Brook Dam Cross-Section.

Analysis Procedure Value of FT

Ordinary Method of Slices 1.85

Bishop's Modified Method 1.93

Finite Element, \) = 0.30 1.97



107

same in the mobilized stress state and the failure stress state. For this reason the stress level factor of safety tends to be larger than the shear stress factor of safety (for values greater than unity). If it is assumed that the shear surface is oriented at the statically correct angle at each point, the stress level factor of safety may be expressed in terms of the shear stress factor of safety by means of the following equation, which is derived in Appendix B:

(24)

As shown in Fig. 45, the difference between the values of F51 and F increases with increasing values of ¢; the values are identical for ¢ = 0.

Because of these two differences in definition, the stress level factor of safety might be either greater or less than the shear stress factor of safety. The values calculated for the Otter Brook Dam crosssection are plotted against the corresponding value of Poisson's ratio in Fig. 46, where it may be noted that the value of F51 exceeds the value of FL at high values of Poisson's ratio and the reverse is true at low values of Poisson's ratio. The values shown for the finite element analysis performed using nonlinear Poisson's ratio are plotted at the average value (0.45). It is interesting to note that the values of Fs1 and FL for this particular analysis are related very nearly as indicated by equation (24) indicating that the assumptions on which this equation is based--that the circular arc is oriented at the statically correct angle throughout its length--is very nearly satisfied for this particular analysis. For smailer, constant values of Poisson's ratio the circular arc orientations differ greatly from the_static_ally correct orienta~ions and the value of FSL is smaller than FL for small values of Poisson's ratio.

108

8

7

6 .J

IJ.."'

.. >-1-w !5 IJ.. < en IJ.. 0 a: 0 4 1-0

~ ..J w > w 3 ..J

en Cl) 1&.1 a: 1-(It

2

0

SHEAR STRESS FACTOR OF SAFETY , F 't

FIG. 45 COMPARISON OF DIFFERENT FACTORS OF SAFETY

109

>-... lLI u.. < fJ)

LL 0

a: .... 0 .... 0 .....

u < LL

3--------~--~--~----------------~------~

2

FAILURE

0 0.1

FIG. 46

FINITE ELEMENT SOLUTIONS F. (NONLINEAR Et)

-(~

~\_ - - ---l J----~-.-1'1"""',.....~~ (CONSTANT v)

/ /

0.2 0.3

PO I S S 0 N R AT I 0, v 0.4

COMPARISON OF FACTORS OF SAFETY FOR

OTTER BROOK DAM CROSS SECTION

0.5

CHAPTER 8

FINITE ELEMENT ANALYSIS OF ZONED EMBANKMENTS

To examine the effectiveness of the finite element analysis procedures described previously for analyses of zoned embankments, analyses were performed for two hypothetical zoned dams. Both dams analyzed had the same cross section, but they differed with regard to the stress-strain characteristics of the core material. The analyses thus provide a means of examining the behavior of embankments having comparatively soft and stiff central cores.

Finite Element Mesh

The cross section analyzed and the finite element mesh employed in the analysis are shown in Fig. 47. The cross section resembles that of Oroville Dam in California in a general way; it is the same height and the slope inclination is approximately the same as the upstream and downstream slopes of Oroville Dam. However, Oroville Dam has an inclined core whereas the hypothetical embankments have symmet~ical sections and centrally located cores. Because the cross sect:ion was symmetrical, it was only necessary to analyze one-half of the section as shown in Fig. 47.

Material Properties

The values of the unit weights and stress-strain parameters employed in the analyses are listed 1n Taoie IO. The same slielT and- transition properties were employed in both of the analyses described subsequently, but the core properties were different. In one analysis the property values listed under "Soft Core" were used, and those listed under "Stiff Core" were used in the other analysis.

The same value of unit weight (150 lb/ft 3) was used for all materials.

The values of the stress-strain parameters for the shell material were the values determined from drained triaxial tests conducted by Marachi (1969) on the Oroville Dam shell material. The values for the transition zone are the same except for the modulus number, which is about 10% lower than for the shell. The parameters shown for the soft core are those determined from the results of unconsolidated-undrained triaxial tests on the Oroville Dam core material by the California Department of Water Resources (1969). The values for the stiff core are the same with the exception of the modulus number, which is nearly 10 times as large as that for the soft core. The values used in the analyses were thus selected to represent drained behavior for the shell and transition zones and undrained behavior for the core.

The values of tangent modulus and tangent Poisson's ratio calculated at the end of construction condition are shown in Figs. 48 and 49; in both figures the values calculat:ed for the stiff core embankment are shown in

111

.... .... N

cROSS -SECTION

SHELL

l 25' -II-

,, I I

I I I I

~v~~ I ~I

I o I I ~I I ....

I OJ I w I ~ I ~

I ll: I u I ~ I

I I

770'

FINITE ELEMENT MESH

(175 ELEMENTS,

204 NODAL POINTS, 12 LAYERS) ~::.._---4---+--+-+-f--f'+HY+fti

FIG. 47 CROSS- SECTION AND FINITE ELEMENT MESH FOR EXAMPLE ZONED EMBANKMENT

Table 10. Unit Weights and Stress-Strain Parameters

Employed in Analyses of Zoned Embankments.

Values Employed in Analyses Soil Parameter Symbol

Shell Transition Soft Core Stiff Core

Unit Weight (lb/ft 3) y 150 150 150 150

Cohesion (T/ft 2) c 0 0 1.32 L32

Friction angle (degrees) cp 43.5 43.5 25.1 25.1

Modulus number K 3780 3350 345 3350

Modulus exponent n 0.19 0.19 Oo76 0.76

Failure ratio Rf 0. 76 o. 76 0.88 0.88

Poisson's G 0.43 0.43 o. 30 0 0 30

Ratio ) F 0.19 0.19 -0.05 -0.05

Parameters d 14.8 14.8 3.83 3.83

113

.CONTOURS ARE IN TS F

STIFF CORE

SOFT CORE

. FIG. 48 CONTOURS OF TANGENT MODULUS IN EXAMPLE ZONED EMBANKMENT

STIFF CORE

SOFT CORE

oAOO

FIG. 49 CONTOUR~ OF TANGENT POISSON RATIO IN EXAMPLE ZONED EMBANKMENT

the upper part of the figure and those for the soft core embankment are shown in the lower part. It may be noted that the final values of modulus at midheight of the embankment are considerably lower than those in the adjacent zones for both the soft core and the stiff c~re embankment. Near the top and bottom of the stiff core embankment, the modulus values are considerably higher, and are approximately the same as those in adjacent zones, whereas they are uniformly low within the soft core embankment. The low modulus values in the center of the stiff core embankment appear to result from the fact that the strength of this material is considerably lower than that of the adjacent zones as indicated by the values of ¢ listed in Table 10. Thus, although the initial tangent modulus values for this material are high, its tangent modulus values decrease rapidly with increasing values of shear stresso

The final values of Poisson's ratio for the two cases shown in Fig. 50 are nearly the same for the shell and the transition. The values for the stiff core, shown in the upper part of the figure, are slightly lower than those for the soft core. In both cases the values of Poisson's ratio in the core are more nearly constant than those in the transition and shell and are higher on the average. It may be noted that the values increase from bottom to top within the shell and transition, whereas they decrease from bottom to top within the core.

Modification of Analysis Procedure

The fact that the stress-strain parameters for the three embankment zones were different led to d~fficulties in the analyses which had not been encountered with homogeneous embankments" Because the stiffer of two adjacent materials does not sectie as much as the softer under the influence of its own weight when first placed, the softer material tends to "hang" on the stiffer material and for chis reason the stresses in the soft material are very low and those in the stiff material quite high. When the next layer is placed the soft material may fail because the confining pressure is low and the shear stress high.

This mode of behavior appeared to result from the fact that the layers employed in the analyses were quite thick. Analyses showed that if very thin layers were employed, the stress conditions within each thin layer immediately after placement corresponded closely to at-rest pressure conditions, and failure did not occur during placement of subsequent layers. Instead the strength of the material increased as the values of a

3 increased,

Because it was not feasible to employ such thin layers in analyses of high embankments, the procedure was adopted of assigning at-rest pressure conditions to each element immediately after placemento The vertical pressure, al, was set equal to the overburden pressure (yh) and the horizontal pressure, a 3 , was set equal to v/(1-v) times the overburden pressure. During the placement of subsequent layers, the additional stresses in the elements were calculated using standard procedures.

116

SETTLEMENTS ARE IN FEET (t+)

STIFF CORE

SOFT COR£

FIG. 50 CONTOURS OF SETTLEMENT IN EXAMPLE ZONED EMBANKMENT I

Results of Analyses

Displacements. Contours of the calculated values of settlement are shown in Fig. SO. It may be noted that the maximum settlement, which occurs near mid-height of the core in both cases, is nearly twice as large for the soft-core embankment as for the stiff-core embankment. The settlements within the shell are nearly the same in both cases. It is thus evident that the stiffness of the core has a large effect on the settlements in the central part of the dam, but not much influence on the settlements in the shell.

Contours of calculated horizontal movements are shown in Fig. 51. Within the shell the values are nearly the same in both cases, reaching a maximum of about one-half foot away from the centerline near the middle of the shell. The movements are uniformly upstream in the case of the stiff core embankment, but are directed toward the centerline near the center of the soft core embankment as shown in the lower part of Fig. 51. These movements towards the centerline are due to the fact that the softer core squeezes inward under the pressures exerted by the adjacent transition and shell.

Stresses. Contours of the major principal stress (o 1) within the dam are shown in Fig. 52. It may be noted that for both the stiff core and the soft core embankment there is some reduction in stress in the core. This effect is fairly small in the case of the stiff core dam but quite pronounced in soft core embankment. At the base of the soft core embankment, where the value of o1 in the transition exceeds 60 tons/ft 2

, the value of cr1 in the adjacent core is only about 35 tons/ft 2

, The scale of overburden _oressures shown in Fig. 52 shows that the values of cr. within the core are ~oughly 80% of the overburden pressure in the stiff c5re, but only about 60% of the overburden pressure in the soft core. Thus in both cases there is appreciable arching within the embankments, increasing the stresses in the shell and reducing them within the core.

Schober (1967) has reported similar variations in the vertical stresses within Gepatsch Dam in Austria, which were observed by means of stress meters placed in the dam during construction. The contours of vertical pressure determined from these measurements, which are shown in Fig. 53, indicate vertical pressures within the core which are roughly 70% of the overburden pressure.

Perhaps the most important aspect of arching of this type is that reduced stresses within the core would result in reduced pore pressures. For conditions where the major principal stress in the core amounts to only 60% or 80% of the overburden pressure, it would be expected that the pore pressures would be accordingly lower than might otherwise be anticipated.

The contours of minor principal stress (o 3) shown in Fig. 54 are quite similar in both cases and do not show any marked reduction due to arching. For both the soft core and the stiff core embankments the values of a

3 range from 25% to 35% of the overburden pressure.

118

DISPLACEMENTS ARE IN FEET (.L)

STIFF CORE

SOFT CORE

FIG. 51 CONTOURS OF HORIZONTAL DISPLACEMENT IN EXAMPLE ZONED EMBANKMENT

... N 0

CONTOURS ARE IN TSF

STIFF C,ORE

SOFT CORE

7h (TSF)

0

20

30

40

50

0

10

20

30

40

50

FIG. 52 CONTOURS OF MAXIMUM PRINCIPAL STRESS IN EXAMPLE ZONED EMBANKMENT

FIG. S3 CONTOURS OF VERTICAL SOIL fRESSURE MEASURED IN GEPATSCH DAM (AFTER SCHROBER 19 6 7 )

.... N N

Th (TSF)

CONTOURS ARE ~ TSF 0

10

20 I

S1'1FF CORE

8 30

'

~ 40

50

0

10

SOFT CORE 20

30

40

50

FIG. 54 CONTOURS OF MINIMUM PRINCIPAL STRESS IN EXAMPLE ZONED EMBANKMENT

Contours of maximum shear stress (Tmax) are shown in Fig. 55. It may be noted that the values of Tmax within the stiff core are nearly twice as large as those within the soft core, because the stiff core carries more axial load. The values of Tmax within the transition zone are somewhat larger for the soft core embankment, but the values within the shell are very nearly the same in both cases. Contours of calculated values of mobilized strength are shown in Fig. 56. It may be noted that the mobilized strength exceeds 90% within a considerable portion of the stiff core, whereas the largest value in the soft core is only slightly more than 60%. The large percentage of mobilized strength in the case of the stiff core appears to arise for the combination of high modulus and low strength; the stiff core tends to carry fairly high loads, and is thus subject to fairly high shear stresses, but its shear strength is fairly low. The percentage of mobilized strength within the shell and transition, however, is affected only a small amount by the stiffness of the core.

123

CONTOURS ARE IN TSF

STIFF CORE

SOFT CORE

Yh (TSF)

0

10

20

30

40

50

0

10

20

30

40

50

FIG. 55 CONTOURS OF MAXIMUM SHEAR STRESS IN EXAMPLE ZONED EMBANKMENT

STIFF CORE

SOFT CORE

FIG. 56 CONTOUR~ OF MOBILIZED STRENGTH IN EXAMPLE ZONED EMBANKMENT

CHAPTER 9

SUMMARY AND CONCLUSIONS

This study was undertaken to develop practical procedures for performing finite element analyses of the behavior of embankments during construction. The procedures developed employ nonlinear, stress-dependent soil stress-strain behavior in analyses which simulate the actual sequence of construction operations. To check the utility and effectiveness of the procedures developed, they were employed in analyses of Otter Brook Dam and two hypothetical zoned embankments.

Simulation of Construction Sequence

The amount of time and the cost of performing finite element analyses increases rapidly with the number of layers used to simulate the placement of embankment fill material. These studies resulted in an improvement in the procedure for calculating displacements, which is accomplished by eliminating initial displacements at the top of each layer resulting from the weight of the layer. They also led to the development of simple graphs, based on the column analogy developed by Clough and Duncan (1969), which may be used to determine the number of layers required for accurate analysis of embankment displacements.

Stress-Strain Behavior of Embankment Materials

The tangent modulus stress-strain relationship developed by Duncan and Chang (1970) has been extended to include volume change characteristics by means of a nonlinear, stress-dependent Poisson's ratio relationship. The values of the required stress-strain parameters were computed from the stress-strain-volume change data available in the literature, providing stress-strain information for a wide variety of soils ranging from cobbles to fat clay. A comprehensive series of triaxial tests was also performed, to determine the influence of dry density and water content on the stressstrain behavior of compacted Pittsburg sandy clay.

Analyses of Otter Brook Dam

The stresses and movements in Otter Brook Dam were analyzed using the finite element analysis procedures and stress-strain relationships developed during this study. The observed and calculated movements were found to be in excellent agreement. The analyses showed that the large amount of bulging deformation of the dam observed during construction may be attributed to the stress-strain characteristics of the compacted fill.

Additional analyses of Otter Brook Dam were also performed using linear elastic properties in both incremental and gravity turn-on procedures. By suitably selecting the values of Young's modulus and Poisson's

126

ratio employed in these analyses, it was possible to calculate displacements in agreement with those observed using these procedures as well. Without prior knowledge of the magnitudes of the displacements, selection of suitable modulus and Poisson's ratio values for use in linear analyses would require considerable judgment. Thus the principal advantage of the nonlinear analysis procedures developed during this study is tha~ values of the parameters required are determined through straightforward interpretation of laboratory test data. Furthermore, because only conventional triaxial tests with volume change measurements are requ~red, and the parameter values are determined using simple techniques, only a small amount of additional effort is involved in the use of these nonlinear procedures.

Evaluation of Embankment Stability

Procedures for analyzing the stability of embankments using finite element stresses were also studied during this investigation. The factor of safety against local failure may be defined as the reciprocal of the largest stress level or fraction of strength mobilized, This factor of safety may be either larg-=r or smaller than the factor of safety against overall slope failure,

Further studies showed that the factor of safety against overall slope failure may be conveniently calculated from finite element stresses by evaluating the shear and normal stresses along a c~rcular ar.::.,. When the factor of safety is calculated in a consistent manner, i's value is very close to that calculated by means of the best limit equilibrium analysis procedures,

A procedure was developed for slmulating_ soil behavior after failure which combines the formulation of the generalized Hooke 1 s law developed by Clough and Woodward (1967) with the procedures for representing nonlinear, stress-dependent stress-strain behavior described hereino Use of th~s procedure appears to result in post-failure behavior which is in better agreement with the behavior of actual soils"

Analysis of Zoned Embankments

In order to analyze stresses and displacements in adjacent zones of dissimilar materials, it was found to be necessary to assign at-rest pressures to newly placed elements. After incorporating this procedural modification, the finite element analysis procedures developed were applied to the analysis of two hypothetical zoned embankments having comparatively soft and stiff cores, These analyses showed that the settlements, horizontal displacements, and stresses within zoned embankments may be influenced appreciably by the relative stiffness of the core and shell materials. On the basis of the analyses conducted it appears to be feasible to use the finite element analysis procedures developed for purposes of calculating differential settlements and load transfer between adjacent embankment zones.

127

Conclusions

The studies conducted during this investigation have shown that an incremental finite element analysis procedure, coupled with the use of nonlinear, stress-dependent stress-strain behavior results in calculated behavior which is in good agreement with that observed, The procedure is well-suited for practical use because the required soil stress-strain parameters may be determined readily from the results of standard laboratory triaxial tests with volume change measurements. Based on the results of the analyses performed, it seems likely that these analysis procedures may be used to predict stresses and movements in embankments during construction, to help in selecting desirable instrument locations in embankments, and to help in interpreting the results of instrumentation studies.

Perhaps the greatest value of these procedures is in connection with instrumentation programs. If the instrumentation results are in agreement with the results of the analyses, it may be concluded that the analyses provide a reliable indication of the actual behavior, The results of the analyses may then be employed to determine information regarding locations where there were no instruments and regarding aspects of the behavior not measured by instruments.

128

LITERATURE CITED

ASCE-USCOLD Committee on Design Criteria for Large Dams (1967) Current United States Practice in the Design and Construction of Arch Dams, Embankment Dams and Concrete Gravity Dams, ASCE, 131 p.

Bechtel Corporation (1969) "Report on Soil Tests for the Proposed New Don Pedro Dam," San Francisco.

Bird, J. M. (1961) "Uncertainties in Earth Dam Design," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol, 87, No. SM3, June 1961, pp. 33-68.

Bishop, A. W. (1952) "The Stability of Earth Dams," Dissertation presented to the University of London in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

Bishop, A. W. (1954) "The Use of Pore Pressure Coefficients in Practice," Geotechnique, Vol, 4, No, 4, December, 195~, pp. 148-152.

Bishop, A. W. (1966) "The Strength of Soils as Engineering Materials," Geotechnique, Vol. 16, No. 2, June 1966, pp. 89-130.

Bishop, A. W., Webb, D. L, and Lewin, P. I. (1965) "Undisturbed Samples of London Clay from the Ashford Common Shaft: Strength-Effective Stress Relationships," Geotechnigue, Vol. 15, No. 1, March, 1965, pp. 1-31.

Bjerrum, L. and Simons, N. E, (1960) "Comparison of Shear Strength Characteristics of Normally Consolidated Clays," Research Conference on Shear Strength of Cohesive Soils, ASCE, Boulder, Colorado, pp. 711-726.

Bricknell, R. G. (1962) "Shear Stresses in Batters," New Zealand Engineering, Vol. 17, No. 1, January 15, 1962, pp. 17-21.

Brown, C. B. and Goodman, L. E. (1963) "Gravitational Stresses in Accreted Bodies," Proceedings, Royal Society of London, Series A, Vol. 276, No. 1367, December 17, 1963, pp. 571-576.

Brown, C. B. and King, I. P. (1966) "Automatic Embankment Analysis: Equi~ librium and Instability Conditions," Geotechnigue, Vol. 16, No. 3, September 1966, pp. 209-219.

Casagrande, A. (1940) "Seepage Through Dams," Contributions to Soil Mechanics, 1925-1940, Boston Society of Civil Engineers, pp. 295-336.

Casagrande, A. (1960) "An Unsolved Problem of Embankment Stability on Soft Ground," Proceedings, 1st Pan-American Conference on Soil Mechanics and Foundation Engineering, Vol, 2, Mexico City, pp. 721-746.

Casagrande, A. (1961) "Control of Seepage Through Foundations and Abutments of Dams," Geotechnigue, VoL 11, No.3, September 1961, pp. 159-181.

129

Casagrande, A. (1965) "Hohe Staudamme~" Mitteilungen des Institutes fur Grundbau und Bodenmechanik, Technische Hochschule, Vienna, No. 6, December, 1965, 32 p.

Casagrande, A., Hirschfeld, R. C, and Poulos, S. J. (1963) "Third Progress Report on Investigation of Stress-Deformation and Strength Characteristics of Compacted Clays," Soil Mechanics Series No. 70, Harvard University, November, 1963, 67 p.

Casagrande, A. and Wilson, S. D. (1953) "Effects of Stress History on the Strength of Clays," Soil Mechanics Series No. 43, Harvard University, June, 1953, 71 p.

Chan, C. K. and Duncan, J. M. (1967) "A New Device for Measuring Volume Changes and Pressures in Triaxial Tests on Soils," Materials Research and Standards, Vol. 7, No. 7, July, 1967, pp. 312-314,

Chang, C-Y (1969) "Finite Element Analyses of Soil Movements Caused by Deep Excavation and Dewatering," Dissertation presented to the University of California, Berkeley, in partial fulfillment of the requirements for the degree of Dor.tor of Philosophy.

Clough, G. W. (1969) "Finite Element Analyses of Soil-Structure Interaction in U-Frame Locks," Dissertation presented to the University of California, Berkeley, in partial fulfillment of the requirements for the degree of Doctor of Philosophy,

Clough, R. W. (1960) "The Finite Element Method in Plane Stress Analysis," -Rror.-eedi.ngs, -2nd -Cnnference _on -Elec_t:_r_unic ~Gomputation~ ASCE~ Pittsburgh~ 1969, pp. 345-377.

Clough, R. W. (1965) "The Finite Element Method in Structural Mechanics," Chapter 7 in Stress Analysis, ed. by 0. C. Zienkiewicz and G. S. Holister, John Wiley and Sons, New York.

Clough, R. W. and Woodward, R. J., III (1967) "Analysis of Embankment Stresses and Deformations," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 93, No. SM4, July, 1967, pp. 529-549.

Crawford, c. B. (1964) "Some Characteristics of Winnipeg Clay," Canadian Geotechnical Journal, Vol. 1, No. 4, November, 1964, pp. 227-235.

da Cruz, P. T. (1963) "Shear Strength Characteristics of Some Residual Compacted Clays," Proceedings, 2nd Pan-American Conference on Soil Mechanics and Foundation Engineering, Vol. 1, Rio de Janeiro, pp. 73-102.

Department of Water Resources (1965) "Oroville Dam--Key Unit of the State Water Project," State of California, August, 1965.

Department of Water Resources (1967) "Instrumentation Results, Oroville Dam," State of California, November 1, 1967,

130

Department of Water Resources (1969) "Report on Unconsolidated-Undrained Triaxial Shear Tests for the Core of Oroville Dam," State of California.

Duncan, J. M. and Chang, C-Y ( 19 70) "Nonlinear Analysis of Stress and Strain in Soils," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 96, No. SM5, September 1970.

Duncan, J. M. and Dunlop, P. (1969) "Slopes in Stiff-Fissured Clays and Shales, 11 Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 95, No. SM2, March, 1969, pp. 467-492.

Duncan, J. M., Monismith, C. L. and Wilson, E. L. (1968) "Finite Element Analyses of Pavements," Record 228--Soil Stresses and Pavement Element Analyses, Highway Research Board, pp. 18-33.

Duncan, J. M. and Seed, H. B. (1965) "The Effect of Anisotropy and Reorientation of Principal Stresses on the Shear Strength of Saturated Clay," Report No. TE-65-3, Soil Mechanics and Bituminous Materials Research Laboratory, University of California, Berkeley, California, November, 1965, 171 p.

Dunlop, P., Duncan, J. M. and Seed, H. B. (1968) "Finite Element Analyses of Slopes in Soil, 11 Report No. TE-68-3, Soil Mechanics and Bituminous Materials Research Laboratory, University of California, Berkeley, California, May, 1968, 232 p.

Durrenberger, R. W. (1967) Patterns on the Land, National Press Books, Palo Alto, California, p. 78.

Engineering News-Record (I968} 11 Muds tone Looks Strange But Makes a Record Dam," April 18, 1968, pp. 52-53.

Engineering News-Record (1968) 11 Russians Fire Off an Instant Rockfill Dam," May 30, 1968, pp. 24-25.

Engineering News-Record (1968) 11Tarbela Dam: Seven Years and 160 Million Cu. Yd. to Go, 11 November 7, 1968, pp. 64-66.

Felippa, C. A. (1966) "Refined Finite Element Analysis of Linear and Nonlinear Two Dimensional Structures," Dissertation presented to the University of California, Berkeley, in partial fulfillment of the requirements for the degree of Doctor of Philosophy,

Finn, W. D. L. (1967) "Static and Seismic Behavior of an Earth Dam," Canadian Geotechnical Journal, Vol, 4, No. 1, Februar~ 1967, pp. 28-37.

Finn, W. D. L. and Troitskii, A. P. (1968) 11 Computation of Stresses and Strains in Dams Made of Local Materials, Earth Slopes and their Foundations, by the Finite Element Method," Hydrotechnical Construction (GIDROTEKHNICHESKOE STROITEL 'STVO), No. 6, June, 1968, pp. 492-499, (Translated from Russian for ASCE by Consultants Bureau.)

131

Fox, A. J., Jr. (1968) "USSR Breaks Records: Earth and Arch Dams," Engineering News-Record, October 24, 1968, pp ,. 32-37.

Gamboa, J. and Benassini, A. (1967) "Behavior of Netzahualcoyotl Dam During Construction," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 93, No. SM4, July, 1967, pp. 211-229. Gianelli, W. R. (1969) "Oroville Dam and Edward Hyatt Powerplant," Civil Engineering, June, 1969, pp. 68-72.

Gibbs, H. J., Hilf, J .. W., Holtz, W. H. and Walker, F. C. (1960) "Shear Strength of Cohesive Soils," Research Conference on Shear Strength of Cohesive Soils, ASCE, Boulder, Colorado, 1969, pp, 33-162.

Girijavallabhan, C" V. and Reese, L. C. (1968) "Finite Element Method for Problems in Soil Mechanics," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 94, No. SM2, March, 1968, pp. 473-496.

Glossop, R. (1968) "The Rise of Geotechnology and Its Influence on Engineering Practice," Geotechnique, VoL 18, No. 2, June, 1968, pp. 107-150.

Goodman, L. E. and Brown, C. B. (1963) "Dead Load Stresses and the Instability of Slopes," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 8g:-N~SM3, May, 1963, pp. 103-134.

Gordon, B. B. (1968) "DWR Experience and Current Practice in Instrumentation," Lecture presented at short course on Recent Developments in Earth and Rockfill Dams, University of California, Berkeley, March, 1968.

Gordon, B. B. and Miller, R. K. (1966) "Control of Earth and Rockfill for Oroville Dam;1i Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 92, No. SM3, May 1966, pp. 1-23.

Gruner, E. (1967) "The Mechanism of Dam Failure," Proceedings,. 9th Congress on Large Dams, Vol. 3, Istanbul, pp. 197-206.

Hall, E. B. and Gordon, B. B" (1963) "Triaxial Testing with Large-Scale High Pressure Equipment," STP 361 - Laboratory Shear Testing of Soils, ASTM, pp. 315-328.

Hathaway, G. A, (1958) "Dams--their Effect on Some Ancient Civilizations," Civil Engineering, January, 1958, pp. 58-63.

Hirschfeld, R, C. and Poulos, S. J. (1963) "High-Pressure Triaxial Tests on a Compacted Sand and an Undisturbed Silt," STP 361 - Laboratory Shear Testing of Soils, ASTM, 1963, pp. 329-339.

Hughes, J. M. 0. (1969) "Culvert Elongations in Fills Founded on Soft Clays," Canadian Geotechnical Journal, VoL 6, No. 2, May, 1969, pp. 111-117.

Insley, A. E. and Hillis, S. F. (1965) "Triaxial Shear Characteristics of a Compacted Glacial Till Under Unusually High Confining Pressures," Proceedings, 6th IntL Conf. on Soil Mech. and Found. Eng., Vol. 1, Montreal, pp. 244-248.

132

Janbu, N, (1963) "Soil Compressibility as Determined by Oedometer and Triaxial Tests," Proceedings, European Conference on Soil Mechanics and Foundation Engineering, Vol. 1, Weisbaden, pp. 19-25.

Kaufman, R. I. and Weaver, R. J. (196 7) "Stability of Atchafalaya Levees," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 93, No. SM4, July, 1967, pp. 157-176.

King, I. P. (1964) "Finite Element Analysis of Two-Dimensional TimeDependent Stress Problems," Dissertation presented to the University of California, Berkeley, in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

Kondner, R. L. (1963) "Hyperbolic Stress-Strain Response: Cohesive Soils," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 89, No. SMl, February, 1963, pp. 115-143.

Kondner, R. L. and Horner, J. M. (1965) "Triaxial Compression of a Cohesive Soil with Effective Octahedral Stress Control," Canadian Geotechnical Journal, Vol. 2, No. 1, February, 1965, pp. 40-52.

Kondner, R. L. and Zelasko, J. S. (1963) "A Hyperbolic Stress-Strain Formulation for Sands," Proceedings, 2nd Pan-American Conference on Soil Mechanics and Foundation Engineering, Vol. 1, Rio de Janeiro, pp. 289-324.

Kondner, R. L. and Zelasko, J. S. (1963) "Void Ratio Effects on the Hyperbolic Stress-Strain Response of a Sand," STP 361 - Laboratory Shear Testing of Soils, ASTM, 1963, pp. 250-257.

Ladd, C. C. and Lambe, T. W. (1963) "The Strength of 'Undisturbed' Clay Determined from Undrained Tests," STP 361 -Laboratory Shear Testing of Soils, ASTM, 1963, pp. 342-371.

Lambe, T. W. (1963) "An Earth Dam for the Storage of Fuel Oil," Proceedings, 2nd Pan-American Conference on Soil Mechanics and Foundation Engineering, Vol. 2, Rio de Janeiro, pp. 257-308.

Lee, K. L. (1965) "Triaxial Compressive Strength of Saturated Sand Under Seismic Loading Conditions," Dissertation presented to the University of California, Berkeley, in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

Linell, K. A. and Shea, H. F. (1960) "Strength and Deformation Characteristics of Various Glacial Tills in New England," Research Conference on Shear Strength of Cohesive Soils, ASCE, Boulder, Colorado, pp. 275-314.

Lowe, J. W., III (1967) "Stability Analysis of Embankments," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 93, No. SM4, July, 1967, pp. 1-33.

Lowe, J. W., III and Johnson, T. C. (1960) "Use of Back Pressure to Increase Degree of Saturation of Triaxial Test Specimens," Research Conference on Shear Strength of Cohesive Soils, ASCE, Boulder, Colorado, pp. 819-836.

133

Lowe, J. W., III and Karafaith, L. (1960) "Effect of Anisotropic Consolidation on the Undrained Shear Strength of Compacted Clays," Research Conference on Shear Strength of Cohesive Soils, ASCE, Boulder, Colorado, pp. 837-858.

Marachi, N. (1969) "Strength and Deformation Characteristics of Rockfill Materials," Dissertation presented to the University of California, Berkeley, in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

Marsal, R. J. (1963) "Triaxial Apparatus for Testing Rockfill Samples," Proceedings, 2nd Pan-American Conference on Soil Mechanics and Foundation Engineering, Rio de Janeiro, 1963, pp. 99-122.

Marsal, R. J. (1967) "Large Scale Testing of Rockfill Materials," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 93, No. SM2, Marc~ 1967, pp. 27-43.

Marsal, R. J., Gomez, E. M., Nunez, A., Cuellar, R. and Ramos, R. M. (1965) "Research on the Behavior of Granular Materials and Rockfill Samples," Comision Federal de Electricidad, Mexico, D. F., February, 1965, 76 p.

Marsal, R. J. and Ramirez de Arellano, L. ( 196 7) "Performance of El Infiernillo Dam, 1963-1966," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 93, No, SM4, July, 1967, pp. 265-298.

Moretto, 0., Bolognesi, A. J. L,, Lopez, A. 0. and Nunez, E. (1963), Propiedades y Comportamiento de un Suelo Limoso de Baja Plasticidad," Proceedings, 2nd Pan-American Conference on Soil Mechanics and Foundation Engineering, Vol. 2, Rio de Janeiro, pp. 1.31.-1.46.

Morgan, G. C. and Harris, M. C. (1967) "Portage Mountain Dam II. Materials," Canadian Geotechnical Journal, Vol. 4, No. 2, May, pp. 142-166.

Polivka, M., Pirtz, D. and Adams, R. F. (1963) "Studies of Creep in Mass Concrete," SP-6 - Symposium on Mass Concrete, ACI, 1963, pp. 257-285.

Richards, R., Jr. and Schmid, W. E. (1968) "Body-Force Stresses in Gravity Structures," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 94, No. SMl, January, 1968, pp. 205-229.

Scherrer, H. U. and Britto, P. A. (1963) "Equipamento e Technica Referentes Aos Ensaios Triaxiais Conjunto de Resultados Obtidos com Amostras de Solos Residuais," Proceedings, 2nd Pan-American Conference on Soil Mechanics and Foundation Engineering, Vol. 1, Rio de Janeiro, pp. 163-179.

Schober, W. (196 7) "Behavior of the Gepatsch Rockfill Dam," Proceedings, 9th Congress on Large Dams, Vol. 3, Istanbul, 1967, pp. 677-699.

Seed, H. B. and Chan, C. K. (1961) "Compacted Clays: Structure and Strength Characteristics," Transactions, ASCE, Vol. 126, 1961, pp. 1344-1385.

134

Seed, H. B., Mitchell, J. K. and Chan, C. K. (1960) "The Strength of Compacted Cohesive Soils," Research Conference on Shear Strength of Cohesive Soils, ASCE, Boulder, Colorado, pp. 877-964.

Sherif, M. A. (1967) "A Method for Determining Effective Stress Envelope from Consolidated-Undrained Tests Without Pore Pressure Measurements," Proceedings, 3rd Pan-American Conference on Soil Mechanics and Foundation Engineering, Vol. 1, Caracas, pp. 29-41.

Skempton, A. W. (1954) "The Pore Pressure Coefficients A and B," Geotechnigue, Vol. 4, No. 4, December, 1954, pp. 143-147.

Squier, L. R. (1967) "A Study of Deformations in Selected Rockfill and Earth Dams," Dissertation presented to the University of Illinois, Urbana, in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

Taylor, R. L. and Brown, C. B. (1967) "Darcy Flow Solutions with a Free Surface," Journal of the Hydraulics Division, ASCE, Vol. 93, No. HY2, March, 1967, pp. 25-33.

Terzaghi, K. (1943) Theoretical Soil Mechanics, John Wiley and Sons, New York, pp. 406-409.

Turner, M. J., Clough, R. W., Martin, H. C. and Topp. L. J. (1956) "Stiffness and Deflection Analysis of Complex Structures," Journal of Aeronautical Sciences, Vol. 23, No. 9, Sept. 1956, pp. 805-823.

Tuthill, L, H., Adams, R. E. and_ Mi_tch~- n_._ R.- (l9b3} "Mass- Concrete- for Oroville Dam," SP-6 -Symposium on Mass Concrete, ACI, pp. 179-197.

Wagner, A. A. (1957) "The Use of the Unified Soil Classification System by the Bureau of Reclamation," Proceedings, 4th International Conference on Soil Mechanics and Foundation Engineering, Vol. 1, London, pp. 125-134.

Wilson, E. L. (1963) "Finite Element Analysis of Two-Dimensional Structures," Dissertation presented to the University of California, Berkeley, in partial fulfillment of the requirements for the degree of Doctor of Engineering.

Wilson, S. D. (1964) "Suggested Method of Test for Moisture-Density Relations of Soils Using Harvard Compaction Apparatus," Procedures for Testing Soils, 4th ed., ASTM, 1964, pp. 160-162.

Wilson, S. D. (1968) Closure to "Investigation of Embankment Performance," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 94, No. SM5, September, 1968, pp. 1184-1185.

Wilson, S. D. and Marano, D. (1968) "Performance of Muddy Run Embankment," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 94, No. SM4, July, 1968, pp. 859-881.

135

Wilson, S. D. and Squier, L. R. (1969) "Earth and Rockfill Dams," Proceedings, 7th International Conference on Soil Mechanics and Foundation Engineering, State of the Art Volume, Mexico City, pp. 137-223.

Woodward, R. J., III (1966) "Analysis of Embankment Stresses and Deformations," Dissertation presented to the University of California, Berkeley, in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

Wright, s. G. (1969) "A Study of Slope Stability and the Undrained Shear Strength of Clay Shales," Dissertation presented to the University of California, Berkeley, in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

Zienkiewicz, 0. C. and Cheung, Y. K. (1967) The Finite Element Method in Structural and Continuum Mechanics, McGraw-Hill, London.

136

APPENDIX A

LABORATORY TESTING PROCEDURES AND RESULTS

A comprehensive series of unconsolidated-undrained triaxial shear tests was conducted on a compacted, partially saturated, sandy clay to measure its stress-strain-volume change behavior during shear under undrained conditions. These tests were conducted because it was shown in Chapters 3 and 4 that there was virtually no data available on the stress-strain-volume change behavior of compacted, partially saturated soils. Therefore a need was evident to establish the range of behavior that might be expected for these soils.

Soil Classification

A moderately expansive, light brown, inorganic sandy clay (CL) from Pittsburg, California was chosen for this study because it is quite similar in gradation and plasticity to many soils presently being utilized for the compacted fill in homogeneous embankments and for the impervious cores in zoned embankments. Seed, Mitchell and Chan (1960) and Seed and Chan (1961) have presented classification data for this soil which is sununarized below

Percent Sand 33% Percent Silt 43% Percent Clay 24% Liquid Limit 35/o Plastic Limit = 19%

Modified AASHO Maximum dry density Modified AASHO Optimum water content =

ll8.9 psf 13.5%

A mineralogical analysis of the clay fraction disclosed that the clay minerals present were predominantly montmorillonoid, with a trace of kaolinite.

Specimen Preparation

All of the specimens used in the tests were prepared from previously processed bulk supplies of soil which were air-dried, crushed and sieved through a No. 30 sieve. For each test series, a sufficient amount of airdried soil was used to give three specimens plus an additional amount for water content determination and trimming waste. To obtain the desired water content, sufficient water was added from a spray bottle during mixing in a rotating pan mixer. The air-dried soil was mixed during the addition of water and for a minimum of ten minutes after the addition of all the water. The moist soil was then removed from the mixer, placed in plastic bags, sealed and cured in a moist room for a minimum of 24 hours prior to compaction. Before compaction, the mixed, cured soil was again thoroughly stirred with a steel ladle.

137

Specimen Compaction

Specimens were compacted in 1.4 inch diameter Harvard miniature compaction molds using the general procedure described by Wilson (1964). A tamper similar to the standard spring-activated Harvard miniature tamper was used, the only difference being that pressure was applied by a constant air pressure supply instead of a spring. Seven layers of soil with fifteen tamps per layer were used for the preparation of all specimens. Compactive efforts were varied by using 12.5, 25, and 50 pounds of force behind the compaction foot, Three specimens were prepared at each water content and compactive effort for a total of 36 specimens. The results indicated (Fig. 57) that the highest compactive effort yielded a compaction curve slightly above that for the modified AASHO curve presented by Seed and Chan (1961), After compaction, the specimens were extruded with a jack and carefully trimmed to insure squareness and smoothness; the final heights therefore ranged from about 3.2 to 3.5 inches.

Specimen Storage

Immediately after trimming, each specimen was placed between a solid lucite cap and base and enclosed within two Trojan prophylactic membranes separated by a thin layer of silicone grease. The membranes were sealed to the cap and base by four rubber "O" rings, and the specimens were stored under water in a moist room for a minimum of seven days to cure before testing.

Unconsolidated-Undrained (UU) Triaxial Shear Test Procedure

After storage, specimens were placed in a triaxial cell developed at the-University o-f -ca:rnornia, 13erke-rey. (See Seed, Mitchell and Chan (1960) for details of the triaxial cell.) The desired chamber pressure (values used were 1, 3 and 6 tons/ft 2

) was applied by air pressure on the chamber water and the specimen was tested 30 minutes later (see section on calibrations for further details) in a Wykeham-Farrance strain-controlled testing machine at a strain-rate of 0.030 inches per minute. At this strain rate, 20% strain could be achieved in about 20-25 minutes. During the test, measurements were taken of the axial load with a load cell, axial deformation with a dial gage and change in volume of chamber water with a volume change device developed at the University of California, Berkeley. (See Chan and Duncan (1967) for further details.) With this equipment, values could be determined to accuracies of 0.01 kilograms for the axial load, 0.0001 inches for the axial deformation and 0.01 cubic centimeters for the change in chamber volume. After shearing, the triaxial cell was dismantled and the water content of the entire specimen was determined,

Equipment Calibration

Because specimen volumetric changes were evaluated by measuring the flow of water into or out of the triaxial cell chamber, it was imperative to evaluate all of the deformations which could occur during application of the chamber pressure and during shearing. For all of the tests, the

138

125

I

' KNEADING COMPACTION

\ ' 7 LAYERS, 15 TAMPS I LAYER

' \

' 120 \ ' 0 12.5 LB. TAMPS \ ' \ ' 6 25 LB. TAMPS

\ ' 0 50 LB. TAMPS \ ' ' ' \ ' • MODIFIED AASHO \ ' ..... , . MAXI MUM DENSITY

u ' Q. 115 \ ' \

' ' >- \;/ ' ' .... ' - ' ' (I)

' ' z w ' ' 0 \ ' ' \ ' ' ..... >- 110 \ ' ' ""' cr \

' ' \0 0 \ '0 ' \ \ ' ' '1\ ' ' ' '~

' ' , ..... ' ' d' '"'o \d' ' ...

,o. 105 \: . ,c;

""· '<Po ' '(a ' 0 ' Vi" ' ,.!. ' ' • ' 100

8 10 12 14 16 IB 20 22 24

WATER CONTENT (%)

FIG. 57 MOISTURE- DENSITY RELATIONSHIPS FOR COMPACTED PITTSBURG SANDY CLAY

same equipment was used to insure uniformity and consistency of all of the calibrations. In addition, each calibration was checked several times to insure that it was reproducible.

To insure that no air bubbles were entrapped in the chamber water, it was necessary to use de-aired water and to let it trickle slowly into the cell through the cell base until the entrance nozzle was completely submerged. After nozzle submergence, the flow rate could be increased to full flow through a 1/8" 0. D. Saran (trade name) tubing until the chamber was completely filled,. This procedure took about 45 minutes to complete but it did not permit any air entrapment in the chamber water.

To evaluate the volumetric creep in the chamber and volume change device, a series of tests was conducted under varying confining pressures, with dummy specimens and with no specimens, to determine the amount and rate of creep. It was found that the creep rate was non-linear during the first 20 to 30 minutes after application of the confining pressure, but within the range of 30 to 80 minutes after the application of the confining pressure, the creep rate was approximately constant. Therefore, all specimens were allowed to sit for 30 minutes after application of the confining pressure" At that time, the specimen was sheared and a correction for chamber volume increase was applied to the measured readings. These values are shown below

Confining Pressure (tsf)

1 3 6

Chamber Volume Increase (cm 3 /min)

0.00225 0.00467 0.00730

Flexibility of the test equipment under load was investigated by setting an invar steel specimen in the triaxial cell and loading it at a constant strain rate. It was found that the axial compression was 0.00007 inches/kg and the chamber volume increase was 0.00025 cm 3 /kg.

Piston friction and volume of chamber water decrease from piston movement into the cell were evaluated by measuring the chamber volume change and load cell response during loading. These values were checked at chamber pressures ranging from 0 to 6 tsf. It was found that the piston friction was 0.31 kg and that the decrease in volume of chamber water was 0.71 cm 3 for one inch of piston movement into the cell.

To determine the validity of these calibrations, checks were made by conducting UU tests on a specimen of slightly porous rubber, 1.383 inches in diameter by 2.840 inches high. Pure, non-porous rubber should have no volume change during loading and, therefore, should have a Poisson's ratio equal to 0.5. Thus, these tests provided a means of checking the calibrations. Results of the tests showed that Poisson's ratio varied from about 0.496 at 1% axial strain to about 0.499 at 10% axial strain. The magnitude of the discrepancy (less than 1%) indicated that the calibration factors could be confidently applied to the data obtained in subsequent tests.

140

Unconsolidated-Undrained (UU) Triaxial Shear Test Results

A total of 30 UU triaxial shear tests were conducted on compacted, partially saturated, Pittsburg sandy clay to determine the variation of the stress-deformation characteristics during shear with compactive effort, water content and confining pressure. The results of these tests are plotted in Figs. 58, 59, and 60 which show the corrected values of deviator stress and volumetric strain versus axial strain for the conditions investigated. It should be noted that the results from three of the tests are not shown because leakage developed during these tests and therefore the results were not valid.

These data were subsequently re-plotted in the form of the transformed hyperbolae discussed in Chapters 3 and 4. From these plots the nonlinear elastic parameters, which show the variation of modulus and Poisson's ratio with confining pressure and stress level, were determined. The variations of these parameters, as well as the strength parameters, were then contoured over the compaction curve shown in Fig. 57. These plots are presented and discussed in detail in Chapters 3 and 4.

141

... 10 b

b- I . en en 6 ..., a:-.... ~ en !:4 a: 0 .... c 2 > ..., 0

.. 8 ... . z ·2 Ci-a: .. .... -en ·4 _j 0 >

... b I

b-.

en

> ..., 0

0 ,. ... 0

z Ci-a: ... , .... _ .,. _j ·Z 0 > 0

FIG. 58

10 ... b tT. • I TSF b- 8 .

CT1 a 3 TSF en 6 en_ .... ~~.

~I!! en- 4 Oj•ITSF Ill: 0 ....

2 IIIUADIIU COMPACTION c > 7 LAYERS, IS •IZ.SLI. TAIIPSILAYU ...,

y~ • 101.1 PC', WI 14,) 'ro 0

25 0 15 20 25 ..r

0 ~ c-c: .. ·2 ...._ en .J ·4 0 > 0 5 10 15 20 25

25 AXIAL STRAIN, ta Pl. I

v.'. i ts•j 2.5 •6TSF

b"' •3 TSF I • I TSF b- 2.0 .

en en _1.5 -cr5-·-:-w - -w II. a: en .... .... en - 1.0 m:

KIIUDIIIG COIIPACTIOII ~ UUDINI COiliPACTIOII !! 0.5 7 UYEIIS, 15•12.5 Ll UIIPS/LiTEit > ? LlYIEitS, 15•12.5 Ll. TAIIPSILAY[II

5

5

y., •101.0 PC,, w I ..... ,. ..., r. •lor.• ,c,, w • ..... ,. c

10 15 20 25 0 5 10 15 20 25

..r 0 . z -o.5 CTs• I TSF c-G:af

CT I I TSF ...._ en -1.0 as·• .J 10 15 20 25 0

AXIAL STRAIN, '• > 0 5 10 15 20 25 ,.,., AXIAL STRAIN, fa c.,. t

UU TRAXIAL SHEAR TEST RESULTS FOR COMPACTED PITTSBURG SANDY CLAY, LOW COMPACTIVE EFFORT

142

20 10 5 ... b ... ..

b b u 5•3TSF 16 I 8 4

b-Us•6 TSF

b-Us •3 TSF

b-

.n 12 en 6 en 3 en_ U:s•ITSF en en- ..... -........ Us • 3TSF ..... ... a:::"-a::: en a::: en t-en ......... ..........

en- 2 en- 4 en.t: 2 a::: U3 •I TSF a::: a::: 0 ~ 0 ..... ..... ~ 4 KNEADING COMPACTION ~ 2 ltNE A DING COMPACTION ~ KNEADING COMPACTION > 7 LAYERS, 15- 25 LB. TAMPS/LAYER > 7 LAYERS, 15-25 Lll. TAMPS/LAYER ·> 7 LAYERS, 15-25 Lll. TAMPS/LAYER .....

y,•llt.7 PCF, W•11.5% ..... w. 14.5 .,. .....

0 0 0 y, • Ill. t PCF, W •17.1-,. .... 0 5 10 15 20 25 ~ 0 5 10 15 20 25 0 5 10 15 20 25

""' i-2 ... 0 ... 0 ... ... ... ...

0 z z c --1 z c;; c -0.5 a:::.., a:::~ ~--2

..... _ en ..... -

en U:s•lTSF en _j -2 _j

...J 0 0 -4 > 0 > 0 5 10 15 20 25 > 0 5 10 15 20 25

0 5 10 15 20 25 AXIAL STRAIN, f:a AXIAL STRAIN, Ea AXIAL STRAIN, fa (\I (\)

I '1.1

FIG. 59 U U TRIAXIAL SHEAR TEST RESULTS FOR COMPACTED PITTSBURG SANDY CLAY, MEDIUM COMPACTIVE EFFORT

20 20 10 .. b,. b"" b

IG I IG 8 b- b-

,_,-------.!!)• 6 TSF b--

12 fiJ 12 '~TSF UJ 6 UJ UJ..,. UJ

UJ- W~a,. w-w&a. a: &a. a:UJ a:UJ

~Oj•ITSF 1-UJ 0"3• I TSF

~=. ......

UJ !:: 4 UJ- c 0"1•1 TSF a: a:

a: 0 0

0 ... ... ~ 4 COM PACTION c ~ ltiiUDII:G COMPACTION ~ 2 lllllUOIIIG COII:PACTIOII

> > f LIYERS, 15 ·50 LB. TAMPSILAYEJI > 7 LAYERS, IS• SO Ll. TAMPSJUYER .... w w 0 Y •liS. 3 PCF, W •11.7 "4 Q r •• 111 ... "'. w ....... .,. a .... 0 0 10 20 25 ~ 0 5 10 15 20 25 •

~z ... ··r; ,., .. ... ... ... ~CT1•1TSF 0 -:! O"J•3TSf ~ z 0 c-o - c-c- a: .. a:,.

a: .. ..... _

..... --1 ... _ UJ O"J•6TSF UJ

UJ -2 ~ ~2 _j CT1•6TSf

~ 0 0 5 10 15 20 25 0 -z 0 > AXIAL STRAIN, 'a > 0 10 20 25 > -4 ,,,

AXIAL STRAIN~ .fa 0 5 10 15 20 25 ",

AXIAL STRAIN, 'a ( ., ,

FIG. 60 U U TRIAXIAL SHEAR TEST RESULTS f'OR COMPACTED PITTSBURG SANDY CLAY 1 HIGH COMPACTIVE EFFORT

APPENDIX B

DERIVATION OF RELATIONSHIP BETWEEN STRESS LEVEL

AND SHEAR STRESS FACTORS OF SAFETY

Values of the factor of safety based upon stress levels differ from the values of factor of safety based upon shear stresses, but these two factors of safety can be related to each other if it is assumed that the shear surface for which the factor of safety is calculated is oriented at the statically correct angle with the principal stresses. In this case the shear stress, Tmf• and the normal stress, an, are related to the principal stresses as shown in Fig. 61.

In calculating the factor of safety based on stress level, it is assumed that the soil is brought to failure by increasing the value of a 1 while holding a 3 constant. Therefore the stress level factor of safety, FSL' is defined as

= (al-a 3) f

(a 1-a 3) m (25)

in which (a1-a

3)f deviator stress at failure and (a

1-a

3) = mobilized

deviator stress. Using the Mohr-Coulomb failure criterion~ the deviator stress at failure can be expressed as

2 (c coscp + a 3 sincjl)

(1-sincjl) (26)

in which c = cohesion and cp = angle of internal friction. A similar equation can be derived for the mobilized stress state by using em and cp~ instead of c and cjl. Substituting these two equations into equation (25) will show

(1-sincjl )(c coscjl + a3

sincjl) F = m

SL (1-sincjl)(c coscjl + a 3 sincjl) m m m

(27)

In defining the factor of safety with respect to the value of shear stress on a given plane, it is assumed that the soil fails with no change in the normal stress on the plane, an. Therefore this factor of safety, called the shear stress factor of safety, F , is defined as

T

(28)

in which Tff =shear stress at failure on the failure plane (i.e.,' shear strength) and Tmf = mobilized shear stress on the failure plane. From

145

FIG. 61 COMPARISON OF MOBILIZED AND FAILURE STATES OF STRESS

the Mohr-Coulomb failure criterion, the shear strength can be expressed as

(29)

A similar equation may be written for the mobilized stress state using c m

and ¢m instread of c and ¢. Substituting these two equations into equation (B-4) will show

F T

= tancp tan¢

m

By trigonometric expansion of equation (30), it can be shown that

sin ¢ = tancp m (F 2 + tan2¢)1/2

T

cos ¢ m

F T = ------~------~

(F 2 + tan2¢)1/2 T

(30)

(31)

(32)

Similar expressions may be developed for sin¢ and cos¢ by substituting unity for FT in equations (31) and (32). Substituting these four equations, plus the equation for em shown in Fig. 51 as expressed in terms of c and F , into equation (29) results in the following equation.

T

(F 2 + tan2¢) 112 - tan¢ T

FSL = 1/2 (33) (L + tan2 ¢-) - tancjL

It may be noted that equation (33) indicates that the two factors of safety are related only by the value of ¢ and that FSL is always greater than or equal to F-r if the shear surface is oriented at the statically correct angle as assumed in the derivation.

147

APPENDIX C

COMPUTER PROGRAM

FOR THE

FINITE ELEMENT ANALYSIS OF EMBANKMENTS

Identification

This computer program which consists of a main program (LSBUILD) and six sub-routines (LAYOQT, LSSTIF, LSQUAD, LST8, BANSOL, LSRESUL) was coded by F. H. Kulhawy (1968-69) using the general programming concepts and solution techniques of Prof. E, L. Wilson (1963 to date), the linear strain element formulation of Felippa (1966) and the incremental loading concepts of King (1965) which were subsequently modified by Woodward (1966).

Purpose

The purpose of this program is to calculate the stresses, strains and displacements in embankments (homogeneous or zoned) by simulating the actual field construction sequence. The analysis is performed by finite element methods, assuming plane strain and isotropic conditions.

Options

The program has the capability of treating linear or hyperbolically nonlinear material properties. Provisions are made to include a foundation with initial anisotropic stresses, if desired.

To simulate the usual practice in large dam construction where cofferdams are constructed prior to the main embankment, the following approach is suggested. First perform a finite element analysis of the cofferdam and have the computer punch out the results using the punch option in the program. If the finite element mesh for the cofferdam differs from the mesh for the cofferdam portion of the cofferdam-embankment mesh, the auxiliary program (FEMINT) may be used to interpolate stresses, strains and material properties to the proper element locations in the cofferdam portion of the cofferdam-embankment mesh. Displacements in the new mesh must be evaluated and punched by the operator using a linear variation between nodal points. These values can then be used as cofferdam input in the analysis of the embankment construction on the cofferdam. An example of this approach is the analysis of Oroville Dam.

Factors of safety can also be computed from the finite element stresses by punching the output from the finite element analysis and using the punched output as input data in the auxiliary program (FEMFS).

148

Sequence of Operations

The main program (LSBUILD) monitors all operations by calling the following sub-routines to perform the analyses for each construction increment:

a) LAYOUT reads and prints the input data, computes the initial foundation stresses and computes the initial elastic properties for the elements.

b) LSSTIF develops the stiffness matrix of the entire assemblage of elements in the structure up to and including those at the specified construction elevationo It also modifies the stiffness matrix for the specified boundary constraints.

c) LSQUAD is called by LSSTIF for each quadrilateral finite element and sets up the stiffness matrix for each element.

d) LST8 is called by LSQUAD for each quadrilateral element and sets up the stiffness matrix for an 8 degree of freedom (4 nodal point) linear strain triangular finite element. LST8 is called twice, once for each of the two triangles comprising the quadrilateral element.

e) BANSOL solves for the unknown nodal point displacements by a Gaussian elimination technique.

f) LSRESUL calculates and prints the stresses, strains and displacements in the structure at the end of each construction increment and evaluates the nonlinear material properties of each element for the next increment. LSQUAD is called by LSRESUL for each quadrilateral element for the stress and strain computations.

Digrammatically, the operations can be shown as below:

once LSBUILD LAYOUT

+ ( each ) ( each element sub-element)

~ each ~ LSSTIF LSQUAD LST8 onstruction t increment BAN SOL

~ ( each ( each ) element) sub-element

LSRESUL LSQUAD LST8

149

INPUT DATA PROCEDURE

1) Control Cards (6 cards required)

a) Card 1 (12A6)

2-76 RED - Title card for program identification

b) Card 2 (6I4)

1-4 NUMELT - Total number of elements in the complete structure (Maximum = 275)

5-8 NUMNPT - Total number nodal points in the complete structure (Maximum = 300)

9-12 NFEL - Number of elements in the foundation portion ~ NUMELT)

13-16 NFNP - Number of nodal points in the foundation portion 0S_ NUMNPT)

17-20 NUMCEL - Number of elements in the cofferdam portion (Maximum = 100)

21-24 NUMCNP - Number nodal points in the cofferdam portion (Maximum = 100)

c) Card 3 (7I4)

1-4

13-16

NUMBC

NZONES NLAY

NUMIT

- Number of nodal points in the structure with a constrained deformation (fixed in x, fixed in y, fixed in x and y (Maximum = 100)

- Number of different material types (Maximum = 10) - Number of construction layers desired (Maximum =

25) - Number of solution cycles per construction layer

(e.g. - for 1 cycle of iteration per layer, NUMIT = 2).

17-20 NONLIN - Code for linear or nonlinear material properties (0 for all linear material, 1 for some or all nonlinear materials)

21-24 NWATER - Code for additional loads (e.g. - water forces) to be placed on the structure after the usual construction sequence is completed (0 for no added loads, 1 if loads are to be added)

25-28 NPUNCH- Code for punching out stresses etc., after last layer (0 = no, 1 = yes)

d) Card 4 (2Fl0.0)

1-10 AKO - Initial earth pressure coefficient in the founda-tion

11-20 REDMOD - Factor used in simulating construction sequences. (0.00001 yields good results)

150

e)

f)

Card 5 (7Fl0. 0) (See following figure for details)

1-10 FNL - X coordinate of foundation surface to the left of the embankment

11-20 TL - X coordinate of embankment toe to the left 21-30 CRL - X coordinate of embankment crest to the left 31-40 CTR - X coordinate of embankment centerline 41-50 CRR - X coordinate of embankment crest to the right 51-60 TR - X coordinate of embankment toe to the right 61-70 FNR - X coordinate of foundation surface to the right

of the embankment

Card 6 (7FlO.O)

Same as Card 5 for the Y coordinates

CRL CTR CRR

FNL TL TR FNR

If the X-coordinates of the following are equal:

FNL = Tt and FNR = TR

then the embankment is considered to be on a rigid foundation.

If only the half-section of a symmetrical embankment is being analyzed, the full section geometry must still be read in above.

2) Material Property Cards (See description following input procedure)

a) Units conversion card (FlO.O)

1-10 CONS - Units conversion constant

b) Weight and elastic constant cards (I 4, 6 F 10. 4) (number of cards required = NZONES)

1-4 N - Material type number 5-14 GAM - Unit weight

15-24 COEF } - Tangent modulus constants 25-34 EXP 35-44 DD } 45-54 GG - Tangent Poisson ratio constants 55-64 FF

151

c) Strength cards (I4,4Fl0,4) (Number of cards required = NZONES)

1-4 N - Material type number 5-14 cc - Cohesion

15-24 PHI - Angle of internal friction (degrees) 25-34 RF - Failure ratio 35-44 CODE - Code for linear or nonlinear material properties

(0 for linear, 1 for nonlinear).

CONS equals a units constant to convert the units that one desires to use into atmospheres, assuming that one uses the modulus and Poisson ratio constants as presented in the main text, Therefore, one of the following units combinations should be used:

GAM - ton/ft 3 - kip/ft 3

- lb./ft 3 (etc,)

CC - ton/ft 2 - kip/ft2

- lb./ft2

CONS - 1.058 - 2.116 - 2116.2

Since the output fields have been made small, it is best to use ton or kip units. For all of these cases, it is assumed that all dimensions are in feet.

When assigning numbers to the different material types in the embankment-foundation system, note that the output contains the principal stresses/yh. Therefore the program has been set up to evaluate yh using the following values:

-for a rigia rounaation, y (Dam) = y (Dam) = y ·(Shell) = y (1) for a flexible foundation, y (Foundation) = y (1)

and y (Dam) = y (Shell) = y (2)

The numbering of the material type should conform to the above.

If NONLIN = 0 on control card lc (all materials are linear elastic), use the following for each material type:

COEF GG EXP

modulus of elasticity =Poisson's ratio = DD = FF = CODE = 0.0

RF = LO

If NONLIN = 1 on control card lc (one or more materials are nonlinear), the tangent modulus (Et) and the tangent Poisson ratio (vt) are automatically calculated after each construction layer according to the following hyperbolic relationships:

V./(1-DD*E ) 2 l. a

152

where:

= CONS * COEF (o3

/CONS)EXP

= GG - FF log (o 3/CONS)

hyperbolic strength= (o1-o3)f/RF

The evaluation of these parameters is discussed in the main text. If the material is nonlinear, CODE = 1, but if the material is linear, evaluate the constants the same way as is done above for NONLIN = 0. If vt becomes greater than 0.49, it is automatically reset to 0.49.

If NONLIN = 1, the initial values of Et and vt in a foundation zone are calculated by assuming that 01 = yh and o 3 = AKO * o 1 . The initial values of Et and vt in an embankment zone are calculated by assuming that 01 = yh and 03 = 01 (v /1-Vt). Iteration is required to assure that the value of vt used to calculate o 3 is equal to the value of vt calculated in the equations above. This is done automatically.

Since the stress-strain relationship is written in terms of the bulk and shear moduli (as calculated from Et and vt) above), simulation of failure can be approximated in the following manner which is done automatically in the program. For tension failure [o 3< 0] or shear failure [mobilized (o1-o3) ~shear strength (ol-a3)f], the shear modulus is set= 0 and the bulk modulus is set = constant = bulk modulus at failure. These values are used in all subsequent computations in the program.

3) Nodal Point Cards (I4,2F8.2)

(Use as many cards as necessary to define the structure.)

1-4 5-12

13-20

MM - Nodal point number ORD(~,l) - X coordinate of nodal point (+ to right) ORD(MM,2) - Y coordinate of nodal point (+ up)

If nodal point cards are omitted, the program generates the omitted information by incrementing MM by one and by calculating ORD (MM, 1 and 2) at equal intervals along a straight line between the two defined nodal points. The first and last nodal points must always be given. (e.g., MM=l and MM=NUMNPT.)

Nodal points must be in numerical sequence from left to right in the finite element mesh and must increase from the foundation up in layers.

4) Constrained Boundary Cards (1814)

(Use as many cards as required to define NUMBC nodal points.)

1-4 NBC 5-8 NFIX

- Number of constrained nodal point. - Code to define type of fixity at this nodal point.

(NFIX 0 for X andY fixity.) (NFIX 1 for X fixity.) (NFIX 2 for Y fixity.)

153

Continue across the card for the constrained nodal points at repeatint eight column intervals as above for a maximum of nine alternating values of NBC and NFIX per card.

Omitted nodal points are considered as freely moving nodal points"

5) Element Cards (6I4)

(Use as many cards as necessary to define the structure")

1-4 N - Element number 5-8 NPN(N,l) - Number of nodal point I for this element 9-12 NPN(N,2) - Number of nodal point J for this element

13-16 NPN (N, 3) - Number of nodal point K for this element 17-20 NPN(N, 4) - Number of nodal point L for this element 21-24 NPN(N,5) - Material type of this element

If element cards are omitted, the program generates the omitted information by incrementing the previous N and NPN (N,l through 4) by one while retaining the same NPN (N,5). Cards must always be supplied for the first and last elements (e,g., N=l and N=NUMELT")

Elements must be numbered consecutively, proceeding counterclockwise around the quadrilateral elements. Nodal point numbers within an element must be ~ 39.

In the finite element mesh, elements are numbered consecutively from left to right in horizontal strips, starting at the bottom of the mesh and proceeding upward.

Triangular shaped elements may be used as long as a fourth nodal point is placed in the center of the "slope side" of these elements. Care must be exercised that the diagonal from nodal point J to nodal point L is not on a straight line including either I or K. Numbering must be done in the following way.

K

~ I J L I

6) Construction Layer Cards (5I4,F8.2)

(One card is required for each layer totaling NLAY cards)

1-4 LN - Number of construction layer, increasing upward from the bottom.

5-8 NOMEL(LN,l) - Smallest element number of the newly placed elements in this layer.

154

9-12 NOMEL(LN,2) - Largest element number of the newly placed elements in this layer.

13-16 NOMNP(LN,l) - Smallest nodal point number of the newly placed nodal points in this layer.

17-20 NOMNP(LN,2) - Largest nodal point number of the newly placed nodal points in this layer.

21-28 HEIGHT - Surface elevation of this layer.

If a foundation is included in the mesh, it must have LN=l. Therefore the first constructed layer= 2.

If NWATER=l, an additional layer card must be added to simulate added loads placed after the embankment is completed. In this case, columns 5 through 28 on the LN=NLAY card (last card) will be identical to those on the LN=NLAY-1 card (last layer of the constructed embankment).

7) Cofferdam Element and Nodal Point Cards

a) Cofferdam element cards (18I4) (Use as many cards as required to define NUMCEL elements.)

1-4 NCEL 5-8 NCEL

etc.

- ·Number of cofferdam element - Number of cofferdam element

Continue across the card for all of the input_ co££e_rdam_elements_ at_ repeating four column intervals for a maximum of 18 values per card.

b) Cofferdam nodal point cards (18I4)

Same as 7a using NCNP for a total of NUMCNP nodal points.

If NUMCEL = 0, these cards are omitted.

8) Force Cards (I4,2F8.2)

(Use as many cards as necessary to define the added loads.)

1-4 MM 5-14 FX

15-24 FY

- Nodal point number where force is applied. - X component of force applied at MM (+ to right) - Y component of force applied at MM (+ up)

If NWATER = 0, these cards are omitted.

If NWATER = 1, these cards must be supplied, in numerical sequence, and the first and last cards must always be supplied, even if there are no forces applied at these points.

155

If cards are omitted, MM is incremented by 1 and FX and FY are set equal to 0.

Care must be exercised to be sure that a force is not applied at a nodal point which is fixed in the direction of the applied force.

9) Cofferdam Existing Property Cards

a) Stress cards (Il0,5Fl0.3) b) Elastic property cards (Il0,5Fl0.3) c) Strain cards (Il0,5Fl0.3)

These cards (a,b,c) are punched out propertly from the auxiliary program (FEMINT).

d) Displacement cards (Il0,4Fl0.3)

1-10 11-20 21-30 31-40 41-50

NCNP - Cofferdam nodal point number ORD(N,l) - X ordinate of nodal point ORD(N,2) - Y ordinate of nodal point DISP(N,l)- X displacement of nodal point DISP(N,2)- Y displacement of nodal point

When the finite element meshes for the cofferdam and the cofferdamembankment systems are different, be sure that the nodal points are in the same locations. If this procedure is followed, only the nodal point numbers will have to be changed on the punched output from the cofferdam analysis before it is used as input in the r~-1lumher~d cofferdam-embankment system.

If NUMCEL = 0, all of the cards for 9a through 9d are omitted.

156

PHOGRA.'4 lJLIUilO I I NIJUTtOV H'UT t TAP( ltf M'l2, T MJUttPUNCHI , ....................................................................... . ft.M NU~·LINLA~ lf-11\A~l.I·:U,.T ANALYSI~ .. F ,11, KlJLHA"'V t 1Y68-69 , ......................................................................... . C.U~"~•"~UI-4 /lo-.1 f./ fiUII lJ I tf'tl.JMt.l.. l tl'fJ•-11~._, f tNF I;.L tid l'tl-' tNIJUNCH (.u"~o"~IJI• /f.tl LL I ~f-'l't I J I~, !J) t\JtUJI JvU tl. t tX(.,I4. 7!Jt tYCP 127~ I C!Jit\MUI~ /CJJAi4/ NU~Ctl t~UI<\(I'tJJtNClL llUOI tNC.NP llOO I CIJMI-~1'4 OW~(/ NoJMIJC tNt" t 1 \JV) tNI- Ill I IOV t tN.<~ATUit, X I JOll t ,ry I JOOI CU~~N IU4f4;,/ MllANUtNUMIJI.kttifUUitACU 1.,tdOI (Ufo4MON /I,. IF fl N\.AY tlNtkOMt t.lz,,z t tN<.iMNP IZ~ tlt tHUC.HTt ZS I •HtJ~Jt tl Y I..UMMUii l~t;.I.H41 tNLilt t TLI21t<.NLI2) •l Uti llt<.NH Ill t 1M lllt'ltMI Zt (\lf'tMUI'II /,..Af 1/ fotVI-fL INtHULA.Il Ul t:aHEANI J7.., J tf'OI ~127) I tC,AMilOhNLUMO&I (\Jf"'MUf.t /MAT I. I C~N~tC.Ut.f flU I tlX.Pf 1u. tOU I .tO J tUC..IlU ltF, llU J tHtUN~:t . CUMMOtc /MAT)/ CC f 101 tPHIIIU I tRF 110 I t0EV111U I tt'IE\1'21lOhCODE 110 hAlO , ...................................................................... .

C AEAO ANO PHINT HtPUT DATA • SET UP INITIAL CONCIITION5 , ......................................................................• c

IUO CALL LAYOUI c , ....................................................................... . C AfttALYlt. fuH (ACti CONSTRUCTION LAYER c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••

c c c

00 «HJO L"t•ltftLA Y , l Pte lNT lUvvt ~~~~~~:~:~ME\. I LN ,,.., tN•It21 'I NOMNP I LNtMI tM•lt21t

00 •ww I T•1tNUMIT CALL ~£<.0/1110 t T IMClJ IF ILN ,Uf• l •UN. Nf[\. •EU. 01 GO TO lOO PRINT l!Jv' GO TO )\lv

O(Yt.LUP ~TIFfN(~$ HATNIXt ~CJLYE [UUATIUNSt EVALUATE RESULTS

2LJO CALL L$:)TIF CALL HAN~OL CALL L!»H[~UL

)UO CALL :,[COHO If IME21 IJM[oii>4El•IIMrJ

•OO PAINT l011Jt LNtl TtTIMt , ...................................................................... . IUUO ""'MAT t lH6 Ill UA6 Ill 20H LAY[R NOt • U II

I 2vH ADDEO ELEMUtT S. • 1• ••H THAU I• II ,Z 2\o!H AODCD ~DAL POINTS• 14t6H THRU 14 II I 2vH WA,AC! [L[VAIIOH • 'IOo) I

100, ,OMMAI 149H•,UUHOAIION LOH[o SEE PREVIOUS PAGES 'OA SIAESSU I

uo

uo

CALL lX II PH I NT 20"u C TN~ • lL12 HI I h112)•TLU II' ICHH I 1-TLt 11 tiC TRil t•fLC U II 00 1bV N•ltNUMNPT IF fUt(L)Ik•11 •l[, C:hU 1J I AUIH•ICfHI lJ-VHDIIhll t/ICUtlJ-TLUU IF IOMOI~tlJ ,c;T. CTHilJ I JIDU•htOMOCNtiJ•CTMfliiiiTRtU•CTRflH YDIM• f OAOt NtZI-CTR, IICTN f Z J -CTR, J ~HINT 1V2Wt NtCUHOCNtMitM•J•Zttli.OIMtYOIM

ME:AU IOU~t INOCIK.)tNFUU,ItK•lt,.Ui4UCI PRINT 20"' DO tau k.•ltNVM8C I' INfiXUI ,Eo, OJ H INFIXtk.l .Eo, 11 IF INFIXIK.I ,t:.o. 21

PRJhT ZO~OtNACIKt PRINT 20,tNBCUI PRINT Z060t.N8C Uti

II~ CON II NUE , ...................................................................•... C N(AU ANo .,HINT tLCMCNT AAHAYt COMPUTE PT, FOR STRE!t\£$ , ......................................................................•

NN•O

c

lOO N(AtJ lOO~t NtfNPNIN•MhM•ltSI J 10 NN•NN+l

IF IN •LE• NNI UO TO i30 DO ZZU 1<.•1tlt

210 NPNt,..Ntk.I•~PNIN"t•ltf.l+l NPN C I\IN t !I) • NPN I NP~""lt !I I

JJO IF' IN ,c;r. NNI r,o TO ZlO JF INUHELT .UT• NNJ UO TO 200

PAINT ZU6:t 00 2,0 N•l tNUMEL T J•NPNINtZJ L•NIJNINtltl X.CPI N l•tJ,)•IORDI Jt II+OROtl tlll YCPI N l•v, ,., QROI Jt ZI+ORn f L, 2 I I IF IXCf.ICNI ,Lt• CTHt 111 X.OIM•ICTNili-ACPthiJifCTRili•TLilJ I IF IJICI-'INI ,~,T, CTNilll XVIM•UCPINI-CTHilll/ITktli•CTJUltl YOIM• t Y(P I NI•CTRF Ill CTR f ZI•C TAF J

ItO PHI NT 2U71Jt Nt tNIJ1'41Nt1<41tM•lt )J tJI.CPINI th.t'tlttltX0""'YUI1'4 , .•.•................................................................•.• C N£AO AND PHINT CUNSTRUCTION ~[iJUEMCE JltfONMATION , ...•................................................................•••

READ 10)1.1 t I Uh I NOf'IEL I LNtN I tN•l tU t I NDMnPfLNtMI•M•J tZ It I HEIGHTfLN)tLN•hNLAYI

PAINT 2"H PNINf 208\.11 CLNtCJ\IOMEltLNtNttN•ltlltiH~tLihMitM•ltZh

1 HEic.HTfLHhLh•ltNLAU li)IO II'UMMAT fTtt-LAYER•IltSXtiiH IYERATION•U•'X•l"H TI,_.EfSECOMOSI•'f•JI C , ....................................................................•.. IF INUMCEL ,[0• 01 GO TO UO

P~INT 2LI82 GO 10 100 !110

SUOHOUT IN( LAYOUT , ....................................................................... . CIJM140N /I Nl T I HU)f 12 I tNU~ll T eNUMNPT tNFtL tNFNPtNPUN(H (-{JIA~N- 1-~U./- NPN+l-!-,-;-5--f-;-Utt-O.f-!UU-;-2-J-i-lt-C-P-! -l-1-~ 1-;-¥(-P--! Z.-7~ ICOMMON /COAM/ NUMCELtNUMCNPtNCfltlOOftltCNPtlOOt (UM11UN /Nt-IJC./ kUMU( tNI.!C. llVU I tN~ I X. I luU I tktoA fi;.Htf )I I )0Uit FYI 3001 COMMON /LiAN5/ foltJANOtNUMUlKtlill601 tAl l6Ut801 (tJi<IMUI• /L I FTI kl.AY tLNt NVMt.L I Z~ tl J tNU/4l~PC 2::. t2 I 1M~ IC.HT 12~ I tNUHI T II f CUM,.,ON /C)[UM/ FNl. C2 I, TL 12 I tCRLl2 J tCTR I 21 t(RN UJ t TRf 2 h ,NA 121 COMMON /MAT 1/ NUNLINtBUliC I 27~1 t$H[ARt 2 7:, I tPU I~ I 27, I tCloAMf 10 I tREDHOD CVMI·1UN /t"!A f 21 CUN~t(.Utf llU I tlXPilu I tiJOilv I tYC..IlO I tFF 110 I tNlONEil CUM14VN /MAT ll CC 110 t ,PHil lUI tNF IIU t tOEV 1 C HH tOE\1'2 110 It COO£ t 10 ltAlO Ulroott.N:,I UN ~ THE.N 127h 3 I t!iTNA INi.! H tJ) tUI !iPI 300t 2 I fwUIYALLNCL. I .~TML..)~ tAtlUIJUII tl ~ f~AI NtAC •1.100 II 1 10 I ::i.Pt A C61JOOI I , ...................................................................... .

( NEAt> AND PrfHH CUNTHOL DATA c••••••••••••••• ••• •••••••••••••••••••••••••••••••••••••••••••••••••••••

REAO lOVUt HEU HEAU lOU,t NvMEL T tHUI"NYf tNFE.L tNFNP tNUM([L tNUMCNP HEAD lOu' t NUMUCtNlONE~tNLAY tNUMJ T tNONLJNtNWATEHtf'iiPUNCH AEAO IOlvt AKOtH(OHOO HtAU 10 lVt I F~L INJ t TLINI tCioiL I hI tC TNI Nl tCNHC HIt TR INI tFNRINt tN•& •,Z I PR PH ZU..J\.It t1t0 f.' HI ~T .:vll'' NUMt:L. T tNU"'-NPT tNFEL ,HfNP,NUHCEL tNUM(NP

c

READ lt.hJh INCELfNJ tN•ltNUMCELI PRINT hJVh INCELCNhN•llfiiUMCELt PRlfH ZUBJ READ lU\oth INCNPINhN•ltNUMCNPJ PAINT lOtl't fPtefrriPtNJtN•ltNUMCMPJ

1".1 IF INWATER ,[0, Ul GO TO ,00 PRiftT zoe' 00--~QU-- NN"-ltNIJHN-P-t-FXINN1•.J,u

400 FYCNNI•O,v NN•U

•10 READ IOZ:tt MMtFXIMMit,YtMMI 420 NN•NN+l

PRINT 11.1~'' NNtFJI:tNNJtFYINNI IF IMM tUT, NNI UO TO 420 IF INUioiNPT .C.T. NNI GO TO 410 , ...................................................................... .

C OETEHI-IINE dAfu.> WIDTH 0' ~TIFf"NE~S MATNI.lt ABORT If TOO LARGE , ...................................................................... . 100 MBAND•O

00 )'l~o~ N•ltNUMELT II•MAX.Of NP"fl N • II tNPNC NtZ I tNPNUh l I tNP .. f Nt•l t JJ•MINU t NPN IN tl I tNPNt Nt2 hNPNI Nt)) tNPitiN•" I u:.•z•t 11-JJ•t 1 IF IKK .GoT. MUANOI FoiOANO•KK IF IMUAND eLE• 801 GO TO SlO PRINT 'OO'- N GO TO 999

110 CONTINUE PHI f"'T 201Ut NUM6C •"'lfJNES•NLAY tNUMl T tNONLINtN-.ATER tNPUfiiCH PAINT 2011)t AKO•AEDMOO ., ....................................................................••• PAINT ~02\ott flll J tTL121tCHLilltCALf 21 tCTAI lltCTRll I tCRRflltCfUIUI t

1 TNflltTH11.) I' UtHL .EO• VI PNPH 2021 IF INFlL ,ur. Ul PHINT JU2lt FNLfl1•'NLI21tFNRilltFNAI2J , ...................................................................... .

( NEAl> AND PRINT MATEMIAL PfiiOPERTV OAfA c••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• ,

READ 1Ulvt CONS. lUAU lOUt I htGAMINJtCOtftfU tElPINitDOINitVC.fNh,'tNI tN•l tNlOHlU Rt:AO 1021.1 t I NtCC INitPHI I HI tRFI Nt tCOOt:UtJ tN• hNlOftES I PRINT 2U2'• CUNS

~=~=~ fg:~, l••tC..A141NI tCUEf I HI tt.XPIHJ tOUINI •(tfifNI ,,., INI tN•hNlOfill•t

::: ~1 fg~~~ I Nt(C INI tPHI UU tAft NltCOOliHttN•ItkZONESt , ....................................................................... . ( Nf:AO AND ..-~«tNT NODAL VOihT ARRAY , ...................................................................... .

ll•..J 100 HEAU ll.l2~t MMtiOROIMMtMitH•ltll

OX• I ORO I P.t-4tli-OA'Of LL tll J IIMM•ll t OY•I ORO I MMt l I .. ORO ILL t2 t 1/ f HM-lL 1

110 LL•LL•J IF IMM-llt l•Otl10tUO

llO OADILLtii•OMOILL•ltlJ+OX OAOILL1Z hOAOILL•J tZ t+DY GO 10 II"

UO I' INI.IMNPI•MMI J•OoUOoiOO UO PAINT SOUOo MM

157

C INITIALIZE VALUES IN FOUNDATION AND IN fMBAHt:;H["T , ...................................................................... .

c

c

PRINT ZU9u 00 '~1..1 N•ltNUMEl T DO 5c~U '-hlt) STRES:)INtMJ•O•U

SIW STAAtNINtM1•UtU 00 ,60 N•ltNUHNPf DO '6U M•lt2

t•O OtSPtNtMI•O•O IF INfEt.. ,[Q, Ul GO TO ,10 00 STO N•ltNFEL STRES~I r•h2 hlJAMf ll*fH[ IG-HT 111-YCPINI I

tTO :aTHt.~ .. IIHlJ•:aTH.[:J!i1Nt2t*AIC.O

tiO ·IF CNUMCEL ,[Ut 01 GO TO 600 READ 103h IN tXCPI Nt tYCPIPO, I $TR(~SI Nt .. ltJrhltJ I tJ•ltfltUfo1CEL I READ l03St t l'hXCPI NltYCP t Nl tl STMAI .. IHtf'IJ tM•lt) hJ•lt,.UMCEL I READ lOnt I NtXCPt Nl tYCPC Nl tOULl II'U t$11(AA Uti tPOI$1 Mit J•1t'*UMCU.. l Rt.AU lU.Ut lrft IUHOCNtMt tM•1t21tiOI!.Pt"•MI •N•hZ, t..l•hiNHC .. PI

•00 " INOHL IN ,[Yo II GO TO ••o 00 620 N• hNUJo("[L T MfYPE•NPNINtSI POl Sf 1111•(-GIMTYP[ I [MOO•CO[FI MTYP[) SHE. ARC Nl•£1400112•0•1 hO+POISI,.t I I

UO BUUINhSH[AA INI/IIoO•Z,OIPOJSINII c

c

640 00 6~u N•ltNlONfS PHI t ~l •PHil N ll 5o 7,29,77951 (OifST•Ze1.11 I IU I NJ I lleO•SI HI PHI Uft, t, OEY II H • •CON.Sf •cc INt 'CUSIPHI fflt I

6,0 DtY2ftt1•CO,...~T•51NIItHIIHII I' INONLIN e[Ue Ot GO TO UO

DU 69..1 Lt~•J tNLAY IF ILN el!Je ~LAY tANO, .. WAT£R •lOt lt GO TO 690 I' f~N .~r, II C.O 10 660 '' '"'n .• oTt "'' no TO 690

660 NFELl•NO~[LfVhlJ 6?0 NfEL2•NOME'LfLNt21

00 bdJ N•NFllltNFf:LZ MfYtJ't•NPNINt5J HT•ftE IUHT I Lfitt •TCPtNt IF IN tGf, NF'ELI eANO• N eLh t<t,E:Llf GO fO 6 .. Je,.PNINtll l•NP~HNt)l L•NPNINt .. l I' IN e(Y, Nfi:Lll HT•IiT•0,, 1 1UHOICt21-uMOCLtZit l' tONOil.tll elU• OlfOILt211 GO TO 610 If IN tlWt NfLlJl IIT•Itl•f.h)lfUMOiktlt•UMOI.ItZI t

610 !.hct:S.:,OI Ntl f •HI •UAMIMTYPt.: 110,1 69\1 CUNfJNU[

POI~l•GGIII 00 1)0 N•l tNUM[lf IF Ut~UI>ICEL ,£Ch 01 GO TO 696 DO bf) M•1tNUMCEL IF IN eEUe N\.LLIMI J GO TO 7JO •n (uNIINV[

696 MTVP[•NPN1Nt51 IF IN eLEt NFI:LJ GO TO 110

700 IF fPUJSl eGTt Ot491JI P01Sl•n, .. 90 :tfN£$!)1 Ntlt ·~ IHE:,SrN,zt•POI !.11 f J,O•POI $1 I

flO OEVUR•SfRES.$CN•lt•:.tA[~SfNt11 OEYfH•O£Vl CMT 't'P( I +O(VltMJ VP£ t•STA!SSC Htll E l•(.t)H I MTYPE t I(.QN~ 1 C I $TIH.SSfNtliiCUMSI••ElCPf M!YPE It [P$•0(V$ fHif E l•t 1,0-1 OEV!IoiAID(V,HI I I PUf Soi•C.C.t Mfl'f.l~ t•F' lfiiiiJYP[ t•ALUGlO I StM£$Stfh 1 IICOff$1 ltOI5T•P01 S.f If C hO•bOIMTYP[ t•lPS t••JeOI

" IPOUt eGJ. "'''"'' POIST•O••to I' IN •I.E• "F£ll GO TO TZO Jr IAd~IPOI$1-POISTI oLfo OoOOOOII 10 TO UO POl $I•POI U+IPOIST•POUI II IOoO GO TO Tu"

UO POI$1NI•POIST IF IN eLEe Nf'~LI GO TO '725 !ifR[SSINtli•U,LI STR£SSUh2 t•O,Q

U' lMOO•U•I I loO-ICOO[IMIYPU•O!VUR/OlV,MII .. ZoOI SHEARUU•EMODII Z.O•f l•0+'01SINJI I OULKINI•SHEAN IMIII h0•2•01P01Sf NJ I

1JO CONTIHVt:

C Uu 00 Hu N•loNVMCLT --[-MOOc-l-.-u•-5-H£-AM-tNi ec hB+POf S-tNII

7!10 PNINT ZUYh NtlCPINitYCPINI•[MUO•tslA.KINitSHEAAtNhPOISINit I ISTIIIESSINtMhM•l•U

IF INUMCEL e[U, Ul GO TO 160 PRI Nf )OUU P~~~; JNT IOJ~t I N,l(PIHitYCPC Nl t C SJRAINffthMI tM•ltJ t tH•ltNUHELTt P~INT lOU' PRINT H14.,., CNtiOROI N•M I tM•ltZI •IOI5Pf N•.,U tM•hZI tfl•hfiVMNPU

160 REWH-10 4 WRITE Ul wNJTE C.t "'HIT£ 14f ~ETURH

I I STkESSI N,MI tM•ltJ I tN•l tNUHtLTI II ~TAAINC NtMhlo(•ltJI tN•ltNUMEL TJ II OISP INeMI•M•I•21tN•hNUMrtPU

c••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• lviJU FO~MAT 1l2A61 100, FURMA T 11814 I J\)lU FORMAT 17Fl0tUI IC,U FONMAT CI••6Fl""•"l lOZU 'QMJotAT ll't, .. Flu, .. t

·l"Z' FVR""AT t t••z,a.zt lUlU FQRM-'f C,U,FIIeZI hUS ,ORMAT lllUe'!JF lo,~,JI lO .. D I"OAMAT 11lJt4Flve41 ZUUD '0RMAT llH6 1111 lZA6t ZUUS 'OAMAT lllti'"'TUUL P'tU, (l(Mf~T~ •I.. 21H TOTAL NO• NODES •I•

l 21H FUUNUATIO"'f Elll"tNTS•I.. ZlH '0UNl\ATION NOnES •I• 2 2lH CuFHMDAI-4 H£14ENTS •14 I 21H CUFFEROAJ.t NODES. •I•

2U1U FORMAT C21HONO. RCSfRAI,..EO NOOES • h I 1 21tt NU, OIFFF.A£NT MATERIALS. • 1 .. I Z 27H NU, CQN:,TRVCTION LAY[~$ • I .. I J Z7H NU• ITCHATIVNS PEA LAYf:A • I .. II 4 .. 9H NUNLINCAR CODE IO•lJN(AN, l•NUNLIN[AMI•••••••-•• 14 ' lt9H CUl.IE tO~ PRtStHC.l OF tlliATllt tl.IMCt!J IO•t.!Otl•YES.I•• I• 6 4YH (UIJE FOH PUNCitiNC, FINAL STRl~S.E~ f •N0t1•Y[SI-·• 1•

JUit POHMAT tJZHU[ ... ~JH PH(SSU!(t. COEFt IN FNUN. • F6,) I l llH 1"\UOULUS R£DUCftON FACTOR • F9e6 II

ZO~O fCJMMAT fllliUtMUAN~t#'II:.NT CUVMOINATt$ I 5le'IH TUIE l.lFTtlOlt 1 llH CHl.:~f LVftYX•llH (.(Nll:.HliN"'9X,llH <.KEST RUitfft9Xt l 1VH h)l AIGitf I ~le1HXt'tX.IHV.YAtiHIIIIU•lHYt9XtiHXt9XtlH'h J 9X•IHXe9XtlHl't9lltiHXt9Xt1Hl' I l1J'l0e) II

Z~o~Zl rORMAT lllH FUUitOATION ASSUMtl> TO Of.: AIC.IO I) 2\IJ~ FUN1ot.t.T IZJ•i t-UVHLIATIUN (WMDI~ATES I 7lt!IH Lt.FTtUXe6H RIC.HT I

1 !tll•lHl<t'IX•lH'ft9X.lHX.'JX.lHV I 11f'l0e) .It l\.IZS tUNf'IAT 12lH-foiAJ[MIAL PIWPt.HTY IJATAII/lJh U~ITS CONSTANT • Fit. lit ZVlO FORMAT lll!lX,aH M00UlUStl3ltl4H POISSON NAfiO I

l )9H MAll. UNIT wT CON.::.JANT [XfiONENf Ot9Jit1HG•9Jit1HP'It tul' FUMI'IAT I .. IH•MATL C PHI fAIL.RATIO N0"LINI1•Y[S.I If 2u40 FOftMAT ll91tl NOOAL POINT AAAAY II

I UM ~p W•OI<O Y•OMO x-o•01~ Y•OR0/14 II 2U., 'OAMAT I )'ll'tlMOOAL POINT~ wUH G(OMETAIC RL:tJRAH•TS II) ZU~O FONMA f I lZH NUOAL POt NT 1 I., 1 UH CAN NOT MOVE t lQ)J ,UHMAT I UH HUOAL ~lNh 14 tllH CAN HUYE UNI..Y IN THl Y•DIAECTIUHI IU .. ,_, IUH - NINTt r. tUM U" 100¥1 IIOO~Y Ill TMI •-oiRlCTIONI

158

20•S FUH,..Af I ~UHl llEM(Nf ARNAY • PTe WHEHf. :.t..:t.SSlS AR[ lVALVATlD Ill l )OH Elf. I J Y. t. MAll •·ORO 't'•ORO t Z ZvH X-OAOIW Y-ORD/H ,,

ZU7\ol FORMAT f6Ut4FlUt31 2UU FUHMAT lllHl CUH~t,.UCTIOPtl LAVI;N JNf'Ujoir4ATIUN II 6H LAY[Rt

1 4tttt INCLU51Vt EU."OH!. INCLU:tl'llt; "WlS tl.lYATION It 2U80 FOH:MAT ll:>ti1Uttltl12•1btfl1tell 2UU fUIH<IAf tl'IH•(VtFlHUAM llt,MlNTS Ill IWII 'OMMAf l~)li•Cm:Ft.MOA~ ~OUAL POINTS Ill Zllll f~MAT U~ttl NUI)AL t-'UitiT 'OHCt:.IIIQ" Nt' IC•r~CI Y•lllottCI. Ill IOWO ,ORMAf ll~Hl INITIAL \IALUF:5 IN [LE~!NIS Ill

1 -~H {L[ X-oRO Y•OAO (LA:, MOO tsUl.C MOOt Z $UH SHEAR MUD PUISSON SIG•JC SIG-Y U.V•II:Y Ill

lCifS fORMAl fl>tlflU•ltJf'lOeh .. FIO•ll JI,)UO FORMAT f 35H1 INITIAL CDfflRUAM tLEMErtT !)lNAI~$ Ill JOH [~h

1 )IJH J•ONO "•ORO [PS-• [P~·Y GA~XY lit JQOI ,ORHAT IUHI I~ITIAL COFHMOAH ~OOAL POINT OI$PLA([H[NT$ Ill

1 '"" pt,p, X•OHO Y•OMD X OUP T•DIS" II. IJOOO FORMAT 117H lteP• £NNOA t It • 1 .. 1 ~001 f()!(HAT llZH OAIW WIDTH TOO LA•GE AT [LlHENT Ul , ......................................................................• H9STOP

ENO

SUOROUTINE LSS.TIF , .••.................................................................... COMMON IINffl HlOilll tNUMt.L T tNUMNPT tNFLL •"FhP,~~UN(H CvH,,.UN /hl-'t.LI tt.,t.ll P$t )t tUNO I JUO tll 1X(PI ~l!tol •'1"(~12 1~ » COMMON /COAHI NUMCELtftU:.tCNP,HClt.tlCOI t"tCNPil OCII CUMHOI't IN.,!JC/ N...N1UCtNUC 1100 I tNI lA I lOil I'"""" H.Me'U JOO h'Y f lei OJ COMMON IUANS./ MdANOtNUMIJLk•Uil60JtAC 16D•IOI CI.H'I,.,Of1 /LIFT/ NLAY•LNtNOt-lt.t.tzhlltHuMNt'l 2~•2 I tHfJCioHT 12) I tlti..I"'IT el T CL114MQN I"'AT 11 HUNLI"ttBUUU 2Uit.S.H(ANf21~1tPIJ15t 21SieC...UU lOI,RtU"OO CU"""''h /L:.T 11 I eJtK.tSfiU•lf.l I1~H hUH t(l ltllePilOI ,,.PUI1VOL

c••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• ( INI riALil.t. tJt.UCk COft~TANf~ A'-0 STIFFNt.,::. ANRAY , ...................................................................... .

c

REWIND 2 P4A•41"' NO• 2•NO ND2•Z•NO NIJMULK•v

00 210 N•l tNUZ UINI•UeU 00 21U M•ltM6AND

ZlU AINtMI•IJ,v , c••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••••••• C FO~M S.tiHN(S.$ MATRIX IN BLOCkS , ..•..................................................................•.

c

JOCJ NUMUL~•NW-tdLK+l NH•NB•I NUi4UlK+lt NM•NH'"'Nd NNL•NM•P48+ 1 KSHI,T•l'NNL•Z

NU•4EL•P.IUM£LI LNeZI IF INUr<tCll ,H(• 01 NUI'tEL•M ... XOUt0."4ELILN•21•1fCLLINUM(E.Lt t OU ,. .. "" N•lthUMEL I(.MIN••·•II'<IIJI l•fl-'iit Nt 1 J tNP/1Cf .. l I tM-'NI Ntl I 1HP.;t "''' r) IF UMI~ eLf• ffNL eUA. K.MIH eGf• ...... G4J TU ... CALL LS.QUAOINJ , ...................................................................... .

C 1• AOIJ ELLMtNT ltf1F'FNC$S. TO TOTAL $TIFfN(SS , ..•.................................................................... 00 .. ,.u 1•1·• 00 ••v K•l•2 J I•Z•NPN I N1l t •Z+k•lSHI fT Klt•2•1·2•K. DIIJI•UIIIt+PCkKI 00 4ftU J•)t4 DO ... o L•l•2 JJ•Z•NPNUhJI •Z+L•II+l•KSH.,f LL•Z•J•2+L IF tJJ •L£• u I GO TO lt)U IF CNU ,c,t,, JJI GO TO •zo PHIMT lUUVt Ml:tANOtN CA~~ Ull

Ol~ AllltJJI•AflltJJIOliUt~~~

lt)J (UNTI~UF ltltV (t,~flfiUt , ....................................................................... .

( J., AlJv LVf-fi.U~II\A1LU f"UH{t.,<;, Wlltil~ t::ACU ULOtJ;. IIF PH(!i[ltTI , ...................................................................... . If IL~ ,Nt,., '"lAV ,QH, N011ATtN ,[Q, 01 C.U TO 6n0 00 ~JJ III•NIIIl tN! .. K•Z'N'"''-:,UI F J Olt..l•llllr. l+fVI Nl

,OJ IIIK•\I•IH<.•11HX1Nt (tl.tll•••• I 1111111 I lllll.llell·l································ffllfll ( J. M.JUifY ;.11tFIIIt.$S E.WUATIUN!. 'UN UUUNUAMY CU~STHAINTS , ...................................................................... .

60U NUMftP •IIIIJ• lr~P f Lfh 21 If 11'4UI~CN~ 1 Nl 1 Ot NUMNP•~AXQCNOHNPILNt2ttNCNPChUMCNPtt 00 6,0 M•l,NUMilC If UWCfiH ,LT• N~LI Go TO 6,0 If INLICIMI ,C,f, NH ,uH, NHCIMI ,GTt NUMNPJ GO TO 700 N• Z•NUC I M l•l•IC.SHIFT If 1Nf-1Xf,.4J 1lY• 0 •UH, NfiXIMI eEUt 1t GO TO 620

6lU 1f fill~ IJtiMI ,t,u, 0 •ORe NFIXIM'l ,[Q, 21 N•N+l 620 OU bltl1 MM•2•M8AND

.,_k.•N-M.Hl IF IU. ,LEe OJ (,0 TO 610 AIKI(,,·I,\II•ueu

6)lol K"'•N+MM•l IF INUJ. •Lft IIOll GO TO 640 Al~•"'·'~l• .. hO

6fllu CUHf INU~ A INtl) •leO RINI•oJ,W JF INf IXIMt ,[u, 0 •A~O• N tE\h ll'NB(IMI•l-ltSHl'Tit GO TO 610

6)-U CONtiNUE , ...................................................................... . t ,., "'~I Tl UL.UC.~ "'" TAf'E.tMUVE U~ LUWLM ttLUCK.tCHE(I(. FOR LAST OLOU. , ...................................................................... .

7UD "HI fC IZ J fOfNJ tiAINtMitM•ltMI:IANOitN•ltNOt 00 11U N•ltNO -.•NHtU RINt•HIItl RllthOeO 00 710 M•ltMOAND AHhMt•ACKtMI

71U AIKtHI•U•v IF IN"' ,LT• NUMNPI ~ TO JOO WUV~N

c••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• luUO '0RMAY llJH BAND 'AIUTH • lh20H EXCEEDED AT 1::LEMf~T Ifill , ....................................................................... .

END

~UUHOU11NE L!IUUAOINI c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••

(C.H'MVN II t•l T I 14[01 U I tNU"'(L T tHUMNPT tNF(L •NfNPtNPUN(tt (.UM•<~Uii /f.ti'LLI N.,Nf .l ''''I tUifUI JOOtll t XCYI .17) I tYCP llH I t_ui .. HON IC_UAHI /lofVMC_LL ,NUUC.NI'tNCLL 1100 I eN<.NPilOOI (Ol"•<IUN IL H T I NLAY tLNtNOMCL.I Z~ ell tNIJMNPI2, 12 l 1HE H,HT 12$ ltNu141T tl t CVI"•"VN lt#.AT 11 NVN ... IN•ttUU.IZf!:t lt$Ht.ANI 21~) ePUISt 2 n I tt.AHilU I tREUMOo CVM,\401'4 /L.)T II ltJtKtSf httlO I t~TI hlOI JC I 3t ,, tPilOit .. Pf •1 t\IOL DATA C 11 •l J t(l 2•lltCf ltll tCf 3t2110tt0t tU,tOel , ....................................................................... .

C INITIALI.ll ANU FORM STAE5$•STRAIN "'ATRIX , ...................................................................... . DO lOu ll•ltlU PCI fl•u,v OU hill JJ•l.tO

lLIU S.llleJJf•..,,u VOL•O• oJ IF IN eLE• NOMCLILr-.tZJ ,oR, NUMCEL eE'Ot 01 (,() TO lJO 00 UU M•leNUMCEL IF IN tEUe NClLfHII GO TO UO

Uv COHtiN\JE At TuRN

lJO ,MOOEhl,.., ·., tNLAY ,[Q, l .oRe NOMCLILNtU tl'O• Nf£L1 CiO TO UO IF IN ,G£• NU"'Et.lllhlt ••NO• N •LE.• NOHt.t.fLNt2ll fMOOEl•RI:DMOO

150 HIYP£-'"PNI Nt~ I C lltli•FMOOL l•tuut.KINI+S.HEAR INI I C ll•Z t •FMUUC t•IUUC..IIOf NI•SHt:AN INI I Cl2tll•<llt21 C12•21•Ciltll Cl )tlJ•Fio4UL>tt•!JH[ARINI , ...................................................................... .

C FVHM 10'1V UUAOt 5TI''~£S5t REOUC£ TO 111t CALCVlAT£ GRAVITY LOADI , ...................................................................... . I•NPNINtll J•NPNitftJ. J IIO•NPNittt4d CALL L~ltltlt:hlt I•NPNINt]l J•N~HUhflll llii!HPNUh21 CALL L:,Oftll ~,h)t IF IVOC.. ,GTe OtUI GO TO 200 PAI~T lUUUt N CALL EJIT

20'-' 00 Jvu It• l •Z IH•lV-1'. IO•IH+l 00 lu"' l•ltiH 5110•1 J•~IIDtltiSf lOtiO I 00 300 J•l•IH

JLI~ SdJtii•SIJ•II-SIJtiOI•SIIOtll If IN 1Lfe N01'4£t.ILNtlt eON• H t6h HOMt:.LfLNtltt RETURN I' IN 1LEt Hf[Lt AETlJAH DO 400 l•loA

159

•oo Pt 2•1 t •Pf z•J 1-GAHt HTYPU•VOL/.,0 RETURN , ...................................................................... .

IUOO ,UI<MAT ll4H lt"O OH HEGAT IVE AHEA AT ELLMENT 141 , ...................................................................... . !NO

SUOROUTINE L5TtiiN)tN2tNll , ...................................................................... . CUMMON INI'I;;LI N~Nt 2 '''' leUHDI lOO tll tJI(Pt ZU lt'I"CPIZH I (Ufo\4<0\U"' /L;..I i I I ,J,.._,:, I I Ut 101 t!tfl J •lUI t(.l J1 Jl ePt lO ltNP( fll J tYUL U IMlN:,toN UAI ltll 1Uf l '" tYI lt_-.1 tWI Jt.4•1_l _ [UUIVALENCE IUVf lltUitiUYI UhVt -, ....................................................................... .

C OH INt. THIANUUL.AH ELEMENT HUOAt. POINTS• GEVMETAYt AHO AREA , ..•.................................................................... NPf li•Nl N~IZI•NZ

NPf lt•Nl NP141 •9 BAt ltli•ORDCJt21-0A0fl(t21 BAIZt li•ORO IK •2 J -OROIItZI BAI ltli•OHOf lt21-0ROI Jt2 I OAf ltZ I•OROIIIO •11•0RDI Jtll lU12t21•0J.IOIItli-OROCittll UAt Je2 J•OHOIJ •li ... OAOIItll A~-tt:A•I UMUI J•li 11 UAUtll +OROiltli•BAt hli•OROflttl1'&At lellll Z.O IF IAH(A ,L[, OeOI GO TO •oo VOL•VOl.+A~EA

c••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• C SET U ... Tt:Ho<t;, FtJtoC STME.$So•ITRAift ANU STMAiti•OI~'I..A(lMEHT MA1AltLll c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••

c

AREA415• .. B,u•AHt.A C ll•Ct 11 1 t IAt<lAittt Cl2•Cilt211AHl:.Afli8 C ll•C I leli/ARC.UI C22•C I 2 • 21/AMl:.Ait$ C2l•Cf2tJJ/AHEA118 Cll•Cilt )IIAAlA•I

OU 2U(J M•lt2 UV'CltltMI• OAil•M• Ul/f2tltMI• I:IA.CltMJ UVIl,ltMI• 8AiltMI UVI lt2t,.O• 8AI2tMI UVC2t2eMI• HAIZtMJ•.Z,O•BAIJtMI UVI )t 21M t •·BA IZtMI UVIlt3tHI• UAI JtMI UVC ltJ eM I ••dAI l•MJ UVI)tJtMJ• dACJeHI•Z•O•BAChM• uvta ......... u.o UVI2tflitMh OAI]tM)efii,O

JOO UVI1t4tMI• 8At2tMI• .. ,o , ...................................................................... . c;; Ut":V'LLUfJ C..UtJLH NeP.• ITNAitt-DUPt MATRIX AHD TRIA"GL£ ~TIFFN[~;, , ...................................................................... . c

AkE.Ati•W•U*AMl:.A 00 lOU l•ltfll ........ .STt lt Ill •$Tf 1t Ill •HUll• U+UUtllliAREAII lfU•II+ll•lltltl .. ll+l fYf 2• I I+VU til IIAMfAit

c

c

.>TIJtiiJ •!>TIJtiiJ +tfVI-ZtiHVIJtUJ/AA£AU

.)f C J• ll+lt•.,~l I J•lltll+ t CUt ltll+UC Jt I I I /AN£ AI)

~UMI.I•\Jf ltl J+lllltlt+Uf hi I ~W4l•~UMIJ+UIItl I :.,uMZ•~UMV+VIZtll ~UV.J•~UMU+UI h f I :,Uir4V•VC l•lhVIZtiHVI )til :i,VMl•S.UMV+VI J 'II $V""Z•~UfotV+VCZtl t ~VMJ•.,~UMV+Vt Jt I I

00 )UJ J•& •• JJ•NPIJI UUU•Uf)tJII!tUMJ + Uf,ZtJI15>UN1 t UlhJII~\JHl VOU•VfltJIISUMl ,f. YthJII!.UMZ + YC3tJII$UMJ YUV•Yt hJJI$VMJ + VIZ.JI'!»VM.l: + Yf3tJII$YM) UUV•t.Jth,JII$VMJ + IJI,hJII$YNZ + UtJtJ)I,WMJ

1-111 •JJ •·~•• 1 •JJ 1 • cu•wu • cu•tvwu uw1 • cu•vw $C II+& •JJ+ll•:tt lhltJJ+JI • CZZ'VOV + (,UifYQtJ UOYJ + CJJIUOU Stll •• IJ+ll•~n• .JJ+ll • czJ•vwv • cu•uuu cu•vou + cu•~

,..,.., $1JJ+hll h~fll •JJ+lt , ...................................................................... . •oo RU~N

tNO

StJBNOUTtNE BAN!JOL , ......................................................................• CUt•MON /BAN.$/ H8AMChr.UMtii..Ktlf1601tAJl60tiOI , ...................................................................... .

C - IH!1-IA.U1E , ...................................................................... . ,.,.. •• 0 NL•NN+l NH•Nf"''+NN REWI~O I RCWINO 2 f4D•U GO 10 IZu , ...................................................................... .

C lo SHifT 8~0CK 0' EIIUAIIOitS ANO RUO NEXf 8l0Cl INfO COR! , ..................................................................... ..

c

100 NB•Ne+J 00 lliJ N•ltNN N""•NI'I+H BIHhtHNMI BINMI•u•"' 00 ll'J M•ltMISAN'D AIPhJr41•AfNMtM)

llU AfNMtHI•~o~,u

IF CNUMBLK ,[th Nnt GO TO 200 UO ~EAO 121 IU(NJtiAINtMhH•hMaANOhN•Nl•,.,l

IF INS tCUt 01 GO JO 100 , ..................................................................... .. C Zt NtWCE I:SLOCI(. OF fWAtiONS t THIN WRJJ£ TK!H Oft TAPE 1 , ...................................................................•••.

100 00 2JO N•leNN IF U.UhU tEO• (hOI Go TO ZJO Bl N IIIUINI lliNtll 00 zzu L•ZtMdANO IF IAINtLf t[Ut OtOI GO TO 220 C•AINtLI/AINtll l•l<t+L-1 JoU 00 210 lt•LtM8ANO J•J+l

llO AI J tJI•AI ltJf•(IAfNtU Bl I l•t:UJI•AIIhLI'8ffU AIPhLI•C

l2U CONI I HUE l)U CONTINUE

JF INUM8llt .EO• NOJ GO fO 30"0 wAITE fll fBINitiAUhMitM•ZtMBANOitN•l•ptiU c;o TO lUoJ

c••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• C 8AC<•SU8H IIUIIOH , ...................................................................... .

JOO DO JZO M•ltNN ... ,.,..., .....

160

DO 310 IC.•2tMBANO L•N•K-l

JIU BINJ•~fN,-~INtKIIflflt N...,•N+N'f BINI·H•OINI

11" AfNMtNUI•BINI NU•NB•l

~~c!~~.~~Qi o • Go ro •o~ READ Ill IOINitUINtflti.H•ZtMAANOJtlt•l•Jrtltl BACKSPACE I GO fO )IJoJ , ...................................................................... .

C OKDE" U~UO.HS IN 8 ARRAY , .....................................................................•. -00 k•O

00 •10 N8•1t"l.o48~k 00 411J N•ltNN NM•N+N,. k•(+J

-10 8fKt•AfNMtN81 RE TUII~ , ...................................................................••.. !NO •

SUUAOUT I NE LSRESUL , •.•.................................................................... (UMMO~ I lit I Tl Ht::l)f lll tNUMlL T tNUMNPT tNFt.L •'tFNP ,~P\;~(H CUMI"1UN /rtt't;;L/ ~._,1112 U t !I; I tOki)f JUOtl l eX(Pil HI t'YCP C ll~ I

-C--L~I'!W/1! -1-CO-AM! -NU<-~Ct~NU~(-N-P~J.tC-£-L-1-HH'H--.'tC~-l-l-'...-N (Uf"'fo.ION /OAH:,/ HUANOtNU1<4dL'-tti f 160 ItA t Uu•BO I (UI-4/<ION /L IF Tl NLAVtUhNU•.((L 12~t21 tNUMNP 12> •21 tH[ IC.f-IT 12!> l, NUM I I tl T CUI"'riUN /(,t.U"''I FNLI ll • ILilltCkLI ll •C. hcllltf..k~C l ltTRIZJ •'frtA 121 CUfo4..t011 /MAf 11 NUNL INttWLK f 27!t J •~HEAR 12 'I) I tPUISf 2 HI tVAMf 101 tAt.IJMO.) (UHoo4(}1t /r'IA Ul (IJNStCUEff 10 t t(XP I ll.l I tOOt lO! tVC,IlO ,.,, 110 I tllllO~E.) (t»>l"ll~ 11'\A. T J/ Cl f 11.1 ltPtH lltJ t tAP: I lUI tiJ(W'l I 101 tCEVZilOhCUCt t lOitAI.U CVMI'\QN IL:;.f 11 ltJelt:;.f lUtlOI t~l I ltlfJitCC ltJ) tPf lOhNPt•t tVOl. Ol•o4EN!,hJN ~I Gl) I ,[PSI Jt tO I~~ •UUU J OIMEN:;.IOtot $IC,I rt Zl~t) l t[PS.I I 12?,_,) l•Ol ~IT I 100•1) OIMt:N~IUN $TN[S!Jf ZJh J It STNAINIZ7!'1tl t tO I $PI lCOtl I EUUI VALENCE I ~rRE~5tAI ZUOOt ttl ST'IAI NtAt•t~OU Itt ID I SPt A 16o0001 t , ....................................................................•..

C INIIIAlllE , ...................................................................•.•.

c

REWINO " AlAO lttl II.:..IKESSINtMitM•hJt•N•ltf4LUIIJ(LJt READ r•t lf!ITRAINHitMitflt•l•lJ•N•ItHUHU.TI READ Ul tt OISP UhMhM•hllt .. •hfitU,.IIPTt IF liT eGf, ll GO TO JOO OU 10"' N•ltNl.H'4E\. T 00 lUO M•lt) SIGI T I NtHI•\Itt.l [P$1JINtMit0aU SIGI T I NtMI•STNE~Sf NtH)

IQO CPSITtNtMJ•:iTNAII''HNtMJ 00 ZOO N•ltNUMNPT 00 ZOU M•ltl OISPIJINtMI•OeO

100 OISPIIINti'II•OISPIHtHt

CTRF• 1~121+11 INIZI·T~I Z II•ICTNIII·f~lll II IIRIII•Illllll tMUHf•C IN 12 I •CTMF S.lL• ICNLI Z 1-TLt 21 I If CALf li•TLilll ~LR•I CHNI Z I•TN f Zl t/1 Tilt I t•CMRI l t I JF CFNLill tEW• TLill tANO. FJIIRIU .£Ue IACUI GO TO Z,O C.AMF•GAHI 11 GAMt>•CiA:.O.IZt EMOO•COEf I 2 I GO TO 28v

lt\1 GAMF•v,Q GAJ40•GAMI11 EMOO•C0fFI11

ItO OIHEN•£MCA>I I C.Aoi40'EMttUT'(MUHt I c•••••••••••••••·••••••••••••••••••••••••••••••••••••••••••••••••••••••• ( CALCVLAT£ frlL OI~PL-'CLMlfriiSt CVN~ltTiriU 1HUSt. lllf tH( Nl"' LA'Y{IIC , ..•.................................................................•..

JOO NUM:f'rwP•r.oM,._.,ILNt21 I' tfrtUMCNt-1 ,lilt.• 01 1\iUMHP•HAICOf~OHHPH,htllt'-ChPfhVM(,.Ptt

()() '"" lt•ltNUMNP 1P llf oLio ltUOIITI <10 tO JIO

X~•N XXN•N/~1.1

IF txXN •ll<~• fXNI)v•Ut eOH, N ,[Q, 1) PMUH 1 .. 0 Jlu '""Vl>l T •1•"

IF I~LAY .(ue 1 •OR• NOf.1NPJLNt21 tUh "-FNPt CtO tO 1U If' IN tU'.e ... ..~,.., .. ,...,lN•II tAHD• H .u:, HUMf1PtLNtZtl 'MODET•M£ONOD

JU OX.•UI.Z•'i•l)•F'h40!"(f O.,..Bil•NI•f':-.uor.T IF tNI.AY ef.Ue 1 eORe N el,.[e NFNPI GO TO JlO I' I"WV•<4'N._. elWt 01 ('il) TU U~ Dv J2>J "1•ltNU~CNJI If IN elY• 'KNPI!~IJ (10 TO ))0

UO CLlNTI,..Ut:. u' ox• .... u

ov.u,o Uu Ol!iVIN•li•I)I':JPIN•li+OX

D I !:.PI~ 1 21•01 :.OP IN •2 J +DY IF liT .ll4• NvMITI OIS.PINtl1•015PITOhl1+01( H Ill ,L\.1• NUI<~Ifl 1Jl!iPHh~I•VI5PIHNtl)+DY TO•!:.WoC T I OJ .>VI Nt11 .. Z+O I !.P t Nt2 I ••21 I' Ill .LT. NUMITI C,O TO 3*0 I' 11..1.-!0INtll ,L[, CfiH)H OU•lOO.O•OI!JPINtltiiCTRfli•TlllU H IVW:LIOhll .ut, CTI<If1Jf OIX•lOOeO•OI!!<PHhltiCTRIH•ctRil»> D I Y•} .... u,v•O I :,PIN •11 I£MOHT OJJt•OI !JP I Ntl t •OIMEN OJY•OI !JP I 'hll •01 M(N PH INf 1\Jv:, 1 N tUX tOYt I 01 !IJII NtMI tH•1 tll t TtJtO IX tO I hOJX tOJYtN

)•u CU"tT I NUf c•••••• ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• C CALC.ULAJ( ht( ~TN[~!tE!t AND STAAJNSt PJcJNT $-THAIN~

, .......... ····························································•

c c

ARE AT f•t\N(Aft:.•AR[ATN•O•O NUMEL•NOM£LI LNt7 I IF UtUMCEL eNE• 01 NUMEL•HAXOifiiOHELILNtZhN(fLtNUMCELil OU fri~U N•ltNUMEL If Ill ,LTe NVMJfl GO TO 3190 YN•N YYN•~USO IF IYYN ei:.!Je tYN/SO•OI tOR• H tEOt 1J PAINT 1•10

J90 MTYPE•NPNI N,, J FMOO£ f •l•u IF INLAY e[U, 1 eOMt NOM[LCLMtZI t(Cole NFEL, GO to 19~ IF IN eCit.e NUt'lt:LILNtU •ANOt N •Lt.. NOfoiLLILfftZ .. '1400ET•MEDM00

,,, oo •oo l•l•l SJGIII•Oeu [PSt t I •O,O 00 "00 J•ltlO

400 S.TIIeJJ•veO

CALL LSOUAOI N I

AREATT•AR(ATT •VOL EHOD•le0 1 UUU:.n•t•t a.o+POI SIN, 1•1 hO•Z•O•POISINI t ARE AT E•AA£AT E •YOL•EMOO ARt ATN•AREA TN+VOLIPOI S I HI

00 fill\) ••••• 11•.2•1 JJ•2•,.PNHh II Pfli•U•QIJJ•lJ

410 PtiiJ•81JJI DO 1t2U 1•9tl0 Pit J•O•U J.k•l-1 DO ,.z.., JC.•ltk.W.

•20 PII J•PIII"'Sf I tKI•PtiO

DO 4]\J l•ltl 00 frl)v K•ltlu

410 EP!tiiJ•f.P~III•!iTIItiC.I•PIIC.J 00 44v l•ltl 00 440 K•lt3 ••o $IC.IIJ•51&1ti•CIItK1•EP51KI IF II~LAY ,[y, 1 ,ONe r-.OKELILPhll •fQ• NFEL, GO TO ••2 IF IN ,LJ, NUMELILNtlt eON, N •GT11 NOMELILNtZtl GO TO ft•J DU •H l•l•J NUMREF •NOM[LI LNt 1 t +1 [LHT •HE IVHT I LNl•YCPI NUMRk.F t S Jlif 2 l •· I ELHT •GAHIHTYP[ t J SIGI 1 J•SI612t•POI SIN III l•O•POISfNt I ••1 SIGIJI•U,O

"ft2 00 ttlt~ l•ltJ IF IFMOO(t ,£Q, "EOHOOI [P~III•O.O IF Ill ,LT• NUMITI $TAESSI1hii•STR[SSINtii•SIGIJttO•!f IF 11T ,[Ue NUMITl STNESSIHtii•SIGIT INtii•:UGCit tF tiT eLT, HVMITt STRAJNIHtiJ•STRAININtlt•EPSIIJ•l00 1 0*FHOOtf JF liT ,t;.~.», NUHITI STRAIPUNtii•EPSIT Uhii•EPStlt•IOO.O•FHOOET

•• , CONTINUE

[•I ~TIU I Nl Nt 2 J +:, TRA I til Ntl t I /leO F•l !J T.tAI Nllhlt-;,TI<A IN I Nt 11 I I leO 6•S.UA I I I !;THAIN IN tl I 12t0f *1 2+F .. 2 J CJOI li•E+G QUI 21 •E·G QQI l l•le..J*G I' liT •Lf• NVMITI GO TO "'0 P.-!INT IOl ~~~ H tE"'WtUIJLKI NJ t~HlAA IIU tPO l$1 N I tl StRAI NCNtMJ tH• hJ It

1 t c.HII Lt tt..•l tl ltN 4~0 CONTI NU£

fMOOAV•AHEATEIANEA Y I POI!:.AV•fiR( ATN/AR(A fT IF Ill ,(Ue NVHITJ PAINT 1020• [MOOAVtPOISAV

c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••"••• ( CALCULATE PRINCIPAL ~TNE:ioJ£S AND PAUtlt CAlCUlATE NEW E At.D HU

, .................. ····················································· 00 470 N•ltNUMEL IF Ill tlfe NUMITI GO TO •SS ZH•H

~:H;~:~o •EQ• tlNISOtOI .OAt N .EO• U ftAJNT 1e2t Ott C•l $1NU~I Noli+>IRlSSI No Ill /ZtO

161

c

O•t ~TH(:., .. d Nt21 ... ;,1H.ES~I N•liiiZ•O UIJI• .. v•HC.,IHL,!ti~•H••z + 0••21 Coli li•C+UC Jl lii21•C .. VI l I QC.I~<J,v

, t:,ll-<l!.l'JINtlJ eECJe o.o eANOe D eC(.I• OeOJ C.0 TO.,. Ul*l•'Ju,.,l J •141 ''12,,•AT ANZ t -~.TRE !»511ft) I tO I

•16 HJYP[•I~PNII~,~ J OCYlT'I•tJfll·t~l21 LH:Yt lt•ULYll 1<1 I '~1: lti,)(Y .. IMT Y..-t IIUt U I' IOlVF•l ,uy, "'•Ot 00 TU friU ~fNLEIJ•Uf;.YI.k.V•..,,O GO TO 4~8

•tf 01;.\ILE.V•OLV~TH/tH:VrH ~ TALEV•OfVL(V I RF f a4TVPE t

,..,,. IF liT .l.T. r-.uMITI GO TO ,.6) IF fCIZI .Eu, (..,OJ 01~1•9?9,999 IF lOili eNEe UeOI Ut,I•Ufli/YIZI VHT•GI1f•,..e.J I, I'HYINJ •C.(, F!\11..1211 YHf•YCPINI•TLilt IF tYCPINI •LT. F~LI211 GHF•GAiotF*ITLI2t•Y(PINit IF IX<.PINl ,...,,, FNLiltl UH•GHF IF l.ll.t..YIPtJ •llle TLIUI UH•<IHF+C.Jt.MCJ•IU\:.Ptllti•H.IlrJ•~LL•YHU IF IXO'INI •'-''• CKLilll UH•UtiF+CJI.Mu•IClNtli•Tlflt•Y'HTI If IX1..I·'INI euJ, CRAIIII <JH•C,tt~•UA"'v•tiTMCII•Il(PINit•S.LR•YHTJ IF IXCPINI ,t,Te TRill! C.H•(tt1F CJl•UI111C.,tl G.!•UI21/lJH GJ•UflJIC.tl PH I r-.T 103..., • Ntl.;t T~E;,Jt N•M t tM•l 'lIt IWI L I tL•ltS) tSfALfY•Gl tU2 tCihN

•• , IF IN .L.E• NO~ELILN•ZI •OR, MJJ<~CEL eECe 01 GO TO •60 00 4t~'l M•l•NUMCEL IF IN •EU• NClLIMII GO TO *6:J

"'' CONTI Nut:. GO TO 1t7J

... 0 IF INUNLIN ,f;.(,l, vi GO TO *7V IF 10121 .Lr. <~t(..l GO TO •62

•U IF l!:.fRLI:.V ,Lf. ltO eAhDe ',H[ARINI eC.T. 0•01 GO TO 46, ••z SHF.ARINJ•u,u GO TO •Tv

"'' l:.IN I T•<.Uit~•CULt IMIYVL I *llut <!! t llOh~I••LA~IHHPU I [MUO•t:: IPtl Y 1 111ev-.OEVLEV'COOE H1TYPf Jt••z.o I PUI ~I•UUII"IfYPt:. I-FF liHYPt.: I•ALOC.lUII.IIll /(IJN~I tPS.AX•OEV:,tR/1 E INJ T•lleO-DEVLEV I I POl SUU •PUJ 51/ t lleO-OOtMT YPE I*EPSAX 1**2•0 I IF IPUISINI eGT, Oe490I POISINI•Q,.90 SHEARIN I•EMOOII2 eO*Il•O+POI$ IN II I BULl!:. t Pti•:,HEAN I Nil t leO•l•O•POI Sl Nil

lt?O CO,..TINU[ , ...................................................................... . IF CJl eLf• NUMIT) GO TO 600 NFEL•NO,..EL llNtll NFHP•NOI<~HPI LNt2 I

600 REWIND 4 WRIT( 141 Cl!.lTRES~INtMitM.•l•lltN•ltNI..:MlLJJ 'WRITE Ul fi;,TRAININtMitM•1dltN•leNU)o;tLfl WRITE 141 II UI!:.P INtMitM•ltlltN•ltNUMNPTI IF-I{.N--,Nlt NL~v--,oR; IT-,I'f[; hU,.HT ,OM, NPur-.(H ,Jit(, 11 RnVM'-PUNCH 1UJ~t I NtXCPI N I tYCP I Nl t I :>THI:;;,.:.I N•1'4l tM•I 1 ) 1 ,,.. i. .~ .... ·.-tl T I PUNCH lOJS• IN' XCPOU tYCP I ttl t I~ INA INI HtM l eM•l t) J eN•I t'-(JV[L T I PUNCH 10l)t I ~,A{PtNI tYCIJihltUULK.t .. l ,,:,rtc..A.HHU 1 P.J; ~lNh·'ll•l t~v.tc..f.. T 1 PUJrrfCH lU'-Vt I 1'-.ti0.-101 NtM 1 tM•l •21 t I Ol ~PHIItMI•M•lt 21t ,., h Nt..~hPY I ~fTUqfrrf

, .................. ····················································· l\JOU fORJo1Al IS,.Hl NP DELTA .. Jt O(LfA ... Y .C-DI~P Y-OJSP fOI.t.L• 1 48H X-OISPIIili Y-OJSPIH l011r4f;.lt YOI"tH "P II

lOOS FORMAT ll4tt7Fioe4t1FllehUI UHO FOfMAT C!JuHlELE ELA$ fo\00 bULK. MOO SHEAIC MOO NV [PS•x

l ltlH I::PS-Y C.AM•JtY EPS•l EP~-J C.A"""'Ax ELE II l~U FOHMAT ll4•lFl\l•l•F6,1t6ffl 1 hl•1 1UZO FURMAT Cttti AVI.I,FlO.&.lOXtf'6tlt IU2' FOM:MAT C)4tHit:.U:: SIG-X SIG•Y TAU•lY SIG•l SIG•h

1 )uH IAv-MAX THETA Slul/3 LEVEL SIVJ/Vt'f $1<1)/<tHt 2 llH TAUH/GH ELE II

lUJO 'URMAT tt•t6FlUeh6FI.lt('l lOU FORMAT 111Ut,f'IV•11 au•o 'OAMAT c uo.-.Flv•••

c ...................................................................... . [NO

APPENDIX D

COMPUTER PROGRAM

FOR THE

INTERPOLATION OF FINITE ELEMENT STRESSES AND STRAINS

FROM KNOWN LOCATIONS TO DESIRED LOCATIONS

Identification

This program which consists of a main program (FEMINT) and one subroutine (SLAE), was coded by F. H. Kulhawy (1969) using the SLAE subroutine coded by S. G, Wright (1969).

Purpose

The purpose of this program is to provide a means of evaluating the stresses, strains and material properties at desired locations when values are known at other locations. This program is primarily intended for usage with finite element embankment analyses where a cofferdam is first built using a fine mesh, but will have a coarser mesh when constructing the embankment on the cofferdam.


The main program (FEMINT) reads and prints the input data then inter~ palates for the stresses and strains at the new locations. With the values of the stresses, the corresponding non-linear material properties are then computed using hyperbolic nonlinear relationships as described in the LSBUILD user's guide.

SLAE is called by FEMINT for each interpolation to solve the simultaneous equations required in evaluating the stresses or strains at the new locations.


1) Control Cards

a) Card 1 (12A6)

2-72 RED - Title card for program identification.

b) Card 2 (3I5)

1-5 NUMELT -Number of elements in cofferdam analysis.

162

6-10 NUMINT - Number of elements to be used for the cofferdam in the embankment construction sequence onto the cofferdam.

11-15 NUMMAT - Number of different material types in the cofferdam.

2) Material Property Cards (15,Fl0.0,8F5.0)

(Number of cards required = NUMMAT)

1-5 6-15

16-20 21-25 26-30 31-35 36-40 41-45 46-50 51-55

N

COEF} EXP

DD } GG FF CCC PHI RF CODE

- Material type number

- Tangent modulus constants

- Tangent Poisson ratio constants

- Cohesion - Angle of internal friction (degrees) - Failure ratio - Code for linear or nonlinear material properties

(0 = linear, 1 = nonlinear)

These values must be in the proper units and are discussed further in the LSBUILD user's guide.

3) Stress and Strain Cards (Il0,5F10.3)

(Number of cards required = NUMELT)

a) Stress cards

b) Strain cards

These cards are punched directly from the cofferdam analysis and are directly used for input at this point.

4) Interpolation Cards (I4,2F8.2,514)

(Number of cards required = NUMINT)

1-4 }. J 5-12 XCTR

13-20 YCTR } - Element number in the new finite element mesh. - X and Y ordinates of element stress and strain

point in the new mesh. When the element nodal points are numbered in the form I,J,K,L, the stress and strain point has ordinates mid-way between J and L.

21-24} 25 - 28 NUMEL (1-4) -29-32

Numbers of the 4 elements in the cofferdam analysis to be used to interpolate for the new element location. These should be the· 4 closest

33-36 37-40 NUMEL (5)

elements, 3 of which cannot be in a straight line. - Material type of this element.

163

PHOGIU,fol: FEI"IffT I I'IPllftOU(PUTtPU-.CHI Cllllllllllllfllt I I I II II lit I 11111111111111111111111111111111111111111111

l)JJoolt.,.. .. tiV" HlLHilltXXf J)U It Yr t l'JOJ • :.,lH[,l.Jil~Ot Jl f!.TRAINI t!JOt!l tJI··II.tf ... dV .... t..IJU f '> ltlXPf) I t~'llJf "I tllul) ltFf. I !.tl t<.C..C.I ~ ltPHII ~ ltMFC iU t

I COllCI'tl OIMU.!.IOr• ,I(JI H t(..P!)I) I tAA.Utft ltUUI C. I tClU ltHUMEl.l tl

(111111111111111 Ill 111111111 lllllllllllllllllllllllllllllllllllllllllfte

( Ht.AIJ ANll PIC(ftf IJ~PUT D&fA c••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••••••••••••••

lltLAI) lU\hlt ttt.a•~'"LL T tNoJMIPtf tNUMI-IAr

c ..,NI,.T .ZOO>Jt Hl.OtHoJ.~£&.TtNVMIHTtftUMMAT

PRI'-T N(AIJ

I PHihT

I

2..1•,, 10 1"' t C NeCOtFI ~I ttl!. PI H ltOOINI t6GI "'It I"F 1"1 tCCC INI tPHI tr.t t

NF ll'tlt(OOE ll'fltft•ltHIJM~AT I 2Uh t t ktCOlf f ttl tllPOtl ttJUHU tUC.t ftl tH I HI tC(( IPtl tPHI UU t

RF l"t ltCVO£ f H t tN•ltNUMMAT I

IJP'IINT l!Jl) lil:t.A~ i"'l"' • I Ht.lllf N I tYYC NJ tl ~1 HE!t!l UhMI tH• It )I '"''ltNUM£LT t P" I t•T 1u2 ... • fHt.IIXf HI tH'I "'' t I :tTNt.:.~t NtMitM•l t.J ltN• hNUMLLT I

Pk I ~T loJl•1 Nl AU lUl"' t tHe.IIXI'-l • YY t HI 1 t .JTNA lt.fhtMt tH•le S leN•ltNuHEL T I PM I "'-f lUlv • ll-4t.UI Nl tYY I HI tl ::O.fAAINt Nt"U •M•l•J ltN•ltmJMtL T I

c••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• C INTlHJ.'uLAfE FUR .Stlf£$$1.,!; AMO. STRAUUt COMPUTE It G AJIIO NU , .•..................................................................•.•

PRI~T 21Jl0 . OU J!JoJ 't£'1i'•l•NUMJttT AEAO lu)_,, JtJI.(Tit.YCTAtiNUM£LIJ.JitJ.J•htl

c C IP4TEAPULA1'E: FUN $TR(SS£S. c

00 Uu ll•ltJ 00 lOU l•lt41 CCC lt•~o~eU II•NUMU.III A811 J•S.TRE.;,S11Itll A All ti 1•1•0 AAIItZI•JI.XIIIJ AAIItlJ•YYIII J

1"0 AAIIt«ti•JilfiJII'IYIJit ltOUE•O CALL ,:,LA[ 14AtUih4•Ut"t4tCCtOE:ftK0011 IF UOOE ,[Q, Ol t".O TO I''J PRINT )Y!J"'t XCTMt'YCHhHtUM[Lfllel•l••t STOP .

110 'IGfli•CCili+CC IZI•.IICTR+CC C J I'YCTIIII+CCI •t IX(T .. •YC'Ut PvttCH 102"' • .JtMCtA,YCTAt U fGCll tk•l t ll

c C CUMPUTE [LAST IC PAUI'ENTIIS 'AOM AIOVE STNISnS

'

c

O•OtS•t SIGI 2 J•SIGil J I R•o,s•tSIGUI•SIGtlll

TAUI"AX•:..tiRTI:..ICdli•!JICii)J • R•RJ .SIGl•C+UUMAX ~16J•I.l•TAoJo.-&X

---or:v~-H,-r.-Jw- .. -JGl 114TYPE•~UMEL I! t AN6•PHI U4f 'YPL 11!7 1 2'151719 U Dt.V l•Z•u•CCC t MT'IP( t•Co~ IAPU,IItl,O•.SINI Altril t Dt:V2•2,\J*~INI ANr.llll •11-S.I PtUirfGII O(VF~oieOfYl+O[. Vl•S JGl OEYltV•OEV!t f RIOF'VFH IF fSIG! ,c;r, v,llt GO TO 170 tsULil•~HEAR•POJ S.•O•O Gv TO U·J

ll'O [I Nl T•<.OfFIMT YP[ 1•1 "'IGJII[XPUU'JP£ I I f.MO~ohUNJ f 1 lll•U-i>EVL[YI(00£fMTYPt t 11'2•01 POl SI•~GI~IYPE 1-f .. MIYPt I•ALOGIOI UOU [P~AJI•Ot Y~TMif[INI Tlll.O-b(VL[YI t POts• POlS Ill ll•U-001 MTYPU I(PSAX t••hOI IF U11 01$ ,Gft u •• 90I POJ~•0,,.9Q S.H£AR•t.MOD'II2 eUifJ,vtPOI $1 I 8UliC.•:tHf.Atll I ltU•l,QIPOI $-I

110 PUNC .. 102"'' JtXC fR tYC fA tBULl•.SH!AihPOI 5

C INTERPOLATE FOR SfAAI"S

' oo ,no K•l ,, 00 lU\J l•lt• CCIII•t.~•"' II•NUMELC II eotl t •S.TRAJNIIItk I &AI ltlt•l•O A&C lt2J•UCtl I ..... ,, . .,,.,,,,

ZlooU AAII••t•Ufiii11'Yfllt I( ()I) [au

CALL !fLAE UA•I::IU•"•"'•"•hCCtO(TtkOO(I IF f IC.OO£ e[U, ..,, GO TO .Z'O PM INT :Juvlt JI(UhY(TAtiH\JM£Lf I ttl•l••l S.TOP

Z!JO EPSfli•CC ll l+CCIZ J •XC UhCC ll )IV( tA•CCf •t •XCtA•YCTiit PUNCH 102"'' JtltC1'RtYCTRti!P~IIlltK•lt31

Jf.JO Pfellltf 1UJ"'•· JtA(fP'It'ICIMtlfrtUMELIJJitJJ•lt,)ti$1GI.IJitJJ•ltJitiUL&t 1 SH(AH,POIStf[PSIJJJtJJ•ltJt

c••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• IJV\.1 FO'ftMAT I UA6 I lt'l ltllU FUAHAT r lhFlO•oJtiF,.OI lU2tJ FORMAT t llUt''H' 1Ut31 lUlO FORMAT tl-.t2Fth2t~llttl)f'l0•JI ZUUO PORMAT t 1H6 Ill llAb Ill

1 ZJH NOt REAO IN [LtMfNT$ • I" Ill 2 ZJI'4 NO, l"tERPt U.£MENT$ • t• Ill J ZJH '-0• OI,F£A!.NT MAlL,S,• • t• lilt

ZOOt PORMAT t,X,,H MATLtl.lltZH ktiXtlH "tiXtJH OtlltiH GtlltlH ,, ••• l ZH Ct6h4H PHIISH RFt'H COO! lit

164

181~ ~z:=:; u~~:l:~~''tt:~tAJ S.fHt.~~E~ , ....... " LlhtiXtl~• ... .~~.zH ,.. I )Vfi S.IG•II: SIG•'1 TAtJ•XY Ill

2UZO I"UHMAf l.iUHIINP\.11 ELEMENT .ifNAJN:, II f:IAt"H t.Ltt1Xt2H JltiAtZH 'YI 1 )..iH [P:,-X [P~·Y 6AM-u Ill

I~JO FORMAl lllHI IHTlRPOUTIOOI INPUT AND OuiPUI Ill l "H [L£t6X•2H X•61tZH YtlOH FONM[N M(SH [l.£eM,U, 2 'OH !if(,•X SIG•Y TAU•li:Y Alti.IC !':·HEARt J 4VH POlS [PS•X fP~-Y GA'4•XY Ill

ICIOO 'OAir•.aT IJH X•,lVtlt5olltJH Y•,IO•)t,Xtii.Ht LLLMLH11J•""'~Ilt,."l'-•'"' t J001 ,UHMAT I ]H .W.•f h•Jt!:lltll'f Y•, lUe.J •~Xt IUH LLtMl#IIT~••&••'h,HI"''"' t , ...................................................................... .

END

, ••••• i~2~~~!! ~!. ~~ ~~.! !: : ~~: ~ :e~~ :~ 1: ~!: ~~: ~~!: ~~2!: ••••••••••••••••••• OIMEN.)JON AAIIU tltZ l t8BIN11 tCC I Nl J

(DIII11il<1; ;;; ; ~ t =; = ;1#6~1111e;Gww•;w;w&55~~1i1i._1J1t-.-.-.-.-.-.-.-.-.-.-.-.-.-.••••••••• C MAlfll l.LIMIHAfiON LOOP• INfERCHAMC.£ RO•~ IF ,.(([!i~ARY , ......................................................................•

(

c

NMl•~-1 OEf•t,u 00 ISO K•ltNMl KPI•IC.•l L•K

OU'IOU l•KPitN I~ IAbSIA411o(ll oLEo ABSIUILollll L•l .

100 CO"T I NUE IF It eLE• ltl GO TO UU 00 110 J•ll•" TEMP•AAIK,JI AAU,Jl•AAtL•Jt

110 &AfLtJt•H..-~ fEAP.iHtH IC.I

::~~::i~!~· DE(•·OET

110 PIYOhAAIIC.tKJ DET•OET•PtVOT If UUS.CPIVOO •LEe ACCI GO TO 999 oo ua I•KPh" FAC•AAI I tiC. II PI VOl

'" r,u.o.o 00 llu J•kPitN

IJO AAIItJI•AAII•Jt-F&C•A .. IltJ, tto netlt•nntti ... FAC•oau.t

GO TO 100

, ..................................................................... .. C CHECK: LA~T ElEMENTt THE.N IIACl•$0\.V[ , ..................................................................... ..

PIVOT•AAUhNI OET•DEf•PtVOf

, IF UUS.frtiVOTJ eLf, ACCI GO TO 9tt

_;1 ~~~=: •em liP I VOT

100 JPI•t+l :,uM•O•I.i 00 ZlU J•IPltN

110 S>UM•SUM+AAfi•Jt•CCt.JI CCII t•t88f ll•$1..it41/AAt I tl) 1•1-1 " II oGTe 01 GO TO ZOO AETuAN

9tt KOOE•I RETUIIN r110

APPENDIX E

COMPUTER PROGRAM

FOR THE

STABILITY ANALYSIS OF EMBANKMENTS OR SLOPES

BASED UPON FINITE ELEMENT STRESSES

Identification

This program which consists of a main program (FEMFS) and four subroutines (CIRCLE, FSSTR, SEARCH, SLAE), was coded by F. H. Kulhawy (1969) using the SLAE sub-routine coded by s. G. Wright (1969).

Purpose

The purpose of this program is to evaluate the factor of safety of embankments on rigid foundations using stresses obtained from finite element solutions. The analysis is performed by evaluating the mobilized stresses and strengths along circular arcs passed through an embankment and then calculating the factor of safety from these values.


The main program (FEMFS) reads and prints the input data and monitors all operations by calling the following sub-routines for each circular arc to be analyzed:

a) CIRCLE computes the intersections of the circular arc with the embankment, as well as the size and subtended angle of the arc.

b) FSSTR computes the factor of safety of the circular arc by evaluating the mobilized stresses and strengths at points along the circular arc.

c) SEARCH is called by FSSTR to select the proper input finite element stresses to use in computing the mobilized stresses and strengths at desired points along the circular arc.

d) SLAE is called by FSSTR to solve the simultaneous equations required to evaluate the stresses at desired points along the circular arc.

1) Control Cards

a) Card 1 (12A6)

2-72 HED


- Title card for program identification.

165

2)

b) Card 2 (315)

1-5 NUMELT - Number of finite element stress cards to be reado (Maximum = 500)

6-10 NCIRC - Number of circular arcs to be analyzedo 11-15 NSLICE - Number of segments along the arc to be used

in evaluating the mobilized stresses and strengthso (i.e • , "number of slices")

Geometry and Strength Cards

a) Card 1 (BFlO.O)

1-10 TLX - X coordinate 11-20 TLY - y coordinate 21-30 CLX - X coordinate 31-40 CLY - y coordinate 41-50 CRX - X coordinate 51-60 CRY - y coordinate 61-70 TRX - X coordinate 71-80 TRY - y coordinate

t +y References axes are -r +x.

of toe at left of embankment of toe at left of embankment of crest at left of embankment of crest at left of embankment of crest at right of embankment of crest at right of embankment of toe at right of embankment of toe at right of embankment

Slopes facing to the left are analyzed with this program.

b) Card 2 (5Fl0.0)

1-10 11-20 21-30 31-40 41-50

CTR CCC PHI XMAX YMAX

- X coordinate of embankment centerline - Cohesion of embankment soil - Friction angle of embankment soil - Twice the height of the highest element - Twice the width of the widest element

If the stresses from a half-section of a symmetrical embankment are read in, the actual value of CTR must be read, If the stresses from a full-section of an embankment are read in, the value of CTR must be set equal to 0.

3) Finite Element Stress Cards

(Il0,5Fl0.3) Number of cards required = NuMBL±

1-10 11-20 21-30 31-40 41-50 51-60

N XX(N) YY(N) STRESS (N, 1) STRESS(N,2) STRESS (N, 3)

- Element number - X coordinate of element stresses - Y coordinate of element stresses - Horizontal stress (ox) in element - Vertical stress (cry) in element - Shear stress (Txy) in element

These cards should be punched out from a finite element solution with the above information on each card. An option is provided for this punch in the finite element program LSBUILD.

166

4) Circular Arc Cards

(2Fl0.0) - Number of cards required = NCIRC

1-10 XY 11-20 YC

- X coordinate of center of circular arc - Y elevation of center of circular arc

167

PMOUHAM HMF!t t II'\IPUT et>UT..-UT J , ......................................................................• COMMON /AcJC./ HLDIUieNUMLLfeN.!.lfCEeCCCtPHI COM1>40~ /Off/ ILJitTLYt(LXt(L'ttCRJte(RYtTRltTAY (Oio\.--tQN /CaHI I XC • YC eNAOtX I N.JL tY I NTL tX I NTNtY INTRtAt.IC.L[ tAR( tOANO CIJ•4MUN IJ..,LI XXI ~001 tYYC 'OOt tSfMtS..SI,OOt)l tXMAX.VHAX (UHJ<o;UN l~ttUI .:.J(,I.)It.UitteftJtOBt•ltCCt•I•~U,..[Lf61 , ......................................................................•

C MEAD AlotO .,MINT (",t;.OHtTIIIYI MAftNUI. lllltOP(ItT'f At.IO ELI!M!friiT DATA , ........................................... , ........................... . 100 lft.AO lOll..,., HLUtr-..UM(l.ftNCIMCtN5\.IC(

ME""' lU t .. ' f LA • YL YtCt.X t(LY tCIU 1(1CY, TRill 1 TMY 1CTA tCCC ,PH ltXMAJttVMAl HlALI lU .h1 • I H tAll NJ 'YY f NJ tl ::.TNE~·I NtMI tM•ltl I tN•ltNUMEL Tl IF tCTR ,[tJ, o.~J GO TO 31)0 OU 1.uv N• leNVMf.L T NNtNIN..SM(l T XII: IN"- f •aJU Nl+l,v•l CTR•IOI.IIU I YYINNI•VYOH !:tlHt:.,::.Udh II•SUH.S.~INt1) 51'R[$!:t I NNtll• ;,Hi£SSIN1ll

lUO .:.lHI:.::.~fNN•JJ•~.,IHES.JINt)l fiiUMELT•Z•rtUM[lf

JUO PRINT zou,~, H[OifLX.TLYtClXICLY'tCRXtCRYtTlltXtTRYtPHitCCC PH1r.l1' 1.1..11'-' PlUtH 10 Jw, I~ 1IIIXUO tYY I Nl t I S.JJIE.S!dHtHI tM•1• 3) 1N•ltNUMlLT I

c••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• ( ANALY~E EACH !.t.IJ.I CINCLE 1'0 Fl~O ITS FACTOM W' $A,ETY , ...................................................................... .

PHI•PHII!J 1 ,l'l) 17Y~1 00 4..JJ "'4•11NC INC CALL ~ECO~O (fiM£11 R[AO l01Ut XCtYC CALL CINCLE CALL f'~$TR

CALL .SECOND IT IME21 T IME•T I MEl-TIME 1

•ou PAIIH Z020t TIME c••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• 1000 FOIIMAT f lZA6 I lUI 1010 FORMAT 18F10t0 I ~FlO,Ol JOJO FORMAT ll10t5FlU,JI ZOCJO FORMAT 1lH61/lZA6 Ill UH ••• [MOAN~C;fo4£NT COORDihATtS ••• II

1 UH TOE LEFT X • no,) t5Xt 4H Y • flO,) I Z 1dH CREST LEFT X • FlO,) ,, ... 4H Y • 'tO,J I J Uti CA[~T RIGHT X • "O•l t~Xt ,.., '( • FlO,J I ll UH TOE RIGHT JC • F10•J ,, .. , .H 'f • Fto.J Ill , ZIH ••• MATtNIAL PNOPERTIE$ ••• II 6 17H FNICTIQN ANGLE • PlOtJ I 1TH COHESION • ,10•1

lOhl FORMAT I , .. H1 [LEM[Nf ll•ORO V•ORD SIGMA•X SIGMA•Yt 1 lvH TAU•XY 1J

lOlO FORMAT I UH- TIME ISECONDSI • ll'lOeJ I c·············································••o••····················· GO TO 100 [ND

!1UUHOVTit\IE CIRCLE c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••

COMMON IDEF/ ILli:ITLY•CLXtC:LYtCIUtCRYefA~tTRY CO/o4Jio\Oirrt /UHI I XC 1 YC tMAO•X I NTL1 Y I Jrt lL •.Ill N lH 1'1 llrrtTRtANGL[ tARCtOANG , ...................................................................•...

C LOCATE L[Ff INT[RSECTION

, ............... ·····················································•·· RAO•YC•TLY lOEANU•Al ANI I C\. Y-Tl Y I lfCI.X•TLX I I AA!'»E•XC-TLX S JU(•SUR T t RAO*RAU+.1ASE•DASE t tU::,A~U•A f ANIMAOIIll\ ~.r t -TOEAirrtC. $LPANU•l• 1ft1 S92&-A!'» 1 "'f SIDE •!.IN I BASAt.IG)INAOJ !,LPHT•!II L>E•CUS f UASANUI+NAO*COS I 5LPAHC.I Y INTL•!lLPHT*$ I Nt TO(ANG .. fL '( X INTL•.:..LPHl*CU~I 'fO(Itt\IC:d +flJt , ...................................................................... .

C LOCATE RIUHJ INTERSECTION , ...................................................................••••

c

VfRhMAU-1 JtC.-TLl I •T ANt TO lANG I .SLAhG•A~I Nl \t(HT*:JI NC l• ~ 70l?63•TOEANGt/RAOt ~LOIS f •YEN f •co:,t 1•) lOT96l-TOCANG I+MAD*C05C SLANG I TUI !,T • .. 1..01 .,~T+ II HAO-VlM T I/!» IN I TU£ANCd f X INTR• TOI !loT •CVJI TOCANG I • TLX Y I filM• fOhT•!.II Nl TOf ANG I +TL Y IF IYINTH elE• (LVI C.O TO 10('1

\'[RhYC-CLV X tNlH•IIC+~YFCT I IIU,IJ•HAI>-V(H T *VERf l Yll'flR•CLY . IF IXIIrrtTM •Lt.• CIUI (•0 TO lnt'l

TQt.ANC..•A fAN I t(NY-TRY »!I TN l•CNX II ::.JOE•I JNX-XCIICO:,t TO(ANG I HE IUHT •I HU•XC I • f ANI TOEAhGI-YC+ TRV $LAfiiV•A;, JltfltLI(,ItT•~Hu ', HoT?f:IJ•TOEANt.l tJHAu• !~LUI ~f •:.doC• I N.AtJ•co:,c ~.LAkU)+HLJt,Hl •COSC 1, !.1707'16J•TOEANGII 't JNUt•!lo\.01 JT•~ t't f TOE.ANG f + lNY X INfR•AU~I.:JLOI !JT•CO!ll TOEANC.I-TRX J

c••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• C COMPUf£ Altl.l PNINT CIRCULAR ARC PROP[Rtf£~ c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••

100 LlHGTH•~QNTt I 'JNTA•Y INTL t *•2+1 XINT~t•X HilL t **ZI ANGLE•ACO!If t 2 ,uiAAD*ilt40-LEMGTH•LCNGTHJit z,oeRAD•AAOI I ARC•R:AOIANGLE PRl~T lOOut RAOtXCtYC,XINTLtYINTLtXlNTihYINTAtAAC

IUUO F~~Al ll'INl••• ~LI"' C)HC.L~ Gt.OM£TW'f ••• II 9H NADIIJS • 'lO•J I 1 19N CENft.R POINT l • '1Q,J •'Xt •H 'I' • rlO•J I l'IH LEFT INTER, llll • '10,1 t~At ltH 'Y • 'lo,J ] 19H RIGHf JNUAt l • '10tJ •'~' 4H "' • nn,J • lJH ARC LF.NCiTH • F JQ,J I ~[TIJIIN uo

168

5UflUOUTIItf. F55ofR , ...................................................................... . (UH•"Uif /.lf]C/ H(Oil211fiiVI~[LTtN:)liCEtCCCtPtif

~3~~:e~ ~e~,) ~ !t: ;t~~A&~: i ~~~ !~7~;t~: ik~~: ~~~TAtAhGLE tAICtDA"O COMMON IJ'-LI AXI~OUitYYf ~UII~fR[_..,niJWtllt.lMAhYMA. CVM•'lOt. /MNIJ/ ~lUI )JIAAC,.tltltttbt•t tC(C•I tNIJMt.t.t•f OIM~N!.ION XSI 1v011YL11001 , ...................................................................... .

C l"lliA~IlC , ....................................................................... . PRINT 101JI) OCV 1•2 tLI*C(C•CO~ I Pit II I lltO•S INI PHI I I DE.V,•,:,o•:.dNIVttt f/tl,O•!>INIPHI I I SLICE•It5<UCl OAP•U•A"t(,li./SLtCr M'IIGo•A~I Nf t YC•Y I NTL I IRAOI ll~C J)•XIt\11'L 'I'll li•YINfL T •NV• T ANC•JII(f • f AUT •5t TAL (VT •0,0 , ..................... ·················································· C C.OMYUft. :,LICt. l.I(O."'t.lH'ft THlN t.VALUATL o~TAL~S. :.TAT(

, .•••.............. ~···················································· 00 "vv NCOur~T•ltN~LICE

c

J•t\ICOV..,f+l TA~C..•TAN<i+OANU XM JJ •li:C-HAIJ•COS.I ANG+UNCi I YL.I J I •YC.-HAO*JI N f At\IC.HANG I 5LAHC•HAD 1 UANU TARC•TAI-IC+5LAHC C T NANC.•AIIIC,+ T ANU-u. ,*OAHU ALF•A T At\11 fYL I J 1-YLI J•ll tl f .1$f.JI•JI$ I ..t-111 t XC T~•JIIC-HAO•CVSI CTRANG I YC TlhYC•MAO•!i IN ICTAA .. GI

C CUMPUTE E~UIVALENT f[H ~TRESSES IXtYtXYI ON BAH 01' SLIC[ c

POJNT•U•U CALL ~t:ARCII IXCTRtYCTRtPOINTf JF IPUINT ,(u, v,OJ GO TO ISO NUM•NUMEll1 I 00 llU 1•1•1 NUMEL f l+li•,.U""ELC 11

UO SIGII t•STRESSIPIUHtll GO TO )OV

Ito co 1,"' tt•le'J oo zao 1•1•• CCfii•U,U

~~;~~:~~~~~Sf ll'tKl AAf ltU•l•U AAflt21••XIIII AAI ltl I•Y'YII I I

100 AAII••I•XXCIII•YYIJII lOOE•O

f~l~K~~:E .l~~ 'Sf'•G~'t&"l\~ •0£ r llOO£ t PRfkT _)..Ql.""·' .UTH.t.Y..(fR!.{N.UM£.t..U-!-e-!~.t«.J STOP

ItO ~IGIK I •CC I li+CCC ZJ *XC fR+CC I J r•vc TR•CCI• t*ICTfi 1 YCTlll c C CllfoWUTE MVOILilEO ~TN[$~(~ ANO ~TNEHC.TH OH BAS£ W' SLI(£t T~[N ,, c

100 DO•SJNf2tU*AlFI E[•COSI z,u•ALF I ~~U~•I u, !1*1.:.1 U12 .. ~IGt llll+t V•Si•t U61 l J-::.16tlf l*l£1•15161 J)•OCII T AU• I U•!J* IS IG 121-S IGtl I 1*001+ I Sf Gl] l *Ell T AUh r AUT • TAtJ*.:.lAAC FF'•U.NIPHII 5TR•CCC+:l-1Ufii•FF ~TH T•~ fR T+::. fH*~LAN( F1•5TRITAU F 1•1 $URT IF l*F l+FF•FF l•fF Ill SORT lleO+FF*F'F I•FF I

~i~~!~:~-~~t~~I; ~L ?~1 ~ ~~~;~gy~; r ~• Grz ~-~IGta • ••••z t .STRLEV•O(Y!'JTR/ f OEY1 +Of.V2•!:. I G) I S TRLEVT•:, I HLE Y r +$ TALEV*!:!L ARC F)el,u/~lRLEV

fltOO PHINf ht.z..,, j'4(UVHTIACIIhYChhiHIJ ... t.LIII•I•l••l•l!tJC:,IIOt.:O•ll)lt~IW.t 1 T AVtF11F l•:i I 6h0[Y;., lRt., lNLl "'' l

c•••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••••••••••• C AFTER ALL .SLICE~t CO~PUT[ OV[RALL F,Se , ...................................................................... .

TAUAV•UUTIUHC F,.l•!iUCTirAUT ft 2•1-Jtnl r I H l*H l+H'H f -H f II.JoiJNl I l•U•H•H I•F'F I S flfLAV•!:tlNLEV fIT ARC Ffl•lelll::.tMLAV PRit\IT 1U)..It JAUAYtFFltFFZtSTRLAVIFFJ , ...................................................................... .

1000 fORMAT tz1H•••• ~LICE YALU[.!t ••• II Z6H :.liCE A-ORO Y-oRO • l !J4H [LEMtlrrtTSI USCO $J(,•I SIC.-Y TAIJ 1Y $IG•N TAv-Mt I •BH F~l[1. ,.!tiTlJ !tiG-J !.dUl•l L[\r'(\. f'SITZI /I

1010 ,URMAT 1311 A•f·1..,tl1~XtlH Y•FlOe]t~lllltHtH lLtM[NTS.••I•ttA•tH lt•t:i• 1010 FONMAT t htZflO•J••I••ll'l•ll 10JO FORMAT t29H•I .. FlftA\. ~AF[TY F ... (TOAS ••• Ill

1 22H AYG, TAU MOtJILIU.O • fiO•J•!a>Xt.H FS .• Flutl II 2 ]S,H F~l TOTALt•CALCULATlO FROM AltO\'[ • rtG,J II l Z2tt AYG, ~TR£:t$ LEVEL • f10,J•~X.t.H rs • r1o,s )

c••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• RETURN END

StJA~OVTPt( SEAHCH IXCTRtYCTA,POINfl , ...................................................................... . CUI-4~:vN /AU(./ •"~lOIUltNUM£loftNUICE•CCG•t-'HI (IJM"'Uft /J,.i../ .UI~OUIIHI~U\IIt.,HC[o~,l~iJUt)ltAM-.x.YMAJl U.IM•"'N /M,~U/ o~Jj,IJttiAihllijtiJUI41t(CI~t•NUMlll61

c••••••••••••••••••••••••••••···~··••••••••••••••••••••••••••••••••••••• C .;,t,L\.Ll l <.LU,l~f E.LtMt:Nf.!t tO OJLIP '-IMCLL PCJINf UCflhYCTJI:I , ................................ , ..................................... .

NIJM[Uli•J DO UO J•ltZ OIST•lOLIUvUU• DU lOu h•ltNUM(LT IF UlJSIU:tfiii ... XCTIIIII ,GE• XMAXI GO TO 100 IF IAIISIYYINI ... 't'CTAI ,c,t;;. YMAXI 60 TO 100 0 I ~fl.,• ... wJt T I I XX IN I •XC TR 1''2+C 't'Y I NJ•VCUC 1112 t IF IDI$TN ,<:.£, Otstl GoO fO 100 IF I'll ,f.Oe ~UH£lfltl GO TO 100 01!"•'-0ISTN N14IN•N

lUU CUk T I N\Jf NUM[LIJI•N~IN IF IXXINHINI ,N£, kCt,_ eORe YYINMIHI efrlte YCTAI GO TO 1,0 POINf•l,v Ml TUQN

UO (C'"tffi'IUF' NA•,.,UMlL Ill NA•NUM(L f 21 NUMELI SI•NUM[I..I 1 I NUM[L I 6-t •NUMEL I 21 , ..................................................................•••..

C SELECT 2 CLOSCSI HOMU• ON V(Mt. PTSt TO ABOV[ PT5e , ..................................................................•.... H IAtt~IIUINAI-lXCNitll •Gft AB~IY\'tNAt•YYCNBttl GO tO 200 J:CHf.CJ.•Je!I•XMAX 'I'CHl(J. •U•l" IV MAX GO TO Z~v

lUO XCHlCII.•u,z!J•xMAX YCHECK. •O, '*YMAX

ltO NU'~'l:LIJI•v 00 HO J•1•" N'41N•O N~flf•N\JM[ll J-21 OIST•loouuoo. 00 JOv H• hNVMEL T I' IAO~txXINt .. XXCNNPfll tGfe XCH£(11 C.O TO JOO IF UU,:,fYfiNI""YVINNNtJ tCITe YCKlCit GO tO )00 IF IN ,[u, NA ,()M, N •E<h fll8t GO TO JOO DISTN•!,QR f f I XX IN I-XC YAJI12+1 'fY INI•'fCTA 11121 J' IDISfN oGfo OISfl GO fO )00 I' IN oEOo NUMELIJII GQ fO )00 IF f J • [0, J t GO TO 290 NVMELI .. •N DO liU JJ•I•" Jo\•JirtUM[LfJJt JB•NUM£L I JJ+l t

JC • NUMfi t-.TJ•t:r p: IXJ.IJAI ,[Q, XXCJOt eANOe XXIJIII e[Oe XXt.J(It GO TO JOO IF IY't'IJAI ,[Q, YYIJBI •AND• YYIJBI eEOe YYtJCII GO fO 100

281.1' CONTINUE NU~ELf~I•V

290 DISf•OISfN NMJH•flf

)00 CONfiNU£ NUMELIJJ•IU4IN

)50 co•fiNV£

JF INUM[LIJt eGTe 0 eANOt HUMElC"t eGTe 01 RETURN DO fi>OO l•le" If INUM[LIII eN£1 l1 GO TO ~00 · If CNVMELUI e(Q, Ot NOM[LfJI•) IF IHUMELt'l e£0, VI NUME&.UI•) R(futiN

000 CONfiMU£ P'WINT hhtwt XCT,.eYCTJhiNUMELCJhJ•lt•t

IOOU 'OII"Af UIH ABOIIf IN UAACHt.l• 'lOoh,ltOH Y• PlOoh,lt l· UH ILl"fNU • OIO I If~ I liD

169

c••••• ~~~~~~!! ~~. ~~:~.!:~: 'e =~: ~~~ :~ !: ~~:~5' ~~!:~~~~ ! •• ••••••••••••••••• OIMEN:JION AAINltNZhBRINJt•CCINIJ

c••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• C M.IN lLJMJ•U JON LOUPt JNfEACHAM6t RO•~ IP hECISSUY , ......................................................................•

c

~~~=~=~ DO l~U K•ltNI'Il KPI•KtJ L•K

DO lUv l•r.PI•N IF tAUSIAAIItKH •L[, A8SIAAfltltll GO TO 100 l•l

100 CONII.UF IF IL •L£• Kl GO TO lZO DO 110 J•lltN flMP•AAU tJI A A I I( • J I • AA I L t J l

110 AAIL,JI•fEMP TEMP•BRIK. I ABII(J•~HHlt

B~«LI•TEMP

DET••DCT

110 PIVOT•AAIIC.tKI DET•OET•PtVOT IF fAU~IPIVOfl •L£, ACCt 60 TO 999 DO 1,(1 I•~PltN

FAC•AAIItKJ/PIVOT AAIItK l•u,u 00 llu J•KPhN

IJO AAIItJI•AAf I•JI-FAC'AAIKtJI hO HUIII 1 lhH IJ•FAC*UBU.t , ..................................................................... ..

C CHECK I..ASl tllMENTt THEN IA(k-SOLYE , ....................................................................•.. PIVOT•AAINtNJ OET•O[ftPIVOT JF IAUSIPIVOT I el!e ACCt GO TO 999 CC lfU •CCUU/PIVOT I•NMI

100 IPI•I+I SUM•O,u 00 210 J•IPltN

110 SU'1•SUM+AAIIeJt•CCIJt CCI II• IR9f lt-~UMtiAAt hi I 1•1•1 IF II oCifo Ul GO fQ ZOO REfUAN

t99 ltOOE•l REf !MIN !NO

DISTRIBUTION LIST

Office, Chief of Engineers ATTN: ENGCW-ES

Library Washington, D. C. 20314

Address

Each Corps of Engineers Division ATTN: Chief, Geology, Soils, and Materials Branch,

Engineering Division

~ach Corps of Engineer.s District ATT'~: Chie£', Foundation and Materials Brax1ch,

Engineering Division

Director, ~uclear Cratering Group 1J. S. Army Corps of Engineers AT'l'R: NCG-ES I,awrence Radiation Laboratory P. 0. Box 808 Livermore, Calif. 94550

District Engineer U. S. Army Engineer District, Jacksonville ATTN: SAJGI .Jacksonville, Fla. 32201

Jvl.r ~ George E~ Bert:cam_ L+701 Kenmore Avenue, Apartment 819 Alexandria, Va. 22304

~. Stanley D. Wilson 1105 North 38th Street Seattle, Wash. 98103

Prof. Arthur Casagrande Pierce Hall, Harvard University Cambridge, Mass. 02138

Mr. John Lowe III Tippetts-Abbett-McCarthy-Stratton 375 Park Avenue New York, N. Y. 10022

Prof. Ralph B. Peck Denartment of Civil Engineering, University of Illinois Urbana, Ill. 61801

1

No. of Copies

lt 2

2 ea

1 ea

2

2

L

1

l

1

1

Address

Prof. Steve J. Poulos Pierce Hall, Harvard University Cambridge, ~;!ass. 02138

Prof. Ronal8 C. Hirschfield Room 1-:no Massachus<ctts Institute of Technology Cambridg(;, Mass. 02139

Prof. H. Bolton Seed Department o:i' Civil Engineering, University of California !.~e:>:>keley, Cal2.f. 91-t T"-0

Prof. James !.Cichael Dunca.11 Depart'Y'.c:r.t o~· Civil Engineerinp;, Fni vers i -:,y of Cali:!:'ornia "·c:rkeley, Ca~-::.~n. 94720

Defense Docu~entation Center A'l'TN': .Y~. :li..eyer Kahn Cameron Station AleXaJ1dTia, Va. 22311-t

2

No. of Copies

1

l

1

25

20

Unclassified Security Claaairlcatlon

DOCUMENT CONTROL OAT A • R & D (Securlt)' cl•••lllcatJott of title, body ol •b•tract and Jnd••ln4 ~t~~notatlon muet b4t entered when th• o"r•ll "___~'!?!' I• ctaeellled)

1. ORIGINATING ACTIVITY (CotpOHta.uthOI) Ia. RI£F'ORT SECURITY CLASSI~ICATION

College of Engineering, Office of Research Services Unclassified University of California, Berkeley, California 2b. GROUP

1. AllPOAT TITLE

FINITE ELEMENT ANALYSES OF STRESSES AND MOVEMENTS IN EMBANKMENTS DURING CONSTRUCTION

4. DESC.-IPTIYE NO TiltS (T}-pa ol report .nd lnclueiY• dlltea)

Final report e. AU THORCSJ (Flret tYme, middle lnlllal, laet name)

F. H. Kulhawy J. M. Duncan H. 13. Seed

e. REPORT OATE 71J. TOTAL NO. O'F PAGES 17b. ;~OF REFS

November 1969 165 a.. CO,.. TRACT OR <;RANT NO. k. ORIGINATOA"I REPORT NU~ER(S)

DAC'tl3')-68-C -0078 ~E-69-4

b. PROJECT NO.

e. Db. OTHER REPORT NO(S) (Any othef' num&.re thllt m.y be aaa/Qned

thl• report) U. S. Arcy Engineer "ria terways

d. Experiment Station Contract Report S-69-8

tO. DISTRIBUTION STATEMENT

This docwnent has been approved for public release and sale; its distribution is uxllimited.

11. SUPPL.EMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

Prepared under contract for U. s. Army Office, Chief of Engineers, u. s. Army Engineer Waterways Experiment Station Washington, D. c. Vicksburg, Mississippi

13. ABSTRACT The objective of this investigation was to develop procedures for conducting finite element analyses of stresses and movements in emba~~ents during construction. The procedures developed involve incremental analyses, simulating successive stages duri~g construction of the emban.~ent, and employ nonlinear stress-strain parameters determined from the results of laboratory tests. Previous studies of the nonlinear, stress-dependent stress-strain behavior of soils were extended during this investiga-tion to include variations of Poisso!1 1 S ratio values as ;rell as modulus values for use in incremental analyses. In order to examine the suitability of these procedures for representing the stress-strain characteristics of a wide variety of soils under both drained and undrained test conditio!ls, the procedures were applied to 46 differe!lt soils, rang:.ne; from cobble sizes to highly plastic clays, for which stress-strain in-formation had been published or was available from other sources. In each case it was found that the simple procedures developed for representing nonlinear, stress-dependent soil stress-strain behavior were convenient and provided reasonably accurate representations of the actual soil behavior. A finite element computer program was developed for incremental analyses of embankment stresses and deformations, incorpo-rating these nonlinear stress-strain characteristics, and this computer program was used to conduct a series of analyses of the deformations in Otter Brook Dam during construction. These analyses showed that the vertical displacements (settlements) within an embankment during construction are affected very strongly by the value of soil modulus, and the horizontal displacements are affected very strongly by the value of Poisson's ratio. The vertical and horizontal displacements calculated using non-linear stress-strain characteristics were in close agreement with those measured

afi'LACII• DO .. ORM 1471, t JAN ... WHICH 1• DD .':'!' •• 1473 Unclassified security Clautiicauon·

Unclassified security Clauifl~tlon ...

Earth movements Earth stresses Embankments Finite element method

13. ABSTRACT (Continued)

I I I I

I I

I

I i I

I

LINK A

"'OLE

I I I I

WT

I I

LINK 8 L.INK C

WT WT

during construction of the dam. Studies of embankment stability shryNed that the values of stress calculated by the finite element method may be used to define a factor of sa.fe~y 'Nith respect to either local over:>tress or overall stability. Provided that the factor of safety with regard to overall stability is defined in a manner co~sistent with that employed in limit equilibriwn analysis procedures, the value of the factor of safety calculated '..tsing finite element stresses is nearly identical to that calculated using the best limit equilibrium procedures of slope stability analysis. Studies were also conducted to determine the effectiveness of these finite element analysis procedures for calculating stresses and displacements in zoned da~s. Analyses •Here performed for t•No hypottetical zoned dams which had the same cross-section, but which differed with regard to the stiffness o:: the core material. These analyses showed that the settlements of embankments are influenced considerably by the stiffness of the core material, and that the stress conditions are strongly affected by the relative stiffnesses of the core and shell.

Unclassified Security ClaaelficaUon ·

cr s-69-8 'finite element analyses of stresses and

Documents