FINITE ELEMENT ANALYSES OF SLOPES IN SOIL

Transcription

1

2 Contrat Report S-68-6 FINITE ELEMENT ANALYSES OF SLOPES IN SOIL A Report of an Investigation by Peter Dunlop, J. M. Dunan and H. Bolton Seed Sponsored by OFFICE, CHIEF OF ENGINEERS U. S. ARMY Conduted for U. S. ARMY ENGINEER WATERWAYS EXPERIMENT STATION CORPS OF ENGINEERS Viksburg, Mississippi under Contrat No. DA CIVENG College of Engineering Offie of Researh Servies University of California Berkeley, California Report No. TE 68-3 May 1968 THIS DOCUMENT HAS BEEN APPROVED FOR PUBLIC RELEASE AND SALE; ITS DISTRIBUTION IS UNLIMITED

3 T/17 W /fo. s-/,g--~ SUMMARY The study desribed in this report was onduted to investigate the behavior of exavated slopes and embankments using the finite element method of analysis. The aspets of slope behavior onsidered inlude stresses, strains, displaements, pore pressures, and progressive development of failure zones. Analyses have been performed using nonlinear stress-strain harateristis as well as linear elasti stressstrain behavior. A method has been developed for simulating analytially the exavation of a slope in a lay layer with arbitrary initial stress onditions. Analyses performed using this proedure have shown that the behavior of an exavated slope depends on the initial horizontal stresses in the soil layer and the variation of strength with depth. Analyses performed using initial stresses and s-trength profiles representatiya. ol normally onsolidated lays indiate behavior whih is ompatible with the results of short-term ( = 0) stability analyses; the failure zones determined strongly suggest development of a urved surfae of sliding, and the fators of safety alulated by the = 0 method were lose to unity at the final stage of the analysis when the failure zone enompassed a large zone between the rest and the toe of the slope. Similar analyses performed using initial stresses and strength profiles representative of overonsolidated lays indiated quite different behavior; failure zones developed whih interseted the bottom of the exavations and the lower portions of the slopes while the fators of safety were still about 2. These failure zones did not resemble the usual urved failure surfaes used in equilibrium analyses of slope stability, but instead suggest the development of the ondition of "loss of ground" wherein the soil in the bottom of the exavation bulges up slowly as additional soil is exavated. Detailed onsideration was given to the stress-strain harateristis of soft, saturated lays under undrained loading onditions. Using the results of horizontal and vertial plane strain tests on San Franiso Bay Mud, an empirial stress-strain relationship was developed for this material. The relationship developed was nonlinear, anisotropi, and

4 dependent on the onsolidation pressure. In spite of its omplexity, the relationship an be represented in a reasonably simple form and ould be onveniently employed in finite element analyses. Using this relationship, analyses were performed to predit the behavior of San Franiso Bay Mud under simple shear loading onditions, and the predited behavior was found to agree quite well with experimental results. Analyses were performed to alulate pore pressures around exavated slopes and in embankment foundations using a variety of possible assumptions regarding the stress-strain behavior and the pore pressure oeffiients of lay soils. The most signifiant fator affeting the alulated distributions were found to be anisotropy with respet to both stress-strain behavior and development of pore-water pressures. Using the finite element method it is possible to inlude realisti soil properties in analyses of slope performane. Nonlinear, anisotropi stress-strain relationships and anisotropi pore pressure oeffiients may be inorporated in analyses in a logial and straight-forward manner, providing a means of making more aurate evaluations of stress distribution and pore pressure distributions in soil masses. 4

5 FOREWORD The work desribed in this report was performed under Contrat No. DA CIVENG-62-47, "Shear Properties of Undisturbed Weak Clays," between the U. S. Army Engineer Waterways Experiment Station (WES) and the University of California. The ontrat was sponsored by the Offie, Chief of Engineers, under Engineering Studies Item ES 525, "Shear Properties of Undisturbed Weak Clays." The general objetive of this researh, whih was begun in February, 1962, is to investigate the influene of pore-water pressure on the strength harateristis of undisturbed weak lays. Work on this projet is onduted under the supervision of H. Bolton Seed, Professor of Civil Engineering, J. M. Dunan, Assistant Professor of Civil Engineering, and C. K. Chan, Assoiate Researh Engineer. The projet is administered by the Offie of Researh Servies of the College of Engineering. The phase of the investigation desribed in this report was performed by P. Dunlop, who is now an Assistant Researh Engineer, and the report was prepared by P. Dunlop, J. M. Dunan and H. Bolton Seed. This is the sixth report on investigations performed under this ontrat. The previous reports are "The Effets of Sampling and Disturbane on the Strength of Soft Clays," Report No. TE 64-1, February, 1964; "The Effet of Anisotropy and Reorientation of Prinipal Stresses on the Shear Strength of Saturated Clay," Report No. TE 65-3 (Contat Report No ) November, 1965; "Errors in Strength Tests and Reommended Corretions," Report No. TE 64-4, (Contrat Report No ) November, 1965; "The Effets of Temperature Changes During Undrained Tests," Report No. TE (Contrat Report No ) November, 1965; and "The Signifiane of Cap and Base Restraint in Strength Tests on Soils," Report No. TE 66-5 (Contrat Report No ) August, The ontrat was monitored by Mr. W. E. Strohm, Jr., Chief, Engineering Studies Setion, and Mr. B. N. Mlver, Chief, Laboratory Researh Setion, Embankment and Foundation Branh, under the general supervision of Mr. W. J. Turnbull, Chief, Soils Division, WES. Contrat offiers were COL A.G. Sutton, Jr., CE; COL J. R. Oswalt, Jr., CE; and COLL. A. Brown, CE. 5

6 TABLE OF CONTENTS Page List of Figures List of Tables List of Symbols CHAPTER 1. CHAPTER 2. INTRODUCTION Fators Influening Slope Behavior Initial Stresses Simulation of Exavation Stress-Strain Behavior Finite Element Method Summary ANALYSES OF SLOPES USING LINEAR ELASTIC STRESS STRAIN BEHAVLOR Stress-Strain Relation~hips Initial Values of Stress and Strain Effets of Initial Stresses and Variation of Modulus Values with Depth Gravity Turn-on Analyses Initiation of Failure Use of Elasti Stresses for Stability Analyses Summary CHAPTER 3. ANALYSES OF SLOPES USING BILINEAR STRESS-STRAIN BEHAVIOR Bilinear Stress-Strain Behavior Stress-Strain Curves and Failure Criteria Shear Strength Profiles Computer Program for Bilinear Slope Analyses Example of Bilinear Analysis The Effet of the Initial Stresses and Strength Variation with Depth Failure Zones Around Slopes in Normally Consolidated and Overonsolidated Clays The Effet of Compressibility on Slope Behavior Slope Stability Summary

7 CHAPTER 4. Introdution UNDRAINED STRESS-STRAIN BEHAVIOR OF SATURATED CLAY Properties of San Franiso Bay Mud Modulus Values Pore Pressure Coeffiients Simple Shear Tests Test Proedures Comparison of Predited and Measured Results Progressive Failure in Simple Shear Simplified Simple Shear Analysis Sunnnary CHAPTER 5. ANALYSIS OF A SLOPE USING NONLINEAR STRESS STRAIN BEHAVIOR -stress---stra:in and Strength Charateristis Analytial Proedure Slope Behavior Summary CHAPTER 6. PORE PRESSURES AT THE END OF CONSTRUCTION OF EXCAVATED SLOPES AND EMBANKMENTS Pore Pressure Coeffiients Changes in Pore Pressure and Pore Pressure Ratios Effet of Bulk Compressibility Pore Pressures Around Exavated Slopes Modulus Variation with Depth Loal Failure Anisotropi Stress-Strain Behavior Anisotropy with Respet to A Pore Pressures in Soft Clay Foundations Beneath Embankments Summary Anisotropy with Respet to A Modulus Variation with Depth Stress-Strain Behavior and Loal Failure

8 CHAPTER 7. REFERENCES CONCLUSION APPENDIX A. REQUIREMENTS FOR BOUNDARY CONDITIONS, ELEMENT SIZES AND ELEMENT SHAPES Lateral Boundaries Boundary Positions Element Size Layer Thikness Element Shapes APPENDIX B. CORRECTIONS FOR SIMPLE SHEAR TEST RESULTS Rubber Membrane Corretion APPENDIX C. Identifiation Purpose Input Data FINITE ELEMENT COMPUTER PROGRAM FOR EXCAVATED SLOPES Sequene of Operations Output Program Listing

9 LIST OF FIGURES Fig. 1 Fig. 2 Fig. 3 Fig. 4 Fig. 5 Fig. 6 Fig. 7 Fig. 8 Fig. 9 Fig. 10 Fig. 11 Fig. 12 Fig. 13 Fig. 14 Fig. 15 Fig. 16 Relationship Between Effetive Stress and Total Stress Coeffiients of Earth Pressure at Rest Analyti Simulation of Exavation (a) Construted Slope Showing the Setion to be Analyzed, (b) Finite Element Idealization of Slope Cross-Setion Typial Finite Element Configuration of a Slope Showing the Constrution Sequene Undrained Stress-Strain Behavior of a Normally Consolidated Clay Idealized Stress-Strain Response of Soil Elements Modulus Variations for Linear Elasti Analyses Effets of Initial Stresses and Modulus Variation on Prinipal Stress Orientations Effets of Initial Stresses and Modulus Variation on Values of T /~H max Effets of Initial Stresses and Modulus Variation on Tensile Stress Comparison of Stresses Calulated Using Gravity Turn-on and Constrution Sequene Analyses with K = V/(1-V) Comparison of Displaements Calulated Using Gravity Turn-on and Constrution Sequene Analyses with K = v/(1-v) Shear Strength and Shear Stress Variations for Linear Elasti Slopes with Modulus Inreasing Linearly with Depth Possible Failure Surfaes Determined from Stress Orientations Using = 0 Bilinear Stress-Strain Relationships Stress-Strain Curves and Strength Criterion

10 Fig. 17 Fig. 18 Fig. 19 Fig. 20 Fig. 21 Fig. 22 Fig. 23 Fig. 24 Fig. 25 Fig. 26 Fig. 27 Fig. 28 Fig. 29 Fig. 30 Strength Variation with Depth for Bilinear Slope Analyses Simplified Flow Diagram for Bilinear Elasti Analyses Example Bilinear Analysis Effet of Initial Stresses on the Development of Failed Regions (Su/P = 0.30) Effet of Strength Variation with Depth on the Development of Failed Regions Development of Failure Zones Around a 3:1 Slope in Overonsolidated and Normally Consolidated Clay Layers Development of Failure Zones Around a L5:1 Slope in Overonsolidated and Normally Consolidated Clay Layers Development of Failure Zones Around 3:1 Slopes in Two Normally Consolidated Clay Layers with S /p =0.30 and S /p = 0.25 u u Effet of Bulk Compressibility on Development of Failure Zones Around a 3:1 Slope in Clay with Constant Shear Strength Effet of Bulk Compressibility on Development of Failure Zones Around a 1.5:1 Slope in Normally Consolidated Clay with Su/p = 0.30 Effet of Bulk Compressibility on Maximum Shear Strains, Prinipal Stress Orientations and Deformations Around a 1.5:1 Slope in Normally Consolidated Clay with S /p = l Effet of Bulk Compressibility on Failure Zones, Maximum Shear Strains and Stress Orientations Around 1.5:1 Slopes in Overonsolidated Clay Correlation Between Failure Zones and Fator of Safety for a 3:1 Slope in a Clay Layer with Constant Shear Strength Correlation Between Failure Zones and Fator of Safety for a 1.5:1 Slope in a Normally Consolidated Clay Layer with S /p = u

11 Fig. 31 Fig. 32 Fig. 33 Fig. 34 Fig. 35 Correlation Between Failure Zones and Fator of Safety for a 3:1 Slope Exavated in an Overonsolidated Clay Layer Slope Failure Observed in Overonsolidated Clay, Exavation for Seatt:'..e Freeway--After Bjerrum (1967) Summary of the Properties of San Franiso Bay Mud (Samples from Hamilton Air Fore Base, Marin County) Stress-Strain Behavior of San Franiso Bay Mud in Vertial Plane Strain Stress-Strain Behavior of San Franiso Bay Mud in Horizontal ~lane Strain Fig. 36 Fig. 37 Fig. 38 Fig. 39 Fig. 40 M~dulus-Strain Variation for San Franiso Bay Mud 109 Variation of Modulus with Diretion of Compression 111 Variation of the Value of Af with Diretion of Compression fo:: San Fran±so Bay Mud- 113 Two Types of Simple Shear Apparatus 116 Variations of Shear Stress and Pore Water Pressure with Shear Strain in Simple Shear Tests on San Franiso Bay Mud 118 Fig. 41 Fig. 42 Normal and Shear Stresses Indued by Shear Loading. After Rosoe (1953) Normalized Shear Stress-Shear Strain and Pore Pressure-Shear Strain Curves for Simple Shear Tests on San Franiso Bay Mud Fig. 43 Fig. 44 Fig. 45 Fig. 46 Fig. 47 Fig. 48 Finite Element Idealization of the Simple Shear Speimen, Showing Initial and Deformed Shape 122 Comparison of Experimental and Analytial Simple Shear Test Results 124 Modified Modulus Variation for San Franiso Bay Mud 126 Comparison of Experimental and Analytial Simple Shear Test Results 127 Development of Failure Zones in Simple Shear 129 Simple Shear and Pure Shear

12 Fig. 49 Fig. 50 Fig. 51 Fig. 52 Fig. 53 Fig. 54 Fig. 55 Fig. -56 Fig. 57 Fig. 58 Fig. 59 Fig. 60 Fig. 61 Fig. 62 Fig. 63 Fig. 64 Fig. 65 Fig. 66 Comparison of Laboratory Simple Shear Test Results with Pure Shear Analyses Variation of the Undrained Strength with Depth Below the Ground Surf ae Variation of Undrained Strength with Diretion of Compression for San Franiso Bay Mud Simplified Flow Diagram for Multilinear Slope Analyses Perent of Shear Strength Mobilized in the Multi Linear Analysis (a) Major Prinipal Stress Diretions, (b) Deformed Shape of Slope Variation of Values of Af with Diretion of Compression for San Franiso Bay Mud Pore Pressures and Pure Pre-s-sure Ratios Effet of Bulk Compressibility on Pore Pressures Around Exavated Slopes in Normally Consolidated Clay Pore Pressure Ratios Calulated for Exavated Slopes Using Linear Elasti Analysis Pore Pressure Ratios Calulated for Exavated Slopes Using Linear and Bilinear Analysis Pore Pressure Ratios for Exavated Slopes Calulated Using Bilinear and Multilinear Analysis Pore Pressure Ratios for Exavated Slopes Calulated Using Constant A and Using A Varying with Stress Orientation Pore Pressure Ratios on a Horizontal Plane Effet of the Value of A on Pore Pressures in Embankment Foundations Effet of Modulus Variation with Depth on Pore Pressures in Embankment Foundations Effet of Stress-Strain Behavior on Pore Pressures in Embankment Foundations Correlation Between Failure Zones and Fators of Safety for Exavated Slopes

13 Fig. 67 Fig. 68 Fig. 69 Fig. 70 Comparison of Stress Conditions Around an Exavated Slope as Determined by Isotropi Bilinear and Anisotropi Multilinear Analyses Comparisons of Pore Pressures Around Exavated Slopes as Determined by Various Methods of Analysis Comparisons of Pore Pressures in Embankment Foundations as Determined by Various Methods of Analysis (a) Finite Element Idealization of a Slope, (b) Cross Setion of Slope Showing Length Fators Fig. 71 The Lateral Extent of Failure Surfa~es Constant Strength in Soil with 185 Fig. 72 Fig. 73 Fig. 74 Fig. 75 Fig. 76 Fig. 77 Fig. 78 Fig. 79 Fig. 80 Fig. 81 Fig. 82 The Lateral Extent of Failure Surfaes in Materials with Strength Inreasing Linearly with Depth Undrained Shear Strength Variations Used in Total Stress Slope Stability Calulations Influene of Lateral Boundary Loation on Horizontal Stress Influene of Lateral Boundary Loation on Vertial Stress and Shear Stress Influene of Lateral Boundary Loation on Horizontal and Vertial Displaements Modulus-Depth Variation and Influene of Lateral Boundary Loation on the Shear Stress The Effet of Layer Thikness on Stresses at 5 ft. on Either Side of a Vertial Cut The Effet of Layer Thikness on Stresses at 40 ft. on Either Side of a Vertial Cut. Simple Shear Box Relative Displaements and Lubriation of Sides Sliding and Rolling Frition in the Simple Shear Apparatus Configuration of Generated Slope

14 LIST OF TABLES Table 1 Table 2 Table 3 Table 4 Table 5 Table 6 Table 7 Table 8 Table 9 Table 10 Effets of Initial Stresses and Modulus Variation with Depth on Stress Orientations, Shear Stress Values, and Tensile Stress Values. 49 Values of S and S /p Required for 3:1 Slopes. 58 u u Material Numbers and Properties for Example Bilinear Analysis. 74 Summary of Plane Strain Test Results on Undisturbed Speimens of Bay Mud. 105 Pure Shear Analysis 132 Values of Af for Normally Consolidated San Franiso Bay Mud. 147 Analyses of Pore Pressures Around Exavated Slopes. 153 Analyses of Pore Pressures in Embankment Foundations. 163 Observed Failures in Cut Slopes" 189 Slopes Analyzed for Boundary Effets

15 LIST OF SYMBOLS ENGLISH LETTERS A A a B B b C C ' E f G G H H HPS h i k k 0 [K] L L L LL pore pressure parameter area as a subsript indiates axial pore pressure parameter, or as a subsript indiates below the toe of the slope as a subsript indiates buoyant as a subsript indiates onsolidation, or a soil index of volume hange shear strength shear stress interept of the effetive stress failure envelope elasti modulus as a subsript indiates failure shear modulus; or speifi gravity as a subsript indiates horizontal, or total slope height or total layer depth horizontal plane strain test depth below the soil surfae as a subsript indiates an instantaneous or tangent value oeffiient of total earth pressure at rest oeffiient of earth pressure at rest total struture stiffness matrix designates a test performed with lubriated ap and base; or length, or as a subsript indiates below slope toe liquid limit 1 distane along the failure surfae from the toe, or 1 as a subsript means lateral m n o as a subsript indiates membrane as a subsript indiates normal to the membrane as a subsript indiates original or initial value 19

16 P PL p p {p} St SS s s Su T t t {u} u VPS V W w w x xy y z z load plasti limit as a subsript indiates perimetral; or major prinipal onsolidation pressure applied load vetor sensitivity simple shear test as a subsript indiates speimen; or as a subsript indiates swelling undrained shear strength as a subsript indiates above slope rest thikness; or time displaement vetor pore pressure vertial plane strain test as a subsript indiates vertial length water ontent, or as a subsript indiates water as a subsript indiates x-ordinate diretion (horizontal) as a subsript indiate shear diretions as a subsript indiates y-ordinate diretion (vertial) depth below the horizontal soil surfae; or as a subsript indiates z-ordinate or vertial diretion 20

17 GREEK LETTERS angle between the major prinipal stress and the horizontal angle between the major prinipal hange in stress and the horizontal E µ \) er T ' as a subsript indiates a value in the t\ diretion total unit weight; or shear strain as a presript indiates a hange in the quantity appended strain visosity poisson's ratio normal stress shear stress angle of inlination of the effetive stress failure surfae 21

18 CHAPTER 1 INTRODUCTION Equilibril.llll methods of slope stability analysis have been widely used and have been found to be satisfatory in many respets for slope analysis and design. These methods are suffiiently simple to be employed routinely for pratial problems, and are apable of aurately prediting the likelihood of atastrophi failure of many types of slopes. Satisfatory slope design, however, should prelude exessive deformation as well as atastrophi failure. It would therefore be desirable to develop methods for prediting slope deformations under design onditions, as well as the degree of safety against sliding. In order to determine in what portions of slopes failure may our first, and how it would progress, it would also be desirable to be able to predit stresses and strains within slopes. In order to predit stresses, strains, and displaements within slopes, it is neessary to employ analytial proedures other than the equilibrium methods. Several suh proedures have already been applied to slope analyses, inluding mathematial analyses, photoelasti analyses, and finite differene numerial tehniques. Beause of the limitations of these methods, and the diffiulties assoiated with their appliation to pratial problems, not many suh analyses have been performed. The availability of high-speed digital omputers, however, oupled with the development of the finite element method of analysis, has made it feasible to analyze problems involving muh greater degrees of omplexity than was formerly possible, while at the same time reduing the effort required. Thus it is now feasible to analyze slope problems involving omplex boundary onditions, heterogeneous material distribution, and nonlinear material properties. The investigation desribed in this report was onduted to examine the usefulness of finite element tehniques for slope analyses. In the following pages the fators affeting the behavior of exavated and embankment slopes are disussed, and the finite element method of analysis is desribed briefly. 22

19 Fators Influening Slope Behavior A omprehensive list of the fators affeting the behavior of embankments and exavated slopes would inlude slope height, slope angle, unit weights, pore pressures, initial stress onditions (prior to onstrution), stress-strain and strength harateristis of the soils, reep harateristis of the soils, drainage onditions, time, and perhaps more. While it may some day be feasible to inlude a wide variety of fators within an integrated approah to slope design, the lass of problems analyzed in this investigation is restrited to short-term (undrained) analyses of slopes in lay. The effets of time on both reep movements and drainage are negleted. The stress-strain relationships used are either linear or nonlinear time-independent relationships expressed in terms of total stresses. The fators onsidered in these analyses are desribed in the following pages. Initial Stresses In a soil deposit with a horizontal surfae, the prinipal stress diretions will be horizontal and vertial. At a depth Z below the surfae of a soil deposit with a unit weight y, the effetive stresses may be expressed as a' yz - u y a' = K a' x 0 y (la) (lb) in whih 0 1 and a' are the effetive vertial and horizontal stresses, u is y x the pore-water pressure, and K is the oeffiient of earth pressure at rest. 0 Brooker and Ireland (1965) have measured values of K for five 0 different soils of varying degrees of plastiity under a wide variety of stress onditions. They found that for normally onsolidated soils, values of K may range from about 0.4 to 0.6, the higher values of K being 0 0 assoiated with more highly plasti lays. The value of K of all the 0 soils tested was found to inrease during unloading to values greater than one. For unloading orresponding to an overonsolidation ratio of 32, the values of K range from about 1.8 for highly plasti Bearpaw shale to about 0 3 for lay of low plastiity. 23

20 24 Unit Weight= K' 1 - QJ '- :::> I/) I/) QJ '- ~ a...: +- '- 0 w i.s,...---r----r-----r-----r _') I ;. I/) I/) QJ ~ o.a1--~~~-+--"'c~c-~1--~~~-t-~~~~i--~~~~ en 0 +- ~ ' QJ u :i: QJ 0 u Coeffiient of Earth Pressure at Rest. K 0 FIG. I RELATIONSHIP BETWEEN EFFECTIVE STRESS AND TOTAL STRESS COEFFICIENTS OF EARTH PRESSURE AT REST.

21 These results indiate that a wide range of in-situ stress onditions may be possible for soils, even where the ground surfae is horizontal; the value of the effetive horizontal stress may range from about 40% of the effetive overburden pressure to about 300%, depending on the soil type and its stress history. It may also be noted that the ratio of horizontal to vertial effetive stress would not neessarily be the same at all depths throughout a soil deposit: In overonsolidated soils, the overonsolidation ratio dereases with depth, sine the amount of unloading is the same for all depths, but the effetive stress after unloading inreases with depth. Therefore, sine the overonsolidation ratio would be expeted to inrease toward the surfae, the oeffiient of earth pressure at rest would logially inrease toward the surfae. Even in normally onsolidated lay deposits whih have a desiated upper rust, the earth pressure oeffiient for the rust may be different from that of the softer lay beneath; and in layered soil deposits, the earth pressure oeffiient may vary from layer to layer. For a slope exavated in the dry, both the earth and water pressures would be redued to zero on the exavated surfae. For analysis of suh slopes, it is neessary to onsider both the hanges in earth and water pressures during onstrution. This may be done onveniently by expressing the total horizontal stress, a, in terms of the total vertial stress, x a, as y a x = KO (2) y in whih K is a total stress earth pressure oeffiient. If the unit weight of the soil deposit is the same at any depth and the water table is at the ground surfae, both the overburden pressure and water pressure will inrease linearly with depth beneath the surfae. Under these onditions, the total stress earth pressure oeffiient, K, may be expressed in terms of K, as 0 K = K 0 yw + (1-K ) ~ 0 y in whih y and y are the unit weights of soil and water. This relationw ship is illustrated in Fig. 1 for two values of soil unit weight. It may (3)

22 be noted that K is always loser to unity than K ; thus while K may range 0 0 from 0.4 to 3.0, K would range from about 0.7 to about 2.0. Simulation of Exavation The initial stress onditions would be expeted to have a onsiderable influene on the magnitudes of the hanges in stress during onstrution and the stresses after onstrution. The initial stresses along a surfae to be exposed by exavation are shown in Fig. 2. During exavation the stresses on this surfae will be redued to zero, whih is equivalent to subjeting the slope to an equal but opposite stress hange along the surfae shown. Thus the larger the initial stresses, the greater will be the hanges in stress during onstrution. Also, sine the stresses following onstrution are the sum of the initial stresses and the hanges in stress, it would be expeted that the postonstrution stresses would be influened to a onsiderable degree by the initial stress onditions. In the analyses desribed herein, exavation of slopes was simulated analytially in the manner illustrated in Fig. 2. Changes in stress, equal in magnitude but opposite in sign to the initial stresses, were applied to the slope along the exavated surfae. Simultaneously the modulus value of the material exavated was redued to a very small value, so that the applied loads indued stress hanges only in the remaining material. The stresses indued by these loads were alulated and added to the initial stresses to determine the post-onstrution stresses. The strains and displaements indued by the hanges in loading were also alulated. Stress-Strain Behavior Beause of the omplexity of soil behavior, onstitutive laws for soil involve many fators and are diffiult to determine. Reent attempts along "these lines have been made by Ko and Sott (1967) for Ottawa sand, and Chang, et. al. (1967) have shown that these relations may be used in onjuntion with finite element tehniques. Various other investigators, inluding Tan (1957), Rosoe and Poorooshasb (1963), and Rowe (1962) have formulated stress-strain relationships for soils. The most straightforward approah for developing stress-strain relations for soils for use in analyses is to test the soil under onditions losely simulating the antiipated onditions, and to derive empirial stressstrain relations diretly from the results of these tests. Considering the 26

23 27 utitil111~ (a ) ttv~tt~ 1:1 (J ( b l FIG. 2 ANALYTIC SIMULATION OF EXCAVATION

24 fat that many types of soil mehanis problems inluding slope behavior, have not been studied thoroughly using even the simplest stress-strain relationships, it seems reasonable to adopt a rational, empirial approah to determination of stress-strain relations for the purpose of preliminary analyses. Consistent with this philosophy, slope analyses have been onduted during the ourse of this investigation using three suessively more omplex types of stress-strain behavior: (1) Linearly elasti isotropi behavior, (2) bilinear, isotropi behavior losely approximating elasti~perfetly plasti behavior, and (3) multilinear, anisotropi stress-strain behavior based on laboratory test results for a soft normally onsolidated marine lay. The stress-strain relations employed are disussed in detail in onjuntion with the various analyses. Finite Element Method As mentioned earlier in this hapter, the finite element method, oupled with the use of high-speed digital omputers, has made possible the analysis of highly omplex stress-deformation problems. The fat that the finite element method may be used for analysis of inhomogeneous strutures makes it possible to approximate nonlinear behavior by adjusting modulus values in aord with stress or strain levels. Therefore the method appears to have a wide variety of appliations in soil mehanis problems. The basi onepts of the finite element method, and some of its previous appliations in soil mehanis are desribed in the following pages. The basi onept of the finite element tehnique is that a ontinuous struture may be represented as an assemblage of elements interonneted at a finite number of nodal points. The elements may be triangles, groups of triangles, or retangles for two-dimensional plane strain or plane stress analyses. Within eah element displaements are assumed to vary in suh a way that ompatibility within the element and along its boundaries is maintained. For triangular elements with three nodal points this may be aomplished by speifying displaements whih vary linearly in two mutually perpendiular diretions within the element. For elements with more nodal points, higher order displaement variations are employed. The finite element analysis of an elasti ontinuum onsists of 4 basi steps: 28

25 (1) The strutural idealization of the ontinuum so that the finite element assemblage simulates the ontinuum. (2) The determination of the stiffness harateristis of eah element to obtain a total strutural stiffness matrix [K). (3) Analysis of the struture by standard strutural methods one the presribed fores and displaements are obtained. The equilibrium equations expressing the relationship between the applied nodal point fores {P} and the nodal point displaements {u} may be represented as {P} = [K) {u} (4) (4) The determination of the element stresses from the nodal point displaements {ul. Beause eah element in the assemblage may have a different modulus value from its neighbors, the method is well-suited to problems involving heterogeneity. Approximate analyses of non-linear strutures are possible either by inremental loading proedures or iterative proedures. The finite element method is also apable of handling virtually any boundary onditions speified in terms of fore.es, stresses (resolved into fores) and displaements; mixed boundary value problems may be handled as easily as problems involving only fore or displaement boundary onditions. The finite element method ensures nodal point fore equilibrium and the assumed element displaement patterns ensure ontinuity of material, but equilibrium of stresses along the element boundaries is not ensured. Wilson (1963) has shown that if the size of elements is redued to infinitesimal proportions the error introdued by approximating stresses by stress resultants is redued and the finite element method will onverge to the exat solution. Thus auray is inreased by the use of smaller elements in any problem. (1) Clough (1965) presented plane, linear elasti analyses of a thik walled ylinder, a plate with an elliptial hole, and a distributed load on a half spae. These analyses were in eah ase ompared to the theoretial solutions and indiated very lose agreement. Also presented were an elasto-plasti plate analysis of a plate with bilinear stress-strain properties. 29

26 (2) Clough and Rashid (1965) analyzed the stress distribution produed by a point load on an elasti half-spae and ompared the stresses and displaements to those omputed by the Boussinesq theory. (3) Wilson (1965) analyzed the steady-state and transient temperature distribution in plane and axisymmetri bodies. (4) King (1965) analyzed two-dimensional time dependent stress problems, and illustrated the use of the analysis for a onrete gravity dam. (5) Zienkiewiz, Mayer and Cheung (1966) analyzed seepage through anisotropi materials. (6) Idriss, I. M. and Seed, H B., Response of Earth Banks During Earthquakes (1967). (7) Dunan, Monismith and Wilson (1967) analyzed layered pavement strutures using linear and nonlinear material properties. The linear finite element analyses results were ompared with theoretial layered system solutions, and the approximate nonlinear analyses were ompared to the behavior of atual pavements. Beause of the large number of simultaneous equations resulting from finite element formulations, solutions by the finite element method require large, high-speed digital omputers. In the finite element analyses of slopes onduted during this investigation, plane strain deformation was assumed. If a slope is exavated in a horizontal deposit, as shown in Fig. 3a, and the longitudinal extent of the slope is suffiiently great, the deformation of a setion far from the end will losely approximate plane strain. A plane strain analysis may be arried out on a unit width of slope (shown dashed in Fig. 3a). For analyses by means of the omputer programs employed in this study, slopes were subdivided into triangles, or quadrilaterals onsisting of four onstant strain triangles, as shown in Fig. 3b. It may be noted that retangular elements were used everywhere exept along the fae of the slope where trapezoids and triangles were used alternately. Whenever possible quadrilateral elements should be employed rather than triangular elements; it has been shown by Wilson (1963) that the stresses obtained 30

27 31 Jl I I I I I I I I (a) _/,/' /9:_ ' " ',I,,,<.". ' ( b) FIG. 3 (a) CONSTRUCTED SLOPE SHOWING THE SECTION TO BE ANALYZED, (b) FINITE ELEMENT IDEALIZATION OF SLOPE CROSS-SECTION.

28 for a triangular element do not represent stresses at any one point in the element, and therefore triangular element stresses are less aurate than quadrilateral element stresses, whih are rationally assoiated with the enter of the elements. Exavation or embankment onstrution may be inorporated in analyses by simulating the addition or removal of elements. Figure 4 shows a typial finite element idealization of a soil deposit and the method of simulating exavation by the removal of elements. At eah stage of onstrution a layer of elements is "removed" by applying stresses to the boundary in the manner previously desribed in the setion on onstrution sequene. The only differene in respet to the finite element onstrution simulation and the true behavior of the slope is an approximation that the stresses are replaed in the analysis by equivalent nodal point fores. The stresses ating aross the boundary are found by an averaging tehnique beause stresses are alulated within elements and not at the element edges, whih form the boundary. An inremental onstrution sequene may be followed if eah sueeding removal of material is based on the stresses resulting from the initial stresses and any previous stress hanges due to onstrution. A onvenient hek may be made of the errors inurred in this tehnique: Brown and King (1966) have shown that for a homogeneous isotropi linear elasti material the resulting stresses are independent of the onstrution sequene; thus analyses involving a single step or a number of steps should give the same results if the material is homogeneous, isotropi and linearly elasti. Comparisons of this type have shown that the stress obtained by single-step and multiple step analyses are nearly idential (a maximum differene of about 6% for triangular elements, less for quadrilaterals). The small differenes in stress are believed to arise from errors assoiated with averaging stresses and omputing nodal point fores for simulation of the exavation proess. Sine the differenes in stress alulated using single-step and multiplestep analyses were very small, it is onsidered that the tehnique for simulating exavation provides an adequate degree of auray. Two different approahes are possible for the solution of problems involving nonlinear stress-strain behavior. These are inremental (stepby-step) onstrution or loading and the iterative proedure. The 32

29 rrn1111ru :::.:-1.::-i:- t= ~ r::i i=i.:;- r: R-. ' '--1- t-,.- t- f-1-,, T f t-_ ~-rfr~i~~~rrrq~..1.~~ I- Il-l.-'H..1.+:Cr.t-i:r I - -f--+ -! -t" +-I f- I-. - i::i i-. :r:.q r R:r. I.,..~, r r1 -, r:' t=-l=..i-1=1:r 0 t- D -t- ~ T I-.::1 +- i- -1- H-l.:r + I-! ~ I I I I - I I I I ' r--i-r 1: +i: ~-1-1 f:cc 1 :J +.I:,_.~ ~ t-: - i:: r'--, i=er f-1--! -..-.J t= ~ i-f- Eri '1--r:r:i--t i 1 I r+ r -t + r:. -,- r f- +- b.t:t - 1 '-'-'- ~::: := 1= :x: :t :. +: _p.:j: F ~.:r. ' r: r:.I _._-1-l-. d "T t-l + ~ ::.i- I-..._ :i:. t"_ I- I -1- f- t--1--t ~ -, FIG. 4 TYPICAL FINITE ELEMENT CONFIGURATION OF A SLOPE SHOWING THE CONSTRUCTION SEQUENCE. 33

30 inremental method is better adapted to slope analysis problems, in whih the loads are applied gradually. Simulating the load appliation by a series of steps makes it possible to modify material properties after eah step with little extra effort. It was therefore deided to use the inremental tehnique in the nonlinear slope analyses onduted for this investigation. The nonlinearity of material properties may be inorporated by evaluating the material properties of eah element prior to eah step of the onstrution proedure. If the onstrution steps are large, or if the material properties undergo large hanges during a single step, an iterative proedure may be used to obtain more aurate material properties for any onstrution step. Non-linearities resulting from hanges in the onfiguration of the struture may be aounted for by hanging the geometry of the finite element idealization to the deformed shape prior to otmnening a subsequent onstrution step. Although Clough and Woodward (1967) note that for low strains (5% to 10%) these nonlinearities are probably not signifiant, the refinement requires little extra effort, and in most of the analyses onduted in this investigation, the geometry of the struture was hanged before eah new step in aord with the previously omputed displaements. Summary The stresses, strains, and displaements of exavated slopes during onstrution and at the end of onstrution are influened by a number of fators, inluding the initial stress onditions in the soil, the stressstrain harateristis of the soil, and the onstrution proedure. The finite element method appears to be readily adaptable to take aount of these fators, and thus may provide a means for more thorough onsideration of slope behavior than was possible previously. Studies onduted to explore the usefulness of the finite element method for slope analyses are desribed in the remaining hapters of this report. 34

31 CHAPTER 2 ANALYSES OF SLOPES USING LINEAR ELASTIC STRESS-STRAIN BEHAVIOR The results of slope analyses based on the assumption of linear elasti stress-strain harateristis provide information whih may be useful for onsideration of the behavior of stable slopes. For example, using linear elasti analyses it is possible to study the influene of various initial stress onditions, and the influene of variations in modulus values with depth on the stress distribution within the slope. The stresses alulated in this way may be useful for determining where, within a slope, failure would be expeted to our first, and through orrelations with the equilibrium method of stability analysis it is possible to determine the value of the fator of safety orresponding to the beginning of failure. Elasti stress distribution theory has frequently been used to estimate the magnitude of stresses in embankment foundations, by representing the embankment as a surfae loading, but only a few previous analyses have been performed to alulate stresses within slopes. Perhaps the first suh analysis was performed by Bishop (1952) using a finite differene tehnique. This method is restrited to slopes in ompletely homogeneous material, however, and has not found wide appliation. Middlebrooks (1936) and LaRohelle (1960) have used photoelasti tehniques for analysis of stresses within slopes. Goodman and Brown (1963) obtained a losed-form solution for the stresses in an infinitely high elasti wedge onstruted in a series of layers; this solution was the first in whih the onstrution method was simulated in the analysis, and demonstrated the importane of inremental loading. More reently Brown and King (1966) and Clough and Woodward (1967) have employed the finite element method for analyses of exavated slopes and embankments. On the basis of the experiene obtained with this method up to the present time (1968) it seems likely that almost any analysis whih may be performed by finite differene methods, photoelasti tehniques, or mathematial analyses may also be aomplished using finite element tehniques, often with muh less effort. Beause 35

32 of its extremely general formulation, the finite element method appears to provide the best means of performing stress analyses of omplex slope problems whih is available at the present time. Stress-Strain Relationships A number of different fators influene the stress-strain behavior of soils in laboratory tests and the orrespondene between laboratory and field behavior. Ladd (1964) has shown that sample disturbane, onsolidation pressure, stress ratio during onsolidation, stress orientations, thixotropy, and strain rate all influene the stress-strain behavior of lays in laboratory undrained tests. If laboratory tests ould be performed whih simulated aurately all of the important fators ontrolling stress-strain behavior under field onditions, it would be expeted that laboratory stress-strain relationships would orrespond well to the field behavior. Preise dupliation of field onditions with regard to loading time and stress orientation is seldom if ever feasible, however, and omplete elimination or prevention of disturbane is nearly impossible. Therefore determination of stressstrain relationships appliable for field loading onditions requires onsiderable are and judgment" Furthermore, if nonlinear stress-strain behavior is to be represented by linear behavior for purposes of analysis, onsideration mus~ be given to the antiipated stress level or perentage of strength mobilized. Typial stress-strain urves for a series of three isotropially onsolidated-undrained triaxial tests performed on San Franiso Bay Mud, a soft, normally onsolidated highly plasti marine lay, are shown in Fig. Sa. Depending on the stress level involved, various modulus values might be seleted to represent any one of these stress strain urves. For example, if the perentage of strength mobilized in the problem analyzed was very low, it might be appropriate to use an initial tangent modulus value; for the upper urve in Fig. Sa, the initial tangent modulus, Ei is approximately 320 kg/m 2 If the average perentage of strength mobilized was SO%, orresponding to a fator of safety of 2, it would be appropriate to use a SO% seant modulus; for the upper urve in Fig. Sa, the SO% seant modulus, ESO' is approximately 180 kg/m 2 For 36

33 ;- a. b,.., U I '--.:: b-1 Q.. E ::::J E ICU Trioxiol Tests on Bay Mud ~ Q..._~_,_~~..._~--.~~--~-- ~ (a) ( b) I I,' I I I / I I / -'--'~~-.(3.~'~~~~~ I I I I / I I I I I I I / 2 I A x i a I St r a i n 1 E' 1 1 /o 10 Cl> u 0 ~ ' ::::J (/) Cl>.: +- 3: 0 Cl> CD.: + a. Cl> Cl Effet: ve Ve rt i a I Pressure 1 p ( ) Q) u Cl> '- ~ 4- Cl (/) (/) E- t; b,.., - I g_ b- u '- Q.. E ::::J E >< 0 ~ (d) Axial FIG. 5 UNDRAINED STRESS-STRAIN BEHAVIOR OF A NORMALLY CONSOLI DATED CLAY. 37

34 higher stress levels and smaller fators of safety, smaller values of modulus would be more appropriate. For onditions where the fator of safety is 1.18, an 8S% seant modulus would be appropriate, sine a fator of safety of 1.18 orresponds to mobilization of 8S% of the ultimate strength. For the upper urve in Fig. Sa, the value of the 8S% seant modulus, E 8 S' is approximately 100 kg/m 2 Thus, as the fator of safety dereases from high values (10 or more) to a low value (1.18), the appropriate modulus value to represent the upper urve in Fig. Sa dereases from 320 kg/m 2 to 100 kg/m 2 For purposes of analyses in whih the perentage of strength mobilized is large, the stress-strain urves shown in Fig. Sa ould be represented by the bilinear approximations shown in Fig. Sb. The slopes of the initial portions of these urves orrespond to 8S% seant modulus values. It is interesting to note that both the modulus values and the undrained strengths determined in these undrained tests are proportional to onsolidation pressure. The ratio S /p, in whih S is the undrained u u strength and p is the onsolidation pressure, is equal to 0,33 for eah of the three tests, and the ratio E /p, where E modulus is equal to 107, 0 0 is the initial tangent Similarly, the SO% and 8S% seant modulus values are proportional to onsolidation pressure, the values being ES 0 /p = 60, and E 8 S/p = 33. Modulus values may also be related to undrained strength, by means of a ratio E/S For the data shown in u Fig. S, E 0 /Su = 320, E 50 ;su = 180, and E 8 S/Su = 100. Bjerrum and Eide (1956) have noted that quite a wide range of values of E /S are possible 0 u for different lays and they found that those for Norwegian marine lays generally range between SO and 200. An interesting onsequene of the proportionality between onsolidation pressure and modulus is the fat that the modulus of normally onsolidated lay would be expeted to inrease with depth. For a normally onsolidated lay deposit with onstant unit weight and water table at the ground surfae, onsolidation pressure would inrease linearly with depth as shown in Fig. S, and therefore the modulus values would also inrease linearly with depth. Thus at a shallow level suh as A where the onsolidation pressure is relatively low, the modulus value would be smaller than at B where the onsolidation pressure is 38

35 relatively high, and the stress-strain urves appropriate for these depths would be related as shown in Fig. 5d. Thus, for analyses of slopes in normally onsolidated lay, it would be appropriate to use modulus values inreasing with depth. It should be noted that while modulus values may be seleted to represent a given stress level or fator of safety, strains alulated using linear approximations for nonlinear behavior annot be expeted to represent strains in atual slopes to a high degree of auray. The perentage of strength mobilized will inevitably vary from point to point within a slope, and no single interpretation will be appropriate for an entire slope within whih various zones are stressed to different portions of the ultimate strength. With regard to the influene of the values of modulus on the results of linear elasti slope analyses, it may be noted that both strains and displaements vary inversely with the value of modulus used in the analyses, but stresses are ompletely independent of the modulus values used. Even where the slope ontains zones having different modulus values, or where the modulus inreases with depth, reduing eah modulus value by a fator of one-half would result in displaement values twie as large but the same values of stress. The same relationship would hold for any problem in whih loads are applied. (For problems in whih displaements were imposed instead of loads, the stresses would be proportional to the modulus values, and strains and displaements would be independent of modulus values.) Initial Values of Stress and Strain For analyses of stresses around exavated slopes, and of strains and displaements resulting from exavation, it is important to onsider arefully the onditions before exavation. The strains and displaements of interest to an engineer onsidering the behavior of a soil mass under the ation of external loads are those whih are indued by the loading. For this reason it is onvenient to onsider the initial strains and displaements, before appliation of loads, as being zero. The stresses of interest, however, are the sum of the initial stresses 39

36 and hanges due to appliation of loads. Therefore typial initial onditions onsist of zero values of strain and displaement, but nonzero values of stress. The stress-strain relationships employed in stress analyses relate hanges in stress with hanges in strain, as shown in Fig. 6a. An element of soil ated on by initial stresses ra and r~ is shown in Fig. 6b. Unless these stresses are equal, the stress differene ra - a~ orresponding to zero strain will not be zero, but ould be either positive or negative, as illustrated by urves A and C in Fig. 6. If subjeted to hanges in stress 6ra and 6r~, the soil element would undergo strains, whih might be either positive or negative. Thus while a linear stress-strain relationship may be represented by a straight line relationship between hange in stress and strain or a single modulus value, the final stress values and the load of the stressstrain urve for the element depend on the initial stresses in the element, as shown in Fig. 6. Effets of Initial Stresses and Variation of Modulus Values with Depth For the purpose of investigating the influene of variations in initial stress onditions and variations of modulus values with depth a number of finite element analyses have been performed using linear elasti stress-strain behavior. These analyses were onduted using two values of the total stress earth pressure oeffiient, K, whih were onsidered to be representative of a normally onsolidated lay (K = 0.75) and an overonsolidated lay (K = 1.25). Two modulus variations were also onsidered, one in whih the modulus value was onstant throughout the depth, and one in whih the modulus values inreased linearly with depth, as shown in Fig. 7. The latter variation was approximated in the analyses using the step-wise variation shown by the dashed lines in Fig. 7. Using these variations of modulus and values of initial stress, analyses have been made of a 3:1 slope exavated in a lay layer with an initially horizontal surfae. The finite element mesh used in these analyses has been desribed previously; a typial mesh is shown in Fig. 4. Care was exerised to insure that the lateral boundaries used 40

37 Q) u Q) = 't- ~ - L.,."'C Q) 't- u ~o <J 0 (/) I.s:: (/) bo u ~ <l -t- - (/) (a) Axia I Strain, a ( ) Axial Strain, 0 Fl G. 6. I DEAL I ZED STRESS - STRAIN RESPONSE OF SOIL ELEMENTS 41

38 / Groun~ Surfae Ground Surfae... Q). +- : 0 u CD. + Cl. u 0 Modulus Variation for Finite Element :r Analyses Constant Modulus Modulus Inreasing Linearly With Depth Below the Surfae Fl G. 7. MODULUS VARIATIONS FOR LINEAR ELASTIC ANALYSES 42

39 in these analyses were suffiiently distant from the slopes so that they did not influene the results; the results are thus representative of slopes exavated in a layer of very large lateral extent. The investigations onduted to determine the influene of various boundary loations, and the riteria developed, are desribed in Appendix A. Although the bottom boundary also influenes the results of the analyses, no effort has been made to minimize or eliminate this boundary effet. It is onsidered that the lay layers in whih the slopes are exavated are underlain by a muh harder layer of soil or rok at a depth equal to twie the height of the slope. The nodal points along this bottom boundary were onstrained from moving either horizontally or vertially, simulating full adhesion between the lay and the harder layer of rok beneath. The prinipal stress orientations determined in these analyses are shown in Fig. 8, where the major prinipal stress orientations are indiated by the longer of the two rossed Iines; ean or tne rosses is tne same size, and the lengths of the lines do not indiate the relative magnitudes of the prinipal stresses. The finite element mesh used in onduting these analyses extended further to the right and left of the slopes than do the regions shown in Fig. 8. However, the stress onditions in the region immediately adjaent to the slope are onsidered to be of primary interest, and only the results for this region are shown in Fig. 8 and subsequent figures. The four ases shown in Fig. 8 illustrate that the initial stress ondit~ons have a large influene on the stress orientations after exavation, but the modulus variation has a relatively minor effet. For initial stress onditions representative of normally onsolidated lays (K = 0.75) the initial major prinipal stresses were vertial everywhere, whereas for onditions representative of overonsolidated lays (K = 1.25), the initial major prinipal stresses were horizontal everywhere. Similarly, it may be noted that the final stress orientations shown in Fig. 8 are in both ases loser to horizontal for high initial horizontal stresses (K = 1.25) than for low initial horizontal stresses (K = 0.75). 43

40 Poisson's Ratio= ,. + + ~ ,It' x )(' )(' )t'...\".,..,..- (a ).- Constant Modulus 1 K = ( b) ""'" ~.)(' k )t' Constant Modulus 1 Jt" K=l.25 + t x x x x () Modulus Inreasing Linearly With Depth 1 K=0.75 (d) Modulus Inreasing Linearly With Depth, K= 1.25 FIG. 8 EFFECTS OF INITIAL STRESSES AND MODULUS VARIATION ON PRINCIPAL STRESS ORIENTATIONS.

41 Near the surfae behind the rest and below the toe of the slopes, the prinipal stresses are oriented approximately vertial and horizontal, and near the sloping surfae the prinipal stresses tend to be roughly aligned with the surfae of the slope. It would be expeted that the stress orientations at these surfaes would oinide with the surfae orientations beause these stress-free boundaries are prinipal planes. The orientations shown do not oinide exatly with the boundaries beause they represent onditions at the entroids of the first rows of elements within the slope, and not at the surfae. Comparing the analyses onduted using the same initial stress onditions but different modulus variations with depth, it may be seen that the stress orientations are almost idential, whether the modulus is onstant or inreases linearly with depth. Only in the area behind the rest of the slope with high initial horizontal stresses is there a signifiant differene: In the ase where the modulus is onstant, the major prinipal stress is approximately vertial, whereas for modulus inreasing with depth, the major prinipal stress is approximately horizontal. In all other regions of the slopes, the stress orientations are influened to a negligible extent by the modulus variation with depth. The influene of initial stresses and modulus variation on the shear stresses around exavated slopes are shown in Fig. 9. The alulated values of maximum shear stress, T, have been divided by the mdx produt of the lay unit weight and the slope height, yh. In an exavated slope, all stress values are proportional to the body fores. Thus the stresses alulated are equally appliable to slopes of any size, provided they are onsidered in terms of dimensionless oeffiients suh as T /yh. max The ontours of T /yh shown in Fig. 9 indiate that the initial max stress onditions have a marked effet on the maximum shear stresses, Whereas the effet of modulus variation with depth is relatively small. For low initial horizontal stresses representative of normally onsolidated lay (K = 0.75), the shear stress ontours are very nearly parallel to the surfae of the slopes and the shear stresses inrease with depth to a maximum at the base of the lay layer. For higher initial horizontal stresses representative of overonsolidated lays (K = 1.25) the shear 45

42 Poisson's Ratio= ( a } Constant Modulus, K=0.75 ~ [. o / ( b) Constant Modulus, K= 1.25 ~o O () Modulus Inreasing Linearly With Depth, K=0.75 (d) Modulus Inreasing Linearly With Depth, K=l.25 FIG. 9 EFFECTS OF INITIAL STRESSES AND MODULUS VARIATION ON VALUES OF "t"max I rh.

43 stress values inrease progressively down the slope, to a value higher than 0.3 yh at the toe of the slope. The magnitude of the shear stresses also inreases with depth, to a maximum at the base of the lay layer. The largest values of shear stresses are somewhat higher for the ases with high initial horizontal stress than for the ases with lower initial stress values, and the values of shear stress at the base of the lay layer are slightly higher for the ases in whih modulus inreases with depth. Perhaps the most signifiant influene of initial horizontal stress values is their effet on the values of maximum shear stress around the toe of the slope. It may be noted that for onditions representative of normally onsolidated lays (K = 0.75), the shear stress values at the toe of the slopes are very small, being less than 0.05 yh for both ases studied. On the other hand, for onditions representative of overonsolidated lays (K = 1.25), the toe of the slope is one of the most ritial loations with regard to shear stress vaiues; the shear stresses near the toes of these slopes are larger than 0.3 yh, and are almost as high as the largest values at the base of the lay layer. For normally onsolidated onditions the largest value of shear stress ours at the base somewhat toward the rest (to the right) from the enter of the slope, whereas for overonsolidated onditions the largest shear stress ours at the base somewhat toward the toe (to the left) from the enter of the slope. Thus the higher horizontal initial stresses appear to ause inreased shear stresses in the region around the toe of the slope and the base of the exavation. Similar results have been determined for slopes of steeper inlinations than the 3:1 slopes shown in Fig, 9, Besides the shear stresses, tensile stresses around slopes are also of interest beause many soils are very weak in tension and many slope failures have been preeded by development of raks, believed to be aused by tension, behind the rest of the slopes. The zones of tension, and the loations and magnitudes of the maximum tensile stress values are shown in Fig. 10. Like the shear stresses disussed previously, these tensile stresses have been nondimensionalized by dividing the alulated values by yh. It may be noted that the size of the tension zones and the magnitudes of the tensile stresses are both muh larger for 47

44 Poisson's Ratio= ~- lh a: ~= I f Co> Constant Modulus, K = 0.75 ( b) Constant Modulus, K= rh = "'3 l'h = ( ) Modulus Inreasing Linearly With Depth, K =0.75 (d} Modulus Inreasing Li nearly With Depth, K =I. 25 FIG.10 EFFECTS OF INITIAL STRESSES AND MODULUS VARIATION ON TENS ILE STRESS.

45 ases in whih the modulus values were onstant throughout the depth than for the ases where the modulus values inreased with depth. The size of the tension zone and the magnitude of the tensile stresses are also larger for higher initial horizontal pressures, or values of K. Additional elasti analyses have shown that the magnitude of the tensile stresses also inrease with inreasing slope steepness. Beause many soils are very weak in tension, it would be expeted that tension raks might develop due to even fairly small tensile stresses. Crak formation would, of ourse, eliminate the tension at the rak and ause redistribution of stresses at other points within the slope. Therefore it seems appropriate to onsider that the tensile zones shown in Fig. 10 indiate loations where tension raks would be most likely to develop, but the tension values shown ould only be developed in materials with appreiable tension strength. The magnitudes of the effets of the initial horizontal stress values and modulus variation with depth on the results of linear elasti analyses of exavated slopes are summarized in Table 1. It may be noted that the magnitude of the initial stresses has a signifiant effet on the stress orientations and the values of shear and tensile stress, whereas only the tensile stress values are signifiantly influened by modulus variation with depth. Table 1. - Effets of Initial Stresses and Modulus Variation with Depth on Stress Orientations, Shear Stress Values, and Tensile Stress Values. Effet on Variable Prinipal Stress Orientations Maximum Shear Stress Values Maximum Tensile Stress Values Initial Stress (K) Major Major Major Modulus Variation (E) Minor Minor Major 49

46 Gravity Turn-on Analyses The analyses desribed previously were onduted by simulating analytially the exavation of slopes in lay layers with initially level surfaes, using the proedures disussed in the previous hapter. Under ertain irumstanes omparable stress values may be alulated by simulating the appliation of gravity to a finite element mesh representing the final slope onfiguration. Suh "gravity turn-on" analyses may often be done quite simply, using standard finite element strutural analysis omputer programs. The orrespondene between gravity turn-on and onstrution sequene analyses may be established quite easily if the gravity turn-on analysis is onsidered to onsist of two separate parts. The first part is appliation of gravity to a finite element mesh with horizontal top and bottom and vertial lateral boundaries. The top boundary is free, the bottom is fixed, and the lateral boundaries are onstrained to move in the vertial diretion only. Under plane strain onditions the stresses resulting from gravity turn-on will be: r v = yz, (Sa) and \) (Sb) in whih rv and rh are the vertial and horizontal stresses, y is the soil unit weight, and Z is the depth beneath the surfae. displaements in the horizontal diretion will be zero. strains will be Both the strains and The vertial vz 2v 2 Ev = E (l - (1 - v)) (6) where Ev is the strain in the vertial diretion, E is Young's modulus, and v is Poisson's ratio. The displaement in the vertial diretion o ' v' is found by integrating the strains over the thikness of the layer, T:.r: l ( 1 zv2 ) z z uv = 2E - (1 - v) (T - z ) (7) To omplete the gravity turn-on analysis following this first step would require use of a proedure whih is exatly the same as the so

47 proedure previously desribed for onstrution sequene analyses; stresses whih are equal in magnitude but opposite in sign from those alulated during the first part of the analysis, would be applied to the surfae being exavated and the resulting stresses, strains, and displaements would be alulated using finite element proedures. Superposition of the results of these two steps would give answers exatly the same as a gravity turn-on analysis performed in a single step, sine the material is assumed to have linear elasti stress-strain harateristis. If a onstrution sequene analysis had been employed instead of a gravity turn-on analysis, the final stress values would be exatly the same provided the initial stress values onformed to those given by the first step of the gravity turn-on analysiso This would be the ase provided the oeffiient of total earth pressure, K, was equal to v/(1-v), The strains and displaements alulated by the onstrution sequene analysis would be the same as those alulated during the seond step of the gravity turn-on analysis, beause the two analyses would in fat be idential. Thus the stresses alulated by adding the initial stresses with the hanges in stress would be the same as those alulated using gravity turn-on proedures, provided K = v/(l - v). The vertial strains alulated by the gravity turn-on proedure would exeed those alulated by the onstrution sequene analyses by the amount shown by equation (6) and the vertial displaements would differ by the amount shown by equation (7). The horizontal displaements and strains would be the same. To illustrate the relationship between the results obtained by the two methods, the same 2:1 slope has been analyzed by both methods for the v ase where K = The alulated stress values (in psf) are shown in 1 - v Fig. 11. To a very high degree of auray, the results of the two analyses are the same. The most signifiant differenes in stress values are assoiated with triangular elements along the ut slope, and may be attributed to differenes in the alulations performed for these elements in the two methods of analysis; initial stress values assigned for onstrution sequene analyses are assoiated with element entroids, but Wilson (1963) has shown that values of a, a, and 1 alulated by finite x y xy element tehniques for triangular elements are eah assoiated with a 51

48 \.11 N Q) u Q) g. en I : -:2.6 0 L ::s ~ LO 1.6 u ::s > 24.7 \,,. ~ ~ en > : \,,. 0 u (.!) $ Q) -a3 O"'x <J"y Lxy E ~l FlG.11 COMPARISON OF STRESSES CALCULATED USING GRAVITY TURN-ON AND CONSTRUCTION SEQUENCE ANALYSES WITH K = 11/( 1-71).

49 different loation within the element. However, beause the maximum differene in stress values alulated using the two proedures is only 16 psf ompared to average stress values of the order of 1000 psf, it may be onluded that the stress values are equivalent for all pratial purposes. The alulated values of displaement are plotted in Fig. 12. It may be noted that the horizontal displaements alulated by the two methods are the same, but the vertial displaements alulated by the gravity turn-on analysis inlude an additional downward omponent. It should be noted, however, that gravity turn-on analyses suffer from three limitations whih limit their usefulness: 1. Beause the initial stress oeffiient K impliit in gravity turn-on analyses is equal to v/(l - v), K values may only range from zero to unity beause the value of Poisson's ratio may only range between zero and one-half. 2. The value of Poisson's ratio, v, appropriate to represent the material behavior may not be onsistent with the value of v appropriate for the initial stress onditions. For example the physial behavior of a saturated normally onsolidated lay would approximately be haraterized by a value of Poisson's ratio lose to 0.5 under undrained onditions, whereas the initial stress onditions might orrespond to K = 0.67, requiring use of a value of v equal to The strains and displaements alulated by gravity turn-on proedures will inlude fititious omponents of vertial ompression and downward displaement. Provided these limitations are taken into aount, gravity turn-on analyses may be useful for many purposes and they an often be performed relatively easily using standard finite element strutural analysis omputer programs. Initiation of Failure The elasti stress distributions desribed in previous setions may reasonably be expeted to be indiative of stress onditions around exavated slopes provided that the alulated shear stress values do not 53

50 Gravity Turn-on Constrution Sequene \ , --- FIG. 12 COMPARISON OF DISPLACEMENTS CALCULATED USING GRAVITY TURN-ON AND CONSTRUCTION SEQUENCE ANALYSES WITH K=ll/(1-11)

51 approah the strength of the lay in any point. They may therefore be useful for onsidering stress distributions for slopes with suffiiently high fators of safety to prevent even loal failure, and they ould be used to determine the fator of safety orresponding to the initiation of loal failure. Inspetion of the dimensionless shear stress values shown in Fig. 9 indiates that for all of the ases shown the largest values of shear stress our at the base of the lay layer. Additional finite element analyses have shown that for steeper slopes and higher values of initial horizontal stress, the largest shear stress may our at the toe of the slope instead of the base of the lay layer. Turnbull and Hvorslev (1967) have also reported results of a photo-elasti study performed by LaRohelle, in whih the maximum shear stress ourred at the toe of a 1:1 slope. Some inrease of undrained shear strength with depth is typial of both normally onsolidated and overonsolidated lay deposits, witlr many strength profiles being intermediate between the extreme onditions of onstant undrained strength throughout the depth of the layer and strength inreasing linearly with depth from zero at the surfae of the deposit. The extremes of onstant strength and linearly inreasing strength with depth may be desribed very onveniently in terms of values of S (for u onstant strength) or S /p (for linear inrease with depth). Using the u shear stress distributions previously desribed, values of S or S /p may u u be determined whih orrespond to failure at a single loation adjaent to the slopes. For this purpose it seems most appropriate to onsider that shear stress values alulated using onstant modulus throughout the depth of the layer would apply best to ases where the shear strength is also assumed to be onstant, beause of the proportionality between undrained shear strength and modulus values. Similarly, it would seem appropriate to onsider that shear stress values alulated using modulus inreasing linearly with depth would apply best to ases where the shear strength also inreases linearly with depth. For the two ases shown in Fig. 9 where the modulus was onstant throughout the depth, the largest shear stress values were: for K = 0,75, T = 0.40 yh. If these shear stress values were equal to the undrained max 55

52 undrained strength of the lay (S = 0.35 yh or S = 0.4 yh) failure would u u our at one loation at the base of the layer. For deposits in whih the shear strength inreases linearly with depth, determination of the values of S /p orresponding to initiation of failure u requires omparison of shear stress and strength values over the full depth of the layer. Comparisons of this type are shown in Fig. 13. The shear stress variations shown in Fig. 13 were determined using the shear stress ontours shown in Fig. 9 for the two ases in whih modulus values inreased linearly with depth. The shear strength variations shown were seleted so that the shear stresses and shear strengths would be equal at the most ritial loations. It may be noted that a onsiderably more rapid inrease of shear strength with depth is required for the ase with high initial horizontal stresses (K = 1.25) than for the ase with low initial horizontal stresses (K = 0.75). For the ase with high initial horizon~l -stresses, -representative of overonsolidated lays, loal failure would our first near the toe of the slope, whereas for the ase with low initial horizontal stresses representative of normally onsolidated lays, failure would our first just below and slightly behind the rest of the slope. To determine values of S /p from the strength profiles shown in u Fig. 13, it has been assumed that the saturated unit weight of the lay is 100 lb/ft 3 and that the water table before exavation oinided with the ground surfae. Under these onditions the initial effetive vertial pressure at any depth Z below the surfae is p = 27.6 Z psf where z is in feet. For the two ases shown in Fig. 13, the values of S /p orresponding u to development of loal failure at one loation are: 0.62 and for K = 1.25, S /p = for K = 0.75, S /p = u u In order to determine the values of fator of safety orresponding to the initiation of failure, these slopes have been analyzed using the " = O" method of analysis. The values of S and S /p orresponding to u u a fator of safety of unity are shown in Table 2, together with the values required so that the elasti shear stresses will not exeed the shear strength in any loation. The ratio of the strengths or strength ratios determined by these two methods is also shown in Table 2. Beause fators of safety omputed by the = 0 analysis are proportional to shear strength, 56

53 5 V1 -...J K= 0.75 / / / / "Strength" Stress ~ ~ p Cf) ~ I K= 1.25 I I I I I, I, I / I / I / / <f I FIG. 13 SHEAR STRENGTH AND SHEAR STRESS VARIATIONS FOR LINEAR ELASTIC SLOPES WITH MODULUS INCREASING LINEARLY WITH DEPTH.

54 Table 2. Values of S and S /p Required for 3:1 Slopes. u u Modulus or Strength Variation with Depth Earth Pressure Coeffiient K Value Required So Elasti Shear Stresses Do Not Exeed Shear Strength Value Required For Plasti Equilibrium, by " = O" Method Ratio s s /p s s /p u u u u Constant yH yH Throughout Depth yH yH Inreasing o Linearly With Depth o the values of the ratio shown in Table 2 represent the fators of safety by the 0 method orresponding to development of the first loal failure. These values range between 1.72 and 2.5, depending on the strength variation with depth and the initial stress onditions. higher the initial stresses before onstrution, the higher the fator of safety orresponding to failure at a single loation within the slope. Similar analyses have shown that the fator of safety orresponding to first loal failure also depends on the slope inlination and the depth to the rigid layer beneath the lay. The For the most ritial ombination of the ontrolling fators investigated (vertial slope exavated ompletely to the rigid base, with a K value of 1,60), the fator of safety orresponding to initiation of failure was found to be greater than 5. It is interesting to note that the smallest value of the ratio shown in Table 2 orresponds to the ase in whih modulus and shear strength values inrease linearly with depth, for initial stress onditions representative of normally onsolidated lays. For these onditions the value of the ratio is only 1.72, indiating that if a 3:1 slope with a fator of safety equal to 1.72 or more was exavated in normally 58

55 onsolidated lay, not even loal failure would develop. For all of the other ases shown, appreiably higher fators of safety would be required to prevent loal failure. Thus it seems likely that normally onsolidated lay deposits are less subjet to loal failure, and to possible subsequent development of progressive failure, than overonsolidated lays having higher initial horizontal stresses and more uniform strength profiles. Use of Elasti Stresses for Stability Analyses Bennett (1951) suggested an interesting proedure for assessing the stability of slopes using the results of stress analyses: Potential failure surfaes are seleted so that at any point the angle between the major prinipal stress orientation (stress trajetory) and the failure surfae is 45 -! degrees. Various trial failure surfaes are seleted by hoosing different starting points on the ground surfae near the toe of the slope and traing out the orresponding failure surfaes in aord with the stress orientations. The average value of shear stress is alulated for eah of the trial failure surfaes, the one requiring the largest value of shear stress for stability being the most ritial. Brown and King (1966) have illustrated the use of this method, employing finite element analyses to determine elasti stress distributions for homogeneous slopes. A value of equal to zero was used in determining the failure surfae loations, so that the failure surfaes investigated were trajetories of maximum shear stress. For a 45 degree slope, and initial stresses orresponding to K = 1.0, Brown and King found that the shear strength required for stability on the most ritial maximum shear stress trajetory was 0.14 yh, slightly less than the required shear strength value whih Taylor (1937) found for the most ritial irle, 0.15 yh. Bennett's stress trajetory method and the irular ar method both employ repeated trials to determine the most ritial failure surfae, whih is the one requiring the largest average value of shear stress to maintain stability. Therefore, of the two methods, the one whih results in the largest value of required shear strength must be onsidered more orret. On this basis the irular ar method appears to be preferable, 59

56 although for the slope mentioned previously the differene in results amounts to only about 7%. Using the stress orientations determined by the finite element analyses desribed previously, failure surfaes have been established using = 0 and = 30. Those established using = 0 are shown in Fig. 14. In eah ase the lowest trial failure surfae shown does not interset the surfae behind the rest of the slope, but instead intersets the bottom boundary. Similarly, all trial failure surfaes beginning further down the slope or further to the left of the toe of the slope also interset the bottom boundary. Suh trial failure surfaes were onsidered not to be kinematially aeptable, and were not used in determining values of shear strength required for stability. For the two ases shown in whih the modulus was onstant throughout the depth, the maximum values of shear strength required for equilibrium were -found to -be 0,064 yh for K-= and O~Dh3 yh for K = Considerably less than required for stability on the most ritial irle (0.16 yh). Even smaller values of required strength were determined for the two ases in whih the modulus values inreased with depth. The same slopes were also analyzed using = 30 to determine failure surfaes ompatible with the stress orientations. The values of shear strength required for stability on these surfaes were also onsiderably smaller than that required for stability on the most ritial irular ar: For the two ases in whih the modulus was onstant throughout the depth, the maximum values of required strength were yh for K = 0.75 and yh for K = For the ases in whih the modulus values inreased with depth, the values of shear strength for equilibrium were again smaller. On the basis of these results it may be onluded that the use of elasti stress distributions for determining failure surfaes and stability oeffiients is not a desirable proedure, beause the values of shear strength required for equilibrium on the most ritial of suh failure surfaes are onsiderably smaller than that required for stability on the most ritial irular ar. It is interesting to note that the use of trial irular failure ars with = 0 onstitutes a valid appliation of the upper bound method of plastiity theory. Aording to the 60

57 , Poisson s Ratio= 0.475,,,,,,, "' ,,, (a) Constant Modulus, K = 0.75 Constant Modulus, K = 1.25 ' ' '... '... \ \ \ \ ' () Modulus Inreasing linearly With Depth, K=0.75 { d l Modulus Inreasing linearly With Depth, K= 1.25 FIG. 14 POSSIBLE FAILURE SURFACES DETERMINED FROM STRESS ORIENTATIONS USING p =O.

58 upper bound theorem, values of shear strength required for stability using upper bound methods will be either unonservative (too low) or orret, and no method indiating lower values of required shear strength than an upper bound method an be orret. Therefore, even though the method suggested by Bennett (1951) satisfies the required relationship between stress orientations and failure surfae orientations, it is apparently less aurate than the onnnonly used irular ar method. Sunnnary The assumption of linear elasti stress-strain behavior provides the simplest possible basis for slope analyses. indiate: The studies performed (1) Values of maximum shear stress and stress orientations around slopes are influened to a major extent by initial stress -orrdrtions (value of -K), but are influerted _only t:o a small degree by the manner in whih the modulus values vary with depth. (2) Values of tensile stress and the size of the tension zone near the rest of slopes is influened signifiantly by both the initial stress onditions and the modulus variation with depth, (3) The values of the largest shear and tensile stresses around exavated slopes inrease with inreasing slope steepness. (4) Gravity turn-on analyses will result in the same values of stress and horizontal displaement as analyses onduted simulating the exavation of slopes, provided the initial earth pressure oeffiient is related to the value of Poisson's ratio by K = v/(l - v). The values of vertial displaement alulated using gravity turn-on proedures ontain an additional downward omponent, however. (5) The fator of safety orresponding to the development of loal failure within slopes is influened to a large extent by the initial stress onditions, and to a lesser extent by the variation of modulus and shear strength with depth 0 62

59 (6) Linear elasti stress distributions and stress orientations annot be used for reliable assessment of slope stability. 63

60 CHAPTER 3 ANALYSES OF SLOPES USING BILINEAR STRESS-STRAIN BEHAVIOR The analyses desribed in this hapter were onduted to investigate the growth of failure zones around slopes during exavation. In order to simulate redued resistane to deformation subsequent to failure, the analyses were onduted in a series of steps representing suessive stages of exavation and the modulus value for any zone where loal failure ourred was redued for subsequent steps in the analyses. any partiular element was assigned one of two possible modulus values depending on the stress onditions: failure, and a relatively small value after failure. a relatively large value prior to Thus The stress-strain urve for any partiular element was -thus trilitre-ar, onsisting of two straight line portions orresponding to the two values of modulus. Bilinear Stress-Strain Behavior The relationship between the modulus values used in these analyses and the undrained stress-strain and strength harateristis of a typial normally onsolidated lay is illustrated in Fig. 15. For a normally onsolidated lay deposit with the ground water table some distane beneath the surfae, the undrained strength would be expeted to inrease with depth in the manner shown in Fig. 15(a). Hypothetial undrained stressstrain urves for speimens from two different depths, and bilinear representations of these urves are shown in Fig. 15(b). If the values of strain orresponding to the breaks in the bilinear urves were the same, then both urves ould be represented by a single normalized stress-strain relationship as shown in Fig. 15(), in whih the ordinate is the ompressive stress divided by the undrained ompressive strength, q The slope u of the initial portion of this normalized urve is equal to the ratio of the modulus to the undrained ompressive strength, E/q, as shown in u Fig. 15. Therefore, if the breaks in the two bilinear urves our at the same value of strain, the modulus value for the initial portion of either urve will be equal to the same onstant times the undrained strength. 64

61 u u ~... :::> en!l: 0 u CD. ~ a I I/) VI u I (/) u > - VI VI u I... a. E 0 u Undrained Compressive Sfrength-qu (a ) Axial Strain- E a :: VI VI u... + (/) 0 ( ) Axial Strain - FIG. 15 BILINEAR STRESS-STRAIN RELATIONSHIPS 65

62 In the bilinear slope analyses onduted during the ourse of this investigation, it has been assumed that the modulus values representing the initial portions of the stress-strain urves are proportional to the undrained strength. Thus the urve representing the variation of modulus with depth is in eah ase similar to the strength-depth profile. The initial modulus values used in the analyses were typially 100 times the undrained shear strength. The modulus values used in the analyses influene the magnitudes of the strains and displaements whih our both before and after failure, but have no appreiable effet on the values of stress alulated. Subsequent to failure, very small modulus values, typially 10 lb/ft 2, were assigned. Stress-Strain Curves and Failure Criteria Although the initial values of strain and displaement at eah point were taken to be zero, the initial stress onditions onsidered were those representative of a lay deposit with a horizontal surfae, in whih both the vertial and horizontal stress values inrease linearly with depth. As in the ase of the linear analyses desribed in the previous hapter, the initial vertial stress at any depth was equal to yz, and the initial horizontal stress was KyZ. Under these onditions the position of the stress-strain urve and the value of strain orresponding to the break in the stress-strain urve for any loation around the slope depend on the orientations of the hanges in stress. Two possible orientations of hange in stress, and the resulting stress-strain urves, are shown in Fig. 16. The initial stress onditions orrespond to the major prinipal stress ating vertially, as shown in Fig. 16(a). The hanges in stress at this point might oneivably be oriented in any diretion; hanges in whih the major prinipal hanges in stress are vertial and horizontal are shown in Fig. 16(a), and the orresponding stress-strain urves are shown in Fig. 16(b). The stress differene (oa - oi) represents the stress differene in the a-diretion, whih is the orientation of the major prinipal hange in stress and the -diretion, whih is the orientation of the minor prinipal hange in stress. 66

63 67 VJ VJ QJ ' -+ (/') : QJ O' : 0. u Before Failure VJ VJ QJ \.. -+ (/) 'L. ' 0 u. (I). Shear Strain Change in Strain () (d) FIG.16 STRESS-STRAIN CURVES AND STRENGTH CRITERION

64 For the urve labelled Vin Fig. 16(b), the stress differene is inreased monotonially to failure when the major prinipal hange in stress is oriented vertially. However, when the major prinipal hange in stress is horizontal, the initial value of stress differene is negative, and its value inreases through zero to a value suffiiently large to ause failure as shown by the urve labelled Hin Fig. 16(b). Beause the slopes of the initial portions of both stress-strain urves are the same, the values of strain orresponding to the breaks in the urves are different, being larger for urve H than for urve V. This variation in the value of strain at failure is representative of the behavior of San Franiso Bay Mud determined using vertial and horizontal plane strain tests; Dunan and Seed (1966) found that the strain at failure under stress hanges of the type labelled V in Fig. 16 were about 3.6% and the strain at failure under stress hanges of the type labelled -H in Fig. 16 were about -1Q_~%_, For stress hange orientations intermediate between those shown in Fig. 16, the stress orientations hange with the addition of eah stress inrement. Beause prinipal stress values may not be added diretly unless their diretions oinide, stress-strain urves for arbitrary stress-hange orientations annot be represented in the manner shown in Fig. 16(b). No matter what the orientation of the prinipal stress hanges, however, the relationship between hange in stress and hange in strain is represented by a straight line as shown in Fig. 16(). Before failure the relationship between stress hanges and strain hanges was represented by a relatively large modulus value (the steeper urve in Fig. 16()) whereas after failure a relatively small modulus value was employed (the flatter urve). Following eah step in the simulated exavation of the slope, the stress onditions in eah element in the remaining onfiguration were evaluated and ompared to the failure riterion. If the maximum shear stress in any element was found to be greater than or equal to the assigned undrained strength value for that element, the smaller modulus value was used in evaluating the stiffness of that element for the next step. Using this proedure it was sometimes found that during a single step the maximum shear stress had hanged from a value less than the undrained 68

65 strength to a value appreiably in exess of the undrained strength. Although redution of the modulus to a small value for subsequent steps was effetive in preventing further inrease in shear stress, one the shear stress in any element exeeded the undrained strength, it remained at that value throughout subsequent steps of the analysis. As illustrated in Fig. 16 (d), therefore, there was inevitably a degree of "overshoot" involved in appliation of the failure riterion in the manner adopted. Although this overshoot might be redued or eliminated by using very small steps or iterative proedures for eah step, it amounted to only about 10% on the average in the analyses performed, and it was not onsidered warranted to use more omplex and time-onsuming proedures to redue it to a smaller value. Shear Strength Profiles For use in the bilinear analyses ondute~ in this investjgation.,_ three different strength profiles were seleted whih are representative of the variations of undrained shear strength with depth in normally onsolidated and overonsolidated lays. in Fig. 17. These strength profiles are shown For many normally onsolidated lays the shear strength inreases approximately linearly with depth beneath the water table; two suh strength variations are represented in Fig. 17, one for S /p = 0.25 u and one for S /p = Both of these strength profiles orrespond to a u saturated unit weight of 125 lb/ft 3 The strength profile representing an overonsolidated lay is haraterized by relatively high strength values near the surfae, where the effet of overonsolidation would be the greatest, and values gradually approahing those for the normally onsolidated lay near the base of the 100 ft. thik lay layer. step-wise strength variations atually used in the finite element analyses are shown in Fig. 17 by dotted lines. In addition to the profiles shown in Fig. 17, some analyses were performed using strength values inreasing linearly from the surfae, or onstant throughout the depth of the lay layer. Computer Program for Bilinear Slope Analyses The proedures desribed for bilinear analyses have been inorporated in a finite element omputer program; the operation of this program (whih The 69

66 ----variations Used in Finite Element Analyses Q) ~ Q) V> CD 60. "O C1' 0:: Q) > 0 : 40 : > u L&.J 20 Normally Consoli dated r = 125 pf Overonsol idated OL-~..l.,_~.L-~.1---~.1...-~'--~'--~'--~u...::.....~ Shear Strength, Su.. psf FIG. 17 STRENGTH VARIATION WITH DEPTH FOR BILINEAR SLOPE ANALYSES. 70

67 is inluded in Appendix C) is illustrated by the simplified flow diagram shown in Fig. 18. Data desribing the positions of boundaries between layers and rows of elements, the unit weights, initial stress onditions, and the elasti and strength properties of the lay serve to define the problem. From these basi input data, the omputer program alulates element numbers, nodal point oordinates and initial stress values for eah element. Using the initial stress values, the omputer program then alulates the nodal point fores required to simulate exavation of the first layer of material. The modulus values of elements in the exavated zone are redued to a very small value (typially 0.01 psf) and the modulus values of all other elements are set equal to the initial value presribed for material at that depth. Using the assigned elasti properties, stiffness values are alulated for eah element in the system, and the individual stiffness values are added into the strutural stiflness matrix, wliidi is appropriately modified for the speified fore and displaement boundary onditions. The simultaneous equations (2 for eah nodal point) represented by the stiffness matrix oeffiients are then solved for the unknown displaements, whih are printed by the omputer. Subsequently, the hanges in strain and stress are evaluated for eah element, added to the initial values, and the resulting values are printed. The exavation of additional layers is simulated in the same way, beginning with evaluation of the neessary nodal point fores to simulate removal of the next layer of elements. Example of Bilinear Analysis To illustrate the operation of this omputer program, an analysis has been performed using a system ontaining 32 elements and 42 nodal points, as shown in Fig. 19. Eah row of elements in this onfiguration was assigned a different material number, 1 for the top row through 4 for the bottom; the material properties assoiated with these material numbers are shown in Table 3. Material 5 is the number assigned to elements after failure, and material 6 is the number assigned to elements whih are being or have been removed. 71

68 ,...-. READ INPUT DATA + CALCULATE FINITE ELEM:NT IDEALIZATION OF. SLOPE + CALCULATE INITIAL STRESS DISTRIBUTION + REf'IOVE ONE LAYER AND CALCULATE NODAL POINT FORCES -.. Cl! EVALUATE THE ELASTIC PROPERTIES OF EACH ELEM:NT.AND FORM STIFFNESS M<\TRIX FOR STRUCTURE UJ o ; z f;l t ll. ~ UUJ SOLVE FOR DISPLACEM:NTS ~~ t <LL PRINT DISPLACEt-ENTS '.... ix: 0 ~ * ~ SOLVE FOR STRESSES AND STRAINS 0. ~ t PRINT STRESSES AND STRAINS FIG.18 SIMPLlflED FLOW DIAGRAM FOR BILINEAR ELASTIC ANALYSES. 72

69 Element Number...: ~ 30 0 CD Cl.> Material Type ~ 20.0 <t ~ 10 Cl.> : 0 62 I\ I ~ 7 6 t t t t..j~s~ 82 llo ~ ~2i54r97 - z-g-4- - ~4~r A ( b) () FIG. 19 EXAMPLE BILINEAR ANALYSIS. 73

70 Table 3. Material Numbers and Properties for Example Bilinear Analysis Elasti Weight Undrained Coeffiient Poisson's Material Modulus Strength of Earth pf psf Ratio psf Pressure The equivalent nodal point fores on the boundary were alulated from the stresses in the elements adjoining the boundary and the reverse of these fores are shown in Fig. 19(b) for the removal of the first layer. The elements above the ut surfae are hanged from material type 1 to material type 6 simulating their removal. Elements below the ut surfae have material types orresponding to their elevations. The onfiguration shown in Fig. 19(b) is then solved to determine nodal point displaements, and element strains and stresses. The values of stress after the first step were alulated by adding the stress hanges from the onstrution step to the original stresses for the horizontal deposit. Maximum shear stress values at the end of the first onstrution step are shown in eah element in Fig. 19(b). It may be seen that the maximum shear stress nowhere exeeds the strength and therefore no elements failed in the first ut step. A seond.onstrution step was undertaken using as an initial ondition the stresses at the end of the first step. The nodal point fores shown in Fig. 19() were alulated, and the seond layer was "removed" (material type hanged). The onfiguration shown in Fig. 19() was solved for displaements, strains and stresses. Again the stresses due to this step were added to those existing prior to the step (Fig. 19(b) and the maximum shear stresses alulated. The resulting values of 74

71 maximum shear stress are shown in the individual elements in Fig. 19(). It may be seen that the values in elements 26, 29 and 30, shown shaded, exeed the speified shear strength values. Thus, at the end of the seond step, failure has ourred in three elements beneath the rest of the slope. Using the onfiguration shown in Fig. 19, it is not possible to ontinue the analysis further, beause the element onfiguration beneath the base of the exavation is not suitable for further extension of the ut surfae. If it were desired to investigate the ontinued development of the failed region as the exavation was made deeper, it would be neessary to use a new onfiguration to represent a deeper exavation. Similar proedures, but using many more layers and elements were used for all the studies desribed in this setion of the report. The Effet of the Initial Stresses and Strength Variation with Depth In order to differentiate between the effets of the initial stresses and the modulus variation it is onvenient to onsider eah independently of the other. The resulting initial stresses and strength variation in any one analysis may not be onsistent (i.e., a normally onsolidated strength profile may not be onsistent with an initial stress ondition with K = 1.25), but, by examining the full range of ombinations of variables it is possible to evaluate separately the effets of the initial stresses and strength profile on the slope behavior. The effet of the initial stress onditions is illustrated by the analyses shown in Fig. 20. The normally onsolidated strength profile shown in Fig. 17, with S /p = 0.3, was used in both of the analyses shown u in Fig. 20, but initial stress onditions orresponding to K = 1.25 were used for the ase shown on the left, while smaller initial stresses orresponding to K = 0.75 were used in the analysis illustrated on the right. The upper onfigurations in Fig. 20 represent an early stage in the exavation proess, when the 1.5:1 slope had been exavated to a depth of 10 ft in the 100 ft thik layer. Even at this early stage, a small region beneath the bottom of the exavation has failed for the ase where the initial horizontal stresses are high. The other slope however, in 75

72 Exavated Depth 10 ft I. 5: I LS :1 ' 10 ft Tension Failed Region 20 ft 20ft Normally Consolidated Strength Profile, K=L25 Normally Consolidated Strength Profile, K=0.75 FIG. 20 EFFECT OF INITIAL STRESSES ON THE DEVELOPMENT OF FAILED REGIONS (Su /P = 0.30)

73 77 whih the initial stresses are small, has not suffered any loal failure at this stage. It may be noted that a small zone of tension has developed in both ases behind the rest, the maximum tensile stress value being somewhat larger for the ase with higher initial stresses. Up to this point in the analyses, i.e. during the removal of the first layer, the analyses orrespond to the linear elasti analyses desribed in the previous hapter, and the same results ould have been determined using the results of those analyses, whih also showed higher shear stress and tensile stress values for a deposit with larger initial horizontal stresses. Even the loation of the first loal failure ould have been determined from the linear analyses. For the following step of the bilinear analyses, however, the modulus values within the small failure zone were redued before the seond layer was removed. The onditions after removal of the seond layer, when the slopes were 20 ft. high, are shown by the lower onfigurations in Fig. 20. It may be noted tliat in ootli ases tnere is a failure zone of onsiderable size beneath the slopes at this stage. The magnitudes of the tensile stresses are also somewhat higher than they were following the first stage. For the ase in whih the initial stresses were larger, the failure zone is also larger and extends to the surfae at the bottom of the exavation, whereas for smaller initial stresses the failure zone is entirely beneath the surfae and extends behind the rest of the slope. Although analysis of the slope on the right ould be ontinued, it is not possible to do so for the slope on the left, beause the failure zone after the seond stage intersets the base of the exavation. Nodal point fores to simulate further exavation may be alulated as before, but appliation of these fores to nodal points in the failed zone, whih now has a very low modulus value, will result in theoretial displaements of an extremely large order of magnitude, perhaps tens or hundreds of feet. Displaements of this order are obviously not realisti, and it is therefore not meaningful to ontinue the analysis beyond the stage shown. In a real slope ontaining a failure zone like the one shown, it would probably be observed that ontinued exavation would result in gradual bulging of soil in the base of the exavation, the phenomena whih Terzaghi and Pek have desribed as "lost ground" (p. 520). Loss of

74 ground would probably result in relaxation of stress around the base until the shear stress values beame ompatible with the strength values, at whih time the bulging would stop. Using the analytial proedures adapted for this investigation, however, it is not possible to simulate this redistribution of stresses, and the analyses must be terminated one the failure zone enompasses nodal points to whih fores must be applied during the next stage. While it is unfortunate that the analyses annot be ontinued to a stage where overall failure of the slope is imininant or has ourred, it is interesting to observe that heaving near the toe of slopes has been observed in exavations in overonsolidated lays ontaining high initial horizontal stresses. For example, Bjerrum (1967) has reported that the operation of a lay pit near Sandnes, Norway was partiularly onvenient for the owners beause the pit, whih was exavated in summers, was always found to -have TrTled its-elf -with -J:ay -squeezing up from the bottom during the winter. During the time the pit was operated a large landslide developed on the adjaent hill, with the lower end of the sliding surfae interseting the bottom of the lay pit. In exavations in overonsolidated lays, bulging near the toe of the slope may preede the full development of a sliding surfa~ and atastrophi failure of the slope. To determine the influene of the variation of strength with depth, two analyses have been performed, the results of whih are shown in Fig. 21. In the ase illustrated on the left, the strength of the lay is onstant throughout the depth of the layer, and for the ase on the right the strength inreases linearly with depth, from zero at the surfae. Both ases were analyzed using the same initial stress onditions, orresponding to K = After the first stage of exavation, when the 3:1 slopes are 10 ft high, a zone of tension has developed behind the rest of the slope in the layer with onstant strength, and a sizable failure zone has formed around the slope in the layer with strength inreasing with depth. Although this zone intersets the surfae of the exavation, it does not wholly enompass any nodal points to whih fores must be applied during the next stage, so the analysis an be ontinued. 78

75 Exavated Depth 10 ft /Tension 10 ft Foiled~ 3:1 Region "'- 20 ft 20 ft Constant Strength, K = 0.90 Strength Inreasing Linearly With Depth, K = 0.90 FIG. 21 EFFECT OF STRENGTH VARIATION WITH DEPTH ON THE DEVELOPMENT OF FAILED REGION~.

76 At the next stage, when the slopes are 20 ft high, the size of the tension zone has inreased, and failure has developed at the bottom of the layer with onstant strength. The size of the failure zone around the slope in the layer with strength inreasing with depth has inreased approximately in proportion with the inrease in slope height. These analyses indiate that the prinipal effets of strength (and modulus) variation with depth are onerned with the development of tensile stresses and the loation where shear failure ours. The results shown in Fig-. 21 are in qualitative agreement with the results of equilibrium methods of stability analysis for the two strength profiles onsidered: Taylor (1937) found that, when the shear strength is onstant throughout the depth of the layer, the ritial irle for flat slopes always extends to the base of the layer, whih is in agreement with the fat that failure ours first at the base of the layer aording to the bilinear analysis. Kenney (1963), on the other hana, round tha:t the fator u-f ~afety nf a slope was independent of the slope height when the shear strength inreases linearly with depth from zero at the surfae; the bilinear analysis shows that the size of the failure zone inreases approximately in proportion to the slope height, and remains about the same shape. that toe irles and slope irles a~e Kenney also found most ritial when strength inrease with depth, whih is onsistent with the formation of a shallow zone of failure in the bilinear analyses. It seems likely therefore that the analytial proedure adopted is apable of providing a means of studying the development of zones of failure around slopes. Failure Zones Around Slopes in Normally Consolidated and Overonsolidated Clays While the analyses desribed in the previous setion indiate very learly the various effets of initial stress onditions and strength variation with depth, the ases analyzed are not neessarily representative of the behavior of slopes in either normally onsolidated or overonsolidated lay layers. For example, although it is lear from the previous analyses that higher values of initial horizontal stress, harateristi of overonsolidated lays, will result in higher values of shear stress, it is also apparent that overonsolidated lays are stronger than normally 80

77 onsolidated lays; thus the effet of higher shear stresses may be offset by higher shear strength. To study the effets of initial stress and strength variations with depth in realisti ombinations, several slope analyses have been performed using initial stress onditions and strength profiles harateristi of normally onsolidated and overonsolidated lays. The strength profiles used in these analyses were the two normally onsolidated profiles shown in Fig. 17 for S /p = 0.25 and S /p = 0.30, u u and the overonsolidated profile shown in the same figure. In all ases initial stress onditions orresponding to K = 0.75 were used with the normally onsolidated strength profiles, and K = 1.25 was used with the overonsolidated strength profile. Four stages during the exavation of 3:1 slopes in overonsolidated and normally onsolidated (S /p = 0.30) lay layers are illustrated in u Fig. 22. During exavation of the first 10 ft layer, small tension zones develop behind the rest of both slopes, but no shear failure ours,, During exavation of the seond 10 ft layer, a small failure zone develops beneath the rest of the slope in overonsolidated lay, but the slope in overonsolidated lay remains ompletely stable. Upon exavation of the next layer, the failure zone in the normally onsolidated layer inreases in size, and a fairly large failure zone extending from the base of the exavation to the bottom of the layer develops in the overonsolidated layer. Beause this zone enompasses only a very small area on the exavated surfae at this stage, however, it was possible to simulate the exavation of an additional 10 ft of lay, as shown in the bottom of Fig. 22; it may be noted that the failure zones in both layers have reahed a onsiderable size by the time the exavations are 40 ft deep. Both failure zones interset the base of the layer at this stage, but they extend upward in different diretions. The one in the overonsolidated lay extends upward to the bottom of the exavation and the lower portion of the slope, but the failure zone in the normally onsolidated layer extends up toward the surfae behind the rest of the slope, but does not interset the surfae. In neither of the ases shown in Fig. 22 does the failure surfae interset the surfae of the slope in suh a way as to suggest development of a urved surfae of sliding. Beause the failure zone enompasses 81

78 Exavated Depth 10 ft 3:1, IOft 3:1 20ft 20ft Foiled~ Region Q) N 30 ft 30ft [ 40ft 40ft Over onsolidated Strength Profile, K = 1.25 Normally Consolidated Strength Profile, Su/P=0.30, K =0.75 f\g. '2.'2. OE\JELOPMEN\ OF FA\LURE 'Z.ONES AROUNO A 3q SLOPE \N O\JERCONSOL\OA\EO ~"~ "QRW..~LL'< C:.Q"SQL\~~\EO C:.L~'< L~'CERS.

79 a onsiderable area in the bottom of the slope in overonsolidated lay, however, it was neessary to terminate the analysis at the stage shown. As mentioned previously, pratial diffiulties might be experiened in extending a real exavation beyond this stage, beause of loss of ground. In determining zones of failure, only shear failure has been onsidered. It may be noted, however, that tension zones of onsiderable size have developed behind both slopes by the time the slopes are 40 ft high. Tension failure in these zones would result in formation of raks whih would also influene the overall stability of the slopeso Comparisons of steeper slopes in normally onsolidated and overonsolidated lay layers are shown in Fig. 23. The development of failure zones around these 1.5:1 slopes is qualitatively very similar to the 3:1 slopes disussed previously, but at eah stage the extent of the failure zones and the tension zones are larger than in the ase of the flatter slopes 0 As before, the analysis of the slope in overonsoridated- ray ould not be ontinued beyond the stage shown beause the failure zone enompasses a large area along the lower portion of the slope and in the bottom of the exavation. In order to study the influene of the value of S /p on the extent u of the failure zone around slopes in normally onsolidated lays, analyses were made of 3:1 slopes exavated in lay layers with the two normally onsolidated strength profiles shown in Fig. 17. The results of these analyses are shown in Fig. 24, for the 6 stages involved in exavation to a depth of 60 ft. The series of steps on the left represent the progressive development of failure in a lay layer with S /p = 0.30, and those u on the right show the development of failure in a lay layer with S /p = u At eah stage in the exavation proess the failure zones are larger for the lower strength values, as would be expetedo In both ases failure ours first below and behind the rest of the slope, then extends down to the bottom of the lay layer and out under the toe of the slope as exavation proeeds. In the ase of the weaker lay layer, the failure zone also extends up to the surfae behind the rest of the slope. The last stage shown for the ase in whih S /p was equal to 0.25 u represents a ondition strongly resembling the development of a urved sliding surfae. 83

80 Exavated Depth 10 ft I. 5: I ' IQ ft I. 5: I Tension Region 20 ft [ 2Ptt, Failed~~ Regio~ 30ft 40ft 40ft Overonsolidated Strength Profile, K=l.25 Normally Consolidated Strength Profile,Su!P=0.30,. K = FIG. 23 DEVELOPMENT OF FAILURE ZONES AROUND A 1_5;1 SLOPE IN...OVERCONSOLIDATED ANO NORMALL'< CONSOL\OATEO CLA'< LA'<ERS.

81 3: I ~ Ex'-a-va-t-ed--De-p-th--IO_f_t.I l_io_f_t Foiled~ _.. Region ~ _ I 2o~f t ~ I 20ft lfil I 30ft 30ft 40ft ~I 40ft -==' I _50_ft I r, -~ I 60_ft -::::::JI -, 50_ft I,60ft~I Norma 11 y Con so I i dated, Su /P =0.30 K = Normally Consolidated, Su/P=0.25 K = FIG. 24 DEVELOPMENT OF FAILURE Z.ONES AROUND 3: I SLOPES IN TWO NORMALLY CONSOLIDATED CLAY LAYERS WITH Su/P=0.30 AND Su/P =

82 The Effet of Compressibility on Slope Behavior The values of bulk modulus, K, and Young's modulus, E, for isotropi elasti materials are related to eah other by the value of Poisson's ratio, as shown by equation 8 below: K E 3(1-2\)) (8) Thus if the value of Young's modulus is redued following failure while the value of Poisson's ratio remains unhanged, the value of bulk modulus is redued in the same proportion as the value of Young's modulus. It has been noted that the use of this proedure often redues the bulk modulus value suffiiently that appreiable volume hanges our in the failure zone during subsequent steps of the analyses. This type of behavior does not appear to be onsistent with the expeted behavior of saturated lays: -rt seems logial to assume that the bulk ompressibility of lay under undrained onditions depends primarily on its degree of saturation and not upon the values of shear stress or shear strain to whih the lay has been subjeted. Therefore it seems likely that the bulk ompressibility of the lay should be the same after failure as before. In order to simulate this aspet of lay behavior in slope analyses it is neessary to use values of bulk modulus in the failure zone whih are several orders of magnitude larger than the values of Young's modulus, and the orresponding values of Poisson's ratio for the failure zones are very nearly equal to one-half. Conventional plane strain finite element omputer programs annot be used with values of Poisson's ratio larger than about 0.49, but speial formulations have been developed (Herman and Toms, 1964) whih may be used with even ompletely inompressible materials, orresponding to Poisson's ratio equal to one-half. These programs are onsiderably more expensive to use than onventional finite element programs, requiring about 70% more time for solution of a given problem. In spite of the additional expense, some of the slope analyses desribed previously were performed using onstant values of bulk modulus, or using values of Poisson's ratio equal to one-half, in order to determine if the hange in bulk ompressibility after failure had any signifiant influene on the results. 86

83 In the analyses performed using a onstant value of bulk modulus, the value seleted was one-tenth of the bulk modulus of water (5 x 10 6 psf), whih would be appropriate to simulate the bulk ompressibility of lay having a degree of saturation between 99% and 100%. The values of Poisson's ratio required to ahieve this onstant value of bulk modulus varied between and A omparison of two analyses of a 3:1 slope in a lay layer with onstant shear strength throughout the depth of the layer is shown in Fig. 25. In the analysis illustrated on the left, the value of Poisson's ratio was onstant throughout, so that the bulk modulus in the failure zone was redued in the same proportion as Young's.modulus after failure. The analysis illustrated on the right was onduted using a onstant value of bulk modulus (5 x 10 6 psf) both before and after failure. noted that the behavior of the two slopes is essentially the same, It may be exept that failure ours at the toe of the slope at an- e-ari:ter- stage- inthe analysis performed using a onstant value of poisson's ratio. This differene is relatively minor when ompared to the extent of the regions already failed, and in the final stage shown in Fig. 25, there is almost no differene between the failure zones as determined by the two analyses. A similar omparison is shown in Fig. 26 for 1.5:1 slopes exavated in normally onsolidated lay layers with S /p = As opposed to the u omparison shown in Fig. 25, in whih the failure zone was somewhat more extensive for the analysis performed using a onstant value of Poisson's ratio, the failure region in this ase is somewhat larger for the ase in whih the bulk modulus was onstant, espeially following the third and fourth stages of exavation. After the last stage shown, when the slopes are 70 ft high, there is little differene between the results of the two analyses exept that the one onduted using a onstant value of Poisson's ratio is ontinuous from the base of the exavation to the surfae behind the rest of the slope, whereas in the analysis performed using a onstant value of bulk modulus, the failure zone emerges on the surfae of the slope near the top and the bottom. of these same two analyses are shown in Fig. 27. Additional results for the final stage It may again be noted that although there are some differenes between the shear strain, prinipal stress orientations and deformed shapes for the two ases, the 87

84 '----I IL---_ I 3: I Exavated Depth 10 ft ~ _IO_f_t.,,, 20ft Fail~d~ Region " 20ft ::: I --30Jt 30ft ~I I _40ft_~ I I 40ft ~I Poisson's Ratio= Constant Bulk Modulus FIG. 25 EFFECT OF BULK COMPRESS! BILITY ON DEVELOPMENT OF FAILURE ZONES AROUND A 3: I SLOPE IN CLAY WITH CONSTANT SHEAR STRENGTH.

85 1.5: I 1.5: I Exavated Depth 10 ft / IOft / I I I 20ft I.Failed~ Region /. 20ft I ~ I I 30ft ~ I 30ft I I I I 40ft 50ft 60ft 40ft I I =- ~ I I ~ 50ft ~ dljj I I 60ft I I I I 70ft ~ I ~ I Poisson's Ratio= ft Constant Bulk Modulus I FIG. 26 EFFECT OF BULK COMPRESSIBILITY ON DEVELOPMENT OF FAILURE ZONES AROUND A 1,5q SLOPE IN NORMALLY CONSOLIDATED CLAY WITH Su/P =

86 90 1.5: I 1.5: I I Maximum Shear Strains -. 0 /o \ &:d]fj] -+ R '.J( / * " * + t t :; f Prini po I Stress Diretions + -I : ;.,. + I ] Final Deformed Shapes Poisson's Ratio= Constant Bulk Modulus FIG. 27 EFFECT OF BULK COMPRESS I Bl LITY ON MAX I MUM SHEAR STRAINS, PRINCIPAL STRESS ORIENTATIONS ANO DE FORMATIONS ARO.UNO A 1.5:t SLOPE IN NORMALLY CON SOLIDATED CLAY WITH Su/P = 0.30

87 behavior as analyzed using either method is essentially the same. It may be noted that the strain ontours in Fig. 27 indiate values of shear strain as large as 60% in the failure zone near the base of the lay layer. Although strains of this magnitude are not onsistent with the small strain theory used in the analyses, these extremely large strains ourred only in the failed regions where hanges in shear stress are very small; therefore it would be expeted that the resultant errors in alulated hanges in stress in the unfailed regions would also be small. Furthermore, the geometry of eah element was generally hanged after eah step in aordane with the displaements alulated on the previous step, thereby aounting for nonlinearity due to hanges in onfiguration as suggested by Clough (1965). Using this approah it is the inremental strain values whih should be ompared to the aeptable values onsistent with small strain theory; the maximum inremental values are only about 10%~ onsiderably smaller than the umulative values shown in Fig. 27. The use of small strain theory in soil mehanis analyses has been disussed by Clough and Woodward (1967) who suggest that the resulting errors are not signifiant for strain values less than 5% or 10%. Thus while some degree of error must undoubtedly arise from the use of small strain theory in these slope analysis problems, it is believed that the error is not very important, beause the largest strains our in the failed regions where stresses remain almost onstant after failure. In order to further study the influene of bulk ompressibility, the behavior of 1.5:1 slopes exavated in a layer of overonsolidated lay were analyzed using values of Poisson's ratio equal to and 0.500, the latter value orresponding to an infinite value of bulk modulus and ompletely inompressible material. The results of these analyses are shown in Fig. 28. The failure zones, shown in the upper part of the figure are similar for both analyses; no failure ourred after exavation of the first layer, but a sizable failure zone developed beneath the toe of the slope following exavation of the seond step. The zones determined in the two analyses are similar in shape but slightly larger for the analysis in whih Poisson's ratio was equal to 0.475, as was the ase for the omparison illustrated in Fig. 26. The values of 91

88 I,, I I ~ l! ;iiti I ~ I... ] Failure Zones 1 Maximum Shear Strains, 0 /o I : :1 I : Prini pol Stress Diretions 'Poisson's Rotio=0.475, K=J.25 Poisson's Ratio=0.500, K=l.25 Overonsol idated Strength Profile Overonsol idated Strength Profile FIG. 28 EFFECT OF BULK COMPRESSIBILITY ON FAILURE ZONES, MAXIMUM SHEAR STRAINS AND STRESS ORIENTATIONS AROUND I SLOPES IN OVERCONSOLIDATED CLAY.. 92

89 maximum shear strain and the stress orientations, whih are also shown in Fig. 28, are virtually the same for both analyses. On the basis of these analyses it was onluded that the differenes between the results of analyses onduted using a value of Poisson's ratio equal to 0.475, and the results of analyses performed with onstant value of bulk modulus or inompressible material were not signifiant. Certainly the overall behavior is not hanged, and it appears that in some ases the failure zones are larger for v = (the 4th and 5th steps in Fig. 25) while in other ases they are larger for onstant bulk modulus (the 3rd, 4th, and 5th steps in Fig. 26). Thus there seems to be little reason to believe that it is neessary to use more omplex and expensive omputer programs formulated for inompressible materials. Slope Stability Although the bilinear analyses- illus-tra-te- in a- graphi manne-r thedevelopment and growth of failure zones around slopes, it is diffiult to assess the degree of stability of the slopes against atastrophi overall failure on the basis of these results. In order to obtain a quantitative measure of the stability of the slopes, they have also been analyzed using the " = O" equilibrium method with irular ar trial failure surfaes. The last two steps in the analysis of one of the slopes desribed previously are shown in Fig. 29; the stages shown represent the extent of the failure zone when the slope was 40 ft high and SO ft high. The slope inlination was 3:1 and the assigned value of shear strength was onstant throughout the depth of the deposit. The bilinear analysis was performed using a onstant value of bulk modulus, K = 5 x 10 6 psf. Beause the shear strength is onstant throughout the depth and the slope is fairly flat, the ritial irular ar intersets the hard layer at the bottom of the lay. The ritial irles for both onditions are shown on the figure. The values of fator of safety shown on the figure were alulated using the mobilized values of shear strength in the failure zone, whih were higher than the assigned shear strength value beause of overshoot. Comparisons of the mobilized and assigned values of shear strength are shown in the lower part of Fig. 29 for the last stage of the 93

90 94 3: I Critial Failure Surf ae {a ) Exavated Depth 40 ft ( b ) Exavated Depth 50 ft ~ 1250.n 0. (/') I.fl <l.>._ ~ (/) I... g 625..:: (/) Shear Stress on Critial Failure Surfae E -~Ass:~ ~hear Strength (625) ::> E ~ 0 L-~~~-1-~~~~.l.-..~~~~~~~~-'-~~~.-..J ~ ( ) Distane Along the Failure Cirle from Toe a R./L FIG. 29 CORRELATION BETWEEN FAILURE ZONES AND FACTOR OF SAFETY FOR A 3: I SLOPE IN A CLAY LAYER WITH CONSTANT SHEAR STRENGTH.

91 analysis. It may be noted that the mobilized shear strength exeeds the assigned value by a onsiderable amount beneath the lower part of the slope, and the average value of mobilized strength is 760 psf ompared to the assigned value, 625 psf. Thus the average amount of overshoot amounts to more than 20%. This partiular analysis was performed using 8 layers, whih are too few to prevent a large amount of overshoot. In subsequent analyses, performed using 10 layers, the amount of overshoot was muh less. The values of fator of safety shown in the figure, 1.12 for the nextto-last stage in the analysis, and 1.04 for the last stage, indiate that the slope is lose to atastrophi failure for both stages even though the failure zone does not enompass the entire length of the ritial ar. This might be expeted, however, beause even outside the failure zone the mobilized shear stress values were very nearly equal to the shear strength. The final two stages in the exavation of a 1.5:1 slope in a normally onsolidated lay layer with Su/P = 0.30 are shown in Fig, 30 The bilinear analysis was onduted using a onstant value of bulk modulus, K = 5 x 10 6 psf. The slope onfiguration onsisted of 10 layers eah 10 ft thik, and at the stages shown in Fig. 30 the slope was 60 ft high and 70 ft high. For both ases the ritial irular ar passes through the toe of the slope, and lies for the most part within the failure zone. The fators of safety shown in the figure were alulated using the mobilized shear stress values whih, as in the ase illustrated in Fig. 29, are somewhat in exess of the assigned shear strength values. It is apparent, however, that the amount of overshoot is onsiderably less in this ase, amounting to only about 5% or 10% on the average. For the next-to-last stage, shown at the top of Fig. 30, the alulated value of fator of safety is only l,01, even though the failure zone beneath the slope has not yet interseted the surfae. In the last stage the failure zone has interseted the surfae at both the top and the bottom of the slope, and the fator of safety against failure on the most ritial irular ar is less than one. A similar analysis for a 3:1 slope exavated in an overonsolidated lay layer is shown in Fig. 31. At the stages illustrated this slope was 95

92 1.5: I 1 Ten$ ion \ Criti'al Tai lure Surfae... Fator of ' Safety=l.01 (a ) Exavated Depth 60 ft Critial Failure Surfae ( b) Exavated Depth 70 ft... J') 2000 f""""'""'i. 0. Shear Stress on Critial Failure Surfae ~ [~=i~ned S~e~r~trengt;--,..., Q)..: (/) E :J E ~ 0 ~ () Distane Along the Failure Cirle from Toe, Q/l FIG. 3G CORRELATION BETWEEN FAILURE ZONES AND FACTOR OF SAFETY FOR A 1.5:1 SLOPE IN A NORMALLY CONSOLIDATED CLAY LAYER WITH Su/P:

93 97 Tension Failed Reg ion Critial Failure Surfae. Fator of Safety = 2.10 (a) Exavated Depth 30 ft Fator of Safety= 1.84 ( b) Exavated Depth 40 ft '+ CJ) a. She a r Stress on C riti o I Failure Surfae ~ 2000 CJ) ~ LA~s~gned Shear Strength ~ ~.r-..j --,,_ ~ I0001--~~~-r-~~~-H~-.::::=:=1-~tt-~~~-t--"'--.,_-'-""-~-... E ~ E )( 0 ~ 0.._~~~-'-~~~--'~~~~~~~~_._~~~ () Distane Along the Failure Cirle from Toe, g; L FIG. 31 CORRELATION BETWEEN FAILURE ZONES AND FACTOR OF SAFETY FOR A 3: I SLOPE EXCAVATED IN AN OVERCON SOLIDATED CLAY LAYER.

94 30 ft high and 40 ft high. In both stages the ritial failure ar passes mainly through intat lay, and the fators of safety are quite high: 2.10 for the next-to-last stage and 1.84 for the last stage. Thus as is suggested by the limited extent of the failure zones, the slope at both stages has a fairly high degree of safety aording to the = 0 analysis. In ontrast to the results for slopes in normally onsolidated lay layers, the position of the ritial irular ar and the failure zone are not in good agreement. The failure zone extends down from the bottom of the exavation and along the base of the lay layer, whereas the ritial irle is fairly shallow and does not interset the bottom of the lay layer. There is some evidene to indiate that the failure zones determined using bilinear analyses may be in better agreement with observed failures than are the ritial failure surfaes determined from = 0 stability analyses: Bjerrum (1967) observed that failure surfaes in overonsolidated lays typially dip down from the base of the exavation, then extend horizontally bak toward the rest of the slope and then rise steeply to the surfae. One suh failure surfaed, observed in the hill adjaent to the Seattle Freeway exavation, is illustrated in Fig. 32. Apparently no hard layer was noted at the elevation of the horizontal portion of the failure surfae, but the bedding may have ontrolled the orientation in this setion. It is interesting that one of the symptoms of trouble in the Seattle Freeway exavation was bulging near the toe of the slope; as shown in Fig. 32, the slide triggered by the exavation ultimately involved a large portion of the hillside. Summary Bilinear analyses appear to provide a simple, useful means of studying the development of failure zones around exavated slopes. studies performed indiate: (1) In slopes in layers having high initial horizontal stresses, the failure zone develops first beneath the toe, subsequently progressing bak beneath the slope. The When the initial stresses are low, however, the failure zone develops first beneath the slope rest, subsequently progressing downward and toward the slope. 98

95 FIG. 32 SLOPE FAILURE OBSERVED IN OVERCON SOLIDATED CLAY, EXCAVATION FOR SEATTLE FREEWAY-AFTER BJERRUM -\ 1967). 99

96 (2) For uniform strength throughout the depth of the layer, failure ours first at the bottom of the layer beneath the slope and extends upward toward the bottom of the exavation and the rest of the slope in subsequent stages. For strength inreasing linearly with depth, from zero at the surfae, a pie-shaped failure zone develops around the slope rest; as exavation ontinues the failure zone grows in proportion with the size of the slope while remaining about the same shape. (3) The failure zones around slopes in layers simulating initial stress onditions and strength profiles representative of normally onsolidated soils begin beneath the slope and extend upward toward the slope rest, and downward toward the toe of the slope and the base of the layer as exavation ontinues. The final shapes of failure zones strongly suggests the possibility of failure on a urved surfae of sliding, For the final stages, when the failure zones enompassed the major portions of the regions beneath the slope, fators of safety alulated by the "p = O" method ranged from to (4) For slopes in layers representing overonsolidated lays, the failure zones begin beneath the base of the exavation, extending upward toward the bottom of the exavation and downward toward the base of the layer as exavation progresses. The analyses ould not be ontinued beyond the stage where the failure zone enompassed several nodal points to whih fores would be applied during the next step. Although these failure zones do not suggest the development of a urved surfae of sliding, aounts of experienes with failures in overonsolidated lays indiate that these analytial results may orrespond to the bulging or loss of ground observed in atual slope failures. For the stages at whih the bilinear analyses were terminated, the fators of safety alulated by the "p = O" method were found to range from to (5) Whether analyses are performed using a value of Poisson's ratio equal to 0.475, or using a onstant value of bulk modulus 100

97 101 representative of saturated lay under undrained onditions, or using a value of Poisson's ratio equal to one-half does not appear to have any major effet on the results.

98 CHAPTER 4 UNDRAINED STRESS-STRAIN BEHAVIOR OF SATURATED CLAY Introdution The full potential of omputers for providing aurate and useful solutions to soil mehanis problems involving stress and deformation under load an only be realized when methods have been established for haraterizing the stress-strain harateristis of soils. The assumption of linear elasti stress-strain behavior, often made for purposes of simplifying analyses, is not ompatible with the nonlinear behavior of many soils. As a result it is very diffiult, or perhaps impossible in some ases, to selet a single value of modulus representative of the behavior of a large mass of soil. Ladd (1964) has shown that seletion of a single modulus value to represent soil behavior involves onsideration of many fators inluding onsolidation pressure, stress level or strain, and stress orientation, and is perhaps highly subjetive. He showed that modulus values determined using isotropially onsolidatedundrained triaxial tests were 2 to 4 times higher than values bakalulated from prototype analyses. In other ases, modulus values determined from unonsolidated-undrained tests have been found to be muh smaller than values inferred from prototype performane, very probably as a result of disturbane effets. With the advent of the finite element method of analysis and the availability of large apaity, high speed digital omputers, it is possible to analyze problems involving nonlinear stress-strain behavior, but it is first neessary to investigate and formulate the nonlinear behavior of the materials involved. Two approahes to this problem may be oneived: the first is to develop relationships of a basi nature whih embody the fundamental relationships between stress and strain (onstitutive laws). For soils this is a formidable task. While it is highly desirable to develop onstitutive laws for soils, their development is likely to require a onsiderable number of years. The seond approah onsists in the development of empirial relationships expressing the stress-strain behavior under speified onditions, suh as the stress-strain harateristis of normally 102

99 onsolidated lay under undrained onditions. This tehnique, whih is simpler, appears to offer the best method for immediate appliations. An important aspet of this task is to develop means of heking and verifying the empirial relationships developed. The study desribed in this hapter was onduted to determine the nonlinear stress-strain harateristis of a normally onsolidated lay under undrained onditions. San Franiso Bay Mud was used for this study beause undisturbed samples of high quality were easily obtainable, and beause a onsiderable amount of valuable data were available from previous studies. Properties of San Franiso Bay Mud San Franiso Bay Mud is a soft, silty marine lay ontaining silt lenses and small amounts of shells and roots. All of the samples used in this investigation were taken from tfie U-nfversity or C-arifOrnfa field test site at Hamilton Air Fore Base in Marin County, California. of the Bay Mud at this site are summarized in Fig. 33. The properties The results 0f a number of plane strain tests onduted on Bay Mud during previous investigations are sunnnarized in Table 4. In eah of these tests the speimens were onsolidated without lateral strain, simulating additional onsolidation to higher values of pressure than existed in the ground where the speimens were obtained. Referring to the in-situ onditions, the major prinipal stress during onsolidation ated in the vertial diretion in all ases. using two tehniques: The undrained shear tests were onduted In the first, the stress ating in the vertial diretion was inreased until the speimen failed. This was termed a vertial plane strain (VPS) test. In the seond, the stress ating in the horizontal diretion was inreased until the speimen failed. This was termed a horizontal plane strain (RPS) test. It may be noted that the values of the pore pressure oeffiient Af, the values of strain at failure, and the values of (ra - oi)f/o~ are quite different for the two types of test. [(o - ojf/o' is the stress differene at failure divided by the a ;v a major prinipal stress during onsolidation, or twie the value of S /p.] u Differenes in the results of the two types of test have been shown by Dunan and Seed (1966) to be due to anisotropy and stress reorientation, 103

100 -.. CD ~ I.. u Cl -... ~... en 0 ~ ill E.. ID s= -Q. CD 0 L k 2 Effetive Overburden Pressure Water Content - perent Shear Strength - II/m k 2 Desription or Preonsolidated Pressure O/m ol--~~~~~~~~~-..!o~l~0:... ;2~0~30i:!...:4~0:L..:5~0~6~0-..!.170~8~0~9TO~IOr0::...,:0;:.-.:0,.~1~0T.2~0~.r3~0~Ar:-:0~5:...::0T~:...;:;Or~~O~B::--,or.:o~.l~OT2:...;:;0r.3~0.r4~0~5=-"'0f.6~0.r7~0.r8~0,~~17.0"'--i Topsoil 5 10 Desiated and weathered brown sandy lay. Stiff to very s!iff, fissured. Soft to very soft grey organi silty lay with isolated silt lenses and a few roots and shells. Average Values w = 90 perent LL= 88 perent PL= 43 perent 15 Cutp= 0.32 St s Cy~ 2x 10 m/se Minimum Vane tests. \ UU tribxial tests \ \ \ Minimum \ \ \ Effetive stress. for hydrostati water pressure with water table~ 8 feet below ground s~rfae. \ zo \ \ AG. 33. SUMMARY OF THE PROPERTIES OF SAN FRANCISCO BAY MUD (SAMPLES FROM HAMILTON AIR FORCE BASE, MARIN COUNTY).

101 Table 4. Summary of Plane Strain Test Results on Undisturbed Speimens of Bay Mud. Test VPS- 3 VPS- 4 VPS- s VPS- 6 VPS- 8 VPS- 9 VPS- 10 VPS- 11 VPS-L-1 VPS-L-2 VPS-L-3 VPS-L-4 AVERAGE RPS- 4 RPS- s RPS- 6 RPS- 7 RPS- 9 RPS- 11 RPS- 12 RPS- 13 RPS- ls RPS-L-1 RPS-L-2 RPS-L-3 RPS-L-4 Average I a r.r. I tf r1' f r3 1 f K a 0 Xf e:af S.2S S s. 7S 0.66 O. ls so S.so SO SO S 4.2S S o. 71 4~00 L3-'Z L L_(ll S SS SS S S o SO S O.S S S o o S o.so S l.sl 4.2S O.S2 1.SS 4.2S o (r~-r~) a f S S84.S S.S8S.S30.61S.S40.S27.S02.S S6 los

102 and signify that the behavior of Bay Mud under unqrained onditions, and its undrained strength, depend on the orientation of the stress hanges ausing failure. The stress-strain urves, as well as the strengths, differ markedly for these two types of test. The stress-strain urves for all of the tests shown in Table 4 have been orreted for the effets of piston frition, end plate frition, filter paper drains and rubber membranes. The urves were then normalized by dividing the stress ordinates by the value of the onsolidation pressure, p = a' Normalized a stress-strain urves for vertial plane strain tests are shown in Fig. 34, and those for horizontal plane strain tests are shown in Fig. 35. stress differene in eah ase is (aa - ai), the differene between the axial and lateral stress values. a ats in the vertial diretion for a the VPS tests, and in the horizontal diretion for HPS tests. These test results show that the stress-strain harateristis of San Franiso Bay Mud depend to a onsiderable extent on the orientation of the stress hanges ausing deformation. The It might be expeted, therefore, that many normally onsolidated lays will be haraterized by similar anisotropi stress-strain relationships, beause there is no reason to believe that Bay Mud possesses an unusually high degree of anisotropy. attempting to develop empirial stress-strain relationships for Bay Mud, it was onsidered imperative to take into aount the influenes of stress orientation and anisotropy. Modulus Values. For purposes of analysis, it is onvenient to represent the normalized stress-strain urves shown in Figs. 34 and 35 by urves representing the variations of normalized modulus values with strain, as shown in Fig. 36. These urves represent variations of the instantaneous tangent modulus, Ei, with the value of maximum shear strain, y Beause the stress-strain urves shown in Figs. 35 and 36 orrespond max to plane strain onditions, the instantaneous modulus values, Ei' are equal to (1 - V 2 ), or 0.75 times the instantaneous slope of the stress-strain urves. Maximum shear strain has been found to be a onvenient strain parameter beause it affords a measure of the total amount of shear distortion, independent of the stress or strain diretions. Beause no test results were available for orientations intermediate between horizontal and vertial ompression, it was neessary to hypothesize In 106

103 , r-----,----., b...;. I 0. 6' 0.8 J , t _ Axial Strain- Ea - perent FIG. 34. STRESS-STRAIN BEHAVIOR OF SAN FRANCISCO BAY MUD IN VERTICAL PLANE STRAIN. 107

104 r-----,----, r--~ -"6 a. u 't: a. E :::t E 0 s \ i :E -0.6 H ~--1--~ ~1-0.8 L L " ' ' ' Axial Strain- Ea- perent FIG. 35. STRESS-STRAIN BEHAVIOR OF SAN FRANCISCO BAY MUD IN HORIZONTA~ PLANE STRAIN. 108

105 ~ , , , M \ \ ' \ \ ' \ en en C1> 0 ;; C1> E i5 10 Horizontal Plane Strain I I Vertial Plane Strain Maximum Shear Strain-Ymax-perent 12 FIG.36. MODULUS-STRAIN VARIATION FOR SAN FRANCISCO BAY MUD 109

106 how the value of Ei might vary with stress orientation at any value of maximum shear strain. Beause the values of E. for horizontal and vertial 1 ompress~on logially represent the extremes, values of Ei for other stress orientations would be expeted to be intermediate between those shown in Fig. 36. Furthermore, beause the variations must be symmetrial about the horizontal and vertial orientations, it seems logial to believe that the variation would be represented by a smooth urve of the type shown in Fig. 37. this shape is by the equation: The simplest means for representing a urve of in whih Bi is the angle between horizontal and the instantaneous diretion of ompression, ES is the instantaneous modulus value in the 1 Si-diretion, and EH and EV are the modulus values for horizontal and vertial ompression, respetively. Using the urves shown in Fig. 36, together with equation 9, it is possible to determine instantaneous modulus values for any value of shear strain in the range of values shown, and for any diretion of ompression. The values of modulus are proportional to the onsolidation pressure p. It should be noted that these relationships would only be expeted to apply to onditions orresponding to the test onditions on whih they are based, i.e. undrained plane strain deformation of Bay Mud normally onsolidated without lateral strain" However, these onditions are satisfied by short-term slope problems involving uts or fills of large longitudinal extent, and the relationships postulated may be useful for analyses of these and similar types of problems. Pore Pressure Coeffiients. (9) In addition to the amounts of strain or deformation, it is desirable for many types of problems to develop tehniques for prediting hanges in pore pressure indued by hanges in total stress under undrained onditions. One onvenient means for making suh preditions involves use of the pore pressure oeffiients A and B suggested by Skempton (1954). Aording to the definition of these oeffiients, the hange in pore pressure indued by hanges in total stress may be expressed as (10) 110

107 .5 E _7 v Angle Between Diretion of Compression and Horlzontal-/3i -degrees FIG. 37. VARIATION OF MODUWS WITH DIRECTION OF COMPRESSION. 111

108 in whih ~u is the hange in pore-water pressure, ~o and 1 ~o are the major 3 and minor prinipal hanges in stress, and A and B are pore pressure oeffiients determined by measuring pore-water pressures under undrained test onditions. It is worthy of note that equation 10 may be applied with equal validity to triaxial and plane strain test onditions, provided the values of A have been measured under appropriate test onditions. The value of the oeffiient B depends primarily on the degree of saturation of the soil, and for soft saturated lays, B is equal to unity for all pratial purposes. The value of the pore pressure oeffiient A has been found to depend on the diretion of ompression as indiated by the plane strain test results summarized in Table 4. The average values of A at failure (Af) were found to be 0.71 for horizontal ompression and 1.08 for vertial ompression. Values for intermediate orientations of the major prinipal stress at failure (Bf) may be derived from the variation of undrained strength with stress orientation at failure. Dunan (1967) has shown that the value of Af orresponding to any value of Bf is given by the expression (11) in whih ' and ' are the effetive stress shear strength parameters, S u is the undrained strength, p is the major prinipal stress during onsolidation, K is the oeffiient of earth pressure at rest, and Bf is the angle 0 between horizontal and the major prinipal stress at failure. Using the variation of undrained strength with Bf determined from unonsolidatedundrained triaxial tests, it was found that the value of Af varies with Bf as shown by the solid urve in Fig. 38. For the purpose of alulating hanges in pore water ~ressure prior to failure, it is more onvenient to relate the value of A to the diretion in whih hanges in stress are applied, Si. Exept for horizontal and vertial ompression (Bf = 0 and Sf = 90 ) the values of Si are different from the values of Sf. For any given value of Bf' however, the orresponding 112

109 1.2 I. I 1.0 A,= fc~1) i--- / " -,,,,.,,,,. v ;,- v / /,I~ 'r...---~ ~ ~,.:; 0.8 ~ '.. v ~ ~- A 1 = f(p,) - Q) u ~ '1i> 0 u e :J en en!? a.!? ~ ' T-/ ~ '~ i It I Range of Calulated Values Angle Between Horizontal and Major Prinipal Stress at Failure - P 1 - or Angle Between Horizontal and Major Prinipal Change In Stress -Pi - degrees FIG. 38. VARIATION OF THE VALUE OF A, WITH DIRECTION OF COMPRESSION FOR SAN FRANCISCO BAY MUD. 113

110 value of Bi may be found by repeated trials, provided failure is aused by a single stress inrement. For example, in order to ause failure with the major prinipal stress oriented 70 from horizontal, beginning from at-rest pressure onditions with the major prinipal stress vertial, the major prinipal hange in stress must be oriented 38 from the horizontal. Thus Bf= 70 orresponds to Si= 38, and the value of Af orresponding to Bf= 70 would also orrespond to Bi= 38. Using this tehnique, the relationship between Af and Bi was determined whih is shown by the dotted line in Fig. 38. In using the relationship shown in Fig. 38 to alulate pore water pressure hanges prior to failure, it is assumed that the value of A does not vary with stress level or strain. Dunan and Seed (1966) found that the value of A was nearly independent of stress level and strain throughout horizontal plane strain tests. Although the value of A was found to inrease with inreasing strain in vertial plane strain test onditions and has also been found to do so in triaxial tests, this may our only beause of inomplete equalization of nonuniform pore water pressures during the early stages of these tests. Bishop, et. al (1960) found that the more slowly undrained tests were onduted, and the higher the degree of equalization in the early part of the test, the less the value of A varied with strain. Therefore it seems reasonable to assume that values of Af may be used to alulate hanges in pore water pressure prior to failure as well as at failure. The nonlinear stress-strain relationship illustrated by Figs. 36 and 37, and the variation of Af with Si shown in Fig. 38 ould be used in onjuntion with the finite element method to alulate stresses, strain, displaements and pore water pressures for slopes in San Franiso Bay Mud and for similar problems involving undrained loading onditions. However, before performing analyses of this type it was onsidered highly desirable to hek the validity of the relationships developed. For this purpose a series of simple shear t~sts were onduted on San Franiso Bay Mud. The measured results of these tests were then ompared to the values predited using the stress-strain and pore pressure relationships disussed previously. The results of this study are desribed in subsequent setions of this hapter. 114

111 Simple Shear Tests Two types of simple shear test devies have been developed in reent years, both of whih afford onsiderable improvement over diret shear apparatus with respet to uniformity of stresses and strains within the test speimens. One of these devies, developed at the Swedish Geotehnial Institute and also manufatured by the Norwegian Geotehnial Institute, employs a short ylindrial speimen enlosed in a rubber membrane. The membrane is reinfored by an embedded wire spiral to restrit lateral expansion, and the top and bottom of the speimen are in ontat with parallel rigid metal diss through whih horizontal and vertial loads or displaements may be applied to the speimen, as shown in Fig. 39. Another devie, developed at Cambridge University, employs a retangular parallelpiped speimen, also enlosed in a rubber membrane. The speimen and membrane are both enlosed within a rigid box having elaborate bearings and hinges suh that the top may move vertially or- horrzonta1.ly9 but always remains parallel to the base. As the top translates, the rigid sides of the box rotate as shown in Fig. 39 so that the sides of the speimen remain straight. The devie has been desribed in some detail by Peaok and Seed (1967). Although both of these devies were available for use, only the Cambridge University devie was used for this study. During previous tests on Bay Mud using the SGI-NGI devie, some of the speimens were observed to deform nonuniformly, as shown in Fig. 39. Furthermore, beause the steel sides of the Cambridge University devie are more rigid than the reinfored membrane used with the SGI-NGI devie, it is better suited for pore pressure measurements. The prinipal drawbak of the Cambridge apparatus is that frition between the membrane and the box sides tends to resist relative movement. Grease was used to lubriate these sliding surfaes, and orretions were made for the remaining frition fores; these orretions are desribed in Appendix B. Test Proedures. The simple shear tests were onduted using undisturbed speimens of San Franiso Bay Mud from Hamilton Air Fore Base. Before testing, the speimens were onsolidated one-dimensionally under higher pressures than those in-situ at the depths from whih the speimens were obtained. Beause the vertial travel of the ap in the 115

112 Reinfored Membrane I I r ; ' _J I I, , I. I I I L I r ,,_ I...J I I r '... J I I ' ' Swedish Geotehnial Institute and Norwegian Geotehnial Institute Cambridge University FIG.39 TWO TYPES OF SIMPLE SHEAR APPARATUS 116

113 simple shear devie is restrited, the speimens were onsolidated initially in a large onsolidometer, then trimmed to the required dimensions (6 m square by 2 m high), and installed in the simple shear devie. They were then onsolidated a small additional amount in the devie, to insure that they were properly seated and normally onsolidated under the desired test pressure. Consolidation within the shear box was one-dimensional, and it was neessary to infer the lateral pressure values from previous tests whih showed that K for Bay Mud is approximately Speimens were 0 allowed to onsolidate for 48 hours at the final pressure to insure that they reahed equilibrium, and a bak pressure of 1.5 to 2.0 kg/m 2 was used to insure that the speimens were fully saturated. After onsolidation the speimens were tested undrained, by applying shear stress at a rate suffiient to ause failure within 1-1/4 to 3-3/4 hours; for these test durations the estimated degree of equalization of nonuniform pore water pressures ranged from 92% to 98%. Variations ofshear stress and pore water pressure with shear strain for the first test are shown in Fig. 40. Shear stress values are shown for both the top and midheight, the two values differing by about 8% at large values of strain. This differene in shear stress values is indiative of some degree of nonuniformity in the stress distribution within the speimen, and the measured amount of differene is in good agreement with values alulated using linear elasti stress analyses presented by Rosoe (1953). results of this analysis are shown in Fig. 41. The stresses shown in Fig. 41 are those indued by applied shear loads only; the stresses at any stage are represented by the sum of the initial stresses, due to vertial loading, and those indued by shear loading. It may be noted that shear loading indues hanges in normal stress 6r and 6r as well as shear stresses T For horizontal x y xy equilibrium, the resultant of the shear stresses at any elevation must equal the resultant of the shear stresses at the top plus the resultant of the hanges in normal stress 6r between the top and that elevation. Thus x the shear stress values inrease from the top and bottom toward the middle, reahing their maximum values at midheight. The These stress nonuniformities result from the fat that although simple shear deformation is enfored by the boundary displaements, omplementary shear stresses are not applied 117

114 ,~------, r r N e ~ Ct ~ I... ~ )( :-----l------t------t Shear Stress at Mldhelght J, f--~~+---.-:::::::;;;--.t-~~=-hllr---=i fl> t!!... Cl) 6S ~~--l'----t r----i -en Test SS-I O"d = 0.90 kg/m 2 0 -o'"---_-,---~~--~3---4~--~s~~~--~7~~a~~g~ ~to 0.40r , , i N E ~ i--::=;;:;-::>-9 I :J I e :J ~ t------:~--t-----r-----t-----f e ~.E G> Ct ~ #--+---t o~--~--=2----=3----1:4----=5=--~s~--1~~e~--9~~0 Shear Straln-yxy - perent FIG. 40. VARIATIONS OF SHEAR STRESS AND POREWATER PRESSURE WITH SHEAR STRAIN IN SIMPLE SHEAR TESTS ON SAN FRANCISCO BAY MUD. 118

115 Txy FIG. 41. NORMAL AND SHEAR STRESSES INDUCED BY SHEAR LOADING. AFTER ROSCOE (1953). 119

116 to the ends of the speimen. Using these stress distributions, the differene between the average values of shear stress at the top and at midheight were alulated to be about 6%, ompared to the measured differene of 8%. Nonlinear finite element analyses indiate that the value may range from 4% to 8% theoretially, with the largest differene orresponding to the early stages of the tests. us~ng The values of shear stress at the top of the speimen were measured an eletroni load ell, whereas those measured at midheight were alulated by applying the required frition orretions to the applied shear loads. The lose agreement between the alulated and measured differenes in shear stress values was onsidered to indiate that both methods of measurement were suffiiently aurate. Two additional shear tests were performed on San Franiso Bay Mud at higher onsolidation pressures (o' = 1.04 kg/m 2 and 1.34 kg/m 2 ) a using the-s~me loading proedures_. ~he IeBults of all three tests have been orreted for frition effets and are shown together in Fig. 42. The stress-strain and pore pressure urves have been normalized by dividing the stress ordinates by a' = p, the major prinipal stress during a onsolidation. The normalized results of all the tests exhibit a degree of satter omparable to the plane strain test results disussed previously. Comparison of Predited and Measured Results. Using the empirial stress-strain relationsnips developed for San Franiso Bay Mud, finite element analyses were performed for the simple shear test onditions employed in the testing program, in order to evaluate the usefulness of the empirial relationship. Fig. 43. The finite element onfiguration used in these analyses is shown in The nodal points on the bottom of the speimen were fixed, and the row of nodal points at the top of the speimen were given a uniform horizontal displaement, simulating full adhesion between the soil and the top and bottom plates. The nodal points at the ends were given horizontal displaements in proportion to their height above the base, and were free to move in the vertial diretion. The analysis was aomplished in a series of inrements; 10 displaement inrements were used from 0% to 5%, and 5 more inrements from 5% to 10% strain. A new value of modulus was alulated for eah element after eah inrement, based on the average 120

117 ,~--...,------r-----r e- >.._,>< I Cl) f (I) ' Cl> (I) "tj Cl> ~ 0 ~ ~ ,-----r ir-----ir------, e- g- io- -e- ::J <J I 0.30 Cl> OI : 0 e ~ t e ~ 0.20 i.!::! ti e 0.10 ~- ~IG Shear Strain-Yxy-perent NORMALIZED SHEAR STRESS-SHEAR STRAIN AND PORE PRESSURE-SHEAR STRAIN CURVES FOR SIMPLE SHEAR TESTS ON SAN FRANCISCO BAY MUD. 121

118 i I I I I t--1-~-j- -r r- - ~- -t-r +~--~-+--L---~-----J--~--~--L---~ I I I / I I I' ~~--r-~--r-1-~-~r-~--r r J -/ ' - -,_ J- - -/- - -r _J_ ,_ J. " I I I I I I I I T -1- ~ + - -,-- -f - -I- - r - -/- - t - "t r - i I - ~I- - t" - -I - -I - - /- - T / - -/- - I ; -f -I- - I /- - -t - I - - I- - -I - -I- - I- r -I J - ~ - - I- -.,_ - - I- - -/- - -/- - """ I- -+ FIG. 43 FINITE ELEMENT IDEALIZATION OF THE SIMPLE SHEAR SPECIMEN, SHOWING INITIAL AND DEFORMED SHAPE. 122

119 value of strain antiipated for the subsequent inrement. Using this proedure, no iteration was neessary even though the analysis was nonlinear, beause the strain values ould be antiipated with a high degree of auray. Displaements were alulated for eah nodal point, and strains and stresses for eah element, after eah displaement inrement. The inremental values were added to the previous values eah time. Using the variation of A shown in Fig. 38, hanges in pore water pressure were also alulated for eah element following eah step. The values of strain, stress, and pore water pressure varied throughout the speimen, as would be expeted on the basis of the linear elasti analysis performed by Rosoe (1953). To develop interpretations onsistent with the experimental results, an average value of shear stress at midheight was alulated using the individual element stresses, and average values of shear strain were alulated from the boundary d±svla~ments. Average values of hange in pore water pressure were alulated assuming no volume hange, and taking into aount the fat that the slopes of the primary onsolidation and rebound urves are not the same. The predited and measured results are ompared in Fig. 44. At all values of strain the predited values of shear stress and pore water pressure are approximately 20% to 30% higher than the measured values, although the measured and predited urves have similar shapes. It therefore appears that the instantaneous modulus values used in the analysis were too large, by about 25% on the average. It seems likely that the modulus values orresponding to horizontal and vertial ompression are aurate, beause these were determined diretly from the plane strain test data. For intermediate diretions of ompression, however, it was neessary to hypothesize the relative modulus values, as shown by the sin 2 variation in Fig. 37. Beause the stress inrement orientations in the simple shear onditions have an average orientation lose to 45 from horizontal, it appears that a more orret variation of modulus with stress inrement orientation would lie below the sin 2 variation over the entral portion of the urve. The basi shape of the urve should logially remain the same, however, with horizontal asymptotes at the ends. 123

120 ir------,-----r-----, Finite Element Simple Shear Analysis Using sln 2 Modulus Variation Laboratory Tests e- :::J Finite Element Simple Shear Analysis Using sin 2 Modulus Variation <J! f!L ~~ I tjt.r::. u Laboratory Tests e :::J e m o. 20 J.---~-+-::~---f t a.. e ~ ~.!::! ~ 0.10 t--+# z Shear Strain -Yxy - perent FIG. 44. COMPARISON OF EXPERIMENTAL AND ANALYTICAL SIMPLE SHEAR TEST RESULTS.

121 A modified modulus variation satisfying these riteria is shown in Fig. 45, together with the sin 2 variation used in the previous analysis. The modified modulus variation may be expressed by the equation in whih S. is the orientation of the major prinipal stress inrement, i ES is the modulus in the Si diretion, and EH and EV are the modulus values in the horizontal and vertial diretions, respetively. An additional finite element analysis of the simple shear test onditions was performed using this modified modulus variation and the same analytial proedure desribed previously. are ompared to the experimental results in Fig. 46. (12) The results of this analysis It may be noted that the predited values of shear stress and pore pressure are less than the measured values at values of shear strain larger than 4%; at 10% strain the predited values are about 10% less than the measured values. Thus, by reduing the modulus values for stress orientations other than horizontal and vertial, fairly good agreement has been ahieved between the predited and measured simple shear test results. While it seems almost ertain that even better agreement ould be ahieved between the predited and measured stress-strain urves, it was not onsidered neessary to ahieve loser agreement to establish the usefulness of the stress-strain harateristis as formulated for these analyses. Although the shape of the predited and measured stress-strain urves in Fig. 46 are very similar, the predited pore pressure-strain urve rosses the measured one, beause the predited values of pore pressure are somewhat too high at low values of strain. As explained previously, it has been assumed that the nonuniform pore-water pressures were ompletely equalized at all stages of the test in omputing the predited values, Thus the predited values orrespond to tests of longer duration than those atually performed, whih lasted between 1-3/4 and 3-1/4 hours. the measured values of pore-water pressure were found to inrease slightly with inreasing test duration, it seems likely that the variation of pore pressure with strain measured in a test of long duration would be more similar in shape to the predited urve in Fig. 46. Sine 125

122 0 +: u Cl> ~ a <:ii" _.5 en ::J :; 'O 0 :E EHPS EVPS. -...;; F':: I I ", sin 1 Variation '!":', ~ ' '.... -Modified Variation~ ' ' '... ~ ' :" ' ' Angle Between the Horizontal and the Major Prlnlpal Change In Stress - ~I - degrees; FIG. 45 MODIFIED MODULUS VARIATION FOR SAN FRANCISCO BAY MUD. 126

123 FIG. 46 COMPARISON OF EXPERIMENTAL AND ANALYTICAL SIMPLE SHEAR TEST RESULTS , , , , Laboratory Tests Finite Element Simple Shear Analysis Using Modified Modulps Variation Shown in Flo , , r , ,... Finite Element Simple Shear Analysis Using Modified Modulus Variation Shown in Fig Shear Strain - Yxy- perent

124 Progressive Failure in Simple Shear During the ourse of these analyses it beame evident that a speimen of lay may fail progressively under simple shear test onditions. Using the finite element method, stresses were alulated for eah element in the onfiguration at eah value of imposed strain. It was thus possible to determine where failure ourred initially and how the size of the failure zones inreased with inreasing amounts of distortion. Six stages of the analysis, orresponding to values of shear strain from 5% to 10%, are shown in Fig. 47; the ross-hathed areas indiate zones in whih the maximtml shear stress has beome equal to the shear strength of the lay. Failure begins, near the ends of the speimen, at 6% shear strain. The size of these failure zones gradually inreases and the zones merge at 8% strain, the size of the failure zone subsequently inreasing still more until only a small stable region remains around the edge of the speimen at 10% strain. It is thus evident that the stress onditions are suffiiently nonuniform to result in early failure in some regions in the speimen. Although the peak of the stress-strain urve has not been reahed even at 10% shear strain, initial failure ours at 5% strain. For a soft lay like San Franiso Bay Mud, with little or no redution in shear resistane beyond the peak, the ourene of loal failure at an early stage of the test is not likely to be signifiant. For a soil with onsiderably redued shear resistane beyond the peak, however, early failure of a portion of the speimen might influene the test results to a onsiderable degree. Determination of the likelihood of progressive failure in simple shear for other soils, however, would require additional analyses using appropriate stress-strain behavior. Simplified Simple Shear Analysis The nonuniform stress onditions whih develop within simple shear test speimens makes full interpretation of the test results quite omplex, requiring use of detailed stress analyses. For some purposes, however, it is feasible to approximate the nonuniform simple shear onditions by uniform pure shear. The linear stress analysis performed by Rosoe (1953) and the nonlinear finite element analyses performed during this study all 128

125 5 /o,._ Perent Shear Strain 6 /o I 7 /o ~ FIG. 47. DEVELOPMENT OF FAILURE ZONES IN SIMPLE SHEAR. 129

126 show that the stress nonuniformities in simple shear are most severe near the ends of the speimen. As illustrated in Fig. 48, simple shear differs from pure shear by the absene of omplementary shear stresses at the ends of the speimen. The magnitudes of the shear stresses on planes parallel to the ends inrease toward the enter, however, and the stresses in the interior of a simple shear speimen orrespond losely to pure shear onditions. Pure shear is a onvenient approximation for the stress hanges in simple shear beause the stresses are uniform and also beause the hanges in stress are extremely simple, as shown in Fig. 48. Pure shear may be represented by equal and opposite hanges in shear stress or by equal and opposite hanges in normal stress on perpendiular planes. The stress state resulting from an inrement of pure shear is shown by the Mohr diagram in Fig. 48. The initial stress onditions are represented by o = o and o = o, as would be appropriate for a normally onsolidated XO 3 yo 1 lay speimen prior to shear. Appliation of hanges in shear stress +6Lxy to the x and y planes, or hanges in normal stress 60 1 and 60 3 to their bisetors results in the same state of stress, as shown in Fig. 48. If it is onsidered that simple shear may be represented by pure shear, it is possible to predit the stress-strain urve for simple shear tests very easily using the stress-strain relationships developed. pure shear the diretion of ompression is at 45 from the horizontal (Si= 45 ), and modulus values for this inremental stress orientation may be determined by evaluating equation 12 for Si= 45. expression is The resulting in whih EH, EV, and E 450 are instantaneous modulus values for horizontal, vertial, and 45 ompression at the same values of strain. To ompute the stress-strain urve for pure shear, the instantaneous moduli EH and EV at inrements of 1% shear strain, beginning with 0.5%, have been evaluated using Fig. 36. listed in Table 5. In (13) These values, and the orresponding values of E 450, are To alulate orresponding stress hanges, the relationships between inrements of stress and strain are onsidered: For pure shear stress inrements with the maximum shear stresses applied to the 130

127 ~--- ff ;'J I 1 f t ~ f--~-- jj..._... ~_.._..., J _ ~... " "f :1 r ~ l 1 ~ ~ l,~ ~,/. L rj rj _,_ ~ ro-... r ~~ Simple Shear Pure Shear Pure Shear Stresses on Vertial After Change Stresses on Horizontal FIG. 48 SIMPLE SHEAR AND" PURE SHEAR. 131

128 Table 5. Pure Shear Analysis Shear Average Values for Inrement Stress Inrements Shear Pore Shear Strain Stress Strain Pressure Inrement EH EV E45o (L\r 1-L\r 3) L\t T L\u y L\y y ~ u _E_ - - % p p p p p p p p % %

129 x and y planes, the various stress hanges are related as shown in Fig. 48, i.e. (14a) and 0 (14b) in whih 6Txy is the shear stress inrement, and 60, 60, and 60 are the major, intermediate and minor prinipal stress inrements. These stress inrements may be expressed in terms of the orresponding strain inrements and the instantaneous modulus values as follows 6T = (G.) 6y xy 1 xy xy (15) in whih G. is the instantaneous shear modulus value and 6y is the shear 1 ~ strain inrement The shear modulus may be expressed in terms of Young's modulus as E G = ~~~- 2 ( l + v) (16) and for v 0.5, appropriate for saturated lay under undrained onditions, E G=- 3 (17) Substituting this expression for G into equation (15), the shear stress inrement may be expressed in terms of the instantaneous value of Young's modulus: (18) and the inremental stress differene may be expressed as (19) Equations (18) and (19) have been used to alulate the stress inrements in Table 5. From these inrements of total stress, inrements of pore-water pressure may be alulated using the pore pressure oeffiients A and B proposed by Skempton (1953): (20) 133

130 For pure shear onditions, the diretion of ompression is 45 from the horizontal (Si= 45 ). The value of A for this stress inrement orientation is 1.01 for San Franiso Bay Mud, as shown in Fig. 38, and the value of B is equal to 1.00 beause the lay is saturated. These values of A and B have been used to evaluate the pore pressure inrements in Table 5. Variations of shear stress and pore pressure with strain were alulated, as shown in Table 5, by summing the inremental values, and the resulting variations are shown in Fig. 49. The values of shear stress alulated using the simplified pure shear analysis are somewhat smaller than the measured values, as in the ase of the finite element analysis shown in Fig. 46, whih was also performed using the modified modulus variation. At 10% shear strain, the shear stress values alulated by pure shear analysis are 13% lower than the measured values, whereas those alulated by the finite element analysis were about 10% smaller. Thus the shear stress-strain urve oetermineo by the pure shear analysis is in very good agreement with the one determined by finite element analysis. As mentioned previously, the analytial and experimental variations ould be brought into loser agreement if a slightly different modulus variation than the one expressed by equation (12) was used in the analysis. The differene between the alulated and measured values of poreu water pressure are muh larger, as shown in Fig. 49. Values of - alulated p by pure shear analysis are 35% lower than the measured values at 10% shear strain. Values alulated by finite element analysis, shown in Fig. 46, are only about 10% lower than the measured values. On the basis of these results it appears to be feasible to represent simple shear onditions by idealized pure shear stress inrements for the purpose of determining stress-strain behavior. Appliation of the same proedures to determine hanges in pore pressure, however, results in large differenes between alulated and measured values. It seems likely that the reason for this disrepany lies in the fat that hanges in pore-water pressure in simple shear are influened to a onsiderable degree by nonuniformities in stresses: As initially nonuniform pore pressures equalize, the equilibrium value is higher than the average beause of the differene between the slopes of the onsolidation and rebound urves. 134

131 tlg.49. COMPARISON OF LABORATORY SIMPLE SHEAR TEST RESULTS WITH PURE SHEAR ANALYSES , ~ , , , Laboratory Tests Pure Shear Laboratory Tests ::Jlo. I 0 :.:: a: Q) ::J Cl) Cl) Q) a.. ~ ~ Pure Shear Shear Strain - Yxy - perent 10

132 Stress and pore pressure nonuniformities are taken into aount in finite element analyses, but they are ignored in pure shear analyses. Summary An empirial nonlinear stress-strain relationship has been developed for San Franiso Bay Mud under undrained onditions whih appears to provide a onvenient and aurate basis for nonlinear analyses of stress and deformation. The relationship evolved enompasses the redution in instantaneous tangent modulus with inreasing degrees of distortion (shown in Fig, 36) and the influene of stress orientation or anisotropy (shown in Fig. 45). The values of modulus are proportional to the onsolidation pressure, p. Finite element analyses of simple shear test onditions have demonstrated the utility of the hypothesized stress-strain relationship, and omparison of the pre-di-ted -and --measured -simpl-e -she-ar results has made it possible to refine the proposed relationship. These simple shear analyses have shown that progressive failure is likely in simple shear beause of nonuniform stress onditions, and that these stress nonuniformities may effet the measured values of pore-water pressure to an appreiable degree. The atual, nonuniform simple shear stress onditions may be adequately represented by uniform pure shear stress onditions for purposes of onsidering stress-strain behavior, but not for onsidering the development of pore-water pressures due to the applied shear stresses. 136

133 CHAPTER 5 ANALYSIS OF A SLOPE USING NONLINEAR STRESS-STRAIN BEHAVIOR If finite element analyses of exavated slopes are onduted using a series of steps representing suessive stages in the exavation, the modulus value of eah of the elements representing the slope may be hanged after eah step of the analysis in aordane with the urrent values of stress and strain. It is possible, therefore, to simulate nonlinear stressstrain behavior with a series of linear inrements. This proedure has been employed in analyzing the behavior of a 50-ft. high slope exavated in San Franiso Bay Mud, to determine the pratiability of the proedure and to examine the influene of gradually hanging modulus values. Stress-Strain and Strength Charateristis The empirial stress-strain relationships developed in the previous hapter were employed in the analysis; the variation of modulus values with shear strain is shown in Fig. 36, and the variation with diretion of ompression is shown by the modified variation in Fig. 45. These are the same relationships employed in the analyses of simple shear onditions shown in Fig. 46. The variation of undrained strength with depth used in the analysis is representative of onditions at the University of California field test site at Hamilton Air Fore Base. As shown in Fig. SO, the ground water table is 10 ft. below ground surfae, and a desiated rust overlies the normally onsolidated lay beneath. The higher strength values shown orrespond to vertial ompression, and the lower values orrespond to horizontal ompression (Dunan and Seed, 1966). The variation of undrained strength with diretion of ompression between these extremes is shown in Fig. 51. This variation was determined using undrained plane strain and simple shear tests on normally onsolidated speimens. The initial values of horizontal stress used were three-quarters as large as the overburden pressure (K = 0.75). 137

134 Or----ir-,-r-~"""'-es~ur~fa~=e-----, r , LWL Y=l25 pf Su. -p=0.37 Vertial Compression ~=0.28 Horizontal Compression Undrained Strength -su-psf FIG. 50 VARIATION OF THE UNDRAINED STRENGTH WITH DEP.TH BELOW THE GROUND SURFACE. 138

135 e- :J en I 0 +: ~.r. -Ct ~ u; "'O Q) e "'O ::> ,,,,,,,., / v- ~ Angle Between Diretion of Compression and Horizontal-degrees FIG.51. VARIATION OF UNDRAINED STRENGTH WITH DIRECTION OF COMPRESSION FOR SAN FRANCISCO BAY MUD. 1.19

136 Analytial Proedure The method of analysis employed was similar to the bilinear slope analyses desribed previously, exept that the modulus for eah element was reevaluated after eah step in the analysis. A simplified flow diagram for the omputer program employed is shown in Fig. 52. It may be noted that eah time a layer is removed, the resulting displaements and stresses are alulated more than one. The first time the modulus values are evaluated using the strains at the end of the previous step. The seond and subsequent times, the modulus values are evaluated using average values of strain during removal of the layer. This repeated solution insures lose orrespondene between the desired nonlinear stress-strain harateristis and the linear segments employed in the analysis. In the analysis desribed in this hapter, the displaements and stresses were alulated only twie, beause preliminary analyses indiated that this resulted in satisfatory simulation of the desired nonlinear behavior. The analysis was performed by simulating removal of 5 layers, eah 10 ft. thik, the final height of the 3:1 slope being 50 ft. The finite element onfiguration employed represented a lay layer 100 ft. thik (50 ft below the toe of the slope) and ontained 330 elements and 368 nodal points. The analysis required 15 minutes of IBM omputer time. Slope Behavior The stress onditions after exavation of the slope to its full 50 ft depth are illustrated in Fig. 53 by ontours showing perentage of shear strength mobilized. It may be noted that no region has failed, the most ritial onditions orresponding to about 80% strength mobilized. The most highly stressed regions are near the toe of the slope and along the ontat between the bottom of the lay and the region layer beneath. This behavior is somewhat different from that established using bilinear analyses with similar strength profiles and initial stress onditions. In those ases it was found that failure ourred first in a zone beneath the rest of the slope. The differene in behavior probably results from the fat that the strength values employed in the present nonlinear analysis vary with diretion of ompression, and would thus be smaller 140

137 READ INPUT DATA CALCULATE FINITE ELEMENT I IDEALIZATION OF SLOPE ' I CALCULATE INITIAL STRESS DISTRIBUTION I REMOVE ONE LAYER AND CALCULATE NODAL POINT FORCES ' I EVALUATE THE ELASTIC PROPERTIES :E ::J OF EACH ELEMENT AND FORM -~ z 0 Cl STIFFNESS MATRIX FOR STRUCTURE Wen -w Wen!:!:...J LL 0: I fdu -w a.. ~ (/)0 frl >- SOLVE FOR DISPLACEMENTS a.. en <t: _, er- w aj t g, I a:: a:: <t LL PRINT DISPLACEMENTS I SOLVE FOR STRAINS AND STRESSES G I PRINT STRAINS AND STRESSES f2w m ~~ fz fhz a:: a:o ~ w a.. w a: FIG.52. SIMPLIFIED FLOW DIAGRAM FOR MULTILINEAR SLOPE ANALYSES. 141

138 - 50 =-----aoo/o --~~~...-~~~--;.,-...;._--=-~~...-~~~~-T-~--~------~~~~--~~~~--~-o Horizontal Distane 1 - feet FIG. 53. PERCENT OF SHEAR STRENGTH MOBILIZED IN THE MULTILIN.EAR ANALYSIS.

139 for the regions near the toe of the slope and along the base, where the diretion of ompression is lose to horizontal. Whereas the linear analyses indiated that loal failure would begin by the time the overall fator of safety had been redued to about 1.7, the nonlinear analysis indiates that no loal failure has developed even though the overall fator of safety is only Thus the use of gradually hanging modulus values, as opposed to abruptly hanging values, tends to prevent the development of stress onentrations and early loal failure. Sine modulus values are lowest in the zones most highly stressed, they are subjeted to less severe stress hanges during subsequent stages of exavation. The values of tensile stress alulated were very small. The values of stress at the entroids of the upper row of elements were all ompressive, but, by extrapolating these stress values to the surfae of the layer, it was determined that tensile stresses on the order of yh would exist at the surfae. These values are smaller than indiated by the previous linear and bilinear analyses, probably beause the modulus values orresponding to horizontal extension (or vertial ompression) are relatively low. The major prinipal stress orientations and the deformed shape of the slope are shown in Fig. 54. The stress orientations beneath the bottom of the exavation are nearly horizontal, whereas those determined using linear analyses with similar modulus variation with depth and initial stress onditions were more steeply inlined, as shown in Fig. 8. The differene in behavior may result from the fat that the stress-strain relationship employed in the nonlinear analysis is anisotropi; at any stress level the modulus values are higher for horizontal ompression than for any other diretion of ompression. Therefore, beneath the base of the exavation, where the diretion of ompression is horizontal, the soil is relatively stiff, and additional horizontal stresses are transmitted to the stiffer region. The deformed shape of the slope shown in Fig. 54 is similar to those determined using linear and bilinear analyses. It is interesting to note that the deformations are quite appreiable, amounting to about 0.6 ft slumping near the rest, and.5 ft bulging near the toe of the slope, even though the fator of safety is about

140 I.,.. ~ ~ ~ ~ I ::;;.-"' J ~ I I I ~ ' -- / _,,,,,,, / / I / / l I I y' / ~ f I I I I I I (a) Horizontal Distane-feet _/' &:>isplaement Sale X (b) Horizontal Distane-feet FIG.54. (a) MAJOR PRINCIPAL STRESS DIRECTIONS, (b) DEFORMED SHAPE OF SLOPE.

141 Summary On the basis of this analysis it appears that gradual redution of modulus values with inreasing strain redues the tendeny for loal overstress to a very signifiant degree. Thus both linear and bilinear analyses probably overestimate the extent of loal failure for stable slopes. Variations of modulus values and strength values with diretion of ompression also influene the stress distribution and the position of the zones where the perentage of strength mobilized is highest. 145

142 CHAPTER 6 PORE PRESSURES AT THE END OF CONSTRUCTION OF EXCAVATED SLOPES AND EMBANKMENTS The analyses desribed in previous hapters of this report simulated short-term, undrained loading onditions for saturated lays, and were therfore performed using stress-strain relationships formulated in terms of total stresses. It is possible, however, to predit pore-water pressures for the end-of-onstrution period using the results of these analyses. This proedure may provide a useful alternative to total stress analyses for the period immediately following onstrution. If pore pressure values are predited for this period, stability analyses may be performed using effetive stress methods. Comparison of the predited pore pressures with values measured during onstrution ould then provide a very onvenient tehnique for evaluating the adequay of the design ontinually during onstrution. Pore Pressure Coeffiients For purposes of expressing pore pressure hanges in terms of hanges in total stress, the pore pressure oeffiients proposed by Skempton (1954) are very onvenient. Aording to the equation proposed by Skempton, the relationship is expressed as (23) in whih 6u is the hange in pore pressure, 60 and 60 are the major and 1 3 minor hanges in total stress, and A and B are pore pressure oeffiients. The oeffiient B reflets the ontribution to the hange in pore pressure aused by equal all-around hanges in total stress, and for soft saturated lays is equal to 1.0. The oeffiient A reflets the influene of the hange in stress differene. The value of A for a partiular lay depends on the degree of overonsolidation and on the diretion in whih the lay is ompressed, among other fators. Values of Af (A at failure) are shown in Table 6 for normally onsolidated San Franiso Bay Mud under onditions of horizontal and vertial ompression. Between these extremes 146

143 ~ the value of Af varies with the diretion of ompression as shown in Fig. 55. (This is the same variation disussed in Chapter 4 of this report and illustrated in Fig. 38). Also shown in Table 6 is the value of Af determined for Bay Mud using isotropially onsolidated-undrained (IC-U) triaxial tests, in whih the speimens are onsolidated under equal all-around pressure and then tested undrained. It may be noted that the value of Af for IC-U triaxial tests (Af = 1.05) is only Table 6. Values of Af for Normally Consolidated San Franiso Bay Mud. Type of Undrained Test Value of Af Horizontal Plane Strain 0.71 Vertial Plane Strain 1.08 IC-U Triaxial (vertial) 1.05 slightly higher than the average for the stress orientations shown in Fig. 55. This value may therefore represent a reasonable value for use with all diretions of ompression. Changes in Pore Pressure and Pore Pressure Ratios The initial pore-water pressure distribution in a saturated, normally onsolidated lay layer with the water table at the ground surfae is shown in Fig. 56; the pore-water pressure inreases linearly with depth from zero at the surfae to 6240 psf at the base of the layer 100 ft below the surfae. To alulate hanges in pore-water pressure during exavation of a slope, equation 23 was employed, using values of 60 and alulated by the finite element method. The values of hange in pore pressure shown by the ontours in Fig. 56 were alulated assuming that the lay is linearly elasti and that A is equal to 1.05 for any diretion of ompression. The values of pore-water pressure for eah element in the onfiguration representing the slope were alulated by the omputer at the same time the hanges in total stress were alulated. Sine the 147

144 I 2 I I I O S 0 7 L/ v / / / v l,.---- ~ l<t -0 u ;:, ~ O I 0 0 I Angle Between Diretion Of Compression and Horizontal -!31 - Degrees. FIG. 55. VARIATION OF VALUES OF Af WITH DIRECTION OF COMPRESSION FOR SAN FRANCISCO BAY MUD. 148

145 ~ I00----,...,~~=~12~5=-~~~~'ij" -~~~~~~~~? I K= b 4 > 5.!! 6 I.Lt 0 nm>»>>>»>>>>>>>>>>>>>>>>>>>>>>,,> 11 >>>> >> > >> > > > >>>>>>> >> Rigid Base -3 Initial Pore Pressures x 10 p sf -3 Changes In Pore Pressure x 10 p sf I Final Pore Pressure x 10 3 p sf = Pore Pressure Rat los, r u FIG. 56. PORE PRESSURES AND PORE PRESSURE RATIOS. 149

146 hanges in stress during exavation are predominantly negative, the hanges in pore-water pressure are also mostly negative. The largest hanges our near the toe of the slope where the amount of unloading is greatest, and only very small hanges our behind the rest of the slope. Addition of the initial values with the hanges in pore pressure results in the final values shown in Fig. 56. Inunediately below the base of the exavation and the surfae of the slope the pore,pressures-, are negative, whereas at greater depths the values are positive. At the extreme right, behind the rest of the slope, the pore pressures have not hanged appreiably during exavation. The same final pore pressures are represented in a different manner in the lower part of Fig. 56. has been divided by the overburden pressureo The value of pore pressure at eah point The resulting ratio, u/yh, has been given the synibol r ror pore pressure ~atio by B~shop u and Morgenstern (1960). It represents the portion of total overburden pressure arried by water, and has been found to be a very onvenient means of expressing water pressure onditions for purposes of stability analyses. For the lay layer shown at the top of Fig. 56, the initial water pressures orrespond to a onstant value of r throughout the layer; u for y = 125 lb/ft 3, the value of r representative of the initial ondiu tions is 62.4/125.0, or Comparing this value to the ontours of ru shown in the lower part of Fig. 56, it may be seen that the value of r u is redued during exavation in the region of the slopeo Effet of Bulk Compressibility The influene of bulk ompressibility on stresses, strains and displaements around exavated slopes has been studied and found to have no great effets. However, sine the prinipal effet of a hange in bulk ompressibility would logially be to alter the value of mean normal stress, it was onsidered that hanges in bulk ompressibility might influene the alulated values of pore-water pressure signifiantly, and a study was made to investigate this possibility. The same slope has been analyzed using two different assumptions onerning bulk ompressibility. The first assumption was that Poisson's 150

147 E ratio was 0.475; thus the bulk modulus of the lay, K = 3 (l-zv), was, in eah element, equal to 6.67 times Young's modulus. When the value of Young's modulus for an element is redued to a small value after failure, the value of the bulk modulus is also redued by the same fator. A better representation of the behavior of saturated lays is represented by the seond assumption made, that the bulk modulus remains onstant (K = 10 6 psf) regardless of the amount of shear stress or strain to whih the lay is subjeted. Both analyses were performed using bilinear stress-strain harateristis with the same strength profile orresponding to S /p = 0.3. u The initial stresses were represented by a value of K equal to 0.75, and the values of A used in the analyses varied with diretion of ompression as shown in Fig. 55. The results of the two analyses are illustrated in Fig. 57 by ontours showing_ the alulated values of ru. The values are virtually idential for both ases, indiating that the influene of bulk modulus value is not large for this slope. On this basis it was deided to perform subsequent analyses using a onstant value of Poisson's ratio (0.475), permitting the use of more rapid omputer programs. Pore Pressures Around Exavated Slopes Several analyses have been made to investigate the effets of modulus variation with depth, stress-strain behavior, and values of A on the alulated pore pressures around exavated slopes. In this study two variations of modulus with depth were onsidered; modulus onstant throughout the depth and modulus inreasing with depth. Three types of stress-strain behavior were onsidered; linear, bilinear, and anisotropi nonlinear. Two possible pore pressure oeffiient values were onsidered; the onstant value determined using IC-U triaxial tests on San Franiso Bay Mud, and the values of A determined using horizontal and vertial plane strain tests, whih vary with diretion of ompression. Of the 12 possible ombinations among these parameters, 6 meaningful ombinations, whih are shown in Table 7, were seleted for study. analyses are disussed in the following setions. The results of these 151

148 -1 0 \trying A Poissons Ratio = K=O 75 Su p =0 30 A Constant Bulk Modulus -1 0 Su -=0 30 p FIG. 57. EFFECT OF BULK COMPRESSIBILITY ON PORE PRESSURES AROUND EXCAVATED SLOPES IN NORMALLY CONSOLIDATED_ CLAY.

149 Table 7. Analyses of Pore Pressures Around Exavated Slopes - Variation of A Analysis with Diretion of Compression Stress-Strain Relationship Variation of Modulus with Depth Results Shown in Fig, No. 1 Constant Linear Constant 58 (top) 2 Constant Linear Inreasing 58 (bot tom) 3 Varying Linear Inreasing 59 (top) 4 Varying Bilinear Inreasing 5 Varying 6 Constant Anisotropi Nonlinear Anisotropi Nonlinear Inreasing Inreasing 59 (bottom) and 60 (top) 60 (bottom) and 61 (bottom) 61 (top) Modulus Variation with Depth. The results of two analyses, performed using linear elasti stress-strain behavior and the same value of A for all diretions of ompression, are shown in Fig. 58. The values of r u shown in the upper part of the figure were alulated using a onstant value of Young's modulus throughout the thikness of the layer, and those shown in the lower part were alulated using modulus values inreasing linearly with depth. The values alulated by the two methods are nearly the same; the value of r for any point in the upper diagram is at most u only a few perent different from that for the same point in the lower diagram, indiating that the influene of modulus variation with depth is small. Loal Failure. The effet of failure on the subsequent development of pore pressures is illustrated by Fig. 59. The values of r shown in u the upper part were alulated using linear elasti stress-strain harateristis, with modulus inreasing linearly with depth. The values shown in the lower part were alulated using bilinear stress-strain harateristis, and a normally onsolidated strength profile orresponding to S /p = 0.30 as shown in Fig. 17. As in all of these analyses, the u unit weights used were 125 lb/ft 3 Both the linear and the bilinear 153

150 A= 1 05 Constant Modulus A=l OS Modulus inreasing Linearly with Depth -I 0 AG. 58. PORE PRESSURE RATIOS CALCULATED FOR EXCAVATED SLOPES USING LINEAR ELASTIC ANALYSlS.

151 Varying A Linear Analysis -1 0 varying A -. --, Bilinear Analysis... ' ' \ / _,,,, / /.,,,,,,..,,,..,.--~Failed / \ \ I I Region AG.59. PORE PRESSURE RATIOS CALCULATED FOR EXCAVATED SLOPES USING LINEAR AND BILINEAR ANALYSIS

152 analyses were performed using values of A varying with inremental stress orientation, or diretion of ompression, as shown in Fig. 55. though a sizable failure zone developed in the slope analyzed using Even bilinear behavior, the alulated values of r are for pratial purposes u the same as those determined on the basis of linear behavior. Anisotropi Stress-Strain Behavior. Perhaps the most realisti representations of the stress-strain and pore pressure harateristis of San Franiso Bay Mud are represented by the nonlinear, anisotropi relationships disussed in Chapter 4. Variations of instantaneous modulus values with inreasing strain are illustrated in Fig. 36, and the influene of stress orientation on modulus values is shown in Fig. 45. These relationships have been employed in an analysis of the post-exavation pore pressures around a 3:1 slope shown in Fig. 60. The overall fator of safety of the slope at its final height of 50 ft was found to be about 1.5, and at this stage no failure zones had developed anywhere around the slope; the largest perentage of strength mobilized was about 80% at the base of the lay layer and near the toe of the slope, based on the strength profiles shown in Fig. 50. Also shown in Fig. 60 are values of r alulated using a similar strength profile and bilinear stressu strain behavior. The variation of A with diretion of ompression shown in Fig. 55 was used in both analyses. The values of pore pressure ratio alulated using multilinear stress-strain behavior are signifiantly higher than those alulated using bilinear behavior, espeially in the region below the lower half of the slope and beneath the bottom of the exavation. The differene apparently results from the differene in instantaneous modulus values for this region in the two analyses. modulus values for horizontal ompression used in the bilinear analysis are relatively large, resulting in larger pore pressures in regions subjeted to horizontal ompression. Anisotropy with Respet to A. Values of r have also been alulated u using anisotropi, multilinear behavior by employing a onstant value of A, independent of stress inrement orientation. The value of A used in these alulations (A= 1.05) was measured in IC-U triaxial tests on San Franiso Bay Mud. The alulated values of r are shown in Fig. 61 u The 156

153 Varying A Bilinear Analysis Pore Pressure Ratio 1 ru Varying A Multilinear Ana~ysis F1G. 60. PORE PRESSURE RATIOS FOR EXCAVATED SLOPES CALCULATED USING BILINEAR AND MULTILINEAR ANALYSIS.

154 A= 1-os Multilinear Analysis A... V1 o Varying A M.iltilinear _Analysis ---A FIG. 61. PORE PRESSURE RATIOS FOR EXCAVATED SLOPES CALCULATED USING CONSTANT A AND USING A \ARYING WITH STRESS ORIENTATION.

155 together with those determined using values of A varying with the diretion of ompression, as is appropriate for Bay Mud. It may be noted that the values of r alulated using A = 1.05 for all regions of the slope are u muh higher, espeially in the region beneath the lower part of the slope. This differene arises from the fat that the stress inrements for this region are nearly horizontal, and the appropriate values of A for this stress orientation are about 0.7 as shown in Fig. 55, muh less than the value 1.05 determined from IC-U triaxial tests. The differenes between the values of r u shown in Figs. 60 and 61 are further illustrated in Fig. 62, where values of r for a horizontal plane u beneath the toe of the slope are plotted. While all three methods result in the same values of r for the region beneath the upper part of the u slope, there are very large differenes for the region beneath the toe. Taking the values alulated using the multilinear analysis and varying values of A as being most orret, it may be onlude-d th-a~ th~- us~- o-fbilinear, isotropi stress-strain behavior results in pore pressures whih are too low, and the use of a onstant value of A determined from IC-U triaxial tests results in pore pressures whih are too high. Analyses performed using isotropi stress-strain behavior (Fig. 50, Fig. 59, upper part of Fig. 60) all result in approximately the same pore pressure distributions. Analyses performed using anisotropi stressstrain behavior, however, result in higher pore pressures near the base of the slope (Fig. 61). Thus it seems reasonable to infer that analysis performed using isotropi stress-strain behavior result in values of r u whih are too low near the lower part of the slope, no matter whih values of A are used in the analyses. Thus it appears that the variation of modulus values with diretion of ompression (anisotropy with respet to stress-strain behavior) is quite important in determining the pore pressures after onstrution. Pore Pressures in Soft Clay Foundations Beneath Embankments The methods of analysis developed for exavated slopes may also be used to determine pore pressures in embankment foundations. These are perhaps of greater pratial signifiane beause the period innnediately following onstrution of an embankment on a soft lay layer may be the 159

156 , ~ (jfilineor Analysis, A= l OS FIG. 62. PORE PRESSURE RATIOS ON A HORIZONTAL PLANE.

157 most ritial period in its entire life, whereas the ritial period in the life of an exavated slope often ours some time after onstrution, when the pore pressures have reahed equilibrium with the ground water onditions in the region of the slope. To illustrate the utility of the methods of analysis for prediting pore pressures in soft lay foundations, several analyses have been performed using different proedures. ase was the same: The embankment onsidered in eah A symmetrial 25 ft high embankment with 3:1 side slopes onstruted on a 35 ft thik lay layer. Constrution of the embankment was simulated by appliation of 5 lifts eah 5 ft thik. unit weight of both the embankment and the foundation was 125 lb/ft 3, the stress-strain harateristis of the embankment soil were in eah ase assumed to be linearly elasti, with Young's modulus equal to 10 5 psf, and Poisson's ratio equal to The initial pore pressures in the foundation layer were in equilibrium with the ground wate-r level ar the surfae, so that the value of r representative of the initial onditions u was 0.50 at all points in the layer. The initial stress onditions in the foundation layer were represented by a value of the total stress earth pressure oeffiient, K, equal to 0,75 (horizontal stress 75% as large as the overburden pressure). Analyses have been made to determine the effets of the properties of the foundation lay layer on the alulated pore pressures beneath embankments. The ombinations of values of A, stress-strain behavior, and modulus variation with depth seleted for study are shown in Table 8, and the results of the analyses are desribed in the following setions. Anisotropy with Respet to A. For the purpose of examining the influene of the values of A on the final pore pressures, two analyses were performed using onstant values of modulus (E = 60,000 psf) for the foundation layers. The results of these analyses are shown in Fig. 63. The values of r alulated using the value of A determined from IC-U triaxial u test results (A = 1.05) are somewhat higher than those alulated using values of A varying with stress inrement diretion as shown in Fig. 55. The largest differene in alulated values of r is in the region beneath u the toe of the embankment and just beyond. The largest value of r u alulated using varying values of A is 2.0, whereas the largest value The and 161

158 ~ ~-----~ E= 100,000 p sf ~--~ E = 60,000 psf j Varying A ---~ ~ ,FIG. 63. EFFECT OF THE VALUE OF A ON PORE PRESSURES IN EMBANKMENT FOU'JDA TIONS

159 Table 8. Analyses of Pore Pressures in Embankment Foundations. Variation of Ā Analysis with Diretion of Compression Stress-Strain Relationship Variation of Modulus with Depth Results Shown in Fig. No. 1 Constant Linear Constant 63 (top) 2 Varying Linear Constant 63 (bottom) and 64 (top) 3 Varying Linear Inreasing 64 (bottom) 4 Varying Bilinear Inreasing 65 (top) 5 Varying Anisotropi Nonlinear Inreasing 65 (bottom) alulated using A= 1.05 is 3.0, some 50% higher. It may be noted however, that these large values of r orrespond to shallow depths, only u 3 ft below the surfae, where the overburden pressure is small (375 psf). Thus the differene in pore pressures at this loation amounts to 375 psf; ompared to pore pressure values of about 3300 psf near the base of the layer this differene may not be extremely signifiant. Another aspet of the values shown in Fig. 63 deserves onsideration; in the regions near the toe of the slope where r exeeds 1.0, the orresu ponding values of effetive vertial stress are negative (tensile). At any depth the total overburden pressure p is equal to yh, and the porewater pressure u is equal tor (yh). Therefore the effetive vertial u stress, p' = p - u, is equal to (1 - r )yh and is negative for values of u r exeeding unity. For soils like soft lays whih have little or no u tensile strength, it is physially impossible to develop s ignif i ant negative effetive stresses, and alulated values of (1 zero may thus indiate only a tendeny for development of - r ) less than u large pore pressures whih annot be realized physially. If values of r greater u than 1.0 are ignored, it may be noted that the remainder of the pore pressure ratios alulated using the two methods illustrated in Fig. 63 are in very good agreement" 163

160 Modulus Variation with Depth. Analyses were also performed to study the influene of variation of modulus of the lay foundation layer with depth. In both analyses, whih are illustrated in Fig. 64, values of A varying with stress orientation were used to alulate values of r One u of the analyses, shown in the upper part of Fig. 64 was onduted using a onstant value of modulus throughout the depth of the layer, whereas the analysis depited in the lower part of Fig. 64 was performed using values of modulus inreasing linearly with depth, the ratio of Young's modulus to initial effetive overburden pressure (E/p) being equal to 100 at all depths. Thus the value of modulus at the bottom of the layer was about 300,000 psf. The values of r shown in Fig. 64 are very nearly u the same for both methods of analysis, indiating that varying modulus values with depth has a very small effet on the pore pressures in the foundationo Stress-Strain Behavior and Loal Failureo To examine the effets of stress-strain behavior and loal failure on the foundation pore pressures, two analyses have been performed using the bilinear and multilinear stressstrain relationships previously desribed. In both analyses, shown in Fig. 65, failure zones developed near the surfae of the layer. The failure zone was restrited to the area beneath the enter of the embankment in the bilinear analysis, whereas failure ourred only in a small region near the toe of the slope in the more realisti multilinear analysis. The pore pressure ratio values are somewhat higher in the failed regions, i.e., the values of r are higher beneath the enter of the embankment u for the bilinear analysis and beneath the toe of slope for the multilinear analysis. It is interesting to note that for the embankment studied, the values of ru alulated using the simplest and most omplex methods of analysis are pratially idential: The values alulated using linear elasti stress-strain behavior with onstant modulus and onstant value of A, shown in the upper part of Fig. 63, are very nearly the same as those alulated using multilinear stress-strain behavior and values of A varying with stress inrement orientation whih are shown in the lower part of Fig. 65. The similarity results from the fat that using a onstant value of A = 1.05 for all stress orientations tends to overestimate pore pressures 164

161 r -----~~~---~~ E= 100,000 psf E= 60,000 p sf Varying A ~ E= I 00,000 psf Modulu'l inreasing 6-~--~ ~~~~~~- FIG. 64. EFFECT OF MODULUS VARIATION WITH DEPTH ON PORE PRESSURES IN EMBANKMENT FOUNDATIONS.

162 I 0 8-6~ ~ E = 100,000 psf Bilinear Stress - Strain Behavior Varying -----~-----~ ~ ~ l O t ~ A I E = 100,000 psf. Multilinear- Stress - Strain Behavior Varying A ~ FIG. 65. EFFECT OF STRESS-STRAIN BEHAVIOR ON PORE PRESSURES IN EMBANKMENT FOUNDATIONS.

163 near the toe, whereas using isotropi stress-strain behavior tends to underestimate pore pressures in the same region. For the onditions analyzed the two fators offset eah other almost perfetly. Sunnnary The analyses desribed in this hapter show that finite element analyses may be readily employed for prediting pore-water pressures at the end of onstrution of exavated slopes and embankments, a proedure whih would be very useful if measured values of pore pressure are used to assess the stability of slopes during onstrution. In order to determine the auray of the analytial methods developed, it is highly desirable to ompare alulated values of pore pressure with values measured in atual slopes and embankment foundations during onstrution. Suh omparisons would be very helpful for determining the effetiveness and pratiality of the vro-edures. 167

164 CHAPTER 7 CONCLUSION The study desribed in the preeding pages was onduted to investigate (1) the stress, strain and pore-water pressure distribution and the possibility of progressive failure in exavated slopes in lay; and (2) the stress, strain and pore-water pressure distribution in soft lay foundations for embankments. The auray of suh a study neessarily depends in large measure on (1) the development of suitable analytial proedures for evaluating the behavior of slopes and embankments omposed of materials with known properties and harateristis and (2) the validity of the proedures used to evaluate and represent in the analyses the properties of the lay soils. The studies have shown that the finite element -n-retho-d of analysis provides a suitable tehnique for investigating the stability and performane of exavated slopes and embankments, taking into aount (1) variations in the stress-strain harateristis of soils, (2) variations in initial stress onditions in the soil, (3) variations in strength in the soil deposit and (4) the onstrution proedure. Criteria have been established for the lateral extent of finite element onfigurations to represent slopes in layers of large lateral extent and proedures have been developed for simulating analytially the exavation of slopes in horizontal layers. Analyses have been performed using two types of finite element omputer programs, the ommonly available type whih annot be used with values of Poisson's ratio muh higher than about 0.49, and a program speially formulated for use with inompressible materials. Using the latter program it was possible to use high values of bulk modulus appropriate for saturated lays under undrained loading onditions, but the program requires approximately twie as muh omputer time as ommonly available finite element omputer programs. Analyses performed showed that the use of values of Poisson's ratio higher than does not lead to greatly different results, and thus use of the speially formulated omputer program is not neessary. 168

165 Previous investigations have shown that most soft lays have urvilinear stress-strain relationships and exhibit a onsiderable degree of anisotropy with respet to strength, stress-strain and pore pressure harateristis. During the ourse of the studies, proedures were developed for representing mathematially the anisotropy and urvilinear stress-strain haralerist1s of soft lays and their validity demonstrated by using them to predit the results of simple shear tests on San Franiso Bay Mud, A stress-strain relationship was developed for Bay Mud from the results of horizontal and vertial plane strain tests; the resulting relationship was non-linear, anisotropi and stressdependent but even so it was suffiiently simple to be inorporated in finite-element analyses. The good agreement between the predited and observed results of simple shear tests on Bay Mud was onsidered indiative of the suitability of the material property representation for making reasonably aurate evaluations of the behavior of soil deposits. For analysis purposes, the omplex harateristis of soft-lays desribed above are often simplified to failitate the analytial proedures. Thus, in order of inreasing omplexity and auray, the stressstrain properties of soils may be represented, for analyses purposes, in the following ways: (1) linear, isotropi stress-strain harateristis (2) bi-linear, isotropi, stress-strain and strength harateristis (3) multi-linear, isotropi, stress-strain and strength harateristis (4) multi-linear, anisotropi stress-strain and strength harateristis Similarly the pore pressure harateristis may be represented by (1) a uniform, isotropi, pore pressure oeffiient A or (2) anisotropi variations in the pore pressure oeffiient A. Clearly different ombinations of the above material property representations are possible, providing results with differing degrees of auray. In general it would be antiipated that the least aurate results would be obtained by analyses based on the assumption of a linear isotropi stress-strain relationship and a uniform isotropi pore pressure oeffiient for a lay soil; similarly the most aurate solutions would be antiipated 169

166 from analyses inorporating multi-linear, anisotropi stress-strain relationships and anisotropi variations in the pore-pressure oeffiient. However it should also be reognized that the introdution of ompensating errors an in some ases lead to reasonably aurate results despite the use of grossly simplified material property representations. Analyses based on the use of isotropi linear elasti stress-strain harateristis have been found to be useful for evaluating some aspets of the performane of stable slopes, and might logially serve as a basis for initial investigations of many pratial slope behavior studies, In the present investigation they were used to examine the signifiane of the initial stress onditions and variations in soil modulus values with depth on the stress distribution within an exavated slope. It was found that values of the maximum shear stress and the prinipal stress orienta- tions in the viinity of slopes are influened to a major extent by the initial stress onditions (represented by the oeffiient of earth pressure at rest, Ko) but only slightly affeted by the variation of modulus values with depth in the deposit; furthermore the fator of safety orresponding to the development of loal failure within exavated slopes was found to vary appreiably with hanges in the initial stress onditions, a result whih may have pratial appliations in slopes omposed of extremely brittle materials. However, in general, it was onluded that analyses based on the use of linear elasti harateristis, would not provide a basis for reliable assessments of slope stability. Analyses of exavated slopes in whih the lay soil is onsidered to have isotropi bi-linear stress-strain harateristis, and in whih onsideration of the strength of the soil may therefore be inorporated, have proved to be useful for determining where loal failure begins and how the failure zones grow as exavation progresses. Analyses of this type for slopes ut in layers having initial stress onditions and strength profiles representative of normally onsolidated lay deposits indiate that failure begins beneath the slope and extends upward towards the slope rest and downwards toward the toe of the slope and the base of the layer as exavation ontinues. The final shapes of the failure zones are ompatible with the development of a urved surfae of sliding and in the final stages, when the failure zones enompassed the major portions 170

167 of the region beneath the slope (see Fig. 66), fators of safety alulated by the = 0 or Su-method of analysis ranged from 0.95 to On the other hand, for slopes in layers with initial stresses and strength profiles representative of overonsolidated lays, failure was found to develop initially beneath the base of the exavation, extending upward toward the bottom of the exavation and downwards toward the base of the layer as exavation progressed. When the fator of safety, based on a = 0 analysis, was still about 2 a large failure zone had developed around the bottom of the exavation, as shown in Fig. 66. This type of behavior is similar to the phenomenon desribed by Terzaghi and Pek (1948) as "loss of ground" wherein the soil in the base of the exavation bulges ontinually as exavation progresses. This bulging may often be the first sign of distress foreshadowing atastrophi failure of slopes in overonsolidated lay. Finally, analyses of the stress distribution within an exavated slope in soft lay taking into aount the multi-linear anisotropi stress-strain and strength harateristis of the soil showed that linear and bilinear analyses tend to overestimate the extent of loal failure in stable slopes. Furthermore the stress distribution determined by this more refined analysis also showed signifiant differenes from that determined by the use of bi-linear material harateristis. A omparison of the stress distributions determined by these two approahes is shown in Fig. 67. The various analytial proedures inorporating different representations of the lay stress-strain harateristis were also used, in onjuntion with different representations of the variation of pore pressure oeffiient within the slope, to evaluate the pore-water pressure distribution, within exavated slopes in lay. Fig. 68 shows a omparison of the results determined by analyses based on the following material harateristis: (1) linear elasti stress-strain harateristis and onstant value of A (2) linear elasti stress-strain harateristis and anisotropi values of A (3) bi-linear stress-strain harateristis and anisotropi values of A 171

168 Tension Zone Failure Zone Critial Failure Surfae L Fator of ~ Safety=0.95, Normally Consolidated Failure Zone Tension Zone Critial Failure Surfae Fator of Safety= l.~4 Overonsolidated FIG. 66. CORRELATION BETWEEN FAILURE ZONES AND FACTORS OF SAFETY FOR EXCAVATED SLOPES. 172

169 IOO Horizontal Dlstan1-f11I 400 FIG. 67. COMPARISON OF STRESS CONDITIONS AROUND AN EXCAVATED SLOPE AS DETERMINED BY ISOTROPIC BILINEAR AND ANISOTROPIC MULTILINEAR ANALYSES. 173

170 LNor Elostl Stress-St"*1 Oualetlstlet And Constant 'AJkJI of A LNor Elostl Stress-Strain Ctaola 1111et Md Al ~IUliopl \tlles f A Bilinear Stress-Strain Chootalstls And, Anllotropl 'kluet of A Wtillnear Stress-Strak'I Qaaletlstlet And Constant 'Wiiie of A Wtfflnear Stress-Strain Charo1911sllet And Anltotropl \tut of A FIG. 68. COMPARISONS OF PORE PRESSURES AROUND EXCAVATED SLOPES AS DETERMINED BY VARIOUS METHODS OF ANALYSIS. 174

171 (4) multilinear stress-strain harateristis and onstant value of A (5) multilinear stress-strain harateristis and anisotropi values of A The differenes in the omputed pore pressure distributions are readily apparent. If the analysis inorporating multi-linear stress strain harateristis and anisotropi values of A is onsidered the most realisti representation of onditions in an atual slope, then it would appear that an analysis inorporating linear stress-strain properties and a onstant value of A under-estimates the pore-water pressures in the lower part of the slope; an analysis based on linear stress-strain relationships but inluding anisotropi values of A gives improved results but the omputed values pore pressure are still too low in the lower part of the slope; an analysis based on bi-linear stress-strain harateristis and anisotropi values of A is still further improved; and an analysis based on multi-linear stress strain harateristis and a onstant value of A over-estimates the pore pressures in the lower part of the slope. The potential variations in pore-water pressure distributions dislosed by these different analyses illustrates the need for the use of the best possible representation of material harateristis in order to evaluate the pore-water pressure onditions in atual slopes and the possible errors resulting from grossly simplified proedures. Fig. 69 shows a similar omparison of the pore pressure distributions omputed by the various analyses in a soft lay foundation for an embankment. In this ase the pore pressure distribution does not vary appreiably whether the analysis is based on the use of linear, bilinear or multilinear stress-strain harateristis, onstant or varying values of the pore pressure oeffiient A, or onstant or varying values of material harateristis with depth. Thus in this ase an entirely satisfatory evaluation of the pore-water pressure onditions ould be obtained using the simplest method of analyses--that is, an analysis based on linear elasti material harateristis and a onstant value of A--even though the atual properties are onsiderably more omplex. It is hoped that these results will help to further the understanding of the behavior of exavated slopes in lays and of soft lay foundations 175

172 Llneor Elasti Stress-Strain Charotet lstls, Constant Value of MolU And Constant ~of A. Linear E1ost1 Stress-Strain O'O'olet ls, Constant Value of. ModU And Anlsolropl ~of A. Linear Elasti Stress Strain ao u1a 1s11s, MoliJ9 Inreasing LNarty With Depth And Anisotropi \tlj8s of A. t Bl&near Stress-Strun Ooolet lslls, Mo:Uus i:reasll io U1ear1y With Depth And Anisotropi \tjlj8s of A. t WtRlnear Stress-Strain ChCJ olei lstls, MoG.4us 1nreaUwJ Llnear'1 With Depth And Anlaot1 opl \l(bs of A. 4 FIG. 69. COMPARISONS OF PORE PRESSURES IN EMBANKMENT FOUNDATIONS AS DETERMINED BY VARIOUS METHODS OF ANALYSIS. 176

173 for embankments; furthermore, that the analytial proedures presented will serve as a basis for improved evaluations of the performane of ut slopes and embankments involving insensitive saturated lays. 177

174 REFERENCES Bennett, P. To (1951), "Notes on Embankment Design," Proeedings of the Fourth Congress on Large Dams, Vol. 1, pp Bishop, A. W. (1952), "The Stability of Earth Dams," Thesis presented to the University of London, at London, England, in 1952, in partial fulfillment of the requirements for the degree of Dotor of Philosophy. Bishop, A. W. (1955), "The Use of the Slip Cirle in the Stability Analysis of Slopes," Geotehnique, Vol. 5, No. 1, pp Bishop, A. W., Blight, G. E., and Donald, I. B. (1960), Closure of "Fators Controlling the Strength of Partly Saturated Soils," Proeedings, ASCE Conferene on Shear Strength of Cohesive Soils, 1960, pp, Bishop, A. W. and Morgenstern, N. R. (1960), "Stability Coeffiients for Earth Slopes," Geotehnique, Vol. 10, No. 4, pp Bjerrum, L. (1955), "Stability of Natural Slopes in Quik Clay," Geotehnigue, Vol. 5, No. 1, pp , Bjerrum, L. (1967), "Progressive Failure in Slopes of Overonsolidated Plasti Clay and Clay Shales," Journal of the Soil Mehanis and Foundations Division, Vol, 93, No, SM5, pp Bjerrum, L. and Eide, 0, (1956), "Stability of Strutted Exavation in Clay," Geotehnigue, VoL 6, No. 1, Marh, 1956, pp Brooker, E. W. and Ireland, H. O. (1965), "Earth Pressures at Rest Related to Stress History," Canadian Geotehnial Journal, VoL 2, No. 1, pp Brown, C. B. and King, I. P. (1966), "Automati Embankment Analysis: Equilibrium and Instability Conditions," Geotehnique, VoL 16, No. 3, pp Chang, T. Y., Ko, H. Y., Sott, R. F. and Westman, R. A. (1967), "An Integrated Approah to the Stress Analysis of Granular Materials," Report of the Soil Mehanis Laboratory, California Institute of Tehnology, Pasadena, California. Clough, R. W. and Woodward, R. J., III (1967), "Analysis of Embankment Stresses and Deformations," Journal of the Soil Mehanis and Foundations Division, ASCE, Vol. 93, No. SM4, pp Clough, R. W. and Rashid, Y. (1965), "Finite Element Analysis of Axi Symmetri Solids," Journal of the Engineering Mehanis Division, ASCE, Vol. 91, No. SMl, pp

175 Clough, R. W. (1965), "The Finite Element Method in Strutural Mehanis," Stress Analysis Ed" Zienkiewiz, O. D., J. Wiley (1965). Collin, A. (1846), Landslides in Clay, University of Toronto Press, Dunan, J. M. (1965), "The Effet of Anisotropy and Reorientation of Prinipal Stresses on the Shear Strength of Saturated Clay," Thesis presented to the University of California, at Berkeley, California, in 1965, in partial fulfillment of the requirements for the degree of Dotor of Philosophy. Dunan, J. M. (1967), "Undrained Strength and Pore-Water Pressures in Anisotropi Clays," Proeedings of the 5th Australia-New Zealand Conferene on Soil Mehanis and Foundation Engineering, August, Dunan, J. M. and Seed, H. B. (1966a), "Anisotropy and Stress Reorientation in Clays, Journal of the Soil Mehanis and Foundations Division, Vol. 92, No. SMS, pp Dunan, J. M. and Seed, H. B. (1966b), "Strength Variation Along Failure Surfaes in Clay," Journal of the Soil Mehanis and Foundations Division, ASCE, VoL 92, No. SM6, pp Dunan, J. M. and Seed, H, B. (1967), "Corretions for Strength Test Data," Journal of the Soil Mehanis and Foundations Division, Vol, 93, No. SMS, pp Dunan, J. M., Monismith, C. L. and Wilson, E. L. (1967), "Finite Element Analyses of Pavements," to be published in the Proeedings of the Highway Researh Board, Dunan, J.M. and Dunlop, P. (1968), "The Signifiane of Cap and Base Restraint," Journal of the Soil Mehanis and Foundations Division, ASCE, Vol. 94, No. SMl, January Goodman, L. E. and Brown, C. B. (1963), "Dead Load Stresses and the Instability of Slopes," Journal of the Soil Mehanis and Foundations Division, ASCE, Vol. 89, No. SM3, pp Herrmann, L. R. and Toms, R. M. (1964), "A Reformulation of the Elasti Field Equation, in Terms of Displaements, Valid for All Admissible Values of Poisson's Ratio," Journal of Applied Mehanis, ASME, Vol, 31, No. 1, pp Idriss, I. M. (1966), "The Response of Earth Banks During Earthquakes," Thesis presented to the University of California, at Berkeley, California, in 1966, in partial fulfillment of the requirements for the degree of Dotor of Philosophy. Ireland, H. 0., (1954), "Stability Analysis of the Congress Street Open Cut in Chiago," Geotehnique, Vol. 4, No. 4, pp

176 Kenney, T. C. (1963), "Stability of Cuts in Soft Soils," Journal of the Soil Mehanis and Foundations Division, ASCE, Vol. 89, No. SMS, pp King, I. P. (1965), "Finite Element Analysis of Two Dimensional Time Dependent Stress Problems," Report No. 65-1, Institute of Engineering Researh, University of California, Berkeley. Kjellman, W., (1951), "Testing the Shear Strength of Clay in Sweden," Geotehnique, Volo 2, No. 3, pp Ko, H. Y. and Sott, R. F < (1967), "Deformation of Sand in Hydrostati Compression," Journal of the Soil Mehanis and Foundations Division, Vol, 93, No. SM3, pp Ladd, ~o C. (1964), "Stress-Strain Modulus of Clay in Undrained Shear," Journal of the Soil Mehanis and Foundations Division, Vol. 90, No. SM5, ppo LaRohelle, P. (1960), "The Short-Term Stability of Slopes in London Clay," Thesis presented to the University of London, at London, ~ngland, in 1960, in partial fulfillment of the requirements for the degree of Dotor of Philosophy. Middlebrooks, T. A. (1936), "Foundation Investigation of the Fort Pek Dam Closure," Proeedings, 1st International Conferene on Soil Mehanis and Foundation Engineering, Vol. 1, pp Peaok, W. H. and Seed, H, B, (1967), "Liquefation of Saturated Sand Under Cyli Loading Simple Shear Conditions," Report No. TE-67-1, Offie of Researh Servies, University of California, Berkeley, California, Rosoe, K. H. (1953), "An Apparatus for the Appliation of Simple Shear to Soil Samples," Proeedings of the 3rd International Conferene on Soil Mehanis and Foundation Engineering, Vol. 1, pp Rosoe, K. H. and Poorooshasb, H. B. ( 1963), "A Theoretial and Experimental Study of Strains in Triaxial Compression Tests on Normally Consoli Dated Clays," Geotehnique, Vol. 13, No. 1, pp Rowe, P. W. (1962), "The Stress-Dilatany Relation for Stati Equilibrium of an Assembly of Partiles in Contat," Proeedings of the Royal Soiety, 269, pp Seed, H. B. and Idriss, I. M. (1967), "Analysis of Soil Liquefation: Niigata Earthquake," Journal of the Soil Mehanis and Foundations Division, ASCE, Vol, 93, No. SM3, pp Seed, H. B. and Wilson, S. Do (1967), "The Turnagain Heights Landslide, Anhorage, Alaska," Journal of the Soil Mehanis and Foundations Division, Vol, 93, No. SM4, pp

177 Sevaldson, R. A., (1956), "The Slide in Lodalen, Otober 6, 1954," Geotehnique, Vol. 6, No. 4, De. 1956, pp Skempton, A. W. (1945), "A Slip in the West Bank of Eau Brink Cut;" Journal of the Inst. of Civil Engineers, No. 7, pp Skempton, A. W. (1954), "The Pore Pressure Coeffiients A and B," Geotehnique, Vol. 4, No. 4, pp Tan, T. K. (1957), Disussion, Proeedings, Fourth International Conferene on Soil Mehanis and Foundation Engineering, Vol. 3, pp Taylor, D. W. (1937), "Stability of Earth Slopes," Journal of the Boston Soiety of Civil Engineers, Vol. 24, No. 3, pp Terzaghi, K. and Pek, R. B. (1948), Soil Mehanis in Engineering Pratie, Wiley, New York. Turnbull, W. J, and Hvorslev, M. J. (1967), "Speial Problems in Slope Stability," Journal of the Soil Mehanis and Foundations Division, ASCE, Vol. 93, No. SM4, pp ff. Wilson, E. L. (1963), "Finite Analysis of Two Dimensional Strutures," Report No. 63-2, Strutures and Materials Researh, Department of Civil Engineering, University of California, Berkeley, June Zienkiewiz, O. C., Mayer, P., and Cheung, Y. U. (1966), "Solution of Anisotropi Seepage by Finite Elements," Journal of the Engineering Mehanis Division, ASCE, Vol, 92, No. EMl, pp

178 APPENDIX A REQUIREMENTS FOR BOUNDARY CONDITIONS, ELEMENT SIZES AND ELEMENT SHAPES The amount of omputer time required to solve a finite element problem depends upon the number of nodal points (and elements) used to represent a struture, For effiient operation, these should be kept to the minimum neessary for aurate representation of the system being studied. The number of elements required is dependent upon the size, shape and distribution of the elements, whih in turn depend on the problem being analyzed. Studies to establish riteria for elements and the loation of arbitrary boundaries may be made for a lass of problems, suh as the stati analyses of slope behavior, one a general sheme for the onfigurations and the type of element is hosen, In the analyses of slopes desribed in preeding setions, the basi finite element used was a quadrilateral omposed of four onstant strain triangles. The arrangement of elements within the slope onfiguration is shown in Fig. 70. It may be noted that retangles are used throughout with the exeption of the region inunediately adjaent to the sloping fae, where triangles and trapezoids are employed alternately. This sheme of elements was onsidered to be well-suited to the simulation of slope exavation by the "removal" of elements outside the slope and it was also onvenient for automati mesh generation within the omputer. Whenever possible however, quadrilateral elements should be employed rather than triangular elements sine it has been shown by Wilson (1963) that the stresses obtained for a triangular element do not represent the stresses ating at any one point in the element. It was assumed that the soil deposit was underlain by a muh more rigid stratum, whih was simulated in the finite element analysis by onstraining the bottom row of nodal points in the onfiguration from moving vertially or horizontally. The nodal points on the artifiial lateral boundaries were allowed to move freely in the vertial diretion but were onstrained in the horizontal diretion. If the exavation being analyzed was bounded left and right by vertial planes of symmetry or rigid boundaries, these ould be simulated 182

179 --,,,,.,... ;,,.. ~, ~ (a) (b) FIG.,7Q. (a) FINITE ELEMENT IDEALIZATION OF A SLQPE, (b) CROSS SECTION OF SLOPE SHOWING LENGTH FACTORS. 183

180 in the analysis without approximation, In these analyses, however, it was desired to simulate a slope in a soil layer of very large lateral extent; therefore studies were undertaken to determine positions of the lateral boundaries whih would aurately simulate a layer of infinite lateral extent. Lateral Boundaries The first step in determining the influene of the artifiial lateral boundaries was to define a "region of interest", within whih it was desired to alulate aurate displaements, strains and stresses. The auray of values alulated for points outside the region of interest was onsidered unimportant,. The purpose of the analyses was to study the behavior of slopes as they approah a ondition of instability and therefore the region of interest was established to ontain the unstable portion of the slope. To determine the size of the region of interest, a study was made of the position of failure surfaes within slopes as determined by slope stability analyses and by field observations. The lateral extent of the failure surfae within a slope may be defined in terms of two leng~hs LB and LT as shown in Fig. 70. It is onvenient to onsider dimensionless ratios LB/HB and LTiHT and values of LB/HB and LT/HT have been alulated from data presented by Taylor (1937), based on analyses of slopes in soil with onstant strength; these values are shown in Fig. 71. It may be noted that the values of the ratios LB/HB and LT/~ inrease with inreasing slope angle to 50 ; beyond this angle the values derease. For a 45 slope LB/HB approahes a value of 2 and LT/HT approahes a value of 1.5. Thus for slopes flatter than 45 in soils with onstant shear strength these analyses indiate that the zone within whih failure ours may extend beyond the toe of a distane equal to twie the depth to rok, and beyond the top for a distane equal to 1-1/2 times the depth to rok. Values of LT/HT alulated from data presented by Kenney (1963), for slopes in soils with strength inreasing linearly with depth, are shown in Fig, 72. In this ase the values of the ratio LT/HT are generally smaller than for the ase when the shear strength is onstant with depth, 184

181 FIG. 71. THE LATERAL EXTENT OF FAILURE SURFACES IN SOIL WITH CONSTANT STRENGTH 185

182 3 ~1~.J :: I 2._ f Ci) u 0 -en a 0. {:. 0 ' ~ ~ Slope( Degrees) I H Depth Fator- ~ H FIG. 72. THE LATERAL EXTENT OF FAILURE SURFACES IN MATERIALS WITH STRENGTH INCREASING LINEARLY WITH DEPTH. 186

183 and the values derease with inreasing slope angle and depth fator HT/H. In all the ases analyzed by Kenney a failure surfae passing through or above the toe was found to be most ritial and the value of LB/HB was therefore zero for all ases, Thus the zone within whih failure may our is smaller for a soil with shear strength inreasing linearly with depth than for a soil with a onstant shear strength, extending above the top of the slope a maximum distane equal to about 1. 2 times the depth to rok and terminating at the toe of the slope. About twenty total stress slope stability analyses were undertaken in onjuntion with the finite element analyses in the preeding setions. In these analyses the undrained shear strength of the soil was used to represent a short-term stability problem and the variations of shear strength with depth used in these analyses are shown in Fig. 73, In all ases the value of LB was nearly zero, and the value of the ratio LT/HT was found to be or less for depth f-rt:urs- HT/H af more, indiating positions of the failure surfae intermediate between those determined from Taylor and Kenney. This is onsistent with the fat that the undrained shear strength variation is intermediate between those used by Taylor and Kenney. The position of sliding surfaes observed within several failed uts are summarized in Table 9. It may be noted that the observed values of LB/HB are somewhat greater than those alulated for the same slopes and the observed values of LT/HT are somewhat smaller, indiating that the atual failure surfae is likely to develop at a shallower depth within the slope than indiated by total stress stability analyses (Skempton, 1945), It is desirable to define the region of interest so that it will ontain the failure surfae as determined by analyses or field observations, From the preeding it appears that this riterion will be satisfied in most ases if the region of interest extends for a distane from the toe equal to the depth to rok (LB HB) and from the top for a distane equal to the distane to rok (LT HT). These riteria were adopted for establishing the size of the region of interest and the neessary boundary loations. Only in ases where the shear strength is onstant or the ratio of HT/H is near unity will LB and LT need to be larger. 187

184 ... ex> (J) - ~..2 u al.s: 80 r t--.., ~~---'----'-~---' Undrained Shear Strength -Su - ps f -- u g 40 t f 't m u.s: - 60 t f ~..2 u al ~80t lr---f u Undrained Shear Strength - Su - psf FIG UNDRAINED SHEAR STRENGTH VARIATIONS USED IN TOTAL STRESS SLOPE STABILITY CALCULATIONS.

185 Table 9. Observed Failures in Cut Slopes. Calulated Slope Slope Depth p = 0 Failure Angle Height Ratio Obi:;erved i H HT/H LB/RB LT/HT LB/RH LT/HT Sevaldson, ' -L Skemp ton, ' LS Ireland, ' Comments Long term failure in slightly overonsolidated lay. Long term failure in layered silts and lays. Short term failure in gritty blue lay LL = 33, PL = 18. Collin, ' ::i Clay shale, short term f ailura Collin, '?.LO Clay, short term failure.

186 In hoosing finite element onfigurations for slope analyses in whih the onstrution proedure was simulated, the extent of the region of interest below the toe was always hosen on the basis of the depth to rok (RB) at any stage in the onstrution, No attempt has been made to establish riteria for speial slope failures suh as regressive failures in quik lays (Bjerrum, 1955), regressive failures aused by liquefation of sand seams or lenses (Seed and Wilson, 1967) or ases involving overonsolidated lays (Bjerrum, 1967), in whih ases the failure surfae may extend a very large distane bak from the rest of the slope. Boundary Positions If the lateral extent of the original horizontal deposit is very large, it may be ineffiient to extend the finite element idealization to the physial extent of the dep-os-i-t, To det:ermine -aeptable bdundary positions, analyses were made using suessively larger distanes to the boundaries (DT) until it was determined that further inreases would have a negligible effet on the values of stress, strain and displaement alulated at the boundary of the region of interest" In these studies linear material properties were used in all ases as eah step of an analysis employing nonlinear material properties is assumed linear. Values of horizontal normal stress a, alulated at the up-slope x boundary of the region of interest are shown in the lower part of Fig. 74 for various positions of the right boundary of the finite element mesh as shown in the upper part of Fig. 74. In this analysis the up-slope boundary of the region of interest was set at Lr = HT behind the rest of the slope. It may be noted that as the right mesh boundary is moved from position l(dt = l,l HT) to position 2(DT = 2.0 HT) the values of ax near the surfae derease by a small amount. Further shift of this mesh boundary produes negligible hange in a. x Values of a alulated at the up-slope boundary of the region of y interest are shown in the upper part of Fig. 75 and exhibit no influene of the mesh boundary loation; the value of a being equal to the weight y of material above the point at whih the sr ess is alulated. Values of T xy are plotted in the lower part of Fig. 75 and indiate appreiable 190

187 Dr = Hr D (g) 11 I 3 20 I 5 60 I. :: 0 +:: > ~ Horizontal Stress - rx - p sf FIG.74. INFLUENCE OF LATERAL BOUNDARY LOCATION ON HORIZONTAL STRESS. 191

188 60 -I 0 :;: 0 l; w Surfae " I ' ~~ '\ ~ "' "' ""' ~ Vertial Stress - uy-psf + - I 0 3() ~~~-t t---i -0 :; 2 0 l--t--f--+--t--t-~:--+---t [ij Shear Stress - TXY - p sf FIG. 75. INFLUENCE OF LATERAL BOUNDARY LOCATION ON VERTICAL STRESS AND SHEAR STRESS. 192

189 inrease in shea~ stress with inreasing distane to the mesh boundary exept from position 3(DT = 3.2 HT) to position 4(DT = 506 HT) where only a small hange in shear stress ours. Horizontal and vertial displaements for nodal points at the upslope boundary of the region of interest are shown in Fig. 76 for various positions of the right boundary of the finite element mesh. The horizontal displaements inrease appreiably with inreasing distane to the mesh boundary for positions 1, 2 and 3; the hange from position 3 to 4 is onsidered negligible. The vertial displaements are affeted to a lesser degree by the mesh boundary loation and no further hange in displaement results for boundary loations beyond position 2. The modulus variation used for the slope analyzed in Figs. 74, 75, and 76 is plotted in the upper part of Fig. 77. This variation was hosen to represent that expeted in the upper portion of San Franiso Bay Mud. On the basis of the analytial t'esults- shown in Figs-. 74, 75 and 76 it was established that the maximum value of horizontal shear stress at the boundary of the region of interest would represent a good riterion of the effet the loation of the mesh boundary. The maximum value of shear stress at the up-slope an down-slope boundaries of the region of interest for various positions of the mesh boundaries are shown in the lower part of Fig. 77. It may be noted that stresses on the up-slope boundary are unaffeted by loating the mesh boundary beyond DT = 3HT; stresses on the down-slope boundary are unaffeted if DB = 4HT or greater. Other analyses were also onduted with various values of slope angle, depth fator, regions of interest and elasti modulus variation as indiated in Table 10. For these onditions it was determined that the distane required to eliminate the mesh boundary effets from the region of interest was 2HT from the up-slope boundary and 3HB from the down-slope boundary; the total distanes to the mesh boundaries may thus be alulated from: 193

190 0 ~ > OJ iii 0 ""--_... _,... _.... O OB Horizontal Displaement-ft. + - (b) 0 oi&-----~--~--a..... ~ io Vertial Displaement - ft. FIG INFLUENCE OF LATERAL BOUNDARY LOCATION ON HORIZONTAL : AND VERTICAL DISPLACEMENTS. 194

191 Surfae I I,Q > Q.I Lil 10 l, L ~ \ Elasti Modulus - E = I0-5 p sf ~---. en a. I ~... ~ u --'-~:.rr H9 ~ 200 J ,." CJ)... 0 OJ.. CJ) _. _ Ls Distane Ratio- - or - FIG. 77. MODULUS-DEPTH VARIATION AND INFLUENCE OF LATERAL BOUNDARY LOCATION ON THE SHEAR STRESS. Ha Lr Hr 195

192 Table 10. Slopes Analyzed for Boundary Effets. Slope Angle, i 18, 90 HT/H. 1. o, 1. 7, 2.0, 3.0 LB/RB 1. 0, 1. 5, 2.0 LT/HT.... o.s, 1. 0, 2.0 Elasti Modulus, E Constant, varied (Fig. 77(a)) In the studies to determine the influene of the loation of the finite element mesh boundaries on the stresses, strains and displaements in the region of interest -the value of P0i.s-son' s ratio was taken to be The influene of the value of Poisson's ratio was not studied beause for the total stress analysis of saturated lays for short term loading there is little volume hange and therefore, if isotropi elastiity is assumed, the value of Poisson's ratio must neessarily approah one-half. Element Size It has previously been mentioned that the most effiient analyses require a minimum number of elements neessary for aurate representation of the system being studied. In ertain problems it has been shown that it is neessary to inlude elements in the regions adjaent to the region of interest for the purpose of eliminating undesirable boundary effets from the region of interest. Beause stresses, strains and displaements in these regions are of little interest, it is desirable to use as few elements as neessary. To determine riteria for the size of elements it is onvenient to onsider the element as a height and a shape fator, where the height is the layer thikness and the shape fator is defined as the ratio of the longest side to the shortest side for retangles and the base to height ratio for triangles and trapezoids. 196

193 Layer Thikness The effet of layer thikness on the auray of the finite element idealization has previously been studied for the stati built-up embankment ase by Clough and Woodward (1967). Clough and Woodward onsidered a onstrution sequene for an embankment whih onsisted of the progressive plaement of layers of soil until the final onfiguration was reahed, eah added layer applying only vertial normal stresses to the previously plaed soil. Goodman and Brown (1963) have shown that suh a onstrution proedure results in stresses, strains and displaements dependent upon the number of onstrution steps taken,. Clough and Woodward (1967) onluded that a 7 step and a 14 step onstrution proedure produed pratially the same results and therefore used a 10 step sequene for their studies. Studies to determine the effet of layer thikness were performed on two vertial slopes ompo&ed of square elements with- 8-- anl 16_ layerb_,_ Beause a diret omparison of stresses, strains and displaements over an area is diffiult the stresses at four vertial setions are ompared; Fig. 78 shows the horizontal, vertial and shear stress ratios at vertial setions 5 feet on either side of the vertial ut fae; Fig. 79 shows the same stress ratios on vertial setions at 40 feet on either side of the vertial ut faeo It may be noted that the stress ratios for the 8 and 16 layet: onfigurations are essentially the same with the exeption of the horizontal and vertial stresses in the region of the toe of the slope (Fig. 78), these being somewhat larger for the 8-layer representation; the shear stresses for the two analyses are virtually the same. Beause the value of the normal stress is relatively unimportant for lays in undrained shear, and beause stress onentrations at lesser slope angles are not as pronouned as for vertial slopes it would appear that an 8 layer onfiguration was adequate for most purposes. Studies of slopes onstruted in materials with bi-linear elasti properties indiated that the stress hanges ouring due to the inremental onstrution proedure in an 8 layer onfiguration were quite large, ausing stresses to exeed the assigned material strengths by appreiable amounts. Subsequent analyses have been onduted on 10 or more layer onfigurations with better results. 197

194 8 Layers ~ Layers t.lioj.# 40...,..., ' ' ' 20 I I ' I Horizontal Stress Ratio - ux ~h a-y Vertial Stress Ratio - 'b'h l O ny Shear Stress Ratio-~ 1 0 FIG. 78. THE EFFECT OF LAYER THICKNESS ON STRESSES AT 5 FT. ON EITHER SIDE OF A VERTICAL CUT.

195 8 Layers 80 n~pj)) Layers 'WY/AV» O'X Horizontal Stress Ratio - 'lfh I O a-y Vertial Stress Ratio-"'ih Shear Stress TXY Ratio - lfh FIG. 79. THE EFFECT OF LAYER THICKNESS. ON STRESSES AT 40 FT. ON EITHER SIDE OF A VERTICAL CUT.

196 Element Shapes The effet of the element shape on the auray of stresses, strains and displaements of the system being analyzed probably depends largely upon the loation of the element within the mesh onfiguration and the fores and/or displaements applied to it. Idriss (1966), in studying the effet of triangular element shape on the response of earth banks subjeted to earthquakes, ~nluded that triangular elements with base-to-height ratios of 5 or less had little effet on the results anywhere in the slope and ratios of 10 away from the slope had little effet on results in the slope region. For retangular elements (omposed of 4 triangular elements) these base-to-height ratios are equivalent to 2.5:1 and 5:1 respetively. Ratios used in the present study are within these limits and no systemati analyses were undertaken to develop limiting ratios. However, it was determined in -the -ourse of -studying -sevet"al ~ertial slop S i:hat element side ratios of 5:1 outside the region of interest did not influene stresses within the region, In general for the slopes analyzed (1.5:1 and 3:1) and the finite element meshes used, the element length to height ratios within the region of interest did not exeed 1.5:1. Ratios greater than 5 were never used within a distane HT or HB of the boundary of the region of interest and a maximum length to height ratio used was 8, at a distane greater than 2HT or 2HB from the boundary of the region of interest. 200

197 APPENDIX B CORRECTIONS FOR SIMPLE SHEAR TEST RESULTS Reent studies have shown that the strength and stress-strain behavior of soft weak lays may be greatly influened by the strength of rubber membranes, filter paper drains and various other fators related to equipment design. For tests on speimens of San Franiso Bay Mud, Dunan and Seed (1967) have shown that negleting to orret for these effets may result in overestimating the soil strength by as muh as 20% at low onsolidation pressures. Sine the simple shear tests are to be performed on Bay Mud speimens at low onsolidation pressures, it is neessary to determine appropriate orretions for the strength and stress-strain urves. Rubber Membrane Corretion high. The simple shear speimens are 6 m square and approximately 2- m. The rubber membranes used to enlose the speimens are approximately 5 perent smaller than the speimen and.010 in. in thikness. This differene in size will result in an initial prestress in the speimen. Beause of the lateral support of the top and base plates only vertial prestress will our in the speimeno This initial vertial stress due to the undersized rubber membrane may be determined as desribed in the following paragraphs. In order to derive orretions for the rubber membrane in the simple shear test, the rubber is asstnned to be an isotropi linear elasti material with the same properties in tension and ompression and to have a Poisson's ratio of one-half. The governing equations may be written as E E r - v (r + r ) m z z p n E E = r - v (r + r ) (B-1) mp p z n E E = r - v (r + r ) m n n z p in whih E = the elasti modulus for the rubber membrane, E = the m z vertial strain, E p the perimetral strain, E = the strain normal to n the membrane surfae, r = the vertial stress, r = the perimetral stress, z p r n = the stress in membrane normal to membrane surfae and v = the 201

198 Poisson's ratio. Dunan (1965) found that E = 14 kg/m 2 for pure latex m rubber membranes, whih were used for the simple shear tests. The initial strain in the membrane is E and opm E p E z = E a = o n opm E opm By substituting these into Equations (B-1) the following expressions for stresses in the membrane may be determined. a 2 E E z m opm a = 2 E E p m opm Only the fore orresponding to a will at on the speimen sine the fore z resulting from a will be transferred to the top and bottom steel plates. p Therefore the vertial stress during onsolidation must be orreted. The vertial fore due to the membrane stress, a may be expressed as z P =A 2E E z m m opm where A = t 4W m m s in whih A = the membrane area and W = the width or length of the square m s speimen. Sine t may be expressed in terms of initial thikness t and m om strains E, opm tm = (1-E ) 2 opm and the speimen area is W 2, the following expression for initial stress s in the speimen due to the membrane is obtained. = 8 t E E o_m m o~p~m- 6.a a ( 1-e:: )2 m w opm s (B-2) During onsolidation under the desired pressure strains will our and produe further stress hanges in the sample, For the onsolidation proess 202

199 e: e: z a e: 0 p a = 0 n where e: = the axial strain during onsolidation. Again substituting a into equations (B-1) results in a z = i E e: 3 m a and the a is again arried by the top and bottom plates. Proeeding in p a similar manner to the above it may be seen that t E e: 16 om m a &u- a 3 -(-l--e:~~)~2 -(l---e:~-)-w~ m opm a s (B-3). The total vertial stress orretion at the end of onsolidation for the simple shear rubber membrane is therefore the sum of equations (B-2) and (B-3). t E e: 16 om m a D.a am 3 (1-e: ) 2 (1-e: ) w opm a s 8 t E e: om m opm (1-e: ) 2 w opm s (B-4) Sine the initial strain e: and the vertial strain during onsolidaopm tion e: are of opposite sign the two portions of the orretion tend to a anel one another as a result the axial stress orretion is usually less than.01 Kg/m 2 Beause predominantly shear strains are indued during the simple shear test the stress-strain orretion due to the deformation of the membrane may be alulated in the following manner. ause a shear fore p xy The shear strains 2W t E s om m (1-e: ) 2 (1-e: ) 2(l+v) opm a (B-5) 203

200 in whih Gm = the shear modulus of the membrane, yxy = the shear strain, and all other symbols are as previously defined, and the shear stress orretion is, therefore, t E 2 om m M xy 3 (1-e: ) 2 (1-e: ) W opm a s 0 Yxy (B-5) This orretion is less than.01 kg/m 2 for strains up to 20 perent with the membranes used in the present study and therefore the orretion was negleted, Frition is present in the simple shear devie as rolling and sliding frition, Beause the steel sides of the simple shear box annot deform in the same manner as the speimen there is relative motion between the rubber membrane and the box sides (Fig. 80). The magnitude of the frition fore resulting from this relative motion depends upon the method of -lubriation, -the -Consolidation pressure ating on the speimen, the bak pressure, the time of onsolidation, the materials in ontat and the onstrution of the simple shear box. Dereased frition results from inreased visosity, lower onsolidation and bak pressures, dereased onsolidation time and smoother materials in ontat. Dunan and Dunlop (1968) have presented an experimental relationship for frition versus theoretial grease thikness between polished luite and rubber with Dow Corning stop ok grease and high vauum grease as lubriants. It is expeted that this relationship should also be appliable to polished steel and rubber. The simple shear box (Fig. 80) as onstruted may be onsidered to represent an infinite strip ontat between the rubber and steel. The relative motion between the rubber membrane and the shear box sides and "drainage paths" for the grease are shown in Fig. 80. During onsolidation the lateral pressure and pore pressure at on the rubber membrane and grease. This pressure will ause the grease to esape at loations A, B, and C sine these points are at atmospheri pressure. (It would be better to have these points at a higher pressure, thereby reduing the pressure gradient in the grease.) Beause of grease esape at point B, the one-half width of the infinite strip was estimated to be O. 75h. 204

201 Shape of Speimen 8 Side Plates 7 Grease _.#'C :::<: ~z \ 'Rubber Membranel Base- f>laj~..u. Membrane Displaement Side Plate Displaement - J1 Relative Displaement FIG. 80. SIMPLE SHEAR BOX RELATIVE DISPLACEMENTS AND LUBRICATION OF SIDES. 205

202 The theoretial thikness of grease between two rigid, perfetly smooth infinite strips is ho 2 h 2 = ----~ ho 2 F t + 4 µ b 3 (B-6) F = b = µ = ho = t - the the the the fore per unit length foring the strips one-half width of the strip visosity of the separating lubriant original grease thikness time during whih the pressure ated. together Using this relationship and the variation of sliding frition with grease thikness (h) determined by Dunan & Dunlop (1968) it is possible to estimate the frition between the simple shear box sides and the rubber mernbraneo Frition is al-s-o --present in the form of rolling ~nd sliding frition of the moving parts of the simple shear apparatus. The amount of frition due to both rolling and sliding was measured at various values of normal pressure using an aluminum speimen in the simple shear devie. The displaements indued by applied shear loads under this test arrangement are shown in Fig. 81a. The shear load differene between the load and unload yles at the same strain is expeted to be double the frition fore sine this frition in eah ase opposes the shear motion. Figure 8lb shows the variation of this frition for various total vertial stresses applied to the sampleo This orretion to the shear stress is applied by determining the frition fore for the test proedure from Fig. 8lb and dividing this by the sample area. A frition alibration for simple shear box frition was also performed using a water-filled membrane, applying a pressure and, after some time, onduting a strain ontrolled test on the water "speimen". In this way the sliding, rolling, and other frition fators will approximate those fators ating during the tests on lay samples. The side frition for this water test is not truly equivalent to that for a soil speimen beause the water sample pressure will be uniform, whereas the lay sample pressure may be nonuniform. A gradient ausing grease flow requires nonuniform pressure, and it would therefore be expeted that 206

203 more grease would be extruded and the frition fores would be somewhat higher for a lay sampleo Although the pressure distribution for the soil and water speimens are not the same, the two methods of orreting for frition result in orretions of almost exatly the same magnitudeo In fat, the orretions estimated from the sliding, rolling and membrane frition and those estimated from the water speimen tests agree within approximately 5%. The total orretion to the shear stress was of the order of.08 kg/m 2 or about 15% of the applied shear stress for the pressures used in the present tests. Equation B-6 and Fig. 8lb show that the total shear stress orretion depends upon the onsolidation pressure and the onsolidation time as well as the thikness and visosity of the lubriant 0 207

204 Cl ~ I "'O J Double Amplitude Frition Fore (a) Shear St rain- 0 /o 1 2 Cl ~ I Cl) () ~ 0 IJ Lt (b) 0 L i Total Vertial Pressure - kg/ m Fl G. 81. SLIDING AND ROLLING FRICTION IN THE SIMPLE SHEAR APPARATUS. 208

205 APPENDIX C FINITE ELEMENT COMPUTER PROGRAM FOR EXCAVATED SLOPES Identifiation This program onsists of a main program (MAIN) and seven subroutines (STIFF, QUAD, TRISTF, STRES, BANSOL, MODI, INIT). Purpose This program alulates stresses, strains and displaements for a ut slope omposed of either homogeneous soils or horizontally layered soils. The program may be used for analysis of slopes in materials with linear, bilinear or multilinear stress-strain properties. Input Data IDENTIFICATION CARD: FORMAT (12A6) Cols Identifying information to be printed with results. SLOPE CONFIGURATION CARD: FORMAT (3I5, Fl0.2, 3I5) Cols. 1-5 Number of layers of elements (NL ~ 23) 6-10 Number of olumns of elements (NI < 450-NT _ - NL+l Number of different materials (NUMMAT $. 20) Vertial aeleration (ACELZ). This is o.o for this program; the initial stresses are generated by 0 vertial = Y depth and 0 horizontal = K a vertial Number of iterations in alulating elasti properties (NPP) Number of the first olumn to the right of the slope top (NX) Number of layers to be removed (NT). These layers are removed 1 at a time. The inlination of the slope is determined by the spaing of vertials and horizontals in the slope (shaded region, Fig. 82) and must be uniform. 1) 209

206 - - N - - x x x z + H z -x Y(I) ~i----r----~--~~~--~~~..._,r------r-----t Y(2) ~i--~+-~~~~~~~~~->t-~r-~--t-~---; NT NL Y(NL+I) FIG CONFIGURATION OF GENERATED SLOPE. 210

207 MATERIAL PROPERTY CARD: FORMAT (1I0,4Fl0.2); NUMMAT ards; one for eah material. Cols Material type (MTYPE) Material unit weight (RO(MTYPE)) Elasti modulus (E(l,MTYPE)) Poisson's ratio (E(2,MTYPE)); this is restrited to <.50 and should perhaps be ~.49 to give aurate results Material shear strength (PRESS(MTYPE)) Total stress oeffiient of earth pressure (CAYO(MTYPE)). The next-to-last ard MTYPE(NUMMAT - 1), designates the properties assigned to the materials after failure; the unit weight and Poisson's ratio should be the same as before failure, but the modulus should be redued. The last ard, MTYPE (NUMMAT), is used to designate the properties assigned to the materials in elements whih have been "removed"; the unit weight should be zero, modulus should be very small (typially 0.01), Poisson's ratio should be the same as before removal, and the strength may be entered as zero. VERTICAL BOUNDARY LOCATION CARD: FORMAT (8Fl0.2) Cols First vertial line, X(l); this represents the left boundary; see Fig. 82, These X values must be whole numbers beause of slope generation, i.e., X = 10.0, 21.0, 57.0; values suh as 10.3, 21.27, eto annot be used for vertial boundaries between elements Seond vertial line, X(2) whih is the boundary between the first and seond olumn of elements Continue to X(number of olumns+ l),x(nl+l) whih is the right boundary. 211

208 HORIZONTAL BOUNDARY LOCATION CARD: FORMAT (8Fl0.2) Cols First horizontal line Y(l); this represents the surfae (must be whole numbers as for vertial boundaries) Seond horizontal line Y(2); this forms the boundary between the top layer and the seond layer Continue to Y (number of layers+ l),y(nl+l); this is the bottom boundary. LAYER MATERIAL TYPE CARD: FORMAT (1215) Cols. 1-5 Material type for top layer MAT(l) = Material type for seond layer MAT(2) = 1 or Continue to number of layers MAT(NL). The material types must inrease onseutively from the top down however, any one type an be repeated in adjaent layerso Sequene of Operations After the input data is read, the elements and nodal points are generated in the main program. The main program then alls subroutine!nit whih alulates the initial stresses in the onfiguration. The stiffness of the removed elements is redued and the main program alls subroutine STIFF (and QUAD and TRISTF) whih assembles the stiffness and nodal point fores. BANSOL is next alled to solve for the nodal point displaements and STRESS is alled to alulate the element stresses and strains. Output The input data and the following quantities are printed. (1) Nodal point displaements. (2) Element stresses, strains and pore pressures. 212

209 Program Listing The program has been oded in FORTRAN IV language and has been used on BC DCS IBM and CDC 6400 omputers. The listing of the program is presented in the following pages. 213

210 $IBFTC MAIN C BOUNDARY REVERSE CUT DOWN POISSON,S RATIO RESTRICTED C COMMON NUMNPtNUMELtNUMMATtACELZ,NPPtNtVOLtMTVPE,NXXtNXPt 1 NLtNTtHED<12ltE<3t20) R0<20)tR(500)tZ(500)tURC500) 2 UZ(50Q),CODE(500)tSIG(l0)tNUMLAYtNUMIT COMMON /ARG/ RRR(5)tZZZ<S),S(l0tl0)tP<lO> tttc4ltlmc4)t 1 DDC3t3ltHHC6tl0)tRR(4)tZZ(4)tCC4t4),HC6tlO)tD<6 6)t 2 F<6 10),TP<6>tXI<lO)tEE<4),JX(450t5)tSTRESS(450t3)t 3 SU(20)tNBACK(50)tSTREG(20)tPOREPR<450)tSTRAIN<450t3)t 4 GAMAX(450)t8ETAI(450) i COMMON /SLP/ CAY0(30),TOPXtTOPYtTOEXtTOEYtNltN2 COMMON I BANARG I MBANDtNUMBLKtBC108)tA<l08t54) DIMENSION XC100)tYC30)tMAT(30) 50 READ(S,1006) HEDtNLtNitNUMMATtACELZtNPPtNX,NT NUMNP=<NL+l)*<NI+l)+NT NUMEL=(NL)*(NI>+2*NT WRITE<6 2000)HEDtNVMNP,NUMELtNUMMATtACELZtNPPtNXtNT Nil=NI+l Nll=NL+l WRITE (6,2011) 56 DO 59 MM=ltNUMMAT READ C5tl001) MTYPEtROCMTYPE)tCE(J,MTYPE)tI=lt2)t 1 STREG<MTYPE)tCAYO<MTYPE) WRITE(6,2010) MTY~E,ROCMTYPE),(ECitMTYPE>tI=lt2)t 1 STREG<MTYPE>tCAYOCMTYPE),MTYPE 59 CONTINUE READ AND PRINT OF NODAL POINT DATA READ(5tl007) CXCI)tialtNll) READ(5tl007) CYft)tl ltnlu READ<5t1008)(MAT(l)tl ltnl) DO 31 N=ltNUMEL GAMAXCN) =Oe BETAJ(N) =O DO 31 K7lt3 31 STRAIN <NtK) =O DO 32 K=ltNUMNP UR(K)=O 32 UZCKJ=Oe C**** LOCATE NODAL POINTS FOR SLOP~ NP=O NS=NL-NT NX2NL=NX-2*NT WRITE C6t2004) I=O 75 I=I+l Nll=NL+l IF (leleenx2nleor. I.GE.NX) THREE GO TO

211 MMM=l-NX2NL+l NNN=MMM/2 IF!MMM.NE.2*NNN) GO TO 77 NS=NS+l Nll=NL+2 77 DO 27 J=l,NLl NP=NP+l IF CMMM.NE.2*NNN> GO TO 24 IF(J-(NL+2-NS))24t Z(NPl=(Y(J-l)+Y(J))/2. GO TO Z(NPl=Y!J-1) GO TO 25 2' l(npl=y(j) 25 CODE(NP)=O IF!I.E0.1.0R.J.EO.Nll) todecnp>=l O. IFIZ<NPl.EO.Y(NL+l)) (00ECNP)=3e0 R CNP l = X < I l 27 WRITE(6,2002) NPtCODECNP)tR(NP)tZCNP)tURCNP)tUZ<NPl IF!NUMNP-NP)100t76t75 66 DO 6 J=l,NLl NP=NP+l Z!NPl=Y(J) CODECNPf=O.- IFC I.E0.1.0R.I.EO.Nil) CODECNPl=l O IF<Z!NP>.EO.Y!NL+lll CODECNP)=3e0 R<NPl=X( J), 6 WRJTE!6,2002) NP~CODECNP)tR(NP)tZ(NP)tUR(NP>tUZ(NP) IF!NUMNP-NP)100,76t WRITE (6,2009) N CALL EXIT C**** GENERATE ELEMENT~ FQR SLOPE 3 76 J=O WRITE!6t2001) NS=NL-NT Nll=NL+l NE=O 70 I=I+l IF ( I.GE.NX2NL.ANO.IeLT.NX> GO TO 39 DO 10 J=l,NL IF <NE.GE.NUMEL) GO TO 190 NF.=NE+l IX(NEtll=NE+I-1 IF(I.GE.NXl JX(NEtll IXCNEtl)+NS-NL.IXCNE,2i=IXCNEtll+l JX(NE,3l=JX(NEt2l+NLl IX!NEt4l=IXCNEt3)-l IX<NEt5l=MAT(J) 10 WRITE(6t2003) NEt(IXCNEtI)tl=lt5) GO TO O= ( ( Y ( 1 l-y C NT+ 1 ) ) I ( X ( NX l -X ( NX2 NL) ) ) * C X ( I ) -X ( NX ) ) +Y C 1 ) DO 11 J=ltNL NY Jl=Y ( J+l) 215

212 NYJ=Y(J) NQ=Q IF (NQ.GE.NYJ> GO TO 49 IF (NQ.GT.NYJl) GO TO 53 P=I PP=NX2NL PPP=<P-PP)/2e+e6 NNN=PPP NE=NE+l IX(NEtl)=NE+I-1-NNN IX(NEt2l=IX(NEtll+l MP=(I-NX2NL) MP2=MP/2 IF (MP.NE.2*MP2) GO TO 57 N22=NL1 GO TO N22=NL IXCNEt3)=IXCN~t2)+N22 IXCNEt4)=IXCNEt3)-l IXCNEt5)=MAT(J) WRITE(6t2003) NEtCIXCNEtl)tl=lt5l IF CNO.NE.NYJl) GO TO 11 NE=NE+l IXCNEtl)=IXCNE-lt2) IX(NEt4>=IXCNE-lt3) IXCNEt3l=IXCNEt4) IXCNEt2)=IXCNEt3)+1 IX (NE, 5) =MA TC J) GO TO NE=NE+l IF (J.NE.1> GO TO 104 IX<NEtll=IX(NE-lt2l+l IX<NEt2)=IX(NEtl)~l IX(NEt3)=JXCNE-lt3)+1 IXCNEt4l=YX(NEt3) IXCNEt5>=MAT(J) GO TO IXCNEtl)=IX<NE-lt2) IX(NEt2l=IXCNE91)+1 IXCNEt3)=lX(NE-lt3~ IX(NEt4)=IXCNEt3) IX(NEt5)=MATCJ) 107 WRJTE(6t2003) NEtCIXCNEtI)tlalt5) NE=NE+l. IX(NEtl)=JXCNE-lt2, JX(NEt2)=IX<NEtl)+l.I X ( NE t 3 ) = I X ( N E-1 t 3 ) + 1 IX(NEt4l=IX<NEt3)-1 IX<NEt5)=MAT(J) GO TO NE=NE+l IF CNQ.EQ.NYJ) GO TO 91 IXCNEtl)mIXtNE-lt2) IXCNEt2)~IX(NEtl)+l ixcnet4) IX<NE-lt3) 216

213 IX(NEt3l=IXCNE,4)+1 I~<lilE,5l==MATCJl GO TO 12?l IXCNEtll=IX(NE-l ll lx(net4l=ixcne-lt2l IX(NE,zl=TXCNEtll+l TXCNEt3l=IXCNE,4)+1 JX(NE,5l=Mi\T(J) 12 WRITE(6,2003) NEtCIXCNEtI),I=l 5l 11 CONTINUE IF CNE.LT.NUMEL) GO TO CONTINUE C**** DETERMINE BAND WIDTH. J=O DO 340 N=ltNUMEL DO 340 I=lt4 DO 325 L=lt4 KKIABS<IXCN,IJ-IXCNtL)) IF CKK.GT.J) J=KK 325 CONTINUE 340 CONTINUE MBAND=2*J+2 CALL INIT DO 339 J=lt4 DO 339 K=l C(J,K)=O.. DO 341 I=lt NBACK(I)=O C**** ALTER STIFFNESS NUMLAY=NL NXX=O NXP=NX+l GO TO DO 164 I=ltNUMNP URCil=O 164 UZ(Il=O 161 NXX=NXX+l TOPX = X(NX) TOPY = Y(l) NN = NX-2>fNXX TOEX XCNN> TOEY = YCNXX+l) WRITE (6,1009) NXP=NXP-2 NNN=NXX-NUMLAY DO 162 I=ltNXP MNN=NUMLAY If(I.GEe(NX2NL+l)) NNN=NNN+MNN MN~=NUMLAY+l 217

214 NBACK(I)=NNN JX(NNNt5)=NUMMAT 162 WRITE (6tl004) NNNtIX<NNNt5) IF <ACELZ.NEeO ANOeNXXeLT.NT> GO TO 16~ IF (ACELZ.GTele) READ (5tl010)MtlNtUR(N)tUZ(N)tNUM=ltM) DO 500 NUMITaltNPP CALL STIFF CALL BANSOL IF CNUMITeLTeNPP,> GO TO. 410 WRITE C6t2006) WRITE l6t2005) (NtB(2*N-l)tBC2*N~tN~l;~uMNP) 410 CONTINUE CALL STRES 500 CONTINUE IF<NXXeLTeNT) GO TO 163 GO ro -so 1001 FORMAT (Jl0t5Fl0e2) 1004 FORMAT (2I5tFl0e0) 1006 FORMAT (12A6/315tFl0 2 3I5) 1007 FORMATC8Fl0e2) 1008 FORMATC12I5) 1009 FORMAT ( // 35H. N MTYPE FOR REMOVED ELEMENTS I ) 1010 FORMAT <I10/(110t2Fl0e2)) 2000 FORMAT (lhl 12A6/ 1 30HO NUMBER OF NOD~L POINTS I 2 30HO NUMBER OF ELEMENTS I 3 30HO NUMBER OF DIFF. MATERIALS I 5 30HO Y ACCELERATION Fl2e4 I 6 30HO NUMBER OF APPROXIMATIONS / 7 30HO NX FIRST COLUMN~ I3/ 8 30HO NT NO. OF LAYERS REMOVED / > 2001 FORMAT <12HlELEMENT N0.6XlHl5XlHJ5XlHK5XlHL4X8HMATERIAL) 2002 FORMAT <Il2;Fl2e2t2Fl2e3t2E24e7tF12.3l 2003 FORMAT (1113t416tlll2) 2004 FORMAT (12HlNODAL POINT ax 28HTYPE x-ordinate Y-ORDINAlE 1 2X 46H XLOAD OR DISPLACEMENT Y LOAD OR DISPLACEMENT 2005 FORMAT ( 112t2Fl2e4) 2006 FORMAT ( 12HlN.P.NUMBER 9X 3H UX 9X 3H UY 2009 FORMAT (26HONODAL POINT CARD ERROR N= IS) 2010 FORMAT ( I9tF8e2tF12e2tFlOe5tFl3e2tF9e4tI3/) 2011 FORMAT C 9HlMATERIAL 8H WEIGHT 4X 7HMODULUS 4X 7HPOISSON 1 5X 8HSTRENGTH SX 4HCAYO //) END 218

215 SIBFTC STIF SU~ROUTINE STIFF COMMON NUMNP,NUMEL,NUMMAT,ACELZtNPP,NtVO~tMTYP~tNXX,NXPt 1 NL,NT,HEDC12),E(3,20),ROC20),RC500)tZl5qO>,URC500), 2 UZC500)tCODEC500)tSIGC10),NUMLAYtNUMIT I COMMON /ARG/ RRRC5)tZZZC5)tSClQ,10)tPC10) ttt(4)tlm(4), 1 DD(3,3)tHHC6tl0)tRRC4>,ZZC4)tCC4,4),HC6,lO)tOC6t6)t 2 FC6tlO),TPC6),XIC10),EEC4)tIXC450t5),STRESSC450,3), 3 SUC20)~NBACKC5Q),STREGC20),pOREPRC450),STRAINC450t3)t 4 GAMAXC450)t8ETAl(450) COMMON I BANARG I MBANDtNUMBLKt8(108)tACl08,54) REWIND 2 NB=27 ND=2*NB ND2=2*ND STOP=O.O NUMBLK=O IF CACELZ.NE.o or.numit.ne.1> GO TO 38 C**** REVERSE BOUNDARY STRESSES 163 DO 16Z f f=l tnxp N=NBACK CI I l 161 J=IX(N,2) J=IXCN,3) DR = RC J) - R ( I ) DZ = l ( J) - Z C I ) JFCII.GE.CNXP-3)) GO TO SIGRY=CSTRESSCN,2l+STRESSCN+lt2))/2 SIGRX=CSTRESSCN,l>+STRESSCN+l,l))/2 SIGRT=<STRESSCNt3)+STRESSCN+lt3))/2e URCJ) = UR(J)+SIGRT*DR*.5 UR(J) = UR(J)+SIGRT*DR* 5 UZCIJ = UZCI)+SIGRY*DR* 5 UZCJ) = UZCJ)+SIGRY*DR* 5 IFCJI.EQ.1) URCJ)::OeO GO TO IFCII.GTel) NN=NBACK(II-1> MM=NBACK (I I+l) NPX= CI J-( NXP-4)) GO TO Cl51,153tl~3tl52J,NPX 152 SIGGN=SGGRN SIGGT=SGGRT SGGRN=O SGGRT=O GO TO IFCNXX.EO.NT) GO TO 168 SIGLY=CSTRESSCNt2)+STRESSCNN+l,2))/2. SIGLT=CSTRESSCNt3)+5TRESS(NN+it3))/2 GO TO SIGLY=SIGRY SIGLX:sSIGRX SIGLT SJGRT 219

216 154 K=N+l M=MM+l IF<NPX.EOalaOR.NPX.E0.2) M=M+l IF(NXX.EO.NT.ANDeNPXeEOal> M=M-1 I F Hl PX E Q 2 ). K = K + 1 IF<NXX.EO.NT.ANDeNPX.EOa2) GO TO 158 GO TO M=MM+2 SIGRV=<STRESS(N,2)+STRESS( M,2))/2. SIGRX=CSTRESS(N,ll+STRESSC M,2))/2. SIGRT=CSTRESSCN,3)+STRESSC M 3))/2e GO TO SIGRY=<STRESS(N,2)+STRESS(K,2l+STRESSCMM,2)+STRESS(M,2))/4. S I GR X = ( STRESS ( N t 1 ) +STRESS K t 1 ) +STRESS ( MM, 1 ) +STRESS ( M t 1 ) ) / 4 SIGRT=<STRESSCNt3)+STRESSCKt3)+STRESSCMMt3)+STRESSCMt3))/4e IF (DZ.Gr.a.) GO TO SIGGN=SIGLY SIGGT=SIGLT SGGRN=SIGRV SGGRT=SIGRT GO TO ALPHA=ATAN2CDZt0R) CC=COSCALPHA) SS=SINCALPHA) SIGGN=SIGLY*CC*CC+SIGLX*SS*SS-2.*SIGLT*SS*CC SIGGT=<SIGLY-SIGLX>*SS*CC+SIGLT*<CC*CC-SS*SS) SGGRN=SIGRY*CC*CC+SIGRX*SS*SS-2.*SIGRT*SS*CC SGGRT=<SIGRY-SIGRX>*SS*CC+SIGRT*CCC*CC-SS*SS) 156 UR<I> = URCI>+DR*<2 *SIGGT+SGGRT)/6e-DZ*<2 *SIGµN+SGGRN)/6. UR(J) = UR(J)+DR*<2 *SGGRT+SIGGT)/6.-DZ*<2 *SGGRN+SIGGN)/6. UZ(J) = UZ(l)+DR*(2e*SIGGN+SGGRN)/6e+DZ*<2 *SIGGT+SGGRT)/6. UZ(J) = UZ<J)+OR*<2 *SGGRN+SIGGN)/6e+DZ*<2 *SGGRT+SIGGT)/6e 167 DO 159 1= STRESSCN,Il=O P.OREPR ( N) =O 162 CONTINUE 38 WRITE (6,3000) DO 40 N=l,NUMNP IF (UR(N).Eo.o ANO.UZ<NleEO.O.) GO TO 40 WRITE (6,2900) NtURCN)tUZ(N) 40 CONTINUE 49 DO 50 N=l,ND2 BCN)=O.O DO 50 M=ltND 50 A<NtMl=OeO NBN=l C**** FORM STIFFNESS MATRIX IN BLOCKS 60 NUMALK=NUMBLK+l NH=NB*<NUMBLK+l) NMmNH-NB NL NM-NB+l 220

217 KSHIFT=2*NL-2 DO 210 N=NBNtNUMEL INDEX=N KMIN=MINO(IX(Ntl)tlXfNt2)tIX(Nt3)tIX<Nt4)J IF <KMINeLTtNL) GO TO 210 IF (KMIN.GT.NM) GO TO CALL QUAD IF (VOLeGT.O.) GO TO 144 WRITE (6t2003) N STOP::sl.O 144 IF <IXCNt3).EQ.JX(Nt4)) GO TO 165 DO 150 II=lt9 CC=S<IItl0)/SC10tl0) P (I I ) =P (I I l-cc*p (f 1 10) DO 150 JJ=lt9 150 S(JJ,JJ)=S<IJ,JJ)-CC*SflOtJJ) DO =1 8 CC=S<II~9)/S(9,9) P(ll)=P<II)-CC*P(9l DO 160 JJ=lt8 160 S<lltJJl=S<IItJJJ-CC*SC9tJJ) C**** ADD ELEMENT STIFFNESS TO TOTAL STIFFNESS 165 DO 166 I=lt4 166 LM<I>=2*IX<Ntl)-2 DO 200 I=lt4 DO 200 K=l 2 II=LM(l)+K-KSHIFT KK=2*I-2+K B ( I I ) = B ( I I > +P ( KK) DO 200 J=lt4 DO 200 L=lt2 JJ=LM(J)+L-II+l-KSHIFT LL=2*J-2+l JF (JJ.LE.O). GO TO 200 IF <ND.GE.JJ) GO TO 195 WRITE (6,2004) N STOP=l.O GO TO A (I I,JJ)=A 0 ( I I 1JJ )+SCKKtLL) 200 CONTINUE. 210 CONTINUE 211 CONTINUE NBN INDEX 221

218 C**** ADO CONCENTRATED FORCES WITHIN BLOCK DO 250 N=NL,NM K=2*N-KSHIFT B(K)=B(K)+UZ(N) 250 B(K-ll=BCK-l)+URCN) C**** DISPLACEMENT BOUNDARY CONDITIONS 310 DO 400 M=NL,NH IF CM.GTeNUMNP) GO TO 400 U=UR(M) N=2*M... I-KSHIFT IF CCODECM)eLTeOe) GO TO 390 IF <CODECM)eEO.Oe) GO TO 400 IF CCODECM)eEOele) GO TO 370 IF CCODECM).E0e2e) GO TO 390 IF CCOOECM)eE0.3e) GO TO CALL MODIFYCA,B,ND2tMBANDtNtU) GO TO CALL MODIFYCAtBtN02tMBANDtNtU) 390 U=UZCM). N=N+l CALL MODIFYCAtBtN02tMBANDtNtU) 400 CONTINUE C**** WRITE BLOCK OF EQUATIONS ON TAPE AND SHIFT UP LOWER BLOCK WRITE (2) (B(N)tCACNtM)tM ltmbano)tn=ltno) DO 420 N=ltND K=N+ND BCN)=B(K) B(K)=O.O o o 420 M=ltND ACNtM)=ACKtM) 420 A<KtM)=O.O C**** CHECK FOR LAST BLOCK IF CNMeLT.NUMNP> IF CSTOP.NE.O.) 500 RETURN GO TO 60 CALL EXIT 2003 FORMAT (26HONEGATIVE AREA ELEMENT NO. 14) 2004 FORMAT (29HOBAND WIDTH EXCEEDS ALLOWABLE I4) 2900 FORMAT CI10t2F.15e2) 3000 FORMAT (IHO 18HNODAL POINT FORCES II ax 2HNP sx 1 lohhorizontal 7X 8HVERTICAL //) END 222

219 SIBFTC QUADF SUBROUTINE QUAD COMMON NUMNPtNUMEL,NUMMATtACELZtNPPtNtVOLtMTYPEtNXXtNXPt 1 NLtNTtHED<12)tEC3t20)tR0(20)tR(500)tZC500)tUR(500)t 2 UZ(500)tCODE(500),SIG(l0)tNUMLAYtNUMIT COMMON /ARG/ RRR(5)tZZZC5)tSC10tl0)tPC10) ttt(4)tlm(4)t 1 DDC3t3)tHHC6tl0)tRR(4)tZZC4)tCC4t4)tHC6t10)t0(6t6)t 2 F<6 10)tTP(6)tXIC10),EE(4)tlX(450t5)tSTRESSC450t3)t 3 SU(20)tNBACKC50)tSTREGC20),POREPRC450)tSTRAINC450t3)t 4 GAMAX(450) tbetai (450). COMMON I BANARG I NDtNUMBLKtBC108)tAC108t54) COMMON /SLP/ CAYOC30)tTOPXtTOPYtTOEXtTOEVtNltN2 I=IX<NtU J=IXCNt2) K=IX(N,3) L=IXCN,4l MTYPE=IXCNt5) 10 EE(lJ = ECltMTYPE) EE(2) = EC2tMTYPE> C**** FORM STRESS-SXRAtN- RELA-1'!-0NSH!P COMM=EEC1)/((l.-2.*EEC2>l*fle+EE(2))) CCltl)=COMM*Cl.-EEC2)) CClt2)=COMM*EEC2) CC2tll=C<l 2) C ( 2' 2 l =C ( 1 1> C<4t4>= 5*EE(l)/(le+EE<2>> C**** FORM QUADRILATERAL STIFFNESS MATRIX 151 RRR(5)=(R(Il+R(J)+R<Kl+R(L))/4. ZZZ(5)=(Z(l)+Z(J)+ZCK)+Z(L))/4e0 DO 94 M=lt4. MM=IX<NtM) 93 RRR<M>=RCMM) 94 ZZZ<M>=Z<MM) DO 100 II=ltlO P<II>=OeO DO 95 JJ=lt6 95 HHCJJ,JI>=OeO DO 100 JJ=ltlO 100 S<IItJJ)=OeO IF 1K.NE.L» GO TO 125 CA~L TRISTFC1t2t3) RRR(5)=(RRR(l)+RRRC2)+RRR(3))/3e0 ZZZ(5)=CZZZ<t>+ZZZC2)+ZZZ(3))/3e0 VOL=XI(l). GO TO VOL OeO CALL TRISTF~4,1,5) 223

220 CALL TRISTF(ltZtS) CALL TRISTF(2t3t5> CALL TRISTF<3t4t5) 131 DO DO 140 JJ=ltlO 140 HHClltJJ) HHClltJJ)/4e 130. RETURN 1000 FORMAT Cl5t6Fl0e3tF15.3tFlOe3) END 224

221 SIBFTC TRJST SUBROUTINE TRISTFCIItJJ,KK) COMMON NUMNP,NUMELtNUMMATtACELZtNPPtNtVOLtMTYPEtNXXtNXPt 1 NLtNTtHED(12)tE(3t20)tR0(20),R(500)tZ(500)tUR(500)9 2 UZ<500)tCODE<500),SJG(l0ltNUMLAYtNUMIT COMMON /ARG/ RRR(5)tZZZ<5hS<lOtl0)tP(10) ttt(4)tlm(4)t 1 DD<3 3ltHHC6tlOltRR(4ltZZ(4)tCC4t4ltHC6,10)t0C6t6)t 2 FC6tlO),TP(6)tXI<tO),EE<4ltIX<450t5)tSTRESS(450t3)t 3 SU<20)tNBACKC50)tSTREG(20)tPOREPR<450)tSTRAIN(450t3)t 4 GAMAX(450)tBETAl(450) COMMON I BANARG I NDtNUMBLKtBtl08)tAC108t54) LM ( 1 J =I I LM<2>=JJ LM(3)=KK RR( 1 )=RRR< I J) RR(2)=RRRCJJ) RRC3>=RRRCKK) ZZ(l)=ZZZ(JJ) ZZ(2)=ZZZ(JJ) zz 3- i = z_z_z_ t KK- i 85 DO 100 I=lt6 DO 90 J=ltlO F( X.Jl=O.O 90 H(l,J)=O.O DO 100 J=lt6 100 Q(I,J)=OeO COMM=RRC2)*CZZt3)-ZZ(l)J+RR<l>*<ZZ<2>-ZZ<3>>+ 1 RRC3)*CZZ(l)-ZZ(2)) XIC1l=COMM/2o0 VOL=VOL+XIfl) DC2t2l=XI(l)*C(ltl) 0(2t6)=XI(l)*C(lt2) 0(3t3l=XI(l)*C(4t4) D(3t5l=DC3t3) DC5t5)=0C3t3') DC6t6)=XI(l)*CC2t2) 108 DO 110 I=lt6 DO 110 J=It6 110 D(J,J)=D(i~J) C**** FORM DISPLACEMENT TRANSFORMATION MATRIX DD(ltl)=(RR(2)*ZZ(3l-RRC3l*ZZ(2))/COMM DD(lt2l=(RRC3l*ZZ(l)-RR(l)*ZZ(3))/COMM 0D(lt3)=(RR(l)*ZZ<2>-RR(2)*ZZ(l))/COMM 00(2tl)=CZZC2)-ZZ(3))/COMM DDC2t2)=(ZZ('3)-ZZ(l))/~OMM DDC2t3)a(ZZ(l)-ZZC2))/COMM 225

222 DD(3tll=<RR<3l-RR<2))/COMM OD(3t2l=<RR(l)-RR(3))/COMM. 0Dl3t3)=(RR(2)-RR(1))/COMM DO 120 I=lt3 J=2*LM<Il-l H(ltJl=DD<ltl) HC2tJl=00(2tI) H(3,J)=00l3tI) HC4tJ+l>=OO<ltl) H(5tJ+ll=00(2tI) 120 Hl6tJ+l)=DOC3tI) C**** FORM ELEMENT STIFFNESS MATRIX J=ltlO K=lt6 IF CH(K,J).EQ.O. ) GO TO I=lt6 129 FCitJl=F<ItJ)+O(ItK)*H(KtJ) 130 CONTINUE DO I--=1-,10 DO 140 K=lt6. IF (H(Ktil.EO.O.) GO TO 140 DO 139 J= 1, sr,j>=s(itj)+h(ktl)*f(k,j) 140 CONTINUE IF ( ACELZeNE 0.1.) GO TO 400 C**** GRAVITY LOADS COMM RO(MTYPE>*XI(l)/3e DO 170 I=lt3 J 2*LMCI)-l 170 P(J+l) PCJ+l>-ACELZ*COMM C**** FORM STRAIN TRANSFORMATION MATRIX 400 DO 410 1=1 6 DO 410 JaltlO 410 HHCitJ) HHCitJ)+H(ltJ) RETURN END 226

223 SISFTC MODI SUBROUTINE MODIFYCAtBtNEQtMBANDtNtU) DIMENSION A(l08t54)t8Cl08) DO 250 M=2tMBAND K=N-M+l IF CK.LE.O> B(K)=B<K>-ACKtM)*U ACKtM>=o.o 235 K=N+M-1 IF <NEOeLT.KJ B<Kl=BCK>-ACNtM)*U A(N,M)=OeO. 250 CONTINUE ACNtll leo BCN) U RETURN END GO TO 235 GO TO

224 SIBFTC BANS SUBROUTINE AANSOL COMMON /BANARG/ MM;NUMBLKtBC108)tAAC108t54) DIMENSION A(54tl08) EQUIVALENCE CAtAA) DIMENSION NABf34) NN=54 NCOUNT = NN*NN JUMPA NCOUNT/ JUMPS = NN/ NTRACK=l NL=NN+l NH=NN+NN REWIND 2 NB=O. GO TO 150 C**** REDUCE EQUATIONS BY BLO(KS 100 NB=NB+l rr.=-1-,-r.m NM=NN+N BCN)=BCNMl BCNM>=o.o DO 125 M=ltMM A(M,N> = ACMtNM) 125 ACMtNM) = OeO C**** READ NEXT BLOCK OF EQUATIONS INTO CORE IF CNUMBLK.EO.NB> 150 READ (2) CBCN)9(A(M IF (NB.EO.O) C**** REDUCE BLOCK OF EQUATIONS N=ltNN IF ( ACltN) ) 225 B(N) = SCN) I ACltN) L=2tMM YF ~ A(LtN) ) 230 C = A(LtN) I A(ltN) I=N+L-1 J=O DO 250 K LtMM J=J+l 250 A(J~I> = A(Jtll - C * ACKtN) B(I) = BCil - ACL,Nl = C 275 CONTINUE 300 CONTINUE ACltN) * BCN> GO TO N ) t M 1 t MM ) t N a GO TO 100 NL 9. NH ) 225, 300t

225 C**** WRITE BLOCK OF REDUCED EQUATIONS ON DISK IF CNUMBLK.EQ.NB> GO TO IFCNCOUNT+NN.GT.(39-MOD(NTRACK,40))*460) 1 NTRACK=CNTRACK/40)*40+40 NAB(NB) = NTRACK CALL WRDISK ( NTRACK, A, NCOUNT NTRACK = NTRACK + JUMPA CALL WRDISK ( NTRACK Bv'NN ) NTRACK = NTRACK + ~UMPB GO TO 100 C**** BACK-SUBSTITUTION 400 DO 450 M=l,NN N=NN+l-M DO 425 K=2,MM L=N+K BCN) = B<Nl - A(KtN) * BCL) NM=N+NN ACNM)=B(N) 450 ACNBtNMl = BlNl NA=NB-1 IF (NB.E0.0) GO TO NTRACK = NAACNB) CALL RDDISK ( NTRACKt At NCOUNT NTRACK = NTRACK + JUMPA CALL RDDISK < NTRACKt Bt NN ) GO TO 400 C**** ORDER UNKNOWNS IN B ARRAY 500.K=O DO 600 NB=l,NUMBLK DO 600 NaltNN NM=N+NN K=K+l 600 B(K) II A(NBt NM) RETURN END 229

226 SIBFTC STRS SUBROUTINE STRES COMMON NUMNPtNUMELtNUMMATtACELZtNPPtNtVO~tMTYPEtNXXtNXPt 1 NLtNT,HEDC12)tEC3t20)tROC20)tRC500)tZC500)tUR(500), 2 UZC500)tCODEC500)tSIG(l0),NUMLAYtNUMIT COMMON /ARG/ RRRC5) tzzz( 5) tsc 10tl0) tpc 10) TTC4) tlm(4)' 1 DDC3t3ltHHC6t10)tRRC4ltZZC4)tCC4t4)tHC6tl0)tDC6t6)t 2 F(6tlO)tTPC6)tXIC10)tEEC4)tIXC450t5)tST~SS(450t3)t 3 SU(20)tNBACKC50)tSTREGC20)tPOREPR(450> STRAINC450t3)v 4 GAMAX(450)tBETAif450) COMMON /SLP/ CAV0(30),TOPXtTO~VtTOEXtTOEYtNltN2 COMMON I BANARG I NDtNUMB~KtB(l08)tAC108t54) C**** COMPUTE ELEMENT STRESSES Nl = 2 MPRINT=O DO 300 N=ltNUMEL MTYPEi=IXCNt5'> IF f MTYPEeEOeNUMMAT) CALL QUAD DO 120 l=lt4 II 2*I JJ=2*IX(Ntl) PCII-l)=B(JJ-1) 120. PC II >=BCJJ) 122 DO 150 I=lt2 RRCI)=PCI+B) K=ltB 150 RRCI>=RRCI>-SCl+BtK>*PCK) GO TO 300 COMM=SC9t9)*5Cl0tl0)-5(9tl0)*SC10t9) IF CCOMM.Ea.o.) GO TO 160 PC9)=CSC10tl0l*RRC1>-S(9tl0l*RR(2))/COMM PC10)=C-SC10t9)~RRCl>+SC9t9)*RRC2))/COMM 160 DO 170 I=lt6 TPCI)=o.o DO 170 K=ltlO 170 TPCI)i=TPCil+HHCitK)*PCK> RRC1l=TPC2) RRC2>=TPC6) RRC.3)=0.0. RRf4) TP(3)+TP(5) I=lt4 SIGCI)aO. DO 180 K=l SIGCil=StG(l)-CfltK)*R~CK) SIG(3) SIGC4) 230

227 BETAICN) = e5*atan2<2e*sigc3ltsig(l)-sig(2)) BTAI = 57e296*BETAl(N) GAMAXCN> a SORT CCSTRAIN<N 2)+RRC2) STRAINCNtl) 1 RR(l)l**2+CSTRAINCNt3)+RRC4l>**2) IF CNUMITeLT.NPP> GO TO 300 C**** INCREMENT STRESSES C**** C**** STRESSCNtl) = STRESSCNtl)+SIG(l) STRESS<Nt2l = STRESS(Nt2)+SIG(2) STRESSCNt3) = STRESSCNt3)+SIGC3) STRAINCNtl) = STRAINCNtl)-RR(l) STRAINCNt2) = STRAINCNt2)-RR(2) STRAINCNt~) = STRAINCNt3)-RRC4) CC= e5*cstresscnt2)+5tresscntlll BB = e5*cstresscnt2)-stresscntl)) CR=SORT<BB*BB+ STRESSCNt3)**2> SlGC4)=CC+CR SIGC5)=CC-CR BETA = 5*ATAN2<STRESSfN,~)t-BB> SIGC6) = *BETA = ABS<BETA) CHECK STRENGTH IF fcr.lt.stregcmtype))go TO 190 MTYPE = NUMMAT-1 WRITE i6t2003)n CONTINUE IF (MPRINTl 110" WRITE (6t2000) MPRINT=50 MPRINT=MPRINT-1 STEP PORE PRESSURE TOP = TOEY IF CRRRC5l.GT.TOEX.ANO.RRR(5l.LT.TOPX) TOP TOEY+CRRRC5) 1 -TOEXl*CTOPY-TOEY)/(TOPX-TOEX) IF CRRRC5).GTeTOPX) TOP =TOPY GAMH = RO<MTYPE)*CTOP-ZZZC5)) BETTA= ABS<BETAICN)) ABAR=l.08 BB = Oe5*CSIGC2)-SIG(l)) CC. Oe5*( SIG( 2)+SIGC1 l J RR = SORTCBB*BB+SIGC3>**2) OSIG3 = CC-CRR OPORE = DSIG3+2e*ABAR*CRR POREPR(N) POREPRCN)+OPORE RU POREPR(N)/GAMH 231

228 305 WRITE l6t200lt NtRRRC5)tZZZl5)tCSTRESS<N t> I lt3)t 1 C SIG f J J, I 4, 6», CR, CST RAJ NC N, I ) I 1 '3 ) G'A AX t N ) t 2 CSJGClltl lt3jtabartoporetbtaftporeprcn> RU 300 CONTINUE Nl 1!20 RETURN 2000 FORMAT f 7HlEL.NO. 7X lhx 7X lhy 4X 8HX-STRESS 3X 1 SHY-STRESS 3X 9HXY-STRESS 3X 3HMAX 9X 3H~IN 4X 4HBETA 2 3X 6HRAOIU5 25H EX EY GAM GAMAXI 27X 6HOELTAX 3 5X 6HDELTAY SX 7HDELTAXY 5X 4HABAR 7X SHDPORE 3X 4 5HBETAI 2X 6HPOREPR 4X 2HRU /) FORMAT (J7t2F8.2t5Fll12~F7.2tF912tF713t2F613tF714/ 23X 2 5F1112tF712tF912tF713 > FORMAT C 17H FAILED ELEMENT 14) 2004 FORMAT C 23X t5fl 1e2 tf 7 2 t F9e2 tf7 e 3/56~ t 2Fll e2 t7x tf9 2 tf7 e3 t 2005 FORMAT Cl7t3F10e3) END 232

229 DISTRIBUTION LIST Address No. of Copies Offie, Chief of Engineers ATTN: ENGCW-ES Library Washington, D. C Eah Corps of Engineers Division ATTN: Chief, Geology, Soils, and Materials Branh, Engineering Division Eah Corps of Engineers Distrit ATTN: Chief, Foundation and Materials Branh, Engineering Division Diretor, Nulear Cratering Group U. S. Army Corps of Engineers ATTN: NCG-ES Lawrene Radiation Laboratory P. O. Box 808 Livermore, Calif Distrit Engineer U. S. Army Engineer Distrit, Jaksonville ATTN: SAJGI Jaksonville, Fla Mr. George E. Bertram 4701 Kenmore Avenue, Apartment 819 Alexandria, Va Mr. Stanley D. Wilson 1105 North 38th Street Seattle, Wash Prof. Arthur Casagrande Piere Hall, Harvard University Cambridge, Mass Mr. John Lowe III Tippetts-Abbett-MCarthy-Stratton 375 Park Avenue New York, N. Y Prof. Ralph B. Pek Department of Civil Engineering, University of Illinois Urbana, Ill ea 1 ea

230 Prof. Steve J. Poulos Piere Hall, Harvard University Cambridge, Mass Address Prof. Ronald C. Hirshfield Room Massahusetts Institute of Tehnology Cambridge, Mass Prof. H. Bolton Seed Department of Civil Engineering, University of California Berkeley, Calif Prof. James Mihael Dunan Department of Civil Engineering, University of California Berkeley, Calif Defense Doumentation Center ATTN: Mr. Meyer Kahn Gameron St-ation Alexandria, Va No. of Copies

231 Unlassified Se uritv ClaHUiatlon DOCUMENT CONTROL DAT A. R & D (S utlty l lllatjon ol title, body ol b trat and lndexln.,,nol llon mu t b nt t d when th overall r potl h l lll d) t. ORIGINATING ACTIVITY (Corpor.t author) za. REPORT SECURITY CLASSIFICATION College of Engineering, Offie of Researh Servies University of California Berkeley, California I. REPORT TITLI: FINITE ELEMENT ANALYSES OF SLOPES IN SOIL Unlassified Zb. GROUP. DlllCRIPTIVIE NOTES (Type ol Hpott Md lnt:lu I~ t ) Final report e. AU THOR(I> (l'ltet NUne, middle Initial, laat name) Peter Dunlop James M. Dunan H. Bolton Seed I fllllport DATC?a. TOTA'- NO. OP' PAGEi Mav CONTJllACT OR GRANT NO. N. ORIGINATOR I REPORT NUM"BIER(SJ DA CIVENG b. PROJl:C: T NO. TE b b. NO. 11 OF48 EFI.. O_T_HCA _Rl.PQ..AJ'_ NOl J--(Aft -~'--rnat1bon---fha-t--may--l;a~,..,,.d lhl repo J U. S. Army Engineer Waterways Exd. periment Station Contrat Report S l:>lltribution ITATEMIENT This doument has been approved for publi release and sale; its distribution is unlimited. 11 IUPPLIEMENTARY NOTEI 12. l~onioring MILITA"Y ACTIVITY Prepared under Contrat for U. s. Army Offie, Chief of Engineers Engineer Waterways Experilllent Station, CE, u. s. Army Viksburg, Miss. 11 Tll CTThe study desribed in this report was onduted to investigate the behavior of exavated slopes and embankments using the finite element method of analysis. The aspets of slope behavior onsidered inlude stresses, strains, displaements, pore pressures, and progressive development of failure zones. Analyses have been performed using nonlinear stress-strain harateristis as well as linear elasti stress-strain behavior. A method has been developed for simulating analytially the exavation of a slope in a lay layer with arbitrary initial stress onditions. Analyses performed using this proedure have shown that the behavior of an exavated slope depends on the initial horizontal stresses in the soil layer and the variation of strength with depth. Analyses performed using initial stresses and strength profiles representative of normally onsolidated lays indiate behavior whih is ompatible with the results of short-term ( = 0) stability analyses; the failure zones determined strongly suggest development of a urved surfae of sliding, and the fators of safety alulated by the = 0 method were lose to unity at the final stage of the analysis when the failure zone enompassed a large zone between the rest and the toe of the slope. Similar analyses performed using initial stresses and strength profiles representative of overonsolidated lays indiated quite different behavior; failure zones developed whih interseted the bottom of the exavations and the lower portions of the slopes while the fators of safety were still about 2. These failure zones did not resemble the usual urved failure surfaes used in equilibrium analyses of slope stability, but instead suggest the development of the ondition of "loss of ground" wherein the soil in the bottom of the exavation bulges up slowly as additional soil is exavated. Detailed onsideration was given to the stress-strain harateristis of soft, (Continued DD.'= ~L.A DO l'oltm t?a, t.iam M, WMICM 18 oaeoi. T ~011 - u Unlassified LCUrity 1 lhat1on