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Prediction of P-Y curves from finite element analyses She, Jairus Lai Yan 1986

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PREDICTION OF P-Y CURVES FROM FINITE ELEMENT ANALYSES By JAIRUS LAI YAN SHE B . A . S c , U n i v e r s i t y of B r i t i s h Columbia, 1983  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in THE FACULTY OF GRADUATE STUDIES Department of C i v i l  Engineering  We accept t h i s t h e s i s as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA November 1986 ©JAIRUS LAI YAN SHE, 1986  3E-6  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the  requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it  f r e e l y a v a i l a b l e f o r reference  and study.  I further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by t h e head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s .  It i s  understood t h a t copying o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain  s h a l l n o t be allowed without my  permission.  Department o f  Civil  Engineering  The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date  (3/81)  December 18, 1986  written  ABSTRACT  The p r e d i c t i o n o f P-Y curves f o r undrained c l a y and sand based on the r e s u l t s o f f i n i t e element  a n a l y s e s i s presented i n t h i s  A h i g h e r - o r d e r e d f i n i t e element a n a l y s e s . The a b i l i t y soil  program was used i n  the  o f the program to a c c u r a t e l y model the  c o n d i t i o n was v e r i f i e d  by comparing p r e d i c t e d  thesis.  undrained  load-deflection  responses with c l o s e d form s o l u t i o n s f o r the c y l i n d r i c a l  cavity  expansion problem. Pressuremeter curves were p r e d i c t e d from plane axisymmetric f i n i t e element  a n a l y s e s . The e f f e c t  strain  o f pressuremeter  size  on the p r e d i c t e d r e s u l t s was examined. P-Y curves were p r e d i c t e d f o r plane s t r a i n and plane conditions.  Values f o r the  initial  s l o p e and P n u  stress  o f the curves were  o b t a i n e d . The curves were n o r m a l i z e d f o r c o m p a r i s o n , and s i m p l i f i e d methods presented f o r d e t e r m i n i n g Finite loaded p i l e  element  P-Y c u r v e s .  p r e d i c t i o n s f o r the pressuremeter and  laterally  problems were a l s o compared. F a c t o r s were determined  these comparisons to generate  from  P-Y curves from pressuremeter c u r v e s .  TABLE OF CONTENTS  Page Abstract  i i  Table of Contents  i i i  List of Tables  vii  List of Figures  viii  Acknowledgements  xiii  Chapter 1  Introduction  1.1  Introduction  1  1.2  Scope of Thesis  4  1.3  Organization of Thesis  5  Chapter 2  Review of Previous Works  2.1  Introduction  7  2.2  Review of Previous Work  8  2.2.1  Empirical Method  8  2.2.2  Centrifuge Tests  10  2.2.3  Finite Element Method  11  Chapter 3  Finite Element Model for Laterally Loaded Pile Problem  3.1  Introduction  14  3.2  Finite Element Mesh  14  3.2.1  Validity of the Disk Concept  18  3.2.2  Mesh Radius  18  3.3  Plane Stress and Plane Strain Analyses  24  3.4  Soil-Pile Adhesion  25  i ii  Page Chapter 4  Interface Elements ~  4.1  Introduction  31  4.2  Properties of Interface Elements  35  4.2.1  Geometry  35  4.2.2  Shearing Behaviour  35  4.2.3  Soil-Pile Separation  38  4.2.4  Failure Criteria  40  Chapter 5  Finite Element Program  Chapter 6  Stress-Strain Relationship  6.1  6.2  41  Introduction  44  6.1.1  44  Stress-Strain Relationship  Plane Strain Condition  46  6.2.1  Stress-Strain Relations  46  6.2.2  Comparison of Finite Element Results  47  with Closed Form Solution 6.3  Plane Stress Condition  51  6.3.1  Stress-Strain Relations  51  6.3.2  Comparison of Finite Element Results  52  with Closed Form Solution Chapter 7  Cylindrical Cavity Expansion  7.1  Introduction  55  7.2  Cohesive Soil  58  7.2.1  Elastic-Plastic Closed Form Solution  58  7.2.2  Finite Element Predictions and Comparison  63  with Closed Form Solution 7.2.2.1 Boundary Conditions  i v  63  Page 7.3  Chapter 8  Cohesionless Soil  68  7.3.1  Elastic-Plastic Materials  68  7.3.2  Elastic-Plastic Closed Form Solution  69  7.3.3  Finite Element Prediction  72  Pressuremeter Expansion  8.1  Introduction  76  8.2  Finite Element Domain Analysed  77  8.3  Cohesive Soil  77  8.4  Cohesionless Soil  80  8.5  Size Effect  83  Chapter 9  Prediction of P-Y Curves  9.1  Introduction  87  9.2  Finite Element Mesh  87  9.3  P-Y Curves for Undrained Clay  89  9.3.1  Results  91  9.3.1.1 Comparison with Empirical P-Y Curves  94  9.3.2  Effect of Pile Diameter  95  9.3.3  Effect of Mesh Radius  96  9.4  Chapter 10  P-Y Curves for Sand  101  9.4.1  Results  103  9.4.1.1 Comparison with Empirical P-Y Curves  107  9.4.2  Effect of Pile Diameter  108  9.4.3  Effect of Mesh Radius  110  Simplified Method for Predicting P-Y Curves  10.1 Introduction  116  10.2 Simplified P-Y Curves for Undrained Clay  116  V  Page 10.2.1 Normalized P-Y Curves  117  10.2.2 Simplified Method for Determining P-Y Curves  117  10.3 Simplified P-Y Curves for Dense Sand 10.3.1 Normalized P-Y Curves  129  10.3.2 Simplified Method for Determining P-Y Curves  130  10.4 Application of the P-Y Curves Chapter 11  129  135  Prediction of P-Y Curves from Pressuremeter Expansion Curves  11.1 Introduction  142  11.2 Cohesive S o i l  143  11.2.1 Pressuremeter Expansion Curves  143  11.2.2 P-Y Curves  144  11.2.3 Comparison of Pressuremeter and P-Y Curves  144  11.2.3.1 Normalized Curves  144  11.2.3.2 Conversion Factors  147  11.3 Cohesionless S o i l  Chapter 12  149  11.3.1 Pressuremeter Expansion Curves  151  11.3.2 P-Y Curves  151  11.3.3 Comparison of Pressuremeter and P-Y Curves  151  11.3.3.1 Normalized Curves  151  11.3.3.2 Conversion Factors  154  Summary and Conclusions  Bibliography Appendix A  158 160  Derivation of Stress-Strain Relationships  164  for Uniaxial Loading Appendix B  Empirical P-Y Curves  vi  168  LIST OF TABLES  Table 3.1  Page Effect of Soil-Pile Adhesion on P - for ult Plane Strain Analysis of Undrained Clay  4.1  30  Proposed Coefficients of Skin Friction Between Soils and Construction Materials  39  6.1  Soil Parameters Used in Stress-Strain Analyses  48  7.1  Material Properties for Undrained Clay  59  7.2  Material Properties for Sand  73  8.1  Material Properties for Undrained N.C. Clay  78  8.2  Material Properties for Dense Sand  81  9.1  Soil Parameters for Undrained N.C. Clay  90  9.2  Soil Parameters for Dense Sand  102  9.3  Results of P-Y Curve Analyses for Dense Sand  106  9.4  Comparison of Theoretical and Predicted P -j_t Values  109  10.1  Simplified Method for Determining P-Y Curves for N.C. Clays 128  10.2  Simplified Method for Determining P-Y Curves for Dense Sand 138  11.1  Conversion Factors for Normally-Consolidated Clays  150  11.2  Conversion Factors for Dense Sand  157  vii  LIST OF FIGURES  Figure  Page  1.1  T y p i c a l P-Y Curve  3  3.1  Zone of S o i l - P i l e Interaction  15  3.2  F i n i t e Element Mesh for the L a t e r a l l y Loaded P i l e Problem  17  3.3  E f f e c t of Mesh Radius on Plane S t r a i n P-Y Curve Predictions  20  3.4  E f f e c t of Mesh Radius on Plane Stress P-Y Curve Predictions  21  3.5  2-D and 3-D Deformations Along the Loading Axis  23  3.6  Assumed F a i l u r e Mechanisms for L a t e r a l l y Loaded P i l e Problem  3.7  26  E f f e c t of S o i l - P i l e Adhesion on Plane S t r a i n P-Y Curve Predictions  3.8  27  E f f e c t of S o i l - P i l e Adhesion on Plane Stress P-Y Curve Predictions  4.1  28  Deformation of F i n i t e Element Mesh During L a t e r a l P i l e Loading  32  4.2  S o i l Pressure D i s t r i b u t i o n s Around L a t e r a l l y Loaded P i l e s  33  4.3  F i n i t e Element Mesh with Interface Elements  36  4.4  Stress-Strain Relationship for Shear Along the S o i l - P i l e Interface  37  5.1  Higher-Ordered Elements  42  6.1  Test Elements  45  6.2  Plane S t r a i n Stress-Strain Relationships for Undrained Clay  49  6.3  Plane S t r a i n Stress-Strain Relationships f o r Sand  50  6.4  Plane Stress Stress-Strain Relationships for Undrained Clay  53  6.5  Plane Stress Stress-Strain Relationships f o r Sand  54  viii  LIST OF FIGURES - Continued  Figure 7.1  Page Finite Element Mesh for Plane Strain Cavity Expansion Analysis  7.2  56  Finite Element Mesh for Plane Strain Axisymmetric Cavity Expansion Analysis  7.3  57  Closed Form Solutions for Cavity Expansion in Undrained Clay  62  7.A  Cavity Expansion Curves for Undrained Clay  64  7.5  Cavity Expansion in a Finite Medium for an Elastic-Plastic Undrained Clay  65  7.6  Variation of Radial Stress with Distance from Cavity  67  7.7  Stress Path for Failed Sand Element  70  7.8  Closed Form Solution for Cavity Expansion in Dense Sand  74  7.9  Cavity Expansion Curves for Dense Sand  75  8.1  Pressuremeter Curves for Non-Linear Elastic Normally-Consolidated Undrained Clay from Plane Strain Axisymmetric Finite Element Analyses  8.2  Pressuremeter Curves for Non-Linear Elastic Dense Sand from Plane Strain Axisymmetric Finite Element Analyses  8.3  84  Comparison of Pressuremeter Curves Predicted Using Different I n i t i a l Cavity Radii  9.1  82  Comparison of Pressuremeter Curves Predicted Using Different I n i t i a l Cavity Radii  8.4  79  85  Finite Element Mesh for Plane Strain or Plane Stress P-Y Curve Analysis  88  ix  LIST OF FIGURES - Continued  Figure  Page  9.2  P-Y Curves for Undrained Clay from Plane Strain Analyses  92  9.3  P-Y Curves for Undrained Clay from Plane Stress Analyses  93  9.A  Effect of Pile Diameter on Plane Strain P-Y Curve Predictions for Undrained Clay  9.5  Effect of Pile Diameter on Plane Stress P-Y Curve Predictions for Undrained Clay  9.6  98  Effect of Mesh Radius on Plane Strain P-Y Curve Predictions for Undrained Clay  9.7  97  99  Effect of Mesh Radius on Plane Stress P-Y Curve Predictions for Undrained Clay  100  9.8  P-Y Curves for Dense Sand from Plane Strain Analyses  104  9.9  P-Y Curves for Dense Sand from Plane Stress Analyses  105  9.10  Effect of Pile Diameter on Plane Strain P-Y Curve Predictions for Dense Sand  9.11  Effect of Pile Diameter on Plane Stress P-Y Curve Predictions for Dense Sand  9.12  114  Normalized P-Y Curve for Undrained Clay from Plane Strain Analyses  10.2  113  Effect of Mesh Radius on Plane Stress P-Y Curve Predictions for Dense Sand  10.1  112  Effect of Mesh Radius on Plane Strain P-Y Curve Predictions for Dense Sand  9.13  111  118  Normalized P-Y Curves for Undrained Clay from Plane Stress Analyses  119 x  LIST OF FIGURES - Continued  Figure  Page  10.3  Normalized P-Y Curves for Undrained Clay  120  10.4  Simplified Normalized P-Y Curves for Undrained Clay  122  10.5  Hyperbolic F i t for Plane Strain P-Y Curves for Undrained Clay  10.6  125  Power Function F i t for Plane Stress P-Y Curves for Undrained Clay  10.7  126  Hyperbolic F i t for Plane Stress P-Y Curves for Undrained Clay  10.8  127  Normalized P-Y Curves for Dense Sand from Plane S t r a i n Analyses  10.9  131  Normalized P-Y Curves for Dense Sand from Plane Stress Analyses  132  10.10  Normalized P-Y Curves for Dense Sand  133  10.11  Simplified Normalized P-Y Curves for Dense Sand  134  10.12  Power Function F i t s for Plane Strain P-Y Curves for Dense Sand  10.13  136  Power Function F i t s for Plane Stress P-Y Curves for Dense Sand  137  10.14  Plane Strain - Plane Stress Transition Zone for Dense Sand  140  10.15  P-Y Curve for 3-Dimensional  141  11.1  Comparison of Pressuremeter and P-Y Curves for  Stress and Strain Condition  Normally-Consolidated Undrained Clay 11.2  145  Comparison of Pressuremeter and P-Y Curves for Normally-Consolidated Undrained Clay  XT  146  LIST OF FIGURES - Continued  Figure 11.3  Page Comparison of Normalized Pressuremeter and P-Y Curves for Normally-Consolidated Undrained Clay  148  11.4  Comparison of Pressuremeter and P-Y Curves for Dense Sand  152  11.5  Comparison of Pressuremeter and P-Y Curves for Dense Sand  153  11.6  Comparison of Normalized Pressuremeter and P-Y Curves for Dense Sand  11.7  155  Comparison of Normalized Pressuremeter and P-Y Curves for Dense Sand  B.l  156  Matlock's Empirical P-Y Curve for Static Loading of Piles in Undrained Clay  169  B.2  Non-Dimensional Coefficients for Soil Resistance  171  B.3  Empirical P-Y Curves for Static Lateral Loading of Piles in Saturated Sand  172  xii  ACKNOWLEDGEMENTS  I thank my supervisor, Dr. P.M. Byrne, for his guidance and interest during this research. Thanks also to Dr. W.D.L. Finn and Dr. Y.P. Vaid, for their review of this manuscript and their helpful comments. My appreciations to my colleagues, U. Atukorala, H. Cheung (where are you?), F. Salgado, and H. Vaziri, for their assistance, and especially to C. Lum, for being my "Vancouver connection". Appreciation i s extended to Ms. E. Seymour-G., for the typing of this manuscript. Financial support provided by the Natural Sciences and Engineering Research Council i s acknowledged with deep appreciation. Special thanks to Maple, for her "distance" support and encouragement. Above a l l , I am grateful to the Lord, for His sovereign control in guiding me to the completion of this research.  xi i i  1  CHAPTER 1 INTRODUCTION  1.1  INTRODUCTION The problem of a p i l e subjected to l a t e r a l loads i s one which  requires the analysis of the i n t e r a c t i o n between s o i l and s t r u c t u r a l member. The behaviour of the s t r u c t u r a l member (the p i l e ) i s governed by i t s strength and s t i f f n e s s properties and those of the surrounding s o i l . Of prime concern are the bending moments, shear stresses, and displacements of the l a t e r a l l y loaded p i l e . The ultimate load i s generally determined by the maximum moment and shear stress that develop i n the p i l e , while the working load i s commonly governed by l a t e r a l displacements. The accurate determination of these quantities i s therefore essential  for p i l e foundation design.  The behaviour of a l a t e r a l l y loaded p i l e i s a three-dimensional problem. A complete analysis of the problem requires an examination of the complex s t r e s s - s t r a i n behaviour of the s o i l surrounding the p i l e . Near the ground surface, displacements of the s o i l are threedimensional. S o i l behind the p i l e may separate from the p i l e surface, leaving a gap. Stresses at these shallow depths are  essentially  two-dimensional. At depths greater than several p i l e diameters, stresses are three-dimensional and strains two-dimensional, with a l l displacements occurring i n the horizontal plane (Scott,  1981). At  intermediate depths, both stresses and strains are three-dimensional. To f a c i l i t a t e  the analysis of t h i s complex problem, simplified  s o i l models were considered. Depending on the s o i l model, methods for predicting the behaviour of l a t e r a l l y loaded p i l e s can be c l a s s i f i e d  2  i n t o t h r e e c a t e g o r i e s ( A t u k o r a l a & Byrne, 1984):  1. The Winkler Foundation a s e t o f independent  Approach: the s o i l i s r e p r e s e n t e d by  l i n e a r or non-linear s p r i n g s  d i s t r i b u t e d a l o n g the l e n g t h of the p i l e . 2. The E l a s t i c Continuum Approach: the s o i l i s i d e a l i z e d as a linear elastic,  i s o t r o p i c and homogeneous  continuum.  3. The F i n i t e Element Approach: the s u r r o u n d i n g s o i l i s d i s c r e t i z e d i n t o f i n i t e elements,  each p o s s e s s i n g the  s t r e s s - s t r a i n p r o p e r t i e s o f the s o i l .  At p r e s e n t , the Winkler approach i s most commonly used f o r a n a l y s i n g the response  o f a l a t e r a l l y loaded p i l e . With t h i s method,  the s o i l s u r r o u n d i n g the p i l e i s r e p l a c e d by d i s c r e t e s p r i n g s . S o i l r e s i s t a n c e t o p i l e d e f l e c t i o n i s r e p r e s e n t e d by the l o a d - d e f l e c t i o n c h a r a c t e r i s t i c s o f the s p r i n g s and a r e s p e c i f i e d by "P-Y" c u r v e s , where P i s the s o i l r e s i s t a n c e t o l a t e r a l p i l e displacement  per u n i t  l e n g t h of p i l e a t a g i v e n depth, and Y i s the c o r r e s p o n d i n g  horizontal  p i l e d e f l e c t i o n a t t h a t depth. A t y p i c a l P-Y curve i s shown i n F i g u r e 1.1. S i n c e P-Y c u r v e s a r e d e f i n e d by s o i l r e s i s t a n c e , they can be expected  t o vary f o r d i f f e r e n t s o i l p r o p e r t i e s . The s i z e and shape o f  the p i l e s e c t i o n , and the roughness o f the p i l e s u r f a c e can a l s o a f f e c t the P-Y  relationship.  The key t o the l a t e r a l l y loaded p i l e problem, then, l i e s i n the a c c u r a c y o f the P-Y c u r v e s . A s i d e from instrumented  p i l e loading f i e l d  t e s t s , which a r e both c o s t l y and time-consuming, methods f o r d e t e r m i n i n g P-Y c u r v e s based  on the r e s u l t s o f pressuremeter  expansion  LATERAL  FIGURE  1.1:  DEFLECTION  TYPICAL  P-Y  CURVE  Y  (FT)  4  t e s t s , c e n t r i f u g e t e s t s , and f i n i t e element  a n a l y s e s have been  proposed. A n a l y s i s o f the l a t e r a l l y loaded p i l e problem approach w i t h the use o f P-Y  by the W i n k l e r  c u r v e s can be performed  by the  finite  d i f f e r e n c e method. The method and i t s f o r m u l a t i o n a r e d e s c r i b e d by Focht & M c C l e l l a n d (1955). Computer programs employing were developed by Reese (1977) and Reese & S u l l i v a n  this  technique  (1980) t o perform  the a n a l y s e s . The use o f the programs are f a c i l i t a t e d  by the r e c e n t  i n t r o d u c t i o n o f i n t e r a c t i v e g r a p h i c s . A m o d i f i e d v e r s i o n o f Reese & S u l l i v a n ' s program (C0M624) w i t h g r a p h i c i n p u t and output  capabilities  i s c u r r e n t l y i n use a t the U n i v e r s i t y o f B r i t i s h Columbia  (Byrne &  Grigg,  1.2  1982).  SCOPE OF THESIS The  purpose  of the r e s e a r c h i s to p r e d i c t P-Y  l a t e r a l l y loaded p i l e s u s i n g f i n i t e element  relationships for  a n a l y s e s . P-Y  curves  p r e d i c t e d f o r both c o h e s i v e and c o h e s i o n l e s s s o i l s u s i n g n o n - l i n e a r e l a s t i c s o i l models a r e p r e s e n t e d . A r e c e n t l y - d e v e l o p e d h i g h e r - o r d e r e d two-dimensional element  program was  used i n the s t u d y . The  program was  finite  not t e s t e d f o r  the type o f a n a l y s e s performed  t o p r e d i c t P-Y  v e r i f y the a c c u r a c y of the P-Y  r e s u l t s , l o a d - d e f l e c t i o n responses f o r  u n i a x i a l compression  o f t e s t elements  c u r v e s . Consequently,  to  and c a v i t y expansion i n i n f i n i t e  e l a s t i c - p l a s t i c media were p r e d i c t e d and compared w i t h c l o s e d  form  solutions. L o a d - d e f l e c t i o n responses f o r pressuremeter p r e d i c t e d u s i n g the plane s t r a i n axisymmetric  expansion were a l s o  f i n i t e element  model  5  used i n t h e c a v i t y e x p a n s i o n a n a l y s e s . The e f f e c t o f p r e s s u r e m e t e r s i z e was examined. P-Y c u r v e s were determined f o r b o t h c o h e s i v e and c o h e s i o n l e s s s o i l s . The s e n s i t i v i t y o f t h e P-Y p r e d i c t i o n s t o v a r i o u s parameters i n the f i n i t e element method were a l s o examined. The p r e d i c t e d were n o r m a l i z e d  curves  w i t h r e s p e c t t o s o i l s t r e n g t h and p i l e s i z e , and  s i m p l i f i e d methods d e v i s e d t o g e n e r a t e P-Y c u r v e s from fundamental s o i l p r o p e r t i e s . F i n a l l y , P-Y p r e d i c t i o n s were compared w i t h p r e s s u r e m e t e r r e s u l t s t o determine f a c t o r s f o r c o n v e r t i n g p r e s s u r e m e t e r c u r v e s t o P-Y  1.3  curves.  ORGANIZATION OF THE THESIS T h i s t h e s i s c o n s i s t s of twelve chapters. A b r i e f review o f  p r e v i o u s r e s e a r c h , h i g h l i g h t i n g t h e methods and t h e r e s u l t s , i s g i v e n i n Chapter 2. Chapter 3 c o n t a i n s a d i s c u s s i o n o f t h e f o r m u l a t i o n o f t h e f i n i t e element model f o r a n a l y s i n g t h e l a t e r a l l y l o a d e d p i l e problem t o p r e d i c t P-Y c u r v e s . The v a l i d i t y o f t h e f o r m u l a t i o n i s c o n s i d e r e d i n l i g h t o f p r e v i o u s r e s e a r c h on t h e problem. The importance o f i n t e r f a c e elements i s d i s c u s s e d i n Chapter 4. R e s u l t s o f p r e v i o u s work i n v o l v i n g t h e use o f t h e s e s p e c i a l elements t o model t h e behaviour  of the s o i l - p i l e i n t e r f a c e are presented. A  s i m p l i f i e d f o r m u l a t i o n f o r t h e i n t e r f a c e elements used i n t h i s is  study  given. The f i n i t e element program used i n t h e r e s e a r c h i s d i s c u s s e d  b r i e f l y i n Chapter 5. R e s u l t s o f u n i a x i a l c o m p r e s s i o n and c a v i t y e x p a n s i o n a n a l y s e s performed t o v e r i f y t h e c a p a b i l i t i e s o f t h e program  6  are presented i n Chapters 6 and 7. Chapter 8 deals with the pressuremeter problem. Analyses were performed by modelling pressuremeter expansion as an axisymmetric cylindrical cavity expansion problem. The P-Y curve problem i s considered i n Chapters 9 to 11. Plane strain and plane stress P-Y curves were predicted for both undrained clay and sand. The effects of pile diameter and mesh size on the predicted P-Y responses were examined. Normalized P-Y curves based on the results of Chapter 9 are shown in Chapter 10. Simplified methods for determining P-Y curves were derived from the normalized curves. In Chapter 11, predicted P-Y curves are compared with pressuremeter curves determined i n Chapter 8. Factors for converting pressuremeter curves to P-Y curves were determined. A summary of the research and the conclusions i s presented in Chapter 12.  7  CHAPTER 2 REVIEW OF PREVIOUS WORKS  2.1  INTRODUCTION The prediction of P-Y curves for the design of laterally loaded  piles has been the subject of much research over the past 10 or 15 years. With the advent of offshore structures for o i l exploration and recovery, and the increasing importance of seismic design for foundations, the laterally loaded pile problem was brought to the foreground of research. Indeed, the interest of o i l companies has led to their funding of much of the research. An annual conference, the Offshore Technology Conference, now in i t s eighteenth year, was established for the exchange of information related to offshore design and construction. Proceedings of the annual conferences f i l l many volumes, a sizeable portion of which deals with offshore piling problems. The successful design of pile foundations subjected to lateral forces, whether they be ice, wave, wind, or seismic, i s contingent on the accuracy of P-Y relationships describing the resistances of foundation soils to lateral pile displacements. Methods for predicting the P-Y curves, based on empirical, mathematical, and analytical solutions, were proposed by various authors. A brief review of these methods and their results i s given in Section 2.2. The finite element method of analysis i s discussed i n greater detail in Chapter 3.  8  2.2  REVIEW OF PREVIOUS WORK  2.2.1  Empirical Method Of the various methods developed to predict P-Y curves, the  empirical approach i s the most widely used in industry. Empirical curves based on P-Y relationships derived from instrumented full-scale pile load tests were developed by Matlock (1970) for soft clays, Reese & Welch (1975) and Reese et a l . (1975) for s t i f f clays, and Reese et a l . (1974) for sands. Though simple to use, these methods require estimates of the ultimate lateral soil resistances, ? ^> and u  reference strain values,^^Q, corresponding to one-half of the maximum deviator stresses. Values forcan  be obtained from  laboratory stress-strain curves, or estimated from tables of representative values i f no stress-strain curves are available (Reese & Sullivan, 1980, and Reese et a l . , 1975). Values of P ^ are calculated from equations derived by Matlock (1970) and Reese et a l . (1974, 1975), assuming passive wedge-type failure near the ground surface and failure by lateral soil flow around the pile at greater depths. Matlock's equation for clay i s  N P  = N cD P 3 + _c_y T" + J H , _  c  undrained strength  D  pile diameter  c  D  2.1 3£ N ^ 9 P  where  overburden effective stress at depth H  9  J  = coefficient ranging from 0.25 to 0.5, depending on the s o i l . A value of 0.5 i s applicable for the soft offshore clay of the Gulf of Mexico, and 0.25 i s valid for stiffer clays.  For sand, the theoretical ultimate lateral resistance for wedge-type failure at shallow depths i s given by  P  ct  = JfH K H Q  taiytf' sin/3 +  tan^-jzf') cos.*  tan/?  (D + H t a n ^ tanot)  tan^-jzf') 2.2  + K H tan/? (tan;*' s i n ^ - tan*) - K^D Q  And for lateral flow at greater depths,  2.3  P . = fHD [K ( t a n J S - l ) + K tan/f' tan*] 3.  CO.  where  O  / = effective unit weight of the sand K  = coefficient of lateral pressure at rest  q  K  = coefficient of active lateral pressure = tan (45° - ^ 12) 2  j6 = internal friction angle of the sand y  &  = 45° + rf /2 f  Agreement between the theoretical P 6  c  values and the values obtained  from the pile load tests was poor, and consequently, P by a factor A according to  £  was adjusted  10  = A P  2.4  c  Values for the adjustment factor, A, were determined by dividing the experimental ultimate resistances by the theoretical P  c  values. The  values for A are shown in Figure B.2 in Appendix B.  2.2.2  Centrifuge Tests The laterally loaded pile problem was also studied under  controlled laboratory conditions. Centrifuge tests on model pipe piles driven in saturated sand were conducted by Barton et a l . (1983). Modelling a prototype pile with a diameter of 25 inches, bending moments (M) were measured at points along the length of the model pile subjected to lateral loads at the pile head. Cubic spline interpolatory functions were fitted to the data and double integrations and differentiations performed to obtain values for Y and P. The mathematical relationships for Y and P are  2.5 and  P  2.6  where EI = stiffness of the pile z  = depth  The accuracy of P values determined according to Equation 2.6 i s questionable. Derivatives of the cubic spline function are very sensitive to the curve shape and the errors are greatly multiplied by the double differentiation. Consequently, considerable errors may  n  e x i s t i n the P-Y  c u r v e s developed by t h i s method.  The r e s u l t s o f the c e n t r i f u g e t e s t s suggest t h a t P -j_ determined  from Reese's e q u a t i o n s (2.2 t o 2.4)  t  overestimate  values soil  r e s i s t a n c e s a t l a r g e depths and underestimate r e s i s t a n c e s near the ground  surface. P  underestimated by f a c t o r s of a p p r o x i m a t e l y  1.9,  1.6  P ^  were not o b t a i n e d from the c e n t r i f u g e t e s t s f o r depths g r e a t e r  u  t  and 1.1  was  a t depths of 2, 4 and 6 f e e t , r e s p e c t i v e l y . V a l u e s of  than 6 f t . The  shape of the c e n t r i f u g e P-Y  curves a l s o d i f f e r s  from the shape o f Reese's e m p i r i c a l c u r v e s . The i n i t i a l  markedly  s l o p e s o f the  e m p i r i c a l c u r v e s are much s t e e p e r than those of the c e n t r i f u g e c u r v e s , and the c u r v e s f l a t t e n out much q u i c k e r than the c e n t r i f u g e p r e d i c t i o n s a t s h a l l o w depths. O v e r a l l , t h e r e i s l i t t l e between the P-Y  2.2.3  c u r v e s p r e d i c t e d by the two  F i n i t e Element  methods.  Method  The f i n i t e element r e s e a r c h e r s t o p r e d i c t P-Y s t r e s s f i n i t e element  resemblance  method o f a n a l y s i s was  used by a number of  c u r v e s f o r sand and undrained c l a y . P l a n e  f o r m u l a t i o n s were used t o a n a l y s e the  problem  f o r s h a l l o w depths w h i l e plane s t r a i n f o r m u l a t i o n s were used f o r g r e a t e r depths. The models and f o r m u l a t i o n s used by the r e s e a r c h e r s a r e s i m i l a r and a r e d e s c r i b e d i n Chapter P-Y  3.  c u r v e s f o r undrained c l a y were p r e d i c t e d by Yegian & Wright  (1973), Thompson (1977), and A t u k o r a l a & Byrne (1984). A wide range o f values f o r P 2 u  t  were o b t a i n e d by the r e s e a r c h e r s .  U s i n g i n t e r f a c e elements  t o model s o i l - p i l e  interface  b e h a v i o u r , and assuming a s o i l - p i l e adhesion f a c t o r , f , o f  0.3  (  c a  = ^  c c  u  '  s  e  e  Section 4.2.2), a P ^ value of approximately 12cD was u  t  obtained by Yegian for plane strain analysis. Similarly, a value of 6.6cD was determined from plane stress analyses. Thompson, following the work of Yegian, performed P-Y analyses for a wide range of soil-pile interface conditions. P ^ values u  t  ranging from about 6cD for complete soil-pile separation behind the pile to llcD for no separation were obtained for plane strain analysis. Likewise, values ranging from 3.1cD to 6.1cD were determined from plane stress analyses. Neither Thompson nor Yegian made any conclusions regarding the i n i t i a l slopes and shapes of the P-Y relationships. P-Y curves for sand were also predicted by Barton et a l . (1983) using the finite element method. The predicted curves were compared with experimental curves from centrifuge tests (see Section 2.2.2). Good agreement exists between the computed curves and centrifuge curves at shallow depths. For depths exceeding 3 f t , however, the finite element predictions were considerably less s t i f f than the centrifuge curves. No values were determined for P ^ > nor was any u  t  conclusion drawn regarding the i n i t i a l slopes of the P-Y curves. Plane strain P-Y analyses performed by Atukorala produced results similar to those of the other researchers. Matlock's empirical curves for soft clay were shown to underestimate P , w h i l e Reese's ult curves for sand drastically overestimate P ^ « perhaps by as much as u  t  6 times. Results of research conducted by the various authors yielded one common observation: Matlock's and Reese's empirical curves for soft clay and saturated sand do not agree with finite element  predictions. Large discrepancies exist in the i n i t i a l slope, shape, and ultimate s o i l resistance of the P-Y relationships predicted by the two methods. Since the empirical curves were developed from limited pile load tests, their applicability for soils other than those in which the tests were conducted i s questionable. On the other hand, many factors that could affect the predicted results (ie: mesh size, boundary conditions, pile diameter, interface properties, s o i l disturbances) were not considered in the finite element analyses. Consequently, the validity of the numerical P-Y curves i s also in doubt. The ultimate proof of the validity of the finite element predictions lies in their ability to predict field data. Bending moments, shear stresses, and deflections of piles determined by using finite element P-Y curves in conjunction with finite difference programs such as C0M624 (see Section 1.1) can be compared with results obtained from field pile load tests. Reasonable agreement between predicted and actual values serves to validate the finite element approach to P-Y prediction. L i t t l e work, however, has been done in this respect. Much additional research i s warranted to fully study the laterally loaded pile problem.  CHAPTER 3 FINITE ELEMENT MODEL FOR LATERALLY LOADED PILE PROBLEM  3.1  INTRODUCTION The prediction of P-Y curves for single laterally loaded piles  from finite element analyses has received much attention in recent years. Previous studies of the problem were conducted by Yegian & Wright (1973), Thompson (1977), Barton & Finn (1983), and Atukorala & Byrne (1984). A review of their works i s contained in Chapter 2. The methods of analysis and the finite element models used by the researchers are similar. An overview of the finite element formulation i s given in the following sections.  3.2  FINITE ELEMENT MESH The finite element method for predicting P-Y curves requires  the analysis of the pile and the surrounding s o i l . A horizontal cross-section of unit thickness is taken of the pile and s o i l as shown in Figure 3.1. At a sufficiently large distance, R, away from the pile, the soil i s generally assumed to be unaffected by the pile in terms of displacements. A displacement boundary can then be inserted at this location and the outlying soil eliminated from further consideration. The selection of the correct value of R, however, i s of importance and i s discussed in Section 3.2.2. The resulting finite element mesh i s a circular disk with the pile located at the centre. The outer boundary of the disk i s fixed, assuming zero displacements. Concentrated loads (P), representing lateral forces on the  Limit  of  Pile  Influence  Section  FIGURE 3 . 1 :  ZONE OF SOIL-PILE INTERACTION  (After Yegian & Wright, 1973, p. 673.)  16  pile, are applied to the pile centre. Pile deflections (Y) resulting from the applied loads produce the desired P-Y curves. Strictly speaking, P i s the s o i l resistance per unit length of a pile subjected to a lateral displacement of Y. For the purpose of the f i n i t e element model, however, i t i s more convenient to consider P as the applied load. In any event, the two quantities are equivalent under an equilibrium load-deflection condition. In a l l of the analyses, piles were assumed to be rigid. Accordingly, elements representing the piles were made 500 times stiffer than the surrounding s o i l elements to prevent significant pile deformations. Since loads are applied to the pile along an axis of symmetry, only half of the mesh needs to be analysed, as illustrated in Figure 3.2. Rollers were placed along the symmetry boundary to ensure zero displacements perpendicular to the direction of loading. In addition to the symmetry boundary, a line of anti-symmetry also exists, but only under the condition that stresses i n the s o i l must not approach levels where tensile failure occurs and causes the s o i l to separate from the pile (Yegian & Wright, 1973). The use of this axis of anti-symmetry permits just one quadrant of the disk to be analysed for the problem. However, the required condition of no soil-pile separation may not be valid for large lateral loads or for analyses of pile sections at shallow depths (see also Sections 3.4 and 4.2.3). Consequently, the boundary of anti-symmetry was not considered and a half-disk was used i n the analyses (Figure 3.2). In analysing only half of the disk, the horizontal load for a corresponding lateral displacement, Y, must be doubled to account for s o i l resistance on the  Y  FIGURE 3.2:  FINITE ELEMENT MESH FOR THE LATERALLY LOADED PILE PROBLEM  omitted half of the pile section.  3.2.1  Validity of the Disk Concept In using a disk with fixed outer boundary to represent the s o i l  in two-dimensional finite element analysis, a finite zone of influence is assumed. In reality, the boundary of this zone is at i n f i n i t y . In three-dimensional analysis where vertical load-spreading and s o i l displacements are possible, the boundary can be moved in from infinity to some finite radius without significant error. But, i f the s o i l i s replaced by uncoupled disks, then pile displacements under lateral loads depend on the size of the disks. The problem then rests in the determination of the appropriate disk radius R to yield the correct pile displacements. In other words, errors introduced by using uncoupled disks can be compensated for by selecting an appropriate disk radius.  3.2.2  Mesh Radius Based on comparisons of actual P-Y curves from field load tests  and P-Y curves predicted using various values for R, a value of R = 8D (D = pile diameter) was determined by Yegian & Wright (1973). Thompson (1977), following the work of Yegian, concluded that zero lateral s o i l displacements beyond 20D, or about half the pile length, would be appropriate. Recent studies by Atukorala & Byrne (1984) attempted to model an outer boundary at infinity, using a disk radius of 20D with "infinity springs" as described by Byrne & Grigg (1980). The use of "infinity springs" for the laterally loaded pile problem i s incorrect, however, since s o i l i s displaced laterally rather than radially.  In his research, Thompson noted that the use of different mesh radii did not affect the predicted value of P ^ » but did affect the u  t  i n i t i a l slope of the P-Y curve. Increases in mesh radius resulted in decreases in the slope, as illustrated in Figures 3.3 and 3.4. It i s apparent from Thompson's results that as R tends to infinity, the slope of the P-Y curve approaches zero. The results of Thompson's research on the effects of varying the mesh radius are supported by theoretical analyses. Baguelin et a l (1977) examined the lateral reaction of piles in an elastic-plastic medium, assuming plane strain condition and perfect soil-pile adhesion. In this two-dimensional study using a rigid circular pile section and a fixed outside boundary at radius R from the pile centre pile displacement (Y) i s given by  Y =  P  1 + u  (3 - 4u) ln  8?rE 1 - u  •JI  R  J2  R  - r2 .  + r  2  3 - 4ju  where P  lateral force (per unit length) on pile Poisson's ratio  r  radius of pile = D/2  Clearly, displacement depends on R, and tends to infinity as R tends to infinity. The results given by Equation 3.1, though valid for a two-dimensional problem, are unrealistic for actual pile behaviour. A three-dimensional study was therefore conducted by Baguelin et a l . to determine the value of R for the two-dimensional model that will give  0 l 0  1  1  1  1  2  4  6  8  JL  FIGURE 3.3:  EFFECT OF MESH RADIUS ON PLANE STRAIN P-Y CURVE PREDICTIONS  (After Thompson, 1977, p.  172.)  10  FIGURE 3.4:  No S e p a r a t i o n Plane S t r e s s  EFFECT OF MESH RADIUS ON PLANE STRESS P-Y CURVE PREDICTIONS  (After Thompson, 1977, p.  173)  22  displacements representative of actual pile behaviour. In this study, the s o i l surrounding the pile was divided into two zones. For soil within a radius r , plane strain condition was a v  assumed. Beyond r , soil behaviour i s three-dimensional. Deformations along the plane of loading for the two-dimensional model (outer radius R) and three-dimensional model were compared. The value of R was chosen to give equal two-dimensional and three-dimensional displacements at the boundary r , as illustrated in Figure 3.5. Values of R thus derived for piles with free heads subjected to horizontal loads at the top are: For flexible piles (h/l > 7/3): R = 71 o o For rigid piles (h/l < 7/3): R = 3h  3.2  where h  = embedded length  1 o  = soil-pile stiffness factor = 4E I /E p p so  E  = pile modulus  I  P P  = moment of inertia for pile section v  = l/4jrr E  so  for circular pile  = i n i t i a l s o i l modulus  Equation 3.2 was used to determine the mesh radius for the P-Y curve finite element analyses. Although the equations were derived for an elastic-plastic medium and assumed perfect soil-pile adhesion, they are, nonetheless, valid for the i n i t i a l elastic behaviour of "real" soils prior to soil-pile separation. Consequently, the i n i t i a l portion of the P-Y responses can be predicted with accuracy. Moreover, as  FIGURE 3.5:  2-D AND 3-D DEFORMATIONS ALONG THE LOADING AXIS (After Baguelin et a l . , 1977, p.  424) l\3 CO  shown by Thompson, the mesh radius does not affect the value of P ]_l-» u  a n  d  n a s  only  a  moderate influence on the shape of the P-Y  curve (see Figures 3.3 and 3.4). Overall, in the absence of soil-pile separation, f a i r l y accurate P-Y curves should be predicted by using values of R determined from Equation 3.2. Where soil-pile separation does occur, softer P-Y responses and lower values of P ^ can be expected.  3.3  PLANE STRESS AND PLANE STRAIN ANALYSES In using two-dimensional finite element models to predict P-Y  curves, a plane strain formulation was used for analyses at large depths, and a plane stress formulation for analyses at shallow depths. At large depths, the plane strain assumption i s justifiable on the basis that pressures from soil above and below are sufficiently large to prevent vertical displacements. Consequently, displacements are restricted to the horizontal plane with soil flowing around the pile as the pile i s displaced laterally under loads. For analyses at shallow depths, the plane strain formulation i s invalid. At the ground surface, vertical stress i s zero and displacements are three-dimensional. Consequently, the two-dimensional plane stress formulation i s appropriate. A transition zone, consisting of three-dimensional stresses and strains, exists between the plane stress condition at the surface and the plane strain condition at greater depths. Reese (1958) used a passive wedge failure condition to estimate P^ u  t  near the surface of a saturated clay. For P  at large  depths, a block flow model was used. These failure conditions are  illustrated in Figure 3.6. A similar method was used to determine P 2_£ for sand (Reese et a l . , 1974). u  Based on Reese's results, Thompson (1977) determined that the depth at which plane strain becomes applicable for saturated clay i s between 1.5 and 3.0 pile diameters, depending on the pile roughness and soil-pile adhesion. Thompson further concluded that the transition from plane stress to plane strain i s gradual and may be approximated by a linear combination of the responses produced by the two deformation conditions.  3.4  SOIL-PILE ADHESION As mentioned in Section 3.2, soil-pile adhesion affects the  displacement and failure characteristics of the laterally-loaded pile and s o i l system. The ultimate soil resistance, ^ ^ , I  s  also a  function of the degree of soil-pile adhesion. Figures 3.7 and 3.8 shows the results obtained by Thompson (1977) for plane stress and plane strain conditions. Using various values of (T/c (ratio of i n i t i a l horizontal stress <T to undrained shear strength c) to represent different degrees of insitu s o i l confinement or soil depths, normalized P-Y curves were predicted for saturated clays. Using a constant value for c, results for adhesion conditions ranging from complete soil-pile separation at zero depth (O*=0) to no separation at large depths were obtained. Separation was assumed when stress changes (decreases) behind the pile exceeded the i n i t i a l confining stress <T. Increases in P 2 with increasing adhesion are u  t  shown by the graphs. Randolph & Houlsby (1984), using plasticity theory, presented  Direction of Pile Movement  )  <?4  • \  •  f 2 \  \  /  s  *  •  x-  '///A//'  /  t  \ Movement  Mud line  /  a)  Lood  ^ Pile of Diometer b  Passive Wedge  FIGURE 3.6:  ASSUMED FAILURE MECHANISMS FOR LATERALLY LOADED PILE PROBLEM  (After Reese et a l . , 1974, p. 481)  ro cn  28  a  FIGURE 3.8:  EFFECT OF SOIL-PILE ADHESION ON PLANE STRESS P-Y CURVE PREDICTIONS  (After Thompson, 1977, p. 81)  solutions for P .. for various values of oc, the coefficient of ult adhesion (ie: c  g  = oic as discussed in Section 4.2.3). Upper and  lower bound solutions determined were shown to be identical, thus indicating an exact solution. Their results for the plane strain deformation of an undrained cohesive s o i l are presented in Table 3.1. Again, a clear trend of increasing P ^  t  with increasing adhesion  ( i e : o c ) i s indicated.  In the finite element analyses performed to predict P-Y curves in Chapters 9 to 11, interface elements were used to model the soil-pile adhesion characteristics of the problem. The interface elements and their properties are discussed in Chapter 4.  30  TABLE 3.1 EFFECT OF SOIL-PILE ADHESION ON P "|t FOR PLANE STRAIN ANALYSIS OF UNDRAINED CLAY u  a 0-0 0-1 0-2 0-3 0-4 0-5 0-6 0-7 0-8 0-9 1-0  P  u 1 t  /cd  9-142 9-527 9-886 10-220 10-531 10-820 11-088 11-336 11-563 11-767 11-940  (After Randolph & Houlsby, 1984, p. 617)  31  CHAPTER 4 INTERFACE ELEMENTS  4.1  INTRODUCTION To properly model the lateral movement of a pile section  through the s o i l , elements exhibiting the appropriate soil-pile interface behaviour are needed. Figure 4.1 illustrates diagrammatically the response of the s o i l to lateral pile displacement. In the absence of interface elements, the soil i s bound to the pile surface (Figure 4.1b). Movement of the pile forces the surrounding s o i l to deform, inducing large shear stresses along the side of the pile and tension stresses behind the pile. The presence of large shear stresses, however, will likely cause slippages along the interface. Tensile stresses may result in the development of a cavity behind the pile, or gapping. If interface slippages and gapping are neglected i n developing P-Y relationships, s o i l resistances and the stiffness of the load-deflection responses may be over-predicted. A better model of the soil-pile interaction allows for s o i l movements at the interface. As the pile i s displaced laterally, high stresses develop in front of the pile while stress reductions occur at the back (Reese et a l . , 1974). This situation i s illustrated in Figure 4.2. At large depths where the i n i t i a l confining pressure i s high (Figure 4.2b), tension stresses will not develop during loading. The resulting stress distribution may be as shown i n Figure 4.2c. Under these circumstances, soil adjacent to the pile flows around the pile from front to back with no separation taking place (Randolph & Houlsby, 1984, and Yegian & Wright, 1973). At shallower depths,  a)  F i n i t e Element Mesh P r i o r to Loading  FIGURE 4.1:  b)  Mesh During L o a d i n g : S o i l Bound to P i l e S u r f a c e Large S o i l Deformation and Resi stance  c)  Mesh During L o a d i n g : S o i l Allowed to Separate and Shear Along Side o f P i l e - Less S o i l Deformation and R e s i s t a n c e  DEFORMATION OF FINITE ELEMENT MESH DURING LATERAL PILE LOADING  A a)  Laterally  Loaded  Pile  Ii,  SECTION A-A  b)  S o i l Pressure D i s t r i b u t i o n P r i o r to Loading (Assuming Perfect P i l e Installation)  FIGURE 4 . 2 :  c)  S o i l Pressure D i s t r i b u t i o n During L a t e r a l Loading - No Tension S t r e s s Development  d)  S o i l Pressure D i s t r i b u t i o n During L a t e r a l Loading With Development o f T e n s i o n S t r e s s e s Behind P i l e  SOIL PRESSURE DISTRIBUTIONS AROUND LATERALLY LOADED PILES  (Adapted  from Reese et  al.,  1974,  p.  481)  however, the development of tension stresses behind the pile i s possible (Figure 4.2d). Subsequent failure of the s o i l in tension leads to the formation of a cavity as illustrated in Figure 4.1c. In this situation, s o i l flows around the front of the pile and separates from the pile at some point along the back (Pyke & Beikae, 1984). A proper representation of the interface behaviour requires the use of special elements. These elements must allow the s o i l to shear along the surface of the pile i f the pile skin friction i s exceeded. There must also be soil-pile separation i f sufficiently large tension stresses develop during loading. Interface, or s l i p , elements were developed by Goodman et a l . (1968) to model the behaviour of jointed rock masses. These elements were subsequently adapted for use in s o i l mechanics. Yegian & Wright (1973) employed curved interface elements in their f i n i t e element analysis of the laterally loaded pile problem. Interface properties were shown to have a noticeable effect on the predicted value of the ultimate s o i l resistance. More recently, the curved interface elements were used by Thompson (1977) in developing P-Y curves for saturated clays (see Figures 3.7 and 3.8). Results similar to those of Yegian were obtained. As a simple alternative to the interface elements developed by Yegian, normal elements with special modulus properties were used. The properties and their formulations are described in the following sections.  35  4.2  PROPERTIES OF INTERFACE ELEMENTS  4.2.1  Geometry In the finite element mesh, the interface s o i l was represented  by a thin ring of elements encompassing the pile. The elements were given a thickness of 0.005D (D = pile diameter) as indicated in Figure 4.3. Instability problems were not encountered in using the thin interface elements despite their high aspect ratio of about 39.  4.2.2  Shearing Behaviour A bi-linear model, shown in Figure 4.4, was used to describe  the shear stress-shear strain relationship at the interface. The shear modulus of the s o i l , G, remains constant during shearing until C , s the maximum allowable shear stress, i s reached. The value of G i s determined from the i n i t i a l elastic and bulk modulii of the s o i l . The value of C  g  i s a function of the properties of the soil and pile  surface, reflecting the maximum skin friction that can develop during loading. The corresponding strain at C  g  i s given by 2^. For  strains beyond / , the s o i l deforms at constant stress. Consequently, G=0. A "zero" value, however, cannot be assigned to G in practice due to instability problems within the finite element program associated with the stress-strain matrix [D]. To maintain stability, the shear modulus at failure was defaulted to 0.001  of i t s i n i t i a l  value. In general, the strength of a soil i s characterized by c and 0", the cohesion and internal friction angle. Similarly, the strength of the soil-pile interface can be represented by the adhesion, c , and  NOT TO  SCALE  DIRECTION OF LOADING  INTERFACE  ELEMENTS  FIGURE 4.3:  FINITE ELEMENT MESH WITH INTERFACE ELEMENTS  CO  FIGURE 4.4:  STRESS-STRAIN RELATIONSHIP FOR SHEAR ALONG THE SOIL-PILE INTERFACE  the friction angle, 8. Using these parameters, the maximum skin friction was determined based on the Mohr-Coulomb failure criterion:  2r  s  = 2c  4.1  cosS + 2 (T sinS 0  1 - sinS  where (T^ i s the minor principal stress. Potyondy (1961) has shown that c  3.  fractions of c and  c  and  S  a  and S can be expressed as  respectively. In general,  = etc  4.2  =/8jf  4.3  Experimental values of oi and /8 determined by Potyondy for various materials under different testing conditions are given in Table 4.1. Based on these recommended values, oi = /S = 0.50 was selected for use with rough steel pile surfaces in clay, and /S = 0.80 for piles in sand.  4.2.3  Soil-Pile Separation The second consideration of interface behaviour i s soil-pile  separation. At shallow depths where the i n i t i a l confining stresses are low, negative stresses may develop behind the pile during loading, resulting in the formation of cavities. Although cohesive soils may be subjected to small tension stresses without failure, with the magnitudes of the stresses limited possibly by the s o i l cohesion (c) or the soil-pile adhesion (c ),  TABLE 4.1 Proposod eoaffldanta of akin hlotluii bttwm •oils and construction materials [/<i->M. A - j . / c " - ^ ^ ; without (actor of safety] 0  Construction material  Sand  Cohesive soil  Cohesionless silt  Clay  gTanular  <H&<D<  2-0 mm  Surface finish of construction material  Dry  Sat. Dense  /* Steel  -j^  Wood  |  50% Clay + 5 0 % Sand  Sat.  Consist. I. = 10-0 5  Dry Dense  Loose  Dense  /*  J*  J*  £><0 08 mm Consist. Index: l-O-fl-73  f*  A  / mu c  Smooth  Polished  0-54  0-64  0-79  0-40  0-68  0-40  —  0-50  0-25  0-50  Rough  Rusted  0-76  0-80  0-95  0-48  0-75  0-65  0-35  0-50  0-50  0-80  0-76  085  0-92  0-55  0-87  0-80  0-20  0-60  0-4  0-85  Parallel to grain At right angles to grain Smooth  Concrete - Grained Rough  Note:  /*  0-002<D<0-06  f  c  0-88  0-89  098  063  0-95  0-90  0-40  070  0-50  0-85  Made in iron form  0-76  0-80  0-92  0-50  0-87  0-84  0 42  0-68  0 40  1O0  Made in wood form  0-88  0-88  0-98  0-62  0-96  0-90  0-58  080  0-50  1-00  0-98  0-90  1-00  0-79  1-00  0-95  0-80  095  0-60  1-00  Made on adjusted ground  5 oc  (After Potyondy, 1961, p. 352)  the stresses likely cannot be sustained for static loadings. Consequently, as a somewhat conservative measure, soil-pile separation was allowed whenever negative stresses developed in the interface elements. To model the possible formation of cavities behind the piles, both the shear and the bulk modulii were reduced by a factor of 1000 upon tension failure. The low shear modulus prevents any further significant changes i n shear stress while the low bulk modulus allows large volume changes to occur.  4.2.4  Failure Criteria To achieve the desired behaviour of the interface elements, the  above criteria were used to define s o i l failure. The interface elements were considered to have failed whenever the maximum shear stress, given by 0^/2, exceeded the skin friction, C , or whenever g  the minor principal stress, (T^', became negative. Upon shear failure, the shear modulus was reduced to 0.001 of i t s i n i t i a l value. Upon tension failure, both the shear and the bulk modulii were reduced to 0.001 of their i n i t i a l values. The low modulii allow large shear deformations and volume changes to occur to model both the shearing of s o i l along the pile surface and the development of a tension cavity behind the pile.  CHAPTER 5 FINITE ELEMENT PROGRAM  A new higher-ordered finite element program was used in the analyses of the cavity expansion, pressuremeter, and laterally loaded pile problems. The program, CONOIL, was developed by Hans Vaziri at the University of British Columbia. The program i s divided into two parts: a geometry program and the main finite element program. The geometry program inputs mesh geometry data, rearranges the order of the nodes to minimize the bandwidth, processes the data, and creates a LINK f i l e to transfer the information to the main program. The advantages of this system i s obvious. Program users can number the nodes the way they desire and the geometry program w i l l do the work to minimize the bandwidth. Moreover, i f the same mesh geometry i s used for more than one analysis, savings in computing time can be achieved by processing the geometry information only once. The main finite element program contains several useful features. The program analyses two types of higher-ordered elements: 6-noded Linear Strain Triangles (LST), and 15-noded Cubic Strain Triangles (CST). Examples of the elements are shown in Figure 5.1. LST elements were found to produce accurate results when compared with theoretical stress-strain and cavity expansion theories (Chapters 6 and 7). These elements were used for a l l subsequent analyses. Load-deflection responses were slightly s t i f f e r than the theoretical predictions but i s to be expected as a result of the incremental elastic method of analysis used in the program. T r i a l analyses  a) Linear Strain Trianaular Element 6 Nodes, 12 D.O.F.  FIGURE 5 . 1 :  b) Cubic S t r a i n Trianqular Element 15 Nodes, 30 D.O.F.  HIGHER-ORDERED ELEMENTS (After V a z i r i , 1 9 8 5 ) ro  performed with CST elements produced better results. The improvements, however, were small and did not warrant the high computing costs incurred by using the CST elements. Another advantage of CONOIL i s i t s ability to handle high Poisson's ratios. Values as high as 0.499 were used without encountering instability problems. The high values were useful i n simulating undrained conditions (no volume change) for cohesive soils.. Other special features of CONOIL are i t s ability to perform consolidation and load-shedding analyses. These options were not used in dealing with the present problems. A complete documentation of the finite element program i s given by Vaziri in his doctoral dissertation (1985).  44  CHAPTER 6 STRESS-STRAIN RELATIONSHIP  6.1  INTRODUCTION The stress-strain relations of s o i l are complex, being  non-linear, inelastic, and stress level dependent. In the finite element program CONOIL, a simple incremental linear elastic and isotropic stress-strain model i s used. The model i s described by Duncan et a l . (1980). To verify the ability of the finite element program to correctly model the complex stress-strain behaviour of s o i l , a group of 4 linear strain triangular elements was tested. The test elements and the boundary constraints are shown in Figure 6.1. distributed pressure loads, A(T^,  Uniformly  were applied to the top of the  elements and the corresponding axial (Y) deflections computed. The elements were tested under both plane strain and plane stress conditions.  6.1.1  Stress-Strain  Relationship  The incremental stress-strain relationship used in CONOIL can be written as follows:  {AC}  = [ D ] {At  }  {A(T}  i s the incremental stress vector  {A&}  i s the incremental strain vector  where  [ D ] i s the stress-strain matrix  6.1  FIGURE 6 . 1 :  TEST ELEMENTS  46  [D] i s a function of the tangent Young's and bulk modulii, E and T  V E B  = E.(I  - R (<r /<r )  t - BtV  W"  T  f  d  k  2  df  where E. 1 k  E  n k  = i n i t i a l Young's modulus = k„P (fl*o/P ) = Young's modulus number  ij s  J  a  = Young's modulus exponent  Bt  = tangent bulk modulus number  m  = tangent bulk modulus exponent  P a  = atmospheric pressure  *3 R  f  f  d  'df  = minor principal stress  failure ratio = deviator  stress  = deviator stress at failure  6.2  PLANE STRAIN CONDITION  6.2.1  Stress-Strain Relations The stress-strain relationship below was derived for the plane  strain uniaxial loading condition:  £  y  = (9B - E )(3B + E ) <T _ s__ sj_ y 36B E 2  s  where = axial strain corresponding to the applied stress 0~y  6.4  E  s  B k  = b u l k modulus = k P ((JT-/P ) oa 3 a  m  R  = b u l k modulus number  B  m  E  = secant Young's modulus  = b u l k modulus exponent  f o r the h y p e r b o l i c s t r e s s - s t r a i n model i s g i v e n by Duncan  g  & Chang (1970) as  E  = E.[l-R ((r /cr  s  f  y  d f  )]  6.5  where (T  J £  = d e v i a t o r s t r e s s a t f a i l u r e = 2c cosfi + 2(To s±ntf 1 - sinaf  The d e r i v a t i o n o f the s t r e s s - s t r a i n r e l a t i o n s h i p i s c o n t a i n e d i n Appendix A.  6.2.2  Comparison o f F i n i t e Element R e s u l t s w i t h C l o s e d Form Solution F i n i t e element t e s t s were performed f o r both c o h e s i v e  frictional  and  (c'=0) m a t e r i a l s . S o i l parameters employed i n these  a n a l y s e s a r e t a b u l a t e d i n T a b l e 6.1. c o h e s i v e m a t e r i a l correspond undrained  (0=0)  c l a y (based  The p r o p e r t i e s l i s t e d f o r the  t o those o f a normally  consolidated  on A t u k o r a l a & Byrne, 1984) w h i l e the f r i c t i o n a l  m a t e r i a l p r o p e r t i e s a r e a p p r o p r i a t e f o r a sand w i t h a r e l a t i v e d e n s i t y of 75% (Byrne & E l d r i d g e , 1982). The r e s u l t s o f the a n a l y s e s a r e shown i n F i g u r e 6.2 f o r the undrained  c l a y , and i n F i g u r e 6.3 f o r the sand. Good agreements e x i s t  TABLE 6.1 SOIL PARAMETERS USED IN STRESS-STRAIN ANALYSES  MATERIAL PARAMETER  COHESIVE SOIL (Undrained Clay)  FRICTIONAL SOIL (Sand)  72.1  750.0  0.0  0.5  24.0  600.0  m  0.0  0.5  f  0.9  0.9  0.0  0.29  k  E  n k  R  B  fi  c  (Psf)  305.0  0.0  (deg)  0.0  39.0  (deg)  0.0  4.0  (deg)  0.0  33.0  ^sat (Pcf)  123.4  122.4  Depth (ft)  20.0  20.0  (Psf)  1220.0  1200.0  Patm . (Psf)  2116.2  2116.2  *1  FIGURE 6.3:  PLANE STRAIN STRESS-STRAIN RELATIONSHIPS FOR SAND  between the finite element predictions and the theoretical curves given by Equation  6.4.  The finite element curves are truncated at stresses corresponding to the failure condition where 0~^f(T^ = 1. Upon failure, the shear modulus was reduced to 0.001  of i t s i n i t i a l value.  This reduction of the shear modulus allows for large deformations on subsequent stress increases and yields the flat portions of the curves shown. The computed stress-strain curves are, in general, slightly s t i f f e r than the theoretical curves predicted by Equation 6.4. This stiffness i s expected, however, due to the inherent nature of the incremental elastic method employed in the finite element program. Better agreements could have been obtained by using smaller stress increments, but was deemed unnecessary. The computed results clearly demonstrate the finite element program's ability to model the non-linear stress-strain behaviour of s o i l under plane strain condition.  6.3  PLANE STRESS CONDITION  6.3.1  Stress-Strain Relations For the loading conditions illustrated in Figure 6.1, a  stress-strain relationship was derived for the plane stress case. As shown in Appendix A, this relationship can be expressed as  y  _y E  s  6.6  _y  u  E.[l  l  -  RJff  f Y  df  y  J  where E^ and ( T ^ are as given in Equation 6.5. The remaining parameters are as defined in Section 6.2.1.  6.3.2  Comparison of Finite Element Results with Closed Form Solution The theoretical and computed stress-strain curves for the  undrained clay and sand (material properties given in Table 6.1) are plotted in Figures 6.4 and 6.5. Again, good agreements exist between the numerical results and the theoretical relationships. As in the plane strain case, the finite element predictions are slightly s t i f f e r than the theoretical curves. The differences, however, are negligible. Once again, the results verify the ability of the program to model the stress-strain behaviour of s o i l under plane stress condition.  FIGURE 6.4:  PLANE STRESS STRESS-STRAIN RELATIONSHIPS FOR UNDRAINED CLAY  CHAPTER 7 CYLINDRICAL CAVITY EXPANSION  7.1  INTRODUCTION The cylindrical cavity expansion problem bears some  similarities to the laterally loaded pile situation. At depths away from the ground surface, soil displacements upon the expansion of a cavity are confined to the radial plane (Hughes et a l . , 1977, and Robertson, 1982). The problem can thus be treated as plane strain. Similarly, plane strain deformations are assumed for the laterally loaded pile problem at large depths. In both instances, the lateral passive resistance of the s o i l i s mobilized. Although no mathematical solution exists for the lateral pile problem, closed form solutions for cavity expansion are readily available. Consequently, to validate the method of analysis for the laterally loaded pile problem, finite element analyses for the expansion of cylindrical cavities were performed. The results are compared with closed form solutions developed for the expansion of infinitely long cylindrical cavities in infinite media. Finite element analyses for the cavity expansion problem were performed using two different mesh geometries, the two-dimensional plane strain quadrant shown in Figure 7.1, and the plane strain axisymmetric domain i n Figure 7.2. Taking into account small discrepancies i n the results due to the different sizes and, to a lesser degree, the pattern or geometry of the elements, both mesh geometries yielded approximately the same pressure-deflection responses. Although the quadrant mesh i s suited for validating the  FIGURE 7 . 1 :  FINITE ELEMENT MESH FOR PLANE STRAIN CAVITY EXPANSION ANALYSIS  FIGURE 7.2:  FINITE ELEMENT MESH FOR PLANE STRAIN AX 1 SYMMETRIC CAVITY EXPANSION ANALYSIS  58  pile results because of i t s similarity to the mesh used in the laterally loaded pile problem (see Figures 3.2 and 4.3), the axisymmetric mesh was chosen because of i t s simplicity, which permitted the use of smaller, and therefore, more elements in the domain without incurring excessive computing costs. Since s o i l failure progresses out radially from the centre of the mesh in the cavity expansion problem, greater accuracy in predicting deflection responses was achieved with the use of smaller elements. Finite element predictions were obtained for both elastic-plastic and non-linear elastic material properties. The results are compared with elastic-plastic closed form solutions. Both cohesionless (sand) and cohesive (clay) s o i l properties were used in the comparisons.  7.2  COHESIVE SOIL Finite element analyses and the elastic-plastic closed form  solution for an undrained clay were compared. The s o i l properties are tabulated in Table 7.1. These values are appropriate for a normally consolidated clay and are based on Atukorala & Byrne (1984). The value of 0.0006 given for the elastic-plastic case serves only as a flag to indicate the material type and i s of no consequence in subsequent calculations. A K  q  value of 1.0 was used for the  isotropic consolidation condition assumed in the closed form solution.  7.2.1  Elastic-Plastic Closed Form Solution The closed form solution for a purely cohesive material was  derived by Hughes (1979) based on the assumptions of expansion in an  TABLE 7.1 MATERIAL PROPERTIES FOR UNDRAINED CLAY  MATERIAL PARAMETER  ELASTIC-PLASTIC  NON-LINEAR ELASTIC  144.1  144.1  0.0  0.0  24021.0  24021.0  0.0  0.0  0.0006  0.9  (Psf)  610.0  610.0  ^2  (deg)  0.0  0.0  Af6  (deg)  0.0  0.0  ^  (deg)  0.0  0.0  0.499  0.499  123.4  123.4  40.0  40.0  2440.0  2440.0  1.0  1.0  2116.2  2116.2  k  E  n k  B  m R  f  c  u  c v  n ''sat  Depth ( f t ) c7*  vo  ' (Psf)  K o Patra ,. (Psf) ' v  infinite medium and a Tresca failure criterion (ie: 1/2 (fX  - (T ) = r 0 c ). In addition, s o i l in the plastic zone was assumed to deform at constant volume. Although no volumetric strain constraint was placed on the deformation of s o i l in the elastic zone, Hughes showed that AIT  = -AO" during cavity expansion. Moreover, in the case of an I*  0  infinitely long cylindrical cavity, A(f = 0. Consequently, the mean z  normal stress, ff^, i s constant during expansion and no change in volumetric strain occurs. The overall effect i s that of constant volume deformation, applicable to the case of an undrained clay. Hughes' solution for small strains i s given below: P  = P  o  + c  u  1 + ln  r or  P  = P  o  + c  u  2G  AT  o  1 + ln Ax r  o  c  W  1  7.1a f 6BE  7.1b  c 9B - E u  where P P  r o  = pressure on wall of cavity = i n i t i a l pressure on wall of cavity  AT  =  deflection of cavity wall  T  =  i n i t i a l radius of cavity  B, G and E = i n i t i a l modulus values  In the equations given above, P  r  tends to infinity as AT  tends to infinity, and no limiting pressure can be determined. Clearly, this relationship breaks down for large deformations. Gibson & Anderson (1961) have derived an equation for large strains as follows:  61  P  r  = P  o  + c  1 + l n 'AT  u  <  2G  7.2  r c where r = current radius of c a v i t y = r  Q  + Ar  F o r t h i s e q u a t i o n , as r t e n d s t o i n f i n i t y , Ar/r  approaches  1, and a  l i m i t i n g p r e s s u r e , based s o l e l y on m a t e r i a l p r o p e r t i e s , i s a c h i e v e d :  P  L T  = P  o  7.3  + c [1 + l n ( 2 G / c ) ] u u L  Assuming E = 5 0 0 c  v  / J  f o r an u n d r a i n e d c l a y (u=0.5), P^ = P  u  The t h e o r e t i c a l  q  + 6.8c u  e l a s t i c - p l a s t i c c u r v e based on E q u a t i o n 7.1a  f o r t h e m a t e r i a l p r o p e r t i e s g i v e n i n T a b l e 7.1 i s shown i n F i g u r e 7.3. F o r AT < 0.001103 f t , o r l n [ ( J r / r )(2G/c ) ] <C - 1 , 4 P = P - P i s o u r o n e g a t i v e . T h i s apparent e r r o r i s caused by t h e ' l o g ' term i n t h e c l o s e d form s o l u t i o n and r e f l e c t s the l i n e a r e l a s t i c b e h a v i o u r o f t h e s o i l p r i o r t o the s t a r t of p l a s t i c f a i l u r e . Consequently, the s m a l l s t r a i n o r i n i t i a l e l a s t i c p o r t i o n o f t h e c u r v e i s g i v e n by t h e l i n e a r e l a s t i c c l o s e d form s o l u t i o n (Byrne & G r i g g , 1980): P  r  = P  o  + 2G(4r/r ) o v  y  7.4  From F i g u r e 7.3, p l a s t i c f a i l u r e can be seen t o b e g i n a t A? = 605 P s f where t h e two c u r v e s meet.  PRESSURE ON CAVITY WALL  29  AP  (psf)  63  7.2.2  Finite Element Predictions and Comparison with Closed Form Solution The result of the finite element analyses are presented in  Figure 7.4. Good agreement with the closed form solution was obtained for the elastic-plastic curves. The cavity expansion curve for non-linear elastic material properties i s also shown in the graph. As expected, the i n i t i a l linear elastic behaviour of the elastic-plastic curves i s absent and the overall pressure-deflection response i s considerably softer.  7.2.2.1 Boundary Conditions Although the inherent nature of incremental elastic analysis i s to predict responses somewhat s t i f f e r than the actual behaviours, the elastic-plastic finite element curve in Figure 7.4 shows an i n i t i a l response slightly softer than that of the close form solution. At larger deflections, the curve stiffens as expected and matches the theoretical curve for deflections greater than about 0.15 f t . The behaviour of the finite element curve can be explained by the boundary conditions. Since undrained clay deforms at constant volume, the expansion of a cavity in a finite medium i s physically impossible. To illustrate, a separate analysis was performed using the mesh in Figure 7.2 but with the outer boundary pinned at nodes 65, 66 and 879 to model a finite medium. The resulting pressure-deflection curve i s shown i n Figure 7.5. Although this curve deviates from an expected vertical line and indicates an increase in deflection with increasing pressure, the response i s much stiffer than that predicted by Equation 7.1. The calculated deflections were found to be the  (X 10 ) 2  « I  0-0  i  i  0.02  i  i  0.04  1  1  0.06  i  1  0.08  1  1  0.1  STRAIN FIGURE 7.4:  r  1  0.12  Ar/r  0.14  0  CAVITY EXPANSION CURVES FOR UNDRAINED CLAY  0.16  0.18  0.2  99  r e s u l t s of s m a l l v o l u m e t r i c s t r a i n s  (£  < 0.5%)  v  i n the mesh  elements owing t o the i n a b i l i t y t o use B = oo, and  the s i m i l a r i t y i n  initial  theoretical  s l o p e between the p r e d i c t e d curve and  the  r e l a t i o n s h i p i s c o i n c i d e n t a l . The use of a f i n i t e element mesh w i t h a d i f f e r e n t r a d i u s would have produced s l o p e . To model expansion was  a curve w i t h a d i f f e r e n t  initial  i n an i n f i n i t e medium, the o u t e r boundary  p e r m i t t e d t o d e f l e c t i n the r a d i a l d i r e c t i o n as shown i n F i g u r e  7.2,  c r e a t i n g a s t r e s s boundary where 4<T.  = 0. The  p r e d i c t e d by t h i s method i s p r e s e n t e d i n F i g u r e In  expansion  curve  7.4.  u s i n g the s t r e s s boundary method, the r e s i s t i n g f o r c e of the  s o i l beyond the r a d i u s of 100 f t i s o m i t t e d . As shown i n F i g u r e the r a d i a l s t r e s s , (T , r  d e c r e a s e s w i t h r a d i a l d i s t a n c e from  c a v i t y a c c o r d i n g t o the e q u a t i o n d e r i v e d by Hughes (T  7.6,  the  (1979):  = P (a /r ) 2  7.5  2  r  where = p r e s s u r e ( o r change i n p r e s s u r e ) on w a l l o f c a v i t y a  = r a d i u s of c a v i t y  r  = r a d i a l d i s t a n c e from c e n t r e of c a v i t y  In  the above a n a l y s i s f o r an i n i t i a l c a v i t y r a d i u s o f 1 f t , the  r e s i s t i n g p r e s s u r e o m i t t e d a t the outer boundary ( r = 100 f t ) i n c r e a s e d from a v a l u e of 0.069 P s f a t the s t a r t o f p l a s t i c at AP  = 690 P s f (a = 1.0034 f t ) t o 0.416  1.166  f t ) . The  = 3060 P s f (a =  o m i s s i o n of t h i s r e s i s t i n g p r e s s u r e i s thought  the cause of the s l i g h t l y low s t r a i n s  P s f a t AF  failure  (Ar/r^  have been reduced  <  0.14)  s o f t response  t o be  of the f i n i t e element curve a t  i n F i g u r e 7.4.  Although  t h i s error could  by extending the r a d i u s of the f i n i t e element mesh,  NOT WALL  OF  TO  SCALE  CAVITY  P = 3060 r  PSF  Q =1FT 0  Q  0.000136  E-P  P  THEOftY  = 7.766  FT  (EQ.Z5)  r  80  RATIO FIGURE 7.6:  120  100 r / a  140  G  VARIATION OF RADIAL STRESS WITH DISTANCE FROM CAVITY  160  180  68  i t was not c o n s i d e r e d  necessary  s i n c e the e r r o r i n t r o d u c e d  by u s i n g a  mesh r a d i u s o f 100 f t i s r e l a t i v e l y s m a l l and i n s i g n i f i c a n t . Volumetric  s t r a i n s c a l c u l a t e d f o r t h e s t r e s s boundary  medium) a n a l y s i s were approximately  (infinite  150 times l e s s than those  obtained  i n t h e f i n i t e medium a n a l y s i s ( F i g u r e 7.5). Consequently, d e f l e c t i o n e r r o r s r e s u l t i n g from volume changes can be d i s c o u n t e d .  7.3  COHESIONLESS SOIL In the case o f a c o h e s i o n l e s s s o i l ,  f i n i t e element  analyses  were performed u s i n g o n l y n o n - l i n e a r e l a s t i c m a t e r i a l p r o p e r t i e s f o r a dense sand.  7.3.1  Elastic-Plastic Materials D i f f i c u l t i e s a r i s e i n the a n a l y s i s o f e l a s t i c - p l a s t i c  due  t o t h e v a r i a t i o n o f shear s t r e n g t h w i t h t h e minor  materials  principal  s t r e s s , cT^', as f o l l o w s :  (T, = 2 C =2 df max  CV 3  f  sin0'  7.6  1 - sinjzf' and  ZTj; = (T^' sinks' costf 1 - sin*  7.7  1  Moreover, tf v a r i e s w i t h (T^' as g i v e n by Duncan e t a l . (1980):  tf  =  fi^  -A0  log«T 7P ) 3  A  As i n c r e a s i n g l o a d s a r e a p p l i e d t o a s o i l element, t7" '  7.8  v a r i e s , r e s u l t i n g i n changes i n t h e shear s t r e n g t h , s t r e s s l e v e l ( ( T ^ / l T ^ ) , and modulus v a l u e s . Problems a r i s e a s t h e s o i l approaches f a i l u r e . The r e d u c t i o n o f modulus v a l u e s a t f a i l u r e t o 0.01 o r 0.001 o f t h e i r i n i t i a l v a l u e s causes  (T./^jr-  d  t o f l u c t u a t e above  di  and below t h e f a i l u r e v a l u e o f 1.0, l e a d i n g t o e r r a t i c s o i l An i l l u s t r a t i o n o f t h e problem was p r e s e n t e d  behaviour.  by E l d r i d g e (1983) and i s  shown i n F i g u r e 7.7. Because o f t h i s e r r a t i c b e h a v i o u r ,  elastic-  p l a s t i c a n a l y s e s were n o t performed. I n s t e a d , a c o m p a r i s o n was made between t h e n o n - l i n e a r e l a s t i c f i n i t e element p r e d i c t i o n and t h e e l a s t i c - p l a s t i c c l o s e d form s o l u t i o n .  7.3.2  E l a s t i c - P l a s t i c C l o s e d Form S o l u t i o n An e l a s t i c - p l a s t i c c l o s e d form s o l u t i o n f o r t h e e x p a n s i o n  of a  c y l i n d r i c a l c a v i t y i n an i n f i n i t e c o h e s i o n l e s s medium was d e r i v e d by Hughes e t a l . (1977) assuming a Mohr-Coulomb f a i l u r e c r i t e r i o n . A complete d e r i v a t i o n o f t h e s o l u t i o n , i n c l u d i n g d i l a t i o n e f f e c t s , was presented  P  R  by E l d r i d g e (1983). The f o l l o w i n g r e l a t i o n s h i p was o b t a i n e d :  = P  Q  (1 + sinpf')  E  Ax  1 + ju and  (l-N)/(l+n)  r nP sinyj' o  N = (1 - s i n 0 ' ) / ( l + s i n 0 ' ) n = (1 - s i n v O / ( l + s i n v )  where ^' = f r i c t i o n a n g l e o f t h e c o h e s i o n l e s s m a t e r i a l V  = dilation  angle  E  = i n i t i a l Young's modulus = k„P ((ToVP ) Ji a J a  n  7.9  M o d i f i e d Mohr Diagram  FIGURE 7.7:  STRESS PATH FOR FAILED SAND ELEMENT  (Adapted from E l d r i d g e , 1983, p. 90)  P  r>  P , r and Ar are as defined in Equations 7.1 and 7.2.  Volumetric strain due to dilation effects i s given by  A£  = -AK sinv*  7.10  For the condition of constant volume deformation, V = 0 and Equation 7.9 reduces to  P  r  E  = P (1 + sin*') o  (l-N)/2  Ar  1 + ji  rP  q  7.11  sin*'  As in the case of the solution for cohesive materials, as Ar tends to infinity, Ar/r approaches 1, and the limiting pressure i s given by  P  L  = P  Q  2G  (1 + sin*') P  o  (l-N)/2  7.12  sin*'  The above closed form solution does not take into account the variation of *' with  given by Equation 7.8, nor does i t consider  the effect of 0"^' on the i n i t i a l modulus values according to  E  G  i " E a«V a> k  =  k  P  G W  /P  /  P  a >  7.13  n  m  Since 0^' may vary considerably during loading, omitting i t s influences may lead to significant errors. Consequently, the  7.14  e l a s t i c - p l a s t i c solution given by Equations 7.9 to 7.12 i s , at best, only an approximation of the r e a l problem. The closed form solution for a dense sand i s shown i n Figure 7.8. The assumption of V = 0 was made for constant volume expansion. Material properties for the sand are based on Byrne & Eldridge (1982) and are tabulated i n Table 7.2. As i n the case of the cohesive s o i l , the i n i t i a l l i n e a r e l a s t i c portion of the curve i s given by Equation 7.4. P l a s t i c f a i l u r e i s shown to begin at A? = 1400 Psf.  7.3.3  F i n i t e Element Prediction The results of a plane s t r a i n axisymmetric f i n i t e  element  analysis and the corresponding e l a s t i c - p l a s t i c closed form solution (j/ = 0) are shown i n Figure 7.9. An outer stress boundary was used i n the a n a l y s i s , as discussed i n Sec. 7 . 2 . 2 . 1 . Material properties are given i n Table 7.2. An unusually high kg value of 1250.0 was used i n order to l i m i t volume changes and to f a c i l i t a t e comparison with the constant volume closed form s o l u t i o n . K was taken as 1.0 to model o the i s o t r o p i c consolidation condition assumed i n the closed form solution. The i n i t i a l slope of the two curves i s 2G, as predicted by the e l a s t i c closed form solution i n Equation 7.4. The expansion curve for non-linear material properties, however, lacks the i n i t i a l l i n e a r e l a s t i c behaviour and exhibits a much softer response. Although a softer response was anticipated, such a large difference between the two solutions was unexpected. The inaccuracy of the closed form solution i n f a i l i n g to take into account the effects of (T^' as discussed i n Section 7.3.2 may be the cause of the large discrepancy.  TABLE 7.2 MATERIAL PROPERTIES FOR SAND  PARAMETER  D k  (%)  r  0.5 1250.0  B  m R  75 750.0  E  n k  NON-LINEAR ELASTIC COHESIONLESS MATERIAL  0.5 0.9  f  c  (Psf)  0.0  *l  (deg)  39.0  A0  (deg)  4.0  (deg)  33.0  (deg)  0.0  ^  c v  '  V  0.4 /sat  <  Pcf  >  122.4  Depth ( f t )  20.0  ' (Psf)  1200.0  (T  vo  K P  v  1.0  o atm  (Psf)  v  2116.2  PRESSURE ON CAVITY WALL  O  ro  20  40  60  AP  80  (psf)  100 (X10 ) 2  PRESSURE ON CAVITY WALL o  5  10  AP  (psf) 15(X10 3)  CHAPTER 8 PRESSUREMETER EXPANSION  8.1  INTRODUCTION The pressuremeter i s essentially an expandable tube which i s  either pushed into the s o i l or inserted into a pre-bored hole i n the ground and inflated under controlled conditions (Robertson, 1982). Plots of Pressure vs. Volume Increase, referred to as pressure expansion curves, are obtained from the tests, from which values for s o i l parameters can be determined. The foregoing plane strain axisymmetric cavity expansion analysis i s generally considered to be a good model for pressuremeter expansion tests. Recent research by Yan (1986), using threedimensional axisymmetric finite element analyses, has confirmed the validity of the plane strain cavity expansion model for pressuremeter analysis. For typical aspect ratios of pressuremeters ranging from about 6 to 8, pressure-deformation curves predicted from threedimensional axisymmetric analyses were nearly identical to those obtained using the plane strain formulation described i n Section 8.2. For simplicity, the cavity expansion formulation can be used to model pressuremeter expansion without significant errors. The load-deflection relationship of pressuremeter expansion was analysed using the incremental elastic finite element method. Pressuremeter curves obtained from the analyses were compared with P-Y curves obtained i n Chapters 9 and 10 to determine a rational method for deriving P-Y curves from pressuremeter curves. The results are presented i n Chapter 11.  77  8.2  FINITE ELEMENT DOMAIN ANALYSED To investigate the pressure-deflection response of the  pressuremeter problem, plane strain axisymmetric analyses were performed using the finite element mesh shown in Figure 7.2. The placement of rollers along the top and bottom boundaries of the mesh ensured deformations only in the horizontal plane. The outer boundary was left unconstrained to simulate a boundary of zero stress change. Errors arising from the omission of pressures exerted along this boundary by s o i l outside the finite element domain were shown in Section 7.2.2.1 to be negligible. Pressure loads were applied to the side of the mesh over a length of one foot as indicated. Soil disturbances and stress changes due to the placement of the pressuremeter probe were ignored. An isotropic consolidation insitu stress condition was assumed for the entire mesh.  8.3  COHESIVE SOIL The results of the analyses for a normally-consolidated  undrained clay are presented in Figure 8.1. Soil parameters used in the study are given in Table 8.1 and were derived as follows:  Bulk unit weight of clay =120  Pcf  Soil depth = H Effective overburden stress, <r ' = (120 - 62.4) H = 57.6 H vo Undrained shear strength, c = 0.265 (T '= 15.25 H u vo I n i t i a l Young's modulus, E. = 200 c = 3050 H ° 1 u w  6  For the undrained condition, n = 0 and E. = k^P ^ ' I E atm therefore, k^ = 1.44 H  TABLE 8.1 MATERIAL PROPERTIES FOR UNDRAINED N.C. CLAY  NON-LINEAR ELASTIC COHESIVE SOIL PARAMETER  DEPTH = 1 0 FT  DEPTH = 20 FT  14.4  28.8  0.0  0.0  1200.0  2400.0  0.0  0.0  0.9  0.9  0.498  0.498  152.5  305.0  (deg)  0.0  0.0  (deg)  0.0  0.0  (deg)  0.0  0.0  120.0  120.0  10.0  20.0  (Psf)  576.0  1152.0  K o P «. (Psf) atm  1.0  1.0  2116.2  2116.2  k  E  n k  B  m R  f  *o c  u  *1  cv * sat  (Psf)  (Pcf)  Depth ( f t )  <r » vo  Taking U = 0.498 for the undrained condition, q  bulk modulus, B = E./3(l - 2u ) = 83.3 E. and B = k„P _ for m=0 B atm therefore, k = 83.3 k p  v  = 120 H  Using H=10 f t and H=20 f t in the above equations yields the values shown in Table 8.1. The i n i t i a l slopes of the expansion curves are as expected. Values of approximately 0.99(2G) were obtained, compared to 2G predicted by the linear elastic closed form solution given by Equation 7.4. The results also show the H=20 f t curve to be merely a scaled-up version of the H=10 f t curve. The scaling factor of 2.0 indicated by the predicted load-deflection values corresponds to the differences in the values of c , kg and kg used i n the two analyses. y  8.4  COHESIONLESS SOIL Finite element analyses were also performed for cohesionless  s o i l using the mesh in Figure 7.2. The predicted pressuremeter curves for a dense sand are shown in Figure 8.2. Properties for the sand were determined from values given by Byrne & Eldridge (1982), and Byrne & Cheung (1984), and are summarized in Table 8.2. Values of about 0.98(2G) were determined for the i n i t i a l slopes of the pressuremeter curves, agreeing well with the theoretical value of 2G. The overall shape of the two curves are similar, the H=20 f t curve being a scaled-up version of the H=10 f t curve. A scaling factor of 1.66 was obtained for the range of strains shown. Small irregularities can be observed in the predicted results.  TABLE 8.2 MATERIAL PROPERTIES FOR DENSE SAND  PARAMETER k  E  n k  B  m R  f  COHESIONLESS SOIL DEPTH=10 FT DEPTH=20 FT 1000.0 1000.0 0.5  0.5  600.0  600.0  0.5  0.5  0.8  0.8  *l  (deg)  39.0  39.0  Atf  (deg)  4.0  4.0  ff y  (deg)  33.0  33.0  (%)  75  75  0.222  0.222  624.0  1248.0  122.4  122.4  330.0*  675.0*  1.0*  1.0*  C  D  r  u  (Psf)  'sat  <  C  (Psf)  K  o  Psf  >  * Values assumed for f i n i t e element analyses  (X10 ) 3  STRAIN  FIGURE 8.2:  Ar/r  D  PRESSUREMETER CURVES FOR NON-LINEAR ELASTIC DENSE SAND FROM PLANE STRAIN AXISYMMETRIC FINITE ELEMENT ANALYSES  00 ro  The load-deflection values plotted in Figure 8.2 do not describe smooth curves, but stray to either side of the best-fit relationships. These irregularities are the direct results of erratic soil behaviour at or near failure, where the strength of the soil varies with changes in 0^', resulting in fluctuations in the stress level above and below the failure condition. A discussion of this problem i s given in Section 7.3.1.  8.5  SIZE EFFECT To simplify the conversion of radial displacements, Av,  strain values, AV/V , Q  into  in the foregoing finite element analyses, an  i n i t i a l cavity radius of 1 f t was assumed. The actual radius of the pressuremeter c e l l , however, i s in the neighbourhood of 1.5 inch. Analyses were performed to determine the existance of any size effect and to assess the validity of the pressuremeter results shown in Sections 8.3 and  8.4.  To examine the pressure—deflection relationship, an analysis was performed for the undrained clay using a mesh with an i n i t i a l cavity diameter, D, of 3 inches. The mesh radius, R, was kept at 50D (150 in) as before. The width of the loaded area was also retained at 1 f t . Other boundary and loading conditions were kept the same as before. Soil parameters given in Table 8.1 for H = 20 f t were used. Figure 8.3 shows the AV vs. Av results of this analysis along with the curve for D = 2 f t (r = 1 f t ) . As expected, smaller displacements were obtained for the D = 3 inches case. A comparison of the 4P vs. Av/v  plots in Figure 8.4, however, shows that the  results of the two analyses are identical. Consequently, size effects  oo  0.0  0.02  0.04  0.06  0.08  0.1  0.12  0.14  RADIAL DISPLACEMENT OF CAVITY WALL FIGURE 8.3:  0.16  Ar  0.18  0.2  (ft)  COMPARISON OF PRESSUREMETER CURVES PREDICTED USING DIFFERENT INITIAL CAVITY RADII 00  FIGURE 8.4:  COMPARISON OF PRESSUREMETER CURVES PREDICTED USING DIFFERENT INITIAL CAVITY RADII  can be e l i m i n a t e d through t h e use o f t h e c i r c u m f e r e n t i a l s t r a i n , Ar/r^,  i n s t e a d o f d e f l e c t i o n , Ar.  be o b t a i n e d f o r p r e s s u r e m e t e r s by m u l t i p l y i n g t h e Ar/r^  Moreover, AY-Ar  r e l a t i o n s h i p s can  o f any s i z e ( a s p e c t r a t i o > 6)  v a l u e s generated  simply  from any a n a l y s i s by t h e  new r v a l u e s , o Based on t h e r e s u l t s shown i n F i g u r e 8.4, c e l l diameter  the i n i t i a l cavity or  can be assumed t o have no i n f l u e n c e on t h e  AV-Ar/r  o r e l a t i o n s h i p . T h i s a s s u m p t i o n , however, i s v a l i d o n l y f o r s e l f - b o r i n g pressuremeters  i n s t a l l e d w i t h no s o i l d i s t u r b a n c e s . I n p r a c t i c e ,  disturbance i s unavoidable relationship d i f f i c u l t  soil  and i t s e f f e c t s on t h e p r e s s u r e - d e f l e c t i o n  to predict.  CHAPTER 9 PREDICTION OF P-Y CURVES  9.1  INTRODUCTION P-Y curves for the laterally loaded pile problem were predicted  using the f i n i t e element formulations described i n Chapter 3. Plane strain and plane stress analyses for both undrained clay and sand were performed. The results are presented i n the following sections. Additional analyses were performed to determine the effects of varying the mesh radius and the pile diameter. The P-Y curves were compared with the pressuremeter curves obtained in Chapter 8 to determine a method for deriving P-Y curves from pressuremeter curves. The results are presented in Chapter 11.  9.2  FINITE ELEMENT MESH The finite element mesh used i n the analyses i s shown in Figure  9.1. As noted i n Section 3.2, rollers were placed along the axis of symmetry to ensure zero displacement perpendicular to the loading direction. The outer mesh boundary was fixed at a radius of R determined from Equation 3.2. The value of R = 22D was calculated as follows:  Assuming flexible piles (h > 7/3 1 )> R = 7 1 Q  q  applies.  To ensure that deformations of the pile elements are insignificant relative to s o i l deformations (ie: rigid pile section), take E /E = 500. p so Also, I = l/47Tr = 1/647TD 4  4  89  And 1 = [4(E /E )I ] o p so p J  1 / 4  = 3.148 D  Finally, R = 7 1 = 22.03 D  Check: Embedded length of pile > 7/3 1 = 7.34 D = 14.7 f t for D = 2 f t . Therefore, assumption of flexible pile i s reasonable.  The mesh shown in Figure 9.1 i s composed of triangular linear strain elements. A brief description of this element i s given i n Chapter 5. The higher-ordered 15-noded cubic strain elements were not used i n the analyses due to the high computing costs involved. A t r i a l analysis performed using the cubic strain elements produced results showing only a slight increase i n sensitivity over the results obtained by using linear strain elements and the mesh i n Figure 9.1. Computing costs, however, were increased by nearly 200%.  9.3  P-Y CURVES FOR UNDRAINED CLAY Finite element analyses were performed for undrained  normally-consolidated clay at various depths. The soil properties are given i n Table 9.1, and are identical to those used i n the pressuremeter expansion analyses in Chapter 8. The pile elements were treated as a linear elastic material. To limit deformations and to prevent failure of the pile elements, parameters 500 times greater than those of the s o i l were used.  TABLE 9.1 SOIL PARAMETERS FOR UNDRAINED N.C. CLAY  PARAMETER k  VALUE 1.44 H  E  0  n k  120 H  B  m  0  n  0.498  f  0.9  R  c  u  (Psf) ' v  *sat < > Pcf  15.25 H 120.0  <T' (Psf)  57.6 H  (T ' (Psf)  57.6 H  V  m  K o  1.0  Note: H = depth (in feet)  91  9.3.1  Results The predicted P-Y curves for various depths are shown in Figure  9.2 for plane strain analyses, and in Figure 9.3 for the plane stress condition. A pile diameter of 2 feet was assumed for the analyses. The i n i t i a l slopes of the four plane strain curves are identical, a l l with a value of 1.57E^. For U=0.498, this i s equal q  to 2.35(2G), considerably stiffer than the slope of 0.99(2G) obtained for the pressuremeter curves. Similarly, a l l three plane stress curves have the same i n i t i a l slope of about 0.98E^, or 1.47(2G). These values were determined by computing theJP/JY ratio for very small load increments, roughly equal to 1% of P ]_ « Slightly steeper u  t  slopes could probably have been obtained by using even smaller A? increments. Hence, for practical purposes, values of 1.6E^  (2.4(2G))  and l.OE^ (1.5(2G)) are appropriate for the plane strain and plane stress conditions respectively. None of the P-Y curves exhibit a well-defined peak value in s o i l resistance corresponding to P ^ « Instead, at large u  t  displacements (Y > 0.7 f t for plane strain, and Y > 0.4 f t for plane stress), the P-Y relationships are linear with P increasing slightly with Y. The load at which the P-Y curve becomes linear i s taken as P ^ u  t<  Using this method, identical values of P ^ u  t  = 12.1cD (c = c^)  were obtained for the plane strain analyses. Likewise, consistent values of P ]_ = 6.1cD were determined for the plane stress curves. t  The continuing small increases in P beyond P -^ u  t  i s caused by  the zero-displacement outer boundary. The use of the fixed boundary restricts soil displacement, which in turn limits the lateral movement of the pile. Consequently, the pile cannot displace infinitely at  o  T  D= 2  0  -0  0.1  i  1  1  r  FT  0.2  0.3  0.4  0.5  Y FIGURE 9.2:  0.6  0.7  0.8  0 9  10  (ft)  P-Y CURVES FOR UNDRAINED CLAY FROM PLANE STRAIN ANALYSES VO  r\3  i  1  1  r^n  1  1  1  i  i  i  i  r  Y (ft) FIGURE 9.3:  P-Y CURVES FOR UNDRAINED CLAY FROM PLANE STRESS ANALYSES vo CO  P^j., and the P-Y curve does not flatten off as expected. The P]_£ value of 12.1cD obtained for the plane strain case u  is in reasonably good agreement with Randolph & Houlsby's (1984) results. As discused in Section 4.2.2, a soil-pile adhesion factor, oi, of 0.5 was used in the finite element analyses for clay. Randolph's theoretical value of ? ^ u  t  corresponding to oi = 0.5 i s 10.82cD (Table  3.1), resulting in an error of about 12% for the finite element prediction.  9.3.1.1 Comparison with Empirical P-Y Curves The finite element P-Y predictions are compared with empirical curves for soft clays. The empirical curves, shown as dashed lines in Figures 9.2 and 9.3, were determined using the method recommended by Matlock (1970). Calculations for determining the curves are contained in Appendix B. P^ u  t  for the empirical curves were determined by assuming a  block flow failure mechanism at large depths (see Figure 3.5b). The value of 9cD obtained i s considerably lower than the predicted value of 12.1cD for plane strain analyses. For three-dimensional deformations near the surface, a passive wedge failure mechanism was used (see Figure 3.5a), giving  P , = N cD ult p N = 3 + (T7c + J H/D p  9.4 ,  3 £ N ^ 9  where (T ' = effective overburden stress v H  = depth of s o i l  p  J  = coefficient ranging from 0.25 to 0.5 depending on s o i l type  Assuming a conservative value of 0.25 for J, P ^ u  t  based on Equation.  9.4 increases with depth, ranging from 3 at the surface to the maximum value of 9 at a depth of  H = c  6D  9.5  • (0-'D/cH) + 0.25 v  For the plane stress curves shown in Figure 9.3, "P ^  values  determined from Equation 9.4 exceed the predicted values. No i n i t i a l slope value i s predicted by Matlock's empirical curves. For lack of better information, the curves are drawn such that their slopes approach infinity for small lateral pile displacements. The overall agreement between empirical and predicted curves i s poor, differing i n both the i n i t i a l slope and the P -^ values. u  9.3.2  t  Effect of Pile Diameter The results shown i n Figures 9.2 and 9.3 were obtained for a  pile diameter, D, of 2 f t . To determine the effect of D on the P-Y predictions, additional plane strain and plane stress analyses were performed using pile diameters of 1 and 4 f t . Soil parameters used i n the plane strain analyses are for the clay at a depth of 20 f t . Parameters appropriate for a depth of 2 f t were used for the plane stress predictions. In keeping with the condition of Equation 3.2, a mesh radius of 22D was maintained for a l l the analyses. The same  96  number of elements were used for a l l the meshes, only the sizes of the elements were varied to accomodate the changing mesh size. The predicted P-Y curves are shown i n Figures 9.4 and 9.5. The results suggest that an increase i n the pile diameter by an arbitrary factor of Fp would have the effect of increasing both P and Y by the same factor. In other words, plots of P/D vs. Y/D would yield a single curve for each of the plane strain and plane stress conditions, regardless of the value of D used i n the analysis. Further discussions on normalized P-Y curves are given in Chapter 10.  9.3.3  Effect of Mesh Radius The effect of the mesh radius, R, on the P-Y responses of  laterally loaded piles were examined. As discussed in Section 3.2.2, previous work by Thompson (1977) has shown that the stiffness, but not the ultimate strength, of the P-Y curves depends on R. P-Y curves were generated for R = 10D, 20D and 50D to verify Thompson's results and to determine the sensitivity of the predicted curves to the mesh radius. The results for plane strain analysis are shown in Figure 9.6. Plane stress P-Y curves for different values of R are presented i n Figure 9.7. Soil parameters for the clay at depths of 20 f t and 2 f t were used i n the plane strain and plane stress analyses, respectively. The results obtained from the analyses confirm Thompson's observations. The stiffness, and hence, the i n i t i a l slopes of the P-Y curves decrease with increasing R. I n i t i a l slopes of 2.16E^, 1.61E^, 1.57E., and 1.22E. were obtained for R = 10D, 20D, 22D, and 50D, respectively, for plane strain analyses. Slopes of 1.33E^, 0.98E^, and 0.77E. were obtained for mesh radii of 10D, 22D, and 50D,  o 1  1  1  i  1  i  1  Y  1  1  1  1  i  r  (ft)  FIGURE 9.4; EFFECT OF PILE DIAMETER ON PLANE STRAIN P-Y CURVE PREDICTIONS FOR UNDRAINED CLAY  §  i  0.0  1  1  0.1  1  1  0.2  i  1  0.3  1  1  0.4  1  1  0.5  1  1  0.6  i  i  0.7  i  i  0.8  i  i  r  0.9  1.0  Y (ft) FIGURE 9.7:  EFFECT OF MESH RADIUS ON PLANE STRESS P-Y CURVE PREDICTIONS FOR UNDRAINED CLAY  o o  respectively, for plane stress analyses. R=22D was perviously determined as the appropriate mesh size for the given condition. Neither P ^ u  n  o  t  r  the shape of the curves appear to be overly  sensitive to the value of R. In fact, only small differences exist between the R = 10D and R = 22D curves, and the R = 22D and R = 50D curves. Since R = 10D and R = 50D can be considered as approximate lower and upper bounds for the zone of influence, the curves predicted for R = 22D can be considered as good representations of actual P-Y relationships.  9.4  P-Y CURVES FOR SAND Plane strain and plane stress P-Y curves were determined for a  dense sand. The s o i l properties are summarized in Table 9.2. The values are identical to those used i n the pressuremeter  expansion  analyses in Chapter 8. As in the analyses for undrained clay, parameter values for the pile elements were 500 times greater than those of the s o i l elements. One exception was the internal friction angle,  . A constant value of  85° was used to ensure high strength. To determine the insitu stresses, the following equations were used:  K  = 1 — sin^'  9.6  tf* = et. * -Jef'logCfT VP ) X  <r ' = [(1 + 2K )/3] <r ' m  Q  v  where K  Q  9.7  3  HI  = coefficient of lateral s o i l pressure at rest  9.8  TABLE 9.2 SOIL PARAMETERS FOR DENSE SAND  PARAMETER k  VALUE 1000  E  0.5  n k  600  B  m R  0.5 0.8  f  0.222 ^ '  (deg)  39.0  W  (deg)  4.0  (deg)  33.0  (%)  75  0  } CV  D  r  122.4 (Pcf)  a K  ' (Psf) vo  60.0 60.0 H 1 - sine*'  o (deg)  <T ' (Psf) mo  ^'-^'log(r VP ) m  (1 + 2K )<r 7 3 o  Note: H = depth ( i n feet)  a  103  internal friction  angle  f r i c t i o n angle a t P  Atf P  a  a  =1  change i n tf f o r a 1 0 - f o l d i n c r e a s e i n 0" ' atmospheric  p r e s s u r e = 2116.2 P s f  mean normal e f f e c t i v e vertical effective  (T  1  atm = 2116.2 P s f  stress  stress  was used as t h e i n i t i a l o v e r a l l c o n f i n i n g p r e s s u r e . The  d e t e r m i n a t i o n o f If ' from E q u a t i o n s 9.6 t o 9.8 i n v o l v e d an i t e r a t i v e m p r o c e s s . A r e a s o n a b l e v a l u e f o r (T * was f i r s t v a l u e i n E q u a t i o n 9.7,  tf  was determined.  from E q u a t i o n 9.6 and a new v a l u e f o r <T m  1  K  q  estimated. Using was then  determined  this  calculated  from  Equation  9.8. The new tf ' was s u b s t i t u t e d back i n t o E q u a t i o n 9.7 f o r a second m i t e r a t i o n . I t e r a t i o n s proceeded to  u n t i l t h e new (T ' was r o u g h l y e q u a l  the o l d value.  9.4.1  Results F i g u r e s 9.8 and 9.9 show t h e P-Y c u r v e s p r e d i c t e d f o r a p i l e  diameter  o f 2 f t . The i n i t i a l  s l o p e s f o r t h e p l a n e s t r a i n c u r v e s range  from 1.065E. t o 1.071E., determined i i l o a d increments  e q u a l t o about  analyses, i n i t i a l  from AP/AY  r a t i o s f o r small  0.5% o f Py-^-* F o r  t  n  e  plane  stress  s l o p e s o f l . O O l E ^ t o 1.043E^ were o b t a i n e d ,  u s i n g l o a d increments  o f about 0.8% o f P -j_ « T u  t  n e  slope values are  t a b u l a t e d i n T a b l e 9.3. Based on these f i g u r e s , an i n i t i a l  slope of  1.08E^ can be r e a s o n a b l y assumed f o r both plane s t r a i n and plane s t r e s s P-Y c u r v e s p r e d i c t e d f o r any depths. F o r JI =0.222, q  i s e q u a l t o 1.32(2G), somewhat s t i f f e r  than t h e i n i t i a l  t h i s slope  slope of  TABLE 9.3 RESULTS OF P-Y CURVE ANALYSES FOR DENSE SAND  PLANE STRESS  PLANE STRAIN DEPTH (ft)  INITIAL SLOPE  LOAD INCREMENT  Y (f£)  2 5  1.071 E. I  0.31% P. ult  1.001 E.  1.05% P  ult  (lb7ft)  Y  950  0.0080  0.080  1.022 E.  0.82% P. ult  2450  0.0225  11000  0.170  1.043 E.  0.65% P. ult  4600  0.0289  0.142  1.041 E.  1.22% P  8150  0.0365  1.065 E.  0.45% P  20  1.065 E.  0.48% P. ult  21000  40  1.066 E.  0.48% P  31500  ult  LOAD INCREMENT  6400  10  ult  INITIAL SLOPE  I  0.118  ult  107  0.98(2G) o b t a i n e d f o r the pressuremeter c u r v e s i n Chapter P ^ u  8.  and t h e c o r r e s p o n d i n g d e f l e c t i o n s , Y , f o r the  t  P-Y  p r e d i c t i o n s a r e a l s o g i v e n i n T a b l e 9.3. As i n t h e a n a l y s e s f o r c l a y , none o f the P-Y  c u r v e s e x h i b i t a peak v a l u e f o r P. R a t h e r , slow l i n e a r  i n c r e a s e s i n P w i t h Y beyond the p o i n t s o f f a i l u r e a r e o b s e r v e d . A c c o r d i n g l y , ~P ^  was t a k e n as t h e l o a d a t which the l i n e a r  l o a d - d e f l e c t i o n behaviour begins.  9.4.1.1  Comparison w i t h E m p i r i c a l P-Y E m p i r i c a l P-Y  Curves  c u r v e s f o r sand a r e a l s o shown i n F i g u r e s 9.8  and  9.9. The c u r v e s were determined a c c o r d i n g t o t h e method recommended by Reese e t a l . (1974). C a l c u l a t i o n s a r e shown i n Appendix P ^ u  B.  f o r the e m p i r i c a l c u r v e s were determined u s i n g a p a s s i v e  t  wedge f a i l u r e mechanism a t s h a l l o w depths, and a f l o w b l o c k model a t l a r g e depths (see F i g u r e 3.5). The e q u a t i o n s d e r i v e d from t h e s e models a r e g i v e n below:  P a s s i v e wedge: P  ult -  A  >'  H  K H tan^' s i n ^ + t a n ^ - 0 ' ) cosoC  +  tan/5  (D + Htan/5 t a n * )  tanfo-0')  tan/J ( t a n ^ ' sin/5 - t a n * ) - IM))  K H Q  9.9  Flow b l o c k : P  ult  =  A D  ^  H  (  K a  (  t a n  / * - 1) + K t a n * ' t a n / * )  where if' = e f f e c t i v e u n i t weight o f the sand H  = depth  9.10  tf = internal friction angle =tf/2  o<  45° + oi  /8  =  K  q  = coefficient of lateral earth pressure at rest  K  a  = Rankine coefficient of active earth pressure = tan (45° - oi) 2  A  = adjustment factor to correct for differences between field and predicted results  Values of A determined by Reese et a l . are shown in Figure B.2a. Using Equations 9.9 and 9.10, theoretical P ^  t  values were  calculated and compared with the finite element predictions. The results are shown in Table 9.4. Equating the theoretical and predicted P^ u  t  values for plane strain analyses, A ranging from 0.0645 to  0.0697 were obtained. A constant value of A = 0.065 appears to be appropriate. For plane stress analyses, the calculated values of A decrease with depth. Such a trend i s expected since the passive wedge failure mechanism i s valid only for shallow depths, and  P^ u  t  predicted for large depths would be overestimated. A correction factor of about 0.35 would be appropriate for P-Y curve predictions at or near the ground surface.  9.4.2  Effects of Pile Diameter Additional P-Y analyses were performed for the dense sand using  pile diameters of 1 and 4 f t . Plane strain and plane stress conditions for the sand at a depth of 20 f t were considered. The radius of the finite element mesh was maintained at R=22D according to Equation 3.2.  109  TABLE 9.4 COMPARISON OF THEORETICAL AND PREDICTED P _ VALUES ult l4  DEPTH (ft)  tf (deg)  <r  1  mo (Psf)  K o  K a  p  ult from  ^ult from  P  THEORY  FEM  f  A  see"] (notej  (Eq. 9.10)  Plan e Strain Analyses  5  43.5  162.5  0.31  0.18  98105 A  6400  0.0652  10  42.5  330.0  0.33  0.20  170467 A  11000  0.0645  20  41.0  675.0  0.34  0.21  286834 A  21000  0.0732  40  39.8  1370.0  0.36  0.22  484331 A  31500  0.0650  (Eq. 9.9)  Plan e Stress Analyses  2  45.1  63.5  0.29  0.17  3033 A  950  0.313  5  43.5  162.5  0.31  0.18  12217 A  2450  0.201  10  42.2  330.0  0.33  0.20  38426 A  4600  0.120  20  41.0  675.0  0.34  0.21  128448 A  8150  0.063  Note:  2T'  = 60.0 Pcf  D  = 2 ft  P _ = 2116.2 Psf atm Values of A were determined by comparing P ^ obtained from u  t  the passive wedge and flow block models with P obtained from the f i n i t e element analyses. The values are not those given by Reese ( i e : Figure B.2a).  no  The results of the analyses are shown in Figures 9.10 and 9.11. As in the case of the undrained clay, changing D by a factor F^ resulted i n changes in both P and Y by the same factor F^. Consequently, plotting P/D vs. Y/D for any pile diameter would yield a unique curve for each of the plane strain and plane stress condition at a given depth.  9.4.3  Effect of Mesh Radius To determine the sensitivity of the predicted P-Y responses to  changes i n the mesh radius, the problem was analysed using mesh radii of 10D, 20D, and 50D. The results for plane strain analyses are shown in Figure 9.12. Plane stress predictions are shown i n Figure 9.13. For  plane strain analyses, the shape of the P-Y curve for loads  approaching P ^ u  i s t  sensitive to the mesh radius used. The R=50D  curve shows a much softer response than the R=10D curve. The i n i t i a l slope and the i n i t i a l portion of the curve, however, are relatively insensitive to R, and only minor differences in P 2 were obtained u  t  for the various mesh radii used. For  plane stress analyses, neither the i n i t i a l slope nor the  shape of the P-Y curve are sensitive to the mesh radius. The curves differ only slightly from each other despite the wide range of mesh radius used i n the analyses. Identical values for ^ ^ were also u  t  obtained from the four analyses. Overall, the results show that the i n i t i a l slope decreases with increasing mesh radius, but at a slow rate. The shape of the curves for loads less than about 1/2 P was also insensitive to the mesh ult radius. Consequently, the curves can be considered as good  o  n  1  i  r  1  \  1  1—n  i  r  r  H-20FT  R = 22D  o CM  D = 4'  CD  ' CM  2'  O H  0 0  i  0.0  0.1  0.2  i  I  i  0.3  0.4  Y  I  0.5  I  I  0.6  L  J  0.7  I  0.8  I  I  L  0.9  (ft)  FIGURE 9.11: EFFECT OF PILE DIAMETER ON PLANE STRESS P-Y CURVE PREDICTIONS FOR DENSE SAND  I.O(XIC)- ) 1  H-20FT D =  Q  CD +  .0  0.04  0.08  0.12  0.16  0.2  0.24  0.28  0.32  2  FT  o R = 10D CD R - 22D + R = 50D  0.36  Y (ft) FIGURE 9.12:  EFFECT OF MESH RADIUS ON PLANE STRAIN P-Y CURVE PREDICTIONS FOR DENSE SAND  C  FIGURE 9.13:  EFFECT OF MESH RADIUS ON PLANE STRESS P-Y CURVE PREDICTIONS FOR DENSE SAND  d e s c r i p t i o n s o f t h e P-Y r e l a t i o n s h i p f o r s m a l l l o a d s . F o r l a r g e r l o a d s near t h e f a i l u r e c o n d i t i o n , use o f t h e p l a n e s t r a i n P-Y c u r v e s may r e s u l t i n e r r o r s . However, s i n c e p l a n e s t r a i n c u r v e s a r e a p p l i c a b l e f o r l o a d - d e f l e c t i o n responses a t l a r g e depths where t h e l o a d i n g conditions are generally l e s s severe, the e r r o r s are minimized.  CHAPTER 10 SIMPLIFIED METHOD FOR PREDICTING P-Y CURVES  10.1  INTRODUCTION The prediction of P-Y curves from finite element analyses is  both costly and time consuming. For many problems concerning laterally loaded piles, P-Y curves derived from empirical correlations are sufficient. Methods were recommended by Matlock (1970), and Reese et a l . (1975, 1974) for determining P-Y curves for soft clay, s t i f f clay, and sand, respectively. These methods were subsequently adopted by the American Petroleum Institute for use in designing laterally loaded piles. Comparisons of these empirical curves with the finite element predictions are shown in Chapter 9. The procedures recommended by Matlock and Reese et a l . for determining P-Y curves are based on correlations with results of pile loading tests. However, load tests were performed at only one site for each of the three s o i l types. The resulting P-Y correlations may therefore be site specific, influenced by local soil characteristics or abnormalities not found in other soils. As an alternative to the empirical methods, simplified P-Y curves based on the finite element predictions were derived. The advantage of the finite element approach lies in its use of fundamental s o i l parameters and stress-strain relationship, and is therefore valid for general applications.  10.2  SIMPLIFIED P-Y CURVES FOR UNDRAINED CLAY P-Y curves were predicted for an undrained normally-  consolidated clay in Section 9.3. The effects of various parameters on  the P-Y predictions were examined and a method devised for normalizing the curves.  10.2.1  Normalized P-Y Curves As discussed i n Section 9.3.1, consistent values were obtained  for the i n i t i a l slopes and P ]_ of the predicted P-Y curves. For t  plane strain analyses, P -j^ = 12.1cD with an i n i t i a l slope of 1.6E.. Values of 6.1cD and 1.0E. were obtained for plane stress 1 • l v  analyses. Moreover, in examining the effects of pile diameter i n Section 9.3.2, plots of P/D vs. Y/D were shown to be identical for a l l values of D. Based on these observations, non-dimensional plots of P/cD vs. Y/D were drawn. The results are as expected. A unique curve was obtained for each of the plane strain and plane stress condition /  as shown in Figures 10.1 and 10.2. The normalized curves are compared in Figure 10.3. The non-dimensional plots are useful for general design purposes. Using the curves shown i n Figure 10.3, P-Y curves can be derived for circular piles of any diameter installed in a normally-consolidated clay with an undrained shear strength c. Care must be taken, however, to ensure that the curves are applied only to problems involving static loadings on single piles. Dynamic loadings and pile groups, or pile interaction effects, were not considered.  10.2.2  Simplified Method for Determining P-Y Curves To further simplify P-Y curve predictions, the normalized  curves in Figure 10.3 were divided into segments as illustrated in Figure 10.4. The steeply rising i n i t i a l portions of the curves reflect  i  0.0  0.05  i  i  0.1  i  i  i  0.15  1  0.2  1  1  0.25  1  1  0.3  1  1  0.35  Y/D FIGURE 10.3:  NORMALIZED P-Y CURVES FOR UNDRAINED CLAY  1  1  0.4  i  1  0.45  r  0.5  121  the s t i f f linear elastic s o i l response for small deformations. The curved middle portions describe progressive s o i l failure and consequently, the softening of the P-Y response. The final horizontal lines correspond to P-j_» at which soil failure i s complete and the u  t  pile deflects at constant load. The curves or curve segments shown in Figure 10.4 can be represented by simple mathematical equations. For the plane strain curve, the i n i t i a l response and the curved centre portion can be described by a hyperbolic equation of the form  _P =  Y/D  cD  10.1  a + R b(Y/D) a  a and b are constants related to the i n i t i a l slope of the curve and the ultimate s o i l resistance:  a = 1/initial slope b = l/(P /cD) ult  R i s an adjustment factor applied to correct for cutting off the a >• curve at (P ^ /cD)=12.1. In other words, the true ultimate u  t  resistance i s at (P , /cD)/R , where R i s less than 1. This ux t a a value i s reached, however, only at Y/D= eo and cannot be used for design purposes. The final segment of the normalized plane strain curve corresponding to complete s o i l failure i s represented by a j  horizontal line. To determine the value of R , a transformed plot of the plane g  strain curve was made. Equation 10.1 can also be written as  (Y/D)/(P/cD) = a + R b(Y/D)  10.1a  3.  The plot of (Y/D)/(P/cD) vs. (Y/D) in Figure 10.5 shows the expected straight line with slope = R b and vertical-axis intercept = a. Although the data for small values of Y/D corresponding to the i n i t i a l segment of the normalized curve do not show a linear relationship, the assumption of a hyperbolic f i t i s nonetheless sufficient. The value of R b determined from the plot i s 0.0731. For P ^ / ^ = 12.1, C  Q  u  t  R = 0.885. The i n i t i a l slope of the curve i s 1.6E./c. For a l r  E^/c = 200 assumed for the P-Y analyses, the theoretical value of a i s 0.003125. This value agrees well with a = 0.003 obtained from the transformed plot. The plane stress curve i s somewhat more complex. Aside from the straight lines describing the i n i t i a l and failure responses, the curved portion i s divided into two segments, a power function and a hyperbola. The hyperbola i s given by a modified form of Equation 10.1:  P = cD  Y/D  10.2  oca + R b(Y/D) a  In Equation 10.1, a i s defined as the inverse of the i n i t i a l slope of the P/cD-Y/D curve. This i s valid, however, only i f the i n i t i a l portion of the curve i s hyperbolic. For the plane stress case, the i n i t i a l segment of the curve i s a power function. A correction factor, oc, i s therefore required for a.  124  The  power f u n c t i o n f o r the i n i t i a l  curve segment i s g i v e n by  the e q u a t i o n  P/cD  = a'(Y/D) '  10.3  b  where a', b' = c o n s t a n t s  To determine  the v a l u e s o f the c o n s t a n t s f o r these e q u a t i o n s ,  transformed p l o t s of the P/cD F i g u r e 10.7  and Y/D  o f (Y/D)/(P/cD) v s . (Y/D)  d a t a were made. The f o r E q u a t i o n 10.2  plot i n  yields  the  expected s t r a i g h t l i n e f o r the h y p e r b o l i c curve segment w i t h s l o p e = R b and i n t e r c e p t =oCa. For R R  a  = 0.145  and P -,.,/cD =  6.1,  = 0.885 as f o r the plane s t r a i n c u r v e . For an i n i t i a l  E^/c = 200 and  a = 0.0028, oi =  s l o p e of  0.56.  F o r the power f u n c t i o n , E q u a t i o n 10.3  can be expressed  as  log(P/CD) = l o g a' + b'log(Y/D)  The p l o t o f log(P/cD) v s . log(Y/D)  10.3a  i n F i g u r e 10.6  likewise yields a  s t r a i g h t l i n e w i t h s l o p e = b' and i n t e r c e p t = l o g ( a ' ) . Values o f b' = 0.693 and a' = 45.31  were o b t a i n e d from the  E q u a t i o n s f o r the f i t t e d a r e summarized i n T a b l e 10.1.  plane s t r a i n and plane s t r e s s  curves  The e q u a t i o n s do not d e s c r i b e the c u r v e s  p e r f e c t l y and s l i g h t d i s c o n t i n u i t i e s may segments. For p r a c t i c a l purposes, is  graph.  occur a t the ends o f the  smoothening out the c u r v e s by hand  s u f f i c i e n t , and would not l e a d t o s i g n i f i c a n t  errors.  0.028  Y/D FIGURE 10.5:  HYPERBOLIC FIT FOR PLANE STRAIN P-Y CURVES FOR UNDRAINED CLAY  -2.2  -2  -1.8  -1.6  -1.4 LOG  FIGURE 10.6:  -1.2  -1  (Y/D)  POWER FUNCTION FIT FOR PLANE STRESS P-Y CURVES FOR UNDRAINED CLAY  -0.8  0.028 0.026 0.024  -  0.022  -  0.02  Q  U  \ \ Q  0.018  -  0.016  -  0.745 = R b a  0.014 0.012  H  0.01 0.008 0.006 H  rhD  .  n  0.0028 ~ oca  0.004  i  -if  1  0.02  1  1— 0.04  —T~  0.06  0.08  0.1  0.12  0.14  0.16  Y/D FIGURE 10.7:  HYPERBOLIC FIT FOR PLANE STRESS P-Y CURVES FOR UNDRAINED CLAY ro  TABLE 10.1 SIMPLIFIED METHOD FOR DETERMINING P-Y CURVES FOR N.C. CLAYS  PLANE STRESS  PLANE STRAIN FROM  TO  EQUATION  (65.5c/E., P = Y/D c/1.6E + 0.073(Y/D) 12.1) cD  (0, 0)  FROM  (1.59c/E.,  (0, 0)  1  1  (65.5c/E., 12.1) X  (», 12.1)  Horizontal Line (P/cD = 12.1)  EQUATION  TO  1.59)  P = E. Y cD c D 1  (0.035, 4 . 4 4 )  P = 45.3(Y/D) cD  (0.035, 4 . A 4 )  (29.6c/E., 6.1)  P = cD  (29.6c/E , 6.1)  (<*>,  (1.59c/E., 1  1.59)  I  6.1)  0 . 5 6 0 ^  0,693  Y/D + 0.145(Y/D)  Horizontal Line (P/cD = 6.1)  Note: Co-ordinates given are for (Y/D, P/cD)  oo  10.3  SIMPLIFIED P-Y CURVES FOR DENSE SAND Plane strain and plane stress P-Y curves were predicted for a  dense sand using finite element analyses. The results are shown and discussed i n Section 9.4. To facilitate the prediction of such curves for other depths and for different soil properties and pile diameters, a simplified method for predicting P-Y curves was developed.  10.3.1  Normalized P-Y Curves The P-Y curves predicted for dense sand were shown to have  similar i n i t i a l slopes. A value of 1.08E^ for the i n i t i a l slopes of both plane strain and plane stress curves at any depth i s a good approximation. In examining the effects of the pile diameter on predicted P-Y curves, P/D vs. Y/D plots at a given depth were shown to be identical, regardless of the pile diameter used. The theoretical equations for P  given by Equations 9.9 and  9.10 are both functions of the pile diameter, D. In the plane strain equation (9.10), P ^ u  t  i s directly proportional to D. In the passive  wedge equation for plane stress deformations (9.9), D i s contained i n only two of the six terms in the equation. For  values of  used in the finite element analyses, the terms  containing D i n Equation 9.9 are relatively insignificant. However, for s o i l near the surface, H i s small, and the four terms not containing D decrease i n magnitude. At a depth of 2 f t , the "D terms" account for about half of the s o i l resistance. At H = 1 f t , the "D terms" account for roughly 2/3 of P ^ » and so on. Since the passive u  t  wedge equation i s valid only for shallow depths, P -  can be  considered as roughly proportional to D. Based on the above conclusions, the P-Y curves can be normalized by plotting P/P ^ u  vs t  « Y/D. And since P ^ u  t  i  s  proportional  to D, the effects of pile diameter are also eliminated. The normalized plots are shown in Figures 10.8 and  10.9.  Although differences can be observed in the shapes of the curves for different depths, a single curve for each of the plane strain and plane stress condition can be estimated. These normalized curves are compared in Figure 10.10. The normalized P-Y curves in Figure 10.10 are useful for design purposes. Given the basic s o i l parameters (ie: /zf', E^, P^ u  t  , etc.),  can be calculated using Equation 9.9 or 9.10 and the adjustment  factor A. Values of A determined by comparing P ^ u  the flow block and passive wedge models with P ^ u  t  t  predicted from obtained from the  finite element analyses in Chapter 9 are given in Table 9.4 and graphed in Figure 10.14. P-Y curves can then be derived for sand at any depth and for any pile diameter. A simplified method for determining these P-Y curves i s presented in the following sections.  10.3.2 Simplified Method for Determining P-Y Curves The normalized P-Y curves shown in Figure 10.10 can be divided into four sections as shown in Figure 10.11. Soil response prior to failure i s represented by three curves to best f i t the results predicted by finite element analyses. The strain-softening behaviour of the sand i s clearly illustrated. At P -j_ , or P/P ]_ = 1» the sand u  t  u  t  is assumed to f a i l completely. The P-Y relationship i s represented by a horizontal straight line, ignoring the small increases in soil  1.2  14 Y/D FIGURE 10.8:  NORMALIZED P-Y CURVES FOR DENSE SAND FROM PLANE STRAIN ANALYSES  y/D  FIGURE 10.9:  NORMALIZED P-Y CURVES FOR DENSE SAND FROM PLANE STRESS ANALYSES  CO  1.2  I  0  1  1  0.02  1  1  0.04 FIGURE 10.10:  1  1  0.06  1  Y/D  1  0.08  1  1  0.10  NORMALIZED P-Y CURVES FOR DENSE SAND  1  1  0.12  l  1  0.14  FIGURE 1 0 . 1 1 :  SIMPLIFIED NORMALIZED P-Y CURVES FOR DENSE SAND  resistance beyond P ]_ » The modelling of s o i l behaviour after u  t  failure by the finite element method i s questionable, and hence, the omission of the post-failure s o i l resistances predicted i n the analyses. The curved portions of the normalized P-Y responses can be described by power functions of the form  P/P _  = a'(Y/D) '  10.4  b  As in Section 10.2.2, plotting log(P/P ) vs. log(Y/D) yields a ult  straight line with a slope = b' and a log(P/P ^ )-axis intercept of u  t  log(a'). The log-log plots for the plane strain and plane stress curves are shown in Figures 10.12 and 10.13, respectively. The equations for the curves obtained from these plots, along with the method for determining the P-Y curves, are summarized in Table 10.2.  10.4  APPLICATION OF THE P-Y CURVES Using the simplified methods recommended in Sections 10.2.2 and  10.3.2, P-Y curves can be predicted for plane strain and plane stress conditions at any depth. For laterally loaded pile analyses, plane strain P-Y curves can be applied to the problem at large depths. Near the surface (ie: H < 2 f t ) , plane stress curves can be used. In the intermediate zone where both stresses and strains are threedimensional, combinations of the plane strain and plane stress curves are appropriate. Thompson (1977) concluded that a linear increase in the value of P ^ u  t  with depth, from the plane stress value at the  surface to the plane strain value at large depths, i s an adequate  TABLE 10.2 SIMPLIFIED METHOD FOR DETERMINING P-Y CURVES FOR DENSE SAND  PLANE STRAIN FROM  PLANE STRESS  TO (0.0028,  (0, 0)  FROM  EQUATION 0.29)  P/P  l  t  = 33.5(Y/D) *  (0.034, 0.87)  P/P  l  t  = 4.0(Y/D)  (0.034, 0.87)  (0.054,  P/P  l  t  = 2.31(Y/D)  (0.054,  (*>,  (0.0028,  0.29)  1.0)  1.0)  1.0)  0  81  0 , 4 5  0 , 2 9  Horizontal Line: P/P , =1 ult  TO  (0, 0)  (0.00136,  0.48)  P/P  l  t  = 190(Y/D)  0.76)  P/P  l  t  = 18(Y/D)  P/P  l  t  = 2.86(Y/D)  (0.00136,  0.48)  (0.00313,  (0.00313,  0.76)  (0.01,  (0.01,  1.0)  EQUATION  (*>,  1.0)  1.0)  0 , 9 1  0 , 5 5  0 , 2 3  Horizontal Line: P/P , =1 ult  Note: Co-ordinates given are for (Y/D, P / P ^ ) P , calculated from Equations 9.9 for plane stress and Equation 9.10 for plane s t r a i n . Values for the adjustment factor A i n the equations are given i n Table 9.4 or can be estimated from Figure 10.14. The i n i t i a l slopes of the curves derived from the above equations should be modified to the value of 1.08E.(D/P ). u  t  u l t  CO  139  approximation f o r r e a l s o i l  b e h a v i o u r . E x t e n d i n g t h i s method t o a l l  v a l u e s of P, p l a n e s t r a i n and p l a n e s t r e s s c u r v e s can then be added l i n e a r l y t o produce P-Y  approximations f o r intermediate s o i l  T h i s method i s i l l u s t r a t e d i n F i g u r e 10.15 h y p o t h e t i c a l depth o f 5 f t w i t h H For  c  depths.  f o r undrained c l a y a t a  = 7.5 f t .  undrained n o r m a l l y - c o n s o l i d a t e d c l a y s , the zone o f  t h r e e - d i m e n s i o n a l s t r e s s e s and s t r a i n s extends t o a depth g i v e n by Equation  H  9.5:  c  =  6D fl* 'D/cH  For  + J  dense sand, the l i m i t o f the t r a n s i t i o n zone can be e s t i m a t e d from  F i g u r e 10.14. The i n v e r s e o f the adjustment 9.4),  factor,  1/A  (see T a b l e  f o r plane s t r e s s a n a l y s e s i s shown t o i n c r e a s e w i t h depth  the plane s t r a i n c o n d i t i o n takes over. A c c o r d i n g l y , H be taken as the l i m i t  c  until  = 19 f t can  of the t r a n s i t i o n zone. The v a l u e o f H , c'  however, i s not c o n s t a n t , but i s a f u n c t i o n o f s o i l p r o p e r t i e s and pile  diameter.  140  FIGURE 10.14:  PLANE STRESS - PLANE STRAIN TRANSITION ZONE FOR DENSE SAND  PL.STRAIN  o.o  Y FIGURE 10.15:  P-Y CURVE FOR 3-DIMENSIONAL STRESS AND STRAIN CONDITION  CHAPTER 11 PREDICTION OF P-Y CURVES FROM PRESSUREMETER EXPANSION CURVES  11.1  INTRODUCTION In recent years, with the refinement of testing techniques and  the  increased sophistication of both the instrument and the data  acquisition system, the pressuremeter has seen increased use as a design tool. One obvious application of the pressuremeter test i s the design of laterally loaded piles. Since loads are applied to the surrounding s o i l i n much the same manner for both the pressuremeter and the lateral pile problem, similarities are expected i n their load-deformation characteristics. Various researchers have attempted to predict or derive P-Y curves from pressuremeter expansion curves. In most instances, the authors have suggested increasing the load component of the pressuremeter curves by some factor to yield P-Y curves for piles (Robertson et a l . (1983), Atukorala & Byrne (1984), and Robertson et a l . (1985)). Factors ranging from 1.9 to 2.6 were suggested for clays, and 1.4 to 1.7 for sands. Having thus determined conversion factors for the load component of the curves, uncertainties s t i l l existed as to the i n i t i a l slopes of the two curves. Using cavity expansion theory, the slope of the  pressuremeter curve was assumed to equal 2G. The i n i t i a l slope of  the P-Y curve, however, was essentially unknown. Values as low as 0.48E (Broms, 1964) and as high as 2.0E. (Pyke & Beikae, 1984) ±  j  were suggested by various researchers. Another uncertainty lies i n the difference i n size, or  143  diameter, between the pressuremeter c e l l and the piles. The validity of applying pressuremeter curves obtained from 3-inch diameter probes to problems involving piles with diameters often in excess of 2 f t was questionable and warranted investigation. To ascertain the i n i t i a l slopes of the curves and to determine the effects of large pile diameters on the load conversion factors, pressuremeter and P-Y curves predicted from finite element analyses were compared. The methods of analysis are as discussed i n Section 8.2 for pressuremeter expansion, and in Sections 3.2 and 9.2 for laterally loaded piles.  11.2  COHESIVE SOIL Pressuremeter and P-Y curves were predicted for a  normally-consolidated undrained clay. Soil properties used in the analyses are given in Tables 8.1 and 9.1. Comparisons were made for curves obtained for the clay at depths of 10 and 20 f t .  11.2.1  Pressuremeter Expansion Curves The predicted pressuremeter curves are shown in Figure 8.1. The  i n i t i a l slopes of the two curves are approximately 0.99(2G). As discussed in Section 8.3, the shape of the curves are similar, and the 20-foot curve i s , in fact, simply a scaled-up version of the 10-foot curve. The scaling factor of 2.0 suggests that a "family" of such curves for different s o i l depths can be normalized to produce a unique curve for the s o i l . Normalizing of the curves are discussed in Section 11.2.3.1.  144  11.2.2 P-Y Curves The P-Y curves predicted i n Chapter 9 were compared with the pressuremeter results. Both plane strain and plane stress P-Y curves were employed i n the comparisons. The curves are shown i n Figures 9.2 and 9.3. As noted i n Section 9.3.2, the i n i t i a l slopes and ultimate resistances of the curves are 1.6E. and 12.1cD for the plane strain I  *  curves, and l.OE^ and 6.1cD for the plane stress curves.  11.2.3  Comparison of Pressuremeter and P-Y Curves In order to compare directly the results of pressuremeter  expansion and lateral pile loading, the pressuremeter curves must be converted to equivalent P-Y plots. Since "P" i n the lateral pile problem represents soil resistance per unit length of pile, pressuremeter curves must be converted to plots of ^PD vs. Ar,  where D  is the diameter of the probe. To be correct, the current probe diameter, equal to D  Q  + 2Ar (D = i n i t i a l probe diameter), should  be used. However, for convenience i n converting pressuremeter results to P-Y curves, D i s taken as the i n i t i a l diameter. The modified pressuremeter curves and P-Y curves are compared i n Figures 11.1 and 11.2.  11.2.3.1  Normalized Curves  The plots in Figures 11.1 and 11.2 are valid for comparison only i f the pile diameter i s equal to the size of the pressuremeter c e l l (ie: about 3 inches). To account for the size difference, normalized plots must be compared. In Section 8.5, size effects for pressuremeters were eliminated by plotting strains, Ar/r  , instead  (X 1 0 ) 2  to  i  CO  1  1  1  i  r  r  i  1  1  r  r = 7 FT D = 2T PILE D = 2FT H = 10 FT 0  o co  0  to CM  Cl-CM Q  CU  o —-o—  to  0.0  0.02  0.04  0.06  0.08  DEFLECTION FIGURE 11.1:  0.1  0.12  Y or Ar  -+  PI.  -0 - O  PI. Stress P-Y Curve Pressuremeter Curve .  0.14  (ft)  Strain  0.16  P-Y Curve.  0.18  0.2  COMPARISON OF PRESSUREMETER AND P-Y CURVES FOR NORMALLY-CONSOLIDATED UNDRAINED CLAY  cn  "i  1  1  1  1  r  i  1  r  r = 7 FT o  D=  2r  PILE  D = 2 FT  H =  Q  20  FT  +-  0.02  0.04  0.06  0.08  DEFLECTION  0.1  0.12  Y or Ar  PI.  Strain  P-Y Curve  —O  PI.  S t r e s s P-Y Curve  •~Q Pressuremeter  ©-0.0  —h  0.14  0.16  Curve  0.18  0.2  (ft)  FIGURE 11.2: COMPARISON OF PRESSUREMETER AND P-Y CURVES FOR NORMALLY-CONSOLIDATED UNDRAINED CLAY  147  of displacements, AT. Since T = D/2, plotting Ar/T) w i l l likewise q  eliminate size effects. Similarly, size effects for P-Y curves were eliminated i n Section 9.3.3 through the use of P/D vs. Y/D plots (D i s the pile diameter). To further simplify analysis, fully normalized plots of P/cD vs. Y/D for the lateral pile problem, andAV/c vs. AT/D for pressuremeter expansion were made. The curves are presented i n Figure 11.3. As shown in Section 10.2.1 and discussed i n Section 11.2.1, the resulting curves are valid for the normally-consolidated clay at any depth and for any value of D.  11.2.3.2  Conversion Factors  Conversion factors were determined for the normalized curves shown in Figure 11.3. The use of normalized curves i s ideal since entire "families" of P-Y curves for different depths can be generated from the results of a single pressuremeter test. In converting pressuremeter curves to P-Y curves, care must be taken to ensure that the correct i n i t i a l slopes are obtained. The slopes obtained from the finite element analyses are as follows:  Plane strain P-Y curves: P/Y = 1.6E  ±  Plane stress P-Y curves: P/Y = 1.0E  ±  Pressuremeter curves from 3-D analysis:  AP/iAr/r^)  = 1.0(2G^)  To convert these values to slopes for the normalized curves, the following relationships were used:  CM  1  o  -  +-  [_  Q  1  i  r -+  ~o o  1  r  i  1  1  r  i  0.7  0.8  1  r  S t r a i n P-Y Curve Stress P-Y Curve Pressuremeter Curve  Pi PI  CL  <a O  CD  a  u C L _  -  CM  0.0  0.1  0.2  0.3  0.4  0.5  0.6  Y/D or Ar/D  0.9  1.0(X10" ) 1  FIGURE 11.3: COMPARISON OF NORMALIZED PRESSUREMETER AND P-Y CURVES FOR NORMALLY-CONSOLIDATED UNDRAINED CLAY 00  = 200c  (previous assumption)  G. = E./[2(l + u)] u = 0.498  I n i t i a l slopes for the normalized curves were thus calculated, yielding the following values:  Plane strain P-Y curves: P/cD = 1.6E. = 320 Y/D " T 1  Plane stress P-Y curves: P/cD = 1.0E. = 200 Pressuremeter curves: A?/c = 2(1.0)(2G.) = 267 AETD  C  X  ~  Conversion factors for the i n i t i a l slopes are therefore 1.20 for plane strain P-Y curves, and 0.75 for plane stress P-Y curves. Load factors were determined by simply taking the ratios of the normalized loads for various values of the normalized displacement. The factors are summarized in Table 11.1. For practical purposes, a plane strain P-Y curve load factor of 2.72 and a plane stress factor of 1.66 can be assumed for Y/D ^ 0.07 without significant errors.  11.3  COHESIONLESS SOIL Pressuremeter and P-Y curves were predicted for a dense sand  using the s o i l properties summarized in Tables 8.2 and 9.2. The method for determining K , tf , and 0" ' for the analyses are described in o ^ mo 3  Section 9.4.1. Curves obtained for the s o i l at depths of 10 and 20 f t were compared to determine factors for converting pressuremeter curves to P-Y curves.  TABLE 11.1 CONVERSION FACTORS FOR NORMALLY-CONSOLIDATED CLAYS  PLANE STRAIN P-Y CURVES  PLANE STRESS P-Y CURVES  I n i t i a l Slope  1.20  0.75  0.005  1.70  1.12  0.01  1.96  1.31  0.02  2.32  1.59  0.'03  2.53  1.72  0.04  2.60  1.74  0.05  2.65  1.72  0.06  2.69  1.69  0.07  Assume 2.72 2. 72  1.68  0.08  2.73  1.66  0.09  2.71  1.65  0.10  2.72  1.65  Y/D or Jr/D  Assume 1. 66  151  11.3.1  Pressuremeter Expansion Curves The predicted pressuremeter curves are shown in Figure 8.2. The  i n i t i a l slopes of the curves are approximately 0.98(2G). As discussed in Section 8.4, the 20-ft curve i s a scaled-up version of the 10-ft curve. A scaling factor of 1.66 was calculated.  11.3.2 P-Y Curves Plane strain and plane stress P-Y curves were predicted in Chapter 9. The i n i t i a l slopes of the P-Y curves are approximately For the value of p. = 0.222 calculated from the given s o i l  I. 08E^.  parameters, this i s equivalent to 1.32(2G), slightly higher than the value of 2G for the pressuremeter curves. The curves are compared in Figures 11.4 and 11.5.  11.3.3 Comparison of Pressuremeter and P-Y Curves As discussed in Section 11.2.3, pressuremeter curves were converted to plots of A V . B  vs. Ar to allow for proper comparisons with  P-Y curves. These curves are shown in Figures 11.4 and 11.5.  II. 3.3.1  Normalized Curves  To facilitate the direct comparison of pressuremeter and P-Y curves, the influences of the pile and pressuremeter diameters must be eliminated. This can be accomplished by normalizing the P-Y plots to give P/P ^ u  v s t  « Y/D as shown in Figures 10.8 and 10.9. For the  pressuremeter situation, size effect can be eliminated by plotting AV vs. AT/T , Q  as illustrated in Figure 8.4, or AV vs. Ar/D  (D = 2 r ) . To normalize the load component of the pressuremeter Q  (X10 ) 3  12 11  -  10 -  9  r =iFT D 2r n  'o  PILE D = 2 FT H - 10 FT  \  JO  .---0  r  Q  •  -<5  <J o  T 0.02  T  •  H O  + PI. Stress P-Y Curve O Pressuremeter Curve  T  0.04 DEFLECTION  o-  —I—  0.06 Y orhr  P I . Strain P-Y Curve  0.08  0.1  (ft)  FIGURE 11.4: COMPARISON OF PRESSUREMETER AND P-Y CURVES FOR DENSE SAND ro  (X10 ) J  20  r -iFT D - 2rc P/L/r D = 2 FT 0  H = 20  FT  0.  <  O  fs 9?  T  —I—  —I—  0.02  0.04  T  D  • P I . Strain P-Y Curve  T O  + PI. Stress P-Y Curve 0 Pressuremeter Curve  —I—  0.06  T  —  (  —  0.08  0.1  DEFLECTION Y or Ar (ft) FIGURE 11.5:  COMPARISON OF PRESSUREMETER AND P-Y CURVES FOR DENSE SAND cn oo  curves, an arbitrary reference value, P-^QI corresponding to a strain o f 1 0 % , was selected. P ^ Q values of 3 6 2 5 Psf and 6 0 7 5 Psf were estimated from the 1 0 - f t and 2 0 - f t curves shown i n Figure 8 . 2 . Fully normalized pressuremeter curves are thus given by plots of .dP/P^Q vs. /Ir/D. The normalized curves are shown i n Figures 11.6 and  11.7.  1 1 . 3 . 3 . 2 Conversion Factors Factors for converting pressuremeter curves to "P-Y" curves were determined using the normalized plots. The recommended values for the conversion factors are given in Table 1 1 . 2 . The i n i t i a l slopes of the normalized curves are as follows:  P-Y curves:  P/P  Pressuremeter curves:  P/P T7i  = 1.38(2G)(D) = 2.76(2G)  in u  = P/P = 2(2G) 772r R o 1n u  For the values of P ]_ and P^Q obtained from the analyses, the U  T  conversion factors, given by the ratio of the slopes above as ^IQ/Pulf  are listed in Table 11.2.  r = 7 FT D = 2r PILE D = 2 FT H = 10 FT Q  Q  •  o +-  O  —+  PI. Strain P-Y Curve P I . Stress P-Y Curve  O Pressuremeter Curve  T 0  0.02  0.04 Y/D  FIGURE 11.6:  orAr/D  COMPARISON OF NORMALIZED PRESSUREMETER AND P-Y CURVES FOR DENSE SAND  1.2  1.1 -+-  o  . <yxy'  Q. X Q.  < O  r  = 7 FT D = 2r  3  0  fi. N  D  o.  P/L£  D = 2 FT  H = 20  a-  o-  FT  —•  PI.  Strain  P-Y Curve  —+  PI.  S t r e s s P-Y Curve  —O  Pressuremeter  Curve  T  0.02  0.04 Y/D o r A r / D  FIGURE 11.7:  COMPARISON OF NORMALIZED PRESSUREMETER AND P-Y CURVES FOR DENSE SAND CJl Ol  TABLE 11.2 CONVERSION FACTORS FOR DENSE SAND  Y/D or  PLANE STRAIN P-Y CURVES  PLANE STRESS P-Y CURVES  H=10FT  H=20FT  RECOMMENDED  H=10FT  H=20FT  RECOMMENDED  Initial Slope  0.455  0.419  0.44  1.088  1.029  1.06  0.002  1.057  0.909  0.98  2.216  2.091  2.15  0.004  1.085  1.000  1.04  2.233  2.202  2.22  0.006  1.100  1.002  1.05  2.145  2.116  2.13  0.008  1.100  1.004  1.05  2.048  2.031  2.04  0.010  1.102  1.004  1.05  1.940  1.946  1.94  0.015  1.081  1.000  1.04  1.679  1.700  1.69  0.020  1.046  0.989  1.02  1.479  1.500  1.49  0.025  1.010  0.979  0.99  1.352  1.360  1.36  0.030  0.967  0.974  0.97  1.256  1.256  1.26  0.035  0.932  0.966  0.95  1.182  1.179  1.18  0.040  0.903  0.950  0.93  1.123  1.109  1.11  0.045  0.882  0.934  0.91  1.074  1.053  1.06  0.050  0.867  0.917  0.89  1.032  1.010  1.02  Jr/D  Assume 1. 00  Assume 2. 15  CHAPTER 12 SUMMARY AND CONCLUSIONS  The prediction of P-Y curves based on the results of finite element analyses was examined. Methods for determining the P-Y relationships for undrained clay and sand are presented. A new higher-ordered finite element program, CONOIL, was used in the analyses. The use of 6-noded linear strain triangular (LST) elements, coupled with the program's ability to handle Poisson's ratios as high as 0.499, permitted the accurate modelling of the undrained s o i l condition. Comparisons of the finite element predictions with closed form solutions for the cylindrical cavity expansion problem showed excellent agreements. The pressuremeter problem was analysed using the plane strain formulation for cavity expansion. Pressuremeter load-deflection curves were predicted for an undrained normally-consolidated clay and a dense sand. Having validated the finite element program's ability to model the cavity expansion problem, which bears some similarities to the laterally loaded pile situation, plane strain and plane stress P-Y curves were predicted for both undrained clay and sand. The i n i t i a l slopes of the plane strain and plane stress curves were confirmed to be approximately 1.6E^ and l.OE^, respectively, for clay, and 1.1E. for sand. P values of 12.1cD and 6.1cD were also obtained 1 ult for plane strain and plane stress loading in undrained clay. The value of 12.1cD i s in reasonably good agreement with the value of 10.82cD obtained from plasticity theory. P .  for sand was shown to be  fractions of the theoretical values determined from assumed failure mechanisms. Normalized plots of the P-Y relationships, P/cD vs. Y/D for clay, and P/P ^ u  v s t  * ^/D for sand, were also shown to reduce  families of curves for a l l pile diameters (D) and s o i l depths to unique curves for each of the plane strain and plane stress conditions. Finite element results for the pressuremeter and laterally loaded pile problems were also compared. Scaling factors were determined from the comparison of normalized curves to convert pressuremeter curves to P-Y curves. Factors ranging from 1.70 to 2.72 were obtained for plane strain curves, and 1.12 to 1.66 for plane stress curves for undrained clay. For dense sand, conversion factors of 0.89 to 1.00 were determined for plane strain P-Y curves, and 1.02 to 2.15 for plane stress curves.  160  BIBLIOGRAPHY  Atukorala, U., and Byrne, P.M., "Prediction of P-Y Curves from Pressuremeter Tests and Finite Element Analyses", Soil Mechanics Series No. 66, Dept. of C i v i l Eng., U.B.C., July, 1984 Baguelin, F., Frank, R., and Said, Y.H., "Theoretical Study of Lateral Reaction Mechanism of Piles", Geotechnique, Vol. 27, No. 3, Sept., 1977, pp. 405-434 Barton, Y.O., Finn, W.D.L., Parry, R.G.H., and Towhata, I., "Lateral Pile Response and P-Y Curves from Centrifuge Tests", Proceedings, 15th O.T.C., Houston, Texas, 1983, Vol. 1, pp. 503-508 Briaud, J.-L., Smith, T.D., and Meyer, B.J., "Using the Pressuremeter Curve to Design Laterally Loaded Piles", Proceedings, 15th O.T.C., Houston, Texas, 1983, Vol. 1, pp. 495-502 Broms, B.B., "Lateral Resistance of Piles in Cohesive Soils", JSMFD, ASCE, Vol. 90, SM2, March, 1964, pp. 27-63 Byrne, P.M., and Cheung, H., "Soil Parameters for Deformation Analysis of Sand Masses", Soil Mechanics Series No. 81, Dept. of C i v i l Eng., U.B.C., June, 1984 Byrne, P.M., and Eldridge, T.L., "A Three Parameter Dilatant Elastic Stress-Strain Model for Sand", Proceedings, International Symposium on Numerical Models in Geomechanics, Rotterdam, Sept., 1982, pp. 73-80 Byrne, P.M., and Grigg, R.F., "OILSTRESS: A Computer Program for Nonlinear Analysis of Stresses and Deformations in Oilsands", Soil Mechanics Series No. 42, Dept. of C i v i l Eng., U.B.C., July, 1980 Byrne, P.M., and Grigg, R.F., "Documentation for LATPILE.G", C i v i l Engineering Program Library, Dept. of C i v i l Eng., U.B.C., 1982 Byrne, P.M., Vaid, Y.P., and Samarasekera, L., "Undrained Deformation Analysis Using Path Dependent Material Properties", Soil Mechanics Series No. 58, Dept of C i v i l Eng., U.B.C., May, 1982 Campanella, R.G., and Robertson, P.K., "Applied Cone Research", Soil Mechanics Series No. 46, Dept. of C i v i l Eng., U.B.C., May, 1981 Craig, R.F., Soil Mechanics, 2nd ed., Van Nostrand Reinhold Co., New York, 1978  161  Denby, G.M., "Self-Boring Pressuremeter Study of the San Francisco Bay Mud", Ph.D. Thesis, Dept of C i v i l Eng., Stanford University, California, July, 1978 Duncan, J.M., Byrne, P.M., Wong, K.S., and Mabry, P., "Strength, Stress-Strain and Bulk Modulus Parameters for Finite Element Analysis of Stresses and Movements in Soil Masses", Report No. UCB/GT/80-01, Dept. of C i v i l Eng., Univ. of Calif., Berkeley, Aug. 1980 Duncan, J.M., and Chang, C.Y., "Nonlinear Analysis of Stress and Strain in Soils", JSMFD, ASCE, Vol. 96, SM5, Sept., 1970, pp. 1629-1653 Eldridge, T.L., "Pressuremeter Tests in Sand: Effects of Dilation", M.A.Sc. Thesis, Dept. of C i v i l Engineering, U.B.C, Feb., 1983 Focht, J.A., Jr., and McClelland, B., "Analysis of Laterally Loaded Piles by Difference Equation Solution", The Texas Engineer, Texas Section, ASCE, 1955 Gibson, R.E., and Anderson, W.F., "In-Situ Measurement of Soil Properties with the Pressuremeter", C i v i l Eng. and Public Works Review, Vol. 56, No. 658, May, 1961, pp. 615-618 Goodman, R.E., Taylor, R.L., and Brekke, T.L., "A Model for the Mechanics of Jointed Rock", JSMFD, ASCE, Vol. 94, SM3, May, 1968, pp. 637-659 Hughes, J.M.O., "Pressuremeter Testing", Lecture Notes for C.E. 577, Dept. of C i v i l Eng., U.B.C, Nov., 1979 Hughes, J.M.O., Wroth, CP., and Windle, D., "Pressuremeter Tests in Sand", Geotechnique, Vol. 27, No. 4, Dec, 1977, pp. 455-477 Janbu, N., "Soil Compressibility as Determined by Oedometer and Triaxial Tests", Proceedings, European Conference on Soil Mechanics and Foundation Engineering, Wiesbaden, Germany, Vol. 1, 1963, pp. 19-25 Matlock, H., "Correlations for Design of Laterally Loaded Piles in Soft Clay", Proceedings, 2nd OTC, Houston, Texas, 1970, Vol. 1, pp. 577-594 Peck, R.B., Hanson, W.E., and Thornburn, T.H., Foundation Engineering, 2nd ed., John Wiley & Sons, New York, 1974 Potyondy, J.G., "Skin Friction Between Various Soils and Construction Materials", Geotechnique, Vol. 11, No. 4, Dec, 1961, pp. 339-353  162  Pyke, R., and Beikae, M., "A New Solution for the Resistance of Single Piles to Lateral Loading", ASTM, STP 835, 1984, pp. 3-20 Randolph, M.F., and Houlsby, G.T., "The limiting Pressure on a Circular Pile Loaded Laterally in Cohesive Soil", Geotechnique, Vol. 34, No. 4, Dec, 1984, pp. 613-623 Reese, L.C, Discussion on "Soil Modulus for Laterally Loaded Piles", by McClelland, B., and Focht, J.A., Jr., Transactions, ASCE, Vol. 123, 1958, pp. 1071-1074 Reese, L.C, "Laterally Loaded Piles: Program Documentation," JGED, ASCE, Vol. 103, GT4, April, 1977, pp. 287-305 Reese, L.C, Cox, W.R., and Koop, F.D., "Analysis of Laterally Loaded Piles in Sand", Proceedings, 6th OTC, Houston, Texas, 1974, Vol. 2, pp. 459-472 Reese, L.C, Cox, W.R., and Koop, F.D., "Field Testing and Analysis of Laterally Loaded Piles in Stiff Clay", Proceedings, 7th OTC, Houston, Texas, 1975, Vol. 2, pp. 671-690 Reese, L.C, and Sullivan, W.R., "Documentation of Computer Program C0M624", Dept. of C i v i l Eng., Univ. of Texas, Austin, 1980. Robertson, P.K., "In-Situ Testing of Soil With Emphasis on Its Application to Liquefaction Assessment", Ph.D. Thesis, U.B.C, December, 1982 Robertson, P.K., Campanella, R.C, Brown, P.T., Grof, I., and Hughes, J.M.O., "Design of Axially and Laterally Loaded Piles Using In-Situ Tests: A Case History", Preliminary Draft, May, 1985 Robertson, P.K., Hughes, J.M.O., Campanella, R.C, and Sy, A., "Design of Laterally Loaded Displacement Piles Using a Driven Pressuremeter", Soil Mechanics Series No. 67, Dept. of C i v i l Eng., U.B.C, May, 1983 Scott, R.F., Foundation Analysis, Prentice-Hall Inc., Englewood C l i f f s , N.J., 1981 Stevens, J.B., and Audibert, J.M.E., "Re-Examination of P-Y Curve Formulations", Proceedings, 11th OTC, Houston, Texas, 1979, Vol. 1, pp. 397-403 Terzaghi, K., and Peck, R.B., Soil Mechanics in Engineering Practice, 2nd ed., John Wiley & Sons, New York, 1967  163  Thompson, G.R., "Application of the Finite Element Method to the Development of P-Y Curves for Saturated Clays", M.S. Thesis, Univ. of Texas, Austin, May, 1977 Vaid, Y.P., "Effect of Consolidation History and Stress Path on Hyperbolic Stress-Strain Relations", Soil Mechanics Series No. 54, Dept. of C i v i l Eng., U.B.C., Dec, 1981 Vaziri, H., Ph.D. Thesis, Dept. of C i v i l Eng., U.B.C., 1985 Yan, L., Forthcoming M.A.Sc. Thesis, Dept. of C i v i l Eng., U.B.C., 1986 Yegian, M., and Wright, S.G., "Lateral Soil Resistance-Displacement Relationships for Pile Foundations in Soft Clays", Proceedings, 5th OTC, Houston, Texas, 1973, Vol. 2, pp. 663-676  APPENDIX A DERIVATION OF STRESS-STRAIN RELATIONSHIPS FOR UNIAXIAL LOADING  A.l  PLANE STRAIN CONDITION The general s t r e s s - s t r a i n relationship for s o i l i s given by the  equations  £  ^  y  = 0" - ulT - u(T  v  = z  (  E  x  z  A.la  z  s -  -  r  r  A  '  l  b  E s where u  = Poisson's  ratio  E  = secant e l a s t i c modulus  For plane s t r a i n analyses, <5 = 0. Hence, z  <r = }i«r z  x  + (T )  A.2  y  Using ff" and £ to represent changes i n stress and s t r a i n , ff" = 0 for x  the u n i a x i a l loading condition considered (see Figure 6.1).  (T = u(T z  and  A.3  y  = r - u(utr) = <T(1 - ;i )  £  2  y  Therefore,  y________y_ E  y_ E  ^^  A.4  165  For a homogeneous, isotropic non-linear elastic material, u varies with the elastic ( E ) and bulk ( B ) modulii according to  ju =  3B -  A.5  E  6B  Substituting E  G  -n = 2  1  for E ,  (9B -  E )(3B + E  s  )  A.6  s  36B  2  Finally,  £  y  = (9B -  _  E )(3B + E  s__  36B E  ) <T s_ y  A.7  2  s  In the hyperbolic stress-strain model for soil given by Duncan & Chang  (1970),  the secant elastic modulus can be expressed as  E = E . [ I - R(<r /<r )] S  f  d  df  where E. l  = i n i t i a l elastic modulus = failure ratio  (T^  = deviator stress = (f^ for uniaxial loading in the Y-direction  ""df  =  deviator stress at failure  A.8  166  The i n i t i a l elastic modulus given by Janbu (1963) i s  E. = k P (<r /P ) a  E  3  A.9  n  a  where kg = elastic modulus number n P  = elastic modulus exponent a  = reference pressure = atmospheric pressure = 2116.2 Psf  (Tg = minor principal stress  And lastly, according to the Mohr-Coulomb failure criterion,  fl^f = 2c cos* +2 0^ sine/  A. 10  1 - sin*  For undrained clay, 0=0  and n = 0. Equations A.9 and  A.10  reduce to  E.=k P E  and  A.9a  a  <Tj£ = 2c  A. 10a  For sand, c' = 0 and  °df V = 20  '  sin|zf  A,10b  1 - sin*'  The complete stress-strain relationship for the plane strain uniaxial loading condition i s therefore given by Equations A.7 to A.10  A.2  PLANE STRESS CONDITION The derivation of the stress-strain relationship for plane  stress loading i s similar to that of the plane strain condition. Using the same general stress-strain equation as before,  E  For the plane stress condition, |T = 0, and for uniaxial loading, z  (T =0, and (T, = rj~ . Therefore, x ' d y  y  -y E  S  iy. E.[i - R (ir /«r )] f  y  d f  E. and U~. are as given by Equations A.9 and A. 10. f  APPENDIX B EMPIRICAL P-Y CURVES  B.l  MATLOCK'S EMPIRICAL CURVES FOR CLAY The empirical curves shown i n Figures 9.2 and 9.3 were  determined according to the method proposed by Matlock (1970) for static loading of single piles. The curves are defined by two parameters, ^ ^ t d Y , given by a n  u  £  P,.=NcD, ult P and  N = 3 + lT'/c + J H/D P  v  Y = 2.5 c 50  fi- D  B.l B.2  n  The value of N l i e s between 3 and 9. P  Using recommended values of <£^Q = 0.010 and J = 0.35 for the s o i l properties used in the finite element analyses (Table 9.1), the following values were determined for a pile diameter of 2 feet:  Depth H (ft)  N  p  P  u l t  (lb/ft)  Y (ft) c  2  7.35  448  0.05  5  7.88  1202  0.05  10  8.75  2669  0.05  20  9.00  5490  0.05  40  9.00  10980  0.05  The empirical curve, as defined by the above parameters, i s shown in Figure B.l. The equation for the curved portion of the P-Y  FIGURE B . l :  MATLOCK'S EMPIRICAL P-Y CURVE FOR STATIC LOADING OF PILES IN UNDRAINED CLAY  (After Matlock, 1970, p.  591) CTl  170  relationship i s  P/P  B.2  l t  = 0.5(Y/Y )  0,333  c  B.3  REESE'S EMPIRICAL CURVES FOR SATURATED SAND The empirical P-Y curves shown i n Figures 9.8 and 9.9 were  determined according to the method recommended by Reese at a l . (1974) for static loading of single piles. The curves are defined by three points, k, m, and u, as shown in Figure B.3. Point u i s given by  P = P = A P u ult c  B.4  where P i s the theoretical ultimate soil resistance determined c according to Equation 2.2 or 2.3. The smaller of the two values i s used. Values for the adjustment factor, A, determined by comparing theoretical values with experimental results, are given i n Figure B.2a. The corresponding deflection at point u i s 3D/80. Point m i s given by  P = B P m c Y = D/60 m  Values for B were also determined experimentally and are given in Figure B.2b. Point k i s defined by the intersection of the i n i t i a l linear segment and the parabolic portion of the P-Y relationship. The  B.5  a) Non-Dimensional C o e f f i c i e n t A for S o i l Resistance  Ultimate  b) Non-Dimensional C o e f f i c i e n t B for S o i l Resistance  Note: x = depth b - p i l e diameter FIGURE B,2;  NON-DIMENSIONAL COEFFICIENTS FOR SOIL RESISTANCE (After Reese et a l . , 1974, p. 482)  B.3:  EMPIRICAL P-Y CURVES FOR STATIC LATERAL LOADING OF PILES IN SATURATED SAND (After Reese et a l . , 1974, p. 482)  equation of the parabola i s  P = CY  1  /  B.6  n  n = P /mY m m C = P /Y m m  1  /  n  m = slope of l i n e between points m and u = (P - P )/(Y - Y ) u rrr u nr x  v  The slope of the i n i t i a l portion of the curve i s k H. Values recommended for k are 20, 60, and 125 l b / i n s and dense sand, respectively.  for loose, medium,  Using the s o i l properties l i s t e d i n Table 9.2 for the dense sand, values for P were calculated: c  Depth H (ft)  P C t  (Eq.2.2) (lb/ft)  P C d  (Eq.2.3) (lb/ft)  A  B  2  3033  5  12217  98105  1.25 0.87  10  38426  170467  0.88 0.50  20  128448  286834  0.88 0.50  484331  0.88 0.50  40  —  2.13 1.55  Values for A and B estimated from Figure B.2 for a p i l e diameter of 2 feet are also tabulated above. The P-Y curves determined from the above equations are shown i n Figures 9.8 and 9.9.  

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