UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1987_A7 S53.pdf [ 6.12MB ]
Metadata
JSON: 831-1.0062921.json
JSON-LD: 831-1.0062921-ld.json
RDF/XML (Pretty): 831-1.0062921-rdf.xml
RDF/JSON: 831-1.0062921-rdf.json
Turtle: 831-1.0062921-turtle.txt
N-Triples: 831-1.0062921-rdf-ntriples.txt
Original Record: 831-1.0062921-source.json
Full Text
831-1.0062921-fulltext.txt
Citation
831-1.0062921.ris

Full Text

PREDICTION OF P-Y CURVES FROM FINITE ELEMENT ANALYSES By JAIRUS LAI YAN SHE B . A . S c , Univers i ty 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 th is thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November 1986 ©JAIRUS LAI YAN SHE, 1986 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of C i v i l Engineering The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date December 18, 1986 3 E - 6 (3/81) ABSTRACT The p r e d i c t i o n of P-Y curves f o r undrained c lay and sand based on the r e s u l t s of f i n i t e element analyses i s presented in t h i s t h e s i s . A h igher-ordered f i n i t e element program was used in the ana lyses . The a b i l i t y of the program to accurate ly model the undrained s o i l condi t ion was v e r i f i e d by comparing pred ic ted l o a d - d e f l e c t i o n responses with c losed form so lu t ions f o r the c y l i n d r i c a l cav i ty expansion problem. Pressuremeter curves were pred ic ted from plane s t r a i n axisymmetric f i n i t e element ana lyses . The e f f e c t of pressuremeter s i ze on the pred ic ted r e s u l t s was examined. P-Y curves were predic ted f o r plane s t r a i n and plane s t ress c o n d i t i o n s . Values fo r the i n i t i a l slope and P u n- of the curves were obta ined. The curves were normalized f o r comparison, and s i m p l i f i e d methods presented for determining P-Y curves . F i n i t e element p red ic t ions fo r the pressuremeter and l a t e r a l l y loaded p i l e problems were a lso compared. Factors were determined from these comparisons to generate P-Y curves from pressuremeter curves . TABLE OF CONTENTS Page Abstract i i Table of Contents i i i List of Tables v i i List of Figures v i i i Acknowledgements x i i i 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 i i 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 41 Chapter 6 Stress-Strain Relationship 6.1 Introduction 44 6.1.1 Stress-Strain Relationship 44 6.2 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 63 i v Page 7.3 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 Chapter 8 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 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 Chapter 10 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 129 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 135 Chapter 11 Prediction of P-Y Curves from Pressuremeter Expansion Curves 11.1 Introduction 142 11.2 Cohesive So 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 So i l 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 Chapter 12 Summary and Conclusions 158 Bibliography 160 Appendix A Derivation of Stress-Strain Relationships 164 for Uniaxial Loading Appendix B Empirical P-Y Curves 168 vi LIST OF TABLES Table Page 3.1 Effect of Soil-Pile Adhesion on P - for ult Plane Strain Analysis of Undrained Clay 30 4.1 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 v i i LIST OF FIGURES Figure Page 1.1 Typical P-Y Curve 3 3.1 Zone of Soil-Pile Interaction 15 3.2 Finite Element Mesh for the Laterally Loaded Pile Problem 17 3.3 Effect of Mesh Radius on Plane Strain P-Y Curve Predictions 20 3.4 Effect 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 Failure Mechanisms for Laterally Loaded Pile Problem 26 3.7 Effect of Soil-Pile Adhesion on Plane Strain P-Y Curve Predictions 27 3.8 Effect of Soil-Pile Adhesion on Plane Stress P-Y Curve Predictions 28 4.1 Deformation of Finite Element Mesh During Lateral Pile Loading 32 4.2 Soil Pressure Distributions Around Laterally Loaded Piles 33 4.3 Finite Element Mesh with Interface Elements 36 4.4 Stress-Strain Relationship for Shear Along the Soil-Pile Interface 37 5.1 Higher-Ordered Elements 42 6.1 Test Elements 45 6.2 Plane Strain Stress-Strain Relationships for Undrained Clay 49 6.3 Plane Strain Stress-Strain Relationships for Sand 50 6.4 Plane Stress Stress-Strain Relationships for Undrained Clay 53 6.5 Plane Stress Stress-Strain Relationships for Sand 54 v i i i LIST OF FIGURES - Continued Figure Page 7.1 Finite Element Mesh for Plane Strain Cavity Expansion Analysis 56 7.2 Finite Element Mesh for Plane Strain Axisymmetric Cavity Expansion Analysis 57 7.3 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 79 8.2 Pressuremeter Curves for Non-Linear Elastic Dense Sand from Plane Strain Axisymmetric Finite Element Analyses 82 8.3 Comparison of Pressuremeter Curves Predicted Using Different Initial Cavity Radii 84 8.4 Comparison of Pressuremeter Curves Predicted Using Different Initial Cavity Radii 85 9.1 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 97 9.5 Effect of Pile Diameter on Plane Stress P-Y Curve Predictions for Undrained Clay 98 9.6 Effect of Mesh Radius on Plane Strain P-Y Curve Predictions for Undrained Clay 99 9.7 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 111 9.11 Effect of Pile Diameter on Plane Stress P-Y Curve Predictions for Dense Sand 112 9.12 Effect of Mesh Radius on Plane Strain P-Y Curve Predictions for Dense Sand 113 9.13 Effect of Mesh Radius on Plane Stress P-Y Curve Predictions for Dense Sand 114 10.1 Normalized P-Y Curve for Undrained Clay from Plane Strain Analyses 118 10.2 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 125 10.6 Power Function F i t for Plane Stress P-Y Curves for Undrained Clay 126 10.7 Hyperbolic F i t for Plane Stress P-Y Curves for Undrained Clay 127 10.8 Normalized P-Y Curves for Dense Sand from Plane Strain Analyses 131 10.9 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 136 10.13 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 Stress and Strain Condition 141 11.1 Comparison of Pressuremeter and P-Y Curves for Normally-Consolidated Undrained Clay 145 11.2 Comparison of Pressuremeter and P-Y Curves for Normally-Consolidated Undrained Clay 146 X T LIST OF FIGURES - Continued Figure Page 11.3 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 155 11.7 Comparison of Normalized Pressuremeter and P-Y Curves for Dense Sand 156 B.l 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 x i i 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 is extended to Ms. E. Seymour-G., for the typing of this manuscript. Financial support provided by the Natural Sciences and Engineering Research Council is 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 pi le subjected to la tera l loads i s one which requires the analysis of the interaction between s o i l and structural member. The behaviour of the structural member (the pile) i s governed by i t s strength and stiffness properties and those of the surrounding s o i l . Of prime concern are the bending moments, shear stresses, and displacements of the la tera l ly loaded p i l e . The ultimate load i s generally determined by the maximum moment and shear stress that develop in the p i l e , while the working load i s commonly governed by latera l displacements. The accurate determination of these quantities i s therefore essential for pi le foundation design. The behaviour of a la teral ly loaded pi le i s a three-dimensional problem. A complete analysis of the problem requires an examination of the complex stress-strain 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 three-dimensional. So i l behind the pi le may separate from the pi le surface, leaving a gap. Stresses at these shallow depths are essentially two-dimensional. At depths greater than several pi le diameters, stresses are three-dimensional and strains two-dimensional, with a l l displacements occurring in the horizontal plane (Scott, 1981). At intermediate depths, both stresses and strains are three-dimensional. To fac i l i ta te the analysis of this complex problem, simplified s o i l models were considered. Depending on the s o i l model, methods for predicting the behaviour of la tera l ly loaded piles can be c lass i f ied 2 into three categories (Atukorala & Byrne, 1984): 1. The Winkler Foundation Approach: the s o i l i s represented by a set of independent l i n e a r or non-linear springs d i s t r i b u t e d along the length 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 l i n e a r e l a s t i c , i s o t r o p i c and homogeneous continuum. 3. The F i n i t e Element Approach: the surrounding 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 possessing the s t r e s s - s t r a i n properties of the s o i l . At present, the Winkler approach i s most commonly used for analysing the response of 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 surrounding the p i l e i s replaced by di s c r e t e springs. S o i l resistance to p i l e d e f l e c t i o n i s represented by the lo 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 of the springs and are s p e c i f i e d by "P-Y" curves, where P i s the s o i l resistance to l a t e r a l p i l e displacement per unit length of p i l e at a given depth, and Y i s the corresponding h o r i z o n t a l p i l e d e f l e c t i o n at that depth. A t y p i c a l P-Y curve i s shown i n Figure 1.1. Since P-Y curves are defined by s o i l resistance, they can be expected to vary f o r d i f f e r e n t s o i l properties. The s i z e and shape of the p i l e section, and the roughness of the p i l e surface can also a f f e c t the P-Y r e l a t i o n s h i p . The key to the l a t e r a l l y loaded p i l e problem, then, l i e s i n the accuracy of the P-Y curves. Aside from instrumented p i l e loading f i e l d t e s t s , which are both c o s t l y and time-consuming, methods for determining P-Y curves based on the r e s u l t s of pressuremeter expansion LATERAL DEFLECTION Y (FT) F I G U R E 1 .1: T Y P I C A L P - Y C U R V E 4 t e s t s , centrifuge t e s t s , and f i n i t e element analyses have been proposed. Analysis of the l a t e r a l l y loaded p i l e problem by the Winkler approach with the use of P-Y curves can be performed by the f i n i t e d i f f e r e n c e method. The method and i t s formulation are described by Focht & McClelland (1955). Computer programs employing t h i s technique were developed by Reese (1977) and Reese & S u l l i v a n (1980) to perform the analyses. The use of the programs are f a c i l i t a t e d by the recent introduction of i n t e r a c t i v e graphics. A modified version of Reese & Su l l i v a n ' s program (C0M624) with graphic input and output c a p a b i l i t i e s i s c urrently i n use at the University of B r i t i s h Columbia (Byrne & Grigg, 1982). 1.2 SCOPE OF THESIS The purpose of the research i s to predict P-Y r e l a t i o n s h i p s for l a t e r a l l y loaded p i l e s using f i n i t e element analyses. P-Y curves predicted f or both cohesive and cohesionless s o i l s using non-linear e l a s t i c s o i l models are presented. A recently-developed higher-ordered two-dimensional f i n i t e element program was used i n the study. The program was not tested f or the type of analyses performed to predict P-Y curves. Consequently, to v e r i f y the accuracy 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 for u n i a x i a l compression of test elements 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 predicted and compared with closed form s o l u t i o n s . Load-deflection responses for pressuremeter expansion were also predicted using the plane s t r a i n axisymmetric f i n i t e element model 5 used i n the c a v i t y expansion analyses. The e f f e c t of pressuremeter s i z e was examined. P-Y curves were determined f o r both cohesive and cohesionless s o i l s . The s e n s i t i v i t y of the P-Y p r e d i c t i o n s to various 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 curves were normalized w i t h respect to 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 devised to generate P-Y curves 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 pressuremeter r e s u l t s to determine f a c t o r s f o r converting pressuremeter curves to P-Y curves. 1.3 ORGANIZATION OF THE THESIS This t h e s i s c o n s i s t s of twelve chapters. A b r i e f review of previous research, h i g h l i g h t i n g the methods and the r e s u l t s , i s given i n Chapter 2. Chapter 3 contains a d i s c u s s i o n of the fo r m u l a t i o n of the f i n i t e element model f o r a n a l y s i n g the l a t e r a l l y loaded p i l e problem to p r e d i c t P-Y curves. The v a l i d i t y of the for m u l a t i o n i s considered i n l i g h t of previous research on the problem. The importance of i n t e r f a c e elements i s discussed i n Chapter 4. Re s u l t s of previous work i n v o l v i n g the use of these s p e c i a l elements to model the 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 the i n t e r f a c e elements used i n t h i s study i s given. The f i n i t e element program used i n the research i s discussed b r i e f l y i n Chapter 5. Re s u l t s of u n i a x i a l compression and c a v i t y expansion analyses performed to v e r i f y the c a p a b i l i t i e s of the program 6 are presented in 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 is considered in 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 in Chapter 8. Factors for converting pressuremeter curves to P-Y curves were determined. A summary of the research and the conclusions is 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 its 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, is 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 is given in Section 2.2. The finite element method of analysis is discussed in 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 is 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 al. (1975) for stiff clays, and Reese et al. (1974) for sands. Though simple to use, these methods require estimates of the ultimate lateral soil resistances, ? u^> and 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 al., 1975). Values of P^ are calculated from equations derived by Matlock (1970) and Reese et al. (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 is = N cD P 2.1 N P 3 + c T " + J H , __y_ c D 3£ N ^ 9 P where c undrained strength D pile diameter overburden effective stress at depth H 9 J = coefficient ranging from 0.25 to 0.5, depending on the soil. A value of 0.5 is applicable for the soft offshore clay of the Gulf of Mexico, and 0.25 is valid for stiffer clays. For sand, the theoretical ultimate lateral resistance for wedge-type failure at shallow depths is given by P c t = JfH KQH taiytf' sin/3 + tan/? (D + H t a n ^ tanot) tan^-jzf') cos.* tan^-jzf') + KQH tan/? (tan;*' s i n ^ - tan*) - K^ D 2.2 And for lateral flow at greater depths, P . = fHD [K (tanJS-l) + K tan/f' tan*] C O . 3. O 2.3 where / = effective unit weight of the sand Kq = coefficient of lateral pressure at rest K = coefficient of active lateral pressure = tan2(45° - ^ 12) j6y = internal friction angle of the sand & = 45° + rff/2 Agreement between the theoretical P values and the values obtained 6 c from the pile load tests was poor, and consequently, P £ was adjusted by a factor A according to 10 = A P 2.4 c Values for the adjustment factor, A, were determined by dividing the experimental ultimate resistances by the theoretical Pc 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 al. (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 EI = stiffness of the pile z = depth The accuracy of P values determined according to Equation 2.6 is 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 2.5 and P 2.6 where n e x i s t i n the P-Y curves developed by t h i s method. The r e s u l t s of the centrifuge t e s t s suggest that P -j_ t values determined from Reese's equations (2.2 to 2.4) overestimate s o i l resistances at large depths and underestimate resistances near the ground surface. P was underestimated by f a c t o r s of approximately 1.9, 1.6 and 1.1 at depths of 2, 4 and 6 feet , r e s p e c t i v e l y . Values of P u^ t were not obtained from the centrifuge t e s t s f or depths greater than 6 f t . The shape of the centrifuge P-Y curves also d i f f e r s markedly from the shape of Reese's empirical curves. The i n i t i a l slopes of the empirical curves are much steeper than those of the centrifuge curves, and the curves f l a t t e n out much quicker than the centrifuge predictions at shallow depths. O v e r a l l , there i s l i t t l e resemblance between the P-Y curves predicted by the two methods. 2.2.3 F i n i t e Element Method The f i n i t e element method of analysis was used by a number of researchers to predict P-Y curves for sand and undrained c l a y . Plane stress f i n i t e element formulations were used to analyse the problem f o r shallow depths while plane s t r a i n formulations were used for greater depths. The models and formulations used by the researchers are s i m i l a r and are described i n Chapter 3. P-Y curves for undrained clay were predicted by Yegian & Wright (1973), Thompson (1977), and Atukorala & Byrne (1984). A wide range of values for P u 2 t were obtained by the researchers. Using i n t e r f a c e elements to model s o i l - p i l e i n t e r f a c e behaviour, and assuming a s o i l - p i l e adhesion f a c t o r , f , of 0.3 ( c a = ^ c c u ' s e e Section 4.2.2), a P u^ t value of approximately 12cD was 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 u^ t values 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 al. (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 ft, however, the finite element predictions were considerably less stiff than the centrifuge curves. No values were determined for Pu^t> nor was any 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 Pu^t« perhaps by as much as 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 soil 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 is questionable. On the other hand, many factors that could affect the predicted results (ie: mesh size, boundary conditions, pile diameter, interface properties, soil disturbances) were not considered in the finite element analyses. Consequently, the validity of the numerical P-Y curves is 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. Little work, however, has been done in this respect. Much additional research is 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 is 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 is 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 soil. A horizontal cross-section of unit thickness is taken of the pile and soil as shown in Figure 3.1. At a sufficiently large distance, R, away from the pile, the soil is 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, is of importance and is discussed in Section 3.2.2. The resulting finite element mesh is a circular disk with the pile located at the centre. The outer boundary of the disk is fixed, assuming zero displacements. Concentrated loads (P), representing lateral forces on the Limit of Pile Influence Sect ion 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 is the soil resistance per unit length of a pile subjected to a lateral displacement of Y. For the purpose of the finite element model, however, i t is 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 soil 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 in the soil must not approach levels where tensile failure occurs and causes the soil 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 in 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 soil 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 soil in two-dimensional finite element analysis, a finite zone of influence is assumed. In reality, the boundary of this zone is at infinity. In three-dimensional analysis where vertical load-spreading and soil displacements are possible, the boundary can be moved in from infinity to some finite radius without significant error. But, i f the soil is 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 soil 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 is incorrect, however, since soil is displaced laterally rather than radially. In his research, Thompson noted that the use of different mesh radii did not affect the predicted value of Pu^t» but did affect the in 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 is 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 al (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) is given by Y = P 1 + u 8?rE 1 - u (3 - 4u) ln •JI 2 R - r J2 . 2 R + r 3 - 4ju where P r lateral force (per unit length) on pile Poisson's ratio 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 al. to determine the value of R for the two-dimensional model that will give 0 l 1 1 1 1 0 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 No Separation Plane Stress FIGURE 3.4: 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 soil 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 is 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: where For flexible piles (h/l > 7/3): R = 71 o o For rigid piles (h/l < 7/3): R = 3h 3.2 h = embedded length 1 = soil-pile stiffness factor = 4E I /E o p p so E = pile modulus P I = moment of inertia for pile section P v = l/4jrr for circular pile E = i n i t i a l soil modulus so 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 Pu]_l-» 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, fairly 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 is 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 is displaced laterally under loads. For analyses at shallow depths, the plane strain formulation is invalid. At the ground surface, vertical stress is zero and displacements are three-dimensional. Consequently, the two-dimensional plane stress formulation is 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 Pu2_£ for sand (Reese et al., 1974). Based on Reese's results, Thompson (1977) determined that the depth at which plane strain becomes applicable for saturated clay is 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 is 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 soil 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 soil 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 u2 t with increasing adhesion are shown by the graphs. Randolph & Houlsby (1984), using plasticity theory, presented Direction of Pile Movement ) ^ Pile of / Diometer b <?4 • f 2 \ \ • / / t \ '///A//' \ * x-s • Movement Mud line Lood a) 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 soil are presented in Table 3.1. Again, a clear trend of increasing P ^ t with increasing adhesion (ie : o c ) is 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 u"|t FOR PLANE STRAIN ANALYSIS OF UNDRAINED CLAY a P u 1 t / c d 0-0 9-142 0-1 9-527 0-2 9-886 0-3 10-220 0-4 10-531 0-5 10-820 0-6 11-088 0-7 11-336 0-8 11-563 0-9 11-767 1-0 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 soil, elements exhibiting the appropriate soil-pile interface behaviour are needed. Figure 4.1 illustrates diagrammatically the response of the soil to lateral pile displacement. In the absence of interface elements, the soil is bound to the pile surface (Figure 4.1b). Movement of the pile forces the surrounding soil 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 in developing P-Y relationships, soil resistances and the stiffness of the load-deflection responses may be over-predicted. A better model of the soil-pile interaction allows for soil movements at the interface. As the pile is displaced laterally, high stresses develop in front of the pile while stress reductions occur at the back (Reese et al., 1974). This situation is illustrated in Figure 4.2. At large depths where the in i t i a l confining pressure is high (Figure 4.2b), tension stresses will not develop during loading. The resulting stress distribution may be as shown in 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 b) Mesh During Loading: So i l Bound to P i l e Surface -Large So i l Deformation and Resi stance c) Mesh During Loading: S o i l Allowed to Separate and Shear Along Side of P i l e - Less S o i l Deformation and Resistance FIGURE 4.1: DEFORMATION OF FINITE ELEMENT MESH DURING LATERAL PILE LOADING A Ii, a) L a t e r a l l y Loaded P i l e SECTION A-A b) So 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 Ins ta l l a t ion ) c) So i l Pressure D i s t r i b u t i o n During Lateral Loading - No Tension Stress Development d) S o i l Pressure D i s t r i b u t i o n During Latera l Loading -With Development of Tension Stresses Behind P i l e FIGURE 4.2: SOIL PRESSURE DISTRIBUTIONS AROUND LATERALLY LOADED PILES (Adapted from Reese et a l . , 1974, p. 481) however, the development of tension stresses behind the pile is possible (Figure 4.2d). Subsequent failure of the soil in tension leads to the formation of a cavity as illustrated in Figure 4.1c. In this situation, soil 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 soil to shear along the surface of the pile i f the pile skin friction is exceeded. There must also be soil-pile separation i f sufficiently large tension stresses develop during loading. Interface, or slip, elements were developed by Goodman et al. (1968) to model the behaviour of jointed rock masses. These elements were subsequently adapted for use in soil mechanics. Yegian & Wright (1973) employed curved interface elements in their finite element analysis of the laterally loaded pile problem. Interface properties were shown to have a noticeable effect on the predicted value of the ultimate soil 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 soil 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 soil, G, remains constant during shearing until C , s the maximum allowable shear stress, is reached. The value of G is determined from the i n i t i a l elastic and bulk modulii of the soil. The value of C g is 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 is given by 2^. For strains beyond / , the soil 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 its i n i t i a l value. In general, the strength of a soil is 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: 2 r = 2c cosS + 2 (T0 sinS 4.1 s 1 - sinS where (T^ is the minor principal stress. Potyondy (1961) has shown that c and S can be expressed as 3. fractions of c and respectively. In general, 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 is 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 soil cohesion (c) or the soil-pile adhesion (c ), c = etc a S =/8jf 4.2 and 4.3 TABLE 4.1 Proposod eoaffldanta of akin hlotluii bttwm •oils and construction materials [/<i->M. A - j . / c 0 " - ^ ^ ; without (actor of safety] Construction material Sand Cohesionless silt Cohesive g T a n u l a r soil Clay <H&<D< 2-0 mm 0-002<D<0-06 50% Clay + 50% Sand £><0 08 mm Surface finish of construction material Dry Sat. Dry Sat. Consist. I. = 10-0 5 Consist. Index: l-O-fl-73 Dense Dense Loose Dense /* /* /* J* J* f* A /cmu Steel -j^  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 Wood | Parallel to grain 0-76 085 0-92 0-55 0-87 0-80 0-20 0-60 0-4 0-85 At right angles to grain 0-88 0-89 098 063 0-95 0-90 0-40 070 0-50 0-85 Concrete -Smooth Made in iron form 0-76 0-80 0-92 0-50 0-87 0-84 0 42 0-68 0 40 1O0 Grained Made in wood form 0-88 0-88 0-98 0-62 0-96 0-90 0-58 080 0-50 1-00 Rough Made on adjusted ground 0-98 0-90 1-00 0-79 1-00 0-95 0-80 095 0-60 1-00 Note: f c 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 in 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 soil failure. The interface elements were considered to have failed whenever the maximum shear stress, given by 0^ /2, exceeded the skin friction, C g , or whenever the minor principal stress, (T^', became negative. Upon shear failure, the shear modulus was reduced to 0.001 of its 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 soil 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 is 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 is obvious. Program users can number the nodes the way they desire and the geometry program will do the work to minimize the bandwidth. Moreover, i f the same mesh geometry is 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 stiffer than the theoretical predictions but is to be expected as a result of the incremental elastic method of analysis used in the program. Trial analyses a) Linear Strain Trianaular Element 6 Nodes, 12 D.O.F. b) Cubic Stra in Tr ianqular Element 15 Nodes, 30 D.O.F. FIGURE 5 .1 : HIGHER-ORDERED ELEMENTS (After V a z i r i , 1 9 8 5 ) r o 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 is its ability to handle high Poisson's ratios. Values as high as 0.499 were used without encountering instability problems. The high values were useful in simulating undrained conditions (no volume change) for cohesive soils.. Other special features of CONOIL are its 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 is given by Vaziri in his doctoral dissertation (1985). 44 CHAPTER 6 STRESS-STRAIN RELATIONSHIP 6.1 INTRODUCTION The stress-strain relations of soil 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 is used. The model is described by Duncan et al. (1980). To verify the ability of the finite element program to correctly model the complex stress-strain behaviour of soil, a group of 4 linear strain triangular elements was tested. The test elements and the boundary constraints are shown in Figure 6.1. Uniformly distributed pressure loads, A(T^, 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 } 6.1 where {A(T} is the incremental stress vector {A&} is the incremental strain vector [ D ] is the stress-strain matrix FIGURE 6.1 : TEST ELEMENTS 46 [D] i s a function of the tangent Young's and bulk modulii, ET and V ET = E.(I - Rf(<rd/<rdf)2 B t - k B t V W" where E. 1 = i n i t i a l Young's modulus = k„P (fl*o/P ) ij s J a kE = Young's modulus number n = Young's modulus exponent kBt = tangent bulk modulus number m = tangent bulk modulus exponent P a = atmospheric pressure *3 = minor principal stress R f failure ratio fd = deviator stress 'df = 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: where £ = (9B - E )(3B + E ) <T 6.4 y _ s__ sj_ y 36B2E s = axial strain corresponding to the applied stress 0~y secant Young's modulus bulk modulus = k RP ((JT-/P ) m o a 3 a bulk modulus number bulk modulus exponent E g for the hyperbolic s t r e s s - s t r a i n model i s given by Duncan & Chang (1970) as E s = E . [ l - R f ( ( r y / c r d f ) ] where (T J £ = deviator stress at f a i l u r e The d e r i v a t i o n of 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 contained i n Appendix A. 6.2.2 Comparison of F i n i t e Element Results with Closed Form  Solution F i n i t e element t e s t s were performed f or both cohesive (0=0) and f r i c t i o n a l (c'=0) materials. S o i l parameters employed i n these analyses are tabulated i n Table 6.1. The properties l i s t e d f o r the cohesive material correspond to those of a normally consolidated undrained clay (based on Atukorala & Byrne, 1984) while the f r i c t i o n a l material properties are appropriate for a sand with a r e l a t i v e density of 75% (Byrne & Eldridge, 1982). The r e s u l t s of the analyses are shown i n Figure 6.2 for the undrained clay, and i n Figure 6.3 for the sand. Good agreements e x i s t E = s B = k B = m = 6.5 = 2c cosfi + 2(To s±ntf 1 - sinaf TABLE 6.1 SOIL PARAMETERS USED IN STRESS-STRAIN ANALYSES PARAMETER MATERIAL COHESIVE SOIL (Undrained Clay) FRICTIONAL SOIL (Sand) kE 72.1 750.0 n 0.0 0.5 kB 24.0 600.0 m 0.0 0.5 R f 0.9 0.9 fi 0.0 0.29 c (Psf) 305.0 0.0 *1 (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 P . atm (Psf) 2116.2 2116.2 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 its 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 stiffer than the theoretical curves predicted by Equation 6.4. This stiffness is 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 soil 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 u_y E E.[l - RJff s l f Y dfy J 6.6 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 stiffer 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 soil 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 al., 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 soil is 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 in Figure 7.2. Taking into account small discrepancies in 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 is 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 its 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 its simplicity, which permitted the use of smaller, and therefore, more elements in the domain without incurring excessive computing costs. Since soil 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) soil 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 soil 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 is of no consequence in subsequent calculations. A Kq 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 k E 144.1 144.1 n 0.0 0.0 kB 24021.0 24021.0 m 0.0 0.0 R f 0.0006 0.9 c (Psf) u 610.0 610.0 ^2 (deg) 0.0 0.0 Af6 (deg) 0.0 0.0 ^ c v (deg) 0.0 0.0 n 0.499 0.499 ''sat 123.4 123.4 Depth (ft) 40.0 40.0 c7* ' (Psf) vo 2440.0 2440.0 K o 1.0 1.0 P ,. (Psf) atra v ' 2116.2 2116.2 infinite medium and a Tresca failure criterion (ie: 1/2 (fX - (T ) = r 0 c ). In addition, soil in the plastic zone was assumed to deform at constant volume. Although no volumetric strain constraint was placed on the deformation of soil 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(fz = 0. Consequently, the mean normal stress, ff^, is constant during expansion and no change in volumetric strain occurs. The overall effect is that of constant volume deformation, applicable to the case of an undrained clay. Hughes' solution for small strains is given below: or where P = P = P + c o u P + c o u 1 + ln 1 + ln AT 2G r c o W f Ax 1 6BE r c 9B - E o u 7.1a 7.1b P = r P = o AT = T = pressure on wall of cavity i n i t i a l pressure on wall of cavity deflection of cavity wall 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: P = P + c r o u 1 + l n 'AT 2G < r c 7.2 where r = cu r r e n t r a d i u s of c a v i t y = r Q + Ar For t h i s equation, as r tends to i n f i n i t y , Ar/r approaches 1, and a l i m i t i n g pressure, 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 achieved: 61 P T = P + c [1 + ln(2G/c )] L o u L v u / J 7.3 Assuming E = 500c u f o r an undrained c l a y (u=0.5), P^ = P q + 6.8c u-The t h e o r e t i c a l e l a s t i c - p l a s t i c curve based on Equation 7.1a f o r the m a t e r i a l p r o p e r t i e s given i n Table 7.1 i s shown i n Figure 7.3. For AT < 0.001103 f t , or l n [ ( J r / r )(2G/c )] <C -1,4P = P - P i s o u r o negative. This apparent e r r o r i s caused by the ' l o g ' term i n the closed 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 behaviour of the s o i l p r i o r to 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 or i n i t i a l e l a s t i c p o r t i o n of the curve i s given by the 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 & Grigg, 1980): P = P + 2G(4r/r ) r o v o y 7.4 From Figure 7.3, p l a s t i c f a i l u r e can be seen to begin at A? = 605 Psf where the two curves meet. PRESSURE ON CAVITY WALL AP (psf) 29 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 is also shown in the graph. As expected, the i n i t i a l linear elastic behaviour of the elastic-plastic curves is absent and the overall pressure-deflection response is considerably softer. 7.2.2.1 Boundary Conditions Although the inherent nature of incremental elastic analysis is to predict responses somewhat stiffer 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 is 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 is shown in Figure 7.5. Although this curve deviates from an expected vertical line and indicates an increase in deflection with increasing pressure, the response is much stiffer than that predicted by Equation 7.1. The calculated deflections were found to be the (X 102) « I i i i i 1 1 i 1 1 1 1 r 0-0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 STRAIN A r / r 0 FIGURE 7.4: CAVITY EXPANSION CURVES FOR UNDRAINED CLAY 99 r e s u l t s of small volumetric s t r a i n s (£ < 0.5%) i n the mesh v elements owing to the i n a b i l i t y to use B = oo, and the s i m i l a r i t y i n i n i t i a l slope between the predicted curve and the t h e o r e t i c a l 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 with a d i f f e r e n t radius would have produced a curve with a d i f f e r e n t i n i t i a l slope. To model expansion i n an i n f i n i t e medium, the outer boundary was permitted to 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 Figure 7.2, creating a stress boundary where 4<T. = 0. The expansion curve predicted by t h i s method i s presented i n Figure 7.4. In using the stress boundary method, the r e s i s t i n g force of the s o i l beyond the radius of 100 f t i s omitted. As shown i n Figure 7.6, the r a d i a l s t r e s s , (T r, decreases with r a d i a l distance from the ca v i t y according to the equation derived by Hughes (1979): (T = P r ( a 2 / r 2 ) 7.5 where = pressure (or change i n pressure) on wall of c a v i t y a = radius of c a v i t y r = r a d i a l distance from centre of c a v i t y In the above analysis f o r an i n i t i a l c a vity radius of 1 f t , the r e s i s t i n g pressure omitted at the outer boundary (r = 100 f t ) increased from a value of 0.069 Psf at the s t a r t of p l a s t i c f a i l u r e at AP = 690 Psf (a = 1.0034 f t ) to 0.416 Psf at AF = 3060 Psf (a = 1.166 f t ) . The omission of t h i s r e s i s t i n g pressure i s thought to be the cause of the s l i g h t l y s o f t response of the f i n i t e element curve at low s t r a i n s (Ar/r^ < 0.14) i n Figure 7.4. Although t h i s error could have been reduced by extending the radius of the f i n i t e element mesh, WALL OF CAVITY 0.000136 Pr NOT TO SCALE Pr = 3060 PSF Q0=1FT Q = 7 . 7 6 6 FT E-P THEOftY (EQ.Z5) 80 RATIO 100 r / a G 120 140 160 180 FIGURE 7.6: VARIATION OF RADIAL STRESS WITH DISTANCE FROM CAVITY 68 i t was not considered necessary since the error introduced by using a mesh radius of 100 f t i s r e l a t i v e l y small 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 ulated f o r the stress boundary ( i n f i n i t e medium) analysis were approximately 150 times l e s s than those obtained i n the f i n i t e medium analysis (Figure 7.5). Consequently, d e f l e c t i o n errors r e s u l t i n g from volume changes can be discounted. 7.3 COHESIONLESS SOIL In the case of a cohesionless s o i l , f i n i t e element analyses were performed using only non-linear e l a s t i c material properties f o r a dense sand. 7.3.1 E l a s t i c - P l a s t i c Materials D i f f i c u l t i e s a r i s e i n the analysis of e l a s t i c - p l a s t i c materials due to the v a r i a t i o n of shear strength with the minor p r i n c i p a l s t r e s s , cT^', as follows: (T,f = 2 C = 2 CV sin0' 7.6 df max 3  1 - sinjzf' and ZTj; = (T^' sinks' costf 7.7 1 - s i n * 1 Moreover, tf varies with (T^' as given by Duncan et a l . (1980): tf = fi^ -A0 log«T 3 7 P A ) 7.8 As increasing loads are applied to a s o i l element, t7" ' v a r i e s , r e s u l t i n g i n changes i n the shear s t r e n g t h , s t r e s s l e v e l ( (T^/ lT^), and modulus values. Problems a r i s e as the s o i l approaches f a i l u r e . The r e d u c t i o n of modulus values a t f a i l u r e to 0.01 or 0.001 of t h e i r i n i t i a l values causes (T./^jr- to f l u c t u a t e above d di and below the f a i l u r e value of 1.0, l e a d i n g to e r r a t i c s o i l behaviour. An i l l u s t r a t i o n of the problem was presented by E l d r i d g e (1983) and i s shown i n Figure 7.7. Because of t h i s e r r a t i c behaviour, e l a s t i c -p l a s t i c analyses were not performed. Instead, a comparison was made between the no 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 the 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 Closed 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 the expansion 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 cohesionless medium was derived by Hughes et 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 of the 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 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 obtained: P R = P Q (1 + sinpf') E Ax 1 + ju r nP sinyj' ( l - N ) / ( l + n ) 7.9 o and N = (1 - s i n 0 ' ) / ( l + sin0') n = (1 - s i n v O / ( l + s i n v ) where ^' = f r i c t i o n angle of the cohesionless m a t e r i a l V = d i l a t i o n angle E = i n i t i a l Young's modulus = k„P ((ToVP ) n Ji a J a Modified Mohr Diagram FIGURE 7.7: STRESS PATH FOR FAILED SAND ELEMENT (Adapted from Eldr idge, 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 is given by A£ = -AK sinv* 7.10 For the condition of constant volume deformation, V = 0 and Equation 7.9 reduces to P = P (1 + sin*') r o E Ar 1 + ji r P q sin*' (l-N)/2 7.11 As in the case of the solution for cohesive materials, as Ar tends to infinity, Ar/r approaches 1, and the limiting pressure is given by P L = PQ (1 + sin*') 2G P sin*' o (l-N)/2 7.12 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 in i t i a l modulus values according to E i " kEPa«V/Pa>n G = k G W / P a > m 7.13 7.14 Since 0^' may vary considerably during loading, omitting its influences may lead to significant errors. Consequently, the e las t ic -p las t ic solution given by Equations 7.9 to 7.12 i s , at best, only an approximation of the real problem. The closed form solution for a dense sand i s shown in 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 in Table 7.2. As in the case of the cohesive s o i l , the i n i t i a l l inear elast ic portion of the curve i s given by Equation 7.4. Plast ic fa i lure i s shown to begin at A? = 1400 Psf. 7.3.3 Fin i te Element Prediction The results of a plane strain axisymmetric f in i te element analysis and the corresponding e last ic -p last ic closed form solution (j/ = 0) are shown in Figure 7.9. An outer stress boundary was used in the analysis, as discussed in Sec. 7.2.2.1. Material properties are given in Table 7.2. An unusually high kg value of 1250.0 was used in order to l imit volume changes and to fac i l i ta te comparison with the constant volume closed form solution. K was taken as 1.0 to model o the isotropic consolidation condition assumed in 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 last ic closed form solution in Equation 7.4. The expansion curve for non-linear material properties, however, lacks the i n i t i a l l inear e last ic 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 in fa i l ing to take into account the effects of (T^' as discussed in Section 7.3.2 may be the cause of the large discrepancy. TABLE 7.2 MATERIAL PROPERTIES FOR SAND PARAMETER NON-LINEAR ELASTIC COHESIONLESS MATERIAL D r (%) k E 75 750.0 n 0.5 k B 1250.0 m 0.5 R f 0.9 c (Psf) 0.0 *l (deg) 39.0 A0 (deg) 4.0 ^ c v ' (deg) 33.0 V (deg) 0.0 / sa t < P c f> Depth (ft) 0.4 122.4 20.0 (T ' (Psf) vo v K o P (Psf) atm v 1200.0 1.0 2116.2 PRESSURE ON CAVITY WALL AP (psf) O 20 40 60 80 100 (X102) ro PRESSURE ON CAVITY WALL AP (psf) o 5 10 15(X10 3) CHAPTER 8 PRESSUREMETER EXPANSION 8.1 INTRODUCTION The pressuremeter is essentially an expandable tube which is either pushed into the soil or inserted into a pre-bored hole in 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 soil parameters can be determined. The foregoing plane strain axisymmetric cavity expansion analysis is generally considered to be a good model for pressuremeter expansion tests. Recent research by Yan (1986), using three-dimensional 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 three-dimensional axisymmetric analyses were nearly identical to those obtained using the plane strain formulation described in 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 in Chapters 9 and 10 to determine a rational method for deriving P-Y curves from pressuremeter curves. The results are presented in 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 soil 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 wvo Undrained shear strength, c = 0.265 (T '= 15.25 H 6 u vo Initial Young's modulus, E. = 200 c = 3050 H ° 1 u For the undrained condition, n = 0 and E. = k^ P ^ ' I E atm therefore, k^ = 1.44 H MATERIAL TABLE 8.1 PROPERTIES FOR UNDRAINED N.C. CLAY NON-LINEAR ELASTIC COHESIVE SOIL PARAMETER DEPTH =10 FT DEPTH = 20 FT k E 14.4 28.8 n 0.0 0.0 k B 1200.0 2400.0 m 0.0 0.0 R f 0.9 0.9 *o 0.498 0.498 c u (Psf) 152.5 305.0 *1 (deg) 0.0 0.0 (deg) 0.0 0.0 cv (deg) 0.0 0.0 * sat (Pcf) 120.0 120.0 Depth (ft) 10.0 20.0 <r » vo (Psf) 576.0 1152.0 K o 1.0 1.0 P «. atm (Psf) 2116.2 2116.2 Taking U q = 0.498 for the undrained condition, bulk modulus, B = E./3(l - 2u ) = 83.3 E. and B = k„P _ for m=0 B atm therefore, kp = 83.3 kv = 120 H Using H=10 ft and H=20 ft 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 ft curve to be merely a scaled-up version of the H=10 ft curve. The scaling factor of 2.0 indicated by the predicted load-deflection values corresponds to the differences in the values of c y, kg and kg used in the two analyses. 8.4 COHESIONLESS SOIL Finite element analyses were also performed for cohesionless soil 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 ft curve being a scaled-up version of the H=10 ft 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 COHESIONLESS SOIL DEPTH=10 FT DEPTH=20 FT k E 1000.0 1000.0 n 0.5 0.5 k B 600.0 600.0 m 0.5 0.5 R f 0.8 0.8 *l (deg) 39.0 39.0 Atf (deg) 4.0 4.0 ffCy (deg) 33.0 33.0 D r (%) 75 75 0.222 0.222 u (Psf) 624.0 1248.0 ' s a t < P s f> 122.4 122.4 C (Psf) 330.0* 675.0* K o 1.0* 1.0* * Values assumed for f in i te element analyses (X103) STRAIN A r / r D FIGURE 8.2: 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 is given in Section 7.3.1. 8.5 SIZE EFFECT To simplify the conversion of radial displacements, Av, into strain values, AV/VQ, in the foregoing finite element analyses, an i n i t i a l cavity radius of 1 ft was assumed. The actual radius of the pressuremeter cell, however, is 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 ft were used. Figure 8.3 shows the AV vs. Av results of this analysis along with the curve for D = 2 ft (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 o o 0.0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 RADIAL DISPLACEMENT OF CAVITY WALL Ar (ft) FIGURE 8.3: 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 the use of the 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 of d e f l e c t i o n , Ar. Moreover, AY-Ar r e l a t i o n s h i p s can be obtained f o r pressuremeters of any s i z e (aspect r a t i o > 6) simply by m u l t i p l y i n g the Ar/r^ values generated from any a n a l y s i s by the new r values, o Based on the r e s u l t s shown i n Figure 8.4, the i n i t i a l c a v i t y or c e l l diameter can be assumed to have no i n f l u e n c e on the AV-Ar/r o r e l a t i o n s h i p . T h i s assumption, however, i s v a l i d only 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 isturbances. In p r a c t i c e , s o i l disturbance i s unavoidable and i t s e f f e c t s on the p r e s s u r e - d e f l e c t i o n r e l a t i o n s h i p d i f f i c u l t to p r e d i c t . CHAPTER 9 PREDICTION OF P-Y CURVES 9.1 INTRODUCTION P-Y curves for the laterally loaded pile problem were predicted using the finite element formulations described in Chapter 3. Plane strain and plane stress analyses for both undrained clay and sand were performed. The results are presented in 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 in the analyses is shown in Figure 9.1. As noted in 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 1Q)> R = 7 1 q applies. To ensure that deformations of the pile elements are insignificant relative to soil deformations (ie: rigid pile section), take E /E = 500. p so Also, I = l/47Tr4 = 1/647TD4 89 And 1 = [4(E /E )I ] 1 / 4 = 3.148 D o p so pJ Finally, R = 7 1 = 22.03 D Check: Embedded length of pile > 7/3 1 = 7.34 D = 14.7 ft for D = 2 f t . Therefore, assumption of flexible pile is reasonable. The mesh shown in Figure 9.1 is composed of triangular linear strain elements. A brief description of this element is given in Chapter 5. The higher-ordered 15-noded cubic strain elements were not used in 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 in sensitivity over the results obtained by using linear strain elements and the mesh in 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 in Table 9.1, and are identical to those used in 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 soil were used. TABLE 9.1 SOIL PARAMETERS FOR UNDRAINED N.C. CLAY PARAMETER VALUE kE 1.44 H n 0 kB 120 H m 0 n 0.498 R f 0.9 c (Psf) u v ' 15.25 H *sat <Pcf> 120.0 <TV' (Psf) 57.6 H (Tm' (Psf) 57.6 H K 1.0 o 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 Uq=0.498, this is equal 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 in 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 Pu]_t« Slightly steeper 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 soil resistance corresponding to Pu^t« Instead, at large displacements (Y > 0.7 ft for plane strain, and Y > 0.4 ft 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 is 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 ]_t = 6.1cD were determined for the plane stress curves. The continuing small increases in P beyond Pu- t^ is 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 i 1 1 r D= 2 FT 0-0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 9 10 Y (ft) FIGURE 9.2: 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 ( f t ) 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 Pu]_£ value of 12.1cD obtained for the plane strain case 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 is 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. Pu^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 is 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 where P , = N cD 9.4 ult p Np = 3 + (T7c + J H/D , 3 £ Np ^ 9 (T ' = effective overburden stress v H = depth of soil J = coefficient ranging from 0.25 to 0.5 depending on soil 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 = 6D 9.5 c • (0-v'D/cH) + 0.25 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 is 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 is poor, differing in both the i n i t i a l slope and the Pu- t^ values. 9.3.2 Effect of Pile Diameter The results shown in Figures 9.2 and 9.3 were obtained for a pile diameter, D, of 2 ft. 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 in the plane strain analyses are for the clay at a depth of 20 f t . Parameters appropriate for a depth of 2 ft 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 in Figures 9.4 and 9.5. The results suggest that an increase in 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 in 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 in Figure 9.7. Soil parameters for the clay at depths of 20 ft and 2 ft were used in 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. Initial 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 1 1 1 1 i r Y (ft) FIGURE 9.4; EFFECT OF PILE DIAMETER ON PLANE STRAIN P-Y CURVE PREDICTIONS FOR UNDRAINED CLAY § i 1 1 1 1 i 1 1 1 1 1 1 1 i i i i i i r 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 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^ t n o 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 soil properties are summarized in Table 9.2. The values are identical to those used in 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 soil 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 ) 9.7 X HI 3 <rm' = [(1 + 2KQ)/3] <rv' 9.8 where KQ = coefficient of lateral soil pressure at rest TABLE 9.2 SOIL PARAMETERS FOR DENSE SAND PARAMETER VALUE k E 1000 n 0.5 k B 600 m 0.5 R f 0.8 0.222 ^ ' (deg) 39.0 W (deg) 4.0 0CV} (deg) 33.0 D r (%) 75 122.4 (Pcf) 60.0 a ' (Psf) vo 60.0 H K o 1 - sine*' (deg) ^ ' - ^ ' l o g ( r m V P a ) <T ' (Psf) mo (1 + 2Ko)<r 7 3 Note: H = depth ( in feet) 103 i n t e r n a l f r i c t i o n angle Atf f r i c t i o n angle at P = 1 atm = 2116.2 Psf a P a change i n tf f o r a 10-fold increase i n 0" ' atmospheric pressure = 2116.2 Psf mean normal e f f e c t i v e stress v e r t i c a l e f f e c t i v e stress (T 1 was used as the i n i t i a l o v e r a l l confining pressure. The determination of If ' from Equations 9.6 to 9.8 involved an i t e r a t i v e process. A reasonable value f o r (T * was f i r s t estimated. Using t h i s value i n Equation 9.7, tf was determined. K q was then c a l c u l a t e d from Equation 9.6 and a new value f o r <T 1 determined from Equation 9.8. The new tf ' was substituted back in t o Equation 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 u n t i l the new (T ' was roughly equal to the old value. 9.4.1 Results Figures 9.8 and 9.9 show the P-Y curves predicted f o r a p i l e diameter of 2 f t . The i n i t i a l slopes f o r the plane s t r a i n curves range from 1.065E. to 1.071E., determined from AP/AY r a t i o s f o r small i i load increments equal to about 0.5% of Py-^-* For t n e plane stress analyses, i n i t i a l slopes of l.OOlE^ to 1.043E^ were obtained, using load increments of about 0.8% of Pu-j_t« T n e slope values are tabulated i n Table 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 reasonably assumed for both plane s t r a i n and plane str e s s P-Y curves predicted f o r any depths. For JI q=0.222, t h i s slope i s equal to 1.32(2G), somewhat s t i f f e r than the i n i t i a l slope of m m TABLE 9.3 RESULTS OF P-Y CURVE ANALYSES FOR DENSE SAND PLANE STRAIN PLANE STRESS DEPTH (ft) INITIAL SLOPE LOAD INCREMENT Y (f£) INITIAL SLOPE LOAD INCREMENT (lb7ft) Y 2 5 10 20 40 1.071 E. I 1.065 E. 1.065 E. 1.066 E. 0.31% P. 0.45% P 0.48% P. 0.48% P ult ult ult ult 6400 11000 21000 31500 0.080 0.170 0.142 0.118 1.001 E. 1.022 E. 1.043 E. 1.041 E. I 1.05% P 0.82% P. 0.65% P. 1.22% P ult ult ult ult 950 2450 4600 8150 0.0080 0.0225 0.0289 0.0365 107 0.98(2G) obtained f o r the pressuremeter curves i n Chapter 8. P u ^ t and the corresponding d e f l e c t i o n s , Y , f o r the P-Y p r e d i c t i o n s are a l s o given i n Table 9.3. As i n the analyses f o r c l a y , none of the P-Y curves e x h i b i t a peak value f o r P. Rather, 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 of f a i l u r e are observed. A c c o r d i n g l y , ~P ^  was taken as the load at 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 Curves E m p i r i c a l P-Y curves f o r sand are a l s o shown i n Fi g u r e s 9.8 and 9.9. The curves were determined according to the method recommended by Reese et a l . (1974). C a l c u l a t i o n s are shown i n Appendix B. P u ^ t f o r the e m p i r i c a l curves were determined using a passive wedge f a i l u r e mechanism at shallow depths, and a f l o w block model at l a r g e depths (see Figure 3.5). The equations derived from these models are given below: Pas s i v e wedge: P u l t - A>' H K H t a n ^ ' s i n ^ + tan/5 (D + Htan/5 tan*) t a n ^ - 0 ' ) cosoC tanfo-0') + K QH tan/J ( t a n ^ ' sin /5 - tan*) - IM)) 9.9 where Flow block: P u l t = A D ^ H ( K a ( t a n / * - 1) + K t a n * ' tan/*) 9.10 if' = e f f e c t i v e u n i t weight of the sand H = depth tf = internal friction angle o< =tf/2 /8 = 45° + oi K q = coefficient of lateral earth pressure at rest Ka = Rankine coefficient of active earth pressure = tan2(45° - oi) A = adjustment factor to correct for differences between field and predicted results Values of A determined by Reese et al. 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 is expected since the passive wedge failure mechanism is 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 ft 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 l 4_ VALUES u l t DEPTH tf <r 1 mo K o K a P u l t from p ^ult from A f see"] (ft) (deg) (Psf) THEORY FEM (notej Plan e Strain Analyses (Eq. 9.10) 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 Plan e Stress Analyses (Eq. 9.9) 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: 2 T ' = 60.0 Pcf D = 2 ft P _ = 2116.2 Psf atm Values of A were determined by comparing P u ^ t obtained from the passive wedge and flow block models with P obtained from the f in i te element analyses. The values are not those given by Reese (ie: Figure B.2a). n o 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 in 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 in 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 in Figure 9.13. For plane strain analyses, the shape of the P-Y curve for loads approaching P u^ t i s 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 u2 t were obtained for the various mesh radii used. For plane stress analyses, neither the in 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 in the analyses. Identical values for ^ u^ t were also obtained from the four analyses. Overall, the results show that the in 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 o CM CD ' CM O H 0 0 n 1 r H-20FT R = 22D i i i i 1 \ 1 1 — n r i r D = 4' 2' I I I I L J I I I L 0.0 0.1 0.2 0.3 0.4 0.5 Y (ft) 0.6 0.7 0.8 0.9 I.O(XIC)-1) FIGURE 9.11: EFFECT OF PILE DIAMETER ON PLANE STRESS P-Y CURVE PREDICTIONS FOR DENSE SAND H-20FT D = 2 FT Q o R = 10D CD CD R - 22D + + R = 50D .0 0.04 0.08 0.12 0.16 0.2 Y ( f t ) 0.24 0.28 0.32 0.36 C FIGURE 9.12: EFFECT OF MESH RADIUS ON PLANE STRAIN P-Y CURVE PREDICTIONS FOR DENSE SAND 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 of the P-Y r e l a t i o n s h i p f o r s m a l l loads. For l a r g e r loads near the f a i l u r e c o n d i t i o n , use of the plane s t r a i n P-Y curves may r e s u l t i n e r r o r s . However, si n c e plane s t r a i n curves are 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 the l o a d i n g c o n d i t i o n s are g e n e r a l l y 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, stiff 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 soil 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 soil 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 in Section 9.3.1, consistent values were obtained for the i n i t i a l slopes and P ]_t of the predicted P-Y curves. For 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 in 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 in 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 i i i i i 1 1 1 1 1 1 1 1 1 i 1 r 0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Y / D FIGURE 10.3: NORMALIZED P-Y CURVES FOR UNDRAINED CLAY 121 the stiff linear elastic soil response for small deformations. The curved middle portions describe progressive soil failure and consequently, the softening of the P-Y response. The final horizontal lines correspond to Pu-j_t» at which soil failure is complete and the 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 10.1 cD a + Rab(Y/D) a and b are constants related to the in i t i a l slope of the curve and the ultimate soil resistance: a = 1/initial slope b = l/(P u l t/cD) R is an adjustment factor applied to correct for cutting off the a >• curve at (Pu^t/cD)=12.1. In other words, the true ultimate resistance is at (P , /cD)/R , where R is less than 1. This ux t a a value is 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 soil failure is represented by a j horizontal line. To determine the value of Rg, a transformed plot of the plane 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 in i t i a l segment of the normalized curve do not show a linear relationship, the assumption of a hyperbolic f i t is nonetheless sufficient. The value of RQb determined from the plot is 0.0731. For Pu^t/C^ = 12.1, R = 0.885. The in i t i a l slope of the curve is 1.6E./c. For a r l E^/c = 200 assumed for the P-Y analyses, the theoretical value of a is 0.003125. This value agrees well with a = 0.003 obtained from the transformed plot. The plane stress curve is somewhat more complex. Aside from the straight lines describing the i n i t i a l and failure responses, the curved portion is divided into two segments, a power function and a hyperbola. The hyperbola is given by a modified form of Equation 10.1: P = Y/D 10.2 cD oca + Rab(Y/D) In Equation 10.1, a is defined as the inverse of the i n i t i a l slope of the P/cD-Y/D curve. This is valid, however, only i f the i n i t i a l portion of the curve is hyperbolic. For the plane stress case, the ini t i a l segment of the curve is a power function. A correction factor, oc, is therefore required for a. 124 The power function f or the i n i t i a l curve segment i s given by the equation P/cD = a'(Y/D) b' 10.3 where a', b' = constants To determine the values of the constants for these equations, transformed pl o t s of the P/cD and Y/D data were made. The pl o t i n Figure 10.7 of (Y/D)/(P/cD) vs. (Y/D) for Equation 10.2 y i e l d s the expected s t r a i g h t l i n e f o r the hyperbolic curve segment with slope = R b and intercept =oCa. For R = 0.145 and P -,.,/cD = 6.1, R a = 0.885 as for the plane s t r a i n curve. For an i n i t i a l slope of E^/c = 200 and a = 0.0028, oi = 0.56. For the power function, Equation 10.3 can be expressed as log(P/CD) = log a' + b'log(Y/D) 10.3a The p l o t of log(P/cD) vs. log(Y/D) i n Figure 10.6 likewise y i e l d s a s t r a i g h t l i n e with slope = b' and intercept = l o g ( a ' ) . Values of b' = 0.693 and a' = 45.31 were obtained from the graph. Equations for the f i t t e d plane s t r a i n and plane s t r e s s curves are summarized i n Table 10.1. The equations do not describe the curves 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 occur at the ends of the segments. For p r a c t i c a l purposes, smoothening out the curves by hand i s s u f f i c i e n t , and would not lead to s i g n i f i c a n t e r r o r s . 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 -1 .2 -1 - 0 . 8 LOG (Y/D) FIGURE 10.6: POWER FUNCTION FIT FOR PLANE STRESS P-Y CURVES FOR UNDRAINED CLAY Q U \ \ Q 0 . 0 2 8 0 . 0 2 6 0 . 0 2 4 -0 . 0 2 2 -0 .02 0 . 0 1 8 -0 . 0 1 6 -0 . 0 1 4 0 . 0 1 2 H 0.01 0 . 0 0 8 0 . 0 0 6 H 0 . 0 0 4 r h D n . 0.0028 ~ oca i 1 1 1— -if 0 .02 0 . 0 4 0.745 = Rab 0 .06 0 .08 — T ~ 0.1 0 . 1 2 0 . 1 4 Y / D FIGURE 10.7: HYPERBOLIC FIT FOR PLANE STRESS P-Y CURVES FOR UNDRAINED CLAY 0 . 1 6 r o TABLE 10.1 SIMPLIFIED METHOD FOR DETERMINING P-Y CURVES FOR N.C. CLAYS PLANE STRAIN PLANE STRESS FROM TO EQUATION FROM TO EQUATION (0, 0) (65 .5c/E., P = Y/D (0, 0) (1 . 5 9 c/E., P = E. Y 112.1) cD c/1.6E + 0.073(Y/D) 1 1 . 5 9 ) cD c 1 D (65.5c/E., X12.1) (», 12.1) Horizontal Line (P/cD = 12.1) (1.59c/E., 1 1 . 5 9 ) (0.035, 4 . 4 4 ) P = 4 5.3(Y/D) 0 , 6 9 3 cD (0.035, 4 . A 4 ) (29 .6c/E., 6.1) P = Y/D I cD 0 . 5 6 0 ^ + 0.145(Y/D) (29.6c/E , 6.1) (<*>, 6.1) 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 in 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 is 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 is directly proportional to D. In the passive wedge equation for plane stress deformations (9.9), D is contained in only two of the six terms in the equation. For values of used in the finite element analyses, the terms containing D in Equation 9.9 are relatively insignificant. However, for soil near the surface, H is small, and the four terms not containing D decrease in magnitude. At a depth of 2 ft , the "D terms" account for about half of the soil resistance. At H = 1 ft, the "D terms" account for roughly 2/3 of Pu^t» and so on. Since the passive wedge equation is 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/Pu^t v s« 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 soil parameters (ie: /zf', E^, , etc.), P u^ t can be calculated using Equation 9.9 or 9.10 and the adjustment factor A. Values of A determined by comparing Pu^t predicted from the flow block and passive wedge models with Pu^t 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 is 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 is represented by three curves to best f i t the results predicted by finite element analyses. The strain-softening behaviour of the sand is clearly illustrated. At Pu-j_t, or P/Pu]_t = 1» the sand is assumed to f a i l completely. The P-Y relationship is 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 1 1 1 1 1 1 1 1 1 1 1 1 l 1 0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Y/D FIGURE 10.10: NORMALIZED P-Y CURVES FOR DENSE SAND FIGURE 10 .11: SIMPLIFIED NORMALIZED P-Y CURVES FOR DENSE SAND resistance beyond Pu]_t» The modelling of soil behaviour after failure by the finite element method is questionable, and hence, the omission of the post-failure soil resistances predicted in 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)b' 10.4 As in Section 10.2.2, plotting log(P/P u l t) vs. log(Y/D) yields a straight line with a slope = b' and a log(P/P u^ t)-axis intercept of 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 ft ) , plane stress curves can be used. In the intermediate zone where both stresses and strains are three-dimensional, 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, is an adequate TABLE 10.2 SIMPLIFIED METHOD FOR DETERMINING P-Y CURVES FOR DENSE SAND PLANE STRAIN PLANE STRESS FROM TO EQUATION FROM TO EQUATION (0, 0) (0.0028, 0.29) (0.034, 0.87) (0.054, 1.0) (0.0028, 0.29) (0.034, 0.87) (0.054, 1.0) (*>, 1.0) P/P l t = 33.5(Y/D) 0 * 8 1 P/P l t = 4 . 0 ( Y / D ) 0 , 4 5 P/P l t = 2 . 3 1 ( Y / D ) 0 , 2 9 Horizontal Line: P/P , =1 ult (0, 0) (0.00136, 0.48) (0.00313, 0.76) (0.01, 1.0) (0.00136, 0.48) (0.00313, 0.76) (0.01, 1.0) (*>, 1.0) P/P l t = 1 9 0 ( Y / D ) 0 , 9 1 P/P l t = 1 8 ( Y / D ) 0 , 5 5 P/P l t = 2 . 8 6 ( Y / D ) 0 , 2 3 Horizontal Line: P/P , =1 ult Note: Co-ordinates given are for (Y/D, P/P u ^ t ) P , calculated from Equations 9.9 for plane stress and Equation 9.10 for plane s tra in . Values for the adjustment factor A in the equations are given in 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 l t ) . CO 139 approximation for r e a l s o i l behaviour. Extending t h i s method to a l l values of P, plane s t r a i n and plane st r e s s curves can then be added l i n e a r l y to produce P-Y approximations for intermediate s o i l depths. This method i s i l l u s t r a t e d i n Figure 10.15 f o r undrained clay at a hypothetical depth of 5 f t with H c = 7.5 f t . For undrained normally-consolidated cl a y s , the zone of three-dimensional stresses and s t r a i n s extends to a depth given by Equation 9.5: H = 6D c fl* 'D/cH + J For dense sand, the l i m i t of the t r a n s i t i o n zone can be estimated from Figure 10.14. The inverse of the adjustment f a c t o r , 1/A (see Table 9.4), f o r plane stress analyses i s shown to increase with depth u n t i l the plane s t r a i n condition takes over. Accordingly, H c = 19 f t can be taken as the l i m i t of the t r a n s i t i o n zone. The value of H , c' however, i s not constant, but i s a function of s o i l properties and p i l e 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 is the design of laterally loaded piles. Since loads are applied to the surrounding soil in much the same manner for both the pressuremeter and the lateral pile problem, similarities are expected in 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 al. (1983), Atukorala & Byrne (1984), and Robertson et al. (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 in 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 2.0Ej. (Pyke & Beikae, 1984) were suggested by various researchers. Another uncertainty lies in the difference in size, or 143 diameter, between the pressuremeter cell 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 ft 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 in 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 in 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 is, 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 soil depths can be normalized to produce a unique curve for the soil. Normalizing of the curves are discussed in Section 11.2.3.1. 144 11.2.2 P-Y Curves The P-Y curves predicted in Chapter 9 were compared with the pressuremeter results. Both plane strain and plane stress P-Y curves were employed in the comparisons. The curves are shown in Figures 9.2 and 9.3. As noted in Section 9.3.2, the in 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" in 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 = in i t i a l probe diameter), should be used. However, for convenience in converting pressuremeter results to P-Y curves, D is taken as the in i t i a l diameter. The modified pressuremeter curves and P-Y curves are compared in 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 is equal to the size of the pressuremeter cell (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 10 2 ) to i 1 1 1 r r0 = 7 FT D = 2T0 PILE D = 2FT H = 10 FT CO o co to CM i r i 1 1 r Cl - C M Q CU o to — - o — - + PI. Strain P-Y Curve. -0 PI. Stress P-Y Curve - O Pressuremeter Curve . 0.0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 DEFLECTION Y or Ar (ft) 0.2 FIGURE 11.1: COMPARISON OF PRESSUREMETER AND P-Y CURVES FOR NORMALLY-CONSOLIDATED UNDRAINED CLAY cn "i 1 1 1 1 r ro = 7 FT D = 2rQ PILE D = 2 FT H = 20 FT i 1 r +-©--— h PI. Stra in P-Y Curve —O PI. Stress P-Y Curve •~Q Pressuremeter Curve 0.0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 DEFLECTION Y or Ar (ft) 0.2 FIGURE 11.2: COMPARISON OF PRESSUREMETER AND P-Y CURVES FOR NORMALLY-CONSOLIDATED UNDRAINED CLAY 147 of displacements, AT. Since T q = D/2, plotting Ar/T) will likewise eliminate size effects. Similarly, size effects for P-Y curves were eliminated in Section 9.3.3 through the use of P/D vs. Y/D plots (D is 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 in Figure 11.3. As shown in Section 10.2.1 and discussed in 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 is 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 - +-C L <a O CD a u C L _ -CM Strain P-Y Curve Stress P-Y Curve o [_ Q o Pressuremeter Curve 1 1 r i 1 r i 1 1 r i 1 r -+ Pi ~o PI 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Y/D or Ar/D 0.7 0.8 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 Initial 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 A E T D 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 soil 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 soil at depths of 10 and 20 ft were compared to determine factors for converting pressuremeter curves to P-Y curves. TABLE 11.1 CONVERSION FACTORS FOR NORMALLY-CONSOLIDATED CLAYS Y/D or PLANE STRAIN PLANE STRESS J r / D P-Y CURVES 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 Assume Assume 0.07 2.72 2. 72 1.68 1. 66 0.08 2.73 1.66 0.09 2.71 1.65 0.10 2.72 1.65 151 11.3.1 Pressuremeter Expansion Curves The predicted pressuremeter curves are shown in Figure 8.2. The in i t i a l slopes of the curves are approximately 0.98(2G). As discussed in Section 8.4, the 20-ft curve is 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 I. 08E^. For the value of p. = 0.222 calculated from the given soil parameters, this is 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/Pu^t v s« 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/TQ, as illustrated in Figure 8.4, or AV vs. Ar/D (D = 2r Q). To normalize the load component of the pressuremeter (X103) 12 11 -10 -9 \ JO Q • <J o rn=iFT D 2r 'o PILE D = 2 FT H - 10 FT . - - - 0 r -<5 o- • PI. Strain P-Y Curve H + PI. Stress P-Y Curve O O Pressuremeter Curve T T T 0.02 —I— 0.08 0.04 0.06 DEFLECTION Y orhr (ft) FIGURE 11.4: COMPARISON OF PRESSUREMETER AND P-Y CURVES FOR DENSE SAND 0.1 ro (X10J) 20 0. < O r0-iFT D - 2rc P /L /r D = 2 FT H = 20 FT fs 9? D • PI . Strain P-Y Curve T + PI. Stress P-Y Curve O 0 Pressuremeter Curve T —I— 0.02 —I— 0.04 T —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 in Figure 8 . 2 . Fully normalized pressuremeter curves are thus given by plots of .dP/P^ Q vs. /Ir/D. The normalized curves are shown in Figures 11.6 and 1 1 . 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 = 1.38(2G)(D) = 2.76(2G) Pressuremeter curves: P/Pin = P/P1n = 2(2G) T 7 i u 772r u R o For the values of P U ]_ T and P^Q obtained from the analyses, the conversion factors, given by the ratio of the slopes above as ^ I Q / P u l f are listed in Table 11.2. rQ = 7 FT D = 2rQ PILE D = 2 FT H = 10 FT • PI. Strain P-Y Curve — + PI . Stress P-Y Curve O O Pressuremeter Curve o +-T 0 0.02 0.04 Y / D o r A r / D FIGURE 11.6: COMPARISON OF NORMALIZED PRESSUREMETER AND P-Y CURVES FOR DENSE SAND 1.2 1.1 -Q. X Q. < O 3 fi. N o. . <yxy' -+- o a-o-r0 = 7 FT D = 2r D P / L £ D = 2 FT H = 20 FT — • PI. Strain P-Y Curve —+ PI. Stress P-Y Curve —O Pressuremeter Curve T FIGURE 11.7: 0.02 0.04 Y/D orAr /D COMPARISON OF NORMALIZED PRESSUREMETER AND P-Y CURVES FOR DENSE SAND CJl Ol TABLE 11.2 CONVERSION FACTORS FOR DENSE SAND Y/D or J r / D PLANE STRAIN P-Y CURVES PLANE STRESS P-Y CURVES H=10FT H=20FT RECOMMENDED H=10FT H=20FT RECOMMENDED I n i t i a l 0.455 0.419 0.44 1.088 1.029 1.06 Slope Assume Assume 0.002 1.057 0.909 0.98 1. 00 2.216 2.091 2.15 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 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 soil 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 is 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/Pu^t v s* /^D for sand, were also shown to reduce families of curves for a l l pile diameters (D) and soil 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 Civil 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 Civil 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 Civil Eng., U.B.C., July, 1980 Byrne, P.M., and Grigg, R.F., "Documentation for LATPILE.G", Civil Engineering Program Library, Dept. of Civil 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 Civil Eng., U.B.C., May, 1982 Campanella, R.G., and Robertson, P.K., "Applied Cone Research", Soil Mechanics Series No. 46, Dept. of Civil 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 Civil 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 Civil 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 Civil 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", Civil 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 Civil 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 Civil 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 Civil Eng., U.B.C, May, 1983 Scott, R.F., Foundation Analysis, Prentice-Hall Inc., Englewood Cliffs, 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 Civil Eng., U.B.C., Dec, 1981 Vaziri, H., Ph.D. Thesis, Dept. of Civil Eng., U.B.C., 1985 Yan, L., Forthcoming M.A.Sc. Thesis, Dept. of Civil 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 stress-strain relationship for s o i l i s given by the equations £ = 0" - ulT - u(T A . l a y v x r z E s ^ z = ( r z - - A ' l b E s where u = Poisson's ratio E = secant elast ic modulus For plane strain analyses, <5z = 0. Hence, <rz = }i«rx + (Ty) A.2 Using ff" and £ to represent changes in stress and s train , ff"x = 0 for the uniaxial loading condition considered (see Figure 6.1). Therefore, (T = u(T A.3 z y and £ = r - u(utr) = <T(1 - ;i 2) A.4 y y________y_ y_ ^ ^ E E 165 For a homogeneous, isotropic non-linear elastic material, u varies with the elastic ( E ) and bulk (B) modulii according to ju = 3B - E A.5 6B Substituting E G for E , 1 - n 2 = (9B - E ) ( 3 B + E ) A.6 s s 3 6 B 2 Finally, £ = (9B - E ) ( 3 B + E ) <T A.7 y _ s__ s_ y 3 6 B 2 E s In the hyperbolic stress-strain model for soil given by Duncan & Chang ( 1 9 7 0 ) , the secant elastic modulus can be expressed as E S = E . [ I - Rf(<rd/<rdf)] A . 8 where E . = in i t i a l elastic modulus l = failure ratio (T^ = deviator stress = (f^ for uniaxial loading in the Y-direction ""df = deviator stress at failure 166 The i n i t i a l elastic modulus given by Janbu (1963) is E. = kEPa(<r3/Pa)n A.9 where kg = elastic modulus number n = elastic modulus exponent P 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 EP a A.9a and <Tj£ = 2c A. 10a For sand, c' = 0 and °df = 20V sin|zf' A , 1 0 b 1 - sin*' The complete stress-strain relationship for the plane strain uniaxial loading condition is 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 is similar to that of the plane strain condition. Using the same general stress-strain equation as before, E For the plane stress condition, |Tz = 0, and for uniaxial loading, (T =0, and (T, = rj~ . Therefore, x ' d y y - y i y . E S E . [ i - R f ( i r y / « r d f ) ] E. and U~.f are as given by Equations A.9 and A. 10. APPENDIX B EMPIRICAL P-Y CURVES B.l MATLOCK'S EMPIRICAL CURVES FOR CLAY The empirical curves shown in 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, ^ u ^ t a n d Y£, given by P,.=NcD, N = 3+ lT'/c + J H/D B.l ult P P v and Y = 2.5 fi-nD B.2 c 50 The value of N lies between 3 and 9. P Using recommended values of <£^ Q = 0.010 and J = 0.35 for the soil 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) Np P u l t (lb/ft) Yc (ft) 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, is 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 is P/P l t = 0.5(Y/Y c) 0 , 3 3 3 B.3 B.2 REESE'S EMPIRICAL CURVES FOR SATURATED SAND The empirical P-Y curves shown in Figures 9.8 and 9.9 were determined according to the method recommended by Reese at al. (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 is given by P = P = A P B.4 u ult c where P is the theoretical ultimate soil resistance determined c according to Equation 2.2 or 2.3. The smaller of the two values is used. Values for the adjustment factor, A, determined by comparing theoretical values with experimental results, are given in Figure B.2a. The corresponding deflection at point u is 3D/80. Point m is given by P = B P B.5 m c Y = D/60 m Values for B were also determined experimentally and are given in Figure B.2b. Point k is 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 a) Non-Dimensional Coef f ic ient A for Ultimate Soi l Resistance b) Non-Dimensional Coef f ic ien t B for Soi 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 = C Y 1 / n B.6 n = P /mY m m C = P /Y 1 / n m m m = slope of l ine between points m and u = (P - P )/(Y - Y ) x u rrr v u nr 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 for loose, medium, s and dense sand, respectively. Using the s o i l properties l i s ted in Table 9.2 for the dense sand, values for P were calculated: c Depth H (ft) P (Eq.2.2) C t ( l b / f t ) P (Eq.2.3) C d ( l b / f t ) A B 2 3033 — 2.13 1.55 5 12217 98105 1.25 0.87 10 38426 170467 0.88 0.50 20 128448 286834 0.88 0.50 40 484331 0.88 0.50 Values for A and B estimated from Figure B.2 for a pi le diameter of 2 feet are also tabulated above. The P-Y curves determined from the above equations are shown in Figures 9.8 and 9.9. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0062921/manifest

Comment

Related Items