a new model for the analysis of laterally loaded piles

A NEW MODEL FOR THE ANALYSIS OF

LATERALLY LOADED PILES

by

DEVANAND V. A. J . KONDUR, B.E., M.S.C.E.

A DISSERTATION

IN

CIVIL ENGINEERING

Submitted to the Graduate Faculty of T^exas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

Approved

Chairperson of the Committee

Accepted

Dean pt the Graduate Sthool

May, 1998

ACKNOWLEDGMENTS

I would like to express my deepest gratitude to Dr. Vallabhan for providing

valuable advice during this research. He has been a mentor and a guide during the course

of my education at Texas Tech University. I would like to thank the members of the

doctoral committee for their support and time in helping me complete this research. I

would also like to thank Texas Tech University and the Department of Civil Engineering

for giving me an opportunity to pursue my higher education.

11

TABLE OF CONTENTS

ACKNOWLEDGMENTS ii

LIST OF TABLES vii

LIST OF FIGURES viii

I. INTRODUCTION 1

1.1 Pile Foundations 1

1.2 Review of Previous Research 4

1.2.1 Early Models Developed by Brinch Hansen and Broms 6

1.2.2 Models based on Winkler Concept 7

1.2.3 Poulos Model 12

1.2.4 Finite Element Models 14

1.2.5 Simple Approximate Variational Models 15

1.3 Why a New Model? 16

1.4 Aims and Objectives of this Research 16

n. A HIGHER ORDER FINITE ELEMENT MODEL 18

2.1 Introduction 18

2.2 Development of the StifiBiess Equations 19

2.2.1 Assumptions of the Pile-Soil System 22

2.2.2 Assumed Displacement Functions 22

2.2.3 Strain - Displacement and Stress - Strain Relationships 26

iii

2.2.4 Application of the Principle of Minimum Potential Energy 28

2.3 Development of the Force Vector 30

2.3.1 Equivalent Nodal Forces for Axisymmetric Axial Force P 33

2.3.2 Equivalent Nodal Forces for Lateral Load H 36

2.3.3 Equivalent Nodal Forces for Applied Moment M 39

2.4 Validation of the Finite Element Solution 41

m DEIOVATION OF A RELATIONSHIP BETWEEN THE

COEFHCIENT OF HORIZONTAL SOIL RESISTANCE AND THE MODULUS OF ELASTICITY OF THE SOIL FOR CIRCULAR PILES 55

3.1 Introduction 55

3.2 Derivation of the Relationship Between the Soil Properties and the Horizontal Winkler Coefficient k 56

3.2.1 Approach 1 57

3.2.2 Approach 2 63

3.2.3 Derivation of the Equation Relating the Winkler CoefiBcient with the Properties of the Pile-Soil System 102

IV. A NEW MODEL FOR THE ANALYSIS OF LATERALLY

LOADED CIRCULAR PILES IN A LAYERED SOIL MEDIUM 123

4.1 Introduction 123

4.2 Assumptions in the New Soil-Pile Model 124

4.3 Development of the System Equations 124

4.3.1 Assumed Displacement Functions of the Soil Layer 126

iv

4.3.2 Strain-Displacement and Stress-Strain Relationships of the Soil 127

4.3.3 Potential Energy of the Pile 128

4.3.4 Potential Energy of the Soil 129

4.3.5 Total Energy Function of the System and Minimization of the Potential Energy 130

4.3.6 Equations Corresponding to the Magnitude of Displacement u 131

4.3.7 Equations Corresponding to the Displacement Parameters <j>

and y/ *" 133

4.4 Numerical Approach to Solve the System Equations 134

4.4.1 Assumed Displacement Functions 134

4.4.2 Strain-Displacement and Stress-Strain Relationships 136

4.4.3 Total Energy Function ofthe Pile Element 136

4.4.4 Application ofthe Minimization of Potential Energy 137

4.4.5 Finite Difference Equations for the Displacement Parameters

(j> and y/ 139

4.5 Iterative Technique to Solve the Matrix Equations 142

4.5.1 Steps Involved in the Iterative Procedure 143

4.6 Validation ofthe Model 144

CONCLUSIONS AND RECOMMENDATIONS 158 5.1 Summary 158

5.2 Conclusions 159

5.3 Recommendations I59

LIST OF REFERENCES 160

VI

LIST OF TABLES

3.1 Data used in deriving Method 1. 57

3.2. Comparison of K^, ^^ and ^*. 121

Vll

LIST OF FIGURES

1.1 Discrete model for laterally loaded piers. 11

2.1. Axi-symmetric pile-soil system subjected to non-axisymmetric loading. 21

2.2. A 9-noded isoparametric element. 25

2.3. Applied load on the pile. 31

2.4. Element used to compute the load vector for the finite element model. 32

2.5. Displacement plots for different harmonics for applied lateral force. 42

2.6. Lateral shear force plots for diflferent harmonics for applied lateral force. 43

2.7. Bending moment plots for different harmonics for applied lateral force. 44

2.8. Convergence plots based on the number of elements in descretization ofthe pile. 45

2.9. Comparison of the I^^ displacement parameter for FEM, Poulos and

Verrujit and Kooijman solutions for L/D ratio of 10. 47

2.10. Comparison of the 1^^ displacement parameter for FEM, Poulos and


2.11. Comparison of the Z ^ displacement parameter for FEM, Poulos and


2.12. Comparison of the Z ;, displacement parameter for FEM, Poulos and


2.13. Comparison ofthe lateral displacement along the depth of a pile for K^

value of 10 between Verrujit and Kooijman's solution and the FEM solution. 52



viu



3.1. Displacement, lateral resistance and coefiBcient of soil resistance plots for K^ = \0-\ 58

3.2. Displacement, lateral resistance and coefiBcient of soil resistance plots for A:, = 10"'. 59

3.3. Displacement, lateral resistance and coefiBcient of soil resistance plots for K^ = 10" . 60

3.4. Displacement, lateral resistance and coefiBcient of soil resistance plots for K^ = \Q-'. 61

3.5. Displacement, lateral resistance and coefficient of soil resistance plots for ^^ = 10" . 62

3.6. Plot of non-dimensional Winkler coefiBcient of lateral resistance A"^ versus K^. 65

3.7. Displacement, shear and bending moment plots for L/D = 10 and K^ = 10' comparing Winkler model using the new coefiBcient of horizontal resistance and the finite element method for an applied lateral load. 66

3.8. Displacement, shear and bending moment plots for L/D = 10 and K^ = 10 comparing Winkler model using the new coefiBcient of horizontal resistance and the finite element method for an applied lateral load. 67

3.9. Displacement, shear and bending moment plots for L/D = 10 and K^ =

10 comparing Winkler model using the new coefiBcient of horizontal resistance and the finite element method for an applied lateral load. 68

3.10. Displacement, shear and bending moment plots for L/D = 25 and K^ = lO' comparing Winkler model using the new coefiBcient of horizontal resistance and the finite element method for an applied lateral load. 69

IX



3.13. Displacement, shear and bending moment plots for L/D = 25 and K^ = lO'* comparing Winkler model using the new coefiBcient of horizontal resistance and the finite element method for an applied lateral load. 72

3.14. Displacement, shear and bending moment plots for L/D = 25 and K^ = 10 comparing Winkler model using the new coefficient of horizontal resistance and the finite element method for an applied lateral load. 73

3.15. Displacement, shear and bending moment plots for L/D = 50 and K^ = lO' comparing Winkler model using the new coefiBcient of horizontal resistance and the finite element method for an applied lateral load. 74



3.18. Displacement, shear and bending moment plots for L/D = 50 and K^ = 10** comparing Winkler model using the new coefiBcient of horizontal resistance and the finite element method for an applied lateral load. 77



10' comparing Winkler model using the new coefiBcient of horizontal resistance and the finite element method for an applied lateral load. 79






lO' comparing Winkler model using the new coefiBcient of horizontal resistance and the finite element method for an applied lateral load. 82


10 comparing Winkler m'odel using the new coefficient of horizontal resistance and the finite element method for an applied lateral load. 83

3.25. Displacement, shear and bending moment plots for L/D = 10 and K^ = lO' comparing Winkler model using the new coefiBcient of horizontal resistance and the finite element method for an applied moment. 84

3.26. Displacement, shear and bending moment plots for L/D = 10 and K^ = 10 comparing Winkler model using the new coefficient of horizontal resistance and the finite element method for an applied moment. 85


10 comparing Winkler model using the new coefiBcient of horizontal resistance and the finite element method for an applied moment. 86


3.29. Displacement, shear and bending moment plots for L/D = 25 and K^ = 10 comparing Winkler model using the new coefiBcient of horizontal resistance and the finite element method for an applied moment. 88


XI

3.31. Displacement, shear and bending moment plots for L/D = 25 and K^ = \0^ comparing Winkler model using the new coefiBcient of horizontal resistance and the finite element method for an applied moment. 90

3.32. Displacement, shear and bending moment plots for LID = 25 and K^ = 10 comparing Winkler model using the new coefiBcient of horizontal resistance and the finite element method for an applied moment. 91





3.37. Displacement, shear and bending moment plots for LID = 50 and K^ = 10 comparing Winkler model using the new coefiBcient of horizontal resistance and the finite element method for an applied moment. 96

3.38. Displacement, shear and bending moment plots for LID = 100 and K^ =

lO' comparing Winkler model using the new coefficient of horizontal resistance and the finite element method for an applied moment. 97





Xll


10"* comparing Winkler model using the new coefficient of horizontal resistance and the finite element method for an applied moment. 100


10 comparing Winkler model using the new coefficient of horizontal resistance and the finite element method for an applied moment. 101


10 comparing Winkler model using the new coefficient of horizontal resistance and the finite element method for an applied lateral load without rotation at the top of the pile. 103

3.44. Displacement, shear and bending moment plots for LID =10 and K^ =

10^ comparing Winkler model using the new coefficient of horizontal resistance and the finite element method for an applied lateral load without rotation at the top of the pile. 104




lO' comparing Winkler model using the new coefficient of horizontal resistance and the finite element method for an applied lateral load without rotation at the top of the pile. 106




10 comparing Winkler model using the new coefficient of horizontal resistance and the finite element method for an applied lateral load without rotation at the top ofthe pile. 108

Xlll


10"* comparing Winkler model using the new coefficient of horizontal resistance and the finite element method for an applied lateral load without rotation at the top of the pile. 109






10 comparing Winkler model using the new coefficient of horizontal resistance and the finite element method for an applied lateral load without rotation at the top ofthe pile. 112




lO"* comparing Winkler model using the new coefficient of horizontal resistance and the finite element method for an applied lateral load without rotation at the top of the pile. 114




10' comparing Winkler model using the new coefficient of horizontal resistance and the finite element method for an applied lateral load without rotation at the top of the pile. 116

XIV






10" comparing Winkler model using the new coefficient of horizontal resistance and the finite element method for an applied lateral load without rotation at the top of the pile. 119



4.1. Pile-soil system for the new model. 125

4.2. Comparison of non-dimensional lateral displacement between FEM and the new model for applied lateral load for L/D =10. 145

4.3. Comparison of non-dimensional lateral displacement between FEM and the new model for applied lateral load for L/D = 25. 146



4.6. Comparison of non-dimensional lateral displacement between FEM and the new model for applied lateral moment for L/D =10. 149

4.7. Comparison of non-dimensional lateral displacement between FEM and the new model for applied lateral moment for L/D = 25. 150


XV


4.10. Cross section of the soil for a stratified medium (Case I). 153

4.11. Displacement, lateral shear and bending moment plots for the layered soil medium shown in Fig. 4.10. 154

4.12. Cross section of the soil for a stratified medium (Case II). 155

4.13. Displacement, lateral shear and bending moment plots for the layered soil medium shown in Fig. 4.12. 156

XVI

CHAPTER I

INTRODUCTION

1.1 Pile Foundations

Pile foundations have been widely used in civil engineering construction for many

years. They are used where shallow foundations are not practical due to the loading

conditions and the soil conditions that may exist at the construction site. Piles are

commonly used in foundations of multistoried buildings, bridge caissons, electric poles,

flag poles, off-shore platforms and other structures. Structures subjected to large lateral

loads usually have pile foundations as shallow foundations cannot sometimes support the

moments on these structures. Pile foundations are also used in situations where the top

layer of the soil does not possess enough strength to bear the load on the structure,

effectively transferring the load to the underlying stronger layer.

The different types of structures in which pile foundations are used vary vastly.

The factors influencing the use of pile foundations are very important considerations

when constructing structures using pile foundations. Multistoried buildmgs are usually

subjected to high wind loading and need to resist a high amount of lateral loading.

Therefore, typically piles are used as foundations for multistoried buildings. Earthquake

loading is another reason to employ pile foundations in such situations. The loading in

such cases is mainly the dynamic loading due to earthquakes. Though axial loading is a

major consideration in designing such structures, large lateral loads necessitate the use of

pile foundations. Another typical type of construction in which pile foundations are used

2

are for offshore platforms. In these structures the piles are embedded into the floor of the

water mass. These piles protrude out of the floor. The superstructure is constructed on

these pile foundations. These type of foundations are typical examples of constructions

where cyclic loading occurs. The load on these piles is axial as well as lateral load. They

are usually embedded in soft clay or sandy types of soils. In both the types of

constructions described above the piles are usually slender, having a high length to width

ratio. Other types of structures which have slender piles are flagpoles and electric poles. f

The primary consideration in these circumstances is the ultimate loading capacity of the

foimdation. The second consideration in the design is the maximum allowable

displacement.

Short piles are also used v^dely in construction. Caissons, retaining walls, etc. are

examples of situations where short piles are used. In the case of retaining walls piles

known as sheet piles are used. These are piles with a large width. In other cases, piles are

placed in large groups or as single piles. Multistoried buildings, offshore platforms, etc.

are examples of structures where pile groups are used. Flagpoles and electric poles are

examples where single piles are used.

One ofthe classification of piles depends on the constraints placed on them, at the

bottom ofthe pile and at the top ofthe pile. Based on the restriction on the top ofthe pile,

they can be classified as having a fixed head, free head or a flexible head condition. They

are said to be fixed head piles when a rigid pile cap is present at top of these piles. These

pile caps can be so rigid, that no rotation is allowed at the top of the pile. Piles are

considered to be free head when they have a flexible pile cap or no pile cap at all. They

3

are also classified as floating or fixed piles depending on the soil stratum on which the

bottom of the pile rests. If the soil is very rigid at the bottom of the pile and has a very

high modulus of elasticity it is considered to have a fixed condition at the bottom. In this

case, there is no rotation or axial displacement at the bottom ofthe pile. If the soil is very

soft or not anchored by a rock stratum at the bottom, then the pile is considered to be a

floating pile. Hence both rotation and lateral motion are possible at the bottom ofthe pile.

Piles can also be classified based on the kind of material used for construction.

Some of the materials used in pile construction are timber, reinforced concrete and steel.

Another form of classification of piles is based on the construction of the pile

foundations. Piles can be cast in-situ wherein a hole is drilled for the depth of the pile.

Cast in-situ piles are usually concrete piles. If necessary reinforcement can be placed in

the drilled hole and concrete is poured. Drilled piers, drilled shafts and drilled caissons

are examples of cast in-situ piles. It is typical in such constructions to have grooves in the

soil at intermittent depths of the soil so that a better anchoring of the pile in the soil

occurs. The other type of construction is to use driven piles. In this case the piles are pre

cast and then they are driven into the ground. The type of pile used depends on various

considerations such as the soil properties, availability of material and design

considerations. Also piles are made in different cross sections. Piles can be circular as is

the case usually with concrete and wood piles. Hollow circular piles and H piles made out

of steel are also used in construction. Square or rectangular piles are usually made out of

concrete.

4

The strength of a pile foundation depends not only on the material of the pile but

also on the soil in which it is placed. The pile foundation can be said to be comprised of

two subsystems, the soil subsystem and the pile subsystem. These two subsystems

interact with each other to contribute to the strength of the pile foundation. So this

problem is very unique and is considered a soil-structure interaction problem. Both the

soil properties and the pile properties need to be considered carefully when analyzing the

behavior of such a system.

As mentioned earlier the material used for the pile is usually concrete, wood or

steel. The material properties ofthe pile rarely vary along the length ofthe pile. The cross

sectional area of the pile might vary along the length of the pile especially for long piles

used in offshore construction. Under working load, linear displacements can be expected

in the pile itself. The soil stratum on the other hand usually varies over the depth of the

pile. This depends on the location of the construction site. The material properties of

these soil strata can vary vastly or gradually. The displacement of the pile system under

horizontal loading is usually nonlinear with the soil material undergoing nonlinear

deformation. In other words the soil usually has linear material properties for small

displacements and nonlinear material properties for large displacements.

1.2 Review of Previous Research

Research of laterally loaded piles has been done by researchers such as Broms [1],

Brinch Hansen [2], Georgiadis and Butterfield [3], Gleser [4], Poulos [5], Matiock and

Reese [6], Vallabhan [7], Randolph [8], Kuhlemeyer [9], Davisson [10], Spillers and Stoll

5

[11], Lenci [12], Maurice and Madinger [13], Matthewson [14] and Banerjee and Davis

[15], Chandrasekharan [16], Verrujit and Kooijman [17] and Sun [18]. These models use

varying techniques towards the solution of this problem and can be broadly classified into

five different categories. In most of these categories the pile is modeled as an elastic

beam. They usually vary in their approach toward modeling the soil continuum. Early

research on laterally loaded piles was done by Broms [1] and Brinch Hansen [2]. Broms

method is a very simplified method and is based on the limiting values of lateral earth

pressure theory. This method is often employed by engineers to design small foundations

for light poles etc. The second category uses the Winkler approach for modeling the soil

behavior. In this approach the pile, in most cases, is modeled as an elastic beam and the

soil is modeled as a set of horizontal elastic springs in which the pile is embedded. Using

this approach Matlock and Reese [6] and Vallabhan [7] developed models for solving

piles under lateral loading. Matlock and Reese [6] modeled the soil as a set of nonlinear

springs. Vallabhan [7] developed a model for rigid piers. The third category adopts an

elastic continuum approach. In this approach the pile and the soil are represented by

elastic material properties. This approach has been used by Poulos [5] to develop a model

for solving laterally loaded piles using a finite difference method. The fourth method is

the finite element solution technique. This technique has been used by Chandrasekharan

[16] and is discussed later in this chapter. Finally various approximate techniques have

been adopted to develop models to solve laterally loaded piles. These include models by

Verrujit and Kooijman [17] and Sun [18] etc.

6

Some ofthe models that have been developed by researchers are presented here to

better understand the behavior of the pile-soil system. The different models presented

here are the early models by Brinch Hansen [2] and Broms [1], Reese Matlock p-y Curve

Method [6], Vallabhan [7] method for the analysis of rigid piers. Also Poulos Method [5]

and the finite element method are studied.

1.2.1 Earlv Models Developed bv Brinch Hansen and Broms f

Brinch Hansen [12] developed a method which is based on the earth pressure

theory. It is applicable to c-(j> soils. It is also applicable to layered soils. The major

disadvantage of this method is that it can be used only for short piles and is very

approximate. This method consists of determining the center of rotation by taking the

moment of all forces about the point of application of the load and equating it to zero.

Using this technique the ultimate soil resistance at any depth is given by the following

equation.

where

= ultimate soil resistance,

c = cohesion ofthe soil,

^,^ = vertical effective overburden pressure, and

K^,K = coefficients for different values of ^.

xu

7

Broms' proposed a method for lateral resistance of vertical piles similar to the

mechanism developed by Hansen. In his method some simplifications are made as given

below.

1. Soil is assumed to be purely cohesionless or purely cohesive.

2. Piles are differentiated into short rigid or long flexible piles.

3. Free-head piles are assumed to rotate about a center of rotation and fixed head piles

are assumed to move as a rigid body and have lateral translation.

1.2.2 Models based on Winkler Concept

The Winkler's subgrade reaction concept of modeling soil behavior is widely used

when deriving a model for laterally loaded piles. In the Winkler's subgrade reaction

approach the soil is modeled as a set of independent elastic springs. The spring

coefficients of these springs reflect the material properties of the soil and are known as

coefficients of subgrade reaction. In these models the pile is modeled as an elastic beam

resting on these springs. The springs are attached to the beam at discrete points, so the

displacement ofthe pile-soil system depends on the soil at discrete points.

In any basic model that uses the Winkler's subgrade reaction concept, the pressure

applied on the pile and the deflection ofthe soil are related by the following equation.

q=-Ku. (1.1)

The pile is assumed to act as a prismatic long beam. Hence the governing equation is

given by

8

V . ^ + *»« = 0. (1.2)

In the above equations

Ep = Modulusof elasticity ofthe pile,

Ip = Moment of inertia ofthe pile,

k^ = Coefficient of horizontal soil resistance,

u = Lateral displacement at the pile-soil boundary, and f

z = Coordinate in the axial direction ofthe pile.

Loads acting on the beam are then applied as boundary conditions to obtain the complete

system equations. These equations can be solved using analytical or numerical methods.

1.2.2.1 Matlock-Reese model

Models for the analysis of laterally loaded piles were developed by Matlock and

Reese considering the pile to be a flexible beam and the soil continuum as a set of

independent nonlinear springs. Their model is based on the classical Winkler model.

Usually this model is discretized along the depth of the pile and at the discretized points

the soil behavior is applied as a set of springs. The nonlinear behavior of the soil can be

modeled by varying the characteristics of these springs, which are represented by p-y

curves. These are curves that show the relationship between the load and the

displacement on the soil. P-y curves are developed using a pseudo empirical method. To

obtain a nonlinear p-y curve for soils, different methods need to be used for clays and

sands. The behavior of the p-y curve until the yield point, which occurs approximately at

9

0.5 times the ultimate load, is usually approximated using a cubic or an exponential

polynomial.

In this technique p-y curves need to be developed at every discretized point of the

soil continuum, depending on the soil profile along the depth. For soils which vary vastly

with depth the method to develop p-y can be time consuming but there are computer

programs available for this purpose. But one has to rely on the computed p-y curves,

which can be developed in different ways according to experts in this area. The other

obvious drawback of this model is that the soil is not considered to be continuous. Also in

this model the shear strain developed in the soil in the horizontal plane is not taken into

account. An application of this model towards laterally loaded piles has also been

developed by Georgiadis and Butterfield [3].

1.2.2.2 Vallabhan Model for Rigid Piers

Vallabhan [6] developed a very simple model for the analysis of rigid piers. This

model is based on the Winkler approach and p-y approach for modeling the behavior of

the soil. The model is shown in Fig. 1.1. In this model the lateral resistance of the soil,

the resisting moment ofthe soil at the bottom ofthe pier, the vertical resistance ofthe soil

at the bottom of the pier and the skin friction between the pier and the surrounding soil

are included and represented by springs. The condition that needs to be satisfied for the

pile to be assumed to be rigid is

T

10

where

L = length ofthe pier,

EI = the bending stiffness ofthe pier, and

k = the horizontal modulus of subgrade reaction.

The system equations for this model are derived using the minimization of the

total potential energy fiinction. ^This model can also be extended to nonlinear soil

properties. This is done by using a concept similar to the p-y curves for representing the

nonlinear soil behavior. The nonlinear/7-j/ curves are taken as the material behavior ofthe

springs and the displacements are calculated.

One of the basic drawbacks of this approach is that the soil is not modeled as a

continuous medium as there is no continuity between these springs, which is

contradictory to the true nature of the soil. Besides, there is no analytical way of deriving

the coefficient of subgrade reaction. The basic source of the coefficient of subgrade

reaction is through experimental and empirical means only. But many experimental

researches have been done and empirical equations have been developed to calculate the

coefficient of horizontal subgrade reaction. Nonlinear soil behavior can be incorporated

into this model empirically based on experimental data. Because of the ease of use, this

concept has been widely used in practice despite of its drawbacks. The experience gained

by engineers in using this method in practice makes this method very attractive for

analyzing laterally loaded piles.

11

SIOC FRICTION

Vf

Mf

•orrow rniCTiONUfev)

•OTTOM VCHTIOU. RCSISTANCC

LATEPUL R£SBTANC£

HNA/WV—^

rAVWV—^

V^ = Applied vertical load.

H, = Applied lateral load.

M, = Applied lateral moment.

SIOC nticrioN ( k . i l

MTTOM MOMCin' ( KtNi )

Fig. 1.1. Discrete model for laterally loaded piers.

12

1.2.3 Poulos Model

Poulos used the elastic continuum approach to develop a model for the analysis of

laterally loaded pile foundations. In this approach the soil is modeled as a an ideal elastic

continuum. Similar approaches have been developed by researchers such as Douglas and

Davis [10], Spillers and Stoll [11], Lenci [12], Maurice and Madigner [13], Matthewson

[14] and Banerjee and Davis [15]. Poulos solved this problem for floating piles and

socketted piles.

In this approach the pile is considered as a thin rectangular strip with constant

width and constant modulus of elasticity. The pile is then divided into a finite number of

elements of equal length except for the top and the bottom elements, which are half the

length of the other elements. Each of these elements is acted upon by a uniform

horizontal stress which is constant across the width of the pile. The soil is assumed to be

an ideal homogeneous, isotropic, semi infinite elastic material. The properties of the soil

are assumed to be unaffected by the presence of the pile, which is a reasonable

assumption. In his basic model, Poulos did not consider pile soil separation, i.e. the soil

was assumed to adhere to the back of the pile near its surface. He also presented an

approximate method which takes into consideration the pile soil separation.

The basic principle of the Poulos model is the Mindlin equation for displacement

of a point in a semi-infinite continuum caused by a point load within the mass. In elastic

conditions the horizontal displacements of the pile and the soil are compatible. So these

displacements are equated at the element centers except for the extreme elements at the

13

top and the tip of the pile. The soil displacements for all points along the pile are

expressed as

{P^-J;MP} (1.3)

where

{/?,} = horizontal soil displacement vector,

{p} = horizontal loading vector between the soil and pile, and

[Z_j] = soil displacement influence factors.

The elements of the \l^ ] matrix are evaluated by integration over a rectangular area of

the Mindlin equation. The pile displacements are calculated using the Euler-Bemoulli

differential equation for bending of a prismatic beam. The beam equation after applying

the finite difference method is given by

where

[pp I = the pile displacement matrix,

{ 4} = vector representing the moment and loads on the pile,

«+l = number of divisions used for discretizint the pile,

\D\ = the matrix offinite difference coefficients,

E = modulus ofelasticity ofthe pile, and

Z = moment ofinertia ofthe pile.

14

Equating the pile and soil displacements from Eq. 1.3 and Eq. 1.4, the system equation is

given by

in^^-'im^A {P} = {B} (1.5)

where {B} represents the vector consisting ofthe load applied on the pile.

One of the basic disadvantages of the Poulos model is that it cannot be extended

to a stratified soil medium as the influence factors cannot be calculated using the Mindlin r

equation. Mindlin equations do not exist for a non homogeneous stratified medium. Also,

the assumption that the pile is a rectangular strip embedded in the soil is approximately

valid only if the pile has a square cross section or an I cross section. In the case of

circular piles this idealization is further approximate, but seems reasonable. This method

has been used in practice by many engineers.

1.2.4 Finite Element Models

The finite element method has been a very popular method for analysis of

problems in soil mechanics. One of the primary advantages of the finite element method

is that it gives a reasonably accurate solution with proper discretization. Also, it can be

easily extended to a stratified soil medium as well as material nonlinearity. One of the

main disadvantages to the finite element method, is that it is computationally expensive

and the development of the system equations can be very cumbersome. There have also

been variations ofthe finite element models developed by researchers. Hybrid techniques

15 using finite element models in conjunction with the boundary element method have been

developed.

There are two approaches in which the problem of laterally loaded pile can be

handled in the finite element method. The first method is to consider the pile soil system

as a 3-dimensional system. Three-dimensional brick elements are used to model the

system. This approach has been adopted by Vallabhan and Sivakumar [19], Rahman [20],

etc. One ofthe disadvantages of this method is that the brick element has a large number r

of degrees of freedom and can be computationally expensive. Also, the brick element

does not accurately model the behavior of the pile as the pile exhibits the behavior of a

beam. In other words to accurately model the behavior of the pile a large number of

divisions need to be employed.

The second and more popular approach towards the analysis of this system is to

use axisymmetric elements on which a non axisymmetric load acts. This method has been

used by researchers such as Wilson et al. [21] for axisymmetric structures. Extension of

this method to the problem of laterally loaded piles has been done by Chandrasekharan

[16]. In the present research this method is used with nine-noded axisymmetric finite

elements. The development of this method is presented in detail in Chapter II.

1.2.5 Simple Approximate Variational Models

Approximate methods are very prevalent in the analysis of laterally loaded pile

foundations. Researchers such as Verrujit and Kooijman [17], Sun [18], etc., have

16

developed approximate method for the analysis of laterally loaded pile foundations. In

this dissertation, a new hybrid model is also developed.

1.3 Whv a New Model?

As can be seen from the previous section, there are many shortcomings in the

research done previously on laterally loaded piles. They do not satisfy the needs for the

prediction of the displacement of a laterally loaded pile. Even though the finite element

technique can solve this problem, it is very laborious and also very expensive to use. Due

to the fact that the pile acts as a beam, more divisions need to be used in the finite

element technique if brick elements are used in a 3-D analysis. The new method

simplifies the solution process by using concepts in continuum mechanics. Also a layered

soil medium can be solved very easily using this technique.

1.4 Aims and Objectives of this Research

The current research intends to develop a model which is very simple and easy to

obtain results for various kinds of foundation problems. A new approach is used here to

develop the model. The finite difference method and Hermitian polynomials to develop

the system equations in this model. Then the model is solved on a computer to obtain

various data which can be used for load deflection prediction for a given kind of soil

problem. One ofthe main features of this method is that it can very easily solve the case

of a stratified soil. Most of the models developed by researchers earlier use empirical

17

methods to consider the case of a stratified soil. However, this research does not intend to

consider the case of material nonlinearity ofthe soil.

The results of the new variational model will be compared with those of the

methods mentioned above, especially those of Poulos and a new axisymmetric finite

element solution developed in this research. The finite element technique will be

discussed later in this dissertation. This finite element program was developed by the

author for the purpose of this research. r

Also, one ofthe major contributions ofthe finite element method in this research

is to develop a coefficient of soil resistance for the Winkler's model that will produce

approximately a result equivalent to that of the finite element model. Here the soil is

assumed to be homogeneous and semi-infinite with a value of Young's modulus of

elasticity and Poisson's ratio. The results are presented using non-dimensional

parameters, whereby knowing the values of £, v of the soil and Ep, Ip and / ofthe pile,

the value of the k representing the modulus of subgrade reaction of the soil can be

calculated. This technique will be discussed in a subsequent chapter in detail.

CHAPTER II

A HIGHER ORDER

FINITE ELEMENT MODEL

2.1 Introduction

Finite element methods usually give accurate results with proper and adequate

discretization. Also the finite element model can easily represent a stratified soil medium r

as the material properties are prescribed as elemental properties. The finite element

models used by most of the researchers, such as Chandrasekharan [16], Desai [22],

Vallabhan and Sivakumar [19] and Rahman [20] can be categorized into two basic

approaches. These approaches differ in the type of element used to model the pile soil

system.

The first approach is to use 3-dimensional solid finite elements. This approach

was adapted by Desai [22], Vallabhan and Sivakumar [19] and Rahman [20]. In this

approach, the entire soil stratum and the pile are modeled using 3-dimensional brick

elements. The disadvantage of this element is the computational cost inherent in the 3-

dimensional element. Here the pile, which actually exhibits the behavior of a beam, is

also modeled using 3-dimensional solid elements which require a large number of

elements to model the pile, so that it exhibits the characteristics ofthe bending of a beam

accurately.

The other method which is used is to represent the entire soil-pile system as an

axisymmetric continuum on which a non-axisymmetric load acts. This approach is

18

19

limited to circular piles only. This finite element model was developed by Wilson [21]

using axisymmetric triangular ring elements. Chandrasekharan [16] in his research used

8-noded rectangular axisymmetric elements to solve this pile-soil interaction problem.

In this research, higher order 9-noded axisymmetric isoparametric quadrilateral

elements are used for modeling the pile soil system. There are two objectives for

developing the finite element method in this research. The first is to use the solutions

obtained wdth tl e finite element model for validation ofthe new simplified model that is

developed and discussed later in Chapter IV. The new finite element model will yield

more accurate values for the maximum displacements, the shears and the moment

distribution in the pile. To the knowledge of the author such information has not been

presented in most of the published researches. The radial and tangential displacement of

the soil can also be studied using the finite element model.

The second objective ofthe development ofthe finite element model is to develop

an equivalent value of the horizontal subgrade reaction for use in the Winkler model. As

discussed in the previous chapter, the subgrade reaction is presented as a fimction of the

geometry and the material properties ofthe pile-soil system. Two different techniques are

suggested to develop the expression for the modulus of subgrade reaction and these

techniques are studied in this chapter.

2.2 Development ofthe Stif&iess Equations

Though the soil stratum and the embedded circular pile are axisymmetric, a

completely axisymmetric formulation cannot be used as the applied loads are not

20

necessarily axisymmetric. As can be seen from Fig. 2.1 only the axial force P^ is

axisymmetric. The horizontal load H^ and the moment Mg are not symmetric about the

axial direction. The approach that has been adopted to solve this problem, taking

advantage of the symmetry of the model, is to use a Fourier expansion for displacements

and forces. Using the infinite Fourier expansion, the loads are split into circular Fourier

components known as harmonics. These Fourier harmonics are comprised of a

summation of amplitudes ofthe loads and sine and cosine terms. The individual terms of

the summation can be uncoupled because of the orthogonal property of the trigonometric

sine and cosine terms. Thus the response of the system to these uncoupled Fourier

harmonics of the applied load can be solved independent of the other harmonics. The

magnitudes of displacement for each harmonic component thus obtained can be

superimposed to obtain the total response to the load. Though the Fourier series is an

infinite series, only a very few harmonics are usually necessary to obtain tolerable levels

of convergence in the displacements, shears and moments as shown later in this chapter.

The number of harmonics required to get accurate results depends on the type of loading.

The harmonic which has the most impact on the displacement can be predicted by the

nature of the loads. From Fig. 2.1 it can be seen that the axial load, which is

axisymmetric, needs only the 0-th harmonic, which represents axisymmetric components.

All the higher order harmonics have very negligible effects on the displacement for such

a type of loading. The majority ofthe contribution to the displacements due to lateral load

come from the 1-st harmonic, as is the case with the moment. Later in the chapter the

effect of the number of harmonics on the results will be presented for illustration. The

21

Non axisymmetric loading

Axisymmetric Domain (Pile and Soil)

L = Length of the pile

R = Radius of the pile

M= Moment on die pile

H= Horizontal force on the pile

P= Axial force on die pile

Fig. 2.1. Axisymmetric pile-soil system subjected to non-axisymmetric loading.

22

theory of the finite element model for axisymmetric solids under non-axisymmetric

loading has been presented by Cook et al. [23]. A 9-noded isoparametric element will be

used here to model the pile-soil system. Chandrasekharan [16] has presented an 8-noded

rectangular axisymmetric finite element for non-axisymmetric loading. A brief

introduction to the problem and the derivations of the elemental stifi&iess and load

matrices is presented in this chapter.

2.2.1 Assumptions ofthe Pile-Soil System

The assumptions made during the representation of the pile-soil system are given

as follows

1. The material properties ofthe soil are assumed to be isotropic, homogeneous and

uniform within each soil layer.

2. The stratified layers of soil are assumed to horizontal and of uniform thickness.

3. The soil is assumed to undergo only linear displacements.

4. The material properties ofthe pile are assumed to isotropic, homogeneous and uniform

5. Both the geometry ofthe pile and the soil and the boundary conditions ofthe system

are axisymmetric except for the loading.

2.2.2 Assumed Displacement Functions

Since Fourier expansion is used to solve die problem, die assumed displacement

fimctions are expressed as a summation of Fourier components. The assumed elemental

23

displacement functions are given as summations of the products of their Fourier

amplitudes and the corresponding trigonometric terms. They are given as follows.

n

u =^Uj(r,z) COS iO

v=^v.(r,z)smi0

/=0

n

^ = Z^/('*'^)cos/^ (2.1) /=0

f

where u = the displacement at any point in the element in the radial

direction,

v" = the displacement at any point in the element in the

tangential ^direction,

w = the displacement at any point in the element in the axial z

direction,

Uj = the amplitude ofthe displacement in the radial direction, V,. = the amplitude ofthe displacement in the tangential direction, Wf = the amplitude ofthe displacement in the axial direction,

/ = the number of the Fourier harmonic, and

n = the total number ofFourier harmonics used.

24

The 9-noded isoparametric element considered to develop the element stifSiess

matrices is as shown in Fig. 2.2. The displacement functions given in Eq. 2.1 are

rewritten for the element as shown below, using the shape functions ofthe element.

9 n

7=1 i=0

9 n

7=1 /=0

^^f,t,NjW,jCosi0 ' (2.2) 7=1 /=0

where the Nj 's are the shape functions for the given element and are given by

• 4

^ 2

^ ^ - l ) ( l - 7 - ) - ~ 2

JV, =( l -^^)( l -7^)

' ~ 2

„ ^^-1M7+1) ' " 4

25

7-.(-i,i)

Fig. 2.2. A 9 noded isoparametric element.

A^, 4

N. = _ ^(1 + )7(1 + 7)

2

where

the natural coordinate corresponding to the r direction

the natural coordinate corresponding to the z direction.

Eq. 2.2 can be expanded and written as

26

(2.3)

u

' V

w \=^]

/=0

lujCosiO

VjSinid

w, cosi0j [=2;

/=0

N^ cosiO

0

0

0

N^ siniO

0

0

0

N^ cosiO g}

W\ \Nl\Ni

(2.4)

where N^,N^..Ng are the shape fimctions given by Eq. 2.3. and \d,] are the nodal

displacement amplitudes given by

{d,]=\_"\i n, ^u "2/ 1 2; ^ 2 - •J. (2.5)

2.2.3 Strain - Displacement and Stress - Strain Relationships

The strain displacement relationship for axisymmetric solids for non-

axisymmetric deformations in cylindrical coordinates is given by the following

relationship [24].

27

^ ^ ^r r

^er

^z

Yzr

Yre

*

Yze\

d

dr 1

—

r

0

d

dz 1 d

r dO

0

0

1 d

r dO

0

0

d 1

dr r d

— dz

0

0

d ~d~z d

dr

0

1 d

r dO

r

"

r TV

(2.6)

which expands to the equation below when the shape fimctions are applied

^ei

^zi

f zri

[rezi]

A j ^ cosiO

^ , cos iO

0

N^^^ cosiO IN,

siniO

0

iN, 0

cost 6

0

0

iV,,-^]sin/^

N,, sinz^

0

0

N^^ cosiO

N^^ cosiO

0

IN, smiO

1 " •

1 • * *

1 . . .

1 • • •

1 • • •

<

/. .\

" i ,

^ 1 /

^ 1 /

«2 /

^2 /

i - J i^2/. l2,3,4 ,...

(2.7)

where s 's and / 's are the axial and shear strains and u, v, and w are the displacements

corresponding to the r, 0, and z axes, respectively.

28

The stress strain relationship is given by the following equation

r -N

'dr

zr

rd

l^zB)

1

V V

E{\-v) (l + vXl -2v)

\-v V

1

V

\-v \-v

0 0

0

0

0

0

0

0 \-v

1 0

\-2v 0

0

0

2(1-v)

0

0

0

0

0

0

\-2v 2(1-v)

0

0

0

0

0

0

l - 2 v 2(1-K)J

Yzr

Yre

Yzd,

(2.8)

where

E = modulusofelasticity ofthe material (Pile or Soil), and

V = Poisson's ratio ofthe material (Pile or Soil),

and the a 's and the x's are the axial and shear stresses corresponding to the r, 0 and z

cylindrical coordinate system, respectively.

2.2.4 Application ofthe Principle of Minimum Potential Energy

The theory of minimization of potential energy has been discussed in may

textbooks and research papers and therefore will not be presented here. Using stress-strain

and strain-displacement relationships, the total energy function ofthe system is written as

n u + v (2.9)

\\]wmDim^d^dA^^ A-fT

29 where

A = area ofcross section ofthe pile,

V = work potential, and

[B] = matrix of differential coefficients.

Applying the principle of minimum potential energy on die above equation the following

stiffiiess matrix is obtained

[k,]=llJ[B,]'[ElB,]r&d0ck (2.10) 0 -«-o

for the / th. harmonic. For isoparametric formulation the element stiffness matrix is given

by the following equation

W= jfj[B,\'[ElB,}rd0\j\d4dr, (2.11)

where

and / is the determinant ofthe Jacobian of transformation given by [23]

y =

Numerical integration techniques can be used to compute the above integral to obtain the

elemental stif&iess matrix. The Fourier harmonics are uncoupled because ofthe

orthogonal property of sine and cosine terms in the equations.

30

2.3 Development ofthe Force Vector

The derivation ofthe load vectors is presented in detail for the convenience ofthe

reader as they are not presented in any text books or research publications. The three

forces acting on the pile are the axial load P,, the lateral load H^, and the moment M^.

In reality, these forces, though expressed as concentrated load vectors, are distributed

forces. Fourier components of these forces need to be applied to the finite element

equations to obtain their corresponding Fourier displacement components. The derivation r

of the Fourier components for each of these loads is considered separately and is

presented here.

The loads act on the head ofthe pile as shown in Fig.2.3a, Fig.2.3b, and Fig.2.3c.

Fig. 2.4 shows the element used in deriving the forces. From this figure it can be seen that

only nodes at the top of the element are considered as the load is applied at these points

only. The pile surface on which the load is applied can be divided into n divisions along

its radius in the radial direction. The displacement fimction used in deriving the

consistent nodal loads is given as n

u - ^{px^\ + 2^2 + I^^u^^osiO

1=0

n

V = 2(iV,v, + N^y^ + N{v^smiO

1=0

n

w = 2(iV,w, + A 2 2 + N^Wj)cosi0 (2.12) j=0

31

a. Apphed axial load on the. pile.

1

b. Applied horizontal load on the pile.

c. Applied Moment on the pile.

Fig. 2.3. Applied load on the pile.

32

Local Coordinates

x=l

Global Coordinates Radius R

- ^

r,x — » ^

Fig. 2.4. Element used to calculate the load vector.

33

where N^, N^ and A3 are the quadratic shape fimctions consistent with the

shape functions ofthe element used in deriving the stiffiiess matrix, and they are given as

below.

JV,= f, 3x 2x']

. X x^^

(

^ 3 =

X 2x - —+

2 ^

VI r J (2.13)

where

local coordinate ofthe top side ofthe element, and

the length ofthe side ofthe element toward the head ofthe pile.

2.3.1 Equivalent Nodal Forces for Axisvmmetric Axial Force P

The load P applied at the head ofthe pile is assumed to be distributed on the

surface ofthe pile head as given in Fig.2.3a. Thus the magnitude ofthe distributed load at

the head ofthe pile is given by a^ = PI ;r R where P is the resultant load applied at the

head ofthe pile. The Fourier components ofthe distributed axial load is given by the

following formula.

m p

7=0 ^ R' (2.11)

34

The work potential Fby the load is given by the following equation.

+/r/J-3 n n

V= JJXiZ{z'}[^]{Wcos/^siny6>^r^^ -;r/J, 7=0 1=0

Using the principle of minimization of potential energy the force vector is obtained as

{P}-rJ^'p{N}rdrd0

where

{p} =

r A

P^

,Pyj

and

{N} =

r -\

N. 3 J

Again due to the orthogonality of the trigonometric terms, the individual terms in

the summation are decoupled. So the consistent load vector is calculated as follows for

different harmonics.

The global coordinates are converted into local coordinates as given below for the

different Fourier harmonics, n.

r = N,R^ + N^R2 + N,R,

and dr = dx

giving

35

@x = 0

@x = l/2

@x = l

i = 0

r = R,,

r = Rj, and

r = R,.

P^ P rr j 3

TV R^ ^''V N,r cosmOcosnO dr dO

P rP^^^i. 3x 2JCM

TV R^ L h\ ~T^1 J r cosmOcosnO dr dO

similarly

PI 15R'

•{4R,+2R2-R,)

2Pl XSR"

{R,+SR,+R,)

i=\

PI

15R' •{-R^+2R2+4R,)

PI

30R^

PI

15R^

PI

30R^

{4R,+2R,-R,)

{R^ -hSRj +R^)

{-R^+2R2+4R^)

36

/ > 1

Pl = • | ^ (^ ,+8J?2+-R3)

2.3.2 Equivalent Nodal Forces for Lateral Load H

The distribution of the shear along the cross section of the pile is given in

Fig.2.3b. The load has components in the r and 0 directions, respectively. The work

potential due to these components is given by the following equation.

^ = 2 { ' " " ^2^"2 + ^3r"3 + ^i^v, + H^.v^ + H,,v,}

- jjj^ {7;[i7]{M}cos i0+T,[N]{w}sin i0}r dr dO 2 -„Q /=o

where

It

T; =Y^Tcosi0, /=0

Tg =2]^s inz^ ,and /=0

, = ^ ( , . _ , . , 0 3 ^ , )

37

The derivation of the horizontal load components is done similar to the method followed

in the previous derivation. The horizontal load components are given as

n = 0

H,^ = [-92RI - 32Rl + IRl - 96R, R] - 96Rl R^

^UR^R^ +4M^R^ +2lRjRf -6R^R^

+ R^{336R,+16SR2 - S4R,) + 24R^R^R,)HI I {945R')

H^, = (- 8y?f -12ml - 8 3 - 24R, R] - 24R^ R^

-24R^Rl -24RlR, +3R,Rf +3R^R,

+ R^{42R, +336R-, + 42 R,) +24 R,R2R,)4 HI I {945 R')

ZZ3, = - ( - IRl + 32Rl + 92Rl -4SR,Rl - URfR^

+ 96R2R^+96R^R, +6R,R^ -21R^R,

+ R^{S4R,-16SR2 -336R^)-24R,R2R,)HII{945R').

n = l

Zr„ = (-927?/ -327^2' + IRl - 96R,Rl -96RfR2

+ \2R2Rl +4SRlR^ +2lR,Rf -6RIR,

+ R^{224R,+U2R, -56R,) + 24R,R2R,]HII{\260R')

H^^ = (- %Rl - 12SRI - SRl - 24R,Rl -24R^R2

- 24R^Rl - 24RlR,+3R,R' + 3RlR,

+ R^{2SR, +224R2 +2SR,) + 24R,R,R,)HI l{3l5R')

3 8

ZZ3, = -{-7Rl+32Rl+92Rl -4SR,Rl-\2R^R,

+ 96R,Rl +96RlR, +6R,Rl -2\RIR,

+ R'{56R,-n2R, -224R,)-24R,R,R,)HIl{\260R')

H,, = {-92Rl - 32Rl + IRl - 96R,Rl - 96R^R,

-^UR^Rl +4SR^R, +2lR,Rf -GR^R,

+ R^{672R, +336R, -16SR,)-24R^R,R,)HII{37S0R')

H^e = (-8^,' - nml -^Rl -24R,Rl -24RfR,

-24R,R} -24R'R,+3R,R! +3R'R, L2^V3 ^-r.iV2.£V3 -r_-.tv3.1v, - r j a v 3 . l v ,

+ R^{S4R^ +6I2R2 +S4R,) + 24R^R2R,)HII{945R')

H,0 = -{-IRl + 32Rl + 92i?3' - 48iJ, 72' " 12i?,' R^

+96 R.R^ + 96 R^R, +6R,R-2\RiR, L2.1V3 -T- ^\J±^2 ^^3 ^ v».iV3.iv . t i i V 3 i v ,

+ R^{\6m, -336R2 -612R,)-24R,R2R,)HII{3nQR')

n>l

H,^ = (- 92Rl +7Rl - 32Rl - 96R,R] - 96RlR^

+ I2R2RI +4SRlR, +2lR,Rl -6RlR,

+ i?^(-84i?, +168i?2 +336i?3) + 24;?,i?2^3)^/(l890;?')

H^r = (- ml -128i?2' - ^Rl - 24R, Rl - 24Rl R^

-24R^Rl -24RlR^ +3R^Rl +3RIRI

R'{42R, +336R2 +42R,) + 24R,R2R,)2Hll{945R') +

http://-r_-.tv3.1v

http://-rjav3.lv

39

H,^ = - (^ IRl + 32Rl + 92i?3' -48i?,R^ -\2RlR

+ 96R2R] +96RlR, +6R,Rl -2\RIR,

+ R'{S4R,-16SR2 -336R,)-24R,R2R,)HIl{lS90R')

H,g = {-92Rl - 32Rl + IRl - 96R,Rl - 96Rl R^

+ I2R2RI +4mlR, +2lR,Rl -6RlR,

+ R^{672R, +336R2 -l6SR^)-24R^R2R,)lIll{37S0R') r

H^g = {-SRl - 12SRI - SRl - 24i?, 2' " 24RIR2

-24R^Rl -24RlR,+3R^Rl +3RIR^

+ R'{S4R, +672R2 +S4R,) + 24R,R2R,)HI l{945R')

H^0 = -{-7Rl + 32Rl + 92Rl - 4SR,Rl - URIR^

+96R2 Rl + 96i?2 R^+6R,R-2 IRl R,

R^{\6SR, -336/?2 -672R,)-24R,R2R,)HIl{37S0R')

2.3.3 Equivalent Nodal Forces for Applied Moment M

The moment distribution along the cross section of the beam is as given in Fig.

2.3c. The work potential due the applied moment is given by the following equation.

V = ~{M^W^+ M2W2+ M^w^]

+;r/?-3 rt n

J \YL^re[N]{^} cosiOsinjO dr dO -rcRy 7=0 '=0

40

where

4M

As in the above derivations for axial load the components ofthe moment are given by

« = 0

M, =

M3

n=l

M, =

«>1

2(40/?,7?2 -I6R2R, -6R,R, +39RI +16RI -3RI)MI

105R'

8(8i?,7?2 +m2R^ -^RiR^ +^Rl +4SRI +5RI)MI

^^ = 105R^

2{-l6R,R2 +4OR2R, -6R^R, -3Rl +l6Rl + 39Rl)Ml 105i?'

(40;?,i?2 -16^2^3 -^RiRs +39RI +\6Rl -3RI)MI

105^

4{SR,Rj +SR2R, -4R,R, +5Rl +4SRI +5RI)MI

^2 - 105R'

(-16ig,i?2 +40/?2i?3 -6R,R,-3Rl +\6Rl +39RI)MI

^3 - 105R'

{40R,R2-16R2R^-6R,R, + 39Rl+l6Rl -3RI)MI

^1 = \05R'

4{SR,R2 +SR2R, -4R,R, +5Rl +4SRI +5RI)MI

^2 = [5iF

M.

41

{-\6R,R2 +4OR2R, -6R,R,-3Rl+\6Rl +39RI)MI

' 105i?' •

The stiffiiess matrices and the load vectors are assembled using compatibility

conditions. The resulting stiffiiess matrix is a symmetric banded matrix and can be solved

using the solution technique for half banded matrices.

2.4 Validation ofthe Finite Element Solution

The solution of the matrfces is coded using a FORTRAN program. The

displacements corresponding to each Fourier component of the loads are obtained. They

are then superimposed and checked for convergence ofthe total displacement. As can be

seen from Fig. 2.5, Fig. 2.6, and Fig. 2.7 only 1 harmonic is required to obtain

convergence in displacements, shears, and moments, respectively.

Also a convergence study based on the number of elements used is presented in

Fig 2.8. From this figure it can be inferred that a reasonable level of convergence has

been obtained by the number of elements used in solving the finite element model. In

analyzing the number of elements used in the solution it has been observed that the

number of elements in the radial direction contributes towards the accuracy of the

solution. Also the mesh density needs to be higher at the head of the pile which is where

the magnitudes ofthe displacements are the highest.

The displacements obtained are compared with the solutions published by other

researchers. The comparison is done using an / ^ parameter. This parameter has been

used by researchers to show the results and is given by the following equation

42

0.00000 0.0

»v

u non 0.00005 0.00010

n =1

«=0, 1

- n =0,1,2

Fig. 2.5. Displacement plots for different harmonics for applied lateral force.

43

-0.4

s e

-0.6

-0.8 -

1.0"-

'^non

1.0 0.5 1.0 1.5 2.0

n =1

n =0,1

« =0, 1,2

Fig. 2.6. Lateral shear force plots for different harmonics for applied lateral force.

44

0.000 0.0

ls>

-0.6

-0.8 -

1.0"-

M_ non

0.005 0.010

Fig. 2.7. Bending moment plots for different harmonics for applied lateral force.

0.050 -

45

a B

J 0.040

0.030

• • ' ' I I I • • ' '

200 400

Number of elements

600

Fig. 2.8. Convergence plots based onthe number of elements used in descretization of the pile.

46

where

E^ = modulusofelasticity of the soil,

H = applied lateral load,

L = length ofthe pile, and

u = displacement ofthe pile.

The comparison is presented in Fig. 2.9 through Fig. 2.12. In tiiese figures die

following notation has been used

^^ EX s

where

Ep = modulus of elasticity ofthe pile,

Ip = moment of inertia ofthe pile,

E^ = modulus ofelasticity ofthe soil, and

L = length ofthe pile.

Also

D = diameter ofthe pile.

It can be seen from Fig. 2.9 through Fig. 2.12 that the finite element solution matches

with the solution developed by Verrujit and Kooijman [17]. Also, there is a slight

discrepancy when it is compared with the solution obtained by Poulos. The finite element

solution produces a softer solution for L/D ratios equal to 10 and 25 and for K^ values

47

10'

'p»

10°

,

lp , 4' '

M

f

11

iuU L iJL

- i

n i 1

J, ...

I 11 • FEM

• Poulos lU - Verrujit

..[ .... -1 -- 1 1 1 1 1

1 1 1

|T[jj

10' 10' 10' 10^ K.

^Qr 10'

Fig. 2.9. Comparison of die /p;,displacement parameter for FEM, Poulos and

Verrujit and Kooijman solutions for L/D = 10.

48

10'

'p*

10'

^ > ^ " '

1

I

1

I I I ^ 1*,

I I I ^ . ' z ' J IT ^ ^

' 1 L t''^

\ ' "

f

- II ^jiS 1 ?^

1 FEM

i . l i Poulos I j

Vrrn r - r -n 1 1 m ^ ti

r^*^'' .''1' '

1 1 1 -

Verrujit |

1 ^ \

11 10' 10 10' 10

K_

10 10'

Fig. 2.10. Comparison of die /p^displacement parameter for FEM, Poulos and


49

10'

'p*

10'

I I I 1 1 IT 1 A 1 I I I ^^ 11 1 1 I I I > ^ 11 1 1 1 1 L .^^^ \ 1 1 1 1 IJK ^ 11

' _ _ .M 1 1 1-• • 7T^ - m + + • 1 V 111 1 1 \ ^ ' I I I 1 I > ^ I I I 1 1 1 Jit^ 111

.:: :::| :::: ""i

1 1 I 1 1 II II -f ^^^ II

j - Poulos

j Verrujit

10' 10' 10' 10' 10 ' 10' K

Fig. 2.11. Comparison of the /p^displacement parameter for FEM, Poulos and


50

10'

p*

10'

1 1 ! 1 1 1' \Mt

^^I^IIH 1 ^ III

•'T . ; ^ ' ]

tith^^ l i t •b^^^^^'i I 1 1 11 1

1 1 1 1 i _..|| _..|| J F E M I I

j Poulos j Verrujit

10' 10= 10 10' 10= 10' K.

Fig. 2.12. Comparison of the/^displacement parameter for FEM, Poulos and

Vemijit and Kooijman solutions for L/D =100.

51

less than 100. Plots showing the comparison ofthe displacement pattem along the depth

ofthe pile for different K^ values are shown in Fig. 2.13, Fig. 2.14, and 2.15. It can be

seen that the finite element solution matches exactly with the solution provided by

Verrujit and Kooijman. From the above comparison it can be inferred that the finite

element model produces a reasonably accurate and convergent solution.

52

-0.2 0.0 I I I O.Oi

A/ l.O"-

u/u 0.2

top

0.4 0.6 0.8 1.0 I I I I

FEM ^ Verrujit

Fig. 2.13. Comparison of die distribution of die displacement along die lengdi 2

of die pile between FEM and Verrujit for ^^ = 10 .

53

u/u top

-0.2 0.0 I I I O.Oi I I I

0.2 0.4 0.6 0.8 1.0 < > > I I I I I I I I I I I I I I

FEM

Verrujit

Fig. 2.14. Comparison of the distribution of the displacement along the length

of the pile between FEM and Verrujit for ^^ = 10 .

54

-0.2 0.0

i.oJL-

FEM

Verrujit

Fig. 2.15. Comparison ofthe distribution ofthe displacement along die lengdi

of the pile between FEM and Verrujit for K^=10\

CHAPTER m

DERIVATION OF A RELATIONSHIP BETWEEN

THE COEFFICIENT OF HORIZONTAL SOIL RESISTANCE

AND THE MODULUS OF ELASTICITY OF THE SOIL

FOR CIRCULAR PILES

3.1 Introduction

The Winkler concept has "been widely used in estimating the displacements of

laterally loaded piles, especially by Matiock and Reese [6]. One ofthe main advantages of

the Winkler model is that the soil characteristics can be represented empirically by the so

called coefiBcient of subgrade reaction k. For stratified soils, diflferent values of ^ are used

to represent the properties of the soil as a set of finite elastic springs applied along the

length of the pile. Various researchers have presented the value of the coefficient of soil

resistance, k and all of them are empirically determined from very sophisticated

experiments. Matlock and Reese have presented curves for non-dimensional deflection and

moment coeflficients. The soil displacement characteristics are extended to include

nonlinear soil behavior and they are represented by p-y curves which are also known as

load deflection curves. These curves are very usefiil but require prior knowledge of the

coeflficient of horizontal subgrade reaction from deformation properties of the soil and

pile. Also, no known correlation exists between the coefiBcient of horizontal subgrade

reaction and the material deformation properties ofthe soil and the pile. In this research

an empirical equation is developed to express the coefficient of horizontal soil subgrade

55

56 reaction as a fimction of the material properties and the geometry of the pile-soil system,

assuming that the soil is uniform with a single value ofk.

3.2 Derivation ofthe Relationship between the Soil Properties and the Horizontal Winkler Coefficient k

In this chapter many problems are studied in order to obtain a relationship

between the Winkler coefficient k and the important parameters of the soil-pile system,

such as length ofthe pile, and modulii ofelasticity ofthe soil and the pile. Poisson's ratio

ofthe soil is kept as 0.3 in this research. Some ofthe work done on this topic has been

presented in a previous publication [25]. Non-dimensional terms used in the analysis are

given as follows.

Kr =

K^ =

LID =

where

(Pile - Soil stiffiiess ratio)

kL'

Epip (Non dimensional coefficient of horizontal soil resistance)

length to diameter ratio ofthe pile

L = length ofthe pile,

D = diameter ofthe pile,

E ,1 = modulus of elasticity and moment of inertia of the pile, and

E^ = modulusofelasticity ofthe soil.

Two different approaches are used here in developing die relationship between die

properties ofthe pile-soil system and the horizontal coefficient of subgrade reaction.

57

3.2.1 Approach 1

In die first approach, different problems are solved for LID ratios varying from 8.3

to 50 and K^ values of 10^ 10', 10', 10' and 10'. The technique used here is to

numerically differentiate the displacements along the lengdi of die pile four times and

equate the result to die soil resistance along the depdi. From this, the horizontal soil

resistance q, is calculated along the length of die pile using the following equation. du'

EJ P P ^ _ 4 dz' = <1 (3.1)

Equating q to the Winkler coeffipient of horizontal subgrade reaction the following

equation is obtained

k = -^ u (3.2)

where k = Coefficient of horizontal subgrade

reaction along the length ofthe pile.

The data used are given in Table 3.1 and the results are presented in Figs. 3.1, 3.2,

3.3, 3.4 and 3.5.

Table 3.1. Data used in deriving Method 1.

L

in.

250

250

500

600

600

r

in.

9

9

12

12

6

Mm

2036.5

2639.4

5213.6

25132

15714

Ep

Mm

2x10'

2x10'

2x10'

2x10'

2x10' 1

L D

8.33

13.89

20.83

25

50

^ .

10

10'

10'

10'

10'

58

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' ' ' ' o o CO

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So ^ II

«n en e)b

63

From the results presented above, die following conclusions can be drawn.

1. The coefficient of soil resistance is not a constant along the lengdi ofthe pile.

2. The coefficient of soil resistance becomes negative in some places, which is

contradictory to the concept of die Winkler model.

For diis reason, diis mediod of determining the coefficient of soil resistance is

abandoned.

3.2.2 Approach 2

In this approach various problems are solved for K^ values ranging from 10',

10^ 10 \ 10', 10' and 10' and for L/D values of 10, 25, 50 and 100 using the finite

element method. Three different loading conditions are considered for the analysis. First,

a lateral force is applied without any moment and the pile is free to rotate. This kind of

behavior is found in free head piles. The second type of loading condition considered is

an applied moment without any horizontal force. Finally, a lateral horizontal force is

applied without rotation at the top of the pile. The reason for considering these loading

conditions is that in reality the loading on laterally loaded piles is a linear combination of

these three loading situations. In each case the maximum value of the displacement at the

head of the pile is obtained. Using this displacement as a reference, a value of the non-

dimensional Winkler coefficient K^ is selected so that the Winkler model produces the

same maximum displacement at the top as that ofthe finite element model. The following

non-dimensional parameters are used in the presentation of the results, which are

obtained using the Winkler coefficients selected above.

_ kL'

p p

EX ' EJn p p

64 u

u = non

non ~

non

^non

Kan

L z

L

D

Eplp

ML

The values of K^ are plotted in Fig. 3.6.

3.2.2.1 Results for Horizontal Force (Case 1)

A lateral load is applied at the top of the pile. The non-dimensional values of

displacement, shear and the moment are presented for different K^ values in Figs. 3.7

thru. 3.24. It can be seen from the figures presented that the Winkler solution obtained

using the value of the new Winkler coefficient K^ matches fairly well with the finite

element solution.

3.2.2.2 Application of Moment

A moment is applied at the head of the pile. The non-dimensional values of


thru 3.42. It can be seen from the figures presented that the Winkler solution obtained

using the value of the new Winkler coefficient K^ matches fafrly well with the finite

element method.

65

10**

10^

10'

w 10^

10^

10'

N . J. t ' f

/.'^ Xf

i n o

r/n-]o T / Uu — z , j

j/D=m z/

11 ..I _. --i , T

I

: — • ; = — > , £ :

/ . -i" '.'XZZM-^'. 7- + '' m

,.._

D=50 9 = 100

"t i ± : ; = ^ = = =

:±;:—iji 1 '^y 1 . (^/-. 1 ' \,^ ' • .f-y

4,

..I : | : :

1 1

I 'm--1 lI

it:: 11

I

1 1 1 E E | | S2 - - • f l r?^?

11 ' / • / I T "y^ "^

1: .i''' 4-1- -<^ - -4-4 '^— '• 4 + «7»^^- + -H- r^y-

' / • • • • '

.,<.

:::: z

- . 1

4 J • "TT

TT

1

• i " 'TT ~

fl 1

..I. __ : : | :

1

..I, ... : : 1 :

1 1

' • ' •

: : s : : : • ^ —

T T

t

" T - . 1 .

E=g •ft

Tf

10' 10' 10' K

10 10= 10'

Fig. 3.6. Plot of non-dimensional Winkler coefficient of lateral

resistance AT versus K^.

66

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U QJ

X CO

4->

G 3 3

O4 CO

. 1 H

Q *

1-H

es en 00

. I H

PLH

, •s 0

1 - H

Is <^i H-2

iH T ) Q;

• I H

' O H O4 CS

a cS I H 0 (^ ,_^ 

• I H

s p

73

8 4-> CO

. 1 — c CO

I H

13 4 - ^

G 0

•a 0

X 0 4->

n (i> . i H

P B Q> 0 0

^ lU G QJ

•5

81

o

fc^

• ' ' I I I J . .

00 .3 CO 3

• - H

P 73 O 3 I H P

^

% 00 G

ari

B" 3 o o m

o 1 - H

II

^:J^ 73

§ —

§ 1 - H

. •i O

I—H

ral

B iH Ti p X OH OH cS G c I H O

( 4 H

13 73 O

3 4-J

G (U 3 p p

B • l H

o I

for

CO 4-.i O O4

H-»

G

ti 0

3 00

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X T )

i ^ 5i

X CO

H - t

G ^ a ^ OH CO

•1H

Q r i <N en 00

• i H

PU

die

73

^ CI 0

H-J CO

• »H CO

I H

3 G 0 N

•c 0 X 0 4-> G P

• 1H 0 la (S p 0 p

p G U •s

82

uou.. d ^ . ' . . . J . . . . ' I • •

oo d

.OE-5

1-H

non

OE-6

in

O.O

EO

n. n

.

^

*\ t \ * \

" t \ t \

• t \ t \

V \ • t \

t \ ^

5 O 1

1 1 1 1

o 1

— 1 1 1 1 L

uou.^ 9

II S II s II /-^

! ^ 5 ^ N ^

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1 1

q 1-H

1

00

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^"H P

73 O

3 I H

a> ^

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ari

& 3 8

• V

o t - H

II

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73

§ 8 1-H

II ^ ^^4 ^^ **^

for

CO 4-> O OH

men

t

o 3 ing

TS

G (D

X 73

u ^ (U

X CO

G ^ 3 p ^ OH CO

• i H

Q en CN en

00 • i H

PU

. - ^ o

13 i-i

B cS

l—H T3 p ;x

OH OH ed G

I H

o c 13 73 O

3 4—>

G

B (U (U

B * ^H

G JC

M 73

^

o 4—>

CO • I-H CO

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•c o X (4-1 o

G Q^

• 1 H

o : ' H (4-1

u o o ^ p G P

B

83

^

l i . i t i 1. I I I 1 I t I I _, 1 *

o VO

»

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oo d p

1-H I

00

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3 73 CS O

> s op ^3

^G 4) e3 'H4

8 § 9 .S " 1)

5< o

G

u 3 p B

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c2 CO

4-J

plo

4-^

G

1 0

3 00

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X 73

ear

-G CO

4-J

G Si 3 p 0 crS OH CO

• I H

Q

B 73

8 4 - > CO

. i H CO

0 I H

'3 G 0

•S 0

X «4H

0 4-J

rl (1) • i H

0

B u 0 0

^ QJ G QJ

B

en

ob . i H

PU

84

s

6 0.

0

g

in

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E-

1 1

jft ^ » • \ »

\ t \ % • \ t

— x **

-0.2

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7 ^non

. 1 . . .

-0.6

*•'

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# /^

, 1 £2>

» • / l - H

/ 1

00

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r"^

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inki

er

^ 00

parin

B o o

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II

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§

4 - ^

omen

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X OH

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n U H

^ ^ p

73 O

3 4—>

G

for

CO 4-^ O O i

Len

t

ti o 3 00

.3 73 G (U

X 73

^ U

X r/1

4 - J

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eme

lac

OH CO

• i H

Q »o <N en 00

. I H

PU

die

73

%

p

4-» CO

• i H CO

talr

c o N •c o X ( M o 4-^ G U

• i H

U iZ3 (t 900

G P

B

85

IT

d - H <N

II s

00

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3 1 3 73 O

3

inki

er

^

pari

ng

S o o

( N

o 1 - H

II

^ ^ T3

u O 1 - H

II p^ ^ ^ • ^

for

CO 4-J O

f - H O i

G P

ti O 3

ing

73 G 73 O

3 H-J

G

3 (U

u ^

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<s M 73 Q

o

4—> W5

• i H CO

I H

3 G O N •c o X <4H

o 4-> G (D

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o £ 8

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86

im-^i^ T"'.—-.-—i>

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3

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er

^

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g

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1

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4 - >

omen

3

apph

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G rrt I H

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3 4 - >

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o ^

for

CO H-t O OH

H-> G P

ti O

3 in

g 73 G (U

X 73

S ^ (U

X CO

4->

G

eme

lac

OH CO

• i H

Q

..27.

e»)

ob • » H

PU

die

T3 3

4 - ^ CO

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I H

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o 4-J G Q

• i H

o s 900

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B

87

00

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er

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die

73

S 0

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88

6 8.

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t

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3

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er

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for

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men

t

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ding

G (U

X 73

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X CO

4 - >

Ul

eme

^ OH CO

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PU

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4—>

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3 0 3 -2 X OH

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0 «4H ,__, P

73 0 3

4 - >

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3 <D 0

i • I H

^

^ 73

s 0

4—>

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4—>

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p iG (I^ 8 0

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90

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d t

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d

_i I i_

00

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3 1—H P o 3 . M g .3 g ^ a • fH i X

5 ^ o o

G cS I H O

O «+H ' - ^ .—H

II U II -o

7 3 *H

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II '^

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CO • rH CO (U I H

OH 4-^ G P

3 o 3 *»0 - H

G O U N

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P G

X ^ CO

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4-i { ^

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91

1-H

l.OE

<N

i S < in

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t \ t \% x»

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00

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3

O

3

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1-H

A

.OEO

1-H 1

5 / ^ ^

P /V'9 / #

^ / / / '

' / '

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y

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-0.4

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q 1

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e

^ 00

pari

n

e o o

<n o T " H

II

k 73 G cS

in <N

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omer

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r—H

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4 - >

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3 l—H

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B • i H

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OH 4->

G

ti 0

3 00 .3 73 G W

X 73

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eme

lac

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PU

p 0

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0 «3 U H 900

^

u c (U

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92

00

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r - H

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3

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er

^

parin

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w^

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II

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for

CO 4-> O OH 4->

G

ti o 3 00

.3 73 G (U

X 73

§ ^ 5:i X CO

4 - J

G

eme

^ OH CO

• i H

Q en en en

00 • I H

PU

4 - ^

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1 O 3 -2 X OH

G cS I H

n C4H

^^ 73 O

3 H-.>

G 5 3 u u i

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S ^ T3 Q

o

4—> CO

• I H CO

I H

3 G O N •c o X C4H O

G to

• 1 H

o ^ C4H

8 p

G U

•B

93

00

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3 13 T3 O

3

.OEO

0 { ;

in

pq

o in S 1 o

of-^ pq o

• 1-H 1

pq i n 1 - H

1 1 I . 1 1 .

I <N X O

\ t

\» ^ \*

yjy ^X*v

^ < 5 ^ ^

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er

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ng

E o o

o 1 - H

II k "^

73

i O in

II ^ ^^^ ^

for

CO H—>

O O I

omen

t

3 00

T3 G (U

X 73

^ (U

X CO

4- .*

(J

eme

? ^ OH CO

• I H

Q T f en en 00

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UH

omen

E

X OH

G c j I H

<a p-*H

T3 O

3 H-^

G

3 0.)

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B

94

d

L l>i-

q 1-H

I

w: .3 CO 3

^^ P

73 O

3

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er

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pari

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O 1 - H

II

: « ' T3

g O in

II p. ^ ^ j

^

for

CO 4-> O OH

omen

t

3 00

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73 G cS

^ (U

X CO

4 - ^

Ui

eme

^ 04 CO

• i H

Q in en en ob

• ^H PU

H-J

G P

3 0 3

X OH

G cS I H

cS ^ - H

•s 0 3

•4->

G

3 (L) U

^ • i H

s ^ T3 3

0

CO • i H

CO

p I H

3 G 0 N

•c 0 X <4H

0 H-^ G P

• 1 H

0 IG QH

8 0

^ QJ G P •s

95

O)

d

e

00

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3 13 73 O

3 G

^ 3

3 X 04

G CJ I H

, 0

I o o ID

II u II 73

^ 3 3 p o 3 in 4) II ' «

t ^ -iH

2 -a

S I 3 -S O -IH

S CO U

;§ 3 G O P N - -c -o o G X TO M H

CO H_>

« G X P

CO •iH

i p

M "OH •iH Qj

Q -a vd en en

00 • I H

PU

8 (J

G

96

l.OE

es § pq

O.O

EO

\\ \\ • \ t

\ i I t

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11

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m \ * \ t 1 t V t

t> •

5

t

^

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o 1

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1 1 1 1 1 1 1

VO

r d 2

FE

M

Win

kler

1 •

-0.8

. . . 1

o .

1-H 1

I i - ' r - - T i i i I ^non

• ' • I I . I I I I . . I

O I

VO «

o I

oo

d

00

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3 13 •o o 3

inki

er

^

pari

ng

E o (J

vn

o 1 - H

II

(^^ 73

§ O >o II

f>. ^*^ ^

for

CO H-» O O H

omen

t

3

ing

73 G 

(-i

eme

lac

O H CO

•»H

Q r-' en en 00

. i H

UH

omen

3

apph

ed

G cS I H

n «4H « - H

73 ora

4 - >

1 3 u l—H

u 3 • i H

s

die

T3

s

ista

nce

CO

talr

G O N

•c o X

(H-l

o H-^ G P

• i H

o { ^ ( H ^ 900

P G P

B

97

Sj 0 < ;

Oc

VD

W O

1 "^ O)

in 1

pq o 1-H

1

7

3s. <N O

1

'K

\ ' \

"•it VO

o o 1 1

' - . , ^ „^^-^''

00

q

/ / //

' y //

//

* § § » /

1 / t 1

* /

t* /

/ / / /

VO 1

pq o ^

o 5 pq

0.0

VO

pq o ^

1

^ N .

S

•

,

$ :

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-

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^^ P

73 O

3 fe

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q ^ 00 G •§ B 3 o p

o 1 - H

II

t<^ 73

§ O O 1 - H

11 ^^ r i i ^ ^ ^

for

CO H-i

' O H .(_» G

3 o 3 00

.3 T^ G X TS

, she

ar

H-^

G Sri Fl P

cS OH CO

. ^ H

Q oo en en 00

. ^ H

PU

H- !

G P

3 i -s . 1 H

"OH O I a § I H

O ( H H

, _ H

&> T3 O 3

4-^

G

3 p p p

H-> • rH

^

% 73

§ p CJ

1 CO

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p I H

G O

•a o X

(HH o

H-J G

•5 Ifi a> o o ^ (1> G (l> •s

98

ot

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^^ P

T3 O

3 fe

^ 00 G

ari

& 3 o p

o 1 - H

II S«J^ 73

i (.-^

o 1 - H

II -^ P ^ l ^^4

M I H

o t M CO

4 - >

plo

4 - >

G

3 o 3 00

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X 73

U

shea

r

4->

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^ 0 4 CO

• » H

Q o\ en en ob

• ^H

PU

I

4-i

p 3 i

T? •1H

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G ci I H 0

U-> , _ H

<L> 73 0 3

4->

G P

3 u l—H

a> p

4 - J • i H

^ (D •s

73

§ V 0

H-» CO

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p I H

13 4-J

G 0

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X

0 H-J G P

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5

99

00

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00 G

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u o o 1 - H

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3 0

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1 3

X 73

§

, she

ar

4.^

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Q 0 ^ en 00

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H-S

p

ti i 'S • I H

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§ I H 0

VM ,_, U

0

3 4-.>

G

3 u p p

H-> • i H

=§ P

:S T3

U CJ

1 CO

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0 I H

3 G 0

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X

0 HH G U

£ (l> 0 0 ^ (l> G (1)

•S

100

CM

d o o o II i

L - - - U - - J - - - 1 -

00

d

00

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3 13 73 O

3

I o > 3 G 4)

e3 "o^

3 ^ O G p c3

O o T-H (^

8 G <0

3 p p

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'OH

G

i o 3 00

.3 73 G U

X 73

O

CO • I H

CO

2^

G O

o X

J3 2 0 G X CO

U

P 0) 3 p

M ' O H

CO

O O

p G

Q 5

en

ob

101

"non

I I • • ± VO

d 00

d

OE-

4

es

ipq s o 1 - H

s

0.0

.\

-\

- \

?

i

o 8 II 2 II I " 9

L. o

• J — I — u ' • • •

VO

d 00

d q l—H

I UOUr.

00

•3 CO 3

ff—M

P 73 O

3 I H

P

^

^ 00 G

ari

& 3 o p

o 1 - H

II i ^ T3

§ 8 T - H

II -^

I ^ A

^ ^ •-^

for

CO H-»

plo

•t-t G

1 O

3 00

din

G

73

i

ear

X CO

H - *

G y 3 p ^

9^ CO • i H

Q es ^ en 00

• i H

PL,

H - '

c 3 o 3

73 (1)

• i H

' O H OH cS

§ I H O

g- l , _ H

CJ 73 o 3

•4-J

G

3 fj

1—H u a> 4 - >

• 1H

^

^ 73

§ a> CJ

1 CO

• i H CO qj VH

13 4—>

G O •a o X

U H

o 4->

«i> • I H

CJ

Iti Q O CJ

^ <1> G «J

•5

102 3.2.2.3 Application of Lateral Shear Without Rotation at the Head ofthe Pile

A horizontal force is applied at the head ofthe pile. No rotation is allowed at the

head of the pile, simulating a fixed-head pile. The non-dimensional values of


thru 3.60. It can be seen fi-om the figures presented that the Winkler solution varies

slightly fi-om the finite element solution for the displacement plots. But the bending

moment and shear match fairly well with the finite element model.

3.2.3 Derivation ofthe Equation Relating the Winkler Coefficient with the Properties ofthe Pile-Soil Svstem

The Winkler coefficients obtained in the above analysis are presented in a

graphical form in Fig. 3.6 using logarithmic scales on both axes. It can be seen from this

figure that the values of the Winkler coefficient roughly form a straight line for different

values of K^ for a give value of LID. Thus an equation ofthe following form is assumed

in order to develop a relationship between the two. This is equation is given as

^. = A^rT (3-3)

Using the above equation and applying linear regression the following values for A and a

are obtained.

(D\ JLV y4 = 0.58571-6.57143

\LJ + 0.00000847

\DJ (3.4)

^ 2

a = 1.111748-0.0010788f—j+0.00000337f^j (3.5)

103

VO 1

pq q 1-H

3 pq o in

.OEO

" =

«

• ^ \

**-«.^ ^ \ ^

*"., *».,

**»,

5 <N ^ . uou^ ^ <» 9 5 O O ^ o O ' - I

1 1 1 > •

01

•3 CO

3 O

mod

I H

u • fc^

^ T ^

^ 00

*5 c3 OH rao

o O 1 - H

II 1^' 73

O 1 - H

II ^ ^ k

S T

**>3

for

^ o 0 4

men

t

0

3 00 G • i H

73 G U

X 73

u 13 0

X CO

ent,

3 p

lac

OH CO

• T H

Q en " ^

en 00

• l H

PU

0

B • I H

»

0 r—1

13 I H P

H - i

73 P

X 04 & ed

§ I H

. 0 C4H

^ i H

P T3 0

3 G

P U P 4 - i

• i H

G la

1 73

s p CJ

CO •1H CO

I H

13 4 - ^

G 0 0 N ^ • T H • I H

5 0 0 ^

CM B 0 U H •M 0

p S-•rt 0 P *-»

0 *_. P cS

« .2 c 3 « S •S 2

104

non

• • ^ ' 1 ' 1 1 1 1—J 1 • • ' r^^v^ty~ - .^ .

o I

VO

d

as

.j__L ' ' _oo. • • • • •

CO

105

p O

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s in

§ g in

1

' ^

S o

• 1-H

:5

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121

Using the above values for ^ and a the following values of A: are calculated and

presented as a comparison with the values of K^ from the graphs.

Table 3.2 Comparison of K^, K^ and AT* •

L/D

5

25

40

50

100

Kr

1000 100 10

100000 10000 1000 100 10

100000 10000 1000 100 10 100000 10000 1000 100 10 100000 10000 1000 100 10

from Fig. 3.6

3075 230 21 225000 16800 1400 120.65 11.93 180000 13500 1145 104 10.3 155900 12500 1053 97 10 119127 10283 931 90 9

fromEqs. 3.4 and 3.5

3863.83 310.22 24.27

232161.09 19006.6 1556.03 127 39 110.43

178945.3 15091.44 1272.74 107.33 9.05 158239.49 13585.62 1166.39 100.14 8.6

113350.85 10395.55 953.39 87.44 8.01

It can be seen from the above table that the equation gives values of K^ which

match with the values of K^ presented in Fig. 3.6. The maximum error in the value of

122 K^ was found to be 17%. The maximum error in the displacements, moments and shears

due to this error was found out to be about 12%, 15% and 15% respectively.

CHAPTER IV

A NEW MODEL FOR THE ANALYSIS OF

LATERALLY LOADED CIRCULAR PILES IN

A LAYERED SOIL MEDIUM

4.1 Introduction

The model developed here adopts the continuum approach to model the soil

material properties. The pile is then assumed to be embedded in the soil medium.

Principles of elasticity are used to model the pile and the soil continuum. This model

results in equations similar to that of the Winkler model except for the fact that the soil

properties are not represented as springs at distinct points as used in the Winkler model.

The present model uses the concept ofthe modified Vlasov model proposed by Vallabhan

and Das [26]. A similar approach was used by Sun [18] in the derivation of his model for

laterally loaded piles. The distinction between this model and that developed by Sun [18]

is that this model uses two distinct parameters to represent the radial and tangential

displacements of the soil while Sun uses a single parameter to represent both. Sun used

an analytical approach, adopting the classical solution technique to solve the differential

equations ofthe system. Also Sun while presenting the results of his model uses a value

for Poisson's ratio of 0.5, which results in a singularity in his model. No explanation is

given in his paper about this problem. The system equations are derived using die

principle of minimization of potential energy^ Here a finite element approach is used to

represent the beam element and die associated soil stratum on die side^ A numerical

123

124

approach is adapted here to solve the system equations. The present chapter deals with

the derivation of the system equations and the numerical model for solvmg diese

equations.

4.2 Assumptions in the New Soil-Pile Model

The following are the assumptions made in this model for the pile-soil system.

1. The cross section ofthe pile is drcular; it can be a hollow or a solid pile.

2. The pile is assumed to act like an elastic beam and is linear and elastic and has

constant material properties along its length.

3. The soil stratum is assumed to be linear, isotropic and homogeneous in each layer.

4. Displacement compatibility is assumed at the pile-soil interface surface and no pile-

soil separation is considered on the tension side ofthe pile.

4.3 Development ofthe Svstem Equations

The laterally loaded pile foundation system can be divided into two components,

one for the pile as a beam and the other for the soil as a semi-infinite elastic stratum.

These subsystems comprise the pile and the soil stratum. The behavior of the pile-soil

system depends on the interaction between the pile and soil in its entirity.

The soil stratum is considered to have a finite depdi as shown in Fig. 4.1. Both die

modulus ofelasticity and the Poisson's ratio are considered to be constant along the depth

/. for a particular soil layer. The displacement notations considered here are as shown in

125

H

m > ^ ^ /

.6

r, u

Diameter of die pile (2R)

U

t . w

Soil Region I

i /- di soil layer

f

L = Length of die pile

R = Radius of die pile

M= Moment on the pile

H= Horizontal force on the pile

Soil Region II

Fig. 4.1. Pile soil system for die new model.

126

Fig. 4.1 and are specifically defined for that soil layer. The radial displacement ofthe pile

is given by w , the tangential displacement is given by v and die vertical displacement is

given by w. The radial, tangential and the vertical axes are represented by r, 0 and z,

respectively. The vertical displacement in die soil is assumed to be very small and is

neglected here.

4.3.1 Assumed Displacement Functions for the Soil Laver

As per the displacement convention shown in Fig. 4.1 the displacement equations

at the pile soil interface can be written as follows.

u{r,0,z) = u{z) (p(r) cos0

v{r,0,z) = -u(z)i/^(r)sm0 (4.1)

w{r,9,z) = 0 .

In the above equation u(z) gives the magnitude of the lateral displacement of the cross

section ofthe pile in the radial direction, which is the same as the displacement ofthe soil

at the pile soil interface. The^(r) and ^(r) fimctions are non-dimensional shape

fimctions that yield the displacement pattems of the soil around the pile in the radial and

the tangential directions, respectively. Each of these parameters has a value of 1 at the

pile soil interface and is equal to zero at infinity. This is consistent with the physical

model, as the displacement gradually decreases towards either expanse of the soil and

finally becomes zero at infinity. The radial or tangential displacement of the soil at any

point is given by the product of the magnitude of the displacement u(z), at the pile soil

interface and the corresponding displacement shape fimctions ofthe soil.

4.3.2 Strain-Displacement and Stress-Strain Relationships ofthe Soil

The strain displacement relationship ofthe soil is given as follows

127

M = 7 re

'Vrz

[r.e

> = <

du

u — + r r de 1 du dv

r de dr du ~d~z dv_ dz

r (4.2)

The stress strain relationships are given by the following equations

{cy\={e\[e\ (4.3)

where

w = <^e

'^ re and \E\ =

:e\

'A, + 2G,

^

0

0

0

^ .

A, + 2G,.

0

0

0

0

0

G^ 0

0

0

0

0

G^ 0

0

0

0

0

G. / _

(4.4)

A. and Gj are Lame parameters for each soil layer and are given by

1 = , ^J."' . a n d a = ^' ••~(l + v j ( l - 2 v j ' " 2 ( 1 + 0

(4.5)

In the above equations

128

^r = Normal strain in the soil in the radial direction,

ê = Normal strain in the soil in the tangential direction,

y re^Yrz^^Yze = Shear strains in the soil,

^r = Normal stress in the soil in the radial direction,

ê = Normal stress in the soil m the tangential direction,

r^Q, r^ and r ^ = Shear stresses in the soil,

E^ = Modulus of elasticity of the soil for a particular

layer, and

y^ "^ Poisson's ratio ofthe soil for a particular layer.

4.3.3 Potential Energy ofthe Pile

First the potential energy of the pile is developed and is superimposed onto the

potential energy of the soil to obtain the total potential energy of the entire system. The

potential energy of the beam can be written corresponding to the lateral displacement

magnitude w, and the displacement parameters ^ and xj/ as follows.

^J. UûY . . . , .~.Idu\ p p I ,o,^dz') dvol - H u\ ^- M

\dzJ (4.6)

z=0

where

Ep = Modulus ofElasticity ofthe pile,

129

Ip = Moment ofinertia ofthe pile geometry,

n ^ = Potential energy of the pile,

Up = Strain energy ofthe pile,

Vp = Work potential by the applied forces on the pile,

H = Prescribed lateral force at the top ofthe pile, and

M = Prescribed moment at the top ofthe pile.

4.3.4 Potential Energy ofthe Soil

The soil is divided into two regions as shovm in Fig. 4.1. The potential energy of

the soil in region II can be written as follows

where

n^ = Potential energy ofthe soil,

U^ = Strain energy ofthe soil, and

V^ = Work by the external forces on the soil = 0.

The strain energy of the soil U^, surrounding the pile can be written using the

strain-displacement and stress-strain relationships in Eq. 4.2 and 4.3, respectively.

^^"11 \{^^^^ ^ ^^^^ " ^''^^ " ^-^^-^ " ^ -^« " ^z^r.^) f dr dz de (4.7) 0 R 0

130

Substituting Eq. 4.2 and Eq. 4.4 into the above equation the following expression

for the potential energy ofthe soil is obtained^

ao <X>2)T

0 0 0

(/I + 2G)< dl dr)

\ 2

+ <!>-¥ V

r J u^ôs'0^2X^^^û'cos'e

dr r

+G (j) y/ ^ ^ 1 ^ 2 ^ Q g 2 ^ ^ | ^ ^ | . . ^ ^ g 2 ^ ^ | ^ ^ l ^ 2 ^

dr

f du

Kd^ f du

rdrdedz (4.8)

4.3^5 Total Energy Function ofthe System and Minimization ofthe Potential Energy

In order to obtain the complete potential energy ofthe system the potential energy

ofthe pile is superimposed onto that ofthe soil. First only region I, the pile and region II,

part of the soil stratum, are considered^ Thus the integration is applied only until the

length of the pile. To obtain the system equations the total potential energy of the system

is minimized. After minimizing the following expression is obtained.

Sn = SUp+SU^+SVp

= Epip j k^2,. f^^..^ d u d u

dz" \ dz" ) dz

+ 41 (X + 2G) d(t> J, d(t>

u—ôl u dr

+ u-——o u-—-drJ r \ r

+ X d(/f J <l>-y/

u—'-oy u-dr \ r J

+ u-——S\u dry

+ Gu\ (!>-¥ I dy/ r dr J

r f u

(p-i// dy/\

V r + dr )

131

+ G du . J du \ du J du — (l>S[—<t>\+ — y/S[ — y/ dz \dz J dz \dz

rdrdz-HSu ^-MS, '=' KdzJ

fdu) = 0 (4.9) z=0

4.3.6 Equations Corresponding to the Magnitude of Displacement u

Collecting the terms with respect to Su from Eq. 4.9 and equating the terms inside

the integrals to 0 the following equation is obtained.

^ , dû ^ dû , ^ EJ,—r-2t—r + ku = 0

' ' dz' dz"-0<=x<=L (4.10)

where

k = 7r{X + 2G)\ (^Xjtll. V^

\dr. + r J ^ 2 X ^ ^ ^ + G(^^^'^'^

dr r

V

dr J rdr (4.11)

and

2t = TT G j{(/>^ + i/^^) r dr (4.12)

In the case of a stratified soil medium the values of A: and 2t need to be calculated for each

layer. These values vary depending on the values of A and 2G for each soil stratum.

Outside the domain ofthe pile, i.e., in soil region II, Eq. 4.10 can be solved and

the following expression for the lateral displacement is obtained using an analytical

solution to the equation.

k -J—z

u(z) = uê * ' (4.13)

132

_*_

u(z) = u,e ^2' . (4.13)

The above equation can comprises a part of the boundary condition of the system. The

corresponding boundary conditions for Eq. 4.11 are as follows.

EÎ^^-2t^-P = 0 (r = 0) (4.14) ^ ' dz dz

EJ„^+M = 0 (z = 0) (free head-pile) (4.15) ^ '^ dz

— = 0 (z = 0) (fixed-head pile) (4.16) dz

w = 0 (2=1) (clamped pile) (4.16)

— = 0 (z = 1) (clamped pile) (4.18) dz

dû dz'

= 0 (z = 1) (floating pile) (4.19)

E J ^ _ 2r — - Î2kiu = 0 (z = 1) (floating pile). (4.20) ^ ^ dz dz

Eq. 4.11 can be solved using the boundary conditions described in the equations above,

and the magnitude of the lateral displacements along the depth of the pile can be

obtained.

133

4.3.7 Equations Corresponding to the Displacement Parameters g and y/

Collecting the terms corresponding to 5<j) in Eq. 4.9 we get the following equation

d^^\d(l> {X^3G)(rV {X-^3G)y/ JX + G) \ dyf

dz" r dr (A + 2 G ) U J ^ (A + 2G)r^ ^ {X^2G)r dr (4.21)

Collecting the terms corresponding to 5\i/'m Eq. 4.9 we get the following equation

d \ ^ \dy/ iX + 3G)fyV {X + 3G) (f> {X + G)\d(f>

dz r dr G V, G G r dr (4.22)

where the /parameter for a single layer is given by

r' = l"'dz + G,\\ •"D'-I".'

f 2 , \2i 2 \u dz + J—u;

(4.23)

and for stratified soil is given by

r' = 'Z'!f^^lo:0)\.^f

2t Y,ju'dz + J-u]

where the u displacement is represented as a summation over the different layers. The

boundary conditions for the above equations are as follows

at X = 0:

^ = 1

v^ = 1

134

and at .J: = oo:

(p = 0

y/ = 0

4.4 Numerical Approach to Solve the System Equations

A numerical approach is introduced in this chapter to solve the system equations

ofthe model given by Eq. 4.11, Eq. 4.21 and Eq. 4.22. An elemental approach is used to

discretize the continuous function given in Eq. 4.11. Hermitian polynomials are used to

develop the system matrices. This approach is preferable over the finite difference

method for solving the differential equations because the finite difference method is a

nodal approach and is very convenient for representing a stratified soil medium. The

finite difference technique is used to solve the differential equations given by Eq. 4.21

and Eq. 4.22 to obtain the values of ^ and y/.

4.4.1 Assumed Displacement Functions

Since the pile needs to be discretized along hs length into elements, a finite

element type of approach is adopted. Hermitian polynomials are used for the shape

fimctions at the nodes of each element. A typical element and its displacement and shape

fimctions are shown in Fig. 4.1. The assumed displaced fimctions for an element are

given by

u =[N{z)\{u} (I){r)cose

v=[A^(z)J{w}^(r)sin(9

135

(4.24)

where {u] are the magnitudes ofthe displacements and the rotations ofthe soil along the

z - axis and ^ and y/ are the mode shapes of the displacements along the radial direction

and tangential direction, respectively. In Eq. 4.24 above

lN(z)j = lN, N, N, N,}

where A ,, A 2' 3 ^^^ ^4 ^^^ Hermitian polynomials ofthe first order and are as follows.

iV,

N2

AT ^ 3

N,

, 3z' 2z'

2z' z' = z + -5-

3z' 2z' r I'

(4.25)

and

/ - -N

{"} <9,

Mn

e 2 J

(4.26)

where w, and ,. are the elemental displacements and rotations at z = 0 and z = /,

respectively. In the above equations z is in the local coordinate system.

136

4.4.2 Strain-Displacement and Stress-Strain Relationships

The elemental strain-displacement relationships are given by the following

equation

{£'}=[<^]{w} where

M=

d dr 1 r

1 d r de

d

dz

0

0

1 d

r de d 1

dr r

0

d

and {w} = u{r,e,z)

(4.27)

dz J

The stress-strain relationships are as given in Eq. 4,4.

4.4.3 Total Energy Function ofthe Pile Element

As described earlier the total energy of the system is obtained by superimposing

the energy ofthe pile on that ofthe soil. The total energy fimction is then minimized and

integrated to obtain the system matrix equations. The development ofthe beam elemental

matrix equations has been widely published in many research papers as well as numerous

text books, so the derivation of the elemental stiffiiess matrix and the load vector for the

beam are not presented here. The development ofthe soil stiffiiess matrix is presented.

The strain energy ofthe soil is given by the following equation

137

^ 4frrH'[ ]M^ ^ - . (4.28)

Substituting into Eq. 4.2 above and integrating with respect to (9 the following equation

obtained

is

'^^=^[[M'W[Es]lBl{u}rdrdz (4.29)

where

[B] =

INJ d^

\_N\

dr

^ \ r a- rJ

Thus the total energy in the system is given by the following equation.

n n^+n.

Up+U^+W,. (4.30)

4.4.4 Application ofthe Minimization of Potential Energy

Minimizing Eq. 4.30. with respect to {w} and equating to 0 the following

equation is obtained

[K]+K]+[^2-]jM = f ) (4.31)

138

The stiffiiess matrices Kp, K,^ and K, can easily obtained by integrating the shape

fimctions and their derivatives. This was done using available software such as Maple.

The different integrations are given as follows.

K]=^U'kJ'K>=

[K,] = klVN^VN\dz = k

420

EI

~ I'

13/

35 11/' 210 9/ 70 13/'

12

6/

-12

. 6

11/' 210 I'

105 13/' 420

I'

6/

4/ '

- 6 /

2/ '

- 12

-61

12

-61

9/ 70

13/' 420 13/ 35 11/'

6/

2 / '

-61

4/ '_

13/' 420 I'

140 11/' 210 I'

140 210 105 J

(4.30)

(4.31)

[K,] = 2tl[_Nj[_N^^\dz = 2t

5/ 1

10 6 5/ 1

10

10 2/ 15

1 10 /

30

5/ 1

10 6 5/

/

30

10 /

30 /

30 2/ 15

(4.32)

m= M. 2)

(4.33)

Minimizing Eq. 4.29 with respect to ^ and ^ t the differential equations for ^ and v are

obtained respectively. They are the same as given by Eq. 4.21 and Eq. 4.22, but die

139

gamma parameter changes because of the discretization of the pile along its depth. The

gamma parameter in terms ofthe discretized displacement fimction is given by

r' =

k , I:JM'MMM^+SGJM'K]'K.]{«K+J:^";

2t

^\{u}\N][N]{u}d. 2t 2 (4.34)

4.4.5 Finite Difference Equations for the Displacement Parameteres (/) and w

The classical finite difference approach is used to discretize continuous fimctions

^ and y/ along the radial direction of the soil stratum. All the terms corresponding to the

magnitude ofthe displacement u are represented by the /parameter. Discrete values of ^

and y/ are stored at the nodes of the finite difference mesh. Boundary conditions given in

Eq. 4.14 thru Eq. 4.20 are applied to solve the algebraic equations formed by the

discretization.

The matrix equations obtained by descretizing Eq. 4.21 and Eq. 4.22 are given by

and [/^^]{H = {/2}

(4.35)

(4.36)

where

\K^ ] = Coefficient matrix for (j) parameter corresponding to Eq. 4.21,

140

[- ^ J = Coefficient matrix for ^^parameter corresponding to Eq. 4.22,

{ } = Vector containing discretized (j) values,

\¥} = Vector containing discretized ^ values,

{/i} = vector representing the right hand side for Eq. 4.21, and

{/j} = vector representing the right hand side for Eq. 4.22.

The coefficient matrix [K\ for Eq. 4.35 and Eq. 4.36 contains linear second-order

differential operators and can be developed as a tri-diagonal matrix with three coefficients

using the central difference formulation. The boundary conditions are applied at the pile-

soil interface and at the end ofthe soil continuum, i.e., at the 1st node and the n+lth node,

respectively.

Thus from / = 1 to « + 1 for Eq. 4.35 the following equation is obtained

A,u,_,+B,u,+C,u,,,=F, (4.37)

where

A;

5.

C

1

2

h'

1

1

2r,/z'

{X

{X

1 . 5

+ 3G)(r + 2G)Vr

and h" 2rh

{X + 3G) y/j (X + G) \dy/,

^ ^ "( / l + 2 G ) r ' ^{X + 2G)r dr

141

Similarly, for Eq. 4.36 from / = 1 to « + 1

^,M,_,+5,M,+C,M,^, =i^. (4.38)

where

A^ J 1_ /?' 2rh

B. 2 (A.+3G,)f/

h' G.

C = -^ + , and /z' 2r^h

^ ^ (/l. + 3 G j ^ , ( A , + G j l ^ ^ , .

G, r ' G, r c/r •

Applying the boundary conditions at r = 0 and at r = i? the following modified equations

at the boundary for both equations are obtained.

for 7=1:

A

B,

c,

E

=

=

=

—

0.0

1.0

0.0

1.0

for / = « + 1:

4 = 0.0

B^ = 1.0

142

C, = 0 0

F, = 0.0.

4.5 Iterative Technique to Solve the Matrix Equations

Two sets of coupled matrices that need to be solved to obtained the solution.

These equations can be decoupled by solving them using an iterative technique. This can

be done by assuming the / parameter, which depends on the magnitude of the

displacement u. Then using this parameter we solve for the displacement parameters ^

and y/. Using the updated values of the displacement parameters, the magnitude of

displacement is calculated and solved iteratively until there is a convergence of the

displacement.

In Eq. 4.31 we see that only the stiffness matrix needs to be updated for each

iteration to solve for the magnitude of displacement. In Eq. 4.35 and Eq. 4.36 both the

load vectors and the stiffness matrix need to be updated for each iteration to solve for the

displacement parameters (p and y/. The linear algebraic equations given by Eq. 4.31, Eq.

4. 37 and Eq. 4.38 can be represented as follows.

[A{k,_,at,.,)]{u]^={A} (4.39)

[B{u,)p}, = {Mr,-,)} (4-40)

[c{um,={fMr ^'-''^

143

To start the iterative procedure the value of / i s assumed to be equal to 1.0. Then

Eq. 4.40 is solved to obtain the values of (j). The new values of (/> are used to update the

right hand side of Eq. 4.41. Then the new y/ vector is calculated by solving Eq. 4.41.

Next these values of ^ and ^^are used to numerically integrate and obtain the values ofk

and 2t. Then Eq. 4.39 is solved using these values to obtain updated value of the

displacement vector u. The convergence from previous iterations is calculated and the

difference is compared to an acceptable tolerance. The calculations are stopped once this

tolerance is satisfied.

4.5.1 Steps Involved in the Iterative Procedure

The following are the steps involved in the iterative procedure.

1. The initial value of / is assumed to be 1.0.

2. Eq. 4.35 and 4.36 are solved using this value of / to obtain the values of { .} and

3. Using these values the parameters k and 2t are calculated using numerical integration.

4. Using these parameters in Eq. 4.31 value of {w,} is obtained.

5. From the new values of ^"' the value of / i s calculated.

6. Convergence is checked using the following equation.

144

'=1 ^ i=l

H("/) IIV ' / m a x I I

where

f = tolerance used for convergence of lateral deflection u.

8. Steps 2 through 7 are repeated until convergence is obtained.

4.6 Validation of the Model

To validate the model it is compared v^th the finite element model. Results

obtained for lateral loads as well as moment loads are compared with the finite element

model. In the case of laterally applied loads the maximum displacement and the

maximum moment are compared with those found with the finite element model. In the

case of laterally applied moment the maximum displacement and the maximum shear are

compared with those of the finite element model. All values presented in the plots are

non-dimensional. It can be seen from Fig. 4.2 thru Fig. 4.9 that there is a slight difference

between the maximum displacement plots with the finite element model. The maximum

moments and the maximum shears show very negligible differences from the finite

element model. The maximum error in the displacements was around 15%. Two cases of

layered medium are considered. The cross section ofthe soil strata is shown in Fig. 4.10

and Fig. 4.12. The displacement, shear and bending moment are compared with the finite

element model in Fig. 4.11 and Fig. 4.13 corresponding to die above cross sections

145

i • l H

Q 73 a o

• l H CO

4^

d o

10'

in-^

10^

10-

\

\ . .

— ^ ^

"1 "" ^ sk H K

\ \ \

11

t|

"ii •

tl

: | : ;

t] 1 1

"1

M

II J

\U M'

1 '

1

: | :

1

4 ' • 1 I I ,

III "~TTT

111 JJl

"Jul 111

- J l l

-l ..in

"1 '"t I

..I Ijj 1 1 1

L/D= 10 I

H,.n = 1 1

10^ 10' 10"= 10 10 10

J

Fig, 4.2. Comparison of die non-dimensional lateral displacement between FEM and the new model for applied lateral load L/D = 10.

146

Non

dim

ensi

onal

Dis

plac

emen

t

P

P

p

q

1 (

L

y 10^

t •

1 III nil ~

jiH

't --M

MKV 11 \v.

Ns

"^^

X--" H I —

lilt

X-: : | | — -

tttj

iHi

tl

.11 T t 1 ]

II • I L/D-

.1 f^non =

im—rm 111 1

II s 1 11

\h^'-'-I I S— t l ^ -tf ^

if—"

II 111

j

- •+

1 1 1 , , .

• 2 5

= 1

[;,.. -x-: : : : = = ^ z : .... ^ 5 ^ -S

11

-41+ 1 1

III -11

1 rim

"[TT

•HI n •ffi

11 TjJ 11

"11 TT 1 1

s i ]

Ii 10 10' 10' 10'

Fig. 4.3. Comparison of the non-dimensional lateral displacement between FEM and die new model for applied lateral load L/D = 25.

147

i I

CO • T H

Q 73 a o

• i H CA

a

T3 a o

;2:;

10'

10-

10'

10"

\ ! . . . _ . .

: : : : : ^ z i : _5v_ _

S^_ s

1 1

::j: :: 1

" L/D =

^,ton =

-•]

1

•ILEEE

- 4 sN --I. -

: : | : : :

..],

1

:::: -:::j T

50 j

:1 1 + •

1 1

1 :::; : : : : |

1 :;;, . . . .

^ ^ ^ ^ • ^ -i™ J \

10 10 10-= 1& 10 10

Fig. 4.4. Comparison of die non-dimensional lateral displacement between FEM and die new model for applied lateral load L/D = 50.

148

a

M CO

73 a o

. i H CO a <p

o

10 10' 10"= ^o' m 10

Fig. 4.5. Comparison of die non-dimensional lateral displacement between FEM and the new model for applied lateral load L/D = 100.

149

10<

a

T—H

CO

o 73 a o

• l H CO

CI

xi a o

10- \ , ^ ,

L/D=10 H =1

non

^ . ^s

-V X

V ^

\ s

10 10' 10' K.

Fig. 4.6. Comparison of the non-dimensional lateral displacement between FEM and the new model for applied lateral moment L/D =10.

150

a

73 a o

• T H CO

a

73 d o

10 10' 10 10 10 10'

Fig. 4.7. Comparison of the non-dimensional lateral displacement between FEM and the new model for applied lateral moment L/D = 25.

151

—H PH CO

73 O

•1H CO

<P

73

10' 10 10' 10" 10= 10'

Fig. 4.8. Comparison ofthe non-dimensional lateral displacement between FEM and the new model for applied lateral moment L/D = 50.

152

Non

dune

nsio

nal

Dis

plac

emen

t

o o

p

K 3'

S "" _^

10=

V

, \ !^ - -

' ^ .

- L/n-

H 11

5 : ' ; ^

;::5 ..._s

100

= 1 ion

V ,

^ ^ - -§ -

\ ;. .

s >

s \

1

10' 10" 10' 10'

Fig. 4.9. Comparison of die non-dimensional lateral displacement between FEM and die new model for applied lateral moment L/D = 100.

153

0

Modulus ofElasticity of die Soil £ 0 5000 10000 ' 15000

~ ^ — I — I — I — \ — I — I — I — I

100

T 1 — I — r T

-200

B

Q -300

-400

-500

Fig. 4.10. Cross section of the soil for a stratified medium(Case I).

154

to

a • l H

o TTS eo

B

1 o CO

73

1—H

c2 CO

I o B e

• T H

73 C

CO

'S

c:

I >-H

CO

• 1H

155

0

Modulus ofElasticity of die Soil £ 0 2500 ' 5000

100

T 1 P

-

-200

B

Q -300

-400

-500 L

I I

Fig. 4.12. Cross section of the soil for a stratified medium(Case II).

156

CS 1-H

^ '

fc a

• T H

a o

B

o CO

•s ; H

CO •*->

JO 'El e I o e c

• 1H

T O

CO

o T2 'El

CO

l—H

bb •TH

fc

157

respectively. It can be seen that there is a negligible difference in the displacement and

shear plots. There is a slight difference, however, in the bending moment plots for the

second case.

CHAPTER V

CONCLUSIONS AND RECOMMENDATIONS

5.1 Summary

The first objective that was set out at the beginning ofthe research was to develop

a relationship between the Winkler coefficient of horizontal soil resistance and the

material and geometric properties of the pile-soil system. The second objective was to

develop a new model for the analysis of laterally loaded piles which includes solution for

stratified soil medium.

A finite element model was derived for solving a axisymmetric solid under non

axisymmetric loading. The solutions obtained using the finite element model were used to

derive a relationship between the Winkler coefficient of horizontal soil resistance and the

properties of the pile-soil system was derived and presented in a graphical form in Fig.

3.6. Also the validity ofthe above mentioned relationship was checked for various soils

and the results presented in Fig. 3.7 through Fig. 3.60. Also a equation for the

relationship was derived and presented. The validity of this equation was checked and the

comparisons presented in Table. 3.1.

A new model for the analysis of laterally loaded piles which can be used to solve

stratified soil medium was developed. A numerical method was applied for the solution

ofthe new model. The validity ofthe model was checked against the finite element model

and was presented in Fig. 4.2 through Fig. 4.13. Also the application ofthe model for

stratified soil medium was presented in Fig. 3.11 and Fig. 3.12.

158

159 5.2 Conclusions

The following are the conclusions that can be dravm from this research

1. A modified Winkler's model wherein a coefficient of horizontal resistance k is

developed. This model is usefiil in predicting the load deformation, shear and bending

moment of a pile embedded in a soil stratum with uniform material properties.

2. A new model is developed using the concepts of the finite element method for the

prediction ofthe load deformation properties of a stratified soil medium.

3. The above mentioned new model has the same size of stiffiiess matrix as that ofthe

Winkler's mode. And hence the computational effort necessary is the very minimal as

compared to the finite element method.

4. The input data to the new model is very simple. Elaborate meshing is not required as

is the case with the finite element model.

5.3 Recommendations

The following recommendations are suggested that can implemented to improve

the new model developed during this research

1. This model can be extended to incorporate nonlinear soil deformation. This can be

done by adopting a p-y curve type of approach towards the nonlinear material

properties ofthe soil system and the interaction ofthe pile with the soil.

2. The model can also be extended to dynamic pile-soil interaction analysis by deriving

the dynamic system equations using the Hamilton's principle.

LIST OF REFERENCES

[I] Broms, B. B., "Design of laterally loaded piles," J. Soil Mech. and Found. Div., ASCE, 91(3), 79-99, 1965.

[2] Brinch Hansen, J., "The Ultimate Resistance of Riged Piles Against Transversal Forces," Danish Geotechnical Institute(Geoteknik Institut) Bull. No. 12, Cohenhagen, 1961, p. 5-9.

[3] Georgiadis, M., and Butterfield, R., "Laterally loaded pile behavior," J. Geotech. Engrg. Div.. ASCE, 108(1), 155-165, 1982.

[4] Gleser, S. M., "Lateral load.tests on vertical fixed-head piles," Symp. Lateral Load Test on Piles: Spec. Publ. No. 154, American Society of Testing Materials(ASTM), Philadelphia, Pa., 75-93, 1953.

[5] Poulos, H. G., "Behavior of laterally loaded piles. I: single piles," J. Soil Mech. And Found. Div., ASCE, 97(5), 711-731, 1971.

[6] Matlock, H., and Reese, L. C, "Generalized solution for laterally loaded piles," L Soil Mech. And Found. Div., ASCE, 86(5), 63-91, 1960.

[7] Vallabhan, C. V. G., "Rigid Piers," J. Soil Mechanics and Foundation Division, ASCE, 1255-1272, 1982.

[8] Randolph, M. F., "The response of flexible piles to lateral loading." Geotechnique, London, England, 31(2), 247-259, 1981.

[9] Kuhlemeyer, R. L., "Static and dynamic laterally loaded floating piles," J. Geotech. Engrg. Div., ASCE, 105(2), 289-304, 1979.

[10] Davisson, M. T., "Lateral Load Capacity of Piles," Highway Research Record, Washington, DC, 1970, pp. 104-112.

[II] Spillers, W. R. and Stoll, R. D. 1964., "Lateral Response of Piles" J.S.M.F.D., ASCE, vol 90, SM6: 1-9.

[12] Lenci, C , "Behavior of laterally loaded piles. I: single piles," J. Soil Mech. and Found. Div., ASCE, 97(5), 711-731, 1971.

[13] Maurice, J., and Madigner, F.(1968) "Pieu Vertical Sollicite Horizontalement." Annales des Fonts et Chaussees, VI, Nov-Dec: 337-383.

160

161

[14] Matthewson, C. D., 1969. "The Elastic Behavior of a Laterally Loaded Pile." Ph.D. Dissertation, Univ. of Canterbury, Christchruch, NZ.

[15] Banerjee, P. K., and Davies. T. G., "The behavior of axially and laterally loaded single piles embedded in non-homogeneous soils," Geotechnique. London, England, 28(3), 309-326, 1978.

[16] Chandrasekharan, V. S., "Numerical design-analysis for piles in sands," L Geotech. Engrg. Div., ASCE, 100(6), 613-635.

[17] Verrujit, A., and Kooijman, A. P., "Laterally loaded piles in a layered elastic medium." Geotechnique. London, England, 39(1), 39-46, 1989.

[18] Sun, K., "Static analysis of laterally loaded piles," Proc, 11* Southeast Asian Geotech. Conf., Singapore, 589-594, 1993.

[19] Vallabhan, C. V. G., and Sivakumar, J., "The AppHcation of Boundary Element Techniques for Some Soil Structure Interaction Problems," Report for U.S. Army Corps of Engineers, WES, Vicksburg, MS, Dec. 1983.

[20] Rahman, K. R., "Coupling of Boundary and Finite Element Methods for a Three-Dimensional Nonlinear Soil Structure Interaction Problem," Ph.D. Dissertation in Civil Engineering, Texas Tech University, May, 1989.

[21] Wilson, E. L., "Structural Analysis of Axisymmetric Sohds," AIAA Journal, L. 3, No. 12, December 1965.

[22] Desai, C. S., "Numerical design-analysis for piles in sands," J. Geotech. Engrg. Div., ASCE, 100(6), 613-635.

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a new model for the analysis of laterally loaded piles

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