Numerical model for laterally loaded piles and pile groups

PILES AND PILE GROUPS

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NUMERICAL MODEL FOR LATERALLY LOADED PILES AND PILE GROUPS

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van

de Rector Magnificus, prof.drs. P.A. Schenck, in het openbaar te verdedigen ten overstaan van een

commissie aangewezen door het College van Dekanen op dinsdag 5 september 1989 te 14.00 uur

door

Arjen Peter Kooijman

Civiel Ingenieur

geboren te Zwijndrecht

TR diss

1745

The research that preceded this thesis was conducted at the Geotechnical Laboratory of the Faculty of Civil Engineering of the Technical University of Delft, under the supervision of Prof. A. Verruijt and Dr. P.A. Vermeer. The support given by the staff of the laboratory is greatly appreciated. Special thanks are directed to Mr. J. van Leeuwen for making the figures in the report.

The project was funded by the Netherlands Technology Foundation (S.T.W.).

1 INTRODUCTION 1

2 DESCRIPTION OF THE MODEL FOR AN ELASTIC SOIL 4

2.1 Introduction 4 2.2 Modeling the pile 5 2.3 Modeling the soil 6 2.4 Coupling routine and iterative process 12

2.5 Validation of the model 16

2.6 Example 24

3 EXTENSION TO ELASTOPLASTICITY FOR COHESIVE SOILS 28

3.1 Introduction 28 3.2 Elastoplastic stress-strain relationship 28

3.3 Elastoplastic response of an isolated layer 31 3.3.1 Finite element analysis and mesh sensitivity 31

3.3.2 The use of interface elements 37 3.3.3 Analysis for reduced adhesion 42 3.4 Extension of the pile-soil model to elastoplasticity 44

~ 3.5 Application to field test 47

4 PILE-SOIL SEPARATION 50

4.1 Introduction 50 4.2 Gap element 50 4.2.1 Flow rule for gap element 50

4.2.2 Consequences of finite load step size 57

4.2.3 The tension strength parameter 60 4.3 Analysis of an isolated layer 62 4.4 P-Y curve versus finite element computations 67

4.5 Cyclic loading 68 4.6 Application to field tests 70

5.2 Additions to the single-pile model 79 5.3 Validation for an elastic soil 82 5.4 Application to field tests 89

5.5 Computer code 98

6 "CLASS A" PREDICTION FOR A SINGLE-PILE FIELD TEST 100

6.1 Introduction 100 6.2 Site layout 100 6.3 Soil investigation and schematization 103

6.4 Loading sequence 106 6.5 Results of computation and experiment 107

6.6 Conclusions 113

PRINCIPAL NOTATIONS 115

REFERENCES 117 SUMMARY IN DUTCH (SAMENVATTING) 123

1 INTRODUCTION

This study on the behavior of laterally loaded piles

originates from offshore engineering. For pile supported offshore structures, the axial pile capacity is of primary importance, due to the weight of the structure and the overturning moment caused by currents, waves and wind. Besides, driving the piles to the required penetration also makes great demands on the pile

dimensions. The level of the lateral loads imposed on the piles is relatively low compared to the failure load. However, because of the unfavorable direction of the load with respect to the pile axis, lateral flexibility still plays an important part in the overall behavior of the structure. Furthermore, damping of the lateral movement by the soil may contribute to the total damping of the system. Of course many other types of structures undergo lateral loading, for example dolphins and wind turbines, and in regions where earthquakes occur even ordinary buildings may be subject to severe horizontal excitation. In practically all of these cases lateral flexibility has to be considered extensively, while the lateral failure load only requires a rough check.

For the analysis of laterally loaded foundation piles various models have been developed. In practically all of these models the pile is modeled as a flexible beam. The main difference in the various models sterns from the schematization of the soil behavior. The majority of models can be grouped into two classes. In the first class the soil behavior is represented by a series of independent nonlinear springs. This makes it possible to closely follow the soil profile, by varying the spring characteristics (often represented by p-y curves). Plastic deformation of the soil can be incorporated by the nonlinear response of the springs. This method is recommended for offshore piles by the American Petroleum Institute (API, 1984) and Det Norske Veritas (DNV, 1977). Both institutions have published procedures to assign p-y curves to a given soil profile on the basis of simple characteristics of the soil. These procedures

originate from papers by Matlock (1970) and Reese et al. (1974, 1975). Although the approach provides a simple and commonly available practical analysis, the fundamental problem is the determination of appropriate and realistic p-y curves. An extension of the p-y curve approach is obtained by introducing shear coupling between the springs. Such a coupled spring model is referred to as a Pasternak foundation model. The application of this model to laterally loaded piles is described for example by Georgiadis & Butterfield (1982). In the second class of models the soil is represented by an elastic continuüm (Poulos, 1971; Banerjee & Davies, 1978; Randolph, 1981; Kay et al., 1986). Although many concessions have to be made in schematizing the soil profile, and plastic soil behavior cannot be taken into account (except for ad hoc modifications, see e.g. Poulos, 1971; Banerjee & Davies, 1979), or an axially symmetrie geometry is demanded (see e.g. Kay et al., 1986), this approach is a very valuable addition to the analysis of laterally loaded piles. The main improvement when a continuüm model is used for the soil is that input variables can be related directly to realistic and measurable soil properties, such as its stiffness and its strength. Furthermore an extension to pile group analysis is possible (Poulos, 1971b; Randolph, 1981), in which the effect of

interaction between the piles and the soil can be taken into account. Both these points are very valuable for foundation engineering practice.

The purpose of this study is to develop a model that combines the advantages of the two approaches. This means that a continuüm model should be developed, allowing for elastoplastic behavior of the soil and for the representation of natural soil profiles, consisting of layers of different properties. The model should be able to predict flexibility and damping of a single pile and pile group foundation under cyclic loads, on the basis of soil

parameters measurable in laboratory tests or in in situ tests. The analysis is restricted to cohesive soil behavior, i.e. undrained clay. Finally, the model should have modest computational reguirements.

In chapter 2 the basic assumptions and equations of the model are described for an elastic soil. The substructuring technique that forms the basis of the model is outlined. The model is validated by comparison of the results with those obtained by using other numerical solutions.

In chapter 3 the model is extended to elastoplasticity for cohesive soil behavior. A special interface element is introduced to describe slip of soil along the pile circumference, and

attention is paid to the mesh requirements of a soil layer. A field test is analyzed, and the results of that analysis are compared with a finite element solution.

Separation of pile and soil at the back of the pile is an important feature of the behavior of laterally loaded piles. In chapter 4 this phenomenon is modeled by using the interface element introduced in chapter 3. A special flow rule is derived for this element type to allow for the existence of a gap behind the pile. The application of this gap-element is demonstrated by studying two large-scale experiments.

The analysis of pile groups is treated in chapter 5. The interaction between piles in a group, both for elastic and

elastoplastic soils, is computed direct with the present method. A comparison is made with other models, and with field tests.

In Chapter 6 a "class A" prediction for a full-scale lateral loading test performed in Delft, is described. The results of the predictions that were made for this test are compared with the data actually measured.

2 DESCRIPTION OF THE MODEL FOR AN ELASTIC SOIL

2.1 Introduction

The major limitation of using three-dimensional numerical models is the huge dimensions of the system of equations that represents the soil. This becomes even more pronounced for pile groups, that reguire extensive meshes with several local

refinements. The basic idea of this study is to subdivide the pile-soil system into several smaller units that can be analyzed one by one, and coupled in an iterative way. To accomplish this, a substructuring technique is employed at two levels. At the first level, the pile-soil system is separated into two subsystems, representing the pile and the soil respectively.

PILE-SOIL SYSTEM

Fig. 2.1. Substructuring of the pile-soil system.

In the analysis the two systems are coupled by satisfying compatibility and equilibrium conditions. At the second level, the soil is subdivided into a number of interacting layers. Consequently, the following components can be distinguished in the model: a model for the pile, a model for a soil layer, and two coupling routines, see Fig. 2.1.

In this chapter the basic assumptions and equations of the model will be presented. The soil behavior will be restricted to

LAYER LAYER

elastic horizontally layered soils, containing a single vertical pile. Although in practice many laterally loaded piles have an inclination, the schematization to a vertical pile is not considered a major limitation of the model. Poulos & Madhav

(1971) examined the influence of the pile inclination on the response to both axial and lateral loading. They found that the axial displacement of a battered pile subjected to axial load, and the normal displacement and rotation of a pile subjected to normal load and moment, are virtually independent of the batter of the pile, for the range of batter angles employed in practice. They propose to analyze the movements of a single battered pile, and pile groups containing battered piles, to a general loading system, by using solutions available for the movements of

vertical piles. Evangelista & Viggiani (1976) add that the

response of a pile embedded in an elastic half-space to axial and lateral load is practically unaffected by inclinations to the vertical up to 30 degrees. However, it should be noted that the correct resolving in axial and lateral instead of horizontal and vertical direction is made.

2.2 Modeling the pile

The pile is considered as a beam supported and loaded by the surrounding soil, and by the boundary conditions at the pile top. The basic equations for such a beam are the familiar equations for a beam on elastic foundation (Hetenyi, 1946), which can be expressed in the bending moment M and the lateral displacement u as follows d2M . _ = k u - f dz d2u M (2.1) (2.2)

P -3 ri­ ff (D Ti n (t> (A (D 3 r t tn rt e a >< ■* P -3 01 TS P -r t ro 0 M> ri­ ff fD < fl> M r t p -O 01 M n 0 3 tn r t M B> P -3 P -3 i Q ri­ ff (D P - M-BJ 3 0> P" >< 0) 0) 01 •* 3 0 T5 M CU 3 (t> 01 r t H 0> P -3 01 P -rt C 0> r t p -0 3 P -01 P -3 TI o 01 (D CL 0 3 ri­ ff (D 3 P -o 0J 3 0) M >< 01 P -01 n ro H 0) rt (0 CL r t 0 (D 0) H rt ff XI e 0) * ■ 0) TI l-i o cr M (D 3 01 • K O s (D < ro H e 3 M P -X (D 3 P -*-^ P> «3 en VO ^-* Qi 3 CL * 01 01 01 CL 0 TS ri­ tu CL cr ►< (D • i Q • W 01 i Q 0) s; 0) B i « M 0) H l r t ^^ P-1D CO H *—' H l O M DJ 3" O M p -N O 3 r t 0> M Ti p " 01 3 fl> • ►3 ff P -01 01 01 01 c 3 Tl r t P -O 3 3 " 0> 01 cr (D ro 3 c ₀₁ fl> a cr (D M i 0 M n> cr >< O 3 (0 3 rt K 01 ff O e l -1 CL cr ro 3 C O ff 01 3 0) M M (6 i i r t ff Üi 3 r t 3 * (1) a P -01 TI M 01 o (t) 3 ro 3 rt 01 c 01 3 CL < tn P -rt P -01 r t O cr (D (D X •o ro 0 ri­ tu a r t 3" 01 rt rt rr ro < _a> I-! r t H -0 Q) M a H -01 •a M 01 n (D 3 (D 3 rt 1 01 O H -M (-■• 3 rt a> M M l 01 O (D £ H -M M •a _I-! 0 & C O (D 3 01 H -3 M >< rr 0 ^ H -N O 3 r t 01 M a <t> Mi o H 3 01 rt H -O 3 01 * 3 rt 3" (D 3 O rt H -O 3 rt 3" 0) rt rt V (t> ff O M M-N O 3 r t 01 M1 M O Q) CL H -3 ^Q O Mi rt ff m 01 o H -M 01 rt rt ff (0 M 3 01 rt H -O 3 O Ml ri­ ff 01 01 0 H -(-■ O 0 3 r t M-3 C C 3 H ff ro 3 w H -3 01 H -3 Ti M P -M l H-O 01 r t H -O 3 01 ^! H -01 01 3 r t M O CL c 0 H -3 i Q tn 0 3 (D 01 H -3 Tl M P -M l H -O 0) r t H -O 3 01 P -3 rt ff a> CL fD 01 0 M H -Ti r t H -O 3 O M i ri­ ff (D H -< P -01 P -O 3 O M l ri­ ff (O 01 O p -p> p -3 r t O p -3 rt m M 0> O r t p -3 i Q M 01 *< _ro i-i 01 0 0) 3 cr ₀₎ 0) o ff p -01 < (D CL 01 tn (D 3 rt ri­ ff 0) p -3 rt (6 M 0) O r t p -o 3 O Mi ri­ ff a> 01 0 p -H CO 3 CL ri­ ff m Ti p -M m • ^ _ff ro M P -3 CL h P -0 01 H ff 0 M 0) ■■ p -3 S ff P -O ff p" 01 r t (6 P . 0> M Ml o l-i O m 01 01 i-s m 01 0 r t p -3 *Q ■* s: ff p -0 ff ff 01 01 0 p -M p -0) 0 0 3 01 p -a (D n 01 a r t 0 O" ro 01 h-> 0i ^ ro M 0) a ro M 01 01 r t p -n o o 3 rt p -3 c: c 3 •* K p -rt ff 3 O CL (D M P -3 i Q ri­ ff 0) 01 O P -M ro 0 r t p -O 3 01 3 CL ri­ ff 01 r t ri­ ff 0) Tl p -P1 0) CL 0 ro 01 3 O r t >~t O r t 01 r t ro 01 O" 0 c _{r t} p -r t 01 01 X p -01 • H ff p -01 3 (0 01 3 01 ri­ ff 01 rt & p -01 n _M 0) O 01 3 0) 3 r t 01 O Ml ri­ ff (0 V p -P1 ro 0 0 0 d p, p -3 O 3 P1 •<; 0 3 0) cr 0) p -3 a p -n 01 rt 0) a M i o p. ri­ ff ro »Q o> 0 3 ro r t M ^< 01 3 CL ri­ ff ro M o Cu CL P -3 ■£> 01 r t rt ff 0) Ti P -P1 (ü > rt ri­ ff p -01 •a 0 p -3 rt p -r t p -01 0) 0) 01 c 3 ro CL ri­ ff 01 r t 0> < 01 l-l r t p -O 0) M Ti M 01 3 0) O Ml 01 >< ₃ 3 ro rt h *< vQ 01 H 3 01 ^< P -O M (H X Ti 01 3 01 P -O 3 Mi O P. ri­ ff ro CL p -01 TS M 01 0 01 3 ro 3 r t 01 p -3 ri­ ff 01 Ti p -P< 01 • 3 n ₀ 01 ro a Q, p -M (D O rt P" ^< • > T l M (0 CL ro ₀₁ n p. p -cr ro CL M 0 r t 01 r t p -O 3 n 01 3 cr ro p -3 •O O 01 ro CL cr »< 0) 0 M i 01 Mi O M O 0) 0 M 3 O 3 0> 3 r t M O 01 CL • > Ti P. 0) CL ro 01 o M p -cr ro CL a p -01 T5 M 01 0 ro 3 ro 3 r t O 0) 3 cr 01 p -3 rt M 0 CL e 0 0) CL cr ^ tn 01 r t p -01 M i ^< P -3 i Q 01 >a e p -M P -cr M p -c 3 O O 3 CL P -ct p -o 3 01 P -3 ri­ ff (0 cr ro M 0 M i TS O H -3 rt 01 • P3 ff 01 cr o c 3 CL 0) M ►< 0 0 3 CL P -rt p -O 3 01 0) r t ri­ ff 01 T( p -M 01 rt O Tl 01 3 a rt p-Tl cr ro 3 & p -3 va 3 0 3 (0 3 r t 3 01 3 CL ri­ ff ro p" 0) r t 01 P. 0) P> CL p -01 Ti M 0) o 01 3 01 3 rt e 01 rt 01 M 0) p. ^Q 0) r t O M i p" P -3 0) 01 « ro 03 c 01 rt P -O 3 01 ■* ro X V M 01 01 01 01 a p -3 r t 01 M 3 01 O Mi ri­ ff 01 < 0) p -c ro 01 o Mi Ml 01 3 CL W > Ml P -3 P -r t ro a p -M l Ml 01 P. 01 3 O ro 01 Tl T3 M 0 01 o ff ff Q) 01 cr ro ro 3 c 01 0) CL rt O CL 0) M P -< ro O 01 M 01 M >Q 0) 3 C 3 cr 01 p, 0 Mi 01 3 0> M H ro M 01 3 ro 3 rt 01 •* 0) 01 0 ff s: p -ri­ ff p -r t 01 O < 3 < 01 M c ro 01 ^^ 01 3 a ^ M • W *— O 01 3 cr ro 01 0 M < ro CL 3 C 3 ro i-( p -0 01 M M ^< cr ^< 01 c cr CL p -< p -CL P -3 vQ ri­ ff ro Ti P -p i 01 ^< 01 p -01 0 M l ri­ ff (0 CL 01 Ml 0 M 3 01 r t P -O 3 O Mi ri­ ff ro tn 0 p -M • P ] ff ro 01 *: 01 r t 0) 3 0 M l ro ö 01 • C 0> M < 01 H c 01 01 0 M i ri­ ff 01 01 0) TS 0) M 0> 3 ro rt 0) M 01 s: P-P1 M cr 01 0 cr rt 01 p -3 01 CL Mi M O 3 ri­ ff 0) 01 M i P -01 01 kQ P -< _ro 3 M 01 rt 0) i-( 01 M M 0 01 CL -» 01 3 a X p -tn 0) 0) T l i-( P -3 l O O O 3 tn rt 01 3 rt P3 ff 01

The second simplification concerns the vertical stress in the soil. It is assumed that during deformation the vertical normal stress will still be determined solely by the overburden weight óf the soil. That is, the vertical normal stress will remain unchanged. For each layer of the soil the basic equations can now be obtained by averaging the equations of horizontal equilibrium over the layer thickness h, and by disregarding all terms

involving the vertical displacement w.

1 h ös ös ös öx öy öz dz = 0 (2.3) 1 h ös Ss öx Ss Öy öz dz (2.4)

This leads to the equations of equilibrium for a plate,

5a Sa x x y x Ö X öy + q = 0 (2.5) Sa Sa x y y y Ö X öy + q = 0 y (2.6)

where er , er and a are to be interpreted as the average

xx xy yy

stresses in the horizontal plane, and where q and q represent

x y

the forces transmitted to the layer by shear stresses from the layers above and below it, i.e.

g. =

(

s

.„

_{zx zx}

- O /

h

q = (s - s ) / h

y zy zy

(2.7)

where the superscripts + and - indicate the values at the bottom and top surfaces of a layer, respectively. These terms that provide the coupling of the layers are indispensable. When an individual uncoupled layer of infinite extent is loaded by a circular pile section, the displacement of this section is indeterminate, as can be seen from the analytical solution for this problem derived by Baguelin et al. (1977).

The average stresses in a layer are directly related to the average displacements in a layer. For layer i these displacements are denoted u and v . The shear stresses s and s at the top

i l zx zy

and bottom of the i layer can be expressed in the shear strains e and e according to Hooke's law. If in the expression for

zx z y

these shear strains the vertical displacement is again

disregarded, it follows that the shear stresses s and s can zx zy be expressed in the horizontal displacements,

s = s = zy G G 5u öz

r sv

— 5z + + Sw — Sx Sw — öy X K SU G — Sz Sv G — 5z (2.9) (2.10)

The derivatives in vertical direction can be obtained by using a finite difference approximation. The average values for two successive half layers can be obtained from:

Su/Sz « 2 (u - u ) / (h + h ) ' v 1+1 i ' ' v i+i i ' Sv/Sz * 2 (v - v ) / (h + h ) ' v 1 + 1 1 ' ' v i + l i ' (2.11) (2.12)

The ~ sign originates from the fact that no distinction has been

displacement of a layer. If the value of the shear modulus G varies from layer to layer, equilibrium at the layer interface demands that s = G (öu/Sz) = s = G (Su/Sz) (2.13) zx i + 1 v ' ' i+1 zx i v ' ' 1 v ' i+1 1 s" = G (öv/öz) = s = G (ëv/öz) (2.14) zy 1+1 v ' i + 1 zy 1 v ' ' 1 l ' i+1 1

Substituting the average values of the derivatives (egs. 2.11 and 2.12) into these expressions, and satisfying compatibility

between the layers, results in the following expressions for the shear stresses: 2 ( u - u ) + 1 + 1 1 , - , r-X s = s « ( 2 . 1 5 ) zxi+i z xi (h / G + h / G ) v l + l ' 1 + 1 V l 2 ( v - v ) = s* « L i ^ Il (2 . i 6 ) "1+1 'i (h / G + h / G ) v l + l ' i + 1 \' i '

The forces q and q , which act as body forces in the system of x y

equations for the horizontal layer, see eqs. (2.5) and (2.6), can be written in a form that distinguishes between the contribution gpres o f t h e l a v e r itself, and the contribution qup and q ow of

the layers above and below:

up pres , low ,^ - _.

= q - q + q (2.17)

X X X

tot up pres , low . ^ - _ .

q = q - q + q (2.18

For l a y e r i : < r e s pres = C U 1 i •res _ p r e s = C V i 1 qu p = cu p U Mx i 1-1 aup = cu p v ^ y i 1 - 1 lOW l O H q = c u Mx 1 1+1 lOW l O H q = c v ( 2 . 1 9 ) ( 2 . 2 0 ) ( 2 . 2 1 ) ( 2 . 2 2 ) ( 2 . 2 3 ) ( 2 . 2 4 ) w i t h : pres low , up C = C + C 1 1 1 Cj = 2 /

("■

f

1 +

7

)

v 1+1 i ' ( 2 . 2 5 ) ( 2 . 2 6 )

c

,U

r -

P _

2

/ (

h

.

h h ( 2 . 2 7 )

The body forces can be determined if an estimate for the

displacements in the various layers is available. The result of • this procedure is that for each layer the system of equations is reduced to the familiar equations for plane stress deformations, with given body forces representing the interaction between the layers. The vertical stresses in the soil are assumed constant.

The analysis of stresses and strains in each layer is

performed by using the finite element method. For the equilibrium of a layer the virtual work equilibrium equation can be derived:

BTD B a dV = HTt dS + HTq dV _(2.28)

where B is the strain interpolation matrix, D contains the elastic material constants in accordance with a plane stress situation, D = 1 v 0 v 1 0 0 0 (l-l>)/2 (2.29)

a are the nodal displacements, H is the displacement

interpolation matrix, t are the boundary stresses, and q are the body forces in the layer. Substituting the expressions for the bodyforces (2.17) and (2.18) into eq. (2.28), and applying the interpolation matrix H, yields:

(BTD B + HTC H) a dV = H t dS + H (q + q ) dV

V (2.30)

where C denotes a diagonal matrix that contains the constants cpres. If I is the identity matrix, then for layer number i:

(2.31)

The body forces qup and q ow are obtained from the nodal

displacements aup and a ow of the layers above and below:

-UP _ , UP

q = (c I) H a (2.33)

Expression 2.30 yields the following system of equations:

A a = AS + Qup + Q1 O H (2.34)

Now the stiffness matrix A also contains a part of the body forces that provide the coupling between the layers. S is the surface load vector, representing the load applied by the pile for a unit pile displacement, and A is the average displacement of the pile over the height of the layer concerned. Qup and Q ow

are the body force vectors, determined by the displacements in the adjacent layers. Both vectors are updated during the

iterative process.

Because of the geometry of the problem, with a soil body of large lateral extent, containing a single pile, the geometry of each layer is the same, and thus the same mesh of finite elements can be used for each layer. This is a great simplification, as the actual finite element calculations can be performed now in a single subroutine, which is called for in the main program, with different values for the soil parameters and the hody forces in each layer, but with the same geometrical data, and the same structure of the system of equations.

2.4 Coupling routine and iterative process

The coupling of the two subsystems is performed in the following way. The response of the soil layers to the forces transmitted to them by the pile can be represented formally as

N N

Fi = 1 K.j Aj = Kn \ + I Kij S (2-35)

where N is the number of layers that surrounds the pile, F is the interaction force between the pile and soil layer i per unit of pile length, and X is the displacement at the pile-soil

interface of layer j averaged over the layer height,

i h

i

u dz (2.36) h.

1

In the case of a linear soil model, as considered here, the stiffness c

written as

stiffness coefficients K are constants. Equation (2.35) can be

Fi = ki \ + fi (2.37)

where k is substituted for K , and where f represents the contribution of all layers except layer i. The value of the spring constants k can be determined from the soil model by running the program with the boundary conditions that \ = 1 and

that the displacements A. of all other layers are zero. This finite element analysis of the layered soil system, with given displacements on the inner boundary, is itself iterative, because the interaction forces between two layers depend upon the

displacements of these layers, see eqs. (2.32) and (2.33). The iterative procedure starts by assuming an estimated displacement field for all layers, for example the uncoupled layer

displacement field. The program then calculates the displacements of all layers using the given boundary conditions and the

•O 0 3 01 (0 H ) e 3 0 r t p -0 3 *-\ W • U) ^J ^^ • <B H • 1-3 ff P -cn M (B 01 & in r t 0 0> 3 C T i & 0) l i ­ ra Cb < CU M c (B M l 0 H r i ­ ff <B •ü 01 H r t H l — P -3 ri­ ff (D c H r t 01 P -3 3 (D * < 0J M e (O 01 >*j — 3 X I-h 0 H r i ­ ff ro P -3 r t ro M. 0i 0 r t P -0 3 i-h 0 H. 0 (1) 01 01 r t (D 0 i O ff ff ro ff Cu 01 p -01 o Ml ri­ ff (B 01 (B Q. 01 r t 01 ri­ ff <B 01 0 P -M 01 c ff 01 ^< 01 r t <B 3 P -01 0) 3 0) P1 >< N ro CL ff P-0 ff Ij — II J T I H 3* ' ^^^ _{M i} | ** c ■ « « • * Q J N <_<^ t o C*) 00 * w p -3 r t <B M-01 O r t p -0 3 M i O M 0 ro ff <B r t s (B ro 3 13 P -P" <B 01 3 & 01 O P -P" • • -0 M 0) O a> 3 a> 3 ri 01 0) r t r t 3 " (D "O P -M <B 1 01 O p -M p -3 r t <& M M i 01 O (B p -3 (B 0) O ff P" 0) K (B N Qi 3 CL Mi 0 M << 01 p -01 0 H l r t 3* (B -d p -M (B 01 << 01 r t (B 3 H (B 01 C P< r t 01 p -3 < 01 M e (B 01 M i O M r t 31 (B a (B r t <B H *3 P -3 (B r t 3 * (B P -3 r t (B M 0> O r t P -O 3 M i 0 M O (B 01 _'q ff (B r t (B (B 3 13 P * M (B 01 3 _CL 01 O P * P1 •-3 ff m •O p -M (B a p -01 •a 0) 0 (B a (B 3 r t 01 O 01 3 ff (B 3 0» a (B ff >< 01 •a *a _M >< p -3 r t 3" (B M l c P1 M 3 0> r t M P -X 01 3 01 M K M (B O. 01 (B •a ₀₁ M 01 r t (B M ► < p -3 01 c 0 0 ro 01 01 p -< (B 01 r t (B 13 01 • > M i P -H 01 r t (B 01 rt p -3 01 r t ro 0 M, « 01 r t p -< (B •o H O o ro a e M (B ^ p -3 « 3* p -O 3* r t 3" (B >a p -M ro 01 3 a r t 3* ro 01 0 p -M 01 c ff 01 *< 01 r t ro 3 01 > H> r t (B H r t 3" 01 r t <■ r t 3" ro n o 3 V P" (B r t ro 01 ><: 01 r t (B 3 f ) 01 3 ff (B 01 3 01 P->< N ro & er ^< 01 3 01 P -01 l_l. C 01 r t K 3" 01 r t p -01 3 ro (B a ro & H l 0 ^ r t 3 " (B 0> 3 01 M >< 01 P -01 0 Ml r t 3 " ro V p -M ro ^ 01 (B ro ro £ t M l p -0 p -ro 3 r t 01 J* 0> N. ro M l C p1 p * *< 3 O C 3 M i H O 3 r t 31 ro Q J p * 01 ^Q O 3 01 M O M l 3 01 r t p -X M (B 01 ro 3 rt ro a ff >< o> 3 (B >Q C 01 r t p -O 3 O M i r i ­ ff (B Ml 0 M 3 ^ - s NJ • U> ^ ] >-^ •* s ff (B M (B r t ff ro fl> K 3 p -3 (B O. • PJ ff ro p -3 r t (B M 0i O r t p -O 3 O M l r t ff (B 01 O P -M 0) 3 a r t ff ro "O p -M (B O 0» 3 3 O S ff ro V H 01 O ro 3 ro 3 r t 01 > r t ff (B ro 3 r t p -i-( (B P -3 r t ro M. 0) 0 r t p * O 3 3 01 r t H p -X ?S n 01 3 ff ro 0 c M 0> r t ro a ^ 01 3 O. ff >< H ro ■o ro 01 r t p -3 <Q r t ff p -01 •a _M 0 0 ro & e M (B M i O M 01 M M P1 01 ^ _ro H H ro CL p -01 •o p< 01 o ro 3 ro 3 r t > • •-3 ff _C 01 01 0 0 p * c 3 3 O M i r t ff (B 3 01 r t •^ p -X « ff 01 01 ff (B (B 3 ff 01 r t rt ff (B M> O H O ro 01 "d ** o 3 01 P1 M *o P -P1 (B ro M ro 3 (B 3 r t 01 01 M. ro a ro _{r t} ro

y

p -3 (B O i M i O H 01 < ro _M >Q (B 3 O ro p -01 H (B 01 O ff ro a < ro N >< >a c p -o w M ^< ^ ff (B H ro 01 e M r t O M l r t ff p -01 01 3 01 M 01 P -01 3 « 01 H & 01 • o 3 P< ^ 3 p -3 O H. 01 & L_l. c 01 r t 3 ro 3 r t 01 0i H (B r t ff ro 3 3 ro 0 ro ₀₁ 01 01 H >< 01 3 a •O c r t (B & & p -01 -a _M 01 0 (B 3 ro ₃ r t M l P -ro M a 0 ff r t 01 p -3 ro a Mi O M r i ­ ff (B "O M. (B < p -o c 01 p" 01 (B >-! (B < (B M 01 H 01 r t ff (B M i a 0 0 o. ro 01 r t p -3 01 r t ro 0 01 3 ff ro 3 01 & _(B ff ^< 01 ff P -H l r t p -3 X I r t ff ro •o p< 01 o ro 3 ro 3 r t 01 s: p -p< p -3 O r t ff ro < ro I-I ^ aa 0 0 a •^ 0 M r t ff ro 3 ro X r t M 01 >< _ro M 01 01 r t M 01 ^ _(B ^ r t 3* 0) r t p -01 01 3 01 M >< N (B CL ■* r t ff ro p -3 p -r t p -01 p -(B 01 r t p -3 0) n (B O H l r t ff (B p -H 0> 01 c H l Mi P -O p -ro 3 r t a ro i f l H (B (B O H l 0> 0 0 c H. 01 0 >< ff 01 01 ff ro ro 3 01 n ff p -ro < ro a •n 0 hj r t ff (B 1 P> ü 1

(START)

estimation parameters

pile model

J

S0IL

\M0DEL

check equilibrium

\1/

(END)

Fig. 2.2. Iteration scheme for elastic analysis.

Then the pile analysis can be repeated, using the same values for the spring constants and updated values for the coupling terms f . This iteration scheme is illustrated in Fig. 2.2. During the analysis of the soil, the body forces that provide the coupling between the layers are adjusted.

The soil layers are analyzed in downward direction. As a consequence, for every layer eq. (2.34) has to be solved according to the following iterative updating process:

k k - l

where k denotes the iteration number, and k is determined by

max J

some convergence criterion. The factor X and the body force vector Qup are computed during the present iteration from the

displacements of the pile and from the displacements of the previously analyzed layer, respectively. The vector Qlow is based

on the displacements of the previous iteration.

There are two convergence criteria to be satisfied. The first criterion concerns the pile-soil equilibrium. Convergence is reached when the difference between the value of the pile-soil interaction force obtained from the pile model and the value obtained from the soil model is small. The second criterion concerns the internal equilibrium of the soil. The difference between the coupling force vectors of the soil layers before and after the iteration should be small.

2.5 Validation of the model

In order to verify the performance of the numerical model it has been validated by comparison of the results with those

obtained by other methods (Verruijt & Kooijman, 1989). Two other (numerical) solutions based on a continuüm approach are

available, both applying to the case of a vertical pile of

constant flexural rigidity in an elastic half-space. The elastic half-space is either homogeneous (Poulos, 1971a) , or its modulus

of elasticity increases linearly with depth (Banerjee & Davies, 1978; Randolph, 1981). Comparisons of the results obtained by the method described in this chapter with the solutions known are presented below.

The finite element grid used for the analysis is shown in Fig. 2.3. The outer boundary is considered as having zero

displacements. The infinite lateral extent of the soil mass is simulated by reducing the modulus of elasticity in the outer ring of elements by a factor 2. When the radial dimension of these elements is one half of the diameter of the entire mesh, such a reduction is in agreement with a decrease of the lateral stresses inversely proportional to the square of the distance in an

Fig. 2.3. Finite element mesh.

infinite soil. The diameter of the outer boundary of the grid has been taken as 30 times the pile diameter. The lower layer of the layered system is to be rigid, thus simulating a fixed boundary condition at a certain depth. The depth of that boundary is taken at one quarter of the pile length below the tip of the pile. At the pile-soil interface complete adhesion has been assumed.

The soil is modeled by a system of 2 5 layers, or, for very flexible piles, 50 layers. In each layer the same finite element grid of 80 eight-noded isoparametric elements is used. Because of symmetry only one quarter of the elements has been taken into account in the actual computations, if the appropriate boundary

50 20 'PH \ \ \ \ \ • 1 - ^ \ Subgrade theory • • Present method 1 1 1 | | L /D =25 ^ = 1 . i - J : - i 2 H - 5 - 4 -3 - 2 - 1 0 1 log(KR)

Fig. 2.4. Displacement influence factor I PH 1000 100 BH \ \ _\ » \ X \ L/D • • \ s _{\ \} \ \ _X^ = 25 Subgra Poulos Presen . IN x x \ X N X de t h e o 1971 ■ methoc ^ ^ y - 6 - 5 -U - 3 - 2 - 1 0 1 l o g ( KD)

Fig. 2.5. Rotation influence factor I

conditions are used. In the analysis a third-order numerical integration scheme is employed. In the finite difference

approximation of the pile, two elements are used for each layer of soil, which means that the pile is subdivided into 40 or 80 elements.

The first verification concerns the case of a pile in a

homogeneous elastic half-space, with modulus of elasticity E and

S

Poisson's ratio v . A comparison will be made with the solutions

s

obtained by Poulos (1971a) . In this analysis the solutions are

expressed in terms of the ratio of pile length to pile diameter (L/D), and a dimensionless factor K , defined by

EI

KR = —Z- (2.41)

E L

which characterizes the stiffness ratio of the pile and the soil. When the pile is loaded at its top by a lateral force H, the displacement p and the rotation 9 of the top of the pile are

expressed in terms of the influence factors

I = pE L/H and IQ = 6E L2/H (2.42)

pH ' s ' 8H s ' v '

The maximum moment in the pile is represented by the dimensionless factor M/HL.

For the case of a pile having a length 25 times its diameter (L/D = 25) the results of the numerical calculations are shown in Figs. 2.4-2.6, as a function of the flexibility ratio K . The value of Poisson's ratio was taken as 0.5. The fully drawn lines have been taken from Poulos. The single dots mark the results

from the present analysis. The agreement is good for intermediate and large values of the flexibility ratio. For small values of K , that is for very flexible piles, the present method results in larger displacements and rotations at the pile top. This is in

/ 1 1 . / ' / / / / / / / / f / l / / / ' / / / / / / / / / / / / / / ' / ^ " 1 ï s / / / L/D = 25 Subgrade theory • • Present method 1 1 -U - 3 - 2 - 1 0 1 log IKR)

Fig. 2.6. Maximum bending moment in the pile M/HL.

agreement with the findings of Evangelista & Viggiani (1976), who reported that the accuracy of Poulos's original analysis is strongly dependent upon the length of the elements near the top of the pile, for piles with a large flexibility. A subdivision into a larger number of elements than the 21 used by Poulos

results in a considerable increase of the displacement at the top of the pile. For the category of medium stiff to very stiff piles the number of elements used by Poulos is amply sufficiënt, and in this region the results of the present analysis are in good

agreement with those of Poulos. For very stiff piles the present method appears to result in displacements that are slightly smaller than those obtained by Poulos. The reason for this

discrepancy may be that the stress transfer in vertical direction takes place over a larger area in these cases. In order to verify this hypothesis the numerical analysis was repeated using a mesh of larger lateral extent, having a diameter of 60 times the pile

diameter. For flexible and medium stiff piles the difference from the former analysis is negligible, but for stiff piles the

displacements are somewhat larger, resulting in deviations from Poulos's results of not more than 3 percent, which is considered to be sufficiently accurate. The fact that no changes are

observed for flexible and medium stiff piles provides support for the consistency of the model. Similarly, it was found that an increase of the depth of the layers below the pile tip had hardly any effect on the behavior of the pile.

The dashed line in Figs. 2.4-2.6 represents the data obtained when the soil is represented by a series of linear springs, with subgrade modulus E . The results for stiff piles are remarkably

s

good. For flexible piles, however, this simple method results in an over-estimation of the deflections.

The maximum bending moment occurring in the pile is shown in Fig. 2.6. Although in general the values are somewhat larger than those obtained by Poulos, the agreement is rather good over the entire range of flexibility factors.

Another aspect that may be compared is the deflected shape of the pile. Randolph (1981) has presented generalized curves for the deflected shape of the upper part of the pile, that is, from the top of the pile to the point of zero deflection, which is termed the critical depth. The displacement profiles obtained from the present analysis show a basically different behavior for flexible piles and stiff piles, and can therefore not be

represented by a single shape function, see Fig. 2.7. It can be seen from the figure that for very flexible piles displacements in the direction of the applied force occur over the entire pile length, whereas for stiffer piles the displacements below a certain depth are negative. Although the generalized curves presented by Randolph give a useful indication of the deflected shape of a flexible pile for engineering purposes, the

differences observed above are worth noticing because they

indicate the fundamental difference between a continuüm approach and a spring model. A behavior with displacements in the

CU o ro • H 3 r t 3* ro V N ro en re 3 r t 3 <D r t 3 * O a r t Ö* P -01 P -01 CU "O T3 M-O X P -3 0> r t (D CL tr >< Cu i Q M 0) CL e CD M P -a n> N (D CL H ) 0 N r t D* (D (!) X r t H (D 3 (D o CU 01 ro o H l 01 rt H -H l H l 3 ro 01 01 N <D M 0 pi f t rt er (D 01 0 P -P1 r t P -O 3 (D CL P -c 3 « P -r t Ö" M P -3 (D CU H M *< P -3 0 M (5 CU 01 P -3 ^Q 3 O CL c M c 01 0 M> ro _M CU 01 r t H -0 p -rt K p -01 > 01 CU 01 (D 0 O 3 CL r t ►< T) (D O H i < (D H P -M l P -o CU r t P -0 3 r t 3" (D 0 CU 01 ro 0 M l CU TS P -M (ü P -3 CU 3 P -O P -(D 3 r t M >< CU O 0 e H CU r t ro < n> _H r t P -0 CU M 01 r t M (D 01 01 r t M CU 3 01 Ml (0 M • c tr CL P -< P -01 P -O 3 O M) r t 3 * p -01 0 T J r t P -3 C 3 a ro -a _{r t} 3 " P -3 r t O NJ O M Cu K 0) M 01 s: H -M H V H 0 < P -a ro CU ro H 3 p -3 p -3 i f l r t V ro < ro M r t p -o CU M CL p -01 O N ro _{r t} p -N CU r t p -O 3 0 M, r t 3" ro 01 0 p -h-> • H 3 i Q ro 3 ro M 01 M "* 3 " ro M O 0 CU r t p -0 3 0 M, r t 3 * ro M 0 z ro H er 0 c 3 a CU M *< p -01 r t 31 ro X ro ^< i ro M ro 3 ro 3 r t P -3 •o _ei r t CU rt p -0 3 Cu M ro M) M l 0 M r t s: Cu 01 K 0) 01 r t (D CL P -3 r t 31 ro t-> 0 s: ro p. M CU »< (D M 01 • H 3" ro 0 3" O p -0 ro 3 * ro M o « ro _M 3 ro 01 3* er o c 3 CL CU M >< S CU 01 r t Cu X ro 3 3 C O 3 " r t O O P1 CU M i Q CD • O O 3 01 P -CL ro M 0) er ro •n M o 3 r t 3" ro M i M ro X p -er M ro >a _p -M ro * Ü i-( 0 M i P -M ro p -r t Cl 01 3 er ro 01 ro ro 3 r t 3 * CU r t r t 3" ro CL ro x> _{r t} 3 " 3" ro V p -M ro P M-ro 3 0 r t CL ro rt 0) i-i 3 p -3 ro CL er >< rt 3" ro M o 0 CU l -1 CL P -01 TJ M CU O ro 3 ro 3 r t 01 0 3 M >< ro 0 r t p -0 3 p -01 < ro * >< t ro M M "Ö O 01 01 p -cr H ro tr ro 0 01 c 01 ro r t 3 -(D P -3 r t ro M 01 o r t p -O 3 M i O M O ro 01 3" 0> CL P -01 "O M CU O ro 3 ro 3 r t Mi P -ro H a s: p -r t ff 01 l -1 H a p -01 -o _M Cu o ro 3 ro 3 r t 01 p -3 r t 3" ro 01 CU 3 ro CL 3 O r t er ro p -3 ro £> c p -P1 p -er H H -c ₃ 0 M i 3 O 3 ro 3 r t 01 • H 3 01 O 0 3 r t p -3 C e 3 0) -a •0 H 0 01 0 3" 3 O r t er ro 0) n 3" p -ro < ro a er K o> t f i *< i 0 c I-i < ro CU V n _M 0 CU O 3* er ro 0 Cu c ₀₁ ro r t V ro t> p -H ro 1 p--J a ro ro o rt ro CL 01 3" Qi V ro co ^ J f t 3* ro •a

p-increase of the (constant) modulus of elasticity of the layers. Results for comparison can be obtained from the papers by

Banerjee & Davies (1978) and Randolph (1981). For flexible piles in this type of medium Randolph suggested that the results can be

* most conveniently expressed in terms of a parameter E / (m R ) ,

p

where R is the radius of the pile, E is the effective modulus of

p

elasticity of the pile,

E = EI / (TTR4/4) (2.43)

p p

and

m* = (1 + 3v/4) dG/dz (2.44)

The displacement of the top of the pile is expressed by the

2 *

parameter uR m /H.

The results of the various methods are shown in Fig. 2.8. The fully drawn line has been taken from Randolph (1981) , the open circles represent values calculated from Banerjee & Davies

(1978), and the fuil dots mark the results from the method

described in this paper. The general agreement seems to be fair, although the deflections predicted from the present method are slightly larger than those predicted by the other continuüm models. The rotation of the pile top and the bending moment, which are not presented here, show a similar agreement.

The dashed line in Fig. 2.8 represents the data obtained by considering the soil as a spring medium, having a subgrade modulus equal to the modulus of elasticity in the continuüm model. As in the case of a homogeneous soil the spring model heavily overestimates the deflections for the more flexible piles, while the results for stiffer piles are good.

In addition to the two type cases reported above, which all apply to a pile having a length to diameter ratio of 25, many other cases were investigated, and compared with results from the literature. In all cases a similar agreement was obtained. The

0.10 0.08 0.06 uR2m*/H 0.04 0.02 0 2 3 4 5 6 7 log(Ep/m*R]

Fig. 2.8. Dimensionless displacement uR2m /H of pile top.

differences between the results obtained from the four types of continuüm models, Poulos's singularity analysis, Randolph's finite element analysis, Banerjee & Davies's two-layer model and the layered model presented here, are small, and can be

attributed to the various approximations that are made in each method for computational reasons.

2.6 Example

In order to demonstrate the typical possibilities of the model presented here, an example involving a non-homogeneous, layered soil is considered. The soil profile consists of a layer of soft soil in an otherwise homogeneous soil. The pile is supposed to have a length to diameter ratio of 25, and a flexibility factor

I I

\ Subgrade theory \ o o Banerjee & Davies, 1978

E/E.

Z/L Z/L

Homogeneous Non-homogeneous

Fig. 2.9. Modulus of elasticity of the soil; example.

Fig. 2.10. Displacement-profile.

K of 10" (expressed in terms of the soil stiffness of the top and bottom layers). The soil stiffness in the layer between depths of 2.5 and 5 times the pile diameter is smaller than that of the other layers by a factor 4. The soil profile is shown in Fig. 2.9. It will be clear that a representation of this type of profile by a homogeneous profile, as some other continuüm models demand, results in a considerable simplification of the problem, while an indication of the consequences of such a simplification can only be made qualitatively. The model described in this paper can be used to analyze this problem without further simplifying assumptions. The results have been compared with a

three-dimensional analysis, made with an axisymmetric finite element program for non-axisymmetric loading, based on a Fourier series expansion in tangential direction (Kay, 1988).

In Fig. 2.10 the displacements of the pile for the homogeneous case are represented by the dotted line. The displacements for the problem with an intermediate flexible layer are represented