NUMERICAL MODEL FOR LATERALLY LOADED PILES AND PILE GROUPS

A.P. KOOIJMAN

Il N J 1

'T 1 ~L<a

^ fl'cUïi >.)h <

NUMERICAL MODEL FOR LATERALLY LOADED PILES AND PILE GROUPS

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

Dit proefschrift is goedgekeurd door de promotor prof.dr.ir. A. Verruijt

To my mother, father and my wife Sylvia

ACKNOWLEDGEMENTS

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.).

CONTENTS

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 PILE GROUPS 78 5.1 Introduction 7 8 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 -

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

- 2 -

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.

- 3 -

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.

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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

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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)

01 0 P -M

P -3

ri­ff (D Ti n (t> (A (D 3 r t

tn rt e a >< ■*

P -3 01 TS P -rt ro 0 M>

ri­ff fD

< fl> M r t p -O 01 M

n 0 3 tn r t M B> P -3 P -3

iQ

P -3

ri­ff (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

ri­ff (D

CL

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o 0J 3 0) M

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n ro H 0) rt (0 CL

r t 0

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0)

TI l-i o cr M (D 3 01

• K O

s (D

< ro H

e 3 M P -X (D

H 0)

LJ. H -3 P -

*- P> «3 en VO

-* Qi 3 CL

* 01 01

01 CL 0 TS ri­tu CL

cr ►<

(D • iQ

• W 01

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Bi

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^ P-1D CO H

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DJ

P -3 01

3" O M p -N O 3 r t 0> M

Ti p" 01 3 fl> • ►3 ff P -01

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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

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<

PJ ff £ tn P -rt P-01

r t O

cr (D

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a H -01 •a M 01

n (D 3 (D 3 rt

Ti H -H 1

(B 1

01 O H -M

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a <t> Mi o H 3 01 rt H -O 3 01

*

Mi M O 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

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ri­ff 01

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H ff ro 3 w H-3

01 H-3

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01

! H -01

01

cr >< H -3 r t M O CL

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iQ

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ri­ff (D

01 C cr CL H -

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ri­ff (O

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p -3 r t O

p -3 rt m M 0> O r t p -3

iQ

M 01

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0 0) 3

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P, 0) T! M 01 tn (D 3 rt

ri­ff 0)

p -3 rt (6 M 0) O r t p -o 3

O Mi

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0>

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■■

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0> M

Ml o l-i O m 01

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p j ff 01

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p -0)

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M

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3 O CL (D M P-3

iQ

ri­ff 0) 01 O P -M

a p -M ro 0 r t p -O 3

01 3 CL

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ri­ff 0)

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0)

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3 O r t

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p -r t 01

01 X p -01

•

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ri­ff ro »Q o> 0 3 ro r t M

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cr 01

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< • >

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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 »<

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01

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M O 01 CL

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01 3 0) p>

< 01 p -01

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ri­ff (0

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ri­ff ro tn 0 p -M

• P] ff ro 01

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•

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c 01 01

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cr 01

0

cr rt 01 p -3 01 CL

Mi M O 3 ri­ff 0)

£ ff 0) M 01

Mi

P -01

01

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M 01 rt 0) i-( 01 M

M 0 01 CL

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a X p -tn

0)

0) T l i-( P-3

lO

O O 3 tn rt 01 3 rt

P3 ff 01

- 7 -

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 xx yx

ÖX öy + q = 0 (2.5)

Sa Sa xy yy

ÖX 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.„ - O / h zx zx

q = (s - s ) / h y zy zy

(2.7)

(2.8)

- 8 -

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 zy 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 made between the displacement half-way a layer and the average

- 9 -

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 zxi (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 y y y y

- 10 -

For layer i :

< res pres

= C U 1 i

•res _pres = C V i 1

q u p = c u p U M x i 1-1

aup = cup v ^ y i 1-1

lOW lOH q = c u M x 1 1+1

lOW lOH

q = c v

( 2 . 1 9 )

( 2 . 2 0 )

( 2 . 2 1 )

( 2 . 2 2 )

( 2 . 2 3 )

( 2 . 2 4 )

with:

pres low , up C = C + C

1 1 1

Cj = 2 / ("■ f1 + 7 ) v 1+1 i '

( 2 . 2 5 )

( 2 . 2 6 )

cr - 2 / (h. , U P _ 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:

- 11 -

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 gu" = (C-K D H a" (2.32)

- 12 -

q = (c I) H a (2.33)

Expression 2.30 yields the following system of equations:

A a = AS + Qup + Q1OH (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

- 13 -

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 estimated body forces, updates the interaction forces, and so on,

N (D UI •O 0 3 01 (0 H )

e 3 0 r t p -0 3

*-\ W

• U) ^J

^ •

H

ro < <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 Ml 0 H

r i ­ff <B

•ü 01 H r t

H l

— P -3

ri­ff (D

M. <B m 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 0i O ff

O 3

r t ff ro ff Cu 01 p -01

o Ml ri­ff (B 01 (B

Q. 01 r t 01

ri­ff <B

01 0 P -M

01

c ff 01

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0) 3 0) P 1

>< N

ro CL

ff P-0 ff

Ij

— II

JTIH

3* ' ^^

Mi

| ** c

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Q J N

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C*) 00

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ri­ff (0 p -3 r t <B M-01 O r t p -0 3 Mi O M 0 ro ff <B r t

s (B ro 3 13 P -P" <B 01 3 & 01 O P -P"

• •

a p -01 -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 Mi 01 O (B

p -3 (B 0) O ff P" 0) K (B N

Qi 3 CL Mi 0 M

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<< 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

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(B

r t O

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 Mi 0 M O (B 01

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a p -01

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O 01 3 ff (B

3 0» a (B

ff >< 01 •a *a M

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r t 3" (B Ml

c P 1

M

3 0> r t M P-

X

01 M (B

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 -

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13 01

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p -r t (B « 01 r t p -

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>a p -M

ro 01 3 a r t 3* ro 01 0 p -M

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*< 01 r t ro 3 01

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rt 3" 01 r t

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ro 01

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P -01

l_l. C 01 r t

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p -01

3 ro (B a ro & H l 0 ^ r t 3" (B

0> 3 01 M

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r t 3 " ro V p -M

ro ^ 01 (B ro ro £ t

n 0 ro Mi Ml p -0 p -ro 3 r t 01

J* 0> N.

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p*

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ro QJ p* 01 Q O 3 01 M

O Ml

3 01 r t

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ro -e M (B 01 ro 3 rt ro a ff >< o> 3 (B >Q C 01 r t p -O 3 O Mi

r i ­ff (B

Ml 0 M 3

^ - s

NJ

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>- •* s ff (B M (B

r t ff ro

& ro rt fl> K 3 p -3 (B O. • PJ ff ro p -3 r t (B M 0i O r t p -O 3 O Ml

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

a p -01 V H 01 O ro 3 ro 3 r t 01

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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

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ff

ro

0 01 p< 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

Mi O M

01 M M

P 1

01

^ ro H

01 p -3

i Q 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 Mi

r t ff (B

3 01 r t

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P -01

r t ff 01 r t

rt ff (B M> O H O ro 01

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M

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(B

ro M ro 3 (B 3 r t 01

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ro a ro r t

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Mi O H 01

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a o K 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

0 0 3 •O c r t (B

& & p -01 -a M 01 0 (B 3 ro 3 r t

Ml 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 >-!

ff O < (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

CL p -01

•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

Mi P -N 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

C 3 rt p -H

0> 01 c Hl Mi P-O p -

ro 3 r t

a ro i f l H (B (B O Hl

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

- 1 5 -

(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

A ak = AkS + Qup + Qlow f o r k = l , 2 . . . k ( 2 . 4 0 )

- 16 -

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

- 17 -

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

18

50

20

'PH

\ \ \ \ \ • 1

- ^ \

Subgrade theory

• • Present method

1 1 1

| | L /D =25

^ = 1 . i -J: - i

2H

-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 theo

1971

■ methoc

^ ^

y

- 6 - 5 -U - 3 - 2 - 1 0 1

log (K D )

Fig. 2.5. Rotation influence factor I 9H'

- 19 -

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

- 20 -

/ 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

- 21 -

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 direction of the applied force over the entire length of the pile

01

e n H l 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

iQ M 0) CL e CD M

n 0 3 0) 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 -P 1

ro M 01 01 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 -Ml P -

o CU r t P -0 3

r t 3" (D

0 CU 01 ro 0 Ml

CU

TS P -M (ü

P -3 CU 3

01 c M l Mi 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

•

CU

01 c tr CL P -

< P -01 P -O 3 O M)

r t 3 * p -01

0 TJ 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

C L

ro r t

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

iQ

ro 3 ro M 01 M

"*

0 Mi

r t 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 3 1

ro X ro < i ro M

ro 3 ro 3 r t

P -3

O O g •o ei r t CU rt p -0 3 Cu M

ro M) Ml 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

0 Mi

r t 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

P 1

CU M

iQ CD

• O O 3 01 P -CL ro M 0) er ro

•n M o 3 r t 3" ro Mi M ro X p -er M

ro >a p -M

ro *Ü i-( 0 Mi 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"

0 3 r t 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

><

a p -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 Mi O M O ro 01

0) c 0 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

K O

c M CL

3 O r t

er ro p -3

ro £> c p -P 1

p -

er H H -

c 3 0 Mi

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"

O 0> 3 3 O r t

er ro 0)

n 3" p -ro < ro a er K

o> tf

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-

- 23 -

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

- 24 -

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

- 25 -

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

- 26 -

by the solid line. Both displacement profiles are plotted relative to the top deflection of the latter case, which exceeds the original top deflection by about 17 percent. The rotation of the top of the pile has increased by 7 percent. The displacement profiles obtained by Kay, which are not shown here, showed a slightly stiffer pile response in all cases. However, the increase of deflection and rotation for the inhomogeneous case are virtually the same as obtained from the present method (18 percent and 7 percent respectively, compared to 17 percent and 7 percent).

In Fig. 2.11a the soil pressure against the pile is drawn for the present method. The same line types have been used to distinguish the homogeneous and the non-homogeneous problem. The average pressure on the upper 2.5 pile diameters of the soil has increased only slightly, in spite of the considerable increase of the pile deflections. Obviously, the influence of the flexible layer exceeds its boundaries due to the stress transfer in vertical direction. This effect clearly demonstrates the difficulties encountered when a similar profile is represented by an eguivalent spring medium. In Fig. 2.11 the soil pressures obtained by Kay are plotted. These pressures are determined from the reactive nodal loads found through the finite element analysis by assuming that nodal loads are shared equally between elements and by assuming a linear pressure depth variation per element.

When the same problem is analyzed using a subgrade model with a subgrade modulus equal to the local modulus of elasticity in the continuüm model, including a reduction of the subgrade modulus at the intermediate layer by a factor 4, an increase of the top deflection of only 6 percent is obtained. The rotation of the top even decreases, by 3 percent. The decrease of the rotation in this model is due to the fact that the stiffness of the upper layer has remained unchanged in spite of the presence of the flexible underlying layer. The considerable difference in behavior of the two types of models can be understood by realizing that in the continuüm model the

- 27

Homogenpous

Non-homogpneous

pLD/H

•2 0 20

Z/L

Fig. 2.11? Soil pressure; present method.

Fig. 2.11. Soil pressure; Kay, 1988.

resistance of the (stiff) top layer is greatly reduced because it rests on a much softer layer, from which it has to derive most of its ultimate resistance. It is believed that this represents the real soil behavior much more accurately than a model of independent springs or a model of coupled springs.

- 28 -

3 EXTENSION TO ELASTOPLASTICITY FOR COHESIVE SOILS

3.1 Introduction

Offshore structures are loaded indirectly by the jacket structure they support. In order to make it possible to examine the part the foundation piles play in the overall structural behavior, the jacket is disconnected and replaced by boundary conditions on the pile heads. These boundary conditions can be resolved into axial and lateral components. Strictly speaking, the response of the nonlinear soil cannot be examined separately for these components. For reasons of simplicity, however, an integrated analysis is avoided in practice, and in fact the distinction between axial and lateral pile response computations forms one of the basic elements in offshore pile foundation design. A practical justification for this distinction is the fact that, for most circumstances encountered in offshore engineering, the lateral load transfer to the soil is concentrated in the upper layers of the soil, while the axial load transfer is concentrated in the lower soil layers. With the computers presently available an integrated analysis of axial and lateral loads based on a continuüm approach can only be made at the cost of accepting other restrictions. For example, an axially symmetrie geometry is demanded. The method presented here does not pursue an integrated approach. On the contrary, it exploits the customary distinction between axial and lateral loading explicitly.

Plasticity for cohesive material behavior will now be introduced into the model. The analysis will again be subdivided into a series of two-dimensional analyses.

3.2 E l a s t o p l a s t i c s t r e s s - s t r a i n r e l a t i o n s h i p

In plasticity theory the strain rates in the soil are

- 29 -

subdivided into an elastic and a plastic component:

ê = èe + êp (3.1)

A yield function is used to distinguish plastic from elastic strains. A change of stress invokes only elastic strain increments if the yield function f is smaller than zero. Both plastic and elastic strain increments may occur for f equal to zero. For the elastic component Hooke's law can be used to relate strain rates to stress rates:

a = D (£ - Êp) (3.2)

The plastic strains cannot be determined from such a direct relationship. It is assumed that the plastic strain rates can be derived from a plastic potential function g as follows

Sg (3.3)

where X is a non-negative scalar that can be determined from the consistency condition, which states that plastic strains can only occur if the yield function remains equal to zero:

SfT . f = — a = 0 (3.4)

8<T

Satisfying this equation, and substituting eqs. (3.2) and (3.3) yields the following expression for the multiplier X:

- 30 -

1 6fT X = D É (3.5)

d Sa

with

5 f Sg d = D — (3.6)

Sa Sa

For values of f smaller than zero, X vanishes. By using expression 3.5, the incremental elastoplastic stress-strain law can be derived:

ög 5fT D D

Sa Sa (3.7)

Computations for finite load increments can be performed by using an implicit integration scheme, assuming that the variation of Sg/Scr is small. If the expression for X is expanded in a Taylor expansion and higher order terms are neglected,

< f (o; + DAe) > ög Aa = D Ac D — (3.8)

d Sa

The term between brackets should be replaced by zero if it is smaller than zero.

In this thesis the elastoplastic analysis will be restricted to cohesive soil behavior, i.e. undrained clay, with perfect plasticity according to the Tresca yield criterion

- 31 -

-Ier - O - l - c £ O (3.9) 2 ' 3 1 ' u x '

where er and er are the major and minor principle stress and c 1 3 u

is the undrained shear strength of the soil. For undrained clay, where no plastic volume changes occur, the yield function can also be used as a potential function for deriving the plastic strains. The numerical implementation of this constitutive model in the finite element method by means of an initial stress type method is described by, for example, Vermeer, (1980).

3.3 Elastoplastic response of an isolated layer

3.3.1 Finite element analysis and mesh sensitivity

As has been stated before, one of the basic elements of the numerical model presented here for a laterally loaded pile is the finite element representation of a soil layer. Before the description of the soil behavior is extended to elastoplasticity in the main program, it is useful to examine the behavior of a single isolated soil layer. For the elastoplastic analysis the finite element representation of the equilibrium condition of an isolated layer (eq. 2.28) has to be put in incremental form. After a loading step the new loads t = t + At and g = q + Aq should correspond with the new stress state er = er + Acr. This leads to the virtual work equilibrium equation:

BTAcr dV = H V dS + r T 1 H q dV

BTa° dV (3.10)

The total loads on the right-hand side and the last term in eq. (3.10) ensure that deviations from equilibrium originating from previous not-fully-converged load steps can be corrected in the present load step. An initial stiffness approach is foliowed to

- 32 -

incorporate the plastic strains cp into the equilibrium equation (3.11). Inserting Acr that

D (Ac - Aep) and Ae B Aa it is found

B DB Aa dV = H V dS + T 1 H q dV - B V dV + BTDAep dV

V V (3.11)

The iterative process can be expressed in the following recursive form:

A Aa § + Q - E1"1 + P1"1 for 1=1,2. (3.12)

where A is the stiffness matrix, S is the surface load vector, Q is the body force vector, E is the vector that accounts for unconverged equilibrium in the previous load step, and P is the pseudo load vector for the plastic strains. The first estimate for the displacement increments in a load step will not be made by an elastic prediction for P=0, but by extrapolating the displacement increments obtained in the previous load step (as described by Vermeer, 1980).

The elastoplastic finite element calculations are carried out for a mesh representing the area within a radius of 5 pile diameters. A 4 point Gaussian integration scheme is used. A local convergence criterion for each integration point (as described by de Borst & Vermeer, 1984) is employed. Within each iteration a statically admissible stress field and a stress field which complies with the yield criterion are calculated. The iteration process is interrupted if the statically admissible field does not deviate from the yield surface by more than a specified norm e, and the maximum difference between the two stress fields is

2 smaller than 4e . The calculations presented here are performed with an accuracy tolerance of e=0.05c .

- 33 -

Before discussing the application of the numerical model described above, it is useful to bring to mind the closed-form solution for the limit load on a laterally loaded circular pile-segment in a cohesive soil under plane strain conditions, obtained by Randolph & Houlsby (1984). Starting from a perfectly plastic material model, they managed to find identical upper and lower bound solutions for the limit load, depending on the roughness of the pile, expressed in the ratio of adhesion and undrained shear strength:

a = c / c (3.13)

The limit load P varies from 9.14 c D for a smooth pile (a=0) to u

11.94 c D for a rough pile (a=l). The general equation is

P = [Ti + 29 + 2cos9 + 4(cos|+sin |) ] CuD (3.14)

where 8 = arcsin a. This closed-form solution will be used for the validation of the numerical analysis presented here.

The first computation was made with the rather coarse finite element grid shown in Fig. 3.1, which consists of 24 eight-noded elements. For a fixed pile-soil connection, the load-displacement curve is shown in Fig. 3.2. The analysis yields a failure load of 14.6 c D, which is about 22 percent higher than the exact value. Since the accuracy was considered insufficiënt, a second computation was made with the fine mesh shown in Fig. 3.3, which consists of 96 eight-noded elements. The load-displacement curve for this analysis is represented by the dotted line in Fig. 3.2. The curve remains almost linear up to about half the limit load; then a strongly nonlinear response is obtained. The analytical limit load of 11.94 c D is overestimated by about 5 percent. Fig.

u 3.4a shows a detail of the velocity field computed at failure. It

- 34 -

Fig. 3.1. Finite element mesh - first computation.

H/CUD

Course mesh

Course mesh with thin row

Fine mesh E = 1 MPa

v =0.3

cu = 1 kPa

O - t i >

0.02 0.03

u/D

Fig. 3.2. Computed load-displacement curves.

Fig. 3.3. Finite element mesh - second computation.'

shows a very realistic flow pattern, which is demonstrated even more clearly when the velocities relative to the pile are plotted, see Fig. 3.4 . The most striking aspect of these velocity field plots, is the large velocity gradiënt near the pile. The inability of the soil to slip along the pile enforces an internal failure mechanism of the soil. Although the quadratic interpolations in the elements dominate the velocity field near the pile, the inner ring of elements, which has a thickness of 5 percent of the pile diameter, is capable of giving a reasonable approximation of the-actually-discontinuous, pile-soil transition. It seems likely, that the thin inner row of elements is the only substantial improvement over the course mesh. To check this conclusion, the course mesh was again used for a computation. However, this time a thin ring of elements was added around the pile, similar to the inner ring in Fig. 3.3. The result was a failure load that was virtually equal to that

- 36 -

Fig. 3.4 . Detail of velocity field at failure.

Fig. 3.4 . Detail of velocity field at failure relative to pile.

- 37 -

obtained with the fine mesh, which confirms that the soil can be modeled relatively coarsely, if sufficiënt attention is paid to the direct vicinity of the pile.

3.3.2 The use of interface elements

When a continuüm model such as the finite element method is applied to a situation where discontinuous displacements may occur, special attention has to be paid to these local effects. At the pile-soil interface such discontinuities are likely to occur in the elastoplastic state. Slip of the soil along the pile will occur if the adhesion at the pile wall is exceeded. In principle there are two ways to deal with the problem of discontinuous displacements.

The first solution procedure, called the constraint approach, is obtained by uncoupling the pile from the soil and adjusting the mutual boundary conditions if necessary, see e.g. Katona (1983). The interface behavior obtained can be described as rigid perfect plastic. The approach requires an updating of the stiffness matrix for every change of mode in a grid node. When computation time plays an important part in the practical applicability of a program, this solution is not very attractive.

The second solution procedure, called stiffness approach, consists in a smoothing of the discontinuities by representing them by continuous displacements in a thin zone. In this zone the stiffness is reduced when a limiting value of the contact stress is reached, see e.g. Goodman et al. (1968). Again a continual updating of the stiffness matrix is required, if the shear stress exceeds the shear strength.

Finally a combination of these two methods, the hybrid approach, can be used, see e.g. Herrmann (1978). The interface is assigned an elastic stiffness. Once a limiting value is reached, the interaction force between the nodes is kept constant.

The use of a thin ring of eight-noded elements in the previous paragraph, can be seen as a variation on this hybrid method. The

- 38 -

Fig. 3.5. Isoparametric six-noded interface element.

elastoplastic model which is used for the nonlinear soil, accounts for the plastic interface behavior. Due to the applied initial stiffness method, no updating of the stiffness matrix is necessary in this case. For reasons of computer time optimization, the last approach is adopted here. However, to prevent a considerable increase of the number of iterations, due to the strain localization in a very thin zone of normal eight-noded elements around the pile, this ring is replaced by special interface elements (Kooijman & Vermeer, 1988). Besides, these elements offer a simple tooi to reduce the pile-soil adhesion to a value lower than the undrained shear strength of the soil. The type of interface element that is used as a starting point is the six-noded isoparametric element (see Fig. 3.5) with an adjusted interpolation in the direction perpendicular to the interface and a simple constitutive relation, expressed in terms of a normal stiffness K , a

n tangential stiffness K (Sharma et al., 1986). This constitutive eguation relates the interaction stress vector er = (o-,x)T to the pseudo-strain vector 8 = (e,?)7, which in fact contains the normal and tangential relative displacements of the interface element,

- 39 -

er = D 5 (3.15)

with

f K O n O K

(3.16)

The geometrical thickness of the element is zero, and therefore the normal and the tangential stiffness have a finite value for computational reasons only. For this six-noded element, only three shape-functions are required.

(s s)/2 2 h = 1 - s

2 h = (s2 + s)/2

(3.17)

The pseudo-strain vector 5 can be expressed in the nodal displacement vector a

a = (u v u v - x ï 1 2 2 u v ) 6 6'

(3.18)

by using the pseudo-strain interpolation matrix B:

5 = B a (3.19)

where B is the product of the transformation matrix T

* - è 5y/5s Sx/ös

-Sx/Ss ' Sy/Ss

(3.20)

- 40 -

with

= / (dx/ds)2 + (dy/ds)2 (3.21)

and the global relative displacement interpolation matrix P.

P = -h 0

ï -h 0

2 -h 0

3 h 0 ï

0 -h 0 -h 0 -h 0 1 2 3

h 0 h 0 ï 2 3

h 0 h 0 h 1 2 3

(3.22)

The element stiffness matrix A can thus be obtained from

Ael = .1 B DB C ds (3.23)

This elastic interface element will now be extended to plastic behavior by introducing a flow rule for the pile-soil contact. The flow criterion for the interface is derived from Tresca's flow criterion, and is adapted to the interface material behavior as follows:

M (3.24)

where T is the shear stress and c the adhesion at the pile-soil interface. This yield function is also used as a potential function for deriving the pseudo-strains in the interface elements. By satisfying the consistency condition that f remains zero in the plastic stage, and employing a Taylor expansion to estimate the shear strain increment, the plastic strain increment

Ayp = < f(T°+ K Ar) > sign(T°+ K A*)/K (3.25)

- 41 -

can be derived. The term between the brackets < > should be replaced by zero if it is smaller than zero. The implementation of the six-noded element is straightforward. The integration scheme for the interface elements is either a 2 point or a 3 point integration. Buragohain & Shah (1977) state that if a 2 point integration is used, the elastic interface element becomes overflexible, leading to erroneous results. They propose reduced integration for the eight-noded elements and a full 3 point integration for the interface elements. This suggestion is adopted here.

The calculations mentioned in the previous section are also performed with the element grid shown in Fig. 3.1. adjusted with a band of interface elements around the pile. The interface stiffness is obtained by assuming a virtual interface thickness t of 5 percent of the pile diameter, i.e.:

K = E / t n soil v

K = G / t s soiï v

The computed load-displacement curve is presented in Fig. 3.2. Compared with the first analysis without interface elements, the differences are small in the elastic range, which indicates that the elastic interface stiffness was taken sufficiently large. Of course, the influence of the finite elastic interface stiffness can be further reduced by increasing the interface stiffness. However, the computer run time and, to a lesser extent, the condition of the system of equations, make opposite demands. Therefore, from this moment on, the virtual interface thickness is always assigned a value of 5 percent of the pile diameter. In the elastoplastic range, however, the mesh with interface elements shows a more flexible behavior, and virtually no difference with the much more refined mesh without interface elements is obtained, but the most important advantage is that

(3.26)

(3.27)

- 42 -

mesh Fig. 3.1 mesh Fig. 3.3 mesh Fig. 3.1 + thin ring mesh Fig. 3.1 + interface

P / c D u

14.6 12.4 12.5 12.4

cpu-time 1 11.7 4.5 2.3

Table 3.1. Computed failure loads and required cpu-time.

the computer run time has decreased by a factor 5 compared with the previous solution using the mesh of Fig. 3.3, and by a factor 2 compared with the coarse mesh with local refinement near the pile, see table 3.1.

3.3.3 Analysis for reduced adhesion

The advantage of the interface elements is not only the short computer run times, but also the possibility of simulating any adhesion at the pile soil interface. Computational results are to be presented for a smooth pile (a=0), a realistic pile (a=0.5) and a rough pile (a=l). For the smooth pile, the tangential stiffness K instead of the shear strength c could have been

s a taken to be zero. For the present analysis only the shear strength is adjusted. In order to get a detailed picture of the sensitivity of the soil to the pile-roughness, the fine grid of elements plotted in Fig. 3.3 is used again. This time, however, 16 interface elements are added to the mesh. The computed load-displacement curves are shown in Fig. 3.6. Because of the very fine grid and the additional interface elements, very accurate predictions for the limit loads are obtained. Indeed, the numerical values deviate less than 2.5 percent from the theoretical solutions by Randolph & Houlsby (1984). Similarly the final velocity fields correspond well with the theoretical slip-line fields. An increase of the roughness not only gives a higher limit load, but also a much larger zone of deforming soil

- 43 -

H/C„D

,'''

/ J/T

f - !//

f 1 rougb a = 1 realistic a = 05 smooth a = 0

E = 1 MPa v = 0.3

cu = 1 kPa D = 1m

u/D

Fig. 3.6. Computed load-displacement curves.

around the pile. This is indicated in Fig. 3.7. From the examples presented above, it can be concluded that

the elastoplastic model is capable of making an accurate analysis of a soil layer loaded by a segment of a pile. However, it should

Fig. 3.7 Deforming soil body around the pile for a smooth pile (a=0), a realistic pile (a=0.5), and a rough pile (a=l).

- 44 -

be noted that the region near the pile, especially the pile-soil connection, needs special attention when the mesh grid is designed. The use of interface elements is advantageous from a cpu-time point of view, and in addition, provides means to adjust the pile-soil adhesion in an elegant manner. This model, including the six-noded interface elements, will now be incorporated into the main computer program for an entire pile.

3.4 Extension of the pile-soil model to elastoplasticity

When the elastic model for a laterally loaded pile is extended to elastoplastic material behavior by incorporating the previously described single layer model into the main program, an extra assumption has to be made. Apart from the internal strains of a layer, there are two other strain components to be considered in the multi-layer soil-model, namely, y and ns ,

zx zy which provide the coupling of the layers. Since these shear strain components are not included in the plastic potential function, the restriction is imposed that they remain purely elastic, i.e.

yp = V = 0 (3.28) zx zy

In other words, it is assumed that the vertical stress is the intermediate principal stress. Near the soil surface this will not be true, due to the small overburden. If the vertical stress er is not sufficiently large, a different failure mechanism may zz occur. The soil will move upwards, and, in terms of the p-y curve procedures, will exhibit a wedge-type failure. Since vertical displacements of the soil are not allowed in the model presented here, such a failure mechanism cannot occur. Therefore the implicit assumption is made that for all pile supporting layers the vertical stress is sufficiently large to prevent an upward movement of a soil wedge. The nonlinear model for the single soil

- 45 -

layer can now be incorporated into the main program. The incremental finite element representation of the eguilibrium condition now becomes:

(BTD B + HTC H) Aa dV = H V dS + HT(Aqup+ Aq low) dV

BTD Acp dV ( B V ° - H T q t o t ° ) dV ( 3 . 2 9 )

o r :

A Aa = AS + (AQup + AQlow) + AP - E (3.30)

A is the average pile displacement for the concerning layer, S is the surface load vector for a unit pile displacement, Q are the body force vectors, P is the pseudo load vector for the plastic strains and E the vector to account for the deviation from converged equilibrium in the previous load step. This deviation is now composed of a contribution due to the error in the estimated plastic strains, and a contribution due to the error in the estimated body forces.

In the pile model the plasticity can be introduced as follows:

= Y K (A - Ap) j = i

K A + Ü 1

f K A - f K Ap j = i j*i

J = l (3.31)

where Ap is the nonlinear displacement component. The subscript i indicates the number of an individual layer. This equation can

- 46 -

again be written as:

F = k X + f (3.32) i 1 i i x '

This means that the coupling procedure used in the elastic model, can be applied to the elastoplastic model without drastie changes. In order to prevent a cumulative error in the pile-soil equilibrium when eq. (3.32) is used in incremental form, Af has to be related to the total interaction force F :

ï

Af = F1 - k (A° + AA ) - f° (3.33) i i i v i i ' i v '

where F. is the total interaction force determined from the soil model. The analysis can now be performed in a way similar to the elastic solution. However, in the elastoplastic analysis, one extra iteration process takes place at the same time. Apart from the coupling terms f and the body forces that provide the coupling between the layers, the plastic strains in the soil have to be determined. It appeared to be advantageous to allow the plasticity subroutine a limited number of sub-iterations within one global iteration. This means that for every layer eq. (3.30) has to be solved according to the following iterative updating scheme:

A Aa"'1 = AkS + AQupk+ AQl0"k_1+ AP"'1"1 - F,""1

f o r 1 = 1 , 2 . . . 1 , f o r k = l , 2 . . . k ( 3 . 3 4 ) max max

where k denotes the global iteration number, and 1 the sub-iteration number. The value of k is determined solely by

max the convergence criteria. The value of 1 is only determined by

max the convergence criteria if that yields a value that is smaller

- 47 -

than a predescribed maximum. For most cases a maximum number of three sub-iterations renders a very efficiënt convergence progress.

3.5 Application to field test

To illustrate the application of the model for practical purposes, a lateral loading test performed in 1960 at the mouth of the Sabine river Texas/Louisiana and used by Matlock (1970) for the soft clay design rules, will be studied. A 42 feet long steel pile, 12.75" in diameter, having a wall thickness of 0.5", is installed in a slightly overconsolidated marine clay containing a sand layer at a depth of 5.1 to 6.1 m. The soil stiffness and strength parameters are taken from Kay et al. (1986), see Fig. 3.8a and 3.8 , who performed finite element computations to analyze this test. Kay et al. used the program

C„IkPa)

• Pressiomefer Kay et al.,1986

v = 0.33

Fig. 3.8a. Modulus of elasticity. Fig. 3.8°. Undrained shear strength.

- 48 -

Fig. 3.9. Finite element mesh.

u Imm)

Fig. 3.10. Load-displacement curves.

- 49 -

Harmony based on axial symmetry of the geometry and Fourier series expansions in tangential direction to introducé non-axisymmetric loading.

For the computations with the present method the soil is subdivided into 24 layers, all having the finite element mesh shown in Fig. 3.9. The measured and computed load-displacement curves are shown in Fig. 3.10. The curve computed by Kay et al. (1986) is also shown. Both the Harmony code and the present method overestimate the stiffness of the pile at large displacements. The initial stiffness is good. The agreement between the two approaches is encouraging, however. In the next sections it will be shown, that the overly stiff predictions were due to the assumption of full soil-pile contact behind the pile.

- 50 -

4 PILE-SOIL SEPARATION

4.1 Introduction

In order to make continuüm models for laterally loaded piles competitive with semi-empirical nonlinear one-dimensional models, an accurate description of the pile-soil contact behavior is indispensable. When a pile is loaded laterally, tensile stresses will develop at the back of the pile. In case of frictional materials the soil behind the pile will collapse. This phenomenon may be described satisfactorily by applying a tension cut-off to the flow rule of the soil, see Lane & Griffiths (1988). However, this is not suitable to describe the behavior of laterally loaded piles in cohesive soils. In an undrained situation, the soil will be able to resist these tensile stresses up to the point where cavitation occurs. However, at a much lower value of the tension stress, separation of pile and soil will occur. A gap will exist at the back of the pile, and when the pile is loaded cyclically, it will be able to move almost freely through a previously formed slot in the upper soil layers. This will have a major impact on flexibility and damping characteristics of the pile.

4.2 Gap element

4.2.1 Flow rule for gap element

In the previous chapter, a very simple flow rule for an interface element, derived from the Tresca flow criterion, was used to describe slip of soil along the pile circumference. The interface element which was introduced for that purpose will now be used as a starting point to model the presence of a gap behind the pile. Gapping is a rather complicated phenomenon to describe. Before a gap is formed behind the pile some tension can be resisted by the pile-soil contact. However, when a limiting value

- 51 -

O"

Fig. 4.1 Decrease of tension stress as crack opens.

is reached, the tension stress decreases to zero and the contact is lost. At the same time the shear stresses in the interface also decrease to zero. On a macro scale these phenomena can be seen as a decrease of the tension strength and the shear strength of the interface, comparable with softening behavior of materials.

In fracture mechanics this process is well known. When a fracture starts developing, the relation between the tension stress and the plastic normal strain observed in experiments is shown in Fig. 4.1. A gradual decrease of strength occurs as the crack widens. Apart from the shape of this curve, the crack behavior is characterized by the tension strength, and the crack energy Ecr. In numerical simulations, this curve is often approximated by a straight line, running from point s on the vertical axis up to point e on the horizontal axis, foliowed by

c a no contact stage. For the separation of pile and soil, a similar schematization will be used for the tension strength. However, in the present study the shear stress in the interface will be included in the softening model. It is assumed that the

- 52 -

decrease of tension and shear strength are due to a gradually diminishing contact area on a micro scale. Therefore, both strength parameters are assumed to decrease in the same measure, dependent on the gap width or plastic normal pseudo-strain cp (Kooijman, 1989a) .

Another important aspect of the pile-soil interface behavior is rebonding, or restoration of the pile-soil contact. When the contact between pile and soil is being restored again, tensile and shear strength will gradually be recovered. In the flow rule, that is also used as a plastic potential function, this is reflected by the hardening stage that is initiated when the gap is closing again:

fT = M (4.1)

ff - a

f = | er I - er ff ■ ' a

if ep = 0 (no gap)

if cp > 0 (gap) (4.2)

where neither c nor ff are constants, but variables which depend on the gap between the soil and the pile:

c_ = c' (1 - ep/e )

ff= = ff^ (1 - e p/c ) if 0 s cp < e (4.3)

e represents the gap width where contact is lost completely. C

Both c and er are real constants representing initial values. For the final residual values at large strains both strength parameters become zero.

c = er if e' (4.4)

- 53 -

A small value can be assigned to the quantity c , which, although C

it is based on experimental observations (see Fig. 4.1), in f act mainly ensures the stability of the numerical process. However, to avoid numerical difficulties, e should be larger than er /K , otherwise there would be a snap-back. The plastic pseudo-strain increments êp and yp can be derived by satisfying the consistency condition:

5 f ~ Sf„ f = _ Z & + — iv = 0 (4.5)

a Sa S e p

Sf Öf f = — - X + — Cp = 0 (4.6) T Sx ë€p

From these equations, the plastic strain increments are obtained:

èp =

Ü£ 5a

5f ög

Sa n Sa

K è

5 f ~ a S e p

Sg a Sa

a Sa

(4.7)

* p ~ -

— - K r + sx s

K

Sfx Sep

Üï

cp

S*x Sx

(4.8)

Sx Sx

For the situations where Sf/Sa is continuous within a load step, the upper term of eq. (4.7) can be approximated by a Taylor expansion:

- 54 -

5f f (a + K Ae, ep) = f (cr/fep) + —-(er ,ep) K Ac + . . (4.9) <JV O n ' O' aKyo' O' _ v O' O' n l '

ÖCT

The first term on the right-hand side vanishes. For the left part of the upper term of eq. (4.8) a similar approximation can be used. This leads to simple expressions for the plastic strains, when cp is smaller than e :

f (er + K Ac, ep ) crv o n ' o '

K sign(cr + K Ac) - er /e n ^ v O n ' a' c

(4.10)

Ayp = \tz(XQ+ KsA*, ep + Acp) y> sign(xo+ KsAy)/Ks (4.11)

The term between the brackets < > should be replaced by zero if it is smaller than zero. When cp is larger than e the stresses i

c and x must be zero, and therefore the plastic strains will be

Aep = Ac (4.12)

A?P = A? (4.13)

There are two special situations worth noticing. When the gap is closing, and when hardening is initiated within a load step, the stresses have to be brought back to the yield surface in two steps. First the normal strain has to be reduced to the point where ep = c . Then the new normal test-stress is returned to the

c yield surface according to expressions (4.10 and 4.11). This two step procedure is expressed in the following equation for the

- 55 -

plastic strain increment:

Aep = Ae + < / f (o- + K (Ae-Ae ) , e ) S. ff O n * c ' ' c '

K signfcr + K (Ae-Ae ) - <ri/c n x O n c' a' c

with

(4.14)

Ae = (e - c ) c c O

(4.15)

The shear stress will be brought back to the new yield surface again, as is indicated by eq. (4.11).

The second situation occurs when the plastic normal strain in the interface is vanishing and the contact is restored completely. The plastic normal strain increment should satisfy

Ae p * -ep o (4.16)

Fig. 4.2. Unloading-reloading behavior of the gap element.

p-

3 rt ro

ua Q) rt P-o 3 P, (D

u3 P-O 3 w H> o p, P-3 ft (D H H) 0) O

(D (D 3 r t

aFl

Y<n

r t H DJ

"O ro N p -c 3 S P -f t 3" f t ff (t)

< (D >-i r t H -0 DJ

>-• t - 1

P -3 fl> W

O O 1 p,

ro ui 73 o 3 a. « p -r t ff

O) 3

ro M DJ cn r t p -o

01 i-i (D 0)

P. (D

73 H. 0) 01 ro 3 r t 01

r t ff (D

(D M OJ oi r t p -O

w r t DJ r t (D

• PJ ff (t>

r t n p -0J 3

iQ £ M DJ 1 01 H (1) DJ

DJ 3

& r t ff CD

& fl> r t (D 1 3 p -3 <D cn ri­ff (D

3 (D X r t

(D 01 r t p -3 OJ r t ro O Bi r t 01 r t ro 0 H>

r t 3 * (D

P -3 r t (T> 11 i-h 0> O (D

• H ff (D

o 0 r t r t

*» • c j

• P J ff CD

M O 0 0) r t p -O 3

O H)

r t ff (D

r t

ro cn r t l cn r t i-i ro 01 01

73 0 p -3 r t

p -3

ri­ff P -01

a p -0)

iQ n 0) 3

H ff (D

p -3 r t (D

iQ

« 0> r t p -O 3

H-ro vQ p -0 3 01

H> O H

ri­ff ro p -3 r t

ro n Hl 0) 0 ro DJ

« ro Cb p 0J s: 3 P -3

T l P -

uO

0-C H p -3 *ö r t ff ro

73 p. 0 0 (D cn 01

O M)

& ro ff 0 3 0-p -3

iQ

0> 3 a. Pi ro ff 0 3 O. p -3

iQ

O H l

•O p -M (D

0) 3 & 01 O p -M

vQ 01 73 • > cn DJ

0 0 3 01 ro A c ro 3 O ro ■•

cn 0 3 ro ro 3 ro I"!

iQ

»< & p -01 01 p -V 0) r t p -o 3

S p -P 1

M

f t 0) X ro w M DJ n ro

M 0 0J a p -3 Q > 3" 0> I"! a ro 3 P -3 ua 0 i i

01 0 H l f t ro 3 P -3 oa & (0 rt­ro H 3 p -3 ro r t V ro a ro < ro M O n 3 ro 3 r t

O H l

r t V ro

T) M 0> cn r t p -

o Oi ro Hl 0 H 3 a> r t p -O 3 01

O O O c p.

-» 0J 3 a •* CL ro 73 ro 3 a ro 3 r t

O 3

r t 3"

ro & p -H (D O r t p -0 3

O H l

r t V ro ua DJ T3 p -cn ro M 0) 01 f t p -0

• s: D" ro 3

f t S* ro ff o c 3 O i 0> H. p -ro cn o H l

r t D" P -01

0) i ro 0)

0) H. ro i i ro QJ n 3"

p -01

01 3" O K 3

P -3

•n p -iQ

• J^

• NJ

• s p -r t ff p -3

r t ff ro r t 1 p -DJ 3

^Q C M 0> 1

DJ P, ro OJ

^ r t ff ro p. ro cn

>D 0 3 01 ro 0 Hl

> u3 P. OJ

rs ff

0 H l

r t ff ro c 3 M O DJ

a p -3

vQ 1

H-ro H 0 o> & p -3

vQ

ff ro ff 0J

< p -0 H

O H l

f t ff ro >Q OJ

-a ro M

ro 3 ro 3 f t

- 57 -

shear stress and a plastic normal stress. The area marked by horizontal lines denotes the slip state where the normal stress is elastic but the shear stress is plastic. Finally the remaining part of the a-T plane represents both plastic shear and normal stress. To the left of the "Gap closure" line the plastic normal strain has vanished. To the right of the "Gap" line contact between pile and soil is lost completely. From this figure it can be seen that a lost-contact strain e smaller than er /K will

c a n yield indeterminate behavior, because the triangle may fold inwards.

From this moment on the actual strength parameters er and c of the interface will simply be called <r and c when they are needed as input parameters.

4.2.2 Consequences of finite load step size

In the elastoplastic analysis, a test stress is used to determine the appropriate integration region for the constitutive relation. This test stress may be the elastic predictor stress, but an appropriate estimate based on the displacement increment determined for the previous load step can also be used. Generally, if due to the finite step size in the numerical analysis, a wrong integration region is chosen, the iterative analysis will correct this mistake. For softening behavior however, this may not be true. Dependent on the stress increment and the steepness of the softening stage, the erroneous indication of softening may be irreversible.

This will be demonstrated in the following example. Two six-noded elements are coupled as shown in Fig. 4.4. The right element is assigned a tension strength of er , and a lost contact strain e . The left element is given a tension strength er ,

c max which is smaller than a . The lost-contact strain of this element

a is taken infinitely large, which means that no softening will occur. Both elements have a normal stiffness K . When the

n right-hand boundary is pulled to the right, the weak element

- 58 -

V-k

G",

max

»

/ h«

© E

F ' -

<V

/®\ hf" \ e

< W = 2 kPa

Kn =1MN/m3

0"a = 2.1 kPa

Kn = 1 MN/m3

Ec = 0.01 m

Fig. 4.4. Mesh containing two six-noded elements, and stress-strain relations belonging to them.

should fail, and the strong element should remain elastic. However, the plastic strains that are computed during the first iteration of a load step for the stronger element, may be larger than that predicted for the weak element, due to the softening stage. This will cause softening, and eventually failure, of the stronger element. For this simple example it can be derived that, if the predictor stress satisfies the equation,

o-test < er (1 + Kc (1/cr - l/a- ) max n c max a

(4.17)

the correct failure mechanism will occur. For the parameters given in Fig. 4.4, this expression yields:

o-test < 2.476 kPa (4.18)

When the load is applied displacement-controlled, and a total displacement of the right boundary of 0.05 m. is predescribed in

- 59 -

N

52 53

Acre i ( kPa )

0 . 4 8 1 0 . 4 7 2

«T7 ( kPa ) el ( '

2 . 4 8 1 2 . 4 7 2

Table 4.1. Predictor stresses for 52 and 53 load steps.

N load steps, the elastic predictor stress increment and the total predictor stress in the seventh load step are summarized in Table 4.1. According to criterion (4.18), undue softening will occur for N=52, but not for N=53. The computed load-displacement curves are shown in Fig. 4.5. The solid line (N=53) shows the proper curve for this problem. The dotted line (N=52) shows a diminishing force, because the strong element fails. The third, dashed, curve represents a similar computation, however, this

2.0 r

H IkN)

N= 52

N = 41 Alternative first prediction

N = 53

15 20 u (mm)

25

Fig. 4.5. Load-displacement curves for example.

- 60 -

time the first prediction of the displacement field is based on the previous load step, as mentioned in chapter 3. Although only 41 load steps were applied in this case, the correct behavior was computed, because the plasticity that already occurred in the weak element at the previous load step also influences the initial estimation. Apart from the improvement of the robustness obtained by using this technique, an enormous reduction of the number of iterations was achieved. All computations were performed with a very small accuracy limit (10~6) . It is noted, that when the error measure is enlarged, a relatively large stress may be accepted in the previous load step. This will cause a high first test stress, due to the incremental computation scheme. Again undue softening may be initiated, before this error can be compensated. From eq. (4.17) it can be seen that defining a very long softening stage will diminish the risk of undue softening. On the other hand, for more complicated cases, a very gentle slope of the softening stage will make a stress redistribution possible in the interface that erroneously precludes progressive collapse due to a peak stress.

Although the problems demonstrated in this paragraph were asked for, they are not entirely academie. These aspects can and will play a part in computations performed with softening models. However, if an acceptable load step size is chosen, no substantial problems will occur. The possible premature opening of a gap behind the pile, can be compared with making a conservative estimate for the tension strength of the pile-soil contact.

4.2.3 The tension strength parameter

In an undrained analysis, the initial situation around the pile is characterized by a horizontal total stress a = a' + p r ■* o o p where <r' is the initial effective horizontal stress, and where p o *P is the pore water pressure. When the pile is loaded, a gap may occur in two different ways. In the first case no water coming

61 -

from the free surface water can penetrate between pile and soil. This means that a gap is formed only when a suction of one atmosphere is reached and cavitation occurs. The tension strength with regard to the initial situation appears to be equal to the pore pressure p plus the atmospheric pressure p . In the second p a case, water can penetrate between pile and soil. The gap is drained. A much lower tension stress can be resisted now. Once the gap has formed, in both situations the hydrostatic water pressure p will be established behind the pile.

In Fig. 4. 6a the normal stress on the pile-soil contact at the back of the pile is plotted. With respect to the initial total stress er , a tension stress s can be resisted. At point D a o t gradual drainage of the pile-soil contact occurs, until pile and soil are separated completely and the free water pressure exists behind the pile. The actual value of the tension strength depends on the suction that is allowed behind the pile before drainage is initiated. If it is assumed that the pore pressure next to the pile equals the hydrostatic water pressure, Fig. 4.6 can be obtained by shifting the curve in Fig. 4.6a upwards. The result is a curve similar to the one presented in Fig. 4.2. When the pile-soil separation is complete, no contact stress remains. In

St

Fig. 4.6 . Normal stress on pile-soil contact.

Fig. 4.6 . Effective normal stress on pile-soil contact.

- 62 -

this figure the input parameters for the model of the gap can be identified. The effective stress should be used as the initial stress in the model. The absolute tension strength er equals s -er'. A fairly conservative estimate for this strength parameter may be made by assuming a fully drained gap-initiation. When the effective stresses in the interface have decreased to zero, pile-soil separation will occur and water will fill the gap. For this situation a tension strength er of zero applies. If the assumption that the porewater pressure is equal to the hydrostatic water pressure is not acceptable or tension is allowed before drainage starts, the same model can be used if the appropriate adjustments of the initial stress and the tension strength are made.

4.3 Analysis of an isolated layer

A small scale test, see Fig. 4.7a, performed by Matlock (1970)

Fig. 4.7a. Laboratory model and Fig. 4.7 . Computed load-typical load-displacement curve, displacement curve. Matlock (1970) .

o Ml

(+ ff <D

r t (D 3 01 H -0 3

0) r t H ro 3 ua r t ff

9 U

*

•n H -

i Q

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I - 1

0 CD ft 1 a H -01 •o M 0) o (D 3 ro 3 r t

o e 3 (D 01

H l O N

& M-H l Ml (D H (t) 3 r t

< 01 M C (D 01

3 3

0 0 3 r> M

n> r t (D 01

r t ff O

H (D 01 (D 3 cr h-i tt 3 O ro 0 Mi

r t ff (D

r t s: 0

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vQ C M ro w • > V

V 01 H ro 3 r t M

< r t ff ro 01 ro

ft ro Ml o H. 3 0> r t H -O 3

O DJ C 01 H -3

ü2

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s; ff ro 3

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T3

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ro (D r t

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iQ

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H -3

ff O . r t ff Mi H -

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■ »

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r t ff ro r t ro 3 01 H -0 3

r t ff ro 3 O < ro 3 ro 3 r t

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r t ff ro •o H-M ro r t ff H O c vQ ff

01

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p -01

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01

r t ro 01 r t

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iQ

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ro to M H -o ro H -01

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O 01

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r t 3" ro H -3 H l H

c ro 3 O ro 0 Hl

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r t 3" ro 01 r t H -Hl H l 3 ro 01 01

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r t 3" ro

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a c H H -3

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iQ

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rt 3" (Il

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0>

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d 3 ft H 0> H -3 (11 ft

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c

H* 01

O ff 0 01 (Il 3

T) H -

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01

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V O 01 ro ft

01 3 ft

01

r t ro 3 01 H -O 3

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iQ r t ff

ro A c 01 M

r t 0

rt ff ro

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M

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n 01 M t-1

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l « ff

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r t ff ro 3 C 3 ro H H-

o 01 1-J

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01

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ft H-01 X

0 0 3 r t 0> H -3 H-3 <a e 0 H-h! O e H» 01 H.

V H-I-1

ro

H -01

01 3 01 H K N (I) ft

02 C 01 h-1

H -r t 01 r t H-

< (D M

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r t ff ro

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ft ro 01 0 H. H-cr ro ft

01 cr 0 < ro

- 64 -

important features of cyclic response can be described and predicted by the finite element model.

The next example concerns a similar circular disk. For different values of the tension strength a the load-displacement curves are computed and plotted in Fig. 4.8. For the value of er of 6c a gap was initiated at some integration

Fig. 4.9. Velocity field at last increment for er = 2c

Fig. 4.10. Principal stresses at last increment for er = 2c

• - 65 -

points. However, due to redistribution of stresses, made possible by the relatively large value of the lost-contact strain, this gap did not develop further. The results of the layer analysis performed in the previous chapter yielded a maximum normal stress of about 6.5c behind the pile. Apparently a tension strength or initial stress larger than this value is sufficiënt to cause failure of the soil before a gap is formed. The final stresses and velocity field for the case er = 2c are plotted in Fig. 4.9. and 4.10. From the slope of the load-displacement curve in Fig. 4.8, it can be seen that the failure load has not been reached yet in this stage.

When very large displacements are imposed on the pile slice after a gap has formed, a very irregular load-displacements curve is obtained from the finite element model, see Fig. 4.11. The cause of the capricious behavior at the upper left and lower right corner of the load-displacement curve can be explained by studying the velocity field at failure for this case, shown in

E = 1 MPa

v = 0.3

<u "- ca = °a

°h - ° "o Ec = S i »

D = 1 m

1 KPa

Fig. 4.11. Load-displacement curve for very large displacements.

- 66 -

Fig. 4.12. Velocity field at failure after gapping has occurred.

Fig. 4.12. When the pile is pushed further and further into the soil, the soil behind the pile will collapse. At the back of the pile the pile-soil contact is restored gradually. From this moment on the normal failure mechanism of flow of soil around the pile without a gap occurs. Although a very flexible response is observed after pile-soil separation, the failure load seems to be unchanged in spite of the presence of the gap. It should be mentioned of course that for displacements of this magnitude, the small deformation approach is guestionable at the least. Randolph & Houlsby (1984) suggest that the failure load after a gap has formed, can be estimated from:

(<r' + 7c )D < P < (er + p + 7c )D (4.19) h u h a u 0 O

This expression is composed of terms for the limiting pressure of a pressuremeter test in front of the pile, a suction of one

- 67 -

atmosphere p or the pore pressure behind the pile, and a side friction of c D. The results of the finite element computations performed here do not seem to confirm this approximation.

4.4 P-Y Curve versus finite element computations

The recommendations of the American Petroleum Institute distinguish between two basic p-y curve types for cohesive soils. For shallow depth, a wedge type failure mechanism is assumed and starting from the failure load for that mechanism, using the strain parameter e to characterize the stiffness of the soil, a p-y curve is composed. The same procedure is foliowed to constitute the p-y curve for large depth, now using a failure mechanism of flow along the pile in a horizontal plane. Applying these procedures without any modifications results in an overestimation of the soil stiffness for intermediate depth, as can be observed in experiments (Matlock, 1970). Therefore Matlock suggests that for intermediate depth the failure load should be reduced. The failure load for wedge-type failure is adjusted using an empirical constant (j), to fit the experimental results. According to this method a wedge-type failure occurs over a relatively large depth. This is in contradiction with field observations, however. The discrepancy between measured p-y curves response and unadjusted Standard curves for intermediate depth may be caused by the existence of a gap. If a gap occurs at intermediate depth, the flexible response of the soil (see Fig. 4.8) is wrongly imputed to wedge type failure with a reduced failure load.

The flexible behavior at intermediate depth is achieved in the model presented in this study in a natural way, without applying empirical adjustment parameters. The initial horizontal stresses in the soil and the tension that can occur at the back of the pile should be estimated, however. Only for a shallow zone where upward movement of a soil wedge occurs the soil reaction will be overestimated at large deformations, because no vertical

- 68 -

displacements are allowed in the model.

4.5 Cyclic loading

Cyclic loading of piles is distinguished from dynamic loading, by considering the influence that inertia effects have on the pile response. When the rate of loading is sufficiently low to allow a guasi-static analysis, an alternating load is interpreted as cyclic instead of dynamic. More or less parallel to this subdivision in types of loading, two types of damping can be distinguished. Firstly, frequency-independent hysteretic damping, composed of material damping and damping due to the soil nonlinearity, will occur. The hysteretic damping ratio of the soil is mainly a function of the shear strain level induced. By choosing an appropriate average shear strain of the soil at a certain depth, and by employing an empirical relation between damping and shear strain, a rough estimate for the damping ratio of the soil can be made. It is noted however, that the effective hysteretic damping ratio at the top of the pile-soil system will be smaller than this value because of the influence of the elastic pile (Gazetas & Dobry, 1984). The second type, frequency dependent radiation damping, caused by the geometrical spreading of waves away from the pile, only contributes to the total amount of damping if the excitation frequency is higher than the first layer resonance frequency of the soil. Radiation will become more pronounced with regard to hysteretic damping when the frequency of the loading is relatively high, for example in the case of earthquake loading. For lower frequencies, such as caused by wave loading, the hysteretic damping due to plastic behavior of the soil will dominate. The model that is presented in this study only predicts the hysteretic damping of the pile-soil system. As a consequence, the damping predicted by this model is less suitable for high frequency phenomena like earthquake loading.

For many cases encountered in soil mechanics, a simple elastic perfect plastic stress-strain relation will only produce

- 69 -

realistic damping ratios at high load levels. For low load levels, hardly any damping will occur due to the long linearly elastic region. Especially axially loaded piles, where the difference of mobilized shear stress along the pile length is small due to the relatively large axial stiffness of the pile, belong to this category. For most laterally loaded piles however, the soil near the ground surface is already loaded heavily at low load levels, due to the low bending rigidity of the pile. Plasticity will occur in the soil even at small lateral loading. For such a situation, where many different stress paths are foliowed in the soil, local errors may compensate each other, leading to an acceptable global response computation, even when a bilinear stress-strain curve is used. This can be compared with the explanation given by Smith (1988) for the small error in his class A predictions for practical situations compared with the error made in calibrating constitutive models for soils. For field problems that do not involve very specific stress paths, Smith assumed an averaging process taking place over the many stress paths present in the soil.

Another aspect of the pile response to cyclic loading is the degradation of both stiffness and strength of the soil. In the present model, this effect should be incorporated by adjusting the stiffness and strength parameters. Therefore, it is proposed to follow the recommendations of Det Norske Veritas (DNV, 1977), to investigate the sensitivity of the soil to cyclic loading by performing special cyclic laboratory tests. On the basis of the observations made, the appropriate soil parameters can be chosen.

The scour of a gap around the pile may also contribute to a decrease of pile stiffness during a storm. Alternating loading on the pile will cause a continual expulsion of water from and consecutively filling of the gap. This may cause scour of the gap wall. However, collapse of the gap during a period of rest may compensate possible scour of the gap wall during a severe loading event. Shortly after the gap has formed, the pore pressure at the gap wall will be egual to the sum of the water pressure and the effective normal stress. The effective isotropic stress has not

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• H

>| r-t U ffl 0) c • H

H

01 X H • 73

dl 01 3 01

• H

*. 0) ü ca i p p 3 01

H - H O 01

01 XI p -p (0

0 p 01 N

e 0 p

l p

D l c • H

4-> P ca p 01

> i rH rH (0 3

73 (0 P D i

(0

O ■p c

• H

73 01

■P P 01

> c 0 u 01

-rH

01 I-t • H l p 0 p ft 01 01 01 p

■p 01

rH ca

• H ■p • H c • H

O i c • H 01 (0 01 p ü c

• H

P 01 > 1 (0

H

(0

p 01

> o 01 01 01 p

4-> 01

4-> c 10

4-> 01 c 0 o (0

c • H

D i c • H

4-1 H 3 01 01 p

». 0) rH • H MH O P ft D> C • H 01 (0 01 P ü c • H

X 01 01

e 0)

X I 4-1

^ 73 O

X I 4-1 0)

e - p c ai 01 01 p ft 01 X +J

X ■p -rH

» 01 c o

• H ■p (0

■p 3 ft e o ü

0) X I • p

p o

6 H

• 01 01 0) c

X u

• H X ■p

73 c 10

73 dl P 3 01 (0 0) g 01 X EH

• c - H (0 D i (0

73 01 01 3 01

• H

-* o> • m

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m P 01

■P ft (0

X ! O

c ■ H

e u o

X I 01

- 71 -

HlkNI

Fig. 4.13. Load-displacement curves, Sabine.

computed load-displacement curves are shown in Fig. 4.13. When a gap is allowed according to the above mentioned parameters, the present method yields much more flexible pile behavior at larger displacements. The measured curve is approximated much better.

More recently, tests have been performed at the Haga site in Norway by Karlsrud and Haugen (1983). They present a very detailed description of laterally loaded pile tests, both for static and for cyclic loading. The piles were installed in a fairly homogeneous clay with a water content of about 38%. The OCR decreases with depth from 20 near the surface to 3 at 5 m. depth. The undrained shear strength profile, obtained from vane tests is shown in Fig. 4.14a, the shear modulus derived by Kalsnes (1988) from a series of direct simple shear tests in Fig. 4.14 , and the initial effective horizontal stress in Fig 4.14c.

The pile that is examined here is numbered as D3 in the

- 72 -

Cu (kPa)

O 50

50 G (MPal CTh I kPa)

Fig. 4.14 . Undrained Fig. 4.14 . Shear Fig. 4.14 . Initial ,50 shear strength. modulus G hor. effect, stress.

Test

Present method

Fig. 4.15. Measured and computed load-displacement field, Haga.

H l H fD >Q e ro 3 O

< CL (D TJ (D 3 C L <D 3 f t

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ff re a fD r t <D K 3 H -3 (D & Mi 0 n 0>

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- 74 -

loading u from:

ceq = 2|3k ( 4 . 2 1 )

For the cycles plotted in Fig. 4.15 the equivalent parameters are summarized in Table 4.2.

keq (kN/m)

ce qu (kN/m)

measured

8 0 0

3 0 4

computed

1100

3 0 8

Table 4.2. Kelvin body parameters for Haga test.

ZO -16 -12 -8 ■U (

U

L

1

2

3

depth (ml

•30 -20

f IkN/m'l

-10 0 10 20

V. 1 1

Present method

Test

. 1

i

2

3 | — 1

depth lm)

5

Fig. 4.16a. Bending moments. Fig. 4.16b Soil react ion.

- 75 -

Bending moments were measured at eight levels during the test. A curve based on the solution of a beam on a Winkler medium was fitted by Karlsrud and Haugen through the measured moments, by assuming that the bending moment is zero at the pile top, and that the soil reaction starts from zero at the ground surface. Two remaining constants were varied until a satisfactory fit was achieved. The moments derived from the test and the computed moments at maximum positive load are shown in Fig. 4.16a. As expected from the load-displacement curves, the measured moments

are somewhat larger than computed, and shifted downwards a bit. The soil reaction profile is shown in Fig. 4.l6b* It is obvious that the computed data do not support the assumption that the soil reaction is zero at ground level, but in general the correspondence is good. The gap that occurs to about 1.4 m. below the ground surface both in reality and in the computations, again plays an important part in the behavior of the pile.

Generally, when field tests are conducted on piles, the available loading equipment is designed to move the pile in only one (including the opposite) direction. When the pile is loaded cyclically, it moves

Fig. 4.17. Deformed mesh top layer for loading in perpen-dicular directions.

73 C (Ö

i X c

-rl

c •H 73 * O

u o

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d) > u 3 O P c 0) 6 01 O (0 r-i a in

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- 77 -

levels, a more severe decrease is observed. However, in general the influence of the perpendicular loading directions is not very large.

- 78 -

5 PILE GROUPS

5.1 Introduction

For single laterally loaded piles the p-y curve method is by far the most popular model used in practical engineering. However, when piles are located at a small distance from each other, so that interaction between the piles may influence their behavior, designers resort to more sophisticated models.

The first major breakthrough in pile group analysis methods was made by Poulos (1971 ) , making use of the Mindlin solution, which gives the stress and displacement field due to a point load acting in the interior of a homogeneous elastic half-space. In spite of the restriction to homogeneous elastic soils, a reasonable estimate could be made for the interaction between the piles, based on a continuüm analysis of the soil. In order to take the nonlinearity of the soil into account, Focht & Koch (1973) proposed to calculate the pile group response by combining Poulos's elastic method and the p-y curve method. The main disadvantage of this method, apart from the questionable theoretical background, is the fact that overlap of plastic zones around the piles, and the non-symmetric character of the soil deformations are disregarded in the analysis. In a real soil the effectiveness of the piles in the front of the group will be much larger than that of the piles in the rear. And besides, failure may occur at a much lower average load level than for single piles. Many practical designs are based on this or similar hybrid methods, however.

Purely empirical methods for the interaction between piles have also been developed. The |3-method described by Dunnavant & O'Neill (1986) provides interaction factors for piles, dependent on pile diameter, spacing, and the relative position of the piles to each other. The shadowing or non-symmetric effect is thus taken into account, but the composition of the soil and the bending rigidity of the piles are not considered.

- 79 -

Another approach is the transfer matrix method of Nogami & Paulson (1985). The expression for the transfer matrix is obtained by idealizing soil medium as horizontally mutually interconnected nonlinear springs. This model provides a pile-soil-pile coupling at one level, but is not capable of modeling the coupling among the responses at different levels.

Finite element solutions for pile groups have also been performed. Randolph (1981) presents a method to derive interaction factors for pile groups in a linearly varying elastic soil, based on the finite element solution of a single pile. Tamura at el. (1982) performed full three-dimensional finite element computations of a laterally loaded pile group. They proposed a simplified method that employs a subdivision of a pile group in inner and outer rows to analyze rectangularly shaped pile groups.

Shibata et al. (1988) present a quasi three-dimensional finite element model for an elastic soil. The soil is modeled by vertically constrained stratified plane strain panels connected with shear springs. This model, still presents the problem of the enormous bandwidth due to the three-dimensional geometry.

When the model described in this study is extended to pile group behavior, it is expected that a very realistic schematization of a laterally loaded pile group will be obtained. Elastic and plastic interaction of piles in a layered nonlinear soil, taking into account the effect of non-symmetric soil response due to gapping, can be computed.

5.2 Additions to the single-pile model

In chapter 2 it was assumed that a vertical plane of symmetry can be indicated for the geometry and the loading at the pile top. As a consequence the only relevant degrees of freedom of the pile are the horizontal displacements in that plane. At the stage of extending the model to pile group behavior, this assumption is abandoned. For it does not prevent an individual pile in a group

- 80 -

from moving in a direction that deviates from the direction of the plane of symmetry of the group. Displacements and loading of a pile in both horizontal directions are allowed. However, the kinematical restriction that a pile does not rotate about its axis remains.

The model for a single pile, based on the eguations for a linearly elastic beam on elastic foundation, can be used to analyze the piles one after the other in x- and y-direction separately:

d2M dz

= k u - f (5.1)

d2u M dz2 EI

d2M o 2 dz

d2v M _ _ _y.

dz2 EI

(5.2)

(5.3)

(5.4)

The complete three-dimensional profiles of the piles are obtained by superposing the computed profiles for the two perpendicular directions. Averaging the deflections of the piles over a layer thickness yields values for the predescribed displacements \ p

and X p of all the pile slices in that soil layer. Similarly, the average soil reactions obtained in both directions, can be compared with the corresponding components of the soil response to the pile slices.

In the nonlinear finite element analysis of the soil layers, the predescribed displacements of all pile slices in x- and y-direction are imposed at the same time of course. However,

81 -

since the ratio of the predescribed displacements of the pile slices is not constant, the right hand side of the system of equations (3.30) nas to be adjusted. The term XS concerning the pile-soil interaction must be split up as follows:

XS = Y Xlp Slp + y X,p Slp (5.5) — L x —x L, y —y I p = l i p = l

where S p and S p represent the load vectors due to the unit — x —y

displacements of a pile in x- and y-direction for the concerning layer. Now the system of equations can be solved efficiently for values of X and X , that change during the iterative process.

The coupling between the piles and a linear soil can be performed by

N P I L E 2N F!p = ï ï <?K < 5 - 6 > j P = i j = i

where F p is the interaction force between pile number ip and the soil for one specific layer in x- or y-direction. N is the number of layers that surrounds the pile. Both the interaction forces and the degrees of freedom X of the piles are now numbered consecutively for the x- and y-direction of a pile slice, resulting in 2N degrees of freedom per pile. By extracting the main diagonal term, eq. (5.6) can again be written as:

Flp = klp Alp + flp (5.7)

where kip is substituted for Klplp. Similar to the case of a single pile, the value of the spring constants k p can be determined from the soil model, by running the program with the boundary conditions that X p = 1 and that all other displacements Xjp are zero. In this way the entire interaction matrix can be

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- 83 -

pC = (1 + a) p (5.8)

For groups of more than two piles, Poulos proposed to calculate the increase in displacement of a pile due to all surrounding piles by adding up the increases in displacement due to each pile in turn, using the interaction factors for two piles. When all piles carry the same load, this yields the following expression:

NP I LE Pl = (1 + I ai^ P (5-9)

j = l

A full superposition would require interaction factors determined with all piles of the group present. In order to calculate these interaction factors, the piles will be loaded 'one by one, while the other (unloaded) piles in the group will stiffen the soil. This effect of stiffening of the soil is not accounted for in the superposition principle proposed by Poulos. For relatively dense pile groups this principle will therefore be conservative.

The model presented in this thesis will be compared with the results of Poulos for a two-pile group. The deflection of the group will be expressed as R I , where I is the displacement influence factor for a single pile (see eq. 2.42), and where R is defined as the ratio of the displacement of the group to the displacement of a single pile carrying the same average load as the group. The comparison has been made for two cases of loading. The angle (3 between the direction of loading and the line joining the pile centers is taken 0° in the first case and 90 in the second case. The piles have a length to diameter ratio of 25 and a flexibility ratio K of 10"5. The Poisson's ratio of the soil

1 R is 0.5. The mesh that is used in the computations is shown in Fig. 5.1. For symmetry reasons only one quarter of the mesh is

co

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- 86 -

be assessed by the studying the following example. For a pile distance of zero, the deflection of the pile group should be equal to that of a single pile with stiffness 2EI under the same load as the pile group. For this "doublé" pile Poulos's single pile graphs (Poulos, 1971a) yield a value for I of about 13.8, which means a value of 27.6 for the response to doublé loading. This point is marked by the large dot on the vertical axis of Fig. 5.2. Extrapolating the dotted and solid lines, seems to confirm the above mentioned assumption. And with that, the explanation for the discrepancy of the results of the two methods for closely spaced piles loaded parallel to the line through their centers, gains some credibility.

The interaction factors can be obtained for the present method from the results in Fig. 5.2 and the analyses of a single pile. The results are shown in Fig. 5.3, together with the values presented by Poulos (1971 ) . In spite of the good correspondence

1.0

0.8

0.6

a P H

0.4

0.2

0 0 1 2 3 4 5

0.2 0.15 0.10 0.05 0 S/D

D/S

Fig. 5.3. Interaction factors from Poulos and present method.

• Poulos

Present method

- 87 -

of the total group response in Fig. 5.2, a large difference is observed between the derived interaction factors. This is caused by the fact that the errors that are made in the computation of the group deflection pG and in the deflection of the single pile p are concentrated in the value of the interaction factor a, since

« = f ~ 1 (5.10)

and p /p is often only slightly larger than 1. As long as the number of piles in the group is small this will not cause significant inaccuracies. However, if the number of piles increases, the term Ta. in eq. (5.9) will become large with respect to the term 1. Then inaccurate results are likely to occur, apart from the consequences of superposition of two-pile interaction factors.

A very useful method for making a quick estimate of the group effect of laterally loaded flexible piles was developed by Randolph (1977,1981), who used the pattern of lateral movement at the soil surface around a single laterally loaded pile to derive interaction factors. The required displacement field was computed by using the finite element method. He deduced interaction factors by relating the magnitude of soil movement at the location of the other piles to the single pile movement. These interaction factors are expressed in simple easy to use algebraic expressions, and can be applied for an elastic continuüm with a linearly varying soil modulus. For the displacement of piles in a homogeneous soil loaded by a horizontal force:

a = 0.25 (E /G*)1/7 - (1 + cos2|3) (5.11)

where E is defined in eq. (2.43) and where

- 88 -

G = G (1 + 3v/4) (5.12)

This expression yields interaction factors that are quite close to those obtained by Poulos, except for very closely spaced piles in line with the applied load. Then Randolph's values are much larger than those of Poulos. Therefore, Randolph suggests replacing a by 1 - (4a )" if the value of a given by eq. (5.11) is larger than 0.5. This modification ensures that, as the pile spacing s tends to zero, the calculated interaction factor tends to unity, which is consistent with Randolph's method of using the surface displacement ratio to determine the interaction factor. However, this value of 1 is not realistic, because now the stiffening effect of the (unloaded) pile for which the interaction factor is calculated is not taken into account. So, even for a two-pile group this is conservative. The correct value of the interaction factor for zero pile spacing should be determined by considering a "doublé" pile, as is described above (for that example a zero spacing interaction factor of 0.84 instead of 1 would be obtained). In spite of the correction applied for closely spaced piles in line with the loads, Randolph observed an overprediction of his interaction factors compared with Poulos's values. As a possible explanation he mentions the fact that Poulos idealizes the pile as a thin strip. This has the effect of increasing the amount of soil between the piles compared with circular piles, thus leading to lower interaction factors at close spacings. The model presented in this thesis does not confirm this explanation. The present method that models the geometry of the piles correctly, yields even lower values than Poulos's analysis. An explanation may be that Randolph even disregards the stiffening effect of the second pile, while Poulos does take into account that effect.

- 89 -

5.4 Application to field tests

Pile group field tests were performed by Matlock et al. (1980). From these experiments a static test on a single pile and on a five-pile group are chosen for comparison with the present method (Kooijman, 1989 ). All piles have a penetration of 11.6 m. below the bottom of a 2.5 m. deep pit, see Fig. 5.4. The piles have a diameter of 0.168 m. and a wall thickness of 7.1 mm.

U.6 m.

>777777777777777777777,

77/7777/7777/7/77/7/7?

1 H - 4 T

„ 0.23 , 77-1 /sss;//.

0.75

1.52

v/ -f //////sssss/s/s

2.5 m.

schemat i za t i on 77/ -> L 77//7777/77/77/T7,

1.6 m.

Fig. 5.4. Lay-out of pile group test, Matlock et al. (1980)

Deflections are enforced at two elevations above the ground surface (at 0.23 m. and at 1.75 m.), to simulate the head-restraint of a cap. The group piles are placed in a circular configuration with a center-to-center spacing of 3.4 pile diameters. That is, an outer group diameter of 6.8 times a pile diameter. As for the soil data, two sources were available.

- 90 -

Cu IkPa)

20 tO

- Leung & Chow,1987

- Nogami & Paulson, 1985

Fig. 5.5 . Undrained shear strength.

Fig. 5.5 . Modulus of of elasticity.

Nogami & Paulson (1985) and Leung & Chow (1987) present rather different soil profiles for this site, summarized in Fig. 5.5. Both profiles were used in the computations. The horizontal effective stress is assumed to increase with depth at a rate of 5 kPa/m., starting from zero at ground surface of the pit. The soil next to the pit has not been taken into account in the computations, nor have the sheet piles that surround the pit. The pile is rough and no tensile stresses are allowed at the back of the pile. The lost contact strain is taken to be 1 mm. The soil was subdivided into 20 layers, all having the mesh shown in Fig. 5.6, which results in 20*860=17200 degrees of freedom for the soil. In Fig. 5.7 the measured load-displacement curves are shown for the single pile and pile group, together with the computed curves. Although the agreement is not very good in the initial stage, the computation based on the input data of Leung & Chow gives the best agreement. The input obtained from Nogami & Paulson results in an overestimation of the stiffness of the piles, except for the initial values. The mutual influence of the

- 91 -

Fig. 5.6. Finite element mesh.

z 30

S 20

E.cu:Uung & Chow, 1967

E.c^Nogami & Paulson, 1985

u Imml

Fig. 5.7. Average load-displacement curve for single pile and pile group.

- 92 -

Fig. 5.8. Plastic integration points top layer.

piles seems to be modeled satisfactorily by the present method. The overlap of plastic zones in the soil, which can also be deduced from the load-displacement curves, is demonstrated in Fig. 5.8. The integration points of the upper soil layer which are in the plastic stage, are represented by a square. Only a small area at the center of the pile group has remained in the elastic stage. Furthermore the extensiveness of the plastic zone in front of the pile group is striking. The computed velocity field relative to the piles is plotted in Fig. 5.9a for the elastic prediction, and in Fig. 5.9 for the final load step. The patterns of soil deformation are entirely different for these two cases, which again emphasizes the necessity of introducing soil-nonlinearity into the analysis of pile groups. A part of the final deformed mesh at ground level is shown in Fig. 5.10. The gaps behind the piles are clearly visible.

An important aspect of lateral pile group behavior is the pattern of load transfer within the group. The distribution of load to the piles in a group was determined experimentally by Brown et al. (1987). A two-way cyclic loading test on a closely

- 93 -

Fig. 5.9a. Elastic velocity field top layer, relative to piles.

Fig. 5.9 . Final velocity field top layer, relative to piles.

- 94 -

Fig. 5.10. Deformed mesh top layer at maximum load level.

spaced pile group was conducted in stiff overconsolidated clay at a site in Houston, Texas. The group consisted of nine steel pipe piles, having an outer diameter of 0.273 m., a wall thickness of 9.3 mm., and a penetration of 13.1 m. Inside these piles, smaller instrumented pipes were placed and connected to the outer pipes by grouting. The piles were installed in a 3 by 3 arrangement with a spacing of three pile diameters on the centers. The loading was applied by using a loading frame that provided moment-free connections to each pile, at 0.3 m. above the mudline. For each pile the applied shear force was measured independently. After a first series of 100 load cycles having an amplitude of 110 kN, four series of 200 load cycles were conducted consecutively, with increasing amplitude. All cycles had a period of 3 0 sec. During the test, gapping and scour occurred. Both phenomena were confined to an area around each individual pile. The scour, due to a rapid expulsion of water from the gaps was significant. The day after the test, a sediment of several inches thick covered the test pit. Although the

- 95 -

C„lkPal

100 200 300

Profi le Brown et a[., 19B7

■ Present schematization • / A*

Y = 20 kN/m3

i - f i kN/m3

Various tests, see

Mahar % O'Neill, 1983

Fig. 5.11 . Undrained shear strength.

Fig. 5.11 . Modulus of elasticity.

Fig. 5.11 K

effects of cyclic degradation and scour during loading, as observed in this test, cannot be reproduced by the numerical model presented in this study, several important features of the measured pile group response can still be used for comparison with the results of the numerical model presented in this study. The distribution of load over the piles in the group will be obtained from the model, for several load levels. The input parameters for the soil that have been derived from the data presented by Brown et al. (1987) and by Mahar & O'Neill (1983), are summarized in Fig. 5.11. The bending rigidity of the compound piles was determined by calibration (Ochoa & O'Neill, 1989). The values are 19.1 MNm for a depth up to 4.9 m., and 13.4 MNm2 below that level. A finite element mesh of 2 0 layers all having the mesh shown in Fig. 5.12 is used, which yields 20*1332=26640 degrees of freedom for the soil. An elastoplastic analysis was made, consisting of 3 0 predescribed displacement increments of 2 mm. The distribution of pile loads measured is characterized as

- 96 -

Fig. 5.12. Finite element mesh.

follows. The back row is the least effective in the entire load range. At larger load levels, the middle row cannot keep up with the front row either. In the direction perpendicular to the direction of loading virtually no differences in pile effectiveness were observed. This load distribution pattern does not resemble the pattern predicted by elasticity based models, in which the corner front and back rows carry an equal load, and the . center piles in a row carry the least load. The computed load distribution per row of piles is summarized in Table 5.1 for the elastic prediction, and at a pile top displacement of 30 and 60 mm. respectively. For the elastic case and the 60 mm. displacement, the complete computed distribution is shown in Table 5.2. The observation of Brown et al. that the load transferred to the individual piles is predominantly a function of row to row position of a pile rather than of the position of a pile in the direction perpendicular to the load direction is clearly confirmed by the elastoplastic analysis of the group. On the other hand an elastic prediction of the load distribution, as

- 97 -

Load-level

Elastic

u = 30 mm.

u = 60 mm.

position of row

Front Middle Back

Front Middle Back

Front Middle Back

Measured distribution

-

36 % 34 % 30 %

40 % 31 % 29 %

Present method distribution

36 % 28 % 36 %

37 % 32 % 31 %

38 % 33 % 30 %

Table 5.1. Row to row distribution of pile loads in a group.

is often used in practice, shows a symmetry with regard to the middle pile, which is not in accordance with reality. The predominant influence of the front row piles is also recognized by Tamura et al. (1982), who performed three-dimensional finite element analyses of pile groups, based on a hyperbolie stress-strain relationship. When the number of piles in the direction of loading increases, and the pile spacing becomes small, this tendency becomes even more pronounced.

Outer Inner Outer

Elastic

Back Middle Front 13 % 10 % 13 % 10 % 8 % 10 % 13 % 10 % 13 %

u = 60 mm.

Back Middle Front 10 % 11 % 12 % 9 % 11 % 13 % 10 % 11 % 12 %

Table 5.2. Computed load distribution per pile.

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5.5 Computer code

The computer program was developed on a IBM-AT personal computer, with a 20 MHz 32 bit DSI-780 coprocessor board, using the motorola 68020 CPU and the 68881 FPU chip, with 4 Mb of internal memory. By using the C programming language, a very efficiënt algorithm could be made for the application of general subroutines to the layers in the soil model. By passing pointers to the start address of the concerning data of a layer, a considerable computational overhead can be avoided. Optimization of the accessibility of the layer stiffness matrices is achieved in a similar manner. First of all, the program determines whether a layer stiffness matrix differs from the previous one. If the following conditions are satisfied,

E = E r\ v = v n i i - i i i - i

K = K n K = K n n n s s i 1-1 1 i -1

c p r e s = c p r e s ( 5 . 1 3 ) i 1-1 l '

and the same boundary conditions are imposed, the stiffness matrices are identical, and only the first matrix has to be stored. As a consequence, the homogeneity of the soil also determines the memory requirements. Thereupon, the available ram space is used to store as many unique stiffness matrices as possible. Swabbing the pointers to the start addresses of the matrices makes it possible to solve the appropriate system of equations. When the ram space is not sufficiënt to store all layers, the amount of memory for one layer matrix is kept as a temporary working space. The layers that are not in core, are read from the harddisk when they are needed. It is clear that this will considerably slow down the performance of the program. However, analyses of very large problems can now be executed on relatively small computers, as is demonstrated in the previous

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paragraph. The computations performed for the 9-pile group case took about 4 0 hours, of which 75 percent was occupied by harddisk-io for reading the decomposed layer stiffness matrices. On a coprocessor board with 16 Mb of internal memory, the harddisk would not have been used by the computer program, and the computation would have taken about 10 hours. The same method of accessing the stiffness matrices is used for the pile model in case of a pile group.

Using the configuration described above, the 9-pile group problem required about 90 percent of the available hardware capacity. Because the program that executes the analysis is a separate unit using only Standard C-functions, much larger problems can be studied, by running this'program on a.larger computer. The pre- and post-processing program developed for a personal computer, and containing many system dependent functions, can still be used to prepare the input data and organize the output. However, the interfacing of the two programs, that takes place by means of binary data files, will need a conversion routine to adjust the internal data representation of the different computer systems.

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6 "CLASS A" PREDICTION FOR A SINGLE-PILE TEST AT DELFT

6.1 Introduction

In October 1988 a field test on a 24" steel tube pile was performed at a site in Delft. The test was conducted by Delft Geotechnics in behalf of a research project on dolphins, carried out for Rijkswaterstaat. The fully instrumented pile and the test site were supplied by the Institute for Wind-energy. This institute is interested in the suitability of single-pile foundations for wind turbines.

Before the execution of the test a prediction was made for the response of the pile. The prediction was based on an extensive soil investigation, comprising cone penetration tests, a soil boring, several laboratory tests, and two pressuremeter tests. Response predicted and response measured are compared in this chapter.

6.2 Site layout

In April 198.7, a 12.5 m. long steel tube pile having an outer diameter of 0.601 m. and a wall thickness of 8.8 mm. was driven open-ended to a penetration of 12 m. After installation the top of the soil plug inside the pile was found at 3.75 m. below the ground surface. Four steel U-profiles had been attached to the upper 12 m. of the pile. These profiles serve the purpose of protecting the strain gauges that are mounted on the pile at ten levels. From these gauges, moments can be measured in two perpendicular directions. A cross section of the pile is shown in Fig. 6.1.

Several days before execution of the test, a pit, with a diameter of about 2.6 m., was excavated around the pile up to 1.5 m. below the ground surface in order to simulate the conditions of a dolphin. During the test the water depth in the pit was

- 1 0 1 -

■U 100x60x6

0.601

Fig. 6.1. Cross section of pile.

0.5

1.5

Fig. 6.2. Set-up of test.

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Fig. 6.3. Picture of the test site at Delft.

about 0.6 m. A horizontal load was applied at the level of the flange that was welded to the pile top, see Fig. 6.2. The reaction construction consisted of a sand body, strengthened with two concrete and two leaden blocks, located at 7 m. distance from the pile. The steel tube that connects the hydraulic jack to the reaction construction rested on a thin sand layer. Displacements were measured at the pile top independently from this construction. A picture of the test site is shown in Fig. 6.3.

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6.3 Soil investigation and schematization

A comprehensive soil investigation program was performed, starting with three cone penetration tests, that were executed about 1 m. from the heart of the pile, see Fig. 6.4. It can be seen in this figure that the foot of the pile is embedded in an intermediate sand layer. The upper portion of the pile is surrounded by soft clay and peat layers. Then, a 066 mm. soil boring was made. From the soil boring, volumetric weight and water content were determined. A series of laboratory vane tests was performed. The plasticity index was determined for the clay layers, and the grain size diagram for the sand layers. Two triaxial tests on clay samples taken at 2 and 5 m. depth

Cone resistance (MPa) Friction ratio (%)

0 5 10 0 10

1 1 1 f^~ i \ t t± 7^

t t ^ ^ v] i^r-3 Wl_ :n : >

! i !

| | LTL

M

!

! i i ! |

1 i 1 1 1 1

"*~hpt 1

/

{ 7 *5

f 5 7 t.

__!___ l__

= * i

i

j ! ■

^

i i 1 ;

Fig. 6.4. Cone penetration test and soil boring.

- 104 -

respectively and one simple shear test on a peat sample, taken at 3 m. depth were executed. Finally, a mini-pressuremeter test and a Retrojet pressuremeter test were performed, both to a depth of 8 m. below the soil surface, i.e. 6.5 m. below the bottom of the pit. The results of the soil investigation were presented by Bijnagte (1988abc) .

In the numerical schematization, the soil above the bottom of the pit was disregarded. Therefore, the depth will be measured relative to the bottom of the pit from now on. For the upper 6.5 m. of soil, the pressuremeter tests procure information on both strength and stiffness. Furthermore, an estimate of the initial horizontal stress can be made. The undrained shear strength and the shear modulus were derived directly from the test. The effective horizontal stress however, was estimated arbitrarily to increase at a rate of 4 kPa per meter depth, starting from zero at the bottom of the pit. The pressuremeter test produced unrealistically low values. Below 6.5 m., only the CPT and the soil boring were available. The undrained shear strength was

E (MPal

0 10 20 30 40 50

4

i U

Cu (kPa)

0 50 100 150 200 250

E;

Fig. 6.5 . Modulus of elasticity.

Fig. 6.5 . Undrained shear strength.

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1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22

23

Fig. 6.6. Vertical discretization of pile and soil.

estimated by simply dividing the cone resistance by 15. The shear modulus was assumed 100 c . For the firm sand layer holding the

u foot of the pile, a shear strength was assigned by computing the API failure load for sand with a friction angle of 35°. By equating this value with the API failure load for clay, an equivalent shear strength can be obtained. The modulus of elasticity of the sand was taken 45 MPa. The soil profile that was derived from these data is shown in Fig. 6.5.

For all soil layers the Poisson's ratio was taken to be 0.5. The pile was assumed rough (c =c ), and no tensile stresses were allowed at the pile-soil contact.

From the summary above it may be concluded that for the upper soil layers a rather complete picture is obtained. For the lower layers a considerable amount of data had to be estimated.

- 106 -

However, the influence of inaccurate parameters for these layers on the pile-top response will not be very large.

Since the mesh generator of the computer program is only suitable for circular piles, the pile is idealized to a circular pile having the same bending rigidity as the pile, but a circumference equal to that of the dotted line in Fig. 6.1. This resulted in a pile diameter of 0.66 m. and a wall thickness of 9.2 mm. Again the mesh of Fig. 3.9, scaled to the correct pile diameter, was used for the computations.

The vertical discretization of the soil is shown in Fig. 6.6. On the right hand side the soil profile is shown. In the computer program the input parameters are specified at the upper and lower boundary of the soil layers that correspond with the dotted lines in Fig. 6.5. A linear interpolation is applied between these boundaries to obtain the appropriate constant values for the finite difference layers of the soil model. The subdivision of the soil in finite difference layers and the discretization of the pile are shown in the left part of Fig. 6.6

6.4 Loading sequence

At first, a static loading test was performed. The load was increased up to the point where 0.1% strain occurred somewhere in the pile. This took place in about 10 seconds. Then the load was kept constant for several minutes, after which unloading was allowed. The next stage of the test, consisting of 10 cycles with a maximum load level of 70% of the maximum static load and a minimum level slightly larger than zero, started after one half hour. Each fuil sine took 9 seconds. Between two consecutive cycles, a 10 minute pause was held. Then, two series of 30 9-second cycles were conducted. The first series again at a maximum load level of 70%, the second series at 85 % of the maximum static load. There was no pause between the cycles. Finally, a static test was performed again. This time the force

- 107 -

250

200

150

H (kN)

100

50

/ / / / . ft: I 1 ■■' II : Il :

1' •'

f" / ' ,

/ / / / / / / •• / • / .'

/ / t • /

' ■■'/

t ■ • / / .•' /

/ / / / /

• / / / . • ■ '

/ / / / / / / / •'

/ / f S / / / / / /

/ // / / / /

^ ^ J-J

. ■ ■ / /

■ ' ' /

/ / ' ' / ■• i• / : ''/ ;' // •

First test

Last test

Class A prediction

20 M 60 80

u (mm)

100 120 140

Fig. 6.7. Load-displacement curves.

was increased up to 90 % of the level of the initial static test in 10 seconds. The load was kept constant for 8 minutes, af ter which unloading took place.

6.5 Results of computation and experiment

The measured load-displacement curves of the first and last static load test are shown in Fig. 6.7 together with the "class A" prediction that was made. Although the predicted response of the pile is more flexible than the response actually measured,

- 1 0 8 -

M (kNm)

200 400 600

+ + Measured

Class A prediction

f (kPa)

-100 0 100 200 300 400

12

7 degree for M

8 degree for M

Class A prediction

Fig. 6.8. Bending moments. Fig. 6.9. Soil reaction.

the correspondence is satisfactory. The maximum load level (0.1% strain) is underestimated by about 10 percent.

Bending moments were measured at ten egually spaced levels. For the upper two levels, which are located above the bottom of the pit, the moments can also be computed from the magnitude of the load at the pile top. These values were used to calibrate the moment measurements. The bending moments that were obtained in that manner for a load of 200 kN, are marked with a + in Fig. 6.8. The solid line in that figure represents a 7th degree polynomial that was fitted by means of a least-sguares approximation. The dotted line shows the bending moment profile that was predicted with the present method. As can be expected

- 109 -

Fig 6.10. Picture of the gap after draining the pit. (Photo: Delft Geotechnics)

from the load-displacement curves, both the magnitude and depth of the maximum bending moment are overestimated in the computation. The polynomial that was fitted through the measured moments, can be used to derive the average soil reaction pressure, by applying doublé differentiation. In Fig. 6.9 two polynomial curves are presented. The solid line was obtained by using a 7 degree polynomial for the bending moment profile. The dashed line is based on an 8th degree polynomial. For the intermediate part of the pile, dependence on the degree of the polynomial is small. At the pile tip and to a lesser extent at the pile top, the curves diverge. The general correspondence of both curves with the computed soil reaction (dotted line) is satisfactory, however.

After the last static load test the pit around the pile was

- 110 -

edge of pit

\ \ \ □ \

Fig. 6.11. Plastic integration points of tcp layer.

drained. A clear gap of about 4 cm. was observed at the surface, see picture in Fig. 6.10. A small weight having a diameter of about 1 cm. was lowered into that gap. It took about 2 m. of wire before the weight was stopped. Shortly after the measuring the gap at the surface collapsed partly. In the calculations a gap width of about 3 cm. was predicted for the top layer after unloading, decreasing with depth and vanishing at a depth of about 2.5 m. At maximum load level the computed gap width was 6 cm., vanishing at about 5 m. below the bottom of the pit.

Fig. 6.11 shows the plastic integration points in the upper layer. The edge of the pit is denoted by the dotted line in that figure. According to this picture, the diameter of the pit was chosen just large enough to prevent the soil around the pit from influencing the plastic behavior of the soil around the pile too greatly.

Part of the "class A" prediction was the calculation of one loading cycle at 70 % of the maximum load. The load-displacement curve of the computed load cycle is represented by the dashed line in Fig. 6.12. The computed cycle was executed directly after the initial loading-unloading curve. That is, the implicit assumption was made that no changes had occurred in the soil

- 1 1 1 -

150

100

HlkH)

50

" 0 20 10 60 80

u Imml

Fig. 6.12. Cyclic load-displacement curves.

during the period of rest after the static test. The shape of the load-displacement curve is determined in a great measure by the existence of the gap. The previously formed slot in the soil also accounts for the small amount of energy dissipated during the cycle (3 30 J). A typical load cycle of the measured response for the first series of cyclic loading is represented by the solid line in Fig. 6.12. The influence of the gap cannot be recognized in this figure, which is surprising because during the test water bubbled up in front of the pile, indicating the closure of a gap. The energy dissipated in one measured cycle (1250 J) is substantially higher than predicted. Since the input parameters of the numerical model were determined from an extensive soil investigation program, it would be improper to try to fit the measured response by adjusting these parameters. There is one exception, however. The tension strength of the pile-soil contact was assumed zero, because no specific information was available on this parameter. By varying the tension strength, an impression is obtained of the influence on the load-displacement curve. The

iy '>'

///

' ■ ' , ' ' '

SS

y

/ '

y /

s s

1

.■■

/

sted (Ta —

- 112 -

maximum amount of dissipated energy was computed for a tension strength equal to the undrained shear strength of the upper soil layer (20 kPa). The cyclic response computed for this case is shown by the dotted line in Fig. 6.12. This computation is no longer a "class A" prediction, of course. The increase of energy dissipation is considerable. The damping ratio obtained for the newly computed cycle is 0.10, compared to 0.04 for the original computation, and 0.24 for the experiment. By studying the new curve, the reason for the relatively high increase of damping compared with the original computation can be explained. At the unloading stage the breaking-away of the soil behind the pile occurs just before the minimum load is reached. Unfortunately, at the reloading stage this phenomenon occurs a bit too early, because the tension strength was not fully restored during unloading. The tension strength that was used apparently fits the applied load-amplitude quite well. It does not seem likely that such an optimum tension strength occurred in the test by chance. A possible explanation is, that the rate of propagation of a gap in downward direction, due to progressive collapse, may be limited. At low loading rates this will not influence the pile response. At higher rates, however, the partial loss of stiffness due to gapping will be concentrated in the slower stages of a load cycle. For a sine-shaped excitation as was used in the test, these stages will be just before the maximum and just before the minimum load level is reached. Unfortunately, no slow cyclic tests were performed in this case, so that an experimental verification of this hypothesis cannot be presented here. An impression of the influence of such a deferred gapping was obtained above by assigning a limited tension strength to the pile-soil contact. Although a slow gap propagation could not be modeled, this gave some idea of the consequences of gradual decrease of the tension strength for the amount of dissipated energy. This observation draws the attention to the necessity of gaining insight into the actual value of the tension strength parameter, and a possible rate dependence. Experimental research is required for this purpose.

- 1 1 3 -

HlkNI

0.04 0.06 u lm)

0.08

Fig. 6.13. Last-series of cycles.

In Fig. 6.13 an overview of the measured load-displacement curves of the last series of cycles is shown. Cyclic mobility is observed in this figure, however, only in a limited measure. Striking is the fact that the stiffness of the loops does not decrease during the loading process.

6.6 Conclusions

From the comparison between measured and computed pile response presented above, it may be concluded that the model developed in this study is capable of providing a reasonably accurate prediction of the flexibility of a laterally loaded pile. Especially, since no experience with the lateral behavior of piles existed for the specific site. As for the damping however, the good results obtained for the Haga case, described in chapter 4, could not be repeated. In the previous paragraph the influence of the tension strength parameter is stressed in

- 114 -

this context. This parameter, which is taken into account in hardly any other continuüm model, appears to have a major impact on the damping of the pile-soil system.

- 115 -

PRINCIPAL NOTATIONS

Scalar:

c Shear strength interface ceq Equivalent viscous dashpot coëfficiënt c Undrained shear strength u

D Diameter of pile E Modulus of elasticity E Equivalent modulus of elasticity of pile p

EI Bending rigidity of pile F Interaction force f Distributed load; Yield function G Shear modulus g Plastic potential function H Horizontal force h Height of layer K , K Normal, tangential stiffness interface n s

K Relative soil-pile flexibility factor k Subgrade modulus keq Equivalent spring constant L Pile length M Bending moment P ,P ,P Atmospheric, pore and hydrostatic pressure a p w

q Body force R Radius of pile s Stress; Center-to-center spacing piles t Virtual interface thickness

V u,v,w Displacement in x-, y- and z-direction x,y,z Cartesian coordinates oc Shear strength ratio of interface and soil;

Interaction factor /3 Damping ratio;

Angle between loading and line joining pile centers

- 116 -

y Pseudo shear strain interface e Strain; Pseudo normal strain interface; error measure e Lost-contact strain interface c

9 Pile top rotation A. Average displacement pile over layer thickness;

non-negative scalar v Poisson's ratio p Pile top displacement er Average stress over layer thickness;

Normal stress interface er ,cr Major and minor principal stress er Tension strength interface T Shear stress interface u Frequency

Vector and matrix:

a Displacement vector A Global stiffness matrix A e Element stiffness matrix B Strain interpolation matrix C Layer-layer coupling matrix D Elastic constitutive matrix E Error equilibrium vector H Displacement interpolation matrix K Pile-soil interaction matrix P Pseudo load vector plastic strains Q Body force vector S Boundary load vector T Transformation matrix t Boundary tractions

- 117 -

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- 118 -

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SUMMARY IN DUTCH (SAMENVATTING)

NUMERIEK MODEL'VOOR LATERAAL BELASTE PALEN EN PAALGROEPEN

In dit rapport wordt een numeriek model voor het gedrag van lateraal belaste palen en paalgroepen beschreven. Deze bijzondere belastingsrichting doet zich met name voor bij offshore constructies. Daarnaast kunnen windmolens en ducdalven genoemd worden als toepassingsgebieden, terwijl de horizontale exitatie van paalfunderingen tijdens aardbevingen kan worden opgevat als een aanverwant probleem.

Het model dat wordt ontwikkeld, is gebaseerd op een continuüm benadering van de grond. Deze aanpak is noodzakelijk om de interactie tussen verschillende palen in een groep te kunnen beschrijven. Het probleem van de enorme afmetingen van het stelsel vergelijkingen dat voortkomt uit het gebruik van drie-dimensionale numerieke modellen wordt ondervangen door gebruik te maken van een "substructuring" techniek. Hierbij wordt het paal-grond systeem onderverdeeld in verscheidene kleinere eenheden, die een voor een kunnen worden geanalyseerd en iteratief gekoppeld. Op het eerste niveau wordt het paal-grond systeem opgedeeld in een paal systeem en een grond systeem. Op het tweede niveau, wordt de grond onderverdeeld in een aantal elkaar beïnvloedende lagen. In de analyse worden deze eenheden gekoppeld door te voldoen aan evenwicht en compatibiliteit.

In hoofdstuk 2 worden de veronderstellingen en vergelijkingen gepresenteerd, die ten grondslag liggen aan het model. In dit stadium is het model beperkt tot elastische horizontaal gelaagde grond, die een enkele paal bevat. De resultaten van het model worden vergeleken met andere numerieke oplossingen.

In hoofdstuk 3 wordt het model uitgebreid naar elastoplasticiteit voor cohesief grondgedrag. Een speciaal interface element wordt geïntroduceerd om slip van grond langs de paal omtrek te kunnen beschrijven. Voorts wordt aandacht besteed aan de eisen die aan de mesh voor een grondlaag worden gesteld.

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Een veld-test wordt geanalyseerd, en de resultaten van de berekening worden vergeleken met een eindige elementen oplossing.

Het verbreken van het contact tussen paal en grond achter de paal speelt een belangrijke rol in het gedrag van lateraal belaste palen. In hoofdstuk 4 wordt dit verschijnsel gemodelleerd door gebruik te maken van het interface element dat in hoofdstuk 3 is geïntroduceerd. Een vloeiregel wordt ingevoerd voor dit element om de aanwezigheid van een "gap" achter de paal mogelijk te maken. De toepassing van dit gap-element wordt gedemonstreerd aan de hand van twee veld-tests.

De analyse van paalgroepen wordt behandeld in hoofdstuk 5. De interactie tussen palen in een groep, voor zowel elastische als elastoplastische grond, wordt direct berekend met de huidige methode. Een vergelijking met andere modellen en met proeven wordt gemaakt.

In hoofdstuk 6 wordt een voorspelling beschreven, die met behulp van het huidige model is verricht voor een laterale belastingsproef uitgevoerd in Delft. De resultaten van de predicties, die voor de uitvoering van de proef zijn gedeponeerd, worden vergeleken met de gemeten waarden.