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INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech., 2000; 24:1109}1138

General elastic analysis of piles and pile groups

K. J. Xu and H. G. Poulos

Department of Civil Engineering, University of Sydney, NSW 2006, AustraliaCo w ey Geosciences Pty. Ltd., and Professor of Civil Engineering, University of Sydney, Australia

SUMMARY

This paper describes the development of a boundary element analysis for the behaviour of single piles andpile groups subjected to general three-dimensional loading and to vertical and lateral ground movements.Each pile is discretized into a series of cylindrical elements, each of which is divided into several sub-elements. Compatibility of vertical, lateral and rotational movements is imposed in order to obtain thenecessary equations for the pile response. Via hierarchical structures, 12 non-zero sub-matrices in a globalmatrix are derived for the basic in#uence factors.

Solutions are presented for a series of cases involving single piles and pile groups. In each case, thesolutions are compared with those from more simpli"ed existing pile analyses such as those developed byRandolph and by Poulos. It is shown that for direct loading e! ects (e.g. the settlement of piles due to verticalloading), the simpli"ed analyses work well. However, for &o! -line' response (such as the lateral movementdue to vertical loading) the di! erences are greater, and it is believed that the present analysis gives morereliable estimates. Copyright 2000 John Wiley & Sons, Ltd.

KEY WORDS: pile groups; 3-D coupled problems; elastic analysis; boundary element method; in#uencefactors

1. INTRODUCTION

In general, a pile group may be subjected to simultaneous axial load, lateral load, moment, and

possibly, torsional load. Various numerical approaches have been employed to estimate the pile

group response to such loadings, and these may be broadly classi"ed into the following

categories:

1. Methods based on the theory of subgrade reaction, including equivalent bent (frame)

analyses that reduce the pile group to a structural system but take some account of the e! ect

of the soil by determining equivalent free-standing lengths of the piles [1}3].

*Correspondence to: K. J. Xu, Department of Civil Engineering, University of Sydney, NSW 2006, AustraliaResearch assistant.Senior Principal.

Contract/grant sponsor: Australian Research Council

Received 8 September 1998Copyright 2000 John Wiley & Sons, Ltd. Revised 13 July 1999

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2. Hybrid analyses that combine load transfer analyses for single pile response and elastic

theory to estimate pile}soil}pile interaction [4}6].

3. Elastic-based analyses that use elastic theory for both single pile response and pile}soil}pile

interaction [7}10].

4. Finite element methods that include a number of ways to analyse pile group deformations

[11}13].

Computer programs for the analysis of pile groups vary in the type of approach used and in the

sophistication of treatment of di! erent aspects of group behaviour. Three programs which have

been used for pile group analysis are PGROUP [10], DEFPIG [7,14] and PIGLET [9], and

these have been compared by Poulos and Randolph [15]. All three programs are based on an

elastic continuum analysis, although DEFPIG can also be extended into the non-linear range by

specifying limiting values of skin friction and lateral pressure along the piles.

Although these programs have been used widely, they all involve a number of simpli"cations

and idealizations, owing to the limitations of computer technology at the time of their develop-

ment.

For these and other similar programs (excepting "nite element analyses), there are simpli"ca-

tions and limitations with respect to at least six aspects:

1. Interaction: Pile}to}pile interaction is widely considered, but a more accurate method

requires consideration of pile element to pile element interaction.

2. Elements: Full cylindrical and/or annular elements are usually used, but the stress distribu-

tion on an element is generally not uniform around the pile circumference for cases of

general loading.

3. ¸oading}deformation coupling: Normally the axial, lateral and torsional responses of the

piles and treated separately, i.e. de-coupled. However, pile}soil interaction is a three-

dimensional problem, and each of the load components has deformation-coupling e! ects,

e.g. vertical loading induces lateral movements as well as vertical movements.

4. Pile cross-section: Normally, the pile cross-sections are taken as uniform. However, for

some cases, such as piles with defects, the cross-sections of at least some of the piles are not

uniform.

5. ¸oadings: Loadings applied to pile groups are usually applied directly to the piles (&on-pile'

loading), but sometimes loadings are induced by ground movements (&o! -pile' loadings),

such as those arising from loading caused by adjacent structures, or from nearby tunnel

construction or excavations.

6. Global pile foundation: Rather than a single group, multiple individual piles and/or pile

groups may exist simultaneously in many engineering projects.

All the e! ects mentioned above may be important if the behaviour of pile foundations is to be

modelled accurately. Because of rapid developments in computer technology, the use of accurate

algorithms can now take precedence over earlier concerns about computer speed and capacity.This paper describes the development of a computer program that eliminates the above limita-

tions and is therefore more general as a tool for the elastic analysis of the behaviour of piles and

pile groups. The accuracy of the program is tested via comparisons with existing solutions. Some

aspects of pile behaviour which cannot be computed by conventional approaches are also

described.

1110 K. J. XU AND H. G. POULOS

Copyright 2000 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech., 2000; 24 :1109}1138

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Figure 1. Schematic global foundation system.

2. BASIC PRINCIPLES

2.1. Introduction

Using the principles of three-dimensional boundary element method, the &analysis platform' is set

up for a half-space with a global foundation system, as illustrated in Figure 1. The analysis

considers the overall foundation performance of multiple single piles and pile groups and can

incorporate the e! ects of such factors as defective piles, soil movements, general on/o! pile

loadings, etc., at the same time.

In the governing equation, a global matrix is derived, which is composed of several-matrices.

Through the introduction of the concept of hierarchical structures and basic in#uence factor

matrices, a set of sub-matrices is developed both for the presentation of the analysis and for

computer program implementation. The development of the global matrix is described below.

2.2. Assumptions and terminologies

The following assumptions are employed for the analysis:

1. The soil is assumed to be an ideal homogeneous isotropic elastic weightless half-space,having elastic parameters E

and

that are not in#uenced by the presence of the piles.

However, it is possible to consider, in an approximate manner, the soil mass as a non-

homogeneous continuum.

2. Piles are assumed to be circular in cross-section and made of a homogeneous isotropic

elastic material with two unchangeable elastic parameters E

and

. The piles are vertical

and perfectly rough. Where the diameter of a pile changes abruptly, no account is taken of

local stress concentrations.

3. The deformation of the piles and soil mass are elastic and the superposition principle is

valid.

4. Two kinds of elements are considered at the soil}pile interface, soil elements and pile

elements. These boundary elements are meshed in partly cylindrical or annular surfaces. The

distributions of the stress components on the boundary elements at the pile}soil interface

are assumed to be uniform over each element.

ELASTIC ANALYSIS OF PILES AND PILE GROUPS 1111


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Figure 2. Elements and stresses of a pile.

5. To simplify the compatibility and equilibrium equations for pile heads and caps, the load

and displacement components at the top of each pile are assumed to be uniform and equal

to the average value.

6. The pile caps above each pile group are assumed to be rigid and not in contact with the soil

surface.

7. There is no slip and gap at the pile}soil interface, i.e. the soil element and the pile element are

in contact at the soil}pile interface.

The term &stress' above and below is used for the load over the area of pile}soil element whose

shape is either plane (at the pile base elements) or cylindrical (at the pile side elements).

A diagrammatic illustration of elements and stresses on a pile is given in Figure 2. Each element

has three stress components (p

, p

, p

) acting on it in the Cartesian co-ordinate system. It should be

noted that the elements, in fact, are subjected to shear tractions,

and

, and a normal traction

in a polar co-ordinate system. However using the Cartesian co-ordinate system it is easier to set up

compatibility and equilibrium equations in the pile group analysis. The computed stresses in the

X, >, Z directions are translated into the real tractions by co-ordinate transformation.

For &structural analysis', each pile is treated as a one-dimensional column which is able to

compress or extend axially, bend in the X}Z and >}Z planes and twist in the X}> plane. E! ects

of Poisson's ratio of piles are involved in torsional behaviour since the shear modulus of the piles

is determined by E

and

. Although the piles are treated as one-dimensional columns, three

basic deformations and considered simultaneously, i.e. the axial, bending and torsional deforma-

tions of piles are produced by the three components of stresses (p ,

p ,

p) on the elements. Forexample, p

results in axial and bending deformations while p

and p

produce bending and

torsional deformations.

In developing this method, stresses, loads and displacements in Cartesian co-ordinates are

assumed to obey the right-hand &srew' rule. In numbering piles, priority is given to piles within

pile groups and then to individual piles.



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where is the zero sub-matrix or vector, A the sub-matrix describing pile and soil behaviour at all

elements ("SIF#PIF), SIF the sub-matrix of displacements due to stresses at soil elements,

PIF the sub-matrix of displacements due to stresses at pile elements, B the sub-matrix of pile

element displacements due to pile tip displacements, D the sub-matrix of pile element displace-

ments due to pile head loads, G the sub-matrix of pile cap loads due to pile cap displacements by

cap-tie-cap beam forces (to allow for pile caps jointed by tie beams), H the sub-matrix of pile caploads due to pile head loads, O the sub-matrix of pile head displacements due to pile element

stresses, P the sub-matrix of pile head displacements due to pile tip displacements, Q the

sub-matrix of pile head displacements due to pile cap displacements, R the sub-matrix of pile head

displacements due to pile head loads, S the sub-matrix of pile head loads due to pile element

stresses,< the sub-matrix of pile head loads due to pile head loads, >

the vector of element stress

o! sets, (">

#>

), >

the vector of pile element displacements due to head loads on

individual piles, >

the vector of pile element displacements due to external soil displace-

ments/stresses/forces, >

the vector of loads on pile caps, >

the vector of pile head loads on

individual piles, X

the vector of pile}soil stresses, X

the vector of pile tip displacements, X the

vector of pile cap displacements and X the vector of pile head forces at the pile cap.

The global foundation is assumed to contain ncap pile groups with any con"guration, npils

non-identical piles which have nipsum individual piles and ncpsum piles in pile groups, and ntotelements of any size which may include partly cylindrical elements on shafts and partly ring

elements on both bases and discontinuities. Then, a coupled global matrix with [3 ntot#

6 npils#6 ncap#6 ncpsum] dimensions is set up, and the following unknowns can be solved:

1. 3 ntot pile}soil element stresses X,

2. 6 npils pile tip displacements X,

3. 6 ncap pile cap displacements X.

4. 6 ncpsum pile head loads for grouped piles X.

After the pile}soil stresses X and pile tip displacements X

are computed from Equation (1),

the head displacements of individual piles X can be determined from the following equation:

X"[O] X#[P] X#[R] > (2)

As a special case, if only individual piles exist, Equation (1) can be reduced to a much simpli"ed

form, as shown in (3)

A

S

B

X

X

"

>> (3)

2.3.2. Sub-matrix classi xcations and known vectors. In the global matrix, there are 12 non-zero

sub-matrices, i.e. SIF, PIF, B, D, G, H, O, P, Q, R, S,

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4. The sub-matrices H, S, < relating pile head loads, loads on the pile caps, or stresses on piles,

are established by force equilibrium.

The detailed expressions for the various sub-matrices are set out in Appendix A.

There are four types of known vectors, i.e. >

, >

, >, >

. The vector of element displacements

due to pile head loads > is determined by structural analysis. Both vectors of pile cap loads> and individual pile head loads >

have six components. Element displacements due to external

soil displacements/stresses/forces are incorporated into the vector>

, which will be described in

detail later in the paper.

2.4. Numerical analysis of the sub-matrices in the global matrix

2.4.1. Basic in yuence factor matrix (BIFM). The term &in#uence factor' in this paper is used

to de"ne the response of one mechanical parameter (such as load/stress/displacement) caused

by imposing another unit mechanical parameter. The concept of in#uence factors was intro-

duced into pile foundation analysis by Poulos et al. [17] for de-coupled cases. However,

it is di$cult to set up a general 3-D fully coupled pile analysis if the single de-coupled in#uence

factors are considered, since the in#uence factors are coupled and are tensors rather thanscalar values. For any two soil elements i and j, for example, there are nine stress}displacement

in#uence factors I

in a general 3-D fully coupled problem, for example, I

"soil dis-

placement in k direction at element i due to a unit stress in l direction on element j (l, k"1,2,3).

The matrix which contains all components of the in#uence factors is called here the basic

in#uence factor matrix (BIFM). The above in#uence factors I

, for example, are composed

of a 33 basic in#uence factor matrix (BIFM) of stress}displacement response. It is found

that an e$cient way to set up the sub-matrices in the global matrix is to consider the basic

in#uence factor matrices (BIFMs) as &basic elements' in sub-matrices, i.e. instead of single

in#uence factors, the sub-matrices are formed by assembling the individual basic in#uence

factor matrices (BIFMs). This important concept simpli"es the setting up of the general 3-D

fully coupled pile analysis. Corresponding to the sub-matrices, there are 12 BIFMs which are

de"ned in Appendix A. Because of space limitations, the expressions of BIFMs are derived inother paper [18].

2.4.2. Hierarchical structure of sub-matrices in the global matrix. Because of the full coupling of

the six stress/displacement components with element}element interaction, expressing these sub-

matrices in the global matrix in full is cumbersome. However, it is found that every sub-matrix

has a hierarchical structure, i.e. a sub-matrix can be divided (or &layered') into some smaller

sub-matrices. The &innermost layer' of a sub-matrix is composed of several basic in#uence factor

matrices (BIFMs), and the outline of the hierarchical structure is as follows:

A"

a

2 a

2 2 2

2 a

2

a

2 a

Na

"

b

2

b

b

Nb

"

c

2

c

c

-

-



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The global matrix A outlined above is assembled from sub-matrices a

(i1"1, m1; j1"1, n1).

The sub-matrix a

is formed from diagonally arranged matrices b

(i2"1, m2) where the

matrix b

is obtained from basic in#uence factor matrices c

(i3"1, m3) which are arranged

vertically.

Each of these BIFMs has clear physical meaning and can be evaluated easily via short

computer subroutines. The sub-matrices in the global matrix can be conveniently built up bycalling the corresponding subroutines. The hierarchical structures of the sub-matrices are pre-

sented in more detail in Appendix A.

2.4.3. Soil element movements. The soil movements in this paper are evaluated via soil in#uence

factors through the integration of Mindlin's equations [16] (displacement in l direction at point

i due to a unit force in the k direction at point j in half-space, l, k"1, 2, 3) on a soil element area. It

should be emphasized that there are nine soil in#uence factors which form a BIFM, because the

loads and deformations are fully coupled in three dimensions.

It is found that most of the computer running time is consumed in generating the "rst type of

sub-matrix, which involves numerical integration of Mindlin's equations. It is obvious that the

result is increasingly insensitive to the number of integration points as the distance between two

elements is increased. Therefore, the &changeable integration point' method has been introduced

here, and involves the use of adjustable integration points, with less integration points being used

as the distance between two elements becomes greater.

2.4.4. Pile element movements. The deformations due to axial and horizontal forces,

moments and torsion in pile elements are coupled, and the diameter of any element

may be arbitrary. Calculation of the second kind of sub-matrix is therefore more compli-

cated than the de-coupled model presented by Poulos [19]. Considering the advantages

of do-loop functions in computation, the expressions for load}deformation characteristics

of pile elements are derived via a nested recursion form which makes the computer program

shorter and easier than in previous methods. Suppose a pile is divided into a number of

cylindrical elements, then, as an example, the nested recursion form for rotation of pile element i isas follows:

"

#¹(¸

/2)

GJ

(4)

where

and

are the rotations of pile elements i and i!1, respectively, ¹, ¸

, G

, J

are

torsion, length, shear modulus and polar moment of inertia, respectively, of element i of the pile.

Derivations of the pile behaviour equations are relatively cumbersome. In de-coupled situations,

Poulos [19] derived pile displacement equations that are expressed by a product of a summation

matrix AD and a pile compression matrix FE. However, following this concept in the full-coupled

situation will be very complicated. The method proposed in this paper uses &adding operations'

instead of &multiplier operations', i.e. pile displacement behaviour is decomposed and expressedby two sub-matrices (sub-matrix of displacements due to stresses at pile elements PIF, and

sub-matrix of pile element displacements due to pile head loads D for pile groups) and a vector

(the vector of pile element displacements due to pile head loads >

for individual piles). The

matrices and vector are relatively simple to express and may be derived from the principles of

structural analyses.



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2.5. Ground movements

One of the most attractive advantages of the proposed analysis is that externally imposed

movements are very easily incorporated into the governing equations if the distributions of these

movements are known. These movements are taken to be &free-"eld' ground movements, i.e.

independent of the presence of the piles, and it is assumed that the principle of superposition can

be applied. The ground movements can then be absorbed into the vector >

in the governingequation (1).

The e! ects of external ground movements can be considered in two ways: as directly imposed

displacements based on the known ground movements, or as induced stresses based on the

known ground movements. In the "rst case, the free-"eld soil movements at the pile}soil interface

are indicated by a vector u

, i.e.

>

"u

(5)

In the second case, the induced stresses are represented by a vector

. The corresponding soilmovements due to the soil stresses are the product of the soil in#uence factor matrix [SIF] and

the soil stress vector

, which is assumed to exhibit an elastic response, i.e.

>

"[SIF]

(6)

2.6. Non-homogeneous soil

In the elastic continuum theory, the soil sti! ness is directly related to Young's modulus of the soil.

If a non-homogeneous soil mass is considered, the following approximation is used by Poulos [7,

19] to evaluate the stress}displacement in#uence factors of soil elements by Mindlin's equation

when the soil variation between adjacent elements is not large or a soil layer is not overlain by

a much sti! er layer:

E

"0.5(E

#E

) (7)

where E

and E

are the values of Young's modulus of the soil at elements i and j, respectively.

As shown in Appendix A, the element (in#uence factor) of BIFM of soil element I

in the

sub-matrix SIF is I

(soil displacement is k direction at element i due to a unit stress in

l direction on element j, k, l"1, 2, 3). It is convenient of I

can be expressed as the value of the

in#uence factor for a unit value of Young's modulus, IF

, divided by the average soil Young's

modulus, E

, i.e.

I

"IF

E

(8)

The solution for the displacement within a uniform semi-in"nite soil mass caused by a pile can

also be used to estimate the displacement of a pile founded within a layer of a system of m

layers of di! erent soils which lie below the pile tip [17]. The displacement of the pile is given approxim-

ately as

I

"

IF

!IF

E

#IF

E

(9)



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Figure 4. Flow chart for program GEPAN.

where IF

"IF

(in Equation (8) with following de"ned j) for layer n, E

"E

(in

Equation (7) with following de"ned j ) for layer n, j is the soil element, if n"1; or refers to the pile

axis at the level of the top of layer n, if n'1.

3. COMPUTER PROGRAM GEPAN

To implement the above analysis, a FORTRAN computer program called general pile analysis

(GEPAN) has been developed.

After reading the source soil and pile data, GEPAN automatically generates a group of uni"ed

geometry and control parameters. The movements of pile and soil of shaft, base and discontinuity

elements can be calculated via appropriate subroutines. Each sub-matrix in the global matrix is

assembled by combining the relevant BIFMs, which are formed by relatively simple subroutines.

Discontinuity elements (annular elements at diameter changes within a pile) can be automatically

generated and labeled. The main chart #ow for program GEPAN is shown in Figure 4.

The cylindrical or annular boundary elements are transformed into rectangular elements by

mathematical transforms. Then the simple rectangular integration method, where the rectangularelements are divided into JJ smaller rectangles, is used for integration of Mindlin's equation.

GEPAN automatically eliminates the singularity of the integration for the displacements of an

element from loads of the element itself via appropriate mathematical processes.

Because the proposed analysis considers element}element interaction within a global founda-

tion system, the global matrix in GEPAN can be very large. GEPAN uses dynamic memory

storage technology for more rational and e! ective use of the computer memory. A matrix with

changeable dimensions is allocated only when needed and de-allocated as soon as it is no longer

needed.



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Figure 5. Comparison of the top settlements S of axially loaded single pile.

4. EVALUATION OF ANALYSIS

Comparisons have been made between the results of the program GEPAN and the widely used

pile analysis programs DEFPIG [14] and PIGLET [9], as well as with other published solutions.

In these comparison, results obtained by the authors using these programs are labelled with the

program names, while results from published papers are indicated by the appropriate source andreference number. In pile parametric studies, two important parameters which describe the

behaviour under axial and lateral loadings, respectively, are the pile sti! ness factor K and the

pile-#exibility factor K

, where K is de"ned as ratio of Young's moduli of pile and soil (E

/ E)

and K

is a dimensionless measure of the #exibility of the pile relative to the soil (E

I

/ E/ ̧ )

[17]. Some of these comparisons are presented below.

4.1. Single pile

Figures 5 and 6 compare various solutions for the top settlement S and distribution of shear stress

p along the shaft, respectively, for a single pile under an axial load. The comparisons for a single

pile with horizontal loading and moment among results via using GEPAN, DEFPIG, PIGLET

and the "nite element analyses from Hull [20] are shown in Figures 7 and 8, where I is de"nedas the in#uence factor for displacement caused by moment for constant E

, and I

, I

are the

in#uence factors for rotation caused by moment and horizontal load, respectively, for constant

E (I

"I

from the reciprocal theorem) [17]. A further comparison, for the rotation of a single

pile subjected to torsion, is shown in Figure 9. Although the principles of the programs GEPAN,

DEFPIG and PIGLET are quite di! erent, the results agree well in most cases. It should be

noted that the solutions from PIGLET for a very compressible axially loaded pile, and for

a rigid laterally loaded pile, are inaccurate because of the inherent assumptions involved in that

analysis [9].



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Figure 6. Comparison of axial load distribution along single pile.

Figure 7. Comparison of the in#uence factor I of horizontally loaded single pile.

4.2. Pile group

Figures 10 and 11 show group reduction factors (group reduction factor"ratio of group sti! ness

to the sum of the sti! ness of individual piles [17]) for a 2 and a 3 pile group, for vertical and



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Figure 8. Comparison of the in#uence factor I

of horizontally loaded single pile.

Figure 9. Comparison of the top rotation of torsionally loaded single pile.

horizontal loading, respectively. Figure 12 compares the load distribution, expressed in terms of

the ratio of pile head load P and average pile head load P

, in a 3 pile group subjected to

a vertical load. The agreement is generally good, although GEPAN gives lower values of the

group reduction factors than DEFPIG or PIGLET for horizontal loading. For a 3 pile group



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Figure 10. Comparison of the reduction factors of vertically loaded pile groups.

Figure 11. Comparison of the reduction factors of horizontally loaded pile groups.

subjected to torsion, Figure 13 shows the load distribution, expressed as the ratio of pile head

torsion load¹

and average pile head torsion load¹

, by GEPAN and PIGLET. GEPANconsiders torsional interaction whereas PIGLET ignores torsional interaction [9] and therefore

does not distinguish between the various piles. Figure 14 illustrates a practical problem presented

by Poulos and Davis [17]. Four methods are compared in Table I in which

,

and are thevertical head displacement of pile no. 3, the horizontal head displacement of the piles, and the

head rotation of the piles, respectively. Good agreement is observed among the three analyses



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Figure 12. Comparison of the load distributions of vertically loaded in 3 group.

Figure 13. Comparison of the load distributions of torsion loading of 3 group.

which consider pile}soil}pile interaction, whereas the equivalent-bent method (which represents

the pile group as a structural frame) gives rather di! erent results.

4.3. Piles embedded in non-homogeneous soils

Figure 15 compares the head settlements of end-bearing piles with di! erent base Young's moduliE

using programs GEPAN and PIES [19]. The latter analyses the axial movement of piles by



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Figure 14. Pile groups considered in comparison of methods.

Table I. Comparison methods of the general analysis on a pile group.

Quantity Equivalent-bent Elastic continuum PIGLET GEPANanalysis [17] analysis [17]

<

(kN) 67.2 50.5 55.7 54.0<

(kN) 200 163.4 155 156.0<

(kN) 332.8 386.1 389.3 390.0H

(kN) 66.8 75.9 80.4 73.7H

(kN) 66.7 48.2 39.3 50.9H

(kN) 66.6 75.9 80.4 75.4

M (kNm) !6.2 !39.6 !42.0 !38.5M

(kNm) !6.2 !23.5 !16.3 !26.1

M

(kNm) !6.2 !39.6 !42.0 !38.6

(mm) 17.5 14.8 9.9 10.8

(mm) 8.9 11.8 11.4 10.5 (rad) 0.00581 0.00248 0.00242 0.00241

consideration of the interaction of annular ring elements at the pile}soil interface. There is a good

agreement between the solutions. Figure 16 compares the computed head settlements of a #oat-

ing-pile embedded in an underlying rigid base below the soil layer. Solutions from GEPAN, PIES

and DEFPIG all agree well. The settlement interaction between two piles in which the soil pro"leis layered is illustrated for two cases in Figure 17, using Equation (7) for the non-homogeneous

soil pro"le. The agreement between settlement interaction factors from the present approach, and

from the solution of Mylonakis et al. [21], is again good (the settlement interaction factor is the

ratio of the additional settlement caused by an adjacent pile to the settlement of an isolated pile

under an equal load).



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Figure 15. Head settlements of an end-bearing pile.

Figure 16. Head settlements of a #oating-pile embedded an underlying rigid base.



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Figure 17. Interaction factors for pile groups embedded in layered soils.

4.4. Numerical e zciency of GEPAN

It is found that, for a typical pile length, when the pile is divided into 20 or more elements

vertically and 4 or more circumferentially, the numerical results for GEPAN become almost

independent of the number of elements.The technique of changeable integration points in GEPAN results is substantial savings in

computer time, without sacri"cing accuracy, as shown in Figure 18 and Table II. For vertical

loading with a symmetrical con"guration of piles, three cases are given in Figure 18(a). In case I,

99 integration points are used, regardless of the distance between two elements. For cases II

and III, changeable integration points are used. The integration point depends on the distance



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Figure 18. Numerical e$ciency on dynamic integration point methods.



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Table II. Comparison of unchangeable and changeable integration points.

Number Case I Case II Case IIIof piles

S

(mm) Time, t

S

(mm) Time, t

S

(mm) Time, t

1 4.7123 19 4.7098 4 4.6394 24 2.7977 501 2.7969 118 2.7763 569 2.1169 3643 2.1164 1722 2.1042 1559

Note: ¸"20m, d"1 m, s"3 m, E

"30 000 MPa, E"20 MPa,

"0.3,

"0.5.

S

"Top vertical movement of piles for case i , t computer running time for case i.

Computer: CP, Pentium 200, 160Mb RAM.

between two elements over pile diameter X/ d, with a smaller of number of integration points

being taken as the distance between two elements increases. In case II, for example, 99 points

are used if X/ d"0, 88 if X/ d"2, 2, down to 22 if X/ d*8. It is seen that, in case II,

GEPAN saves about 75 per cent computer running time as compared with the time for case I,

while the relative error is only about 0.05 per cent.

5. SOME APPLICATIONS

The GEPAN analysis appears to have potential application to a variety of pile analysis problems.

While some of these applications will form the subject of future papers by the authors, three

simple applications are described below to illustrate the utility of the analysis.

5.1. O w -line e w ects of piles

&O! -line' response exists between piles. For example, two piles with vertical loads may also

experience some horizontal movement. Since GEPAN is a fully coupled load}deformationprogram, it can, as a matter of course, describe the o! -line e! ects. Figure 19 shows an example of

the horizontal pile head movements due to vertical load on a group of to piles. It is found that the

o! -line horizontal movements are signi"cant for axially loaded piles that are highly compressible

and closely spaced. It should also be noted that the horizontal o! -line movement in case II

(anti-symmetrical) is larger than one in case I (symmetrical). As a further example, for two

horizontally loaded piles, Figure 20 shows the torsional twist at the pile head due to the

horizontal loads. When the angle (between the direction of loading and the piles) is about 453,the o! -line twist reaches a maximum value.

5.2. Di w erence between positive and negative interaction factors

The concept of intraction factors is widely applied to pile foundation problems [17,20,21}23] tocompute group e! ects on deformations. The interaction factor , which is de"ned as the ratio of additional settlement caused by adjacent pile and settlement of pile under its own load [17], is

generally determined for two piles with loads in the same direction (symmetrical). If two loads are

opposite (anti-symmetrical) this interaction factor is generally assumed to have the samemagnitude but to be opposite in sign. However, the magnitude of the positive and negative



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Figure 19. O! -line e! ects of vertically loaded piles.

Figure 20. O! -line e! ects of horizontal loaded piles.



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Figure 21. Positive and negative interaction factors of vertical loaded piles

in#uence factors are, in fact, di! erent, as can be discovered indirectly from some two-dimensional

analyses [17,21]. The GEPAN analysis reveals that the interaction factor in the anti-symmetrical

case is larger, especially when the two piles are closely spaced, Figures 21 and 22 show this for the

interaction factors

and

which apply for a free-head pile subjected to horizontal load only

[17]. This phenomenon is mainly attributed to the e! ects of the di! erent o! -line responses, as

already discussed in relation to Figure 19.

5.3. Interaction between two pile groups

GEPAN can analyse the interaction behaviour of multiple groups of piles. Group interaction

factors can be de"ned in a similar manner to the interaction factors for individual piles. For

example, the group interaction factors (

) can be de"ned as the ratio of additional vertical

(horizontal) movement caused by an adjacent pile group over vertical (horizontal) movement of

pile group under its own vertical (horizontal) load. Figures 23 and 24 show group interaction

factors for two 2 groups, for the "xed head condition, for vertical and horizontal loadings,

respectively. As can be seen, the interaction factors between the two groups have a similar trend to

those for two individual piles.

5.4. Other potential applications

The method presented in this paper has the potential to analyse complex pile foundation

problems and is capable of being developed to consider more general pile foundation analyses

such as the following:



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Figure 22. Positive and negative interaction factors of horizontal loaded piles

Figure 23. Group interaction factor

of two 2 vertical loaded groups



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Figure 24. Group interaction factor

of two 2 horizontal loaded groups

(1) Ground}pile interaction problems: In practical engineering problems, ground movements,

such as those from soil shrinkage/expansion, tunneling, excavation, embankment construc-

tion, and installation of adjacent piles, may a! ect the pile behaviour. If the directly imposed

&free-"eld' ground displacements or stresses are known, the complex ground}pile interac-

tion problems may be assessed by employing Equation(5) and/or Equation (6).

(2) Behaviour of pile groups containing defective piles: At present, most available computer

programs appear to have no capability for analysing the e! ects of defective piles. The

program GEPAN may be used to examine the e! ects of various defects and numbers of

defective piles on a pile group because each pile in the group can be speci"ed as having

di! erent properties and/or geometry.

(3) ¸oad-transfer analysis: A general load-transfer analysis may be carried out by representing

the sub-matrix SIF by the &t-z1 and &p-y1 concepts. In this case, only diagonal elements are

considered in the matrix SIF.

(4) Non-linear and elastoplastic analysis: A general non-linear and elastoplastic anal-

ysis for piles and pile groups may be developed by modifying the matrices SIF andPIF.

(5) Pile foundation}structure analysis: Since multiple pile-groups can be analysed by this

analysis, it can be extended further to consider pile foundation}structure interaction if theglobal matrix is expanded into a &global pile foundation}structure matrix' by assembling

the pile foundation matrix presented in this paper and the structure matrix developed via,

for example, a conventional direct sti! ness method.

At present, some of above applications are being developed and will be published in future

papers.



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6. CONCLUSIONS

This paper presents a general three-dimensional load}deformation analysis for pile foundations

using the boundary element method.

For a &global' foundation which includes multiple pile groups and the e! ects of soil movements,

the analysis uses the concepts of hierarchical structures and a basic in#uence factor matrix(BIFM), to derive the sub-matrices for the global matrix. The analysis is implemented via

a computer program, GEPAN.

Through extensive comparisons, it is shown that, for direct loading e! ects, the three di! erent

pile analyses, PIGLET, DEFPIG and GEPAN generally agree well, although each of these

approaches has a di! erent basis.

However, the present analysis is more general and allows proper considerations of &true'

three-dimensional situations, as it considers all 6 load components and 6 displacement compo-

nents for each of the piles, and also incorporates full coupling e! ects. Arbitrary dimensions of

piles and general 3-D soil movements can be analysed, enabling problems involving both direct

loading and ground movements to be analysed. GEPAN exhibits a high numerical e$ciency and

the method of changeable integration points enables signi"cant savings to be made in computa-

tion times.The present analysis reveals some subtle phenomena, which conventional analysis cannot, such

as o! -line e! ects of piles, di! erent interaction factors for compression and uplife loadings,

interaction among pile groups, and interaction within groups subjected to torsion. It is also

capable of being extended to cover more complex ground}pile interaction problems, such as

those involving ground movements and defects within some of the piles in a group.

ACKNOWLEDGEMENTS

The work described in the paper forms part of a project on the behaviour of pile groups containing defectivepiles, which is supported by a grant from the Australian Research Council.

APPENDIX A: HIERARCHICAL STRUCTURES AND BIFMs

In the governing equation the global matrix has 12 non-zero sub-matrices which are hierarchical

and assembled by several basic in#uence factor matrices (BIFMs). The hierarchical structures of

and the BIFMs of the sub-matrices are introduced below.

1. Sub-matrix SIF:

SIF"

I

I

I

I

I

I

2 2

I

2 I

2

I

2 2 I

3 ntot3 ntot

I

is the 33 BIFM of soil element displacements due to soil element stresses, which is expressed

as I

(soil displacement in the k direction at element i due to a unit stress in the l direction on

element j ).



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2. Sub-matrix PIF:

PIF"

J

J

J

J

3 ntot3ntot

J"

J

2 J

2 J

2 2

J 2 J

2

nept(i)nept(i)

where nept(i) is the total number of elements of pile i, neps(i) the total number of shaft and

discontinuous elements of pile i. J

the 33 BIFM of pile elements displacements due to pile

element stresses, which is expressed as J

(displacement in the k direction at pile element i due

to a unit stress in the l direction at pile element j ).

3. Sub-matrix B:

B"b

b

2

b

b

3 ntot6 npil

b"

b

b

2

b

b

3 nept(i)6

b

is the 36 BIFM of pile element displacements due to pile tip displacements, where the

elements are b

(displacement in the k direction at element j of pile i due to a unit tip

displacement in l component at pile i).

4. Sub-matrix D:

D"d

d

d

d

2

3ntot6 ncpsum

d"

d

d

d

d

3 nept(i)6

d

is the 36 BIFM of pile displacements due to pile head loads, with the elements

d

"displacement in k direction at element j of pile i due to a unit head load in l component on

pile i .



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5. Sub-matrix G:

G"

g

g

2 g

g

2 2 2

2 2 g

2

g

2 2 g

6 ncap6ncap

g

is the 66 BIFM of cap loads due to load displacement. If cap i and cap j are independent,

g

is a zero sub-matrix. If cap i and cap j are joined by a tie beam, g

is expressed as g

"load

in k component of cap i due to a unit displacement in l component of cap j .

6. Sub-matrix H:

H"

h

h

h

h

6 ncap6 ncpsum

h"(h

h

h

h

)66 ncpil(i)

where ncpil(i) is the total number of the piles under pile cap i and h

the 66 BIFM for cap loads

due to pile head loads, and the elements are h

"head load (k component) on cap i due to

a unit head load (l component) on pile j.

7. Sub-matrix O:

O"

o

o

o

2

o

,

6ncpsum

3ntot

o"(o

o

o

o

)6nect(i)

where nect(i) is the total number of elements of the piles under pile cap i, necs(i) the total number

of shaft and discontinuous elements of the piles under pile cap i, and o

"the 63 BIFM of the

pile head displacements due to pile element stresses, the element is o

"the head displacement

(k component) of pile i due to a unit stress (l direction) on element j of pile i .

8. Sub-matrix P:

P"

p

p

p

p

6 ncpsum6 npils

p is the 66 BIFM of the pile head displacements due to pile tip displacements, and the elements

are expressed as p

"the head displacement (k component) of pile i due to a unit tip

displacement (l component) of pile i.



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APPENDIX B: NOMENCLATURE

In principle and Appendix A parts

Control values

ncap "total number of pile capsnpils "total number of piles

ntot "total number of elementsnipsum "total number of individual piles

ncpsum "total number of piles of pile-groups

ncpil(i) "total number of the piles under cap i

nept(i) "total number of elements of pile i

neps(i) "total number of shaft and discontinuous elements of pile i

nect(i) "total number of elements of the piles under pile cap i

necs(i) "total number of shaft and discontinuous elements of the piles under pile cap i

Basic in -uence factor matrices

I "displacements at soil element i due to unit stresses on soil element jJ

"displacements at pile element i due to unit stresses on pile element j

b

"displacements at element j of pile i due to unit tip displacements at pile i

d

"displacements at element j of pile j due to unit head loads on pile i

h

"loads on pile cap i due to unit head loads on pile j

o

"head displacements on pile i due to unit stresses on element j of pile i

p

"head displacements of pile i due to unit tip displacements of pile i

q

"head displacements of pile j due to unit displacements of cap i

r

"head displacements of pile i due to unit head loads on pile i

s

"head loads on pile i due to unit stresses on element j of pile i

v

"head load on pile i due to unit loads on pile i

In evaluation and application parts

Pile and soil parameters

E

"Young's modulus of pile

E

"Young's modulus of soilE

"base Young's modulus

"Poission's ratio of pile

"Poission's ratio of soilP, H, M, ¹ "vertical, horizontal, moment, torsional loadings, respectively

¸, d "length and diameter of pile, respectivelyI

"moment of inertia of pile section

s, c "spacing (c/c) between two piles and pile groups, respectively

Geo- factors of pile parameter studies

K "pile sti! ness factorK

"pile #exibility factor

I

, I

, I

, "in#uence factorsR

R

"group reduction factors



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,

,

"interaction factors

,

"group interaction factors

Others

GEPAN "general pile analysis

BIFM "basic in#uence factor matrix "zero sub-matrix or vector

REFERENCES

1. Hrenniko! A. Analysis of pile foundations with batter piles. ¹ransactions of the ASCE 1950; 115 :351}374.2. Kocsis P. ¸ateral loads on piles. Bureau of Eng., Chicage, 1968.3. Nair K, Gray J, Donovan N. Analysis of pile group behavior. AS¹M, S¹P 444, 1969; 118}159.4. Focht JA, Koch KJ. Rational analysis of the lateral performance of o! shore pile groups. Proceedings of 5th O + shore

¹echnical Conference, Houston, vol. 2, 1973; 701}708.5. O'Neil MW, Ghazzaly OI, Ha HB. Analysis of three-dimensional pile groups with nonlinear soil response and

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6. Chow YK. Analysis of vertically loaded pile groups. International Journal for Numerical and Analytical Methods inGeomechanics, 1986; 10(1):59}72.

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25(2):159}174.12. Desai CS. Numerical design * analysis for piles in sands. Journal of Geotechnical Engineering Division, ASCE 1974;

100(GT6):613}635.13. Pressley JG, Poulos HG. Finite element analysis of mechanisms of pile group behavior. International Journal for

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Australia, 1990.15. Poulos HG. Randolph MF. Pile group analysis, a study of two methods. Journal of Geotechnical Engineering Division,ASCE 1983; 109(GT3):355}372.

16. Mindlin RD. Force at a point in the interior of a semi-in"nite solid. Physics 1936; 7 :195}202.17. Poulos HG, Davis EH. Pile Foundation Analysis and Design. Wiley: New York, 1980.18. Xu KJ, Poulos HG. Principle of program GEPAN for general pile elastic analysis. Research Report No. R782,

Department of Civil Engineering, The University of Sydney, 1999.19. Poulos HG. PIES ;sers Guide. Centre for Geot. Res. Univ. of Sydney, Australia, 1989.20. Hull TS. The static behaviour of laterally loaded piles. Ph.D. Dissertation, University of Sydney, Australia 1987.21. Mylonakis G, Gazetas G. Settlement and additional internal forces of grouped piles in layered. Ge Hotechnique 1998;

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