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A Continuum-Based Model for Analysis ofLaterally Loaded Piles in Layered SoilsD BasuUniversity of Connecticut - Storrs

Rodrigo SalgadoPurdue University, [email protected]

Monica PrezziPurdue University, [email protected]

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Basu, D., Salgado, R. & Prezzi, M. (2009). Geotechnique 59, No. 2, 127–140 [doi: 10.1680/geot.2007.00011

127

A continuum-based model for analysis of laterally loaded piles inlayered soils

D. BASU*, R. SALGADO† and M. PREZZI†

An analysis is developed to calculate the response oflaterally loaded piles in multilayered elastic media. Thedisplacement fields in the analysis are taken to be theproducts of independent functions that vary in the verti-cal, radial and circumferential directions. The governingdifferential equations for the pile deflections in differentsoil layers are obtained using the principle of minimumpotential energy. Solutions for pile deflection are obtainedanalytically, whereas those for soil displacements areobtained using the one-dimensional finite differencemethod. The input parameters needed for the analysisare the pile geometry, the soil profile, and the elasticconstants of the soil and pile. The method producesresults with accuracy comparable with that of a three-dimensional finite element analysis but requires much lesscomputation time. The analysis can be extended toaccount for soil non-linearity.

KEYWORDS: elasticity; piles; theoretical analysis

Dans la presente communication, on concoit une analysepermettant de calculer la reaction de pieux a charge lateraledans des milieux elastiques multicouches. Les champs dedeplacement dans cette analyse sont censes etre les produitsde fonctions independantes variant dans les sens vertical,radial et circonferentiel. Les equations differentielles deter-minantes pour les fleches des pieux dans differentes couchesde terrain s’obtiennent en appliquant le principe de l’ener-gie potentielle minimale. Les solutions s’obtiennent de dif-ferentes facons : de facon analytique pour la fleche despieux, et avec une methode des differences finies unidimen-sionnelles pour les deplacements du sol. Les parametresd’entree necessaires pour l’analyse sont la geometrie despieux, le profil du terrain, et les constantes elastiques du solet des pieux. Cette methode permet d’obtenir des resultatsdont la precision est comparable a celle des analyses tridi-mensionnelles aux elements finis, mais qui necessite destemps de calcul beaucoup plus courts. En outre, il estpossible de renforcer cette analyse pour tenir compte de lanon linearite du terrain.

INTRODUCTIONPiles subjected to lateral forces and moments at the head areanalysed in practice with the p–y method (Reese & Cox,1969; Matlock, 1970; Reese et al., 1974, 1975; Reese & VanImpe, 2001). In the p–y method the pile is assumed to behaveas an Euler–Bernoulli beam with the soil modelled as a seriesof discretely spaced springs, each connected to one of the pilesegments into which the pile is discretised. The springs modelthe soil response to loading through p–y curves (p is the unitresistance per unit pile length offered by the springs, and y isthe pile deflection), which are developed empirically byadjusting the curves until they match actual load–displace-ment results (Cox et al., 1974; Briaud et al., 1984; Yan &Byrne, 1992; Brown et al., 1994; Gabr et al., 1994; Briaud,1997; Wu et al., 1998; Bransby, 1999; Ashour & Norris,2000). However, the p–y method often fails to predict pileresponse (Anderson et al., 2003; Kim et al., 2004), for it isnot capable of capturing the complex three-dimensional inter-action between the pile and the soil.

The continuum approach is conceptually more appealing;however, in order to model the soil as a continuum, the use ofnumerical techniques such as the three-dimensional (3D) finiteelement (FE) method, finite elements with Fourier analysis, theboundary element (BE) method or the finite difference (FD)method is often required (Poulos, 1971a, 1971b; Banerjee &Davis, 1978; Randolph, 1981; Budhu & Davies, 1988; Brown

et al., 1989; Verruijt & Kooijman, 1989; Trochanis et al., 1991;Bransby, 1999; Ng & Zhang, 2001; Klar & Frydman, 2002).The 3D FE or FD method can capture the most importantfeatures of the complex pile–soil interaction, but three-dimen-sional analyses are computationally expensive for routine prac-tice. The BE method accounts for the pile–soil interaction bydiscretising the pile into small strips and modelling the inter-action between these strips with the soil continuum throughnumerical integration of Mindlin’s solution (Mindlin, 1936) fora point force within a continuum.

Considering the soil surrounding the pile as a continuum,Sun (1994), Zhang et al. (2000) and Guo & Lee (2001)developed closed-form solutions based on linear elasticitythat can be used to obtain lateral pile deflection with depth.Their analyses capture the three-dimensional aspects of theinteraction of the pile–soil system and produce resultsquickly, which is advantageous in practice. However, theseauthors made an assumption that the variation of displace-ments within the soil mass depends on the same displacementfunction for both the radial and the circumferential directions.This leads to a soil response that is stiffer than it is in reality.

Most continuum analyses of laterally loaded piles do notconsider soil layering. Soil heterogeneity with depth has beenapproximately taken into account in the BE and FE analyses byassuming (typically) a linear variation of soil modulus withdepth (Poulos, 1973; Randolph, 1981; Budhu & Davies, 1988).The BE analysis has also been used to analyse two-layersystems (Banerjee & Davies, 1978; Pise, 1982). However, BEanalysis of laterally loaded piles is strictly not applicable tolayered systems, because Mindlin’s solution used in BE analy-sis is valid only for homogeneous continuums. Verruijt &Kooijman (1989) solved a layered elastic system by discretis-ing the soil layers using FE and the pile by FD methods.

In this paper, an advanced continuum-based method ofanalysis of laterally loaded piles is proposed by assumingthe soil displacement field to have a shape that is consistent

Manuscript received 18 February 2007; revised manuscript accepted24 September 2008. Published online ahead of print 3 December2008.Discussion on this paper closes on 1 August 2009, for furtherdetails see p. ii.* Department of Civil and Environmental Engineering, Universityof Connecticut, Storrs, USA† School of Civil Engineering, Purdue University, West Lafayette,USA.

with the drop in displacement expected as distance from thepile increases, and with the fact that the displacement isexpected to depend on the direction of the load with respectto the point considered in the soil. The analysis considers apile embedded in a multilayered elastic soil (continuum),and rigorously takes into account the three-dimensionalpile–soil interaction. The governing differential equationsfor the pile and soil displacements are developed usingvariational principles. Closed-form solutions are obtained forpile deflection, and soil displacements are obtained using theone-dimensional (1D) FD method. Pile response obtainedusing this method compares favourably with 3D FE analysis,although the computation effort required by this method issmall. Because soil displacements and strains can be calcu-lated alongside pile deflection using this method, the analysisforms the basis for future analysis that can model the inter-action of piles in a group, and can account for soil non-linearity by relating the progressive degradation of soilstiffness to induced soil strains.

ANALYSISProblem definition

We consider a pile with a circular cross-section of radiusrp and length Lp embedded in a soil deposit that has n layers(Fig. 1). Each layer extends to infinity in all radial direc-tions, and the bottom (nth) layer extends to infinity in thedownward direction. The vertical depth to the base of anyintermediate layer i is Hi, which implies that the thicknessof the ith layer is Hi � Hi�1 with H0 ¼ 0 and Hn ¼ 1. Thepile head is at the ground surface, and the base is embeddedin the nth layer. The pile is subjected to a horizontal forceFa and a moment Ma at the pile head such that Fa and Ma

are orthogonal vectors. In the analysis, we choose a cylind-rical (r–Ł–z) coordinate system with its origin coincidingwith the centre of the pile head and the positive z-axis(coinciding with the pile axis) pointing downwards. The goalof the analysis is to obtain pile deflection as a function ofdepth caused by the action of Fa and/or Ma at the pile head.

The soil medium is assumed to be an elastic and isotropiccontinuum, homogeneous within each layer, with Lame’sconstants ºs and Gs. There is no slippage or separationbetween the pile and the surrounding soil, or between thesoil layers. The pile behaves as an Euler–Bernoulli beamwith a constant flexural rigidity EpIp.

Potential energyThe total potential energy of the pile–soil system, includ-

ing both the internal and external potential energies, is givenby

— ¼ 1

2Ep Ip

ð Lp

0

d2w

dz2

� �2

dz þð1

0

ð2�

0

ð1rp

1

2� pq� pq rdrdŁdz

þð1

Lp

ð2�

0

ð rp

0

1

2� pq� pq rdrdŁdz

� Faw

��z¼0

þ Ma

dw

dz

��z¼0

(1)

where w is the lateral pile deflection, and � pq and � pq are thestress and strain tensors (see Fig. 2) in the soil (summation isimplied by the repetition of the indices p and q in the productof corresponding stress and strain components). The firstintegral represents the internal potential energy of the pile.The second and third integrals represent the internal potential

2rp

r

r0

Fa

MaPile

H1

Hn�2

Hi

Hi�1

H2

r0Ma

… ………

Hn�1

Layer 1

Layer 2

Layer i

Layer 1n �

Layer n

z

Fa

Pile

Lp

θ

�

�

�

Fig. 1. Laterally loaded pile in layered elastic medium

(a)

uθ

r0

z

ur

uz

Faθ

(b)

τzr

τθr

τθz

σrr

r0

τrz

σzz

z

τrθ

τzθ

θ

σθθ

Fig. 2. (a) Displacements and (b) stresses within soil mass

128 BASU, SALGADO AND PREZZI

energy of the continuum (note that the third integral repre-sents the energy of the column of soil with radius rp startingat the pile base and extending to infinity downward, while thesecond integral represents the energy of the soil surroundingboth the pile and this column of soil). The remaining twoterms represent the external potential energy.

Soil displacementWe assume the following displacement fields (Fig. 2) in

the soil:

ur ¼ w zð Þ�r rð Þ cos Ł (2a)

uŁ ¼ �w zð Þ�Ł rð Þ sinŁ (2b)

uz ¼ 0 (2c)

where w(z) is a displacement function (with a dimension oflength), varying with depth z, representing the deflection ofthe pile axis; �r(r) and �Ł(r) are dimensionless displace-ment functions varying with the radial coordinate r; and Ł isthe angle measured clockwise from a vertical referencesection (r ¼ r0) that contains the applied force vector Fa.Equation (2c) is based on the assumption that the verticaldisplacement of the pile caused by the lateral load andmoment applied at the pile head is negligible.

The functions �r(r) and �Ł(r) describe how the displace-ments within the soil mass (due to pile deflection) decreasewith increasing radial distance from the pile axis. We set�r(r) ¼ 1 and �Ł(r) ¼ 1 at r ¼ rp (this ensures compat-ibility at the pile/soil interface) and �r(r) ¼ 0 and �Ł(r) ¼0 at r ¼ 1 (this ensures that displacements in the soildecrease with increasing radial distance from the pile). Thus�r and �Ł vary between 1 at the pile/soil interface and 0 atinfinite radial distance from the pile.

Stress–strain–displacement relationshipsThe strain–displacement relationship, considering equation

(2), leads to

�rr

�ŁŁ

�zz

ªrŁ

ªrz

ªŁz

26666666664

37777777775¼

� @ur

@ r

� ur

r� 1

r

@uŁ

@Ł

� @uz

@z

� 1

r

@ur

@Ł� @uŁ

@ rþ uŁ

r

� @uz

@ r� @ur

@z

� 1

r

@uz

@Ł� @uŁ

@z

266666666666666666666664

377777777777777777777775

¼

�w zð Þd�r rð Þ

drcosŁ

�w zð Þ�r rð Þ � �Ł rð Þ

rcos Ł

0

w zð Þ�r rð Þ � �Ł rð Þ

rþ d�Ł rð Þ

dr

� �sinŁ

� dw zð Þdz

�r rð Þ cosŁ

dw zð Þdz

�Ł rð Þ sinŁ

26666666666666666666664

37777777777777777777775

(3)

The strains in equation (3) are related to stresses using theelastic stress–strain relationships, which allow expression ofthe soil potential energy density 1

2� pq� pq in terms of the

displacement functions w(z), �r(r) and �Ł(r), and the soilelastic constants ºs and Gs (see Appendix 1). Substitutingthis expression for the potential energy density into equation(1), we obtain

—¼ 1

2Ep Ip

ð Lp

0

d2w

dz2

� �2

dzþ�

2

ð10

ð1rp

ºsþ2Gsð Þw2 d�r

dr

� �2"

þ2ºsw2 d�r

dr

�r��Łð Þr

þ ºsþ3Gsð Þw2 �r��Łð Þ2

r2

þGsw2 d�Ł

dr

� �2

þ2Gsw2 �r��Łð Þr

d�Ł

drþGs

dw

dz

� �2

�2r

þGs

dw

dz

� �2

�2Ł

#rdrdzþ�

2r2

p

ð1Lp

Gs

dw

dz

� �2

dz

�Faw

��z¼0

þMa

dw

dz

��z¼0

(4)

Principle of minimum potential energyA system in equilibrium exists with its potential energy at

a minimum. Hence minimising the potential energy of thepile–soil system (i.e. setting the first variation of the poten-tial energy �— equal to 0) produces the equilibrium equa-tions. We apply �— ¼ 0 to obtain an equation of the form(see Appendix 1)

�— ¼ A wð Þ�w þ B wð Þ�dw

dz

� ��

þ C �rð Þ��r

� þ D �Łð Þ��Ł

� ¼ 0

(5)

Since the variations �w(z), �(dw/dz), ��r(r) and ��Ł(r) ofthe functions w(z) (and its derivative), �r(r) and �Ł(r) areindependent, the terms associated with each of these varia-tions must individually be equal to zero (i.e. A(w)�w ¼ 0,B(w)�(dw/dz) ¼ 0, C(�r)��r ¼ 0 and D(�Ł)��Ł ¼ 0) inorder to satisfy the condition �— ¼ 0. The resulting equa-tions produce the optimal functions wopt(z), �r,opt(r) and�Ł,opt(r) that describe the equilibrium configuration of thepile–soil system.

While considering the terms of the variation of thepotential energy related to w, we do so for the followingsub-domains: 0 < z < H1, H1 < z < H2, . . ., Hn�1 < z <Lp, and Lp < z , 1. Accordingly, w is forced to satisfyequilibrium within each of these sub-domains, and henceover the entire domain. For �r and �Ł the domain overwhich the potential energy and its variation are calculated isrp < r , 1.

Soil displacement profilesWe first consider the variation of �r(r). Referring back to

the equation �— ¼ 0, represented by equation (5), we firstcollect all the terms associated with ��r and collectively setthem equal to zero to obtainð1

rp

�ms1 rd2�r

dr2þ d�r

dr

� �þ ms2 þ ms3ð Þ

d�Ł

dr

�

þ ms4

�r

r� ms4

�Ł

rþ ns r�r

��rdr

þ ms1 rd�r

drþ ms3�r � ms3�Ł

� ��r

��1

rp

¼ 0

(6)

ANALYSIS OF LATERALLY LOADED PILES IN LAYERED SOILS 129

where

ms1 ¼ ºs þ 2Gsð Þð1

0

w2dz ¼Xnþ1

i¼1

ºsi þ 2Gsið Þð Hi

Hi�1

w2i dz

(7)

ms2 ¼ Gs

ð10

w2dz ¼Xnþ1

i¼1

Gsi

ð Hi

Hi�1

w2i dz (8)

ms3 ¼ ºs

ð10

w2dz ¼Xnþ1

i¼1

ºsi

ð Hi

Hi�1

w2i dz (9)

ms4 ¼ ºs þ 3Gsð Þð1

0

w2dz ¼Xnþ1

i¼1

ºsi þ 3Gsið Þð Hi

Hi�1

w2i dz

(10)

ns ¼ Gs

ð10

dw

dz

� �2

dz ¼Xnþ1

i¼1

Gsi

ð Hi

Hi�1

dwi

dz

� �2

dz (11)

The subscript i in the above equations refers to the ith layerof the multilayered continuum (Fig. 1); wi represents thefunction w(z) in the ith layer with wijz¼Hi

¼ wiþ1jz¼Hi. Note

that the nth (bottom) layer is split into two parts, with thepart below the pile denoted by the subscript n+1; therefore,in the analysis, Hn ¼ Lp and Hnþ1 ! 1.

The last term on the left-hand side of equation (6) is amultiple of the subtraction of the value of ��r at r ¼ rp

from the value of ��r at r ¼ 1, and is therefore identicallyzero for the boundary conditions of our problem (�r ¼ 0 atr ¼ 1 and �r ¼ 1 at r ¼ rp) because a known (orprescribed) �r implies that its variation ��r ¼ 0. After thisterm is made equal to zero, what is left is an equation ofform C(�r)��r ¼ 0. The function �r(r) has a non-zerovariation (i.e. ��r 6¼ 0) for rp , r , 1 because �r is notknown a priori in this interval, so C(�r) ¼ 0, which meansthe integrand in equation (6) must be set to zero, leading tothe differential equation

d2�r

dr2þ 1

r

d�r

dr� ª1

r

� �2

þ ª2

rp

� �2" #

�r ¼ª2

3

r

d�Ł

dr� ª1

r

� �2

�Ł

(12)

where the ªs are dimensionless constants given by ª21 ¼

ms4=ms1, (ª2=rp)2 ¼ ns=ms1 and ª23 ¼ (ms2 þ ms3)=ms1.

When solved, equation (12) yields �r,opt.We now consider the variation of �Ł(r). We collect the

terms containing ��Ł in the equation �— ¼ 0 (equation (5))and, proceeding similarly as for �r, we get the followinggoverning differential equation for �Ł:

d2�Ł

dr2þ1

r

d�Ł

dr� ª4

r

� �2

þ ª5

rp

� �2" #

�Ł¼�ª26

r

d�r

dr� ª4

r

� �2

�r

(13)

with the following boundary conditions: �Ł ¼ 0 at r ¼ 1and �Ł ¼ 1 at r ¼ rp, where ª2

4 ¼ ms4=ms2, (ª5=rp)2 ¼ns=ms2, and ª2

6 ¼ (ms2 þ ms3)=ms2.

Pile displacementFinally, we consider the variation of the function w and its

derivative. We again refer back to the equation �— ¼ 0,collect all the terms associated with �w and �(dw/dz) andequate their sum to zero, to obtain

Xn

i¼1

ð Hi

Hi�1

Ep Ip

d4wi

dz4� 2ti

d2wi

dz2þ kiwi

� ��widz

þð1

Lp

�2t nþ1

d2wnþ1

dz2þ k nwnþ1

� ��wnþ1dz

þ Ep Ip

d3w1

dz3� 2t1

dw1

dz� Fa

� ��w1

��z¼0

� Ep Ip

d2w1

dz2� Ma

� ��

dw1

dz

� ��z¼0

þ � Ep Ip

d3w1

dz3� 2t1

dw1

dz

� ��w1

��z¼H1

"

þ Ep Ip

d3w2

dz3� 2t2

dw2

dz

� ��w2

��z¼H1

#

þ Ep Ip

d2w1

dz2�

dw1

dz

� ��z¼H1

"

� Ep Ip

d2w2

dz2�

dw2

dz

� ��z¼H1

#þ . . . :

þ � Ep Ip

d3wn

dz3� 2tn

dwn

dz

� ��wn

��z¼Lp

"

� 2t nþ1

dwnþ1

dz�wnþ1

��z¼Lp

#

þ Ep Ip

d2wn

dz2�

dwn

dz

� ��z¼Lp

þ 2t nþ1

dwnþ1

dz�wnþ1

��z¼1

¼ 0

(14)

where

ti ¼

�

2Gsi

ð1rp

�2r þ �2

Ł

�rdr

" #; i ¼ 1, 2, . . ., n

�

2Gsn

ð1rp

�2r þ �2

Ł

�rdr þ r2

p

" #; i ¼ n þ 1

8>>>>><>>>>>:

(15)

ki ¼ � ºsi þ 2Gsið Þð1

rp

rd�r

dr

� �2

dr þ Gsi

ð1rp

rd�Ł

dr

� �2

dr

"

þ 2ºsi

ð1rp

�r � �Łð Þd�r

drdr

þ 2Gsi

ð1rp

�r � �Łð Þd�Ł

drdr

þ ºsi þ 3Gsið Þð1

rp

1

r�r � �Łð Þ2dr

#(16)


We first consider the domain below the pile, that is,Lp < z , 1. The terms associated with �w and �(dw/dz) inequation (14) for Lp < z , 1 are equated to zero. Sincethe variation of w(z) with depth is not known a priori withinthe interior of the domain Lp , z , 1, �wnþ1 6¼ 0, and sothe integrand in the integral between z ¼ Lp and z ¼ 1must be equal to zero in order to satisfy equation (14). Thisresults in the differential equation

2t nþ1

d2wnþ1

dz2� k nwnþ1 ¼ 0 (17)

The displacement in the soil must vanish for z equal toinfinity. We use this as our boundary condition:

wnþ1 ¼ 0 (at z ¼ 1) (18)

The above equation implies that �wnþ1 ¼ 0 at z ¼ 1,making the term associated with �w at z ¼ 1 equal to zero(which is of course required to satisfy equation (14)).

The solution of equation (17) satisfying boundary condi-tion (18) is

wnþ1 ¼ wnjz¼Lpe�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik n=2 t nþ1ð Þ

pz�Lpð Þ (19)

We now consider the function w for the domains 0 < z <H1, H1 < z < H2, . . ., Hn�1 < z < Lp. The terms containing�w and �(dw/dz) in equation (14) are equated to zero foreach domain. Considering the integrals associated with eachindividual layer (or each domain Hi�1 , z , Hi), theintegrand for each of these integrals must equal zero,because �wi 6¼ 0 (as the function wi(z) within the domainsis not known a priori). This gives us the differential equationfor the ith layer, which, expressed in terms of normaliseddepth ~zz ¼ z=Lp and displacement ~ww ¼ w=Lp, is given by

d4 ~wwi

d~zz4� 2~tti

d2 ~wwi

d~zz2þ ~kki ~wwi ¼ 0 (20)

The terms associated with the boundaries (i.e. z ¼ Hi) ofeach domain in equation (14) must also each be equal tozero. For each boundary, there are two terms: one multi-plying �wi and another multiplying �(dwi/dz). Setting eachseparately equal to zero yields the boundary conditions forthe differential equations represented by equation (20). Theseterms can be seen to be a product of an expression and thevariation of the displacement or of its derivative. If thedisplacement or its derivative is specified at the boundary,then its variation is equal to zero; otherwise, the expressionmultiplying the variation of the displacement or of itsderivative is equal to zero. The boundary conditions at thepile head (z ¼ ~zz ¼ 0) are

~ww1 ¼ constant (21a)

or

d3 ~ww1

d~zz3� 2~tt1

d ~ww1

d~zz� ~FFa ¼ 0 (21b)

and

d ~ww1

d~zz¼ constant (21c)

or

d2 ~ww1

d~zz2� ~MMa ¼ 0 (21d)

At the interface between any two layers (z ¼ Hi or ~zz ¼ ~HHi),

~wwi ¼ ~wwiþ1 (22a)

d ~wwi

d~zz¼ d ~wwiþ1

d~zz(22b)

d3 ~wwi

d~zz3� 2~tti

d ~wwi

d~zz¼ d3 ~wwiþ1

d~zz3� 2~ttiþ1

d ~wwiþ1

d~zz(22c)

d2 ~wwi

d~zz2¼ d2 ~wwiþ1

d~zz2(22d)

At the pile base (z ¼ Lp or ~zz ¼ 1), the boundary conditionsare

~wwn ¼ constant (23a)

or

d3 ~wwn

d~zz3� 2~tt n

d ~wwn

d~zz¼ �2~tt nþ1

d ~wwnþ1

d~zz(23b)

and

d ~wwn

d~zz¼ constant (23c)

or

d2 ~wwn

d~zz2¼ 0 (23d)

Equation (23b) is further simplified and expressed solelyin terms of ~wwn by differentiating wnþ1 in equation (19) withrespect to z, normalising the expression, and then substitut-ing it back into equation (23b) to yield

d3 ~wwn

d~zz3� 2~tt n

d ~wwn

d~zz�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2~kk n~tt nþ1

q~wwn ¼ 0 (23b9)

The dimensionless terms in the above equations aredefined as ~tti ¼ ti L

2p=Ep Ip, ~kki ¼ ki L

4p=Ep Ip, ~FFa ¼ Fa L2

p=Ep Ip, ~MMa ¼ Ma Lp=Ep Ip and ~HHi ¼ Hi=Lp.

The governing differential equation for the pile (equation(20)) resembles that of an Euler–Bernoulli beam resting onan elastic foundation (the soil mass). The parameter ki (withdimensions of FL�2, where F ¼ force and L ¼ length) isrelated to the modulus of subgrade reaction (or to the‘spring constant’ proposed by Winkler, 1867) and determinesthe portion of the soil resistance due to compressive stressesin the elastic medium (Fig. 3). On the other hand, the

Pile (deformedconfiguration)

Shear resistance betweensoil ‘columns’ owing todifferential lateralmovement (accounted forby )ti

Infinite soil‘columns’ ofinfinitesimalthicknessprovidingresistance to pilemovement

Soil ‘columns’ getcompressed (orextended) to pilemovement from, say,point A to point B(accounted for by )ki

owing

A

B

Pile(undeformedconfiguration)

……

.

Groundsurface

Shear resistances betweensoil ‘columns’ below thepile produce pile base shear(occurs only if pile basedeflects laterally)

……

.

�

Fig. 3. Illustration of the two sources of soil resistance: soilcompression and shear


parameter ti (with dimension of F) determines the fractionof the soil resistance due to the shear stresses that developbetween soil layers of infinitesimal thickness displaced dif-ferentially in the lateral direction (Vlasov & Leont’ev, 1966).

The boundary conditions (equations (22a)–(22d)) at theinterface of any two layers (~zz ¼ ~HHi) ensure the continuity ofpile deflection, slope of the deflection curve (¼ dwi/dz),bending moment (¼ Ep Ip(d2wi=dz2)) and shear force(¼ Ep Ip(d3wi=dz3) � 2ti(dwi=dz)) (the shear force has con-tributions from both pile flexure and soil deformation). Atthe pile head (~zz ¼ 0) the shear force equals the applied force(equation (21b)), and either the slope of the pile deflectioncurve is a known constant (equation (21c)) (this boundarycondition is generally used with the value of slope takenequal to zero when fixed-head conditions are used to idealisethe case of a pile that is part of a group of piles joined atthe head by a cap) or the pile bending moment is equal tothe applied moment (equation (21d)) (free-head case). Equa-tion (21a) must be used in the analysis instead of equation(21b) to estimate the magnitude of an applied force requiredto produce a given (known) head deflection. Similarly, if, forboth the free- and fixed-head cases, it becomes necessary toestimate the magnitude of an applied moment that producesa given (known) slope at the head, equation (21c) must beused.

At the pile base (~zz ¼ 1), either the pile deflection is set(equation (23a)) (used for the ideal fixed-base case, forwhich the deflection is taken equal to zero, which may beused with satisfactory results if the pile is socketed into avery firm layer, like hard rock) or the shear force at asection infinitesimally above the pile base is equal to thatinfinitesimally below (equation (23b9)) (free-base case). Theother boundary condition active at the pile base is that eitherthe slope is a constant (equation (23c)) (assumed to be zerofor the fixed-base case) or the pile bending moment is zero(equation (23d)) (free-base case).

Expressions for the ªs in terms of dimensionless deflectionsThe ªs appearing in equations (12) and (13) are expressed

in terms of the dimensionless pile deflection and slope asfollows:

ª1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

i¼1

ºsiþ3Gsið Þð ~HHi

~HHi�1

~ww2i d~zzþ ºsnþ3Gsnð Þ

ffiffiffiffiffiffiffiffiffi~ttnþ1

2~kk n

s~ww2

nj~zz¼1

Xn

i¼1


~HHi�1



2~kk n

s~ww2

nj~zz¼1

vuuuuuuuut(24)

ª2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

ł2

Xn

i¼1

Gsi

ð ~HHi

~HHi�1

d ~wwi

d~zz

� �2

d~zzþGsn

ffiffiffiffiffiffiffiffiffiffiffi~kk n

8~tt nþ1

s~ww2

nj~zz¼1

Xn

i¼1


~HHi�1


ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~tt nþ1

2~kk n

~ww2nj~zz¼1

s

vuuuuuuuut(25)

ª3 ¼


i¼1

ºsiþGsið Þð ~HHi

~HHi�1

~ww2i d~zzþ ºsnþGsnð Þ

ffiffiffiffiffiffiffiffiffi~tt nþ1

2~kk n

s~ww2

nj~zz¼1

Xn

i¼1


~HHi�1



2~kk n

s~ww2

nj~zz¼1

vuuuuuuuut(26)

ª4 ¼


i¼1


~HHi�1



2~kk n

s~ww2

nj~zz¼1

Xn

i¼1

Gsi

ð ~HHi

~HHi�1

~ww2i d~zzþGsn


2~kk n

s~ww2

nj~zz¼1

vuuuuuuuut(27)

ª5 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

ł2

Xn

i¼1

Gsi

ð ~HHi

~HHi�1

d ~wwi

d~zz

� �2

d~zzþGsn

ffiffiffiffiffiffiffiffiffiffiffi~kk n

8~ttnþ1

s~ww2

nj~zz¼1

Xn

i¼1

Gsi

ð ~HHi

~HHi�1

~ww2i d~zzþGsn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~ttnþ1

2~kk n

~ww2nj~zz¼1

s

vuuuuuuuut (28)

ª6 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

i¼1

ºsiþGsið Þð ~HHi

~HHi�1

~ww2i d~zzþ ºsnþGsnð Þ


2~kk n

s~ww2

nj~zz¼1

Xn

i¼1

Gsi

ð ~HHi

~HHi�1

~ww2i d~zzþGsn


2~kk n

s~ww2

nj~zz¼1

vuuuuuuuut(29)

where ł ¼ Lp/rp. These expressions can be directly used inthe computations.

ANALYTICAL SOLUTION FOR PILE DEFLECTIONThe general solution of equation (20) is given by

~wwi(~zz) ¼ C(i)1 �1 þ C

(i)2 �2 þ C

(i)3 �3 þ C

(i)4 �4 (30)

where C(i)1 , C

(i)2 , C

(i)3 and C

(i)4 are integration constants (for the

ith layer), and �1, �2, �3 and �4 are individual solutions(functions of ~zz) of the differential equation. The functions �1,�2, �3 and �4 are standard trigonometric or hyperbolic func-tions that arise in the solution of the linear ordinary differentialequations (Table 1). The integration constants for each layercan be determined using the boundary conditions. The bound-ary conditions given in equations (21)–(23) lead to a systemof linear algebraic equations (see Appendix 2) of the form

¨½ � C½ � ¼ F½ � (31)

where [¨]4 n34 n is a matrix containing the functions �1, �2,�3 and �4 calculated at the boundaries of the soil layers,[C]4 n31 is the vector of unknown integration constants of allthe layers, and [F]4 n31 is the right-hand side vector contain-ing the applied forces and/or displacements (the subscript 4ndenotes the number of equations, which is four times thenumber of soil layers). Simultaneous solution of the systemof equations represented by equation (31) produces the valuesof the integration constants C

(i)1 , C

(i)2 , C

(i)3 and C

(i)4 , which,

when substituted in equation (30), produce the particularsolution of pile deflection (i.e. the pile deflection profile) fora given set of boundary conditions and applied loads. Theslope of the deflected pile axis, and the bending moment andshear force in the pile, can be obtained as a function of depthby successively differentiating equation (30) and using thevalues of the integration constants.

FINITE DIFFERENCE SOLUTION FOR SOILDISPLACEMENTS

The differential equations (12) and (13) for �r and �Ł aresolved using the FD method. The equations are interdepen-dent and must, as a result, be solved simultaneously. Usingthe central-difference scheme, equations (12) and (13) canbe respectively written as


� jþ1r � 2� j

r þ � j�1r

˜r2þ 1

r j

� jþ1r � � j�1

r

2˜r

� ª1

r j

� �2

þ ª2

rp

� �2" #

� jr ¼

ª23

r j

� jþ1Ł � � j�1

Ł

2˜r� ª1

r j

� �2

� jŁ

(32)

� jþ1Ł � 2� j

Ł þ � j�1Ł

˜r2þ 1

r j

� jþ1Ł � � j�1

Ł

2˜r

� ª4

r j

� �2

þ ª5

rp

� �2" #

� j

Ł ¼ � ª26

r j

� jþ1r � � j�1

r

2˜r� ª4

r j

� �2

� jr

(33)

where j represents the jth node, which is at a radial distancerj from the pile axis; and ˜r is the distance betweenconsecutive nodes (discretisation length). The total numberof discretised nodes m should be sufficiently large that theinfinite domain in the radial direction can be adequatelymodelled (Fig. 4). The discretisation length ˜r should besufficiently small to maintain a satisfactory level of accu-racy.

Table1.Functionsin

equation(30)forpiles

crossingmultiple

soil

layers

Rel

ativ

em

agn

itu

des

Co

nst

ants

aan

db

Fu

nct

ion

san

dth

eir

Ind

ivid

ual

solu

tio

ns

of

equ

atio

n(2

0)

of~ kk

and

~ tta

bd

eriv

ativ

es�

�1

�2

�3

�4

~ kk.

~ tt2ffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2(

ffiffiffi ~ kkpþ

~ tt)

qffiffiffiffiffiffiffiffi


1 2(

ffiffiffi ~ kkp�

~ tt)

q�

sin

ha~zz

cos

b~zz

cosh

a~zz

cos

b~zz

cosh

a~zz

sin

b~zz

sin

ha~zz

sin

b~zz

�9

a�

2�

b�

4a�

1�

b�

3a�

4þ

b�

2a�

3þ

b�

1

�0

(a2�

b2)�

1

�2

ab�

3

(a2�

b2)�

2

�2

ab�

4

(a2�

b2)�

3

þ2

ab�

1

(a2�

b2)�

4

þ2

ab�

2

�90

a(a

2�

3b

2)�

2þ

b(b

2�

3a

2)�

4a

(a2�

3b

2)�

1þ

b(b

2�

3a

2)�

3a(a

2�

3b

2)�

4�

b(b

2�

3a

2)�

2a

(a2�

3b

2)�

3�

b(b

2�

3a

2)�

1

~ kk,

~ tt2ffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffi~ ttþ


(~ tt2�

~ kk)

qr



~ tt�


(~ tt2�

~ kk)

qr

�si

nh

a~zz

cosh

a~zz

sin

hb~zz

cosh

b~zz

�9

a�

2a�

1b�

4b�

3

�0

a2�

1a

2�

2b

2�

3b

2�

4

�90

a3�

2a

3�

1b

3�

4b

3�

3

� Pri

me

(9)

ind

icat

esd

iffe

ren

tiat

ion

.

∆r∆r∆r

m � 12

r

Pile

1 3 … j j 1� mNode number

∆r

rj

…

Fig. 4. Finite difference discretisation for �r and �Ł

d

d

w

zi

�

�Calculate ,wi�

� �| 0·001γ new1/2/.../6|γ old

1/2/.../6

Check if

d

d

w

zi

�

��Store wi,

No

Yes

Input , , , , , , , ,L r E n H G F Mp p p s s a ai i iλ

Normalise input parameters: calculate , , ,F M Ha a i ψ

Choose initial guess for ( ), ( ), ..., ( )γ γ γ γ γ γ1old1 2

old2

old6� � �6

Calculate andk ti i

Input

....,

as the final values

End

Calculate γ γ γ γ γ γ1new1 2

new2 6

new6( ), ( ), ..., ( )� � �

Calculate andφ φr θ

� � �

γ γold1

new1 ,�

γ γold2

new2� ,

γ γold6

new6�

Fig. 5. Solution flow chart


Equation (32), along with the boundary conditions�(1)

r ¼ 1 (at r ¼ rp) and �(m)r ¼ 0 (at r ¼ 1), is applied to

the discretised nodes, yielding the equation

The non-zero elements of the left-hand side matrix[K� r ]m3m in the above equation are given by

K� r

j, j�1 ¼ 1

˜r2� 1

2r j˜r(35)

K� r

j, j ¼ � 2

˜r2� ª1

r j

� �2

þ ª2

r p

� �2" #

(36)

K� r

j, jþ1 ¼ 1

˜r2þ 1

2r j˜r(37)

in which the subscript j is valid for nodes 2 to m � 1, withthe exception that K

� r

2,1 ¼ 0 and K� r

m�1,m ¼ 0 (as is evidentfrom equation (34)).

The elements of the right-hand side vector fF� rgm31 inequation (34) are given by

F� r

j ¼ ª23

r j

� jþ1Ł � � j�1

Ł

2˜r� ª1

r j

� �2

� jŁ (38)

where j represents nodes 3 through m�2. The elementscorresponding to node 2 and m�1 are given by:

F� r

2 ¼ � 1

˜r2þ 1

2r2˜rþ ª2

3

r2

�(3)Ł � 1

2˜r� ª1

r2

� �2

�(2)Ł (39)

F� r

m�1 ¼ ª23

rm�1

��m�2Ł

2˜r� ª1

rm�1

� �2

�m�1Ł (40)

Using equation (33) and the boundary conditions that�(1)Ł ¼ 1 (at r ¼ rp) and �(m)

Ł ¼ 0 (at r ¼ 1), a matrixequation (similar to equation (34)) for �Ł can also beformed for the discretised nodes:

K�Ł½ � �Łf g ¼ F�Łf g (41)

The number and positioning of the non-zero elements of[K�Ł ]m3m in equation (41) are exactly the same as that of[K� r ]m3m of equation (34). The expressions of the off-diagonal elements of [K�Ł ]m3m and [K� r ]m3m are also thesame (i.e. K� r

p,q ¼ K�Łp,q for p 6¼ q). The diagonal elements of

[K�Ł ]m3m for j ¼ 2 to m � 1 are given by

K� r

j, j ¼ � 2

˜r2� ª4

r j

� �2

þ ª5

r p

� �2" #

(42)

The structure of fF�Łgm31 in equation (41) is also similarto fF� rgm31 of equation (34) with F

�Ł1 ¼ 1 and F�Ł

m ¼ 0.The remaining elements of fF�Łgm31 are given by

F�Łj ¼ � ª2

6

r j

� jþ1r � � j�1

r

2˜r� ª4

r j

� �2

� jr

j ¼ 3, 4, . . . , m � 2ð Þ

(43)

F�Ł2 ¼ � 1

˜r2þ 1

2r2˜r� ª2

6

r2

�(3)r � 1

2˜r� ª4

r2

� �2

�(2)r (44)

F�Łm�1 ¼ � ª2

6

rm�1

��m�2r

2˜r� ª4

rm�1

� �2

�m�1r (45)

Since the right-hand side vectors fF� rgm31 andfF�Łgm31 contain the unknowns �Ł and �r, iterations arenecessary to obtain their values. An initial estimate of � j

r ismade and given as input to fF�Łg, and � j

Ł is determined bysolving equation (41). The � j

Ł values are then given as inputto fF� rg to obtain � j

r from equation (34). The newlyobtained values of � j

r are again used to obtain new valuesof � j

Ł, and the iterations are continued until convergence isreached. The criteria 1

m

Pmj¼1 � j

rprevious � � j

rcurrent

�� < 10�6

and 1m

Pmj¼1 � j

Łprevious � � j

Łcurrent

�� < 10�6 are used (a strin-gent value of 10�6 is used because this iterative solutionscheme is central to another set of iterations described next)to ensure that accurate values of �r and �Ł are obtained.

SOLUTION ALGORITHMIn order to obtain pile deflections by solving equation

(20), the soil parameters ki and ti must be known. However,these soil parameters depend on �r and �Ł, which are notknown a priori. Hence an iterative algorithm (separate fromthe iterations between �r and �Ł described in the previoussection) is necessary to solve the problem. First, initialguesses for ª1 to ª6 are made, and for these assumed values�r and �Ł are determined using the iterative techniquedescribed in the previous section. Using the calculatedvalues of �r and �Ł, ki and ti are calculated by numericalintegrations (with ˜r of Fig. 4 as the step length). Using thevalues of ki and ti, the pile deflection is calculated. Fromthe calculated values of pile deflection and slope of thedeformed pile, ª1 to ª6 are obtained. The new values of ª1

to ª6 are then used to recalculate �r and �Ł, and so on. Theentire process is repeated until convergence on each ofthe ªs is attained. The tolerance limit prescribed onthe ªs between the pth and (p+1)th iteration isjª( pþ1)

1=2=...=6� ª( p)

1=2=...=6j , 0:001. The details of the solution

steps are given in the form of a flow chart in Fig. 5. Wechose an initial guess of ‘one’ for all the ªs, but any other

1 0 0 � � � � � � 0

0 K� r

2,2 K� r

2,3 0 � � � � � � 0

0 K� r

3,2 K� r

3,3 K� r

3,4 0 � � � � � � 0

0 0 K� r

4,3 K� r

4,4 K� r

4,5 0 � � � � � � 0

..

. . .. . .

. . .. . .

. . ..

� � � ...

0 � � � � � � 0 K� r

j, j�1 K� r

j, j K� r

j, jþ1 0 � � � 0

..

. . .. . .

. . .. . .

. . .. ..

.

0 � � � � � � 0 K� r

m�2,m�3 K� r

m�2,m�2 K� r

m�2,m�1 0

0 � � � � � � 0 K� r

m�1,m�2 K� r

m�1,m�1 0

0 � � � . . . 0 0 1

266666666666666666664

377777777777777777775

�(1)r

�(2)r

�(3)r

�(4)r

..

.

�( j)r

..

.

�(m�2)r

�(m�1)r

�(m)r

2666666666666666664

3777777777777777775

¼

1

F� r

2

F� r

3

F� r

4

..

.

F� r

j

..

.

F� r

m�2

F� r

m�1

0

26666666666666666664

37777777777777777775

(34)


choice would produce results with the same level of accu-racy at approximately the same computation time.

BENEFITS OF THE PRESENT ANALYSISThis analysis is an improvement over the analysis of Sun

(1994) for laterally loaded piles in homogeneous soil on atleast two accounts: (a) our assumption of the displacementfield is more general and more realistic than that assumedby Sun (1994), who chose �r(r) ¼ �Ł(r) ¼ �(r) for boththe displacements ur and uŁ (equation (2)); and (b) weobtained solutions for a multilayered soil, whereas the solu-tion of Sun (1994) is valid only for a single soil layer.

The need for an improved form for the displacement field(equation (2)) arises from the fact that the displacementassumption of Sun (1994) produces zero displacement in thesoil mass perpendicular to the direction of the applied forceFa (Fig. 6). Consequently, the resultant displacement vectoru ¼ erur þ eŁuŁ (er and eŁ are the unit basis vectors in theradial and tangential directions respectively) at any pointwithin the soil mass is forced to be parallel to the appliedforce Fa. Thus the displacement field in the soil mass(which, in general, has a component perpendicular to thedirection of Fa) is artificially constrained, the normal strainin the circumferential direction �ŁŁ (equation (3)) becomeszero, and the pile response is stiffer than what it is in reality.In fact, Guo & Lee (2001) found that the Sun (1994)analysis produces unreliable pile response, particularly if the

Poisson’s ratio of soil is greater than 0.3. This artificiallystiff pile response is not observed in our analysis. Toillustrate this point, we present below two examples compar-ing the results of our analysis, the analysis based on thedisplacement assumption of Sun (1994), and 3D FE analysis.

We consider as an illustration of use of the analysis a15 m long drilled shaft, with a diameter of 0.6 m and pilemodulus Ep ¼ 24 3 106 kN/m2, embedded in a four-layersoil deposit with H1 ¼ 2.0 m, H2 ¼ 5.0 m, and H3 ¼ 8.3 m;Es1 ¼ 20 MPa, Es2 ¼ 35 MPa, Es3 ¼ 50 MPa and Es4 ¼80 MPa; �s1 ¼ 0.35, �s2 ¼ 0.25, �s3 ¼ 0.2 and �s4 ¼ 0.15(Es i and �s i are the soil Young’s modulus and Poisson’s ratiofor the ith layer; Es i and �s i are related to ºs i and Gs i byºsi ¼ Esi�si=(1 þ �si)(1 � 2�si) and Gsi ¼ Esi=2(1 þ �si)). Ahorizontal force Fa ¼ 300 kN acts on the pile. The pile headand base are free to deflect and rotate. Fig. 7 shows the piledeflection profile obtained using our analysis, the analysisbased on the displacement assumption of Sun (1994), and a3D finite element analysis (FEA). The pile response obtainedfrom our analysis closely matches that of the 3D FEA (adifference of 9.6% in the head deflection was observedbetween our analysis and the FEA); the analysis of Sun(1994) produces a stiffer pile response.

Next, we consider a large-diameter drilled shaft, 40 mlong, with a diameter of 1.7 m and Ep ¼ 25 3 106 kPa,embedded in a four-layer soil profile with H1 ¼ 1.5 m, H2 ¼3.5 m, and H3 ¼ 8.5 m; Es1 ¼ 20 MPa, Es2 ¼ 25 MPa,Es3 ¼ 40 MPa and Es4 ¼ 80 MPa; �s1 ¼ 0.35, �s2 ¼ 0.3, �s3

¼ 0.25 and �s4 ¼ 0.2. A 3000 kN force acts at the pile head,which is free to deflect and rotate. Fig. 8 shows the piledeflection profiles, as obtained from our analysis, the analy-sis based on the displacement assumption of Sun (1994),and 3D FEA. As before, our results match those of the FEAmore closely than the results based on the Sun (1994)assumption; the difference in the head deflection obtainedfrom our analysis and FEA is 6.6%.

The 3D FE analyses were performed using ABAQUS. Thedomain for these analyses can be visualised as a cylinder ofsoil mass containing the pile at its centre as a concentriccylinder. The top (horizontal) surface of the soil cylinderwas flush with the pile head, and the bottom (horizontal)surface was located at a finite distance below the pile base(thus the soil mass below the pile base participating in thepile–soil interaction was incorporated in the analysis). Thehorizontal force Fa (acting at the pile head) was applied as auniformly distributed shear stress (i.e. force per unit pilecross-section area) acting on the pile-head surface (the

Ma

Fa

uθ

uruy

y r

r x0,

Displacementvectors

Pileux

θ

θ

Ma

Fa

uθ

uruy

y r

r x0,

Displacementvectors

Pile ux

θ

θ

(a)

(b)

Fig. 6. Displacement field in soil: (a) according to Sun (1994)assumption; (b) according to assumption made in this paper

1086420�2

Pile deflection: mm

15·0

12·5

10·0

7·5

5·0

2·5

0

Dep

th,

: mz

Present analysisSun (1994)Finite element analysis

Fig. 7. Deflection profile of a 15 m long pile


distributed shear stress multiplied by the pile cross-sectionarea produced Fa). The vertical plane passing through thepile axis parallel to Fa is a plane of symmetry (the planecontains the Fa vector) and divides the cylindrical domaininto two equal and symmetrical halves. Only one such halfwas used as the analysis domain. Different boundary condi-tions were prescribed at different boundaries of the FEdomain: all components of displacements were assumed tobe zero along the bottom (horizontal) surface and along theouter, curved (vertical) surface of the soil domain; on the(vertical) boundary surface created by the plane of symme-try, the displacement perpendicular to the boundary wasassumed to be zero. A perfect contact (with no slippage orseparation) between the pile and the surrounding soil wasassumed. The radial distances of the outer curved (vertical)boundary of the soil domain from the pile axis were takenas 20 m and 25 m for the 15 m and 40 m piles respectively;the corresponding vertical distances from the pile base tothe bottom (horizontal) boundary of the soil domain were5 m and 20 m. Twenty-noded brick elements were used torepresent both the pile and the soil for both the problems.The element size in the pile and at the pile/soil interfacewas approximately 0.1 m for both the examples, and wasincreased gradually with increasing radial distance from thepile axis to 2.0 m (for the 15 m pile) and 3.8 m (for the40 m pile) at the outer curved boundary of the soil domain.The number of degrees of freedom used for the 15 m pilewas 56 653, and that used for the 40 m pile was 90 564. Theoptimal domains and meshes described above were obtainedby ensuring that there were no boundary effects and byperforming convergence checks.

The CPU run times of the 3D FE analyses (run in a 16-core x86 server containing eight 2.6 MHz dual-core Opteron8218 processors with 32 GB RAM) were 9 min (for the15 m pile) and 14 min (for the 40 m pile), while the CPUrun time for our analysis (performed with a Fortran code runin an Intel Centrino Duo 2.0 GHz processor with 2 GBRAM) was 9.75 s for both the examples. Considering thefact that construction of the geometry (domain) and optimalmeshing for a FEA requires considerable time, our analysisis much more efficient than FEA because, in addition tobeing faster, the input to our analysis (the dimensions andelastic properties of pile, and the thickness and elasticconstants of soil layers) is accomplished through a simpletext file.

Finally, we consider the field example of a laterally loaded

pile load test performed by McClelland & Focht (1958). Thelength (Lp) and radius (rp) of the pile are 23 m and 0.305 m,and the pile was embedded in a normally consolidated clay.The pile was acted upon by a lateral force Fa ¼ 300 kN anda negative moment Ma ¼ �265 kNm at the head. Randolph(1981) back-calculated the pile modulus Ep as 68.42 3106 kN/m2 from the reported pile flexural rigidity. Randolph(1981) further suggested, based on back-calculation of testresults to match his FEA (coupled with Fourier series), thatthe soil shear modulus profile for this soil deposit can berepresented as Gs ¼ 0.8z 3 103 kN/m3 with �s ¼ 0.3. Wedivided the soil profile into four layers and calculated theshear modulus at the middle of each layer, which wereconsidered the representative values for each layer (Table 2).Using these values of soil modulus, we calculated the piledeflection profile using both our analysis and that based onthe assumption of Sun (1994). Fig. 9 shows the pileresponses. Also plotted are the measured pile response andthat obtained by Randolph (1981). Our analysis produces apile deflection profile that closely matches the measuredprofile.

We now investigate how an explicit incorporation of soillayering can be useful in obtaining proper pile response. Forthat purpose, we studied the response of two piles – a shortstubby pile with Lp ¼ 10 m, rp ¼ 0.5 m and Ep ¼ 25 3106 kN/m2 and a long slender pile with Lp ¼ 20 m, rp ¼0.25 m and Ep ¼ 25 3 106 kN/m2 – for various soil profiles.Both piles are subjected to a horizontal force Fa ¼ 1000 kN,and both are assumed to be free at the head and base.

For the short pile (Lp ¼ 10 m), we consider the followingcases:

(a) a homogeneous soil layer with Gs ¼ 25 MPa(b) a two-layer system with H1 ¼ 2 m, Gs1 ¼ 25 MPa and

Gs2 ¼ 50 MPa

2520151050�5Pile deflection: mm

40

30

20

10

0D

epth

,: mz

Present analysis

Sun (1994)

Finite element analysis

Fig. 8. Deflection profile of a 40 m long drilled shaft

Table 2. Soil properties at the pile load test site of McClelland& Focht (1958)

Depth: m Extent of soil layers: m Shear modulus, Gs: MPa

2.0 0 to �4.0 1.66.0 �4.0 to �8.0 4.810.0 �8.0 to �12.0 8.017.5 �12.0 to great depth 14.0


25

20

15

10

5

0

Dep

th,

: mz

MeasuredPresent analysisSun (1994)Randolph (1981)

Fig. 9. Deflection profile for the pile load test of McClelland &Focht (1958)


(c) a two-layer system with H1 ¼ 2 m, Gs1 ¼ 25 MPa andGs2 ¼ 100 MPa

(d ) a two-layer system with H1 ¼ 2 m, Gs1 ¼ 50 MPa andGs2 ¼ 25 MPa.

The Poisson’s ratio was kept constant at 0.25 for all thecases and for all the layers. Figs 10(a), 10(b) and 10(c) showthe pile deflection, bending moment and shear force profiles.

Next, we consider the long pile (Lp ¼ 20 m) and obtainthe pile response for the following cases:

(a) a homogeneous soil layer with Gs ¼ 10 MPa(b) a four-layer system with H1 ¼ 1 m, H2 ¼ 3 m, H3 ¼

5 m, Gs1 ¼ 10 MPa, Gs2 ¼ 20 MPa, Gs3 ¼ 40 MPa andGs4 ¼ 80 MPa

(c) a four-layer system with H1 ¼ 1 m, H2 ¼ 3 m, H3 ¼5 m, Gs1 ¼ 10 MPa, Gs2 ¼ 40 MPa, Gs3 ¼ 40 MPa andGs4 ¼ 80 MPa

The Poisson’s ratio was again assumed to be 0.25 for all thecases and for all the layers. Figs 11(a) and 11(b) show thepile deflection and bending moment profiles for the abovecases respectively.

The effect of soil layering on lateral pile response isevident from Figs 10 and 11. The modulus and thickness ofdifferent soil layers (particularly those near the pile head)have a definite effect on lateral pile response. The examplesshow that proper characterisation of soil deposits and expli-cit accounting for the different layers are necessary foraccurate prediction of pile response and optimal design oflaterally loaded piles. Using our analysis, the three-dimen-sional interaction between pile and soil can be explicitlyaccounted for with full consideration of soil layering. Theassumptions made in the estimation of soil displacements(that the displacements can be represented as products ofseparable variables, and that the vertical displacement iszero), albeit reasonable, do not strictly represent the exactdisplacement field for a pile in an ideal elastic soil: conse-quently, the pile response obtained from this analysis willdeviate, even if slightly, from the actual pile response inelastic soil. Notwithstanding the limitations of these assump-tions, pile response comparable with those obtained fromFEA can be produced at much less time and cost.

In addition to pile deflection, the analysis produces thesoil displacement field surrounding a pile (using equation(2)). Thus, if additional piles are present in the neighbour-hood of a loaded pile, the effect of the loaded pile on theneighbouring piles can be determined by modifying theanalysis. Such an analysis can be further extended to devel-op a method of analysis of pile groups.

The analysis described in this paper is valid for linearelastic soils. As a result, its use is restricted to thoseproblems for which an equivalent elastic soil modulus canbe obtained from field sites. Given that the analysis matchescarefully performed FEA rather well, it can be used as abenchmark in future studies. Additionally, the analysis servesas the basis for more elaborate analysis that can take intoaccount soil non-linearity, because the degradation of soilstiffness resulting from progressive yield due to loading ofthe pile can be obtained from the soil strain field surround-ing the pile (equation (3)), which is available as a result ofthis analysis (in general, modulus degradation of soil de-pends on the strains induced and on the shear strength ofsoil).

CONCLUSIONSAn advanced method of analysis for a single, circular pile

embedded in a multilayered elastic medium and subjected toa horizontal force and a moment at the head was presented.

1086420�2

Pile deflection: mm

8

6

4

2

0

Dep

th,

: mz

(1) / 1·0G Gs2 s1 �

H L1 p/ 0·2�

(2) / 2·0G Gs2 s1 �

(3) / 4·0G Gs2 s1 �

(4) / 0·5G Gs2 s1 �

800 1000600400200�200

Bending moment: kNm

H L1 p/ 0·2�

(1) / 1·0G Gs2 s1 �

(2) / 2·0G Gs2 s1 �

(3) / 4·0G Gs2 s1 �

(4) / 0·5G Gs2 s1 �

(b)

12008004000

(c)

H L1 p/ 0·2�

(1) / 1·0G Gs2 s1 �

(2) / 2·0G Gs2 s1 �

(3) / 4·0G Gs2 s1 �

(4) / 0·5G Gs2 s1 �

Dep

th,

: mz

10

0

10

8

6

4

2

0D

epth

,: mz

�400

Shear force: kN

10

8

6

4

2

0

(a)

Fig. 10. Effect of layering in a two-layer soil: (a) pile deflection,(b) bending moment and (c) shear force of a 10 m long pile


The differential equations governing pile deflection and soildisplacements were derived using energy principles. Theequation of pile deflection was solved analytically, and theone-dimensional FD method was used to solve the soildisplacement equations. The solution is fast, and producesresults comparable with 3D FEA. Using this method, piledeflection, slope of the deflection curve, bending momentand shear force for the entire length of the pile can beobtained if the following are known: the pile radius andlength, the thicknesses of the soil layers, the Young’s mod-ulus of the pile material, the elastic constants of the soil inthe various layers, and the magnitudes of the applied forceand moment.

The solution depends on a set of parameters ª1 to ª6 that

determine the rate at which the displacements in the soilmedium decay with increasing radial distance from the pileaxis. These parameters are not known a priori and must bedetermined iteratively. Hence an iterative scheme was devel-oped and coded to obtain solutions for a variety of boundaryconditions and soil profiles. Notwithstanding the iterationson the ªs, the solutions are obtained in seconds.

Illustrations of use of the analysis for layered soils showthat soil layering has a definite impact on pile response.Hence proper site characterisation and explicit accountingfor the different layers are necessary to predict lateral pileresponse accurately. The present analysis has the capabilityto produce pile response with full consideration of soillayering. The analysis can be further extended to account forsoil non-linearity, and to analyse pile groups.

ACKNOWLEDGEMENTSThis material is based in part upon work supported by the

National Science Foundation under Grant No. 0556347.Tanusree Chakraborty assisted in the finite element analyses,for which we are grateful.

APPENDIX 1The potential energy density of the soil can be expressed in terms

of the displacement functions and elastic constants as

1

2� pq� pq ¼ 1

2ºs þ 2Gsð Þw2 d�r

dr

� �2

cos2 Ł

"

þ 2ºsw2 d�r

dr

�r � �Łð Þr

cos2 Ł

þ ºs þ Gsð Þw2 �r � �Łð Þ2

r2cos2 Łþ Gsw2 �r � �Łð Þ2

r2

þ Gsw2 d�Ł

dr

� �2

sin2 Łþ 2Gsw2 �r � �Łð Þr

d�Ł

drsin2 Ł

þ Gs

dw

dz

� �2

�2r cos2 Łþ Gs

dw

dz

� �2

�2Ł sin2 Ł

#(46)

Substituting equation (46) in equation (1) and performing integra-tions with respect to Ł produces equation (4). Applying the principleof minimum potential energy to equation (4) results in

(ð Lp

0

Ep Ip

d2w

dz2�

d2w

dz2

� �dz:þ �

ð10

ð1rp

�ºs þ 2Gsð Þw �w

d�r

dr

� �2

þ 2ºsw�w�r

r

d�r

dr� 2ºsw�w

�Ł

r

d�r

dr

þ ºs þ 3Gsð Þw�w�2

r

r2þ ºs þ 3Gsð Þw�w

�2Ł

r2

� 2 ºs þ 3Gsð Þw�w�r�Ł

r2þ Gsw�w

d�Ł

dr

� �2

þ 2Gsw�w�r

r

d�Ł

dr

� 2Gsw�w�Ł

r

d�Ł

drþ Gs

dw

dz�

dw

dz

� ��2

r þ Gs

dw

dz�

dw

dz

� ��2Ł

�rdrdz

þ �r2p

ð1L p

Gs

dw

dz�

dw

dz

� �dz�Fa�w

��z¼0

þ M a�dw

dz

� ��z¼0

)

(1) G G G Gs1 s2 s3 s4� � �

(2) 0·5 0·25 0·125G G G Gs1 s2 s3 s4� � �

(3) 0·25 0·25 0·125G G G Gs1 s2 s3 s4� � �


20

16

12

8

4

0

Dep

th,

: mz

H L1 p/ 0·05�

H H

H H1 2

1 3

/ 0·33

/ 0·20

�

�

6004002000

(1) G G G Gs1 s2 s3 s4� � �

(2) 0·5 0·25 0·125G G G Gs1 s2 s3 s4� � �

(3) 0·25 0·25 0·125G G G Gs1 s2 s3 s4� � �

Dep

th,

: mz

H L1 p/ 0·05�

H H

H H1 2

1 3

/ 0·33

/ 0·20

�

�

(b)

(a)

�200Bending moment: kNm

20

16

12

8

4

0

Fig. 11. Effect of layering in a four-layer soil: (a) pile deflectionand (b) bending moment of a 20 m long pile


þ(�

ð10

ð1rp

"ºs þ2Gsð Þw2 d�r

dr

� ��

d�r

dr

� �þºsw2 1

r��r

d�r

drþºsw2�r

r�

d�r

dr

� ��ºsw2�Ł

r�

d�r

dr

� �þ ºs þ3Gsð Þw2�r

r2��r

� ºsþ3Gsð Þw2��r

�Ł

r2þ Gsw2 1

r��r

d�Ł

drþGs

dw

dz

� �2

�r��r

#rdrdz

)þ(�

ð10

ð1rp

"�ºsw2 1

r��Ł

d�r

drþ ºs þ3Gsð Þw2�Ł

r2��Ł

� ºsþ3Gsð Þw2�r

r2��ŁþGsw2 d�Ł

dr�

d�Ł

dr

� �þGsw2�r

r�

d�Ł

dr

� ��Gsw2 1

r��Ł

d�Ł

dr�Gsw2�Ł

r�

d�Ł

dr

� �þGs

dw

dz

� �2

�Ł��Ł

#rdrdz

)¼0

(47)

Simplifying further, and considering a layered system (Fig. 1), we get(ð H1

0

Ep Ip

d4w1

dz4�2t1

d2w1

dz2þ k1w1

� ��w1dz þ

ð H2

H1

Ep Ip

d4w2

dz4�2t2

d2w2

dz2þ k2w2

� ��w2dzþ . .. :þ

ð Lp

H n�1

Ep Ip

d4wn

dz4�2tn

d2wn

dz2þ k nwn

� ��wndz

þð1

Lp

�2tnþ1

d2wnþ1

dz2þ k nwnþ1

� ��wnþ1dz þ Ep Ip

d3w1

dz3�2t1

dw1

dz� Fa

� ��w1

��z¼0

� Ep Ip

d2w1

dz2�Ma

� ��

dw1

dz

� ��z¼0

þ � Ep Ip

d3w1

dz3�2t1

dw1

dz

� ��w1

��z¼H1

þ Ep Ip

d3w2

dz3�2t2

dw2

dz

� ��w2

��z¼H1

" #þ Ep Ip

d2w1

dz2�

dw1

dz

� ��z¼H1

� Ep Ip

d2w2

dz2�

dw2

dz

� ��z¼H1

" #

þ . . . :þ � Ep Ip

d3wn

dz3�2t n

dwn

dz

� ��wn

��z¼Lp

�2tnþ1

dwnþ1

dz�wnþ1

��z¼Lp

" #þ Ep Ip

d2wn

dz2�

dwn

dz

� ��z¼Lp

þ2tnþ1

dwnþ1

dz�wnþ1

��z¼1

)

þð1

rp

�ms1 rd2�r

dr2þd�r

dr

� �þ ms3 þms2ð Þd�Ł

drþms4

�r

r�ms4

�Ł

rþ ns r�r

� ��rdrþms1 r

d�r

dr��r

��1

rp

þms3�r��r

��1

rp

�ms3�Ł��r

��1

rp

( )

þð1

rp

�ms2 rd2�Ł

dr2þd�Ł

dr

� �� ms2 þms3ð Þd�r

drþms4

�Ł

r�ms4

�r

rþ ns r�Ł

� ��Łdrþms2 r

d�Ł

dr��Ł

��1

rp

þms2�r��Ł

��1

rp

�ms2�Ł��Ł

��1

rp

( )¼0

(48)

APPENDIX 2The expanded form of the matrix [¨] in equation (31) is given by

¨½ �4n34n ¼

¨½ �(Head)

~zz¼0 0½ �234 � � � � � � 0½ �234

¨½ �(1)

~zz¼ ~HH1� ¨½ �(2)

~zz¼ ~HH10½ �434 � � � � � � 0½ �434

0½ �434 ¨½ �(2)

~zz¼ ~HH2� ¨½ �(3)

~zz¼ ~HH20½ �434 � � � 0½ �434

..

.0½ �434

. .. . .

. . .. ..

.

..

. ... . .

. . .. . .

.0½ �434

0½ �434 0½ �434 � � � 0½ �434 ¨½ �(n�1)

~zz¼ ~HH n�1� ¨½ �(n)

~zz¼ ~HH n�1

0½ �234 0½ �234 � � � � � � 0½ �234 ¨½ �(Base)

~zz¼1

2666666666666664

3777777777777775

(49)

where [¨](i)

~zz¼ ~HH l(i ¼ 1, 2, . . . n) is a 4 3 4 matrix valid for the ith soil layer. It contains the parameter ~tti, the functions �1, �2, �3, �4 and

their derivatives (Table 1) calculated at the layer interface ~zz ¼ ~HH l (l ¼ i�1 or l ¼ i). In its expanded form, [¨](i)

~zz¼ ~HH lcan be expressed as

¨½ �(i)

~zz¼ ~HH l¼

�1 �2 �3 �4

�91 �92 �93 �94� 01 � 02 � 03 � 04

�1 � 2~tti�-1 �1 � 2~tti�-2 �1 � 2~tti�-3 �1 � 2~tti�-4

2664

3775

~zz¼ ~HH l

(50)

The matrix [¨](Head)~zz¼0 has a dimension of 2 3 4 and is expressed in its expanded form as:

¨½ �(Head)

~zz¼0 ¼

�1 � 2~tt1�-1 �2 � 2~tt1�-2 �3 � 2~tt1�-3 �4 � 2~tt1�-4�91 �92 �93 �94

� �~zz¼0

; fixed-head condition

�1 � 2~tt1�-1 �2 � 2~tt1�-2 �3 � 2~tt1�-3 �4 � 2~tt1�-4� 01 � 02 � 03 � 04

� �~zz¼0

; free-head condition

8>>><>>>: (51)

in which the functions �1, �2, �3, �4 and their derivatives are calculated at ~zz ¼ 0 (pile head). The 2 3 4 matrix [¨](Base)~zz¼1 is given by

¨½ �(Base)

~zz¼1 ¼

�1 �2 �3 �4

�91 �92 �93 �94

� �~zz¼1

; fixed-base condition

�-1 � 2~tt n�91 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2~kk n~ttnþ1

q�1 �-2 � 2~tt n�92 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2~kk n~ttnþ1

q�2

� 01 � 02

"

�-3 � 2~ttn�93 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2~kk n~ttnþ1

q�3 �-4 � 2~ttn�94 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2~kk n~ttnþ1

q�4

� 03 � 04

#~zz¼1

; free-base condition

8>>>>>>>>><>>>>>>>>>:

(52)

in which the functions �1, �2, �3, �4 and their derivatives are calculated at ~zz ¼ 1 (pile base). The matrices [0]234 and [0]434 haverespectively two rows and four columns and four rows and four columns, and contain ‘0’ as all the elements.


The vector [C] (with a dimension of 4n) in equation (31) is givenby:

C½ � ¼ C(1)1 C

(1)2 C

(1)3 C

(1)4 C

(2)1 C

(2)2 C

(2)3 C

(2)4 � � �

h

� � � C(n)1 C

(n)2 C

(n)3 C

(n)4

T

(53)

The superscript T implies the transpose of a matrix.The right-hand side vector of equation (31) is given by

F½ �4n31 ¼ ~FFa F2 0 � � � � � � 0� T

(54)

with F2 ¼ 0 for the fixed-head condition and F2 ¼ ~MMa for the free-head condition.

REFERENCESAnderson, J. B., Townsend, F. C. & Grajales, B. (2003). Case

history evaluation of laterally loaded piles. J. Geotech. Geoen-viron. Engng ASCE 129, No. 3, 187–196.

Ashour, M. & Norris, G. (2000). Modeling lateral soil–pile re-sponse based on soil–pile interaction. J. Geotech. Geoenviron.Engng ASCE 126, No. 5, 420–428.

Banerjee, P. K. & Davies, T. G. (1978). The behaviour of axiallyand laterally loaded single piles embedded in nonhomogeneoussoils. Geotechnique 28, No. 3, 309–326.

Bransby, M. F. (1999). Selection of p–y curves for the design ofsingle laterally loaded piles. Int. J. Numer. Anal. MethodsGeomech. 23, No. 15, 1909–1926.

Briaud, J.-L. (1997). Sallop: simple approach for lateral loads onpiles. J. Geotech. Geoenviron. Engng ASCE 123, No. 10, 958–964.

Briaud, J.-L., Smith, T. & Meyer, B. (1984). Laterally loaded pilesand the pressuremeter: comparison of existing methods. InLaterally loaded deep foundations (eds J. A. Langer, E. T.Mosley and C. C. Thompson), ASTM STP 835, pp. 97–111.Philadelphia, PA: ASTM.

Brown, D. A., Shie, C. & Kumar, M. (1989). P–y curves forlaterally loaded piles derived from three-dimensional finite ele-ment model. Proc. 3rd Int. Symp. on Numerical Models inGeomechanics (NUMOG III), Niagara Falls, 683–690.

Brown, D. A., Hidden, S. A. & Zhang, S. (1994). Determination ofp–y curves using inclinometer data. Geotech. Test. J. 17, No. 2,150–158.

Budhu, M. & Davies, T. G. (1988). Analysis of laterally loadedpiles in soft clays. J. Geotech. Engng Div. ASCE 114, No. 1,21–39.

Cox, W. R., Reese, L. C. & Grubbs, B. R. (1974). Field testing oflaterally loaded piles in sand. Proc. 6th Offshore Technol. Conf.,Houston, TX 2, 459–472.

Gabr, M. A., Lunne, T. & Powell, J. J. (1994). P–y analysis oflaterally loaded piles in clay using DMT. J. Geotech. EngngASCE 120, No. 5, 816–837.

Guo, W. D. & Lee, F. H. (2001). Load transfer approach forlaterally loaded piles. Int. J. Numer. Anal. Methods Geomech.25, No. 11, 1101–1129.

Kim, B. T., Kim, N.-K., Lee, W. J. & Kim, Y. S. (2004).Experimental load-transfer curves of laterally loaded piles in

Nak-Dong river sand. J. Geotech. Geoenviron. Engng ASCE130, No. 4, 416–425.

Klar, A. & Frydman, S. (2002). Three-dimensional analysis oflateral pile response using two-dimensional explicit numericalscheme. J. Geotech. Geoenviron. Engng ASCE 128, No. 9, 775–784.

Matlock, H. (1970). Correlations for design of laterally loaded pilesin soft clay. Proc. 2nd Offshore Technol. Conf., Houston, TX 1,577–594.

McClelland, B. & Focht, J. A. Jr (1958). Soil modulus for laterallyloaded piles. Trans. ASCE 123, 1049–1063.

Mindlin, R. D. (1936). Force at a point in the interior of a semi-infinite solid. Physics 7, May, 195–202.

Ng, C. W. W. & Zhang, L. M. (2001). Three-dimensional analysisof performance of laterally loaded sleeved piles in slopingground. J. Geotech. Geoenviron. Engng ASCE 127, No. 6, 499–509.

Pise, P. J. (1982). Laterally loaded piles in a two-layer soil system.J. Geotech. Engng Div. ASCE 108, No. GT9, 1177–1181.

Poulos, H. G. (1971a). Behavior of laterally loaded piles: I – singlepiles. J. Soil Mech. Found. Div. ASCE 97, No. SM5, 711–731.

Poulos, H. G. (1971b). Behavior of laterally loaded piles: III –socketed piles. J. Soil Mech. Found. Div. ASCE 98, No. SM4,341–360.

Poulos, H. G. (1973). Load–deflection prediction for laterallyloaded piles. Aust. Geomech. J. G3, No. 1, 1–8.

Randolph, M. F. (1981). The response of flexible piles to lateralloading. Geotechnique 31, No. 2, 247–259.

Reese, L. C. & Cox, W. R. (1969). Soil behavior from analysis oftests of uninstrumented piles under lateral loading. In Perform-ance of deep foundations, ASTM STP 444, pp. 160–176.Philadelphia, PA: ASTM.

Reese, L. C. & Van Impe, W. F. (2001). Single piles and pilegroups under lateral loading. Rotterdam: A. A. Balkema.

Reese, L. C., Cox, W. R. & Koop, F. D. (1974). Analysis oflaterally loaded piles in sand. Proc. 6th Offshore Technol. Conf.,Houston, TX 2, 473–483.

Reese, L. C., Cox, W. R. & Koop, F. D. (1975). Field testing andanalysis of laterally loaded piles in stiff clay. Proc. 7th OffshoreTechnol. Conf., Houston, TX 2, 671–690.

Sun, K. (1994). Laterally loaded piles in elastic media. J. Geotech.Engng ASCE 120, No. 8, 1324–1344.

Trochanis, A. M., Bielak, J. & Christiano, P. (1991). Three-dimen-sional nonlinear study of piles. J. Geotech. Engng ASCE 117,No. 3, 429–447.

Verruijt, A. & Kooijman, A. P. (1989). Laterally loaded piles in alayered elastic medium. Geotechnique 39, No. 1, 39–46.

Vlasov, V. Z. & Leont’ev, N. N. (1966). Beams, plates and shells onelastic foundations. Jerusalem: Israel Program for ScientificTranslations.

Winkler, E. (1867). Die Lehre von der Elasticitaet und Festigkeit.Prague: Dominicus.

Wu, D., Broms, B. B. & Choa, V. (1998). Design of laterally loadedpiles in cohesive soils using p–y curves. Soils Found. 38, No. 2,17–26.

Yan, L. & Byrne, P. M. (1992). Lateral pile response to monotonicpile head loading. Can. Geotech. J. 29, 955–970.

Zhang, L., Ernst, H. & Einstein, H. H. (2000). Nonlinear analysisof laterally loaded rock-socketed shafts. J. Geotech. Geoenviron.Engng ASCE 126, No. 11, 955–968.


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