The Japanese Geotechnical Society

Soils and Foundations

Soils and Foundations 2013;53(6):789–803

0038-0http://d

nCorE-m

p_ravanhelnaggPeer

806 & 201x.doi.org/

respondinail addreshenas@ar@eng.ureview un

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Interaction between inclined pile groups subjectedto harmonic vibrations

Mahmoud Ghazavia,n, Pedram Ravanshenasb, M. Hesham El Naggarc

aCivil Engineering Department, K. N. Toosi University of Technology, Tehran, IranbK. N. Toosi University of Technology, Tehran, Iran

cGeotechnical Research Centre, Faculty of Engineering Science, University of Western Ontario, London, Ont., Canada N6A 5B9

Received 31 December 2011; received in revised form 16 June 2013; accepted 24 July 2013Available online 21 November 2013

Abstract

Vertical and inclined piles are used in seismic areas where they could be subjected to oblique harmonic vibration loads. The effect of closelyspaced battered piles on the pile–soil–pile interaction has not yet been fully recognized. A simple analytical method, based on the elasto-dynamictheory by Novak and his associates, is used in the present study to characterize vertical and inclined isolated cylindrical piles subjected to inclinedharmonic vibrations. The free field movement of the ground in the vicinity of the piles is determined using an approximate approach based on theinterference of the cylindrical wave field originating along each pile shaft and spreading radially outward. In calculating the interaction factorbetween two battered piles, an analysis has been carried out to demonstrate the effect of the presence of a neighboring pile (receiver) while thefirst pile (source) is loaded. In this situation, it has been found that the movement of the source pile head is decreased when a receiver pile ispresent. Also, the effects of the pile–pile distance, the group geometry, the length of the piles, and the inclined angle for each or all of the pileshave been studied and the corresponding results will be presented.& 2013 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved.

Keywords: Inclined pile group; Pile–soil–pile interaction; Closed-form solution; Elasto-dynamic theory; Harmonic vibration

1. Introduction

Vertical piles are used in foundations to carry vertical andlateral loads. When the horizontal load per pile exceeds thevalue suitable for vertical piles, batter piles are used along withvertical piles to enhance the lateral stiffness and the capacity ofthe foundation.

3 The Japanese Geotechnical Society. Production and hosting by10.1016/j.sandf.2013.08.009

g author.sses: ghazavi_ma@kntu.ac.ir (M. Ghazavi),dena.kntu.ac.ir (P. Ravanshenas),wo.ca (M.H. El Naggar).der responsibility of The Japanese Geotechnical Society.

Applications of batter piles include offshore structures,bridges, and towers. These types of structures are usuallysubjected to overturning moments due to wind, waves, and theimpact of ships. It is thought that the use of batter and verticalpiles in a pile–soil system increases its overall efficiency. Forexample, Gazetas and Mylonakis (1998) reported that inparticular cases, the use of batter pile groups is advantageousin supporting lateral loading.During the last few decades, attempts have been made to

investigate the behavior of batter piles using both laboratory testsand theoretical studies. Laboratory tests have been performed byTschebotarioff (1953), Murthy (1964), Yoshimi (1964), Prakashand Subramanyam (1965), Kubo (1965), Awad and Petrasovits(1968), Ranjan et al. (1980), Lu (1981), and Sastry et al. (1995).Theoretical investigations have been performed by Alizadeh andDavisson (1970), Poulos and Madhav (1971), and Meyerhof and

Elsevier B.V. All rights reserved.

Fig. 1. Source and receiver batter piles.

M. Ghazavi et al. / Soils and Foundations 53 (2013) 789–803790

Ranjan (1973). Veeresh (1996) conducted model tests on batterpiles subjected to cyclic lateral loading. The seismic response ofbatter piles has been investigated by Lam and Martin (1986).Centrifuge tests have been performed by Juran et al. (2001),Tazoh et al. (2010), and Tamura et al. (2012). Moreover, Gazetasand Mylonakis (1998) and Berrill et al. (2001) have examinedseveral case histories and have highlighted the potential helpprovided by inclined piles. More recently, several researchershave investigated the seismic response of batter piles by Sadekand Shahrour (2004 and 2006), Okawa et al. (2005), Poulos(2006), Deng et al. (2007), Ravazi et al. (2007), Gerolymos et al.(2008), and Padron et al. (2009). The characteristics of dynami-cally loaded vertical piles under vertical vibrations have beeninvestigated using various methods involving lumped masses (ElNaggar and Novak, 1994), the continuum approach (Nogami,1980), the boundary element method (Kaynia and Kausel, 1982),and finite element solutions (Chow, 1985).

Rational analyses of pile group displacements were pio-neered by Poulos (1968), who introduced the concept of‘interaction factors’ and showed that the effects of pile groupscan be assessed by superimposing the effects of only two piles.The interaction between piles vibrating harmonically has alsobeen investigated using various methods. Kaynia and Kausel(1982) used the elastic boundary element method to determinethe interaction factors. Naylor and Hooper (1975) and Ottaviani(1975) used finite element formulations. Randolph and Wroth(1979) used an analytical solution for the deformation of verticalpiles. The results of this simple method were verified by Nogami(1980, 1983), Kaynia and Kausel (1982), Sheta and Novak(1982), El Sharnuby and Novak (1985), and Novak (1984). Allthis research work has focused on vertical piles under lateral andvertical static and harmonic loading (Ghazavi and Ghadimi, 2006;Ghazavi et al., 2009). The behavior of the pile–soil–pile interac-tion of batter piles under various types of loading, however, hasscarcely been reported in literature.

2. Objectives and scope of work

The main objective of this paper is to present a simpleanalytical solution to the problem of dynamic pile–soil–pileinteraction for battered pile groups. It is assumed that pilegroups embedded in a half-space medium with no slippagebetween them are subjected to inclined harmonic forces or toseismic excitation. In addition, an investigation of the applic-ability of Poulos' superposition scheme to evaluate the effect ofdynamic pile groups is presented. The method presented herecalculates the vertical and the horizontal dynamic pile–soil–pile interaction of soil, considering the flexibility of thereceiver pile, and clarifies the shortcomings of the existingmethods that ignore the presence of the receiver pile while thesource pile is loaded.

3. Displacement of source pile

Fig. 1 shows a pair of batter piles inclined at angles θ1 andθ2 with respect to the vertical axis.

An external harmonic load, P¼Po exp(iωt), is applied at thesource pile head at an angle θp with respect to the verticaldirection. Here, Po denotes the load amplitude, ω representsthe excitation frequency, t stands for time, and i¼

ffiffiffiffiffiffiffi�1

p.

The loaded source pile induces the displacement of thesurrounding soil, and the soil induces the displacement of theload-free pile (receiver pile). The presence of the receiver pileusually reduces the soil displacement. If the receiver pile doesnot follow exactly the displacement induced in the surroundingsoil, the reaction of the soil to the pile movement isproportional to relative displacement US(s/d, z)�U21(s/d, z),where US(s/d, z) is the soil displacement and U21(s/d, z) is thereceiver pile displacement produced by loading the source pile.An inclined load, P, can be broken down into two

components (Fig. 2): an axial component, Pv¼PV exp(iωt),and a lateral component, Ph¼PH exp(iωt). The definition ofload angle ζ is given by

ζ¼ θpþθ1 ð1Þ

The excitation amplitude of the two load components may bedivided by the corresponding terms for the complex values ofthe dynamic impedances (lateral stiffness Kh and axial stiffnessKv) and the response amplitude, in which u and w stand forlateral and axial displacements, respectively, as depicted inFig. 2. Then we have

Ph ¼ Kh � u¼ P sin ζ

Pv ¼ Kv � w¼ P cos ζ)

Kh ¼ Ph=u¼ khþ iðchωÞKv ¼ Pv=w¼ kvþ iðcvωÞ

((

ð2Þ

where k and c are interpreted as equivalent ‘spring’ and‘dashpot’ coefficients at the head of the pile and are a functionof the dimensionless frequency, ao¼ωd/Vs, where d and Vs arethe pile diameter and the soil shear wave velocity, respectively.The load distribution along the batter pile length, under axialload and lateral load, is assumed to be similar to that for thevertical pile. This assumption is acceptable for practicalinclination angles of less than 301 (Poulos and Davis, 1980).

Fig. 2. Source batter pile subjected to harmonic force P¼Poexp(iωt) under load angle ζ.

Fig. 3. Illustration of influence of laterally loaded source pile on adjacent free-loaded receiver pile (‘S’ and ‘R’ stand for Source pile and Receiver pile,respectively).

M. Ghazavi et al. / Soils and Foundations 53 (2013) 789–803 791

3.1. Evaluation of horizontal displacement of source pile

The lateral response of a single source pile subjected tohorizontal loading or to deformation due to the surroundingsoil may be calculated using a beam-on-Winkler-foundationformulation. For a two-pile group, with only one pile (sourcepile) loaded laterally, a general approximation method isproposed that involves modeling the soil based on the Winklermethod (Fig. 3).

The pile is considered to be linear elastic with a circularcross-section of AP, a diameter of d, a second moment of areaIP, a Young's modulus of EP, a damping ratio of DP, a massdensity of ρP, a length of L, and a mass per unit length of thepile mP¼ρPAP. The soil parameters are assumed to be Young'smodulus ES, shear modulus GS, shear wave velocity VS,damping ratio DS, and Poisson's ratio ν. The soil reaction onthe pile motion is modeled using continuously distributedfrequency-dependent springs k1 and dashpots k2. The complexhorizontal stiffness per unit length of the pile is given by(Novak, 1974) as

ku ¼ k1þ ik2 ¼ πGa20T ¼G½SU1ða0; υ;DSÞþ iSU2ða0; υ;DSÞ�ð3Þ

where SU1 and SU2 are real and imaginary parts of the layerreaction in the horizontal direction and T is given by

T ¼� 4K1ðbn0ÞK1ðan0Þþan0K1ðbn0ÞK0ðan0Þþbn0K0ðbn0ÞK1ðan0Þbn0K0ðbn0ÞK1ðan0Þþan0K1ðbn0ÞK0ðan0Þþbn0a

n0K0ðbn0ÞK0ðan0Þ

ð4aÞ

bn0 ¼a0i

ηffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ i2DS

p ; η¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð1�υÞð1�2υÞ

s; an0 ¼

a0iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ i2D

pS

ð4bÞ

where K0 and K1 denote the zero-order and the first-ordersecond kind of modified Bessel functions, respectively.The differential equation governing the soil–pile system is

given by (Novak, 1974), namely,

EPIPd4U11ðzÞ

dz4þððk1�mpω

2Þþ iðk2þcωÞÞU11ðzÞ ¼ 0 ð5Þ

For horizontal harmonic vibrations, the horizontal displace-ment assumes u11(z)¼U11(z).exp(iωt), and the solution forEq. (5) is given by

U11ðzÞ ¼ AU11e

RðaþbiÞzþBU11e

�RðaþbiÞzþCU11e

RðcþdiÞzþDU11e

�RðcþdiÞz

ð6ÞConsidering appropriate boundary conditions gives

z�1 )U11 ¼ 0

θ11 ¼ dU11dz ¼ 0

( )) AU

11 ¼ dU11 ¼ 0;

z¼ 0 )θ11 ¼ dU11

dz ¼ 0

V11 ¼ EPIPd3U11dz3 ¼ Ph

8<:

9=; ) ða¼ d; b¼�cÞ

ð7aÞ

BU11 ¼ Ph

EPIPR3½ðaþbiÞðcþdiÞ2�ðaþbiÞ3� ;

CU11 ¼ Ph

EPIPR3½ðcþdiÞ3�ðcþdiÞðaþbiÞ2�ð7bÞ

The final solution is obtained as

u11ðzÞ ¼U11ðzÞeiωt ¼ fBU11e

�RðaþbiÞzþCU11e

RðcþdiÞzgeiωt ð8Þwhere

R¼ ðk1�mpω2Þ2þðk2þcωÞ2ð4EPIPÞ2

� �1=8;

(

M. Ghazavi et al. / Soils and Foundations 53 (2013) 789–803792

θ¼ arctan ðk2þ cωÞðk1�mpω2Þ ; 0oθoπ=2

a¼ cos θ4� sin θ

4 ; b¼ cos θ4 þ sin θ

4

8<: ð9Þ

Soil )

k1 ¼G� SU1k2 ¼G� SU2

cmsω¼ 2DSKS;with : G� SU2ða0; υ;DS ¼ 0ÞKS ¼ δES

δðFixed� head pilesÞ ¼ 3:1097� ðlog EP=ESÞ�0:7413

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

Pile )cmpω¼ 2DPKP

KPðFixed� head pilesÞ ¼ 12EPIPL3

(ð10Þ

Damping constant c represents the energy losses due to theradiation of waves (cr) and the hysteretic dissipating of the soiland the pile (cms, cmp), in which cm incorporates the internalenergy dissipating in the soil, and thus, is related to thedamping ratio (DS). The value for cr is computed from

c� cmþcrcrω¼GSU2 ¼ k2

ð11Þ

3.2. Evaluation of vertical displacement of source pile

For a two-pile group, with only one pile (source pile) loadedvertically, a general approximation method is proposed, asillustrated schematically in Fig. 4.

The complex vertical stiffness per unit length of the pile is(Novak and Aboul-Ella 1978)

kw ¼ 2πGð1þ i2DSÞan0K1ðan0ÞK0ðan0Þ

; an0 ¼ia0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ i2DSp ð12Þ

The complex base stiffness of the pile, kb, is defined as(Bycroft 1956)

kb ¼Gðd=2ÞðCw1þ iCw2Þ

ν¼ 0:25 ) Cw1 ¼ 5:33þ0:364a0�1:41a20Cw2 ¼ 5:06a0

(

Fig. 4. Illustration of influence of vertically loaded source pile on adjacentfree-loaded receiver pile.

ν¼ 0:5 )Cw1 ¼ 8:00þ2:18a0�12:63a20þ20:73a30�16:47a40þ4:458a50Cw2 ¼ 7:414a0�2:986a20þ4:324a30�1:782a40

(

ð13ÞThe equilibrium of the pile–soil system at a certain depth canbe expressed as (Novak, 1974)

mpd2w11ðz; tÞ

dt2þcp

dw11ðz; tÞdt

�EPAEd2w11ðz; tÞ

dz2þkww11ðz; tÞ ¼ 0

ð14ÞFor vertical vibrations, the vertical displacement assumesw11(z)¼W11(z)exp(iωt). By substituting the vertical displace-ment into Eq. (14) and solving this equation, the verticaldisplacement amplitude is given by

W11ðzÞ ¼ AW11expðΛzÞþBW

11expð�ΛzÞ ð15Þwhere

Λ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkw�mpω2þ icpωÞ

EPAP

sð16Þ

where AW11 and BW

11 are the integration constants determinedfrom the boundary conditions. They are

p11ðzÞ ¼ �EPAPdW11ðzÞ

dz ¼�EPAPðΛÞðAW11expðΛzÞ�BW

11expð�ΛzÞÞp11ðz¼ LÞ ¼ kbw11ðz¼ LÞ

ð17Þ

4. Pile–soil–pile interaction

The pile–soil–pile interaction may be determined using thefollowing definition:

α¼ Additional head displacement of second pile caused by first pileHead displacement of first pile considered individually

ð18ÞIn the proposed method, four steps are used to determine theinteraction factor between two batter piles, as shown in Fig. 5.These steps will be described subsequently.

Fig. 5. Four steps for determination of interaction factor for two batteridentical piles.

Fig. 6. a) Illustration of a two-pile group with spacing s/d and departure angle β; b) geometry of batter pile groups with four piles and batter angle θ.

M. Ghazavi et al. / Soils and Foundations 53 (2013) 789–803 793

In Fig. 5, u and w are lateral and axial displacements of thesource batter pile along its own axis, and ut and wt are itsdisplacements along the global horizontal and vertical axes.The attenuated horizontal and vertical ground displacementsadjacent to the receiver pile are u′t and w′t, respectively, andfinally, u′ and w′ are the lateral and axial displacements of thereceiver batter pile along its own axis. The calculation of thesedisplacements is given in detail in the following section.

4.1. Transfer of response of source batter pile to groundresponse [(u,w) to (ut,wt)]

The first step involves the calculation of lateral and axialdisplacements of the source batter pile along its own axis. Inthe second step, these values are projected along the globalaxis of the free field in order to determine the displacement ofthe ground adjacent to the pile (Fig. 5(b)). The horizontaldisplacement, ut, and the vertical displacement, wt, areobtained from

þ cos θ1 � sin θ1

þ sin θ1 þ cos θ1

" #u

w

� �¼

utwt

" #ð19Þ

The resultant displacement of the ground adjacent to the sourcepile, δ, is given by

δ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2t þw2

t

qð20Þ

As illustrated in Fig. 5(b), this resultant displacement isinclined at angle γ with the normal axis (deflection angle),which is given by

tan γ ¼ utwt

ð21Þ

Considering the loading direction of the source pile (θp), thedirection of the soil stiffness (ϕ), and the inclination of thesource pile (θ1), the deflection angle (γ) is given by

tan γ ¼ tan ðθpþθ1Þ= tan ϕ� tan θ11þ tan ðθpþθ1Þ= tan ϕ� tan θ1

ð22Þ

where tanϕ¼Kh/Kv.

4.2. Attenuation of ground displacement away from sourcepile [(ut,wt) to (u′t,w′t)]

The displacement of the ground affects the response of thereceiver pile. Thus, the displacement field around the source

pile is evaluated in order to calculate the displacement of thereceiver pile. This displacement field can be evaluatedconsidering the attenuation of the ground displacement fromthe location of the source pile to the location of the receiverpile located at distance s from the source pile, as shown inFig. 5(c).

4.2.1. Lateral movement [(ut) to (u′t)]The displacement field around the loaded source pile is

approximately evaluated considering an attenuation function.This approximation yields a variation in lateral soil displace-ment US with distance s from the lateral source loaded pile.When a pile located along a line perpendicular to the

direction of loading (β¼901) is affected essentially only byS-waves with phase velocity Vs, the S-waves propagate fromthe active pile. However, when β¼01, the pile is affected bycompression-extension waves coming from the active pile. Forβ¼01, such waves propagate with an apparent phase velocityand they are approximately equal to VLa (Gazetas and Dobrv,1984). In this approach, the displacement of a segment of thereceiver pile at depth z can be estimated from the displacementof the source pile using an interaction factor, ψH. Thisinteraction factor depends on the spacing to diameter ratiobetween the two piles, s/d, the damping ratio of the soil, DS,and the angle β between the line connecting the two piles andthe direction of the horizontal applied force (Fig. 6), i.e.,

usðs=d; β; z; tÞ ¼ ψHðs=d; βÞU11ðzÞexpðiωtÞ ð23ÞIt may be sufficient to compute ψH(s/d,β) for any departureangle of β between piles by using ψH(s/d, β¼01) and ψH(s/d,β¼901). These are given by Gazetas and Dobrv (1984) andexpressed as

αh ¼ ψHðs=d; βÞ � ψHðs=d; 0Þ cos 2βþψHðs=d; π2Þ sin 2β ð24aÞ

ψHðs=d; 0Þ ¼ 2s

d

� ��0:5exp

�DSa0ρ

s

d�0:5

� �� �exp

�ia0ρ

s

d�0:5

� �� �ð24bÞ

ψH s=d;π

2

� �¼ 2

s

d

� ��0:5exp �DSa0

s

d�0:5

� �� �exp �ia0

s

d�0:5

� �� �ð24cÞ

where

a0 ¼ωd

VS; ρ¼ VLa

VS¼ 3:4

πð1�υÞ ð24dÞ

M. Ghazavi et al. / Soils and Foundations 53 (2013) 789–803794

4.2.2. Axial movement [(wt) to (w′t)]In order to calculate the interaction factor between two piles,

the dynamic displacement around a vibrating pile is describedby the following expression, initially introduced by Morse andIngard (1968), as reported by Dobry and Gazetas (1988):

αv ¼ ψVðs=dÞ ¼ ð2ðs=dÞÞ�0:5expð�DSa0ðs=dÞÞexpð�ia0ðs=dÞÞð25Þ

Fig. 7. Batter angle (θ2) of receiver batter pile.

4.2.3. Resultant ground movementThe horizontal and vertical ground displacements at a

distance s from the source pile are given by

u′t ¼ αhut w′t ¼ αvwt ð26ÞThe resultant displacement at distance s from the source pileand departure angle β is δ′ that is inclined at an angle γ′, withrespect to the global vertical axis, as illustrated in Fig. 5.

δ′¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu′2t þw′2t

qð27Þ

tan γ′¼ u′tw′t

ð28Þ

The relationship between the directions of movement δ andmovement δ′ is explained with tanγ and tanγ′, respectively(Fig. 5), as expressed by

tan γ′¼ u′tw′t

¼ αhutαvwt

¼ αhαv

� utwt

ð29Þ

or

γ ¼ γ′þα ð30Þwhere α can be obtained from

tan α¼ tan γ � ð1� tan ðαh=αvÞÞ1þ tan ðαh=αvÞ tan 2γ

ð31Þ

Ιt is noted that αv and αh are not equal; thus, based on Eq. (31),it is concluded that γ′aγ.

4.3. Ground displacement along local axis of receiver batterpile [(u′t,w′t) to (u′,w′)]

Following the calculation of the ground displacementsadjacent to the receiver pile, these values must be projectedto the local axis of the receiver batter pile, as shown in Fig. 7.

Lateral displacement u′ and axial displacement w′ at thereceiver pile head are obtained from

þ cos θ2 � sin θ2

þ sin θ2 þ cos θ2

" #u′tw′t

" #¼ u′

w′

� �ð32Þ

The interaction factor for two batter piles along their ownhorizontal and vertical axes

ααh ¼ u′=u and ααv ¼w′=w ð33Þ

and along the resultant direction is defined as

αt ¼ δ′=δ¼4αt ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu′2t þw′2t

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2t þw2

t

p ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiαh2 � ðutÞ2þαv2 � ðwtÞ2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðutÞ2þðwtÞ2

qð34Þ

By combining Eqs. (19) and (26), the lateral and axialmovements of the receiver batter pile are given by

þαh cos θ1 �αh sin θ1

þαv sin θ1 þαv cos θ1

" #u

w

� �¼

u′tw′t

" #ð35Þ

By combining Eqs. (32) and (35), the following equation canbe obtained:

þ cos θ2 � sin θ2

þ sin θ2 þ cos θ2

" #þαh cos θ1 �αh sin θ1

þαv sin θ1 þαv cos θ1

" #u

w

� �

¼ u′w′

� �or ½I�½II� u

w

� �¼ u′

w′

� �ð36Þ

In Eq. (36), matrix [I] represents the influence of the batterangle of the receiver batter pile and matrix [II] shows both theinfluence of the batter angle of the source pile and the presenceof the receiver pile. For the two batter piles with an equal

M. Ghazavi et al. / Soils and Foundations 53 (2013) 789–803 795

inclination angle (i.e., θ1¼θ2¼θ), we have

þαh cos 2θ�αv sin2θ � αh þαv

2

sin 2θ

þ αh þαv2

sin 2θ þαv cos 2θ�αh sin

2θ

" #u

w

� �¼ u′

w′

� �

orαu′u αu′w

αw′u αw′w

" #u

w

� �¼ u′

w′

� �ð37Þ

Parameters αh and αv, defined in Eqs. (24) and (25), are usedin the absence of the receiver batter pile to determine theinteraction between the two piles.

4.4. Interaction for receiver pile

Consider now, for lateral loading, the second (receiver) pilelocated at distance r¼s from the first (source) pile. Soildisplacement field US(z,t), due to the load on the first pile,excites the second pile, and the horizontal displacement of thereceiver pile is equal to U21(z,t). Considering the dynamicequilibrium for the second pile under lateral loading and usingUS(z) from Eq. (23), the horizontal amplitude of the receiverpile displacement, U21(z), is given by

EPIPd4U21ðzÞ

dz4þððk1�mpω

2Þþ iðk2þcpωÞÞU21ðzÞ¼ ððk1Þþ iðk2þcpωÞÞψHðs=d; βÞU11ðzÞ ð38ÞThe particular solution, UP(z), and the homogeneous solu-

tion, UG(z), are obtained in a similar manner to that used forcomputing U11(z). The horizontal amplitude of the receiverpile displacement, U21(z), is obtained from

U21ðzÞ ¼ ψHðs=d; βÞðUPðzÞþUGðzÞÞ

)UPðzÞ ¼ zðBU

21e�RðaþbiÞzþCU

21eRðcþdiÞzÞ

UGðzÞ ¼ ðB′U21e�RðaþbiÞzþC′U21eRðcþdiÞzÞ

(ð39Þ

where B′U21 and C′U21 are new integration constants determinedfrom the boundary conditions of the receiver pile (i.e., zeroforce and rotation at the pile head). These are

BU21 ¼� ððk1Þþ iðk2þcpωÞ=EPIPÞ

4R3ðaþbiÞ3 BU11;

Fig. 8. Interaction factors for horizontal displacement of pile 2 due to horizon

CU21 ¼

ððk1Þþ iðk2þcpωÞ=EPIPÞ4R3ðcþdiÞ3 CU

11 ð40Þ

Considering Eq. (40), B′U21 and C′U21 are determined from

�RðaþbiÞ RðcþdiÞ�R3ðaþbiÞ3 R3ðcþdiÞ3

" #B′U21C′U21

" #

¼�1 �1

�3R2ðaþbiÞ2 �3R2ðcþdiÞ2" #

BU21

CU21

" #ð41Þ

It is useful to express the pile-to-pile interaction in terms of thedynamic interaction factor. From Eq. (41) and for z¼0, it isobvious that

α′h ¼U21

U11ð42Þ

The horizontal dynamic interaction factor, defined by Eq.(42), is valid when the receiver pile is assumed to bepresent, whereas Eq. (24a)–(24d) is applied when thereceiver pile is assumed to be absent. For verification andto explain the difference between the interaction factors forthe presence and the absence of the receiver pile conditions,a group of two vertical piles with s/d=5 are considered.Other results showed similar behavior. The dynamicresponse results for the fixed-head piles, subjected tohorizontal oscillation, are presented in Fig. 8, where theseinteraction factors are obtained assuming the presence or theabsence of the receiver pile. The results for these two casesare compared with the rigorous solutions presented byKaynia and Kausel (1982). As seen, with the presence ofthe receiver pile, the real and the imaginary parts of thehorizontal interaction factor decrease.Ghazavi and Ravanshenas, 2008a, 2008b found that for a

two-pile group subjected to horizontal loading, consideringthe presence of the receiver pile, the interaction factor issometimes reduced to 25% compared with the conventionalanalysis in which the presence of the receiver pile isignored.Consider now, for vertical loading, a second (receiver) pile

located at a distance r=s from the first pile. The first pile,w21(z,s,t), induces displacements in the soil; in turn, thesoil induces displacements in the second pile (w21(z,t)).

tal force on pile 1 for θ¼01, s/d¼5, L/d¼15, EP/ES¼1000, ρS/ρP¼0.7.

M. Ghazavi et al. / Soils and Foundations 53 (2013) 789–803796

The dynamic equilibrium for the second pile under verticalloading is

mpd2w11ðz; tÞ

dt2�EPAP

d2w11ðz; tÞdz2

þkvðw21ðz; tÞ�w21ðz; s; tÞÞ ¼ 0

ð43Þ

Using w21(z,t)=w21(z)exp(iωt), the particular and generalsolutions for W21(z) are obtained from

W21ðzÞ ¼ WPðzÞþWGðzÞ ¼ΛψVzðζ′Þ

2ð�AW

11expðΛzÞ

þBW11expð�ΛzÞÞþðA′W21expðΛzÞþB′W21expðΛzÞÞ

ð44Þwhere

ðζ′Þ ¼ ðkvþ icvωÞðkv�mPω2þ icvωÞ

ð45Þ

A′W21 and B′W21 are new integration constants determined fromthe boundary conditions of the receiver pile which are

ð1Þ p21ðz ¼ 0Þ ¼ p0 ¼ 0

ð2Þ Kb � w21ðz ¼ LÞ ¼ p21ðz ¼ LÞ ð46Þ

The pile-to-pile interaction for z=0 is expressed as

α′v ¼w21

w11ð47Þ

The vertical dynamic interaction factor, defined by Eq. (47),is applied to consider the presence of the receiver pile,whereas Eq. (25) is used for the case when the receiver pileis absent. For verification, a two-pile group with s/d¼5 isconsidered and the dynamic response results for the fixed-head vertical piles subjected to vertical oscillation arecomputed, as shown in Fig. 9. Other results showed similarbehavior. As seen in Fig. 9, the interaction factors aredetermined assuming the presence or the absence of thereceiver pile and compared with the rigorous solutions givenby Kaynia and Kausel (1982).

Fig. 9. Interaction factors for vertical displacement of pile 2 due to horizon

5. Parametric results and comparisons

The various components of the interaction factor in matrixform, represented by Eq. (37) (for example, αu′u is the portiondisplacement u′ of displacement u), varying with the departureangle (β), the distance ratio (s/d), the dimensionless frequency(a0), the pile diameter (d), and the batter angles of the sourceand the receiver pile (θ1, θ2), are shown in Figs. 10 and 11where either the presence or the absence of the receiver batterpile is considered.In Eq. (37), if parameters αh and αv, defined in Eqs. (24) and

(25), are used, the various components of the interaction factorbetween a two-pile group in matrix form are for the absence ofthe receiver batter pile. If parameters αh and αv, defined in Eqs.(42) and (47), are used, (instead of αh and αv), thesecomponents are for the presence of the receiver batter pile todetermine the interaction between a two-pile group. Otherresults showed a similar trend.From Figs. 10 and 11, it is observed that the absence or

presence of the assumption of the receiver pile leads todifferent interaction factors. If the presence of the receiverpile is ignored, greater vibration amplitudes are computed.This gives greater displacements, and thus, lower pile groupstiffness. As a result, the design of pile groups may beuneconomical. In practice, when the pile group is loaded, allpiles are usually present and installed close to each other.Thus, in the computation of the pile–soil–pile interaction, thepresence of the receiver pile must be taken into consideration.Therefore, the findings of this paper could be used for designpurposes.Fig. 12 shows the influence of the inclined angle of the

receiver pile on the interaction factor, while the presence of thereceiver pile is considered. Fig. 13 shows the variation in theinclined angle of either the source pile or the receiver pile,while the presence of the receiver pile is considered. Changesin the interaction factors are more pronounced with anincreasing batter angle of the receiver batter pile, while thesource pile is vertical. In addition, changes in the interactionfactors are more remarkable with increasing batter angles ofboth source and receiver piles in a two-pile group, as expressedby Eq. (37).

tal force on pile 1, θ¼01, s/d¼5, L/d¼15, EP/ES¼1000, ρS/ρP¼0.7.

Fig. 10. Comparison between various components of interaction factor (I.F) in matrix form represented by Eq. (37) with presence and absence of receiver batter pileand for β¼451, θ1¼θ2¼θ¼151, and S/d¼8.

M. Ghazavi et al. / Soils and Foundations 53 (2013) 789–803 797

Fig. 11. Comparison between various components of interaction factor (I.F) in matrix form represented by Eq. (37) with presence and absence of receiver batter pileand for β¼451, θ1¼01, θ2¼151, and S/d¼8.

M. Ghazavi et al. / Soils and Foundations 53 (2013) 789–803798

Fig. 12. Comparison between various components of interaction factor (I.F) in matrix form represented by Eq. (37) with presence of receiver batter pile and forβ¼451, θ1¼01, θ2¼θ¼01, 10 and 201, and S/d¼5.

M. Ghazavi et al. / Soils and Foundations 53 (2013) 789–803 799

Fig. 13. Comparison between various components of interaction factor (I.F) in matrix form represented by Eq. (37) with presence of receiver batter pile and forβ¼451, θ1¼θ2¼θ¼01, 101 and 201, and S/d¼5.

M. Ghazavi et al. / Soils and Foundations 53 (2013) 789–803800

M. Ghazavi et al. / Soils and Foundations 53 (2013) 789–803 801

6. Conclusions

In this paper, the behavior of a two-batter pile groupsubjected to an inclined external force has been investigatedusing an analytical solution. Parametric studies have beenperformed to determine the interaction factor with various pileproperties, pile geometries, pile distances, soil properties,dimensionless frequencies, and batter angles for the sourceand receiver piles. The effect of the absence or presence of thereceiver pile in a two-pile group has also been investigated.The general findings in this research may be outlined asfollows:

�

For a given batter angle, with an increasing pile spacing anddeparture angle, the interaction between two batter piles decreases.

�

The interaction decreases with an increasing batter anglewith respect to the vertical direction.

�

Assuming the presence of the receiver pile in a two-pile group,the interaction factor between the two piles decreases, resultingin lower vibration amplitudes, greater stiffness, and a moreeconomical design. This reduction with a variation in dimen-sionless frequency, for both real and imaginary components ofinteraction factors for the battered pile group, is non-uniform.

The changes in interaction factors are more pronouncedwith increasing batter angles in a two-pile group.

Appendix

M. Ghazavi et al. / Soils and Foundations 53 (2013) 789–803802

References

Awad, A., Petrasovits, G., 1968. Considerations on the bearing capacity ofvertical and batter piles subjected to forces in different directions. In:Proceedings of the 3rd Budapest Conference on Soil Mechanics andFoundation Engineering. Budapest, pp. 487–497.

Alizadeh, M., Davisson, M.T., 1970. Lateral load test on piles-Arkansas riverproject. Journal Soil Mechanics and Foundation, ASCE 96, 1583–1603.

Berrill, J.B., Chris Tensen, S.A., Keenan, R.R., Okada, W., Pettiga, R.J., 2001.Case study of lateral spreading forces on a piled foundation. Geotechnique51 (6), 501–517.

Bycroft, G.N., 1956. Forced vibration of a rigid circular plate on a semi-infiniteelastic space and on an elastic stratum, Philos. Trans., R. Soc. Lond. Ser. A,248, 327–368.

Chow, Y.K., 1985. Analysis of dynamics behavior of piles. InternationalJournal for Numerical and Analytical Method in Geomechanics 9 (4),383–390.

Deng, N., Kulesza, R., Ostadan, Farhang, 2007. Seismic soil-pile groupinteraction analysis of a battered pile group. In: Proceedings opf the 4thInternational Conference on Earthquake Geotechnical Engineering. 25–28June. Thessaloniki.

Dobry, R., Gazetas, G., 1988. Simple method for dynamic stiffness anddamping of floating pile groups. Geotechnique 38 (4), 557–574.

El Sharnuby, B., Novak, M., 1985. Static and low-frequency response of pilegroups. Canadian Geotechnical Journal 22, 79–94.

El Neggar, M.H., Novak, M., 1994. Non-linear model for dynamic axial pileresponse. Journal of Geotechnical Engineering Division, ASCE 120 (2),308–329.

Gazetas, G., Dobrv, R., 1984. Horizontal response of piles in layered soils.Journal Geotechnical Engineering Division ASCE 110, 2–40.

Gazetas, G., Mylonakis, G., 1998. Seismic soil-structure interaction: newevidence and emerging issues. In: Proceedings of the Conference onGeotechnical Earthquake Engineering and Soil Dynamics. 3–6 August.Geo-Institute ASCE, Seattle.

Gerolymos, N., Giannakou, A., Anastasopoulos, I., Gazetas, G., 2008.Evidence of beneficial role of inclined piles: Observations and summaryof numerical analyses. Bulletin of Earthquake Engineering 6 (4), 705–722.

Ghazavi, M., Ghadimi, M., 2006. A new approach for pile–soil–pile interactionunder static vertical loading. In: Proceedings of the International Con-ference on Coasts, Ports and Marine Structures. Tehran.

Ghazavi, M., Ravanshenas, P., 2008a. Pile–soil–pile interaction under hor-izontal loading: a simple approach. In: Proceedings of the 12th Interna-tional Conference of International Association for Computer Methods andAdvances in Geomechanics. IACMAG, Goa, India, pp. 3399–3407.

Ghazavi, M., Ravanshenas, P., 2008b. “Pile–soil–pile interaction underhorizontal harmonic vibrations: a simple approach. In: Proceedings of8th International Conference on the Application of Stress Wave Theory toPile. Lisbon, pp. 341–351.

Ghazavi, M. Khazaie S., Ravanshenas, P., 2009. A numerical modeling forpile-soil-pile interaction under static horizontal loading. In: Proceedings ofTC International Conference on Deep Foundations-CPRF and EnergyPiles. ISSMGE. Frankfurt.

Juran I., Benslimane A. and Hanna, S. (2001). Engineering Analysis ofDynamic Behavior of Micropile Systems. Transportation Research Record.No. 1772. Soil Mechanics, pp. 91–106.

Kaynia, A.M., Kausel, E., 1982. Dynamic behavior of pile groups. In:Proceedings of the 2nd International Conference on Numerical Methodsin Offshore Piling. Austin, Texas, pp. 509–532.

Kubo, J., 1965. Experimental study of the behavior of laterally loaded piles?In: Procedings of the 6th International Conference SM and FE, vol. 2.

Lam, P.I, Martin, G.R., 1986. Seismic design of highway bridge foundations,vol. II. Design procedures and guidelines. Report No. FHWA/RD-86/102.Federal Highway Administration. Virginia.

Lu, S.S., 1981. Design load of bored pile laterally loaded. In: Proceedingsof the l0th International Conference of Soil Mechanics andFoundation Engineering, vol. 2. Balkema, Rotterdam, The Netherlands,pp. 767–770.

Meyerhof, G., Ranjan, G., 1973. The uplift capacity of foundations underoblique loads. Canadian Geotechnical Journal 10 (1), 64–70.

Morse, P.M., Ingard, K.U., 1968. Theoretical Acoustics. McGraw-Hill, NewYork.

Murthy, V.N.S., 1964. Behavior of battered piles embedded in sand subjectedto lateral loads. In: Proceedings of the Symposium on Bearing Capacity ofPiles. CBRI, Roorkee, India, pp. 142–153.

Naylor, D.J., Hooper, J.A., 1975. An effective stress finite element analysis topredict the short and long-term behavior of a pile raft foundation onLondon clay. In: Proceedings of Conference Settlement Structure, April1974. Cambridge, England, pp. 394–402.

Novak, M., 1974. Dynamic stiffness and damping of piles. CanadianGeotechnical Journal 11 (4), 574–598.

Novak, M., Aboul-Ella, F., 1978. Impedance functions of piles in layeredmedia. Journal of Geotechnical Engineering, ASCE 104 (6), 643–661.

Novak, M., 1984. Evaluation of dynamic experiments on a pile group. Journalof Geotechnical Engineering, ASCE 110, 738–756.

Nogami, T., 1980. Dynamic stiffness and damping of pile groups ininhomogeneous soil. Dynamic response of pile foundation. In: O'Neill,M., Dobry, R. (Eds.), ASCE Special Technical Publication. ASCE,NewYork.

Nogami, T., 1983. Dynamic group effect in axial responses of grouped piles.Journal of Geotechnical Engineering, ASCE 109, 225–243.

Okawa, K., Kamei, H., Zhang, F., Kimura, M., 2005. Seismic performance ofgroup-pile foundation with inclined steel piles. In: Proceedings of the 1stGreece–Japan Workshop on Seismic Design, Observation, and Retrofit ofFoundations. Athens, pp. 53–60.

Ottaviani, M., 1975. Three-dimensional finite element analysis of verticallyloaded pile groups. Geotechnique 25 (2), 159–174.

Padron, L.A., Aznarez, J.J., Maeso, O., Santana, A., 2009. Dynamic stiffnessof deep foundations with inclined piles. Journal Earthquake Engineeringand Structural Dynamics, 1–6.

Prakash, S., Subramanyam, G., 1965. Behavior of battered piles under lateralloads. Journal Indian National Society of Soil Mechanics and FoundationEngineering, New Delhi 4, 177–196.

Poulos, H.G, 1968. Analysis of the settlement of pile groups. Geotechnique,18, 449–471.

Poulos, H.G., Madhav, M.R., 1971. Analysis of Movement of Battered Piles.Research Report No: RJ73. University of Sydney. Sydney, Australia, pp.1–18.

Poulos, H.G., Davis, E.H., 1980. Pile Foundation Analysis and Design. JohnWiley and Sons, New York.

Poulos, H.G., 2006. Raked piles-virtues and drawbacks. Journal of Geotechni-cal and Geoenvironmental Engineering, ASCE 132, 795–803.

Ranjan, G., Ramasamy, G., Tyagi, R.P., 1980. Lateral response of batter pilesand pile bents in clay. Indian Geotechnic Journal, New Delhi 10 (2),135–142.

Randolph, M.F., Wroth, C.P., 1979. An analysis of the vertical deformation ofpile groups. Geotechnique 29 (4), 423–439.

Ravazi, S.A., Fahker, A., Mirghaderi, S.R., 2007. An insight into the badreputation of batter piles in seismic performance of wharves. In: Proceed-ings the 4th International Conference on Earthquake Geotechnical Engi-neering, Thessaloniki, June 25–28 (CD-Rom).

Sadek, M., Shahrour, I., 2004. Three-dimensional finite element analysis of theseismic behavior of inclined micropiles. Journal of Soil Dynamics andEarthquake Engineering 24, 473–485.

Sadek, M., Shahrour, I., 2006. Influence of the head and tip connection on theseismic performance of micropiles. Journal of Soil Dynamics and Earth-quake Engineering 26, 461–468.

Sastry, V.V.R.N., Koumoto, T., Manoppo, F.J., Sumampouw, J.E.R., 1995.Bearing capacity and deflection of laterally loaded flexible piles. BulletinNo. 79. Faculty of Agriculture. Saga 840. Japan, pp. 77–85.

Sheta, M., Novak, M., 1982. Vertical vibration of pile groups. JournalGeotechnical Engineering Division. ASCE 108, 570–590.

Tamura, S., Adachi, K., Sakamoto, T., Hida, T., Hayashi, Y., 2012. Effect ofexisting piles on lateral resistance of new piles. Soils and Foundations 52(3), 381–392.

M. Ghazavi et al. / Soils and Foundations 53 (2013) 789–803 803

Tazoh, T., Sato, J., Jang, Y., Taji, Y., Gazetas, G., Anastaspoulos, 2010. In:Chouw, Pender (Eds.), Kinematic Response of Batter Pile: Centrifuge Test.Soil-foundation-Structure Interaction-Orense. Taylor & Francis Group,London, pp. 41–47.

Tschebotarioff, G.P., 1953. The resistance to lateral loading of single piles andpile groups. Specific publisher. No. 154. ASTM, West Conshohocken,pp. 38–48.

Veeresh, C., 1996. Behavior of Batter Piles in Marine Clays. Indian Institute of

Technology, Madras, India (Ph.D. thesis).Yoshimi, Y., 1964. Piles in cohesionless soils subjected to oblique pull.

Journal Soil Mechanics and Foundation Design, ASCE 90 (6), 33–41.