nonlinear analysis of torsionally loaded pile groups

275

i) Associate Professor, MOE Key Laboratory of Soft Soils and Geoenvironmental Engineering, Zhejiang University, China (klg＠zju.edu.cn)(formerly Postdoctoral Research Associate, Hong Kong University of Science and Technology).

ii) Associate Professor, Hong Kong University of Science and Technology, Hong Kong (cezhangl＠ust.hk).The manuscript for this paper was received for review on April 3, 2008; approved on January 29, 2009.Written discussions on this paper should be submitted before November 1, 2009 to the Japanese Geotechnical Society, 4-38-2, Sengoku,Bunkyo-ku, Tokyo 112-0011, Japan. Upon request the closing date may be extended one month.

275

SOILS AND FOUNDATIONS Vol. 49, No. 2, 275–286, Apr. 2009Japanese Geotechnical Society

NONLINEAR ANALYSIS OF TORSIONALLY LOADED PILE GROUPS

L. G. KONGi) and L. M. ZHANGii)

ABSTRACT

An empirical approach is developed to analyze the nonlinear torsional behavior of free-standing pile groups withrigid pile caps. In this approach, the lateral and torsional responses of individual piles in a pile group are modeled byp-y and t-u curves; the interaction among lateral resistances of the individual piles is predicted through Mindlin's elas-tic solutions; the interactions between the torsional and lateral resistances of the individual piles are described throughRandolph's solution; and the coupling eŠect of lateral resistance on torsional resistance of the individual piles is quan-tiˆed using an empirical factor ``b''. The proposed approach is capable of capturing the most signiˆcant aspects ofpile-soil-pile interactions and coupling eŠect in pile groups subjected to torsion. The proposed approach is veriˆed us-ing results of centrifuge model tests. In general, the applied torque-twist angle response and the transfer of applied tor-que in pile groups can be reasonably well predicted and are sensitive to the pile group conˆguration.

Key words: centrifuge tests, nonlinear analysis, pile foundations, pile group, pile-soil-pile interaction, torsionalresponse (IGC: E12)

INTRODUCTION

Foundations for oŠshore platforms, bridge bents, andtall buildings are subjected to signiˆcant torsional loadsby virtue of eccentric lateral loading from wind and waveaction, ship impacts or high-speed vehicles. In the past,researchers have developed numerical solutions for singlepiles subjected to torsion (e.g., Poulos, 1975; Randolph,1981; Chow, 1985; Georgiadis, 1987; Guo and Randolph,1996). Zhang and Tsang (2005) studied the behavior of atorsionally loaded 2×2 bored pile group using a three-dimensional ˆnite diŠerence method. Recently, Kong(2006) and Kong and Zhang (2007b) reported a series ofcentrifuge model tests to investigate load sharingmechanisms and pile-soil-pile interactions in three-diameter spaced 1×2, 2×2 and 3×3 ˆxed-head pilegroups subjected to torsion. From these studies, the fol-lowing attributes of torsionally loaded pile groups wereobserved:1. A pile group subjected to torsion simultaneously

mobilizes lateral and torsional resistances of individ-ual piles, as shown in Fig. 1. The torsional resistancesresist 20¿50z of the applied torque; the mobilizationof lateral resistance is closely related to pile locationswithin the group and pile-soil-pile interactions.

2. The eŠect of horizontal movement of a pile on thelateral behavior of its adjacent piles or ``shadowing''eŠect is also present in the pile groups subjected to tor-sion. However, the interaction eŠects in pile groups

subjected to torsion and lateral loading are signiˆcant-ly diŠerent, because the directions and magnitudes ofhorizontal movements of individual piles in pilegroups subjected to torsion are diŠerent (see Fig. 1).In addition, the horizontal movement of a pile aŠectsthe torsional behavior of its adjacent piles.

3. The lateral soil reaction on a pile in a pile group tendsto increase the torsional resistance of the pile, which isreferred to as de‰ection-torsion coupling eŠect.To date, few computer programs for routine pile de-

sign simulate the behavior of pile groups subjected to tor-sion considering the pile-soil-pile interactions (alsoknown as ``PSPI'') and the de‰ection-torsion couplingeŠect. MPILE, a program originally developed by Ran-dolph (1980) under the name of PIGLET, uses approxi-mate analytical solutions for the lateral and torsionalstiŠnesses of individual piles in a group, and takes intoaccount the eŠect of lateral resistances of the individualpiles on the torsional stiŠness of the pile group using in-teraction factors. The program is considered applicablefor the analysis of pile groups under small deformations.GROUP (Reese et al., 2000) and FB-Pier (Hoit et al.,2001), based on Winkler's spring idealization of soil, em-ploy load-transfer functions to represent the relationshipbetween a load at any point along a pile and the associ-ated soil deformation at that point. The load-transfer ap-proach has been widely adopted for routine design, espe-cially where nonlinear soil behavior has to be consideredand/or soil stratiˆcation is complex. In the load-transfer

276

Fig. 1. Load distribution in a pile group subjected to torsion and dis-cretization of piles

Fig. 2. Pile-soil-pile interactions

276 KONG AND ZHANG

approach, pile-soil-pile interactions are taken into ac-count using empirical factors, which are usually obtainedby back-calculating pile load test results. For example, toanalyze laterally loaded pile groups, a p-multiplier factor,ˆrst suggested by Brown et al. (1988), is employed inGROUP and FB-Pier to quantify the loss of soilresistance due to ``shadowing'' eŠects (i.e., the geometricnonlinearity that in‰uences the lateral response of a pileas lateral support is withdrawn from the soil in front ofthat pile as the pile positioned forward of it moves in thesame direction as the piles in the group). As reported byKong (2006), ``shadowing'' eŠects in pile groups subject-ed to torsion and lateral loading are signiˆcantly diŠer-ent, so empirical factors back-calculated from tests onlaterally loaded pile groups may not be applicable to tor-sionally loaded pile groups.

This paper describes a rational numerical approach totake into account several attributes of torsionally loadedpile groups; namely, (1) nonlinear behavior of soil adja-cent to piles in pile groups subjected to torsion, (2) pile-soil-pile interactions, and (3) the de‰ection-torsioncoupling eŠect. Formulations for the approach will bepresented ˆrst. Next, the experimental results reported byKong (2006) and Kong and Zhang (2007b) are used toverify the accuracy of the proposed approach. Finally, in-‰uences of the pile-soil-pile interactions and the de‰ec-

tion-torsion coupling eŠect on response of pile groups arestudied.

PILE-SOIL SYSTEM

A pile-soil-pile interaction problem, as represented inFig. 2, can be decomposed into two domains; namely(1) the pile domain, i.e., the group piles subjected to ex-

ternal loads sQtand pile-soil interaction forces act-ing on the piles, sPpt;

(2) the soil domain, i.e., the soil mass acted on by a sys-tem of pile-soil interaction forces sPst, at the bound-ary of the pile-soil interface.

The pile domain and the soil domain interact with eachother through the pile-soil interaction forces, as shown inFig. 2, and the compatibility of the deformations of thesoil and pile domains.

Pile DomainThe pile shafts are assumed to be linear elastic and

obey the small deformation assumption; the piles are ver-tical and ˆxed to the pile cap; the pile toes are subjectedto torsional and vertical restraints; and the pile cap isrigid and not in contact with the ground. Thus, an in-dividual pile in a pile group under torsional loading issubjected to a lateral load, a bending moment, and a tor-sional load at the pile head ( see Fig. 1). Assuming eachpile works as a simple beam, the governing fourth andsecond order diŠerential equations for lateral de‰ection yand twist angle u of the pile can be given by,

277277NONLINEAR ANALYSIS OF TORSIONALLY

EpIp&4y&z4＋khy＝0 (1)

GpJp&2u&z2－kuu＝0 (2)

where Ep and Gp are the elastic modulus and shear modu-lus of pile shaft, respectively; Ip and Jp are the second mo-ment and polar second moment of area of pile section, re-spectively; z is depth; and kh and ku are the moduli of sub-grade reaction for lateral loading and torsional loading,respectively. Each pile is modeled by a number of discretebeam elements. Figure 1 shows the node numberingmethod employed in the proposed approach. Based onthe ˆnite element method (e.g., Smith, 1982), the load-deformation relationship is written as

[Kp]sWpt＝sQt＋sPpt (3)

where [Kp] is the global stiŠness matrix of all elements ofthe group piles; sWptis the vector of deformations at thepile nodes.

Soil DomainIt is assumed that nonlinear behavior of a pile group is

due to the nonlinear soil response in the near êld (i.e., atindividual piles in the group, represented by p-y and t-ucurves, where p is the lateral soil reaction; y is the lateralpile de‰ection; t is the torsional shear stress; and u is thelocal twist angle of pile shaft) but the far-êld interac-tions (i.e., pile-soil-pile interactions) are linear elastic. Inthe present approach, a ``lumped'' formulation in whichsoil stiŠness is lumped at the pile nodes is adopted, whichis adequate in most practical problems. Thus, the soildeformation at node i due to its own loading as well asloadings at other nodes, si, can be obtained by superpo-sition:

si＝n

Sj＝1

fijPsj (4)

where fij is the ‰exibility coe‹cient denoting the deforma-tion at node i due to a unit load at node j; Psj is the pile-soil interaction force acting on the soil at node j; and n isthe total number of nodes. Equation (4) can be writtenfor each node, leading to the following ‰exibilityrelationship for the soil

sWst＝[Fs]sPst (5)

where sWstis the vector of soil deformation; [Fs] is thesoil ‰exibility matrix; and sPstis the vector of pile-soil in-teraction forces acting on the soil.

In torsionally loaded pile groups, the pile-soil interac-tion forces acting on the soil at node j, Psj, include a later-al force and a torsional force, denoted as Hsj and Tsj, re-spectively. The corresponding ‰exibility coe‹cients arealso divided into two parts, denoted as f H

ij and f Tij. Thus,

Eq. (4) can be decomposed as,

si＝j＝n

Sj＝1

( f Hij Hsj＋f T

ijTsj)＝j＝n

Sj＝1

[ f Hij f T

ij]{Hsj

Tsj}. (6)

Since the soil deformation at node i, si in this case, in-cludes a lateral component, H

si , and a torsional compo-nent, T

si, Eq. (6) is further decomposed as,

{HsiTsi}＝

j＝n

Sj＝1 « f HH

ij

f THij

f HTij

f TTij ${Hsj

Tsj} (7)

where f HHij and f TH

ij are the two components of f Hij ; f HT

ij andf TT

ij are the two components of f Tij. The physical meaning

of each of these ‰exibility coe‹cients is:f HH

ij : the lateral component of the soil deformation atnode i due to a unit lateral force at node j;

f THij : the torsional component of the soil deformation at

node i due to a unit lateral force at node j;f HT

ij : the lateral component of the soil deformation atnode i due to a unit torsional force at node j;

f TTij : the torsional component of the soil deformation at

node i due to a unit torsional force at node j.In terms of sources of loading, the soil deformation at

a node consists of the one due to the forces at the nodeand the one due to the forces at other nodes. So H

si andTsi are each further divided into two components:

Hsi＝

Hsii＋

Hai (8)

Tsi＝

Tsii＋

Tai (9)

where sii represents the soil deformation at node i as aresult of forces at the same node; ai represents the addedsoil deformation at node i as a result of forces at nodesother than i. Referring to Eq. (7), Eqs. (8) and (9) can berewritten as

Hsi＝

Hsii＋

Hai

＝( f HHii Hsi＋f HT

ii Tsi)＋j＝n

Sj＝1j$i

( f HHii Hsj＋f HT

ii Tsj) (10)

Tsi＝

Tsii＋

Tai

＝( f THii Hsi＋f TT

ii Tsi)＋j＝n

Sj＝1j$i

( f THii Hsj＋f TT

ii Tsj) (11)

The matrix form of Eqs. (10) and (11) is in essence thesame as Eq. (5). Inherent in the load-transfer approachfor modeling soil behavior in a single pile is the assump-tion that the soil reactions are uncoupled; that is, the dis-placement at a particular node will only aŠect the soilreaction at that node. Thus, for loadings at node j whichis associated with the same pile as node i, and for j»i, thevalues of the ‰exibility coe‹cients, f HH

ij , f HTij , f TH

ij and f TTij

are zero. The calculation of the non-zero ‰exibilitycoe‹cients in Eqs. (10) and (11) is presented in the nextsection.

Flexibility Coe‹cients of Soilf HH

ii and f TTii

f HHii and f TT

ii are used to model soil response in the nearêld. f HH

ii denotes the lateral component of soil deforma-tion at node i due to a unit lateral force at the same node;f TT

ii is the torsional component of soil deformation atnode i due to a unit torsional force at the same node. Inthis paper, nonlinear p-y and t-u curves are used to calcu-late the two ‰exibility coe‹cients:

278

Fig. 3. Modeling of soil responses and de‰ection-torsion couplingeŠect

278 KONG AND ZHANG

f HHii ＝

1khd

(12)

f TTii ＝

2p

1kuD2d

(13)

where D is the pile diameter and d is the pile segmentlength. The soil reaction moduli kh and ku, which varywith load, are obtained using p-y and t-u curves. In theliterature, many p-y curves were recommended for diŠer-ent soils (e.g., Reese et al., 1974; Yan and Byrne, 1992;Kong, 2006); several t-u curves for sand were alsoproposed (Dutt and O'Neill, 1983; Hoit et al., 2001;Kong, 2006).

f THii and f HT

ii

f THii and f HT

ii re‰ect the eŠect of the lateral force/tor-sional force on the torsional force/lateral force in an in-dividual pile, which are named `coupling eŠect' by Kong(2006). f TH

ii represents the torsional component of soildeformation at node i due to a unit lateral force at thesame node. f HT

ii represents the lateral component of soildeformation at node i due to a unit torsional force at thesame node. Kong (2006) found from centrifuge modeltests that the latter eŠect is minor, so f HT

ii is ignored in thepresent approach.

To further clarify the physical interactions behind f THii ,

Tsii in Eq. (11) is rewritten as

Tsii＝f TH

ii Hsi＋f TTii Tsi＝Ø1＋f TH

ii Hsi

f TTii Tsi

»f TTii Tsi＝

f TTii

aTH(i)Tsi (14)

where aTH(i) is a modiˆcation factor (or interaction factor)at node i and expressed as

aTH(i)＝f TT

ii Tsi

f THii Hsi＋f TT

ii Tsi＝1－

f THiiTsii

Hsi. (15)

If one employs the same numerical technique to analyzethe response of a torsionally loaded single pile, the tor-sional soil deformation at node i, si, is then

si＝fiT?si (16)

where fi and T?si are the soil ‰exibility coe‹cient and thetorsional force at node i in the single pile. Assuming T

sii

＝ si and f TTii ＝fi, one obtains

Tsi＝aTH(i)T?si. (17)

Equation (17) reveals that aTH(i) is the ratio of the tor-sional force at node i of a group pile to that at the corre-sponding node on a single pile at the same twist angle.aTH(i) is a function of Hsi ( see Eq. (15)). Therefore, f TH

ii

re‰ects the eŠect of the lateral force Hsi on the torsionalforce Tsi at the same node.

Deˆning a new coupling coe‹cient, b,

b＝－f TH

iiTsii

paDd (18)

where pa is the atmospheric pressure, Eq. (15) becomes

aTH(i)＝1＋Ø bpaD» Hsi

d. (19)

Because lumped formulations are used in the approach,Hsi/d is the average soil reaction on the pile elementwhose central point is i, denoted as pi. If the two sides ofEq. (17) are divided by pD2d/2, aTH(i) can be expressed ina stress form as,

aTH(i)＝šts

št?s＝1＋Ø b

paD»pi (20)

where šts and št?s are the average torsional shear stresses atthe pile element on the group pile and the single pile, re-spectively. In Eq. (20), b quantiˆes the contribution ofsubgrade reaction to the increase in the torsional shearresistance at the soil-pile interface. Figure 3 illustrates thephysical meaning of b.

f HHij and f TT

ij (i»j )f HH

ij (i»j ), the ‰exibility coe‹cient due to the lateral in-teraction between two piles, is obtained from Mindlin(1936)'s solutions for the in‰uence of a unit lateral pointforce in a homogeneous, isotropic elastic half-space. Thistechnique has been used by Poulos (1971), Leung andChow (1987) and others for the analysis of laterallyloaded pile groups. Non-homogeneous soils can be tack-led by means of an averaging procedure described byChow (1986). However, for the case of pile groups sub-jected to torsion, the angle between the lateral loads ontwo arbitrary piles could be of any value, so a moregeneral analysis of lateral interaction between two piles isneeded.

Based on Mindlin's solutions, the lateral displacementof a given point in an arbitrary direction due to a unitlateral load in an interior elastic half space is derived inAPPENDIX. In the present approach, the soil ‰exibilityis condensed at the nodes of the pile elements, so the soildisplacement at node i on a pile due to a unit lateral forceat node j on another pile, f HH

ij (i»j ), therefore, can be cal-culated using Eq. (A5),

f HHij ＝uj cos gij＋nj sin gij (i»j ) (21)

where uj and nj are the soil displacements at node j on apile in the same direction and in the perpendicular direc-


tion of the unit force at node i, respectively; gij is the an-gle between the directions of the two lateral forces atnode i and node j.

f TTij (i»j ) represents the torsional component of soil

deformation at node i on a pile due to a unit torsionalforce at node j on another pile. This eŠect is omitted inthis paper since both Poulos (1975) and Kong (2006)found that the interaction between two three-diameterspaced piles, if any, is negligible.

f HTij and f TH

ij (i»j )f HT

ij (i»j ) represents the lateral component of soildeformation at node i due to a unit torsional force atnode j on another pile. f TH

ij (i»j ) represents the torsionalcomponent of soil deformation at node i due to a unitlateral force at node j on another pile. f HT

ij (i»j ) can becalculated using Randolph (1981)'s analytical solutionfor torsionally loaded single piles. In Randolph's solu-tion, the soil was treated as independent horizontal lay-ers, so, at a particular depth, the relationship between thecircumferential soil displacement at a point, r, and thetorsional shear stress on the soil-pile interface t is

&&r Ø r

r »＝ tD2

4Gsr3 (22)

where r is the distance from the centre of pile to the point(rÆD/2); Gs is the shear modulus of soil. Integrating theabove equation yields the circumferential soil displace-ment at the point,

r＝tD2

8Gsr. (23)

In the present approach, torsional shear stresses on pileelements are condensed at nodes. Assume the adjacentpiles near the pile subjected to torsion follow exactly thefree-ˆeld soil displacement. Given a lumped torque atnode i, Ti＝p štsD2d/2, the induced circumferential soildisplacement at node j on another pile rj is,

rj＝Ti

4pGssd(24)

where s is the center-to-center spacing between the twopiles. Thus, f HT

ij can be expressed as:

f HTij ＝

rj

Tisin c＝

sin c4pGssd

(i»j ) (25)

where c is the angle between the line jointing nodes i andj and the direction of the lateral loading at node j. Equa-tion (25) is valid only when nodes i and j are in the samedepth. Otherwise, f HT

ij (i»j ) is zero.Consider now the in‰uence of the horizontal loading of

a pile on the rotation of a neighboring pile. From thereciprocal theorem, the rotation at node i on pile 1, due toa unit lateral force at node j on pile 2, must be equal tothe soil displacement at node j on pile 2 in the loadingdirection due to a unit torsional force at node i on pile 1,that is

f THij ＝f HT

ij ＝sin c

4pGssd(i»j ). (26)

GOVERNING EQUATION AND CONSTRAINTCONDITIONS OF PILE-SOIL SYSTEM

The ‰exibility matrix [Fs] in Eq. (5) is inverted to givethe following stiŠness relationship for the soil

sPst＝[Ks]sWst (27)

where [Ks] (＝[Fs]－1) is the soil stiŠness matrix. Equilibri-um of the interaction forces acting at the pile-soil inter-face yields

sPst＝－sPpt. (28)

Assuming no separation between the soil and the piles,the compatibility of the deformations of the soil and thepiles yields

sWst＝sWpt. (29)

Using Eqs. (3) and (27)–(29), the load deformationrelationship of the pile group system is expressed as

([Kp]＋[Ks])sWpt＝sQt. (30)

In Eq. (30), if [Ks] is assumed constant, the pile groupsystem becomes linear elastic. If nonlinear p-y and t-ucurves are used, [Ks] will vary with soil displacements andthe system becomes non-linear. It is noted that the soilstiŠness matrix [Ks] obtained should be augmented withzeros in appropriate rows and columns corresponding tothe rotational degrees-of-freedom of the pile to be com-patible with the pile stiŠness matrix [Kp].

In this paper, the pile cap in a pile group is assumedrigid and the pile-cap connection is assumed ˆxed, so ro-tation of all the pile heads in the vertical plane should bezero, and the twist of all the pile heads should be equal tothe twist of the pile cap. The pile-head lateral displace-ment of an individual pile is equal to the twist angle of thepile group times the distance from the pile to the torsionalcentre of the group. In addition, the force equilibrium ofthe pile cap should be satisˆed:

N

SI＝1

(TI＋HIsI)＝Tg (31)

where TI and HI are the pile-head torque and shear forceon pile I; sI is the distance between the torsional centre ofthe pile group to pile I; Tg is the applied torque on the pilecap; N is the total number of piles in the pile group.

An iteration technique is used to solve Eq. (30). First,assume an initial value of [Wp], denoted as [Wp](0). Then,substitute the initial value into [Ks] and solve Eq. (30) toobtain the ˆrst iterative solution. After that, the diagonalelements of [Ks] are calculated with the secant moduli ofsoil subgrade reaction with respect to the ˆrst solution of[Wp]. Repeating the steps m times, the mth iterative solu-tion is

[Wp](m)＝([Kp]＋[Ks](m－1))－1sQt (32)

280

Fig. 4. Pile group dimensions and conˆgurations (not to scale)

280 KONG AND ZHANG

where [Ks](m－1)＝[Ks(W (m－1))]. The convergence criterionis

n

Si＝1

mpi－

n

Si＝1

m－1pi

n

Si＝1

m－1pi

ºe (33)

where pi is the pile deformation at node i and e is the al-lowable tolerance, which is taken as 10－3 in the numericalcalculation. When two consecutive iterative solutionssatisfy a convergence criterion, the iteration is terminat-ed. The iteration technique is convergent as long as thep-y curves and t-u curves are convex. An in-house pro-gram, NATLPG, was developed using Mathematica 5 forthe present approach.

COMPARISON OF NUMERICAL ANDEXPERIMENTAL RESULTS

Centrifuge Model TestsResults of the centrifuge model tests reported by Kong

(2006) and Kong and Zhang (2007b) are utilized to verifythe proposed approach. Aluminum model piles 19 mm indiameter and 300 mm in length were employed in the cen-trifuge model tests. The model piles were rigidly connect-ed to aluminum pile caps at three-diameter spacing.Three pile group conˆgurations, 1×2, 2×2 and 3×3,were tested as shown in Fig. 4. Before starting the cen-trifuge, each of the pile groups was ˆrst jacked into thesand bed to 90 mm using a pile jacking device. After thecentrifuge was gradually accelerated to the desired g level,the pile group was jacked at 1 mm/s to the ˆnal embed-ment depth of 270 mm. Then, the pile group was tor-sionally loaded in increments using two horizontal actua-tors. At each load level, the load was kept constant untilno further variations in twist angle were observed. Fourlaser sensors were used to measure the movements of thepile cap at four points. Any three of the four displace-ment measurements could be used to calculate the twistangle and the two horizontal displacements of the pile

cap. These tests were conducted at 40 g, thus the modelpile groups simulated three-diameter spaced closed-endpipe pile groups with a prototype outside diameter of0.76 m and an embedded pile length of 10.8 m, as shownin Fig. 4. The ‰exural stiŠness of piles is 220.5 MN.m2,the Poisson's ratio is 0.3, and the torsional rigidity of thepiles is 169.9 MN.m2. The simulated prototype pile capsare 1.2 m thick and 1.2 m above the ground surface inprototype. Details of the pile group tests have been de-scribed by Kong and Zhang (2007b).

Leighton Buzzard sand was used in these tests. Thesand was a quartz-based uniform sand, with the grain sizeranging from 0.09 to 0.15 mm. The average grain size,which is the grain diameter at which 50z of the soil isˆner by weight, was 0.14 mm. Two dry soil densities(13.76 kN/m3 and 14.83 kN/m3) were used, with relativedensities of 35z for the loose sand and 75z for the densesand after considering the settlement of the sand bed atthe 40 g acceleration.

Comparison of Numerical and Test ResultsTwo sets of soil parameters are required. One set is the

soil subgrade reaction parameters for calculating f HHii and

f TTii ; another set is soil modulus, Poisson's ratio and shear

modulus of soil for calculating f HHij , f HT

ij and f THij .

Kong (2006), based on results of laterally loaded andtorsionally loaded single pile tests (Kong and Zhang,2007a; Zhang and Kong, 2006), proposed a series of ex-ponential p-y curves, hyperbolic t-u curves, and hyper-bolic toe torsional resistance curves:

pD

＝kA0lz Ø zD »

0.5

Ø yD »

0.5

(34)

t＝Atfu

Au＋tf(35)

Tt＝AtBtut

Atut＋Bt(36)

where k is a ˆtting coe‹cient, 0.01; z is the embeddeddepth; A0 is a factor related to soil density and soil stressstates; lz is a reduction factor for considering the eŠect ofground surface; Tt is the toe torsional resistance; ut is thelocal twist angle of the pile toe; A and At are the initialslopes of a hyperbolic t-u curve and a hyperbolic toeresistance curve, respectively; and tf and Bt are the ulti-mate shaft torsional shear stress and toe torsionalresistance, respectively. Factors A0, lz, A, At, and Bt areexpressed as,

A0＝2G0(1＋y0)(2.97－e0)2

1＋e0Ø2K0＋1

3Dg?0pa»

0.5

(37)

lz＝Ø zzc»

0.5

(zºzc) (38)

A＝2cG0(2.97－e0)2

1＋e0Ø2K0＋1

3Dg?0pa»

0.5

(39)

At＝23

D3Gt (40)

281

Fig. 5. Comparison of single pile tests and numerical analyses

Table 1. Key parameters for p-y curves, t-u curves, and toe torsionalresistance curves

n0 e0g?0

(kN/m3) K0pa

(kPa)Gt

(MPa)ttf

(kPa)

Loose sand 0.2 0.897 13.76 0.4 101.3 123 509Dense sand 0.3 0.761 14.83 0.5 101.3 282 2192


Bt＝112

pD3ttf (41)

where G0 is a material parameter, taken as 387, which isthe average value for the Leighton Buzzard sand meas-ured by Cai (2001) with the maximum discrepancy within10z; n0, e0 and g?0 denote the initial Poisson's ratio, ini-tial void ratio, and eŠective unit weight of sand beforepile jacking, respectively; K0 is the coe‹cient of earthpressure at rest, recommended to be 0.4 for loose sandand 0.5 for dense sand by Terzaghi and Peck (1948); zc isthe critical depth, Baguelin et al. (1978) found the criticaldepth for granular soil is of the order of 4D; lz is 1.0when z is larger than zc; c is an empirical factor; and Gt

and ttf are the initial shear modulus of pile toe-soil inter-face and the ultimate torsional shear stress at the pile toe,respectively. c, tf, Gt and ttf are curve-ˆtted from the ex-perimental t-u curves and the toe torsional resistancecurves reported by Zhang and Kong (2006). Figure 3shows the schematic shape of the p-y and t-u curves; fur-ther details of the p-y curves, t-u curves, and the toe tor-sional resistance curves are described by Kong (2006).Figure 5 compares the simulated and measured horizon-tal force-displacement curves and torque-twist anglecurves for the single piles, which demonstrates the valida-tion of the proposed load transfer curves. Some keyparameters for the p-y curves, the t-u curves, and the toetorsional resistance curves used to simulate the single piletests, as well as the pile group tests, are summarized inTable 1. Distributions of c and tf are available fromKong (2006).

The shear modulus of soil is a key parameter for pilegroup interactions. It is calculated from the soil modulus

and Poisson's ratio. The modulus of cohesionless soilsproposed by Poulos (1971) is used in the study. Theproposed values of soil modulus are 0.9–2.1 MPa with anaverage of 1.7 MPa for loose sand, 2.1–4.1 MPa with anaverage of 3.5 MPa for medium dense sand, and 4.1–9.7MPa with an average of 6.9 MPa for dense sand. Aspointed out by Poulos (1971), the use of a constant valueof soil modulus with depth in sands is highly questiona-ble, and it must be considered as being somewhat artiˆ-cial ˆtting parameters rather than meaningful soil modu-lus. The moduli for the loose and dense sands are taken as2.1 MPa and 5.0 MPa in this study, respectively. For thesoil Poisson's ratio, Budhu (2000) suggested typicalvalues from 0.15 to 0.25 for loose sand and from 0.25 to0.35 for dense sand. In this study, the Poisson's ratios of0.2 and 0.3 are used for the loose and dense sands, respec-tively.

Kong and Zhang (2008) reported values of couplingcoe‹cient b of 0.4 and 0.8 for the loose and dense sands,respectively, which were back calculated from the 1×2pile group tests in the loose and dense sands and the 2×2pile group test in the dense sand.

In the present study, each pile is divided into 40 ele-ments. An investigation of the in‰uence of element num-ber found that the use of twenty elements generallyproduces reasonable accuracy and there is little diŠerencein solutions using forty and ˆfty elements.

Figure 6 shows the experimental torque-twist anglecurves of 1×2, 2×2, and 3×3 pile groups subjected totorsion and the corresponding numerical predictions us-ing the present approach. Good agreement with the testdata is achieved for the 1×2 and 2×2 pile groups. Forthe 3×3 pile group, agreement is favorable at the initialloading stages, whereas the analysis over-predicts the pileresistance at higher loading stages. In addition, the pro-gram was employed to calculate the responses of the pilegroups without considering the pile-soil-pile interactionsand the de‰ection-torsion coupling eŠect. This calcula-tion was performed by setting a large far-ˆeld soil modu-lus, 103 MPa, and a b value of zero. Kong (2006) foundthat when the far-ˆeld soil modulus is larger than 10MPa, the eŠect of the pile-soil-pile interactions on thepile group response is negligible. The calculated resultsare also shown in Fig. 6. It is found that the curveswithout considering the pile-soil-pile interactions and thede‰ection-torsion coupling eŠect are slightly lower thanthe present numerical results considering all the interac-tion and coupling eŠects for the 1×2 and 2×2 pilegroups, but are slightly higher for the 3×3 pile group.EŠects of pile-soil-pile interactions and coupling eŠect

282

Fig. 6. Torque-twist angle curves of 1×2, 2×2 and 3×3 pile groupsFig. 7. Torsional resistances of individual piles in 1×2 and 3×3 pile

groups

Fig. 8. Pile-head shear forces in individual piles in 1×2 and 3×3 pilegroups

282 KONG AND ZHANG

will be explained in detail later.The calculated and experimental torsional resistances

of individual piles in the 1×2 and 3×3 pile groups arecompared in Fig. 7. In Fig. 7(a), the numerical predic-tions ˆt the test data well, especially at large twist angles;while the analysis not considering the interaction andcoupling eŠects appears to underestimate the torsionalresistance at large twist angles. The latter analysis curvesreach their ultimate values at a twist angle of 49, whichare similar to the torque-twist angle curves for the singlepiles in Fig. 5. The comparison between the test data andthe calculated results demonstrates that the de‰ection-torsion coupling eŠect indeed exists in the pile groupssubjected to torsion and coupling coe‹cient b is su‹cientto quantify this coupling eŠect. Figure 7(b) shows thetorsional resistances of the individual piles in the 3×3pile group. In Fig. 7(b), the numerical predictions ˆt thetest data well at small twist angles, but somewhat deviatefrom the test data at large twist angles. The center pile inthe 3×3 pile group is only subjected to a torque at thepile head like a single pile subjected to torque only, but itsresponse from the numerical analysis is weaker than thatwithout considering all the eŠects because the pile-soil-pile interactions are taken into account in the numericalprediction.

Figure 8 shows the calculated and experimental pile-head shear forces in the individual piles in the 1×2 and 3×3 pile groups. In Fig. 8(a), the numerical predictions ˆtthe test data very well. In Fig. 8(b), the numerical predic-tions tend to over-predict the shear forces, but are betterthan the results without considering the interaction and

coupling eŠects. The diŠerence between Figs. 8(a) and (b)may be attributed to the overlapping of the zones of plas-

283

Fig. 9. Percentages of torsional contribution in applied torque

Fig. 10. Distributions of bending moment and torque along depth in 1×2 pile group in dense sand


tic ‰ow around the closely spaced individual piles. Theplastic ‰ow in‰uences the soil reaction around the pileshafts, and in turn, the torsional resistances shown in Fig.7(b) through the coupling eŠect.

Figure 9 shows the variation of torsional resistancecontribution with twist angle, in which the torsional con-tribution of a pile group is the sum of the torsionalresistances of all the individual piles in the pile group. InFig. 9, the numerical predictions ˆt the torsional contri-bution well over the entire range of twist angle for the 1×2 and 2×2 pile groups; while for the 3×3 pile groups, thepresent approach slightly underestimates the torsionalcontribution. The results without considering all eŠects inthe pile groups appear to slightly overestimate the tor-sional contribution at small twist angles but considerablyunderestimate the torsional contribution at large twist an-gles.

The numerical predictions for torque and bending mo-ment distributions along the depth using the present ap-proach in the 1×2 pile group test in dense sand, as an ex-ample, are also compared with the test data, and areshown in Fig. 10. A reasonably good agreement betweenthe predictions and measurements is obtained.

EVALUATION OF INTERACTIONS ANDCOUPLING EFFECTS

The present approach uses the empirical factor b toquantify the de‰ection-torsion coupling eŠect and usesthe Mindlin (1936) solutions and the Randolph (1981) so-lution to calculate pile-soil-pile interactions assuminglinear elastic soil medium. The in‰uences of the couplingeŠect and pile-soil-pile interactions on response of pilegroups subjected to torsion are evaluated in this section.

Figure 11 shows the de‰ection-torsion coupling eŠectwith b＝0, 0.4, and 0.8 on the responses of the 1×2, 2×2

284

Fig. 11. De‰ection-torsion coupling eŠect on response of pile groupsin loose sand (soil modulus 2.1 MPa)

Fig. 12. Pile-soil-pile interaction eŠects on response of pile groups inloose sand (b＝0)

284 KONG AND ZHANG

and 3×3 pile groups in the loose sand. The group tor-sional resistance increases with b for all the three groupconˆgurations. As b increases from 0 to 0.8 the grouptorsional resistance increases by 9–15z. Such an increaseis clearly due to the fact that the de‰ection-torsioncoupling eŠect increases the torsional resistances of in-dividual piles in a pile group.

In the present approach, separate stiŠness relationshipsare used for the near-êld soil (i.e., p-y curves and t-ucurves) and for the far-êld soil (i.e., pile-soil-pile inter-actions), which permits independent adjustments of thetwo relationships. To study the in‰uence of the pile-soil-pile interactions on response of pile group subjected totorsion, three far-êld soil moduli, 0.9, 2.1 and 5.0 MPa,are used for a parametric study. As reported above, 0.9MPa is the lower bound of the recommended soil modu-lus for loose sand by Poulos (1971). The calculated resultsare shown in Fig. 12. It is clear that the far-êld soilmodulus has a signiˆcant in‰uence on the response of thepile groups. An increase of soil modulus tends to decreasethe torsional stiŠness of the 1×2 and 2×2 pile groups,but increase the torsional stiŠness of the 3×3 pile group.Because of the linear elastic assumption of soil medium, amovement of a pile in the 1×2 pile group subjected totorsion induces the second pile to move in the oppositedirection. As the far-êld soil modulus increases, thebackward displacement decreases, so the lateral displace-

ments of the two piles in the pile group, and in turn, thetorsional stiŠness of the pile group decreases. In the 2×2pile group, the displacements of a pile induced by the ad-jacent piles and the pile at the opposite corner counteract,so the in‰uence of far-êld soil modulus on the responseof the pile group is much smaller than that on theresponse of the 1×2 pile group. For a particular pile inthe 3×3 pile group, the interaction with most of the otherpiles tend to increase its lateral displacement; so the tor-sional stiŠness of the pile group increases as the far-êldsoil modulus increases. Figure 12 indicates that the pile-soil-pile interactions in pile groups subjected to torsionare sensitive to the group conˆguration.

In the present approach, the interaction between thelateral resistances of the individual piles and the interac-tion between the torsional and lateral resistances of theindividual piles are quantiêd separately. A further para-metric study indicates that the latter interaction alwaysleads to reduced group torsional stiŠness. Therefore, theformer and the latter interaction eŠects counteract in the1×2 and 2×2 pile groups. In the 3×3 pile group, bothtypes of interaction eŠects produce a reduced group tor-sional stiŠness.

Experimental results for torsionally loaded pile groupsgreater than 3×3 conˆgurations are not yet available. Aspile-soil-pile interactions are sensitive to group conˆgura-tion, more research is needed if the results of this paperare to be extended to larger pile groups subjected to tor-sion.

SUMMARY

Previous studies show that piles in a pile group subject-ed to torsion simultaneously mobilize lateral and tor-sional resistances. These lateral and torsional resistancesinduce not only complex pile-soil-pile interactions (i.e.,the interactions between the lateral resistances of individ-ual piles and the interaction between the torsional andlateral resistances of the individual piles), but also de‰ec-tion-torsion coupling eŠects that the lateral loading hason the torsional resistance of individual piles. In thispaper, a nonlinear approach is proposed to predict theresponse of pile groups subjected to torsion. Nonlinearsoil response in the near êld is modeled using load-trans-fer curves (i.e., p-y and t-u curves). The far-êld pile-soil-pile interactions are predicted using analytical solutions:the interaction between lateral resistances of the individ-ual piles is considered through Mindlin (1936)'s solutionsand the interactions between the torsional and lateralresistances of the individual piles are considered throughRandolph (1981)'s solution. An empirical couplingcoe‹cient is proposed to take into account the de‰ection-torsion coupling eŠect. The proposed approach not onlysimulates nonlinear behavior of individual piles in a pilegroup but also captures major pile-soil-pile interactionsand the coupling eŠect in the pile group.

The proposed approach was applied to predict resultsof centrifuge model tests on 1×2, 2×2, and 3×3 pilegroups subjected to torsion. In general, the applied tor-

285

Fig. A1. Unit point load in interior of elastic half-space


que-twist angle response and the transfer of applied tor-que in the pile groups can be predicted with reasonableaccuracy, except for the 3×3 pile groups where themethod tends to underestimate the pile twist angle. In ad-dition, this approach was used to study the in‰uences ofpile-soil-pile interactions and the de‰ection-torsioncoupling eŠect on the response of pile groups. It wasfound that the de‰ection-torsion coupling eŠect can in-crease the torsional resistance of pile groups by 9–15z.The pile-soil-pile interactions are found to be sensitive togroup conˆguration. More research is needed if theresults of this paper are to be extended to pile groups larg-er than 3×3 conˆgurations.

ACKNOWLEDGMENTS

The authors acknowledge the ˆnancial support fromresearch grant HKUST 6037/01E provided by theResearch Grants Council of Hong Kong SAR and grant50809060 provided by the National Natural ScienceFoundation of China.

REFERENCES

1) Brown, D. A., Morrison, C. and Reese, L. C. (1988): Lateral loadbehavior of pile group in sand, Journal of Geotechnical Engineer-ing, ASCE, 114(11), 1261–1276.

2) Budhu, M. (2000): Soil Mechanics and Foundations, John Wiley &Sons, New York.

3) Cai, Z. Y. (2001): A comprehensive study of state-dependentdilatancy and its application in shear band formation analysis, PhDThesis, The Hong Kong University of Science and Technology,Hong Kong.

4) Chow, Y. K. (1985): Torsional response of piles in nonhomogene-ous soil, Journal of Geotechnical Engineering, ASCE, 111(7),942–947.

5) Chow, Y. K. (1986): Analysis of vertically loaded pile groups, Inter-national Journal for Numerical and Analytical Methods in Geo-mechanics, 10, 59–72.

6) Dutt, R. N. and O'Neill, M. W. (1983): Torsional behavior ofmodel piles in sand, Geotechnical Practices in OŠshore Engineer-ing, American Society of Civil Engineers, Austin, 315–334.

7) Georgiadis, M. (1987): Interaction between torsional and axial pileresponse, International Journal for Numerical and AnalyticalMethods in Geomechanics, 11, 645–650.

8) Guo, W. D. and Randolph, M. F. (1996): Torsional piles in non-homogenous media, Computers and Geotechnics, 19(4), 265–287.

9) Hoit, M., Hays, C., McVay, M. and Williams, M. (2001): FB-PierUsers Guide and Manual for the Analysis of Group Pile Founda-tions, Florida Department of Transportation and the Federal High-way Administration, Tallahassee, Fla, Contract # DTF61–95–00157.

10) Kong, L. G. (2006): Behavior of pile groups subjected to torsion,PhD Thesis, The Hong Kong University of Science and Technolo-gy, Hong Kong.

11) Kong, L. G. and Zhang, L. M. (2007a): Rate-controlled lateral-loadpile tests using a robotic manipulator in centrifuge, GeotechnicalTesting Journal, ASTM, 30(3), 192–201.

12) Kong, L. G. and Zhang, L. M. (2007b): Centrifuge modeling of tor-sionally loaded pile groups, Journal of Geotechnical and Geoen-vironmental Engineering, ASCE, 133(11), 1374–1384.

13) Kong, L. G. and Zhang, L. M. (2008): Experimental study of inter-action and coupling eŠects in pile groups subjected to torsion,Canadian Geotechnical Journal, 45, 1001–1017.

14) Leung, C. F. and Chow, Y. K. (1987): Response of pile groups sub-

jected to lateral loads, International Journal for Numerical andAnalytical Methods in Geomechanics, 11(3), 307–314.

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

16) Poulos, H. G. (1971): Behaviour of laterally loaded piles: II-pilegroups, Journal of Soil Mechanics and Foundation Division,ASCE, 97(SM5), 733–751.

17) Poulos, H. G. (1975): Torsional response of piles, The Journal ofGeotechnical Engineering Division, ASCE, 101, 1019–1035.

18) Randolph, M. F. (1980): PIGLET: A computer program for theanalysis and design of pile groups under general loading conditions,Cambridge University Engineering Department Research Report,Soils TR91.

19) Randolph, M. F. (1981): Piles subjected to torsion, Journal of SoilMechanics and Foundation Division, ASCE, 107(GT8), 1095–1111.

20) Reese, L. C., Cox, W. R. and Koop, F. D. (1974): Analysis of later-ally loaded pile in sand, Proc. 6th OŠshore Technology Confer-ence, Houston, TX, paper OTC 2080, 473–483.

21) Reese, L. C., Wang, S. T., Arrellaga, J. A. and Hendrix, J. (2000):Computer Program GROUP for Windows User's Manual, Version5.0, Ensoft, Austin, Texas.

22) Smith, I. M. (1982): Programming the Finite Element Method, withApplication to Geomechanics, Wiley, Chichester.

23) Yan, L. and Byrne, P. M. (1992): Lateral pile response to monoton-ic head loading, Canadian Geotechnical Journal, 29, 955–970.

24) Zhang, L. M. and Tsang, C. Y. M. (2005): Three-dimensional anal-ysis of torsionally loaded large-diameter bored pile groups, Proc.6th International Conference on Tall Buildings (eds. by Y. K.Cheung and K.W. Chau), World Scientiˆc, 311–317.

25) Zhang, L. M. and Kong, L. G. (2006): Centrifuge modeling of tor-sional response of piles in sand, Canadian Geotechnical Journal,43(5), 500–515.

APPENDIX

Referring to Fig. A1, given a point A in the half space,the displacements at A in the X- and Y-directions, u andv, due to a unit point load in the interior of a semi-inˆnitesolid are as follows (Mindlin, 1936):

u＝1

16pGs(1－m)

×

3－4mR1

＋1R2

＋x 2

R31＋

(3－4m)x 2

R32

＋2czR3

2Ø1－3x 2

R22»

＋4(1－m)(1－2m)

R2＋z＋c Ø1－ x 2

R2(R2＋z＋c)»

(A1)

v＝xy

16pGs(1－m) « 1R3

1＋

3－4mR3

2－

6czR5

2－

4(1－m)(1－2m)R2(R2＋z＋c) $

(A2)

286286 KONG AND ZHANG

where z is the depth of A; Gs is the shear modulus of soil;m is Poisson's ratio of soil; c is the distance from theground surface to the loading point; R1 and R2 are ex-pressed as

R1＝ r 2＋(z－c)2 (A3)

R2＝ r 2＋(z＋c)2 (A4)

where r 2＝x 2＋y 2. The lateral displacement in any direc-tion, w, deˆned by angle g in Fig. A1, can be calculatedby

w＝u cos g＋v sin g (A5)

nonlinear analysis of torsionally loaded pile groups

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