TORSIONAL RESPONSE OF PILE GROUPS

Transcription

1 TORSIONAL RESPONSE OF PILE GROUPS Franceco Baile, Geomarc Ltd, Meina, Italy Tall building play a key role in current urban trategie and regeneration. Development of thee building preent everal geotechnical problem related to the deign and aement of pile foundation. Among thee, the tranmiion of torional force to the pile due to the eccentricity of wind action i of particular interet. In thi paper, a numerical method of analyi i introduced for the determination of the non-linear repone of ingle pile and pile group to torional loading. Application of the propoed method, a implemented in the computer program, i illutrated through comparion with alternative numerical analye, analytical olution, centrifuge model tet and publihed cae hitorie. INTRODUCTION Pile foundation of ome tructure, uch a tall building, bridge pier, offhore platform and electric tranmiion tower, can be ubjected to ignificant torional force due to eccentric lateral loading from hip impact, high-peed vehicle, wind and wave action, and other ource of loading. Inadequate deign of the pile againt torional load may eriouly affect the erviceability and afety of thee tructure with catatrophic conequence. The literature report two cae of tall building in Miami and in Lubbock (Texa) which have uffered eriou damage due to wind action and exhibited marked permanent deformation from torion (Vickery, 1979). Another cae, decribed by Barker & Puckett (1997), report the collape of a upport pier of the 6.82km long Sunhine Skyway Bridge in Florida caued by the eccentric impact of a bulk carrier. About 395m of the bridge fell into the ea, reulting in thirty-five death. It i therefore important that the trength and deformation characteritic of the foundation pile are properly addreed in deign in order to enure afety and cot-effectivene. When a pile group i ubjected to torion, it repone i mainly governed by the interaction between the torional and lateral behaviour of the individual pile (ee Fig. 1). Thu, the total applied torque on the group (T) will be hared by the pile torional component (T i ) plu the lateral contribution from the pile hear force H i S i (where H i i the pile hear force and S i i the ditance of the pile from the torional centre of the group). Several invetigator have carried out model teting and developed numerical olution for the analyi of pile foundation under torion. However, thee tudie are mainly concentrated on the torional repone of ingle pile while little attention ha been paid to the effect of group interaction between pile. Torional movement Lateral movement =3d 3 2d Applied torque Figure 1: Movement in 3x3 group under torque Poulo (1975) preented a continuum-baed olution, uing the boundary element method, for the elatic repone of ingle pile under torque. Uing a load-tranfer approach, Randolph (1981) derived cloed-form olution for the torional tiffne of a ingle pile baed on a imple aumption concerning the tre field around a pile undergoing torion. The analyi wa then extended by Guo & Randolph (1996) in order to include a more general non-homogeneity of the oil profile and the non-linear oil repone uing a hyperbolic tre-train law. Chow (1985) preented a dicrete element approach in which the pile i modelled a a erie of element and the oil i treated a a erie of independent layer, each with a modulu of ubgrade reaction. A for experimental tudie, everal invetigator carried out conventional 1-g model tet of ingle pile ubjected to torion. Poulo (1975), a well a Georgiadi & Saflekou (199), carried out model tet of ingle pile in clay to invetigate the relationhip between axial and torional repone of a pile. The reult of the tet indicated that there wa a good correlation between the pile-oil adheion from each type of tet and that the oil hear modulu calculated from axial load tet could reaonably be ued to predict the repone of pile under torion, at leat at ordinary working

2 load level. Another intereting inight reported by Poulo (1975) i that no ignificant interaction effect between the torionally loaded model pile wa oberved, even at a pile pacing of about three pile diameter. Recently, Kong (26), Zhang & Kong (26) and Kong & Zhang (27, 28) reported a erie of centrifuge model tet on torionally loaded ingle pile and pile group in and. From thee tudie, it wa found that a pile group ubjected to torion imultaneouly mobilize lateral and torional reitance on the individual pile (ee Fig. 1), and the torional contribution to the total applied torque i in a range of 2-5%. The tet alo howed no evidence of ignificant interaction with repect to the effect of torional movement on the torional behaviour of adjacent pile located at a ditance of three pile diameter, thereby confirming the previou finding by Poulo (1975). In addition, Kong (26) and Kong & Zhang (29) propoed an empirical method to analye the repone of pile group to torion in which load-tranfer curve are ued for ingle pile repone, while Mindlin (1936) and Randolph (1981) olution are adopted to evaluate group interaction. No full-cale tet on torionally loaded pile group have been reported in literature. The only available reult from field tet refer to ingle pile and have been decribed by Stoll (1972), who devied a imple device for teting cylindrical pile and conducted tet on two teel pipe pile. Computer program for pile-group analyi For pile group, the complexity of the problem depicted in Fig. 1, which i fully three-dimenional, generally require the ue of computer-baed method of analyi. To date, everal computer program are available in order to etimate the deformation and load ditribution among the individual pile of a group ubjected to general loading condition, including torion. A decribed by Baile (23), thee program may be broadly claified into two categorie: (1) continuum-baed approache; (2) load-tranfer (or ubgrade reaction) approache. The latter category, including program uch a GROUP (Reee et al., 2), i baed on the ocalled Winkler idealiation of the oil (i.e. the oil i modelled a a erie of independent pring). Thi method i attractive in it flexibility, enabling nonlinear and inhomogeneou oil condition to be incorporated eaily. In the cae of torional loading, the program i baed on τ-θ curve, where τ i the torional hear tre and θ i the local twit angle of the pile haft. The approach i imilar to that adopted for the axial and lateral repone, baed on t-z and p-y curve, repectively. However, by diregarding oil continuity, uch an approach overimplifie the problem and make it impoible to find a rational way to quantify the interaction effect between pile in a group. Thu, in evaluating group effect, recoure i made to an entirely empirical procedure in which the ingle pile load-tranfer curve are modified on the bai of empirical parameter (e.g. the "p-multiplier" for lateral loading) derived from a limited number of mallcale and full-cale experiment. Thu, many uncertaintie remain on the general ue of the approach in routine deign (Rollin et al, 1998, 2; Baile, 23; Finn, 25). Another limitation i the election of oil parameter in that the key input parameter (i.e. the modulu of ubgrade reaction, k) i not a fundamental oil property but i alo dependent on the dimenion of the pile. Thu, the modulu of ubgrade reaction i an empirical parameter which can only be determined with ufficient confidence by back-figuring from the reult of a field tet on an intrumented pile. In concluion, the load-tranfer approach may be regarded a a link between the interpretation of full-cale pile tet and the deign of imilar ingle pile rather than a a general tool for pile group deign. A more rational approach i offered by oil continuum-baed olution which make ue of finite element (FEM), finite difference (FDM) or boundary element (BEM) method. Thee olution provide an efficient mean of retaining the eential apect of pile interaction through the oil continuum and hence a more realitic repreentation of the problem. Further, the mechanical characteritic to be introduced into the model have now a clear phyical meaning (e.g. the oil Young modulu, E ) and they can be meaured directly in oil invetigation. Finite element and finite difference olution are the mot powerful numerical tool for the analyi of pile group. However, when applied to routine deign problem, thee method uffer from ignificant limitation, mainly related to the high computational cot and their complexity (e.g. high meh dependency and uncertain in aigning tiffne/trength propertie to the pile-oil interface), particularly if non-linear oil behaviour i to be conidered. By contrat, BEM provide a complete problem olution in term of boundary value only, pecifically at the pile-oil interface. Thi lead to a dratic reduction in unknown to be olved for, thereby reulting in ubtantial aving in computing time and data preparation effort. Thi feature i particularly important for threedimenional problem uch a pile group.

3 Among the pile-group program baed on the boundary element method, PIGLET (Randolph, 198), CLAP (an extenion of DEFPIG by Poulo, 199) and GEPAN (Xu & Poulo, 2) are able to deal with torional loading, a well a other type of loading. PIGLET and CLAP adopt the implified olution by Randolph (1981) for the ingle-pile repone to torion, while group effect due to axial and lateral loading are modelled uing the interaction-factor approach (i.e. by calculating the influence coefficient for each pair of pile and by merely uperimpoing the effect). However, thi approach produce a number of limitation, including approximation in the evaluation of load and moment ditribution along the pile, and an overetimate of interaction effect between pile (a the tiffening effect of intervening pile in a group i neglected). The effect of torional interaction between pile in a group are not conidered. A more rigorou BEM analyi, which take into account the imultaneou preence of all element of all pile within the group (i.e. the o-called complete analyi), i performed by the numerical code GEPAN. In thi approach, the cylindrical element at the pile-oil interface are in turn divided into partly cylindrical or annular ubelement. The program i retricted to the linear elatic range. In thi paper, a new method of analyi for ingle pile and pile group ubjected to torional loading i introduced. The propoed method i an extenion of the boundary element formulation decribed by Baile (23) for axially and laterally loaded pile group, and implemented in the computer program (the calculation engine of the pile-deign oftware Repute by Geocentrix, 29). The main feature of lie in it capability to provide a complete nonlinear BEM olution of the oil continuum (while retaining a computationally efficient code), thereby overcoming the approximation which occur with the traditional interaction factor and load-tranfer approache. It hould be emphaied that ue of a non-linear oil model i of baic importance in pilegroup deign a it avoid exaggeration of tree at pile group corner (a common limitation of linear elatic model), and therefore reduce conequent high load and moment, even at typical working load level. Benefit of thi include an improved undertanding of pile group behaviour and thu more effective deign technique. METHOD OF ANALYSIS The propoed approach, baed on the complete boundary element method, employ a ubtructuring technique in which the pile and the urrounding oil are conidered eparately and then compatibility and equilibrium condition at the interface are impoed. The method involve dicretiation of the pile-oil interface into a number of cylindrical element, each element being acted upon by an unknown uniform tre, a hown in Fig. 2a. If the pile bae i aumed to be mooth, the effect of the torional component of tree over the bae area can be ignored. Such an aumption i in line with previou finding by Poulo (1975) and Randolph (1981) who found that very little torque i tranferred to the pile bae (even for a hort rigid pier). (a) Dicretiation of pile-oil interface into element (b) Zoom of Element 3: elevation Element 1 Ground urface dp δ z Element 2 Partlycylindrical ub-element Element 3 Soil torional tree (t ) y du x φ Element 4 du φ du y φ dp Element 5 Element 6 R x (c) Zoom of Element 3: plan Figure 2: Definition of problem

4 Soil domain The boundary element method involve the integration of an appropriate elementary ingular olution for the oil medium over the urface of the problem domain, i.e. the pile-oil interface. With reference to the preent problem, the welletablihed olution of Mindlin (1936) for a point load within a homogeneou, iotropic elatic half pace ha been adopted to correlate the oil rotation and the correponding oil torional tree: { } = [ G ]{ t } φ (Equ. 1) S where φ are the oil rotation, t are the oil torional tree and G i the oil flexibility matrix obtained from the integration of Mindlin olution over a cylindrical urface. Conidering a mall, partly-cylindrical ub-element at the pile-oil interface (ee Fig. 2b and 2c), the oil rotation dφ can be calculated from the oil deflection du φ tangential to the pile urface a dφ = du R (where R i the pile radiu). The φ / tangential deflection du φ i then expreed a the um of it component in the x and y direction (Poulo, 1975), i.e.: du x φ duφ + x y φ = du (Equ. 2) y where du φ and du φ are the component of deflection in the x and y direction, repectively, and are evaluated from the equation of Mindlin for a horizontal uburface point load dp. By integration of Equ. (2) over a cylindrical element, the tangential deflection, and thu rotation, for all the element at the pile-oil interface are obtained. It hould be oberved that Mindlin olution i trictly applicable to homogeneou oil condition. However, non-homogeneou oil profile may be treated uing the claical averaging procedure propoed by Poulo (1979) and widely adopted in practice (i.e. in the evaluation of the influence of one loaded element on another, the value of oil modulu i taken a the mean of the value at the two element). Thi procedure i adequate in mot practical cae. Pile domain If the pile are aumed to act a imple beamcolumn which are fixed at their head to the pile cap, the rotation and torional tree over each element can be related to each other via the elementary beam theory, yielding: { } = [ G ]{ t } { B} p p p + φ (Equ. 3) where φ p are the pile rotation, t p are the pile torional tree, B are the pile rotation due to unit boundary diplacement and rotation of the pile cap, and G p i a matrix of coefficient obtained from the claical beam theory for a cylinder. Solution of the ytem The oil and pile equation (1) and (3) may be coupled via compatibility and equilibrium contraint at the pile-oil interface. Thu, by pecifying unit boundary condition, i.e. unit value of vertical diplacement, horizontal diplacement and rotation of the pile cap, thee equation are olved, thereby leading to the ditribution of tree, load, torque and moment in the pile for any loading condition. Limiting pile-oil tree The foregoing procedure i baed on the aumption that the oil behaviour i linear elatic. In practice, however, a the pile i twited, the torional tre on the pile-oil interface will reach a limiting value, t lim. In the preent analyi, thi limiting value i aumed to be equal to the available pile-oil haft friction. For coheive oil, following a total tre approach, the limiting torional tre may be expreed a: tlim = αc u (Equ. 4) where C u i the undrained hear trength of the oil and α i the adheion factor. For coheionle oil, following an effective tre approach, the limiting torional tre i taken a: ' v t = σ tanδ (Equ. 5) lim K where K i the coefficient of horizontal oil tre, ' σ v i the effective vertical tre and δ i the angle of friction between pile and oil. Extenion to non-linear oil behaviour Non-linear repone of the oil i modelled, in an approximate manner, by auming that the oil Young modulu varie with the tre level at the pile-oil interface. Similarly to the approach employed for the axial repone, the hyperbolic tre-train law introduced by Duncan & Chang (197), and applied to the torional cae by Guo & Randolph (1996), ha been adopted: E 2 R 1 f t Ei (Equ. 6) t = tan lim where E tan i the tangent oil modulu, E i i the initial tangent oil modulu, R f i the hyperbolic curve-fitting contant, t i the pile-oil torional

5 tre and t lim i the limiting value of pile-oil torional tre defined in Equ. (4) and (5). Thu, the boundary element equation decribed above for the linear repone are olved incrementally uing the modified value of oil Young modulu of Equ. (6) and enforcing the condition of yield, equilibrium and compatibility at the pile-oil interface. The hyperbolic curve fitting contant R f define the degree of curvature of the tre-train repone and it value can range between (an elaticperfectly platic repone) and.99 (R f = 1 i repreentative of an aymptotic hyperbolic repone in which the limiting pile-oil torional tre i never reached). The mot reliable method to determine the value of R f i by backfitting the torque-twit angle curve at the pile-head with the meaured data from a fullcale torional pile load tet. In the abence of any tet data, the value of R f can be aeed baed on experience and, from the reult illutrated in thi paper, a value of R f =.99 appear to provide a reaonable etimate. It i worth noting that a imilar high value (i.e. R f =.95) i recommended by Guo & Randolph (1996) uing a imilar nonlinear model, thereby confirming that, in the cae of torional loading, the effect of non-linearity cloe to the pile i much more localied than in the axial cae (where value of R f in the range -.5 are generally appropriate). COMPARISON WITH NUMERICAL AND EXPERIMENTAL RESULTS Numerical reult from the preceding analyi are validated through a comparion with alternative numerical analye, analytical olution and centrifuge model tet. Comparion with Xu & Poulo (2) In order to invetigate the twit angle (φ T ) at the head of a torionally loaded ingle pile, Figure 3 compare reult with thoe of the complete BEM analyi of GEPAN, a reported by Xu & Poulo (2). The figure alo how the reult obtained from PIGLET, which adopt the analytical olution by Randolph (1981). The pile ha a diameter of 1 m, i embedded in a linearelatic homogeneou oil (E = 2 MPa, where E i the oil Young modulu) and i ubjected to a torional moment of 1 MNm. Reult are expreed a a function of the pile lenderne ratio L/d (where L i the pile length and d it diameter) and the pile-oil relative tiffne K= E p /E (where E p i the pile Young modulu). A very good agreement between the complete BEM olution of GEPAN and i oberved, while PIGLET give higher value of the twit angle, particularly for rigid pile. Turning to pile-group behaviour, Figure 4 how the torque ditribution, expreed a the ratio of pile-head torque T to pile-head average torque T av, predicted by, GEPAN and PIGLET for a 3x3 pile group under torional loading (with L/d = 25, K= E p /E =15 and oil Poion ratio of.5). 1 1 MNm K =1 φt (1-3 rad) 1 K =1 PIGLET GEPAN K = L/d Figure 3: Comparion of the twit angle at head of torionally loaded ingle pile

6 Reult are expreed a a function of the normalied pile pacing (/d), where i the centre-to-centre pile pacing. Figure 4 alo include the prediction from a verion of which ignore the effect of torional interaction between the individual pile of the group (imilarly to the approach followed by PIGLET), thereby reulting in a uniform torque ditribution between pile. When uch interaction effect are taken into account within the GEPAN and analye, thi lead to different torional contribution among the pile, depending on the pile location within the group. Similarly to the cae of axially and laterally loaded pile group, the corner pile carry the greatet proportion of torque, the central pile carrie the leat, and the ide pile are in between. Some dicrepancie between the value predicted by and GEPAN are oberved in Figure 4, owing to the different analyi method employed by the two program. It i found that the percentage of torional contribution to reit the total applied torque i higher in than in GEPAN (in which the contribution of the lateral component will be higher). Table 1 compare the main reult obtained from PIGLET and for the 3x3 pile group of Figure 4, uing a typical pile pacing /d = 3 and applying a total torque of 9 knm. While the value of twit angle and maximum torque are imilar, ome difference in the lateral load and bending moment prediction are oberved. Thee may be explained with the different approache adopted by the two program: PIGLET, a well a ignoring torional interaction effect, i baed on uperpoition of two-pile interaction factor, wherea perform a complete analyi of the entire group, thereby conidering in a more rigorou fahion the coupling between lateral and torional loading (thi feature of behaviour i uually referred to a the load-deformation coupling effect). It i worth noting that conideration of torional group effect within the analyi lead to an increae of the twit angle, maximum lateral load and maximum bending moment of about 7%. Finally, it hould be emphaied that the above PIGLET and GEPAN olution are retricted to pile embedded in a linearly elatic oil. However, that i far from realitic for real oil, whoe behaviour i highly non-linear, even at low load level. The effect of oil nonlinearity on the torional repone of pile will be examined in the next ection. Table 1: Comparion of olution for 3x3 pile group under torque Quantity PIGLET (no torional group effect) Twit angle (radx1-4 ) Maximum lateral load in x dir. (kn) Maximum lateral load in y dir. (kn) Maximum bending moment in xz plane (knm) Maximum bending moment in yz plane (knm) Maximum torque (knm) ,4 Pile 1, Pile 2, Pile 3,3 Pile 1 Pile 3 PIGLET T / T av,2 Pile 2 Pile 1 Pile 2 GEPAN,1 Pile 3 (no torional group effect), /d Figure 4: Load ditribution of torion loading in 3x3 pile group

7 Comparion with centrifuge tet of Kong & Zhang (29) Kong (26), Zhang & Kong (26) and Kong & Zhang (27, 28) reported a erie of centrifuge model tet on torionally loaded ingle pile and pile group in looe and dene and. Although ome problem related to cale effect may till be preent, centrifuge teting i an improved method of phyical modelling over the conventional model tet in 1-g condition in that it i able to better reproduce the tre condition that would exit in a full-cale intallation. A ummary of the main input parameter for the ingle pile and the pile group i reported in Table 2 below. For convenience, input data and reult are reported in prototype cale. Figure 5 and 6 how a good agreement between tet data and prediction for the torquetwit angle curve and torque ditribution of the ingle pile. It i of interet to oberve that the non-linear oil model adopted by i capable of capturing the main non-linear feature of behaviour from the centrifuge tet. Figure 5 alo include the numerical prediction obtained from the empirical approach by Kong & Zhang (29), baed on back-analyi. In thi approach, ingle-pile repone i modelled by mean of load-tranfer curve (i.e. p-y curve for the lateral repone and τ-θ curve for the torional repone), including an empirical factor (β) to model the interaction between the lateral and torional repone, while group effect are modelled uing the linear elatic olution of Mindlin (1936) and Randolph (1981). It hould be emphaied that, in addition to the parameter hown in Table 2 (which follow thoe reported by Kong & Zhang), the oil parameter adopted for the analye include an uniform oil modulu (E ) of 7 MPa and a coefficient of horizontal oil tre (K ) of.6 for the looe and, with the correponding value for dene and being equal to E = 11 MPa and K = 1.1. The hyperbolic curve fitting contant (R f ) for the lateral and torional repone were both taken a.99, while the interface angle of friction between pile and oil (δ) i taken a.8φ for both looe and dene and. Pile embedded length, m Pile freetanding length, m Table 2: Input parameter for comparion of Fig. 5-1 Pile diameter, m ingle pile pile group Pile torional rigidity, MNm 2 ingle pile pile group Pile flexural tiffne, MNm 2 Pile Poion' ratio Soil Poion' ratio Soil friction angle, degree Soil buoyant unit weight, kn/m 3 Looe and Dene and Torque 4 Torque at pile head (knm) Dene and Looe and Kong & Zhang (29): Tet data Kong & Zhang (29): Prediction Twit angle (degree) Figure 5: Torque-twit angle curve for ingle pile

8 Torque along pile haft (knm) Torque along pile haft (knm) Torque Depth (m) 6 Depth (m) Kong & Zhang (29): Tet data 1 (a) Looe and 1 (b) Dene and Figure 6: Torque ditribution along depth for ingle pile It i noted that the above value of oil modulu are relatively low when compared to the value normally adopted in the analyi and deign of fullcale pile in and. However, a reported by Kong (26), pile jacking in centrifuge tet ha ignificant effect on the oil adjacent to the pile and therefore the value of oil modulu can be ignificantly different from the value before pile jacking. In addition, cale effect remain an important iue in centrifuge modelling and a higher influence of pile-oil interface propertie in a model than in a prototype can be expected. Thu, in the analye for ingle pile, the above value of E and K were elected in order to fit the initial portion and the failure load of the torque-twit angle curve obtained from the centrifuge tet. Then, in the ubequent pile group analye, the ame value of E and K ued for the ingle-pile analye have been kept (i.e. without any curve fitting with the tet data) in order to imulate the application of in normal deign (when only ingle-pile tet data, if any, i available). An intructive application of uch deign philoophy i decribed by Hardy & O'Brien (26) for the deign of pile foundation under general loading condition for the new Wembley Stadium in London. Turning to pile-group behaviour, the centrifuge tet were carried out on pile arranged in 1x2, 2x2, and 3x3 configuration (ee inet to Fig. 7-8), with a centre-to-centre pacing of three pile diameter and connected by a rigid cap (1.2m tick) with a clearance of 1.2m from the groundline in prototype. Figure 7 and 8 how the experimental torque-twit angle curve and the correponding numerical prediction from Kong & Zhang (29) and from. A good agreement between tet data and numerical prediction i oberved for the 1x2 and 2x2 pile group, while the numerical analye (particularly that from Kong & Zhang) tend to overpredict the pile reitance at high load level for the 3x3 pile group. A dicued by Kong (26), the effect of pile intallation may be a poible reaon for thi dicrepancy; namely, pile jacking denifie the oil inide and near the pile group in looe and but looen the oil inide and near the pile group in dene and. Thi effect become more pronounced a the number of pile in the group increae. It hould be emphaied that, for both ingle pile and pile group, the numerical prediction of Kong & Zhang (29) are baed on a back-analyi of the data from the centrifuge tet, including the ue of a back-calculated parameter (i.e. the coupling coefficient, β) in order to reproduce the experimental curve for the pile group. By contrat, the analye for the pile group were carried out uing the ame oil parameter adopted for the ingle-pile analye, without uing any additional curve-fitting parameter to improve the agreement with the centrifuge tet data. It i noted, however, that the reult are of comparable accuracy to thoe obtained from the numerical prediction of Kong & Zhang, thereby confirming the validity of the propoed approach a a practical tool for pile group deign. Figure 7 and 8 alo how that incluion of interaction effect with repect to the effect of torional movement on the torional behaviour of adjacent pile lead, in the analyi, to a maximum increae of the twit angle of about 4% for the 3x3 group in dene and.

9 2 LOOSE SAND Kong & Zhang (29): Tet data =3D 1x2 group Applied torque on group (knm) x3 group Kong & Zhang (29): Prediction (no torional group effect) 2x2 group =3D 2x2 group 3x3 group 1x2 group =3D Twit angle (degree) Figure 7: Torque-twit angle curve for pile group in looe and 25 DENSE SAND Kong & Zhang (29): Tet data =3D 1x2 group Applied torque on group (knm) x3 group Kong & Zhang (29): Prediction (no torional group effect) =3D 2x2 group 3x3 group 1x2 group Twit angle (degree) Figure 8: Torque-twit angle curve for pile group in dene and =3D A hown previouly in Figure 1, the utained torque by each pile in the group i hared by the pile torional component (T i ) plu the lateral contribution from the pile hear force H i S i (where H i i the pile hear force and S i i the ditance of the pile from the torional centre of the group). Figure 9 how the decompoition of the torional reitance component for a pile of the 1x2 group in dene and, a an example. It i worth noting that the torional contribution i largely mobilied at a twit angle of about 2.5 degree, while the lateral contribution continue to increae with the twit angle. Thi feature of behaviour (alo found by Kong & Zhang in the other pile group tet) implie that, at mall twit angle, the torional reitance take larger proportion of the utained torque and that the proportion decreae at large twit angle. Although ome dicrepancie (up to about 2%) between the prediction and the tet data are oberved, it i worth noting that i capable of capturing the above feature of behaviour. For the ame 1x2 pile group in dene and, Figure 1 how the bending moment and torque profile for two value of the total applied torque (T = 119 knm and T = 2246 knm). A fair agreement between and tet data i found. Finally, it hould be oberved that intrumentation and meaurement error in the centrifuge tet data up to 13% were reported by Kong & Zhang (27).

10 2 1x2 group Lateral reitance Applied torque Torque contribution (knm) Total utained torque Lateral contribution Torional contribution Torional reitance x Kong & Zhang (27): Tet data Twit angle (degree) Figure 9: Component of utained torque in 1x2 pile group in dene and Bending moment (knm) Torque (knm) x2 group 2 2 x Depth (m) Depth (m) Kong & Zhang (29): Tet data - T=119kNm (T=119kNm) Kong & Zhang (29): Prediction - T=119kNm Kong & Zhang (29): Tet data - T=2246kNm 1 1 (T=2246kNm) Kong & Zhang (29): Prediction - T=2246kNm Figure 1: Ditribution of bending moment and torque along depth in 1x2 pile group in dene and COMPARISON WITH FIELD TESTS BY STOLL (1972) A further application of i illutrated by the analyi of full-cale torional load tet reported by Stoll (1972). The tet were conducted on two teel pipe pile, backfilled with concrete, of external diameter.273 m, 6.3 mm wall thickne, and a torional rigidity (GJ) p of 12.8 MNm 2. The pile were driven into depoit coniting of layer of oft organic ilt, overlying clayey ilt, and, and gravel. Pile A-3 wa driven to a penetration depth of 17.4 m, while the other pile, pile V-4, wa driven to 2.7 m. Following the SPT N value reported by Stoll, the analye are baed on a uboil idealiation into two layer, a ummaried in Table 3 below. For the top layer of oft clay, a contant value of N equal to 4 ha been aumed for both pile. The underlying andy layer ha been modelled on the bai of an N value of 4 at the top, increaing linearly to 2 at the bottom of the layer, for pile A-3, while value of N equal to 5 at the top, increaing linearly to 5 at the bottom of the layer, have been adopted for pile V-4. Baed on the above value of N, the initial value of oil modulu (E ) hown in Table 3 have been etimated from the following empirical correlation, generally ued for the axial repone of pile (refer to Poulo, 1994):

11 [ MPa] E = 14N clay (Equ. 7).9 [ MPa] E = 16.9N and (Equ. 8) In order to define the limiting pile-oil tree of Equ. (4)-(5) for the analye, a value of undrained hear trength (C u ) equal to 18 kpa for the top layer ha been etimated from the common correlation C u = 4.5N (Stroud, 1975), while the friction angle (φ) of 3 o and 34 o for the underlying layer have been etimated from the SPT N-value, a decribed by Peck et al. (1967). The value of the coefficient of horizontal oil tre (K ) have been elected in order to match the ultimate torque meaured in the field tet, i.e. T u = 29.3 knm and T u = 52.1 knm for pile A-3 and V-4, repectively. Finally, the hyperbolic curve fitting contant (R f ) for the torional repone ha been taken a.99. Figure 11 how a generally good agreement between and the field meaurement for the torque-twit angle curve (where r o i the pile external radiu, equal to.1365m), particularly at low load level. It i of interet to oberve that i capable of reproducing with good accuracy the initial portion of the meaured curve. Thi initial portion i mainly governed by the value of oil modulu defined in Equation 7-8, thereby providing further upport to the concluion of Poulo (1975) and Randolph (1981), i.e. the oil modulu adopted for the axial repone can be ued with reaonable confidence to predict the repone of pile ubjected to torion. Thickne (m) Pile A Pile V Table 3: Input oil parameter adopted for the analye Soil modulu, E (MPa) Rate of increae of oil modulu with depth, m E (MPa/m) Poion' ratio, υ Soil layer 1 (undrained clay) Undrained hear trength, C u (kpa) Rate of increae of undrained hear trength, m Cu (kpa/m) Adheion factor, α Buoyant unit weight, γ ' (kn/m 3 ) Soil layer 2 (drained and) Thickne (m) Soil modulu, E (MPa) Rate of increae of oil modulu with depth, m E (MPa/m) Poion' ratio, υ Friction angle, φ (degree) Interface friction angle, δ (degree) Coefficient of horizontal oil tre (K ) Buoyant unit weight, γ ' (kn/m 3 ) Pile A φ Pile V PILE A-3 5 PILE V-4 4 Torque hear T/r o (kn) 2 1 Torque hear T/r o (kn) 3 2 Stoll (1972) 1 Stoll (1972),,2,4,6,8,1,12 Twit angle (radian),,2,4,6,8,1,12,14 Twit angle (radian) Figure 11: Comparion with torque-twit angle curve meaured by Stoll (1972) for ingle pile

12 CONCLUSIONS A numerical method of analyi, baed on a complete BEM olution of the oil-continuum and implemented in the computer program, ha been introduced for the determination of the non-linear repone of ingle pile and pile group to torional loading. The propoed method ha been uccefully validated through comparion with alternative numerical analye, analytical olution, centrifuge model tet and a publihed cae hitory. It i found that the imple hyperbolic oil model adopted by i capable of capturing the main non-linear feature of pile behaviour under torion, thereby offering the propect of more realitic prediction and thu more effective deign technique. Incluion of torional interaction effect in a group of pile (i.e. the effect of torional movement of a pile on the torional behaviour of adjacent pile) ha reulted, within, in a relatively low increae of twit angle (up to a maximum of about 7%). However, the torque ditribution in the individual pile i affected by torional interaction effect, leading to a non-uniform torque ditribution among the group pile. The comparion with the load-tranfer analye (baed on p-y and τ-θ curve) of Kong & Zhang (29) ha provided further evidence that, by taking into account the continuou nature of pile-oil interaction, remove the uncertainty of empirical load-tranfer approache and provide a imple deign tool baed on conventional oil parameter. The analyi of the field tet by Stoll (1972) ha hown that reaonable prediction of pile repone under torque may be obtained uing oil parameter derived from the ite invetigation data, thereby confirming the uefulne of the approach for practical problem, particularly when pile tet reult are not yet available. It i alo found that the value of oil modulu normally adopted for the axial repone can be ued with ufficient confidence to predict the torional repone of pile. ACKNOWLEDGEMENTS The author i grateful to Prof. Harry Poulo, Coffey Geotechnic, for reading the manucript and providing helpful comment. REFERENCES Barker, R.M. and Puckett, J.A Deign of Highway Bridge - Baed on AASHTO LRFD Bridge Deign Specification, John Wiley and Son, New York, 1169 p.. Baile, F. 23. Analyi and deign of pile group. In Numerical Analyi and Modelling in Geomechanic (ed. J.W. Bull), Spon Pre (Taylor & Franci Group Ltd), Oxford, Chapter 1, pp Chow, Y.K Torional repone of pile in non-homogeneou oil. J. Geotech. Engng. Div., ASCE, Vol. 111, No. 7, pp Duncan, J.M. and Chang, C.Y Non-linear analyi of tre and train in oil. J. Soil Mech Fdn Diviion, ASCE, Vol. 96, No. SM5, pp Finn, W.D.L. 25. A tudy of pile during earthquake: iue of deign and analyi. Bulletin of Earthquake Engineering, Springer Netherland, Vol. 3, No. 2, pp Geocentrix 29. Repute 2., Pile group deign oftware. < Georgiadi, M. and Saflekou, S Pile under axial and torional load. Computer and Geotechnic, Vol. 9, No.43, pp Guo, W.D. and Randolph, M.F Torional repone of pile in non-homogeneou media. Computer and Geotechnic, Vol. 19, No. 4, pp Hardy, S. and O'Brien, A.S. 26. Non-linear analyi of large pile group for the new Wembley tadium. Proc. 1th Int. Conf. on Piling and Deep Foundation, Amterdam, The Netherland, Publ. by Deep Foundation Intitute, USA, pp Kong, L.G. 26. Behaviour of pile group ubjected to torion. PhD Thei, The Hong Kong Univerity of Science and Technology, Hong Kong, 339 p. Kong, L.G. and Zhang, L.M. 27. Centrifuge modelling of torionally loaded pile group. J. Geotech. and Geoenv. Engng., ASCE, Vol. 133, No. 11, pp Kong, L.G. and Zhang, L.M. 28. Experimental tudy of interaction and coupling effect in pile group ubjected to torion. Canadian Geotech. J., Vol. 45, pp Kong, L.G. and Zhang, L.M. 29. Nonlinear analyi of torionally loaded pile group. Soil and Foundation, Vol. 49, No. 2, pp Mindlin, R.D Force at a point in the interior of a emi-infinite olid. Phyic, Vol. 7, pp

13 Peck, R.B., Hanon, W.E. and Thornburn, T.H Foundation Engineering, John Wiley, New York, 31 p. Poulo, H.G Torional repone of pile. J. Geotech. Engng. Div., ASCE, Vol. 11, No. GT1, pp Poulo, H.G Settlement of ingle pile in nonhomogeneou oil. J. Geotech. Engng, ASCE, Vol. 15, No. GT5, pp IAHR/IUTAM Practical Experience with Flowinduced Vibration Sympoium, Karlruhe, Germany, pp Xu, K.J. and Poulo, H.G. 2. General elatic analyi of pile group. Int. J. Numer. Anal. Meth. Geomech., Vol. 24, pp Zhang, L.M. and Kong, L.G. 26. Centrifuge modelling of torional repone of pile in and. Canadian Geotech. J., Vol. 43, No. 5, pp Poulo, H.G Uer' guide to program DEFPIG - Deformation Analyi of Pile Group, Reviion 6. School of Civil Engineering, Univerity of Sidney. Poulo, H.G Settlement prediction for driven pile and pile group. Proc. Conf. on vertical and horizontal deformation of foundation and embankment, College Station, Texa, Geotechnical Special Publication, Vol. 2, No. 4, pp Randolph, M.F PIGLET: A computer program for the analyi and deign of pile group under general loading condition. Cambridge Univerity Engineering Department Reearch Report, Soil TR91. Randolph, M.F Pile ubjected to torion. J. Geotech. Engng. Div, ASCE, Vol. 17, No. GT18, pp Reee, L.C., Wang, S.T., Arrellaga, J.A. and Hendrix, J. 2. Computer program GROUP for Window uer manual, verion 5.. Enoft, Inc., Autin, Texa. Rollin, K.M., Peteron, K.T. and Weaver, T.J Lateral load behaviour of full-cale pile group in clay. J. Geotech. and Geoenv. Engng, ASCE, Vol. 124, No. 6, pp Rollin, K.M., Spark, A.E. and Peteron, K.T. 2. Lateral load capacity and paive reitance of full-cale pile group and cap. Tranportation Reearch Board Record 1736, Paper No , pp Stoll, U.W Torque hear tet of cylindrical friction pile. Civil Engineering, ASCE, Vol. 42, No. 4, pp Stroud, M.A The Standard Penetration Tet in Inenitive Clay and Soft-Rock. Proc. Europ. Symp. on Penetration Teting, Vol. 2, pp Vickery, B.J Wind effect on building and tructure-critical unolved problem. In