DYNAMIC BEHAVIOR OF PILE GROUP CONSIDERING SOIL-PILE-CAP INTERACTION

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1 October 1-17, 8, Bejng, Chna DYNAMIC BEHAVIOR OF PILE GROUP CONSIDERING SOIL-PILE-CAP INTERACTION A. M. Halaban 1 and M. Malek 1 Professor, Faculty of Cvl Engneerng, Isfahan Unversty of Technology, Isfahan, Iran Grad. Student, Faculty of Cvl Engneerng, Isfahan Unversty of Technology, Isfahan, Iran Emal: mahd@cc.ut.ac.r, 14wcee@gmal.com ABSTRACT : It s well known that the earthquake response of a ple-supported structure s strongly affected by the dynamc sol-ple nteracton. In the case of weak sols or severe ground motons, group ples are often the sutable soluton. In most practcal cases the ple cap s sttng on the ground as well as the ples n the group. The contact between the ple cap and supportng ple s smply neglected n practce. However, wth advent of fast computers, geotechncal researchers would lke to reach precse and rgorous solutons. Therefore, n ths research, a fnte element model to take nto account sol-ple-cap nteracton for dynamc loads s developed. The proposed model for the sol-ple-cap (SPC) system conssts of structural elements to represents ples and concrete cap, brck elements representng the sol and nterface element to model the contact surfaces between sol-ple and sol-cap. Sesmc response of the system s found usng an explct successve-couplng ncremental soluton scheme. Materal nonlnearty of sol s also ntroduced nto the model. Proper absorbng boundary condtons, smulatng radaton effects are used. Developng a comprehensve study, a seres of numercal analyses usng dynamc fnte element method are conducted to nvestgate affectng parameters on dynamc behavor of group-ple foundaton. Along wth the effect of ple-sol-cap nteracton, the ples slenderness, type of ples, ples space and embedment of the ple cap s also studed. The results showed that neglectng the ple-sol-cap nteracton n dynamc analyss of ple groups could lead to non-exact results n terms of the ple group response. KEYWORDS: ple groups, sol-ple-cap nteracton, dynamc behavor

2 October 1-17, 8, Bejng, Chna 1. INTRODUCTION For the last three decades, a large of number of studes has been conducted on dynamc sol-ple group nteracton. Dynamc behavor of sol-ple group systems n dynamc loadng processes s hghly affected by nonlnear behavor of sol, sol plastcty and also gappng and sldng durng extreme exctatons. Ths hghlghts the necessty of conductng more precse analyss. Wth advent of fast computers, focus of researchers s ncreasng towards rgorous sol-ple-structure nteracton analyses. The most promnent developed methods for dynamc analyss of sngle ples can be lsted as Wnkler model, emprcal non-lnear p-y, t-z curve models, analytcal and sem-analytcal formulatons (Novak plane-stran approach) and Fnte-element formulatons. In these methods, t has been found that the most affectng parameters on response of sngles ples are: sol-ple stffness rato, the ple slenderness, type of ple, sol shear modulus and the ple head boundary condton. For the ple groups, snce each ple s response s affected not only by ts own load, but also by the load and deflecton of the neghborng ples, therefore the dynamc response of the group ple should be evaluated accountng for the frequency dependent ple-to-ple nteracton phenomenon. A varety of numercal and analytcal methods have been developed to compute the dynamc response of ple groups. Wolf and Von Arx (1978) employed an axsymmetrc fnte element formulaton to establsh the dynamc dsplacement feld due to rng loads. Wass and Hartmann (1981) formulated an effcent sem-analytcal method, whch uses rng loads and was well suted for layered meda for near feld. Kayna and Kausel (198) further mproved the accuracy by combnng the cylndrcal loads wth the consstent stffness matrx of layered meda usng a boundary element formulaton. He ntroduced the dynamc nteracton factors, whch are complex and frequency dependent. The ple group response to earthquake exctatons not only s depended to the man affectng dynamc parameters of sngle ples but also s nfluenced by relatve dstances of ples due to ple-to ple nteracton effect. The sol nonlnear behavor can play an mportant role n both sngle ple response and ple-to-ple nteracton resultng more deformaton to the group ple. Heavy damage reported on ple foundatons durng recent earthquakes (e.g. Ch-Ch 1999 and Kocael 1999) showed a need to take nto account the nonlnear behavor of sol for desgnng ple-supported structures. To adequately account for the sol non-lnearty, dynamc analyss should be performed n the tme doman. Nogam [6] ntroduced materal and geometrcal non-lneartes n the analyss usng dscrete systems of mass, sprng and dashpots. However, proper representaton of dampng and nerta effects of contnuous sol meda s dffcult wth such dscrete systems. The ncluson of materal non-lnearty due to sol plastcty requres the analyss to be performed usng fnte element approach. Wu and Fnn (1997) presented a quas-3d method for the analyss of nonlnear ple response. Bentley and El Naggar () nvestgated the knematc response of sngle ples to account for the sol plastcty usng the Drucker Prager sol model and gappng at the sol ple nterface. No work hardenng for sol meda was taken nto account. Ca et al. ncluded sol plastcty wth the work hardenng n a fnte element model for sngle ples. However, they assumed fxed boundary condtons and neglected dampng n the foundaton subsystem. Usng the herarchcal sngle surface (HSS) sol model, Maheshwar et al. (3) examned the effects of materal non-lnearty of sol on the free feld response as well as on the knematc response of sngle ples. Maheshwar et al. (4) extended t for ple groups. The ple cap may affect the knematc and nertal dynamc response of ple-supported structures. In some cases of ple groups, ple caps could be restng on the ground surface or may be even embedded. In both cases, an nteracton between the flexble cap, ple and supportng sol s antcpated. Furthermore, embedded caps could provde an excessve stffness and dampng durng the dynamc exctaton as the embedded shallow foundatons do. To clarfy the sgnfcance of the dynamc sol-ple cap nteracton effects n non-lnear sesmc group ple responses a 3D nonlnear model was developed. In ths paper, the effect of ple-sol-cap nteracton along wth the other mentoned affectng parameters on sesmc response of a group wth shallow and embedded cap are nvestgated. The model was subjected to transent exctaton.

3 October 1-17, 8, Bejng, Chna. SOIL-PILE SYSTEM In ths study three sol-ple systems are consdered. One nvolves a x ple group shown n Fgure 1. havng the ple cap located above the ground surface (Fgure a). The other two systems are usng the same confguraton and geometry of ples referred as the reference model (Fgure 1) but havng the ple cap rested on the ground surface (Fgure b) and embedded n a sol sde layer (Fgure c). The dameter and length of ples n the reference model are.4m and 5m, respectvely. The ple cap n all three systems s havng m x m x.5m dmensons. Fgure 1. The reference ple group plan a b c Fgure. Three ple-sol-cap systems Full three-dmensonal fnte element models are developed to represent the sol-ple-cap systems. The sol and ples are modeled usng eght-node brck elements wth each node havng three translaton degrees of freedom. Vscous boundares as artfcal transmttng boundares are attached n all three drectons along the model boundares (Fgure 3) n order to model the far feld condtons and allow for wave propagaton. The coeffcents of the dashpots are derved separately for the horzontal and vertcal drectons based on sol shear and compresson veloctes (V s, V p ). For computatonal reasons, sol materal dampng s taken as a Ralegh dampng. Therefore, the materal dampng s assumed to be proportonal to stffness and mass matrces. The coeffcents of the lnear combnaton are obtaned usng two natural frequences of the sol-ple-cap model, provded that the correspondng dampng coeffcents are gven as: = + / / (1)

4 October 1-17, 8, Bejng, Chna Fgure 3. Fnte element model for the ple group system: (a) 3D (b) cross secton where F, G are the mass proportonal and stffness proportonal coeffcents of the dampng matrx. Also H, s the natural frequency of the th mode of the system. In the plastc zone the Drucker-Prager falure consttutve model was adopted where the falure envelope s based on Mohr-Coulomb crtera. Accordng to ths theory, falure along a plane n the sol occurs by a crtcal combnaton of normal and shear stresses and not by normal or shear stress alone. Usng the Drucker-Prager crtera, the yeld functon can be defned as: c = - +p M JP = tan F( 4, k) J () In the present analyss, the sol-ple and sol-cap nterfaces are modeled usng an nterface element. In the nterface element, t s assumed that separaton occurs n the drecton of loadng only and the sol and ple are Fgure 4. The frcton coeffcent varaton versus sldng rate stll n contact n the other horzontal drecton. For the normal drecton to the cap-sol nterface surface and base of end-bearng ples hard contact s used. Frcton at the sol ple and sol-cap nterfaces s consdered usng tangental knematcs nteracton algorthm. In ths defnton, due to developng the nteracton forces,

5 October 1-17, 8, Bejng, Chna the frcton coeffcent of the nterface surfaces decreases from a constant statc value to a dynamc lmt value. Fgure 4 shows the varaton of ths coeffcent wth respect of surface sldng rate. In ths fgure µ s, µ k are the statc and dynamc nteracton coeffcents chosen n ths research as.4 and., respectvely. 3. EQUATIONS OF MOTIONS AND SOLVING ALGORITHM As the transent loadng s represented n the tme ncremental form, therefore, the governng equaton of moton at tme t+ t can be wrtten as: [ M ]{ u&& } + [ C] { u& } + [ K] { u} = R (3) n whch [M] s the mass matrx of the sol-ple-cap system. [C] and [K] are the total dampng matrx ncludng both materal dampng and lumped boundary dampng due to radaton effect and stffness matrx determned assumng full couplng n all three drectons of moton contanng materal non-lnearty due to sol plastcty.{ u}, { u& }, { u& } are relatve nodal dsplacement, velocty and acceleraton, respectvely at t t t+ t due to sesmc exctaton. R + s the external load at the tme step t+ t and s calculated as: where [ ] R = [ M ][ R ] { u& } R F s the pseudo-statc response nfluence coeffcents matrx that s updated at each step of tme for the nonlnear sol model; and{ u& & g } s the bedrock acceleraton vector. In ths study, the explct approach as a computatonal effcent approach was adopted to solve the governng equatons. The statc materal non-lnear analyss under the statc stuaton s essental as a startng pont for the non-lnear sesmc analyss usng explct algorthm, takng the ntal condtons at rest for the sol and accountng the ntal nduced strans to the ple system. The explct central-dfference operator satsfes the dynamc equlbrum equatons at the begnnng of the ncrement, t; The acceleratons calculated at tme are used to advance the velocty t soluton to tme t + and the dsplacement soluton to tme t + t as N N t t {} u & {} u& {} u& N 1 = 1 + (5) + N N { u} { } { } N + 1 u + t+ 1 u F 1 + g = & (6) The subscrpt refers to the ncrement number n an explct dynamcs step. {u} N s the dsplacement vector, { u& } N the velocty vector and { u& &} N s the acceleraton vector, where N s the number of degrees of freedom n the model. The explct ntegraton rule s qute smple but by tself does not provde the computatonal effcency assocated wth the explct dynamcs procedure. The acceleratons at the begnnng of the tme ncrement usng D Alambert s prncple are computed by N { } [ ] 1 NJ J u& & = M ({ P} { I } J ) (7) where [M] -1 NJ s the nverse mass matrx, { P} J s the appled load vector, and { I} J s the nternal force vector ncludng stffness and dampng forces and J s a numerator. A lumped mass matrx s used because ts nverse s smple to compute and because the vector multplcaton of the mass nverse by the nertal force requres only N operatons. The explct procedure requres no teratons and no tangent stffness matrx. The nternal (4) s

6 October 1-17, 8, Bejng, Chna force vector, { I} J s assembled from contrbutons from the ndvdual elements such that a global stffness matrx need not be formed. The explct procedure ntegrates through tme by usng many small tme ncrements. The central-dfference operator s condtonally stable, and the stablty lmt for the operator wth dampng s gven n terms of the hghest frequency of the system as t ( 1 + max max ) (8) max where max s the hghest natural frequency and max s the fracton of crtcal dampng n the mode wth the hghest frequency. 4. NUMERICAL RESULTS The sol-ple-cap systems were subjected to the El Centro exctaton and the response of the ple cap s evaluated. In ths study, the non-lnear analyses of ple-sol-cap systems are carred out accountng for sol plastcty. The effects of nteracton between cap, sol and ples, sol stffness, ple s slenderness and cap s flexblty and embedment on the response of ple group are nvestgated n detal n a form of seres of analyses. In each analyss only the effect of one parameter s nvestgated, whle the remanng parameters are kept constant. a) Ple-sol-cap nteracton As the man am of ths study, the effect of dynamc nteracton between sol, cap and ple on the response of ples head s consdered. To acheve ths mportant goal, models (a) and (b) are used to calculate the response of a ple group wth and wthout the effect of ple-sol-cap nteracton. In both models, the ples are assumed to be end bearng-floatng ples. The ple spacng rato (S/D) and the slenderness rato (L/R) s taken equal to 3 and 5, respectvely. A flexble cap wth thckness of 5 cm connects all the ple-heads. Fgure 5 shows a comparson of the ples head response for the models wth and wthout cap nteracton. It s seen that the response of ples, n general, ncreases as the ple-sol-cap nteracton s neglected. b) Ple-spacng rato To consder the effect of ple-spacng rato on the response of the ple group when the ple-sol-cap nteracton s taken nto account, model (b) s assumed to have two dfferent ple-spacng ratos equal to 3 and 3.5. Fgure wth cap nteracton wthout cap nteracton Fgure 5. Tme hstores of ples head wth and wthout cap nteracton S/D= S/D=3 Fgure 6. Tme hstores of ples head wth cap nteracton for two ple spacng ratos

7 October 1-17, 8, Bejng, Chna shows the comparson between the ples head responses. It can be noted that the ple spacng rato makes not a remarkable dfference (less than 15 percent) on lateral dsplacement of the ple group system. Depends of the ples type ths parameter could affect dfferently. c) Sol stffness It has been found that dynamc response of ple groups depend prmarly on the type and stffness of the sol profle. The sol shear modulus of elastcty affects not only on the knematc response factors of sem-nfnte sol medum but also on vscous constants of FEM boundares. In ths study, two values for sol shear wave veloctes, 1 and 15 m/sec are used. Fgure 7 shows the lateral response of model (b) usng two dfferent mentoned sol stffnesses when the sol-ple-cap s taken nto account. As t can be seen more sol stffness results less lateral ples dsplacements E=5e7Pa E=1e8Pa Fgure 7. Tme hstores of ples head wth cap nteracton for two sol stffnesses L/R= L/R=5 Fgure 8. Tme hstores of ples head wth cap nteracton for two slenderness ratos d) Ple slenderness rato The model (b) was analyzed wth two dfferent slenderness ratos (L/R= and 5) to examne the effect ple slenderness along wth sol-ple-cap nteracton on lateral dynamc response of ple groups. The results are llustrated n Fgure 8. It s shown that the slenderness s among the less sgnfcant parameters affectng the total response of ple groups accountng for sol-ple-cap nteracton. S33(Pa) End bearng Floatng Fgure 9. Tme hstores of ple cap s nduced prncpal stress wth cap nteracton Floatng End bearng Fgure 1. Tme hstores of ples head wth cap nteracton for two type of ples

8 October 1-17, 8, Bejng, Chna e) Type of ple Ple Group dynamc behavor also depends on the ples propertes wth the depth. Assumng two sets of ples (end-bearng and floatng) model (b) was subjected to the gven transent ground moton. The tme hstores of the prncple stress nduced to the ple cap and tme hstores of lateral dsplacement of ples head, assumng the sol-ple-cap s taken nto account, for two dfferent cases are shown n Fgures 9 and 1, respectvely. Results showed that the floatng ples nduce more stress nto the ple cap. Ths could be due to more nonlnear behavor of sol adjacent to ples n floatng ples resultng some torsonal moments at the ples head. f) Cap flexblty The cap flexblty s examned usng two dfferent thcknesses for the ple cap (4 cm and 5 cm) n model B. Accountng for sol-ple-cap nteracton, the results of ples head dsplacements are shown n Fgures 11 and 1 for two sets of end bearng and floatng ples, respectvely. Rotaton of ples n the group depends on the cap s rgdty. Rotaton to the ples head results reducton to the nduced stresses and ncreasng to the ples head dsplacements. The results obtaned from Fgures 11 and 1 llustrate that the effect of ths parameter s more pronounced for floatng ples than end bearng ples Cap Thck.= 5cm 5= 4= Cap Thck.= 4cm Fgure 11. Tme hstores of ple s head dsplacement wth cap nteracton for end bearng ples Cap Thck.= 4cm 4= Cap Thck.= 5cm 5= Fgure 1. Tme hstores of ple s head dsplacement wth cap nteracton for floatng ples g) Cap embedment effect Some ple caps do not rest on the surface of the sol and are partly embedded. Embedment s known to ncrease Embedded =>?@A BCDGF No embedment =>?@A BCD =E@F Fgure 13. Tme hstores of ple s head dsplacement wth cap nteracton and embedment effect

9 October 1-17, 8, Bejng, Chna both stffness and dampng but the ncrease n dampng s more sgnfcant. These effects of embedment are observed n Fgure 13, when the effect of sol-ple-cap nteracton n model (c) s consdered. The results showed the embedment effect could also decrease the ple group response up to 1 percent. REFERENCES Bentley, K. J. and El Naggar, M. H. (). Numercal analyss of knematc response of sngle ples. Canadan Geotechncal Journal 37, Ca, Y. X., Gould, P.L. and Desa, C. S. (). Nonlnear analyss of 3D sesmc nteracton of sol-ple-structure system and applcaton. Engneerng Structures :, Kayna, A. M. and Kausel, E. (198). Dynamc behavor of ple groups. Proceedng of nd Internatonal Conference on Numercal Methods n Offshore Plng, Austn, Maheshwar, B. K., Truman, K. Z., Gould, P.L. and El Naggar, M. H. (3). Three-dmensonal nonlnear sesmc analyss of sngle ples usng FEM: effects of plastcty of sol. Internatonal Journal of Geomechancs, ASCE. Maheshwar, B. K., Truman, K. Z., El Naggar, M.H., Gould, P. L. (4). Three-dmensonal fnte element nonlnear dynamc analyss of ple groups for lateral transent and sesmc exctatons. Canadan Geotechncal Journal 41, Nogam, T., Otan, J., Konaga, K. and Chen, H.L. (199). Nonlnear sol-ple nteracton model for dynamc lateral moton. Journal of Geotechncal Engneerng, ASCE 118:1, Wass, G. and Hartmann, H.G. (1981). Ple foundatons subjected to dynamc horzontal loads. European Smulaton Meetng on Modelng and Smulaton of Large Scale Structural Systems. Italy, 17. Wolf, J. P. and Von Arx, G. A. (1978). Impedance functon of a group of vertcal ples. Proceedng of the specalty conference on Earthquake Engneerng and Sol Dynamcs, ASCE, Pasadena, Wu, G. and Fnn, W.D.L. (1997). Dynamc nonlnear analyss of ple foundatons usng fnte element method n the tme doman. Canadan Geotechncal Journal 34, 44-5.