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.2.3. Modeling of a flexible pile capThe stiffness method [17] can consider neither couplingbetween the pile group and the column nor flexibility ofthe pile cap because the pile cap is assumed to be a rigidbody in the stiffness method. Much work has been doneto analyze this coupling and the effect of pile cap flexibility.Typically, a plate element has been used as a pile cap, inseveral numerical methods [14,20,21]. These methods, how-ever, have some limitations in that the horizontal behaviorof the pile cap cannot be considered, because horizontaldegrees of freedom (in the x- and y-directions) areexcluded. These limitations can be overcome by using aflat-shell element [22]. In the present study, a four nodeflat-shell element was adopted; this was developed by com-bining a Mindlin plate element and a membrane elementwith torsional degrees of freedom, as shown in Fig. 5. This element, having six degrees of freedom per node, permitsan easy connection to other elements such as beams andfolded elements. The stiffness matrix of a flat-shell elementKsE
, which is of order 24 · 24 per element, is con-structed by combination of the stiffness matrices of a plateelement ðKeplateÞ and of a membrane element ðKemembraneÞ in alocal coordinate system as follows:KsE ¼ Keplate 00 Kemembrane : ð2ÞHere, the stiffness matrix of a plate element Keplate is ex-pressed in the following matrix form:Keplate ¼ZVBeTb DebBeb dV þZVBeTsDesBesdV ; ð3Þwhere Beb is the bending-strain matrix and Besis the shear-strain matrix. For an isotropic material, Deb and Desare asfollows:Deb ¼ Et312ð1 m2Þ1 m 0m 1000 ð1 mÞ=226 437 5; ð4aÞDes¼ WEt2ð1 þ mÞ1001 ; W ¼ 56; ð4bÞwhere E is the Young’s modulus, m is the Poisson’s ratio, tis the constant thickness of the plate, and W is the shearcorrection factor. Next, the stiffness matrix of a membraneelement Kemembrane is as follows:Kemembrane ¼ Kemembrane þ cXe heheT; ð5ÞKemembrane ¼ZXe½BemGeRe T C ½BemGeRe dX; ð6Þhe¼ZXehbe; ge; reiTdX; ð7Þwhere C is the constitutive modulus and c is taken as theshear modulus. Bem, Ge, and Reare the strain matrices rep-resenting the relationship between the displacements (the membrane displacement, the rotation, and the midsideincompatible displacement, respectively) and the strains;be, ge, and reare the strain matrices for the infinitesimalrotation fields.
2.4. Computational procedureFor given conditions, such as the geometry, the load,and the properties of the structures (piles, cap, and column) and of the soil layers, the deformations and theforces on members (the stress in the cap, and the bend-ing moments and shear forces in the piles and column)can be calculated by the proposed method (YSGroup).The properties of the structures here include the pilecap flexibility. Before constructing the stiffness matricesof the flat-shell elements (for the cap) and of the beamelements (for the column), our method calculates 10load–displacement curves for each pile head by repeatedload transfer analyses using t–z/q–z and p–y curves andsaves them. In this step, reduction factors, typically p-multipliers for the lateral response, are incorporated intothe load transfer curves. In this study, the p-multiplierswhich have been experimentally derived from full-scaleload tests or from tests in centrifuge are used. These ini-tially calculated load–displacement curves are not chan-ged during the iterative procedure. For a nonlinearanalysis, all external forces are first pided by the incre-ment number (N), and then the current stiffness matricesof the inpidual piles KpE ½ are calculated from the storedload–deflection curves using the tangential slope in thefirst iteration (j = 1) and the secant modulus for j >1(Fig. 6). Here, the tangential slope (df(u)/du) and theload (f(u)) are estimated using a cubic-spline method.Combining KpE ½ with the stiffness matrix of the flat-shellelement KsE and beam element ½KbE , the coupled stiff-ness matrix of the pile group supported column [KE]iscalculated as follows:½KE ¼ KsE þ½KbE þ KpE ½ : ð8ÞFrom the global stiffness matrix [KG] and the global loadincrement vector [DFG] obtained by combining [KE] and[DFE], the total displacement increment vector [DDG]isgiven by½DDG ¼½KG 1½DFG ð9ÞThe procedure described above is iterated until the errorbetween the assumed and calculated displacements fallswithin a tolerance limit. The displacement increments atthe target step are obtained by adding the calculated re-sults to those from the previous step. The internal forcesin the pile cap and columns and in the inpidual piles arethen calculated