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2.1. Structural systemFig. 1 illustrates a schematic diagram of a typical pilegroup supported column (a piled pier) and the numericalmodel of it used in the present study. The pile group con-sists of four piles and a flexible pile cap, which is not indirect contact with the soil. In the numerical method, theinpidual piles are modeled using beam–column elements.The soil around the inpidual piles is represented by a setof load transfer curves, and the interaction between piles isrepresented by a p-multiplier. For the modeling of the flex-ible pile cap and the column, three-dimensional finite ele-ments, such as four-node flat shell elements for the pilecap and three-dimensional beam elements for the column,were used.In this study, unlike the case of a general three-dimensional structural analysis (FBPier 3.0, SAP2000,etc.), the inpidual piles were analyzed one by one to keepa total of 10 load–displacement relationships (axial direc-tion = 1; lateral direction = 8; torsional direction = 1) foreach pile head in the first instance. The stiffness matricesKpE ½ of the inpidual piles which were incorporated intothe structural analysis were derived from the load–displace-ment curves of the inpidual piles.2.2. Modeling of pilesFig. 2 shows the coordinate system (X,Y,Z) of the gen-eral structure and the local coordinate system (u, v,w)ofa pile. An arbitrary pile is connected to the pile cap at point1. Point 2 indicates the pile toe, and point 3 is a referencepoint, used for transformation between local and globalcoordinates. The equation for equilibrium at the pile head(point 1) in the local coordinate system is as follows: where KpE ½ iis the stiffness matrix of the pile head, {d}i is thedisplacement vector, and {F}i is the vector of the force atthe ith pile head. The pile head stiffness matrix KpE ½ iis oforder 6 · 6, representing three displacement constants,three rotational constraints, and four couplings betweenthe displacement and rotational constraints.In this study, a nonsymmetric stiffness matrix KpE ½ i, i.e.one where k26 ¼ k62 and k35 ¼ k53, was used to considerthe nonlinear problem, because in nonlinear analysis theentries of KpE ½ ivary at each iteration and the coupling con-stants are not required to be equal, even though in linearanalysis the coupling constants are identical, by Maxwell’sreciprocal theorem.
Each entry of KpE ½ iwas obtained fromthe four principal modes of pile head movement (Fig. 3);this procedure was suggested by Reese et al. [10]. In thepresent method, the p-direction in Fig. 3 agrees with thev-direction in the u–v plane and corresponds to the w-direc-tion in the u–w plane. The Fp–dp and Mq–dp curves (ModeI) were obtained by repeated single-pile analysis, based onthe load transfer method using p–y curves with increasingdp, with the boundary condition aq =0.The Fp–aq andMq–aq curves (Mode II) were obtained by a similar lateralanalysis with increasing aq, with the boundary conditiondp = 0. The relationships (Modes I and II) between lateralload (moment) and displacement (rotation) in u–v plane arenot always identical to those in u–w plane. Thus, c1, c2, c3,and c4 should be computed in both u–v plane and u–wplane. In the case of Mode III, for increasing settlementof the pile head, represented by du, the Fu–du curve wasobtained by continuous load transfer analysis using t–z/q–z curves. Mode IV, representing the torsional behavior,was not considered in this study.Here, all of the load–deformation functions which wereused for estimating KpE ½ igenerally showed nonlinear char-acteristics (Fig. 3). In the present method, a mixed incre-ment–iteration method, with iteration within each loadincrement, was used to adjust the deflections and reactionsto ensure compatibility and was compared with other non-linear techniques such as purely incremental solutions [12] and the secant modulus method (an iteration method) [10].Fig. 4, which represents load–settlement curves at the bot-tom of the column (point A), shows that for a purely incre-mental method the curves are significantly dependent onthe increment number N, whereas for a mixed increment–iteration method identical load–settlement curves are esti-mated regardless of N. On the basis of this result, onlyone increment (N = 1) with iteration, which is identical tothe secant method, was used for further analysis