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. Fig. 7 shows a flowchart of the presentanalysis.3. Validation by comparison with other numerical studiesA series of linear pile group analyses were performed toverify the structural-analysis routine used in the presentmethod by comparison with other numerical methodswhich have been used in the design of pile groups. Theresults of a nonlinear analysis of a typical pile group sup-ported column (a piled pier) were then compared withresults obtained from Group 6.0 and FBPier 3.0. This sec-tion focuses on the verification of the present method. Inaddition, the load–deformation relationship of a piled pierwas investigated.3.1. Linear analysesA schematic diagram of a 2 · 2 pile group structure isshown in Fig. 8. This structure consists of a pier, a pilecap, and four identical vertical piles, which are spaced by3 m (=6D, where D is the pile diameter). A p-multiplierof unity was used because the interaction between piles inthis configuration (s/D = 6) seems not to be significantand because this study focuses on the pile–cap interactionrather than on interactions between piles. The four pileshave an embedded length of 10 m, a diameter of 0.5 m,and a flexural rigidity (EI) of 147,264 kN m2. The thicknessof the pile cap is 0.75 m, and the pile head conditions arefixed. The pier is 10 m in length and 1 m in diameter, andFig. 8.
Schematic diagram of a piled pier.Table 1Material properties for pile groupsElements Properties ValuesPier Elastic modulus (E) (MPa) 40,000Moment of inertia (Ix, Iy)(m4) 0.04909Area (A)(m2) 0.7864Polar moment of inertia (J)(m4) 0.09817Shear modulus (G) (MPa) 17,391.3Pile cap Elastic modulus (E) (MPa) 40,000Poisson’s ratio (m) 0.18Thickness (t) (m) 0.75Pile Elastic modulus (E) (MPa) 48,000Moment of inertia (Ix, Iy)(m4) 0.003068Area (A)(m2) 0.19635 has a flexural rigidity of 1,963,600 kN m2. Table 1 showsthe material properties used in this study.The results of linear analysis by the present method havebeen compared with results from three different numericalmethods to verify the present structural analysis routine fora piled pier. Fig. 9 shows the soil conditions used for thelinear analyses. The same axial soil spring constants wereused along the pile depth, with a constant value of2000 kN/m2, which includes the pile perimeter, corre-sponding to loose sand with an SPT-N value of 9. Theend-bearing spring constant was 10,000 kN/m2, which cor- responds to an SPT-N value of 40, and the tension part wasneglected. The constants of the horizontal soil springs wereincreased from 0 to 100,000 kN/m2along the pile depth.The pile group was subjected only to a lateral load, of1000 kN at the pier top.
The pile head stiffness (c1 c5) cal-culated by the load transfer method using the presentmethod, shown in Fig. 3, is described in Table 2 (case 1),where the results are identical to those obtained fromFBPier 3.0.Table 3 shows the estimated displacements and distrib-uted forces when a lateral load of 1000 kN in the x-direc-tion is applied to the pier top. The displacement methodand Group 6.0 cannot model pier elements; therefore anequivalent moment was applied to the pile cap, equal tothe lateral load multiplied by the length of the pier.As shown in Table 3, the axial and lateral displacementsat the pile head estimated by the present method, by Group6.0, and by the displacement method are approximately thesame, while the values obtained from FBPier 3.0 aregreater by about 22%. This suggests that FBPier 3.0 givesmore flexible results for pile groups in the case of lateralloading, because of the difference in the modeling methods,whereby the former methods model piles using stiffnessmatrices and the latter method does so using three-dimensional elements. The axial forces and moments at the pile head predicted by each method show some discrep-ancies, whereas the same lateral forces are distributed toeach pile head in all the numerical methods. The presentmethod predicts smaller axial forces and larger momentsat the pile head than does FBPier 3.0 and estimates resultssimilar to those of Group 6.0. For all the linear analyses, itis found that the present method gives results similar tothose obtained from other numerical methods, especiallyGroup 6.0.3.2. Nonlinear analysesThe behavior of axially and laterally loaded single pileswas predicted by the present method (YSGroup) by consid-ering nonlinear load transfer curves and then comparingthem with the results of TZPile [23] and LPile 4.0 Plus[24], respectively. It was found that the predictions werealmost identical.After these single pile analyses, a series of group pileanalyses were performed on the pile group configurationshown in Fig. 8. The inpidual piles were assumed to bePHC driven piles installed in a sandy soil. The soil aroundthe inpidual piles was modeled with nonlinear load trans-fer curves. The types of soil considered were three sandysoils, which were a loose, a medium, and a dense sandaccording to density, and had SPT-N values of 7, 20, and40, respectively. The axial load transfer curves (t–z and q–z curves) were estimated using an equation developed byMcVay et al. [25] suitable for driven piles; the lateral loadtransfer curves (p–y curves) were used as an API model[26]. The properties used for estimating the axial and lateralload transfer curves are shown in Table 4. The ultimate skin friction (sf) and ultimate end-bearing capacity (qp) were esti-mated from Meyerhof’s equations, to obtain sf =2N (kPa)and qp = 400N (kPa). The end bearing shear modulus Giwas applied together with the values of the modulus forloose, medium, and dense sand used in the analysis ofZhang and Small [21]. The internal friction angles for thep–y curves were estimated from the mean value of the pre-dictions of Dunham’s equation, / ¼ffiffiffiffiffiffiffiffiffi12Npþ 15 , andPeck’s equation, / = 0.3N +27 , using SPT-N values.The values of the p–y modulus k used were the proposedvalues in the LPile 4.0 Plus manual.Fig. 10 shows the results of a nonlinear analysis of apiled pier, subjected to only a lateral load at the pier top.Very large lateral displacements are estimated at the piertop in Fig. 10(a), which can be predicted only by a coupledanalysis method such as FBPier 3.0 and the presentmethod. Note that the lateral displacement at the piertop obtained by the present method is almost the same asthat obtained from FBPier 3.0. Fig. 10(b)–(d) representsthe lateral displacement, the settlement of the pile headA, and the settlement of the pile head B, respectively.The lateral displacements of the pile heads obtained bythe present method are a little smaller than the valuesobtained from FBPier 3.0, but are quite similar to thoseobtained from Group 6.0.