simple expression of the dynamic stiffness of grouped piles

16 SIMPLE EXPRESSIONS OF THE DYNAMIC STIFFNESS OF GROUPED PILESWith assumptions (1) and (4), there are only two degrees of freedom for each cut-end ofall slices of the soil-pile composite, namely, sway and rocking motions respectivelyu ( { u u L } Tw ( = { w w L } T) (Figuredesignated as { }= ) and { }12u N L2.2b). The rocking motions are expressed in terms of the anti-symmetric vertical motion{ w } at the outermost edge ( r = R0) of the equivalent beam with respect to the beam’scentroid. In sway motions, all nppiles are equally displaced (assumption (1)), causingthe bending stiffness, EI , of the equivalent beam to be simply nptimes as large asthe bending stiffness of an individual pile. Assumptions (3) and (4) imply that axialmotions of the piles control the overall anti-symmetric rocking motion of the equivalentbeam just as reinforcements in a concrete beam do. Therefore, another bending stiffnessparameter, EI G , is introduced to describe the rocking motion of the beam. Thisstiffness parameter EI G is evaluated following the same procedure as that used for theevaluation of bending stiffness of a reinforced concrete beam (See APPENDIX I).Lateral external forces { p x} and moments { M } are finally described in matrixnotation in terms of {} u and { w } as specified in Equation (A12) in APPENDIX I:12w NL−1−11st column of[ ][ ] [ ][ L][ D]/R ⎤0andL D L M⎥zeros for other columns ⎧ u ⎫⎥⎪⎪LLLLLLLL ⎥⎨L⎬−1−1[ D] [ L] /R and [ Q] with D added to the ⎥⎪⎪ ⎭⎡⎧ ⎫⎢⎪px ⎪⎢⎨L⎬ = ⎢ LLLLLLLL⎪ M ⎪ ⎢1st row of0⎪ ⎪ ⎢⎩R0⎭ ⎢⎣zeros for other rowswhere, [ L ], [ D ] and [ ]M1,1upper - left corner⎥⎩w⎥⎦(2.1)Q are assembled global matrices corresponding to theindividual layer parameters of 1 / hj( hj= thickness of the j-th layer), h j/ EI andG 2EI / R0hj, respectively, (See Equations (A2), (A4) and (A10) in APPENDIX IQ , respectively).defining [ L ], [ D ] and [ ]“TLEM” has been upgraded for evaluation of the behaviors of an equivalent singlebeam (Ver. 1.2). Figure 2.3 shows pile cap stiffnesses k xxfor sway motions of 2×2and 3×3 steel pile groups (Table 2.1) plotted as functions of frequency. The results forthe equivalent beams are shown as open circles. Each pile group is embedded in thesame homogeneous soil deposit (Table 2.2) equally divided into 20 slices. Downwarddips in these plots of k xxoccur at essentially the resonance frequencies of the soilstratum for vertical shear wave propagation. As a whole, however, every real part of thepile cap stiffnesses decreases slowly as the frequency increases, whereas its imaginarypart representing the damping of a soil-pile group system shows a general upward trendto the right. The curves for the equivalent single beams agree well with rigoroussolutions from “TLEM” (Ver. 1.1).