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Mylonakis et al. 1997): The dynamic stiffness matrix of thesuperstructure is attached to an additional impedance matrix representing theunderlying unbounded soil region and the superstructure is then excited bythe response history (denoted as Foundation Input Motion – F.I.M.) of a hypo-thetical soil-foundation sub-system lacking the superstructure mass.In case of deep (pile) foundation, the procedure can be summarized in fourindependent steps:1. Analysis of the free-field soil response (i.e. without the presence of piles) tovertically incident S waves.2. Analysis of the interaction of the single pile or pile group with the surroundingsoil, driven by the free-field response of step 1.3. Computation of the dynamic impedances (“springs” and “dashpots”) at the pilehead or the pile-group cap, associated with the swaying (Rx and Ry), rocking Rryand Rrx) and cross-swaying-rocking (Rx,ry and Ry,rx) motion of the foundation.It is noted that specifically for pile groups, the dynamic impedance of thefoundation cannot be computed by simply adding the dynamic stiffness of theinpidual piles. This pile-to-pile interaction is frequency-dependent, resultingfrom waves that are emitted from the periphery of each pile and propagate tostrike the neighbouring piles (Mylonakis et al. 1997). A variety of numerical andanalytical methods have been developed (Nogami et al. 1992; Makris andGazetas 1992; El-Naggar and Novak 1996; Gazetas and Mylonakis 2002)tocompute the dynamic response of this interaction. Especially for the rotationalstiffness of foundations supporting bridge piers or building columns that areexpected to develop plastic hinge at their base, a non-linear moment-rotation relationship can be implemented (Sextos et al. 2003a) combining the rotationalcompliance of the foundation with a lumped plasticity model for the RC section.In particular, by combining the flexibility of the non-linear pier-base inelasticspring and the linear rotational foundation spring that was calculated in theinertial soil-structure interaction stage, the final rotational spring (Fig. 2.47)isderived, being characterized by a first branch (uncoupled rotational) stiffnessequal to=Y and a second branch stiffness =0yequal to:=0y ¼ 11=yþ ypMu My¼ 1ReKdynHH KdynHM eKdynHH KdynMM KdynHM2þKdynyV KdynHH þ ð0 08Lþ0 022fyl dblÞ ðfu fyÞMu My(2.21)where yp,Mu,My are the plastic rotation, the ultimate and the yield moment ofthe pier base RC section respectively, KdynHH KdynMM KdynHM are the horizontal,rocking and coupled modes of vibration terms of the dynamic stiffness matrix,which for the case of pile groups are functions of the damping coefficients zΗΗ,zMΗ, zMM and the dynamic interaction factors adynij, Kdynyv is the (static) rotationalstiffness component attributed to the antisymmetric vertical loading of the piles,e ¼ H/M is the foundation eccentricity, L is the distance from the critical piersection to the point of contraflexure, fyl is the yield strength of the longitudinalbars and dbl is the diameter of the longitudinal reinforcement. The above approach is similarly applied for the case of surface foundationsaccording to the corresponding four steps. A summary of closed-form solutions ofdynamic stiffness of spread footings that have been derived by regression analysis,based on Finite- and Boundary-Element data can be found elsewhere (Mylonakiset al. 2006a). Commonly, due to their small embedment, the cross-swaying-rockingstiffness of the spread footings can be neglected.In general, for each of the above analysis steps, several alternative formulationshave been developed and published in the literature, including Finite Element,Boundary Element, semi-analytical and analytical solutions, as well as a varietyof simplified methods (reviewed by Pender 1993 and Gazetas andMylonakis 1998).In practice however, especially when specialized software is not available, it isconvenient that the above inertial and kinematic interaction uncoupled sub-systemsare split into two FE models using any standard structural analysis software and thecorresponding damping and stiffness coefficients proposed in the literature.2.6.2.1 Considering the Characteristics of Seismic Ground MotionFrom the four steps described above, it is believed that the highest level ofuncertainty is related to the identification of the incoming wavefield (as describedin step 1). 源1自37518.论~文'网·www.751com.cn