Equivalent-linear stiffness and damping in rocking of circular and strip foundations

An approximation is developed for obtaining the nonlinear stiffness K_R and damping C_R of a shallow circular or strip footing undergoing rocking oscillation on a homogeneous but inelastic undrained clayey stratum. Based on the parametric results of 3-D and 2-D finite-element analyses, equivalent-linear K_R and C_R are expressed in readily usable dimensionless forms. K_R, normalized by its linear elastic value, is shown to be a unique function of: (1) the vertical factor of safety F_s against static bearing capacity failure, and (2) the angle of rotation θ{symbol} normalized by a characteristic angle θ{symbol}_s. The latter is approximately the angle for which uplifting usually initiates at one edge of the foundation. Three sources contribute to the value of the dimensionless damping ratio ξ_R (derived from C_R): wave radiation, which is a function of frequency but is shown to amount to <3 %; soil inelasticity (hysteresis), for which graphs are developed in terms of θ{symbol}/θ{symbol}_s and F_s; and energy loss due to impact and the collateral vertical motion when severe uplifting takes place, which is tentatively determined from dynamic M: θ{symbol} loops. Comparative parametric seismic time-history analyses provide an adequate validation of the iterative equivalent-linear approximation which implements the developed equivalent K_R and ξ_R, but they also highlight its limitations.