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 shallowcircular 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. normalized by a characteristic angle nu(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 xi(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 nu/nu(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 : nu 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 xi(R), but they also highlight its limitations.