Complex Response of a Rocking Block to a Full-Cycle Pulse

The rocking response of rigid, free-standing bodies to earthquake pulses is revisited. A two-dimensional rectangular block resting on a rigid base is considered, subjected to an idealized ground acceleration pulse composed of two constant half-cycles of equal amplitude, equal duration, and opposite sign. Closed-form expressions are obtained for the dynamic response, whereas rigorous overturning criteria are established for conditions with and without impact. The solutions are expressed in terms of three dimensionless parameters, namely, pulse duration, uplift strength, and restitution coefficient. Despite the apparent simplicity of the problem, the response can exhibit complex—even counterintuitive—patterns, a trait attributed to the possibility of overturning in two distinct modes (forward and backward), the nonlinear nature of the impact, the real-valued (positive) pole of the differential operator, and the presence of multiple immobility points in a particular response branch. The bifurcation behavior associated with the existence of two overturning modes is highlighted. Comparisons against idealized pulses of other shapes and actual near-fault recorded ground motions are presented and commented.