Rocking response and overturning criteria for free standing rigid blocks to single—lobe pulses

The classical problem of rocking of a rigid, free-standing block to earthquake ground shaking containing distinct pulses, as is the case of near-fault earthquake motions, is revisited. A rectangular block resting on a rigid base is considered, subjected to a range of idealized single-lobe ground acceleration pulses expressed by a generalized function controlled by a single shape parameter. The problem is treated analytically in the realm of the linearized equations of motion under the assumption of slender block geometry and rocking without slipping. Peak rocking response and overturning criteria for different waveforms are presented in terms of dimensionless closed-form expressions and graphs. Two parameters are employed to this end: dimensionless pulse duration f (i.e., actual pulse duration times characteristic block frequency) and dimensionless uplift strength η (i.e., ratio of minimum required acceleration for initiation of uplift over peak pulse acceleration). The linearized response is compared analytically with the fully non-linear one using an ad hoc energy formulation leading to an approximate closed-form solution. It is shown that the non-linear equations of motion yield more stable response than their linearized counterparts. A brief discussion on scaling laws is provided. ► Novel solutions are presented for rocking to an infinite number of pulse shapes. ► Overturning is controlled by dimensionless pulse duration f and uplift strength η. ► Analytical formulae are provided for the safe (S) and overturning (O) regions. ► Critical line separating (S) from (O) i.e., “safety wall” is described in closed form. ► Other parameters being equal, nonlinear analysis leads to more stable rocking response.