Sliding Blocks under Near-Fault Pulses: Closed-Form Solutions

Analytical and numerical solutions are presented for the rigid-plastic response of geo-structures to idealized ground acceleration pulses. These shock-like waveforms are typical of near-fault earthquake motions generated by forward fault-rupture directivity and may inflict large permanent displacements in the absence of substantial residual soil strength. The geo-structures are modeled as rigid blocks resting on inclined frictional planes. Although idealized, these models are widely accepted by geotechnical engineers, for simulating a variety of structures including retaining walls, embankments and slopes. Four basic simple pulse waveforms are examined: (1) rectangular; (2) sinusoidal; (3) triangular; (4) exponential. An analytical study is presented on the effect of frictional strength and number of excitation cycles on peak displacements. Results are presented in the form of dimensionless graphs and closed-form expressions that elucidate the salient features of the problem. It is shown that Newmark approaches based on conventional motions may under- or over-estimate peak displacements depending on the circumstances. It is also shown that all three time histories of ground motion (i.e., acceleration, velocity, and displacement) control peak response — contrary to the widespread view that ground velocity alone is of leading importance. Issues related to scaling laws of peak displacement are discussed.