Explicit solutions for the ground displacement of a sliding model simulating both the frictional and rotational effects under idealized acceleration pulses

For the first time, the present work derives explicit equations predicting the downward ground displacement of a sliding model simulating both the frictional and rotational effects under idealized acceleration pulses, in the form of simple formulas. Explicit equations allow not only accurate predictions in all cases, but also analysis of the solutions and derivation of expressions for limit cases. Half and full cycles of (i) rectangular, (ii) triangular, (iii) trapezoidal, and (iv) sinusoidal pulses and slopes both under static stability and instability are considered. For this purpose, for pulse cases (i)–(iii) recently proposed implicit analytical solutions are used, while for case (iv), first the analytical equations predicting the sliding displacement and velocity of a sinusoidal pulse in terms of time are obtained and then the time duration of motion is estimated, by using the Bhaskara approximation of the sine and cosine functions. Then, from these, solutions for the particular limit cases corresponding to the “conventional” sliding-block model (Case A) and the post-failure run-off movement without any applied pulse (Case B) are derived. The results for Case A provide a useful tabulation of sliding-block solutions, some of which are not reported in the literature. The results for Case B provide novel predictions of the time duration of motion in the case of post-failure movement. The general solutions are analyzed graphically and the deviation from the solutions of Cases A and B is illustrated. Finally, the explicit solutions are compared to solutions of actual accelerograms.