Analytical solutions for the sliding displacement of model slopes simulating both the frictional and rotational effects under piecewise linear acceleration pulses

Simplified methods predicting earthquake-induced displacement of slopes are based on the Newmark sliding-block model. This model does not consider the decrease in inclination of the sliding mass as a result of its downward motion and this effect is usually modeled by improving the above model by assuming that the critical acceleration value for relative motion of the block increases linearly with the distance moved. This work, for the first time, derives analytical solutions predicting the sliding displacement under idealized acceleration pulses of the model described above simulating both the frictional and rotational effects. First, a general solution is derived in dimensionless form for the motion of the sliding mass under a piecewise linearly varying pulse for the cases of both static stability and instability. The solution is obtained in each interval by (a) solving for the displacement and velocity in terms of these quantities at the end of the previous interval, (b) estimating the time corresponding to a sliding velocity equal to zero, and (c) investigating if the estimated time solution is consistent with the time interval and thus if it can be considered as the global solution. Then, based on this solution and the conditions where downward and upward motion starts, a procedure predicting the final displacement of the sliding mass for a piecewise linearly varying pulse is proposed. Finally, analytical solutions are derived, given in graphical form, partly validated, and discussed for the particular cases of half and full cycle pulses of rectangular, triangular, and trapezoidal shapes.