## 3 Numerical Verification
Thus, the overlap tolerance was 0.05% width of the block for all runs. If this overlap tolerance is exceeded during a run, UDEC issues an error message and stops the run. Several preliminary runs were performed to check values of overlap and time increment. During these runs, the block was allowed to rock freely for several cycles after the cessation of the cyclic base motion in order to verify the period of free vibration as compared with the analytical values derived by [PERSON] (1988) where the period is strongly dependent on the block rotation at the beginning of free vibrations cycles. As a result, the spring stiffness was set to 200 GPa/m for the 1 m block in both normal and shear directions and was scaled with the size of the block to maintain similar overlap values. The calculation cycle steps for the distinct element method are shown in Figure 3b. The loads applied to the model where of two types: (i) the self-weight of the block and (ii) a repeated periodic pulse base velocity with a prescribed amplitude and period. For each applied pulse at a given period the acceleration was incremented until the block overturned within 25 cycles. At overturning, the critical combinations of period and acceleration of the applied pulses were noted. The application of up to 25 cycles was selected to produce realistic estimations of the critical combinations of accelerations and periods for a given pulse as it was noted by previous researchers ([PERSON], 1998; [PERSON], 1998) that the critical acceleration found by UDEC for short harmonic pulse periods decreases as the number of applied cycles of the forcing function increases from 1 to 5 stabilizing beyond that. For long harmonic pulse periods the critical acceleration found is independent of the number of cycles applied. In order to visualize the rocking movement, the applied excitation at the base block and the response displacement at the top of the rocking block were plotted versus time. When both plots fall on top of each other it means the block is moving in one piece without rocking. When the two plots are different that means the block is rocking, and the difference in displacement between the two points can be used to measure the magnitude of rocking, the time interval between successive impacts and the angle of rotation. Tables 1 and 2 show typical results for the 0.5 m and the 1 m sized blocks at critical combinations found by numerical simulation. Both blocks show the same results qualitatively except for an expected shift in the corresponding periods. Tables 1 and 2 show two regimes of rocking for the 0.5 m and 1 m block at critical signals:  
1. For periods greater roughly than 0.4 s for the 0.5 m block and 0.6 s for the 1 m block, the base and the top of the block move together. Inception of rocking is indicated by a minor lag between the two displacements followed by rapid increase over one or two impacts leading to overturning. 2. For periods smaller than 0.4 s for the 0.5 m block and 0.6 s for the 1 m block, multiple rocking takes place followed by overturning. The rocking response exhibits an apparent period that is  
Figure 3: (**a**) Schematic of the numerical model. (**b**) Calculation cycle steps for the distinct element method. different from the period of the applied excitation with an increase in the magnitude of rocking, akin to resonance, which leads to overturning after a few cycles. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**T (s)** & \begin{tabular}{c} **Velocity** \\ **Amplitude** \\ **(m/s)** \\ \end{tabular} & \
\begin{tabular}{c} **Acceleration** \\ **Amplitude** \\ **(g)** \\ \end{tabular} &
\begin{tabular}{c} **Block, 0.5 m, 25 cycles,** \\ **Displacement History (Base and Top)** \\ \end{tabular} \\ \hline
0.2 & 0.091 & 0.2914 \\
0.3 & 0.086 & 0.1836 \\
0.4 & 0.065 & 0.1041 \\
0.6 & 0.096 & 0.1025 \\
0.9 & 0.144 & 0.1025 \\
1.5 & 0.243 & 0.1038 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Critical combinations for 0.5 m block UDEC simulation.