## 4 Discussion  
The responses of 0.5 m and 1 m blocks to harmonic pulses presented in Tables 1 and 2 show clearly two modes of response as predicted by the theory. The numerically observed transition periods, around 0.4 and 0.6 s for the 0.5 and 1 m blocks, respectively, fall consistently below the theoretically predicted transition periods of \(\overline{T}\)(0.5 m column) = 0.580 s, \(\overline{T}\)(1 m column) = 0.821 s; for comparison \(\overline{T}\)(20 m column) = 3.67 s. This is also shown in the normalized graphs of Figure 4 where the normalized transition period falls around pT = 2 for the numerical simulation while pT = \(\pi\) theoretically. Periodic pulses with periods less than the transition period of the block have high acceleration that can drive rocking; this rocking becomes unstable if the minimum energy condition is met. At higher periods (that are larger than the transition period), the energy of the pulse is much higher than the energy required to overturn the block, therefore whenever the rocking condition is met, overturning takes place immediately without rocking or within one or two impacts, i.e., within less than one rocking cycle. In Figure 4, the time/scale normalized curves fall on top of each other showing a very good match. The normalized UDEC boundary for all three block sizes is bi-modal, all three curves show two regimes that correspond to the two conditions of the simplified safe-unsafe theoretical boundary defined by Equation (7) for the minimum acceleration and minimum kinetic energy required for overturning. The transition period separates the flat part of the UDEC boundary at higher periods from the sharply rising part below the transition period value where higher accelerations are required to overturn the block. It is important to note here that both theoretical and numerical curves are truncated when the normalized acceleration required for overturning reaches a value of \(\mu/\alpha\) at which point sliding takes over and the conditions for rocking do not hold any more. For the columns used in the numerical model \(\mu\) = 0.5 and \(\alpha\) = 0.11 setting \(\mu/\alpha\) slightly above 4.5. Above the transition period the UDEC results fall on or just below the flat line which corresponds to a = \(\alpha\)g (Equation (1)), where \(\alpha\) = tan\({}^{-1}\) (b/h) is the critical stability angle and \(\alpha\)g is the minimum theoretically required acceleration for overturning. The normalized transition period between the two modes of response is the same for all block sizes used in the numerical simulation. Its value is however smaller by about one third than the  
Figure 6: Normalized numerical results for all three types of pulses for the 1 m block. theoretically calculated period shown by the intersection of the block overturning impulse curve giving the minimum kinetic energy and the rocking line defined by the minimum critical acceleration value. In Figure 6, the pulse energy normalized curves for the 1 m size block are bi-modal, fall close to each other and show satisfactory agreement with the simplified safe-unsafe boundary following closely Equations (1), (5), and (7) for the minimum acceleration and minimum kinetic energy necessary to cause overturning. The normalized transition period between the two modes of response is the same for all periodic pulses used in the numerical simulation. As noted above, its value is smaller by about one third than the theoretically calculated transition period. In their numerical analysis of columns subjected to harmonic pulses, [PERSON] et al. [80] and [PERSON] et al. [81] reported the same observations as in this work with respect to the decreasing trend of the safe-unsafe boundary in the ground acceleration-period space and its bi-modal shape. [PERSON] et al. [81] also reported that the safe-unsafe boundary exhibits a discontinuity in slope for a given transition pulse duration reflected also in a change in the mode of collapse.