## 4 Discussion
They also reported that independently of the type of excitation applied scalability with respect to the square root of R was observed in agreement with the results presented in the previous paragraph. The findings of the theoretical and numerical approach also agree well with the findings of [PERSON] and [PERSON] [82] who have distinguished size and slenderness as the two competing parameters in the stability of free-standing rocking columns. Slenderness determines the theoretical minimum acceleration required for overturning while the size determines the minimum pulse energy required for overturning. The results of shaking table experiments on a single block stone column of 0.5 m height representing a simplified reduced model of the Baalbek column (\(\alpha=\tan(\text{b/h})=0.11\)) under harmonic excitations [79] are compared to the theoretical and numerical models in Figure 7. Up to 25 cycles were applied and the experiment was noted as a green circle when the block did not overturn. Overturning at any time before the 25 cycles were elapsed was noted as a red cross. The data presented was obtained by systematic sweep in the acceleration-period space and confirmed repeatable results were reported noting the sensitivity of the shake table test results around the critical stability values [79]. The survival and overturning zones are clearly separated. The separation line between the green circles and the red crosses, representing each a single experiment, falls below both the theoretical and the numerical models. This is probably caused by the multiple imperfections in the experimental set up such as eccentricities and surface roughness as well as the differences between the actual situation of a cylindrical block rocking in 3D and the 2D modeling approach adopted [83]. Wobbling, or out-of-plane movement, can take place under transitional conditions in actual experiments when the conditions for rocking are met. The simple theoretical model proposed cannot predict out-of-plane response and a DEM 2D simulation maintaining the same geometric conditions was used for numerical verification. A comparison of the theoretical model and the 2D DEM results with 3D DEM simulations has been carried out by the authors showing no major differences with respect to the stability chart [84; 85]. The stability chart of Figure 2a can be plotted in the acceleration-frequency domain nothing that the frequency of the half sine impulse signal, \(\omega\), is given by \(\omega=2\pi\)/T, where T is the period of the impulse. The dimensionless boundary is plotted in Figure 8 on top of the overturning acceleration spectrum for a freestanding rocking column with \(\tan\alpha=0.25\) when it is subjected to a one-sine acceleration pulse from the work of [PERSON] and [PERSON] [63] and [PERSON] [82]. The double lobed numerical solution is presented in a dimensionless form that can be used for any column size and shape. The safe-unsafe boundary is plotted on top of the overturning spectrum as per Equations (1), (5), and (7). The periods shown on the abscissa axis are calculated for the 8 m tall column with \(\tan\alpha=0.25\) studied [82]. Noting that rock friction factors vary typically from 0.25 to 0.85 [86], the sliding limit is set for discussion at a maximum friction factor of \(\mu=0.85\) giving \(\mu\alpha=3.4\). In reference to Figure 8: (i) no rocking takes place under the green line when the pulse acceleration is smaller than \(\alpha\)g; (ii) rocking can take place when the pulse acceleration is greater than \(\alpha\)g and smaller than \(\mu\)g, i.e., between the green and the blue lines; and (iii) sliding takes place above the blue line when the pulse acceleration is greater than \(\mu\)g. The red line plots the critical pulse energy required for overturning. All pulses above the line contain sufficient energy for overturning the block. Overturning can only take place when both acceleration and energy conditions are met, i.e., above both red and green lines and before sliding take over, i.e., below the blue line.