## 4 Discussion
The example presented in Figure 8 shows a practical application to an 8 m high column with \(\alpha\) = 0.25, with \(\overline{T}\)(8 m column) = 2.32 s and other relevant periods calculated. It is seen that the relevant domain concerned with real physical conditions is the same as the one in the previous section. Figure 8: Safe–unsafe boundary for half sine impulse drawn on top of the overturning acceleration spectrum of the freestanding rocking column with tan \(\alpha\) = 0.25 when it is subjected to a one-sine acceleration pulse after Makris and Kampas [82]. Figure 7: Experimental results for 0.5 m solid cylinder compared to theoretical and UDEC model for 0.5 m block after [79]. values is rather limited compared to extent of numerical modeling. The domain of large periods for the 8 m column [2.32 s; \(\infty\)) corresponds to \(\omega_{\text{p}}\)/p in the interval of \((0;2]\) and the domain of the small periods (2.32 s; 0.68 s) corresponds to \(\omega_{\text{p}}\)/p in the interval of \((2;6.8)\). Similarly to Figure 8, Figure 9 compares the original numerical data of [PERSON] and [PERSON] [8] for a block with tan \(\alpha=0.25\) under a harmonic acceleration forcing function to the predicted safe-unsafe boundary for half sine impulse under the same maximum friction factor limitation. The simple energy based approach proposed in this work provides a practical, reliable, and conservative alternative to the presentation and study of the stability and overturning of blocks carried by others in the frequency spectrum domain [8,63,82]. The proposed stability chart corresponds to the \"dynamic identity card\" of [PERSON] and [PERSON] [87] and to the \"contour plot\" of [PERSON] et al. [88]. It offers a lower bound to the \"safety wall\" judiciously proposed by [PERSON] and [PERSON] [20] and to the \"overturning acceleration spectrum\" of [PERSON] and [PERSON] [89] and a linearization of the solution proposed by [PERSON] [90] for a rocking-sliding block. [PERSON] and co-workers [63,82] have proposed to use the characteristic length scale, \(L_{\text{p}}=a_{\text{p}}T_{\text{p}}{}^{2}\), of the ground excitation as an intensity measure for the near-fault single pulse noting that they still needed one more size related variable to properly characterize the rocking instability. [PERSON] et al. [91] investigated the effect of 23 ground motion intensity measures (IMs) on the stability of ancient rocking columns trying to identify the most relevant ones. They found that acceleration based parameters could predict rocking response only if combined with a frequency content parameter and that free-standing columns exhibited increased stability due to increasing size and rocking behavior. They only collapsed under long period extreme seismic records. [PERSON] et al. [92] investigated the efficiency of using Peak Ground Acceleration (PGA) and Peak Ground Velocity (PGV) as earthquake IM for the seismic vulnerability assessment of monolithic rocking bodies. To do so, a three-dimensional model of an existing free-standing ancient column of the Roman Agora of Thessaloniki was analyzed numerically with a three-dimensional DEM. A strong influence of the mean frequency of the ground motions, fm, on the structural response was observed and the authors concluded that it is deemed more efficient to consider dual Intensity Measures such as, (PGA, fm) and (PGV, fm) pairs, instead of single  
Figure 9: Safe–unsafe boundary for half sine impulse drawn on top of the overturning acceleration spectrum of the freestanding rocking block with tan \(\alpha=0.25\) under a harmonic acceleration forcing function after Figure 4 of [PERSON] and [PERSON] [8] (permission requested). PGA and PGV amplitudes. Some of their numerical results can be interpreted and proved using the simple energy based critical pulse approach presented above.