## 2 Theoretical Model
Therefore, the critical impulse curve or relationship is given by:  
\[v=\frac{\alpha g}{p}=\beta t_{p}a_{p}\]  
or:  
\[\frac{a_{p}}{\alpha g}=\frac{1}{p\beta t_{p}} \tag{5}\]  
and the point of intersection with the rocking limit, or the transition point: \(a_{p}=\alpha\)g, is at \(\tilde{t}_{p}\):  
\[\frac{1}{2}I_{\mathrm{\alpha}}\mathrm{\alpha}^{2}=WR(1-cos\mathrm{\alpha}) \tag{2}\]0  
A normalized curve describing the Safe-Unsafe space for a pulse characterized by \(a_{p}\) and \(t_{p}\) is then given by: \(\tilde{a}_{p}=\frac{a_{p}}{\alpha g}\), where:  
\[\frac{1}{2}I_{\mathrm{\alpha}}\mathrm{\alpha}^{2}=WR(1-cos\mathrm{\alpha}) \tag{2}\]1  
For the case of a half-sine pulse of period T = 2 \(t_{p}\), we have:  
\[\frac{1}{2}I_{\mathrm{\alpha}}\mathrm{\alpha}^{2}=WR(1-cos\mathrm{\alpha}) \tag{2}\]2  
Therefore, \(\beta t_{p}a_{p}=a_{p}\frac{2t_{p}}{\pi}\) and \(\beta=\frac{2}{\pi}\). For the cases of step and linear pulses \(\beta\) is equal to 1 and 0.5, respectively. Similarly, \(\beta\) is equal to 2/3 for a parabolic pulse. For a given periodic pulse, the intersection point between Equations (1) and (5) is given by:  
\[\frac{1}{2}I_{\mathrm{\alpha}}\mathrm{\alpha}^{2}=WR(1-cos\mathrm{\alpha}) \tag{2}\]3  
where \(\tilde{t}_{p}\) is the pulse duration at the intersection of the two curves and \(\overline{T}\) is the characteristic period of the block, or transition period, given by \(p\overline{T}=\pi\) for the sinusoidal pulse. The safe-unsafe boundary in the acceleration-pulse period as per Equations (1), (5), and (7) is plotted in Figure 2a showing that at low periods the relationship between maximum impulse acceleration and pulse period is hyperbolic as per Equation (5) and at larger periods the condition of Equation (1) dominates. The intersection or transition at the calculated critical normalized pulse period is also shown. Figure 2b shows the shape of the overturning half sine impulses for different periods. The proposed approach allows for a simple determination of the unsafe domain boundary based on the identification of a critical pulse energy value that is a characteristic of the rocking block. Strong-motion earthquake accelerations may be thought of as a train of pulses and the identification of a critical pulse in the earthquake signal could help determine the overturning effect of that earthquake. This is of course a first order approximation as there may be pulses which are not critical but are sufficient to cause stable rocking opening up the possibility for kinetic energy build up over a period of time under a number of sub-critical pulses leading to overturning of the block. The conditions for these critical combinations require further refinement of the model. Previous theoretical and numerical works [5; 6; 8; 16; 17] have shown that there are various rocking response modes prior to overturning under a single pulse with or without impact. They have also shown that as the magnitude of the acceleration is increased for the same pulse frequency the rigid block may overturn in one mode at a given acceleration, undergo stable rocking at a greater acceleration then overturn again in another mode as the magnitude of the acceleration is further increased. Hence, there may be non-critical combinations of acceleration and period that fall inside the identified unsafe zone, the locus of these combinations is however limited and may be conservatively ignored given the first order approximation proposed in this work. Equation (7) presents a dimensionless parametric formulation of the model that allows scaling between blocks of different dimensions to be established. The larger of two geometrically similar blocks would be more stable than the smaller block. For the same critical acceleration the 1/N reduced block would have a critical period 1/N\({}^{0.5}\) smaller. This similarity rule should be used in scaling up the experimental results to the full size columns as well as in scaling down recorded earthquakes signals before their application to small scale structural models [77].