## 2 Theoretical Model  
Based on the seminal works of [PERSON] [2], [PERSON] and [PERSON] [3], [PERSON] [4], [PERSON] [1], [PERSON] [7], [PERSON] [14], [PERSON] et al. [15], [PERSON] and [PERSON] [16], [PERSON] and [PERSON] [17], and noting more recent derivations and contributions [18; 20; 21; 22] including the authors' [76], the stability of a rigid rectangular block subjected to a single pulse can be determined using five parameters: (i) the size of the block, 2h, (ii) the shape factor or slenderness of the block, b/h, (iii) the amplitude of the pulse or maximum acceleration, a, (iv) the duration of the pulse, \(\mathrm{t_{p}}\), and (v) the shape of the pulse. With reference to Figure 1 adopted from [PERSON] [1]: the block height is 2 h, its width is 2 b, its mass is m, its weight is W = mg with g being the acceleration of gravity and its rotational moment of inertia is \(\mathrm{I_{o}}\) = (4/3) mR\({}^{2}\); with R = \(\mathrm{\sqrt{(}h^{2}+b^{2})}\). As the block rotates around its corner O, the block rotation, \(\mathrm{\theta}\), increases from zero to a limit value at which the block overturns. The limit value of \(\mathrm{\theta}\) is the angle \(\mathrm{\alpha}\) = tan\({}^{-1}\) (b/h). Values of (b/h) are relatively small for slender blocks and \(\mathrm{\alpha}\) \(\approx\) b/h. In order for rocking to start, the overturning moment due to the horizontal acceleration, \(\mathrm{M_{O}}\) = (ma)h, must exceed the restoring moment of the block, \(\mathrm{M_{R}}\) = (mg)b. Hence for inception of rocking it is necessary that \(\mathrm{M_{O}}\) \(\geq\)\(\mathrm{M_{R}}\) or the horizontal acceleration caused by the applied pulse, a, exceeds at some point the minimum value of \(\mathrm{\alpha}\)g:  
\[\mathrm{a}\geq\mathrm{\alpha}\mathrm{g}. \tag{1}\]  
Another necessary condition for rocking to take place is that no sliding takes place before inception of rocking and afterwards, i.e., the friction between the block and its base is sufficiently high to prevent sliding. In other terms, the coefficient of friction between the block and its base, \(\mathrm{\mu}\), is such that a \(\mathrm{\leq\mu}\) at all times. In order to rotate around its edge until eventually reaching the limit position of stability beyond which overturning takes place, the rotational kinetic energy of the block must be equal or greater than the change in potential energy necessary to reach the unstable position at \(\mathrm{\theta}\) = \(\mathrm{\alpha}\), i.e., \(\mathrm{\Delta}\)(kinetic energy) = \(\mathrm{\Delta}\)(potential energy) at the limit:  
\[\frac{1}{2}I_{\mathrm{\alpha}}\mathrm{\alpha}^{2}=WR(1-cos\mathrm{\alpha}) \tag{2}\]  
where \(\frac{1}{2}I_{\mathrm{\alpha}}\mathrm{\alpha}^{2}\) is the rotational kinetic energy of the block and \(\mathrm{\omega}\) the angular velocity reached at the end of the pulse. Linear momentum is imparted to the block by the horizontal ground movement pulse. It is equal to mv with v being the linear horizontal velocity reached at the end of the pulse. When the block rotates the linear momentum is transformed into angular momentum due to momentum conservation and we have:  
\[\frac{W}{g}vh=I_{\mathrm{\alpha}}\mathrm{\omega} \tag{3}\]  
Figure 1: Rocking Block after [PERSON] [1]. By combining Equations (2) and (3) and eliminating \(\omega\) we obtain:  
\[v^{2}=\left(\frac{gI_{o}}{Wh}\right)^{2}\frac{2 WR}{I_{o}}(1-cos\alpha)=\frac{8 gR} {3}\frac{(1-cos\alpha)}{cos\alpha^{2}}\]  
or:  
\[v=\sqrt{\frac{8 gR}{3}\frac{1-cos\alpha}{cos^{2}\alpha}}\]  
For small angles \(\alpha\), \((1-cos\alpha)=\alpha^{2}/2\) and \(cos^{2}\alpha=1\), hence:  
\[v=\sqrt{\frac{8 gR}{3}\frac{\alpha^{2}}{2}}=\alpha\sqrt{\frac{4 gR}{3}}=\frac{ \alpha g}{p} \tag{4}\]  
where \(p\) is the frequency parameter of the rocking block given by \(p^{2}=\text{WR}/\text{I}_{o}\), or \(p=\sqrt{\frac{3g}{4R}}\) For any pulse:  
\[v=\beta t_{p}a_{p}=\int_{0}^{t_{p}}a(t)dt\]  
where: \(t_{p}=\) duration of pulse, \(a_{p}=\) peak acceleration of the pulse, and \(\beta=\) fullness factor according to the above equation [19; 22; 68].