## 2 Methodology
### Non-Linear Dynamic (Rocking) Analysis  
Once the geometry of the collapsing macro-block is defined, the rocking dynamic analysis can be performed. Following the approach presented in Reference [20], the rocking equation of motion assumes the following general linearised form:  
\[\widetilde{I}\phi-\widetilde{K}(\phi-\phi_{cr})=-\widetilde{B}\tilde{u}_{g}+ \widetilde{M} \tag{5}\]  
where \(\widetilde{I}\) is the moment of inertia of the macro-block computed about the axis of rotation (hinge), \(\widetilde{K}\) is the rotational stiffness of the structure, \(\phi\) is the rotation, while \(\phi_{cr}\) is the critical (unstable equilibrium) rotation of the macro-block, about which the equation of motion is linearised in order to obtain dynamic equivalence with the single rocking block. Additionally, \(\widetilde{M}\) is the moment due to external static forces (if any), while \(\widetilde{B}\tilde{u}_{g}\) is the moment provided by the ground motion.  
Through the following transformation of variables (where \(g\) is the acceleration due to gravity):  
\[\theta=\phi\frac{\widetilde{K}}{g\widetilde{B}} \tag{6}\]  
Equation (5) can be re-written as:  
\[\ddot{\theta}=p_{eq}^{2}\left(\theta-\widetilde{\lambda}-\frac{\tilde{u}_{g} }{g}\right) \tag{7}\]  
where \(p_{eq}\) is the equivalent rocking frequency parameter and \(\widetilde{\lambda}\) is an approximation of the load multiplier that activates the mechanism (i.e., not explicitly taking into account frictional forces) as opposed to the minimum frictional load multiplier \(\lambda\) as defined in the previous sub-section. In this work, rocking is defined to initiate when \(\tilde{u}_{g}>g\widetilde{\lambda}\).  
Note that both \(p_{eq}\) and \(\widetilde{\lambda}\) are the equivalent rocking parameters defining the rocking equation of motion, and depend on the kinematic constants \(\widetilde{I},\ \widetilde{K},\ \widetilde{B},\ \widetilde{M}\) and \(\phi_{cr}\), which in turn are dependent on the geometry of the macro-block(s) and the type of mechanism taking place (i.e., single/two/multiple block). For more detailed (i.e., mechanism-specific) expressions of these terms, the reader is directed to Reference [26].  
Furthermore, the rotation \(\theta_{ov}\) upon the exceedance of which the macro-block will overturn and collapse also needs to be computed. As in the case of \(p_{eq}\) and \(\widetilde{\lambda}\), this rotation depends only on the kinematic constants:  
\[\theta_{ov}=\left(\phi_{cr}\widetilde{K}-\widetilde{M}\right)\frac{1}{g \widetilde{B}} \tag{8}\]  
Numerical integration of the equation of motion is then conducted using the explicit Runge-Kutta method of order 5(4), whereby the error is controlled assuming the accuracy of the fourth-order method, while steps are taken using the formula with fifth-order accuracy. The solution procedure is iterative: starting from a given set of initial conditions, the algorithm computes the rotation and angular velocity at each time-step, which are subsequently used as input (i.e., the initial conditions) for the following time-step. In the case of impact, the energy dissipated by the block(s), which results in a reduction in angular velocity, is accounted for through the coefficient of restitution \(\eta_{COR}\), which depends both on the block geometry as well as on the type of rocking (i.e., one-sided or two-sided). More detailed expressions for this term can also be found in Reference [26].