## 2 Materials and Methods
### Numerical Model Characteristics
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \
\begin{tabular}{c} **Medium-Quality** \\ **Masonry Properties** \\ \end{tabular} &
\begin{tabular}{c} **Low-Quality** \\ **Masonry Properties** \\ \end{tabular} & **Concrete** \\ \hline Young’s Module (MPa) & & 2800 & & 31,500 \\ Poisson’s module (\(-\)) & & 0.25 & & 0.3 \\ Mass (kN/m\({}^{3}\)) & & 22 & & 25 \\ Strength in compression (MPa) & 8.0 & & 5.6 & - \\ Strength in tension (MPa) & 1.6 & & 0.46 & - \\ \hline \multicolumn{4}{c}{**Parameters to define the CDP Yield surface**} \\ \hline Dilatation angle, \(\psi\) & \(10^{\circ}\) & - \\ Correction parameter, \(k_{c}\) & 0.666 & - \\ Eccentricity, \(e_{eccen}\) & 0.1 & - \\ Biaxial strength ratio, \(f_{b0}/f_{c0}\) & 1.16 & - \\ Viscosity & 0.01 & - \\ \hline \multicolumn{4}{c}{**Non-Smooth Contact Dynamic Method**} \\ \hline Friction masonry structure & 0.50 & \\ Friction inner core & 0.30 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Materials’ parameters used in the continuous and the discontinuous model. the concrete, considering it in the analysis as over-resistant and, therefore, avoiding the post-elastic behavior characterization. The mono-axial tensile and compressive laws are shown in Figure 7. The yield surface parameters employed are presented in Table 1.  
On the other hand, in the discrete model, all the complex is assigned with the density of the cut stone, 2200 kg/m\({}^{3}\), except for the inner core and the roof. The inner core was given a density of 1900 kg/m\({}^{3}\), while the roof was assigned a density of 2500 kg/m\({}^{3}\), equal to concrete. The joints' friction was specified as follows: 0.9 between the blocks and the ground; then, the value of 0.3 is applied both between the inner core stones of the tower and at the contact surface of the inner core and the external layers of the tower (including the blocks crossing the entire wall's thickness). Finally, the value of 0.5 is used between the remaining blocks. Those values were chosen based on previous studies [35, 36, 37]. The CM and DM underwent nonlinear dynamic analyses to assess the complex's response during the Central Italy seismic sequence. The three main shocks that occurred on 24 August, 26 October, and 30 October 2016 were considered. The station of Amatrice (AMT), nearby Accumoli village, was selected only for the first two shocks, since, in the Accumoli station (ACC), the quality of the recordings is good enough only for the last shock considered. The accelerations and velocities recorded were applied at the CM and DM foundations, respectively. Table 2 lists the main characteristics of these earthquakes. Class EC8 indicates the type of ground on which the earthquake was recorded; \(R_{ph}\) is the Joyner-Boore distance or, rather, the smallest spacing between the rupture site and the rupture surface projection; \(R_{trip}\) is the shortest distance between the rupture site and the rupture surface; and \(R_{epi}\) is the distance calculated by the geometric swap. To expedite the computation, only 10 s of each earthquake's strong motion were considered. These were applied in a series, every two seconds of rest, for a total time of 34 s (Figure 8). All three directions were considered. A load step of 0.005 s was set to perform the analysis. As regards the damping, in the CM model, it was determined using the Rayleigh method considering the 1 st and 77 th modes and a damping of 3% [23]. The 77 th mode was selected so that, in the X and Y directions, the participation mass reached a value of 85%. The DM model neglected the damping effects, which are crucial to the continuum model, and the dissipated energy was related to the contribution of friction.