## 3 Numerical Analysis
### Retrofitted Configuration: Intervention on the Roof Structure  
In addition to the retrofitting interventions described above concerning the improvement of the load-bearing capacity of the masonry walls and the realization of the connection between orthogonal walls and the diaphragm floors, in view of the need to replace the roof, seriously damaged during the 2016 Centro Italia earthquake, a cross-laminated roof solution has been proposed. In particular, a new cross-laminated roof characterized by a thickness equal to 14 cm (composed by seven layers each, 2 cm thick) has been realized. To evaluate the effects of the cross-laminated roof on the seismic behavior of the retrofitted configuration of the building, a 3D Finite Element model has been implemented using a commercial software [31], where the masonry walls have been modelled with solid elements (Figure 13). The presence of the reinforced slabs has been considered through the introduction of rigid diaphragms in correspondence to each floor. The cross-laminated panels are represented by bidimensional elements, whereas the panel-to-panel and wall-to-panel connections are introduced by inelastic springs (Figure 14) [32]. Table 8 summarizes the mechanical properties of the cross-laminated panel considering its orthotropic behavior. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Mechanism** & **PGA\({}_{\text{CLM}}\)** & **PGA\({}_{\text{D}}\)** & **RI\({}_{\text{PGA}}\)** & **RP\(  
The hysteretic behavior of the connections has been defined according to what is reported in [33] (Figure 15) based on Clough constitutive model [34] and considering cylindrical connectors with a diameter equal to 10 mm and elastic stiffness 6200 N/mm. Figure 14: Modeling of the panel–panel and panel–curb connections. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{Y}\) & \(\mathbf{E_{x}}\) & \(\mathbf{E_{y}}\) & \(\mathbf{E_{z}}\) \\
**[kN/m\({}^{3}\)]** & **[MPa]** & **[MPa]** & **[MPa]** \\ \hline
5 & 7059 & 5387 & 370 \\ \hline
**G** & \(\mathbf{\
u_{xy}}\) & \(\mathbf{\
u_{xz}}\) & \(\mathbf{\
u_{yz}}\) \\
**[MPa]** & **[-]** & **[-]** & **[-]** \\ \hline
690 & 0.4500 & 0.0138 & 0.0138 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Mechanical properties of the cross-laminated panel. Figure 13: (**a**) Finite element model implemented using MIDAS FEA software; (**b**) Mesh. A non-linear time history analysis considering Rayleigh damping has been performed using the seven spectrum-compatible accelerograms reported in Figure 16, obtained using REXEL software [35] by considering the site seismic parameters listed in Table 2.  
Figure 16: _Cont._  
Figure 15: Connection hysteretic behavior. Figure 16: _Cont._  
Starting from the masonry mechanical properties reported in Table 5, a Concrete Damage Plasticity (CDP) constitutive law has been adopted considering the fundamental parameter listed in Table 9, where \(\psi\) is the dilation angle, e is the dimensionless eccentricity, \(\mathrm{f_{bo}/f_{co}}\) is the ratio of biaxial compressive to uniaxial compressive yield stress, \(\mathrm{Kc}\) is the coefficient determining the shape of the deviatoric cross-section, and v is the viscosity parameter [36]. Figure 17 shows the results in terms of X and Y displacement histories comparing the cross-laminated roof structure with a classic wood roof solution (with 4 cm-thick single planks). The results show the displacements of the roof for the first 25 s of the time history analysis representing the most significant part of the seismic events. The trends shown in Figure 17 represent the average value among the seven spectrum-compatible accelerograms combined according to what is reported in [25]. Figure 16: Seismic records. \begin{table}
\begin{tabular}{c c c c c} \hline \(\psi\) & **e** & \(\mathbf{f_{bo}/f_{co}}\) & \(\mathbf{Kc}\) & \(\mathbf{v}\) \\ [\({}^{\circ}\)] & [-] & [-] & [-] & [-] \\ \hline
10 & 0.100 & 1.160 & 0.667 & 0 \\ \hline \end{tabular}
\end{table} 
Table 9: Concrete Damage Plasticity parameters used in this work.