## 3 Numerical Analyses
### Solid FE Model  
The last FE model of the tower was implemented in Midas FEA [55] using solid elements (Figure 4c). The nonlinear behavior of masonry was described using the Concrete Smeared Crack (CSC) approach (Figure 10), adopting the parameters reported in Table 6 according to [56]. For the shear behavior, a linear model was adopted with a shear reduction factor \(\beta\) = 0.1. As in the shell model, an equivalent static lateral load distribution was taken into account for the seismic actions (Table 3). Figure 10: Compressive (**a**) and tensile (**b**) inelastic behavior adopted for the masonry in the solid model. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & \multicolumn{2}{c}{**Modal Distribution**} & \multicolumn{2}{c}{**Uniform Acceleration**} & \multicolumn{2}{c}{**Modal Distribution**} & \multicolumn{2}{c}{**Uniform Acceleration**} \\  & \multicolumn{2}{c}{**X Direction**} & \multicolumn{2}{c}{**X Direction**} & \multicolumn{2}{c}{**Y Direction**} & \multicolumn{2}{c}{**Y Direction**} \\ \cline{2-9}  & **CSM** & **N2** & **CSM** & **N2** & **CSM** & **N2** & **CSM** & **N2** \\ \hline
**S\({}_{\text{a}}\) [g]** & 0.305 & 0.276 & 0.411 & 0.412 & 0.307 & 0.280 & 0.411 & 0.412 \\
**S\({}_{\text{d}}\) [mm]** & 35.14 & 33.30 & 17.23 & 18.77 & 35.78 & 31.80 & 16.49 & 18.01 \\
**V [kN]** & 4901 & 4495 & 6597 & 6700 & 5030 & 4645 & 6740 & 6836 \\
**d [mm]** & 52.23 & 44.82 & 25.61 & 25.26 & 52.55 & 42.54 & 24.22 & 24.09 \\
**q** & 1.08 & 1.37 & 1 & 1 & 1.07 & 1.39 & 1 & 1 \\ \hline \end{tabular}
\end{table} 
Table 5: Comparison of results obtained with N2 and CSM methods. Figure 11 shows the results of the nonlinear static analysis in terms of tensile stress and cracking paths at the beginning of the inelastic phase for the different loading directions. Cracks are mainly located around the openings and on the vaults, especially when considering the \(-\)X direction. The stresses configuration and the horizontal loads scale factors \(\lambda\) are comparable with the ones obtained from the shell model previously presented. A further pushover analysis was performed to obtain the capacity curves of the structure, using a horizontal load profile distribution proportional to the first vibration mode in the X direction and to the second vibration mode in the Y direction (Figure 12). The uniform acceleration load profile was neglected since, from the previous analyses, the tower subjected to a uniform acceleration load profile turns out to be within the elastic range. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(\mathbf{f_{C}}\) & \(\mathbf{G_{C}}\) & \(\mathbf{h}\) & \(\mathbf{f_{t}}\) & \(\mathbf{G_{f}^{t}}\) \\  & [MPa] & [N/mm] & [mm] & [MPa] & [N/mm] \\ \hline Walls & \(1.25\) & \(0.1\) & \(440\) & \(0.15\) & \(0.1\) \\ Vaults & \(1.67\) & \(0.1\) & \(440\) & \(0.15\) & \(0.1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameters adopted for the definition of the CSC model. Figure 11: Results of the nonlinear static analysis of the solid FE model at the beginning of the inelastic phase in terms of tensile stresses (**left**) and crack openings (**right**) for the following loading directions (**a**) +X, (**b**) \(-\)X, (**c**) +Y, and (**d**) \(-\)Y with \(\lambda\) horizontal load multipliers. According to the N2 method, the behavior factor q was obtained by transforming the MDOF capacity curves into the equivalent SDOF capacity curves. Nonlinear force-displacement relations are then simplified into elastic-plastic relations, whose main parameters are reported in Table 7, where M* is the mass of the equivalent SDOF system, \(\Gamma\) is the transformation factor, and F\({}_{\text{u,MDOF}}\) and d\({}_{\text{u,MDOF}}\) are the ultimate load and displacement of the MDOF system, while F\({}_{\text{u,SDOF}}\) and d\({}_{\text{u,SDOF}}\) are the ultimate load and displacement of the SDOF system.