## 4 The Numerical Case Studies
### Santa Maria and San Giovenale Cathedral Bell Tower
Table 3 lists all the scenarios; the corresponding first natural frequency is also reported to highlight the frequency shift induced by the presence of damage. \begin{table}
\begin{tabular}{l l l l} \hline \hline \multicolumn{1}{c}{**Fixed Parameters**} & \multicolumn{3}{c}{**Constrained Parameters**} \\ \hline return period \(T_{R}\) & 475 years & TLVL & 10–25 s \\ rated life \(V_{N}\) & 50 years & Damping coefficient & 0.05–0.07 \\ Probability of exceedance \(P_{V_{R}}\) & 0.9 & & \\ Undrained shear strength \(C_{U}\) & 1.0 & & \\ topographic category & T1 & & \\ stratigraphic category & C & & \\ peak ground acceleration \(a_{g}\) & 0.109 \(g\) & & \\ seismic input duration & 35 s & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Seismic intensity parameters. Figure 5: **Left:** scheme of the sensor placement and direction of the acquisition channels (red, green, and yellow arrows). **Right:** the updated Finite Element (FE) model. Figure 6: (**a**) to (**f**): Damage Scenarios 05–10, in the same order. The scenarios have been derived by observation of similar structures damaged by earthquakes; areas with reduced Young’s Modulus are portrayed in grey. \begin{table}
\begin{tabular}{r l l} \hline \hline
**Case** & **Name** & **Description** \\ \hline
00 & Undamaged & As it is, no Young’s modulus reductions \\
01 & 1 st Damaged & 5.00\% reduction of the \(E\) modulus at Level 0 (all four facades) \\
02 & 2 nd Damaged & 5.00\% reduction of the \(E\) modulus at Level 1 (all four facades) \\
03 & 3 rd Damaged & 5.00\% reduction of the \(E\) modulus at Level 2 (all four facades) \\
04 & 4 th Damaged & 5.00\% reduction of the \(E\) modulus at Level 3 (all four facades) \\
05–10 & 5 th–10 th Damaged & 15.00\% local reduction of the \(E\) modulus at specific, realistic points (as reported in Figure 6) \\
11 & 11 th Damaged & 10.00\% global reduction of the \(E\) modulus of all levels, all facades \\
12 & 12 th Damaged & 50.0\% reduction of spring stiffness at the linking with the Cathedral, \(x\)- and \(y\)- directions \\
13 & 13 th Damaged & 50.0\% reduction of spring stiffness at the linking with the Cathedral, \(x\)-direction only \\
14 & 14 th Damaged & 50.0\% reduction of spring stiffness at the linking with the Cathedral, \(y\)-direction only \\
15 & 1 st Unaltered & +0.25\% global increase of the \(E\) modulus of all levels, all facades \\
16 & 2 nd Unaltered & +1.00\% global increase of the \(E\) modulus of all levels, all facades \\
17 & 3 rd Unaltered & \(-0.25\%\) global reduction of the \(E\) modulus of all levels, all facades \\
18 & 4 th Unaltered & \(-1.00\%\) global reduction of the \(E\) modulus of all levels, all facades \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Baseline model and other scenarios for numerical validation (Fossano bell tower). Cases 01 to 04 only present a 5% reduction of \(E\) along one of the four stories of the structure, on all facades. Cases 05-10 are realistic post-earthquake scenarios, which are based on the Authors' previous experience and intended to model typical damage pattern encountered in similar cases; they are portrayed in Figure 6. Case 11 is a global stiffness reduction of 10% throughout the whole structure. Cases 12-14 emulate damage to the linking with the adjacent cathedral, with 50% of the spring stiffness. Finally, four 'unaltered' scenarios were also accounted for: as for case 11, a global variation of \(E\) was applied; specifically, a reduction/increase equal to 1% and 0.25%. This was intended to simulate statistical fluctuations of the material properties, which cannot be unequivocally related to the presence of occurring damage.