## 4 The Numerical Case Studies
### Santa Maria and San Giovenale Cathedral Bell Tower  
The Finite Element Model (FEM) of the Santa Maria and San Giovenale Cathedral bell tower (as visible in Figure 4) has been used as a numerical case study for the proposed damage-sensitive feature, as previously proposed for similar alternatives in [22]. The baseline FE Model of the structure 'as is' has been calibrated accordingly to the results of a campaign of ambient vibration tests [29], aimed to establish the mechanical properties of the deteriorated (or cracked) masonry walls. Table 1 provides material properties, with [PERSON]'s moduli defined accordingly to the story level, as also pictorially described in Figure 2. The story levels are labelled as follow. Level 0: from ground floor to 9.9 m; level 1: from there to 28.2 m; level 2: from the previous level top m to 35 m, and level 3 corresponds to the belfry (35 m to 46 m high). The octagonal belfry at the top of the bell tower has a wooden rooftop and 0.5 m thick masonry walls, while the thickness of the external perimeter is 1.5 m for the lower levels. By assuming a [PERSON]'s viscous model, the damping parameters \(\alpha\) and \(\beta\) were defined from the experimentally determined first two flexural eigenfrequencies, \(f_{1}=1.297\) Hz and \(f_{2}=4.253\) Hz. \(8\)-noded, \(6\) DoF-per-node rectangular shell elements were used everywhere in the FEM. The mesh size was \(0.5\times 0.5\) m wherever possible. The distributed connections with the Cathedral were modelled by means of three-dimensional (3D) spring elements at the contact points [29]. For all simulations (both on the baseline model and on the damaged scenarios), the acceleration THs were recorded at the nodes closest to the actual sensor positions visible in Figure 5, along with the 20 directions of the corresponding acquisition channels. The sampling frequency \(f_{s,num\ 1}\) was set to 100 Hz. Seismic \begin{table}
\begin{tabular}{c c c} \hline \hline
**Structure-Wise Properties** & **Story-Wise Properties** \\ \hline
**Rayleigh’s Damping Parameter [-]** & **Young’s Modulus E [MPa]** \\ \hline \(\alpha\) & 0.0993 & Level 0 & 2.690 \(10^{3}\) \\ \(\beta\) & 0.0180 & Level 1 & 1.320 \(10^{3}\) \\ Density [kg/m\({}^{3}\) ] & & Level 2 & 1.250 \(10^{3}\) \\ \(\rho\) & 2000 & Level 3 & 2.470 \(10^{3}\) \\ Poisson’s ratio [-] & & & \\ \(\
u\) & 0.3 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fine-tuned Bell Tower baseline Finite Element Model’s (FEM’s) mechanical properties. Figure 4: Structural scheme of the bell tower (to the right in the image) and picture of its front view (to the left); autofrettage interventions are clearly visible on the external facades. strong ground motions were applied as input; 300 spectrum-compatible earthquakes were artificially generated for this aim accordingly to the Italian normative requirements for earthquake engineering at the time of the research (_Norme Tecniche per le Costruzioni_ - NTC, D.M. 14 th January 2008, Chapter 7) and in compliance with the Eurocodes. The limit State of safeguard of life was considered. The seismic intensity parameters are reported in Table 2. Some of the values were fixed, while the remaining parameters were left free to vary in between realistic ranges, generating 10 combinations. Additive white Gaussian noise (AWGN) was then inserted to obtain 10 slightly different dynamical inputs from any of these 30 cases. Elastic spectra were employed, with the smallest period of the response being equal to 0.02 s, the largest period 4.00 s, the beginning of the stationary part of the accelerogram at 2 s, and the duration of the stationary part TLVL (NTC SS3.2.3.6) free to float in a given range. The baseline FE model was modified to obtain several damage scenarios. These are, again for comparison' sake, the same as utilised in [22]. In all cases, the damage was simulated as a reduction of the Young's modulus \(E\) at given locations to emulate damage in the masonry external walls.