# Operational Modal Analysis and Safety Assessment of a Historical Masonry Bell Tower
## 4 Seismic Assessment
### EL1 Assessment
Next, it is possible to calculate the acceleration factor \(f_{a,ULS}\) as the ratio between the collapse acceleration and the maximum expected peak ground acceleration at the site (\(a_{g,ULS}\)):  
\[f_{a,ULS}=\frac{a_{ULS}}{a_{g,ULS}} \tag{10}\]  
Given the exposure characteristics of the tower (nominal life of 50 years and use class III), the return period of the seismic action at the Ultimate Limit State is assumed to be 712 years. The elastic response spectrum (ULS) adopted for the EL1 structural check is shown in Figure 11 (soil type C, flat soil), and it is characterized by \(a_{g,ULS}=0.103\) g.  
The fundamental period (\(T_{1}\)) was determined according to the AVT results. In this sense, it should be remarked that both first and second vibration periods (0.39 s and 0.32 s, respectively), which correspond to translational modes in the two principal directions of the tower, are included in the constant branch of the design response spectrum. In the presence of limited tests on materials, the Italian code [25] suggests a procedure that, starting from some typology-based ranges for mechanical properties, updates their median values accounting for the results of the tests. Applying this methodology to the type of masonry under consideration, an average compressive strength \(f_{m}=1.23\) MPa was obtained. Nonetheless, given that the tests consistently returned lower compressive strength values, in the next calculation, \(f_{m}=0.7\) MPa was assumed, based on the results obtained from the sclerometer, double flat jack test and compressive test on masonry sample. This value was then reduced due to the partial safety factor for masonry for seismic actions (\(\gamma_{\text{M}}=2.0\)) and a confidence factor \(F_{C}=1.20\), with the latter defined according to the performed investigation activity. Thus, \(f_{d}=0.29\) MPa was adopted. As specified in equation (3), both a linear displacement and mode-proportional force distributions along the height of the bell tower were considered. The mode-proportional distributions were defined according to the AVT results. In particular, the profiles of normalized modal displacements (\(\phi=\)\(\varphi_{i}/\varphi_{max}\)) related to the 1 st (y-direction) and 2 nd (x-direction) vibrational modes shown in Figure 12 were adopted. Figure 11: Elastic response spectrum for EL1 analysis. For the EL1 analysis, the tower was thus divided into sectors, whose base sections were considered for the structural checks, according to the scheme proposed in Figure 13.  
The seismic weight of each sector was estimated by assuming a density of the masonry \(\gamma=1600\) kg/m\({}^{3}\), as returned by laboratory tests. The summary of the weight estimation is reported in \[F_{h}=\frac{0.85\cdot S_{e}(T_{1})\cdot S\cdot W}{q\cdot g} \tag{4}\]0 \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sector** & **Volume** & \(\mathbf{z_{k}}\) & \(Y\) & \(\mathbf{W_{k}}\) \\  & **[m\({}^{3}\)]** & **(m)** & **(kg/m\({}^{3}\))** & **(kN)** \\ \hline
1 & 43.12 & 2.63 & 1600 & 620.9 \\
2 & 38.85 & 7.00 & 1600 & 559.4 \\
3 & 33.87 & 10.46 & 1600 & 487.7 \\
4 & 20.56 & 13.61 & 1600 & 296.1 \\
5 & 10.21 & 16.20 & 1600 & 147.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Seismic weight estimation. Figure 12: Normalized mode-proportional displacement profiles adopted for the EL1 checks. Figure 13: Subdivision of the tower in sectors and sections adopted for the EL1 checks. Based on the total weight of the tower (\(W\) = 2111.18 kN), the base shear \(F_{h}\) was determined by assuming \(S\) = 1.5 (due to the soil type and topographic conditions) and a behavior factor \(q\) = 2.8, according to the minimum value of the range proposed in [4]. The calculation returned \(F_{h}\) = 269 kN. Thus, the seismic forces to be applied at the centroid of each tower sector were determined according to equation (3). The results are shown in Tables 2 and 3 for the linear displacement and mode-proportional acceleration profiles, respectively.