# Operational Modal Analysis and Safety Assessment of a Historical Masonry Bell Tower
## 4 Seismic Assessment
### EL1 Assessment  
According to EL1 methodology, the structural check against combined compression and bending in a generic section is performed under the following hypotheses:  
* conservation of plane sections;
* zero tensile strength of masonry;
* distribution of compressive forces according to a stress-block model. The structural check is carried out along the two principal directions of inertia of the section, in both directions, and at different heights, considering the roof only as a superimposed load. The design moment can be evaluated by considering a system of forces distributed along the height of the structure, assuming either a linear or a modeproportional acceleration profile. The force to be applied at the centroid of each block is given by  
\[F_{i}=\frac{W_{i}\cdot\xi_{i}}{\sum_{k=1}^{n}W_{k}\cdot\xi_{k}}F_{h} \tag{3}\]  
where the following apply:  
* \(F_{h}\) is the total shear base determined according to Equation (4);
* \(W_{i}\) and \(W_{k}\) are the weights of sectors \(i\) and \(k\), respectively;
* \(\xi_{i}=z_{i}|\phi_{i}\) is either \(z_{i}\) the height of the centroid of sector \(i\) with respect to the foundations (linear acceleration profile) or \(\phi_{i}\) the normalized modal displacements at the centroid along the considered principal direction (mode-proportional acceleration profile). \[F_{h}=\frac{0.85\cdot S_{e}(T_{1})\cdot S\cdot W}{q\cdot g} \tag{4}\]  
In Equation (4), the following apply:  
* \(S_{e}(T_{1})\) is the ordinate of the elastic response spectrum, a function of the first period \(T_{1}\) of the structure in the considered direction;
* \(S\) is the amplification factor due to soil and topographic conditions;
* \(W\) is the total weight of the tower;
* \(q\) is the behavior factor;
* \(g\) is the gravity acceleration. The resultant of the seismic forces \(F_{hi}\) acting on the \(i\)-th section is given by Equation (5), while the relative height \(z_{Fi}\) at which \(F_{hi}\) is applied is calculated as per Equation (6). \[F_{hi}=\frac{\sum_{k=i}^{n}W_{k}\;\xi_{k}}{\sum_{k=1}^{n}W_{k}\;\xi_{k}}F_{h} \tag{5}\]  
\[z_{Fi}=\frac{\sum_{k=i}^{n}W_{k}\;\xi_{k}\;z_{k}}{\sum_{k=i}^{n}W_{k}\;\xi_{k} }-z_{i}^{*} \tag{6}\]  
where in the above equations \(z_{i}\)* is the height of the \(i\)-th verification section relative to the base. By imposing the equality between the resisting bending moment \(M_{u,i}\) and the design bending moment \(M_{ed,i}\):  
\[M_{u,i}=F_{hi}z_{Fi} \tag{7}\]  
it is possible to derive the value of the ordinate of the elastic response spectrum corresponding to the achievement of the Ultimate Limit State (ULS) in the \(i\)-th section (taking into account the confidence factor \(F_{C}\)):  
\[S_{e,ULS,i}(T_{1})=\frac{q\cdot g\cdot M_{u,j}\sum_{k=1}^{n}W_{k}\;\xi_{k}}{ 0.85W\left(\sum_{k=i}^{n}W_{k}\;\xi_{k}\;z_{k}-z_{i}^{*}\sum_{k=i}^{n}W_{k}\; \xi_{k}\right)F_{C}} \tag{8}\]  
Based on this value, the corresponding Peak Ground Acceleration (PGA) a\({}_{\text{ULS}}\) is evaluated by considering the parameters of the design spectrum at the ULS and the fundamental period of the structure. In cases of hollow rectangular sections, such as the ones typical of the tower under consideration, the ultimate moment of resistance at the base of the \(i\)-th sector can be calculated as per Equation (9):  
\[M_{u,i}=\frac{\sigma_{0i}A_{i}}{2}\left(b_{i}-\frac{\sigma_{0i}A_{i}}{0.85a_ {i}f_{d}}\right) \tag{9}\]  
where the following apply:* \(a_{i}\) is the length of the side perpendicular to the direction of the considered seismic action for the \(i\)-th section, excluding any openings;
* \(b_{i}\) is the length of the side parallel to the direction of the considered seismic action for the section under analysis;
* \(A_{i}\) is the total area of the section under analysis, excluding any openings;
* \(\sigma_{0i}\) is the average normal stress in the section under analysis (\(W/A_{i}\), where \(W\) is the weight of the structure above the section under analysis);
* \(f_{d}\) is the design compressive strength of the masonry.