# Operational Modal Analysis and Safety Assessment of a Historical Masonry Bell Tower
## 4 Seismic Assessment
### EL2 Assessment  
In order to estimate the load multiplier \(\alpha_{0}\) provoking the activation of the local damage mechanism according to the EL2 methodology, the following forces acting on the rigid blocks of the kinematic chain must be considered:  
* the self-weight of the blocks applied at their centroid and other carried vertical loads (self-weight and superimposed loads from floors and roofing, and other masonry elements not considered in the structural model);
* horizontal forces proportional to the carried vertical loads, if these are not effectively transferred to other parts of the building;
* any external forces (e.g., those transmitted by tie roads);
* any internal forces (e.g., actions related to the interlocking between masonry blocks). Given a virtual rotation to the generic \(k\) block, based on the geometry of the structure, the displacements related to each considered load are calculated. Thus, by applying the Principle of Virtual Works, the value of \(\alpha_{0}\) can be calculated according to Equation (11):  
\[\alpha_{0}\left(\sum_{i=1}^{n}P_{i}\cdot\delta_{x,i}+\sum_{j=n+1}^{n+m}P_{j} \cdot\delta_{x,j}\right)-\sum_{i=1}^{n}P_{i}\cdot\delta_{y,i}+\sum_{k=1}^{o}F_ {k}\cdot\delta_{k}=L_{fi} \tag{11}\]  
where the following apply:  
* \(n\) is the number of all weight forces \(P_{i}\) applied at the centroid of the different blocks of the kinematic chain;
* \(m\) is the number of weight forces \(P_{j}\) not directly acting on the blocks, whose masses generate horizontal forces on the elements of the kinematic chain due to the seismic action, as they are not effectively transferred to other parts of the building;
* \(o\) is the number of external forces \(F_{k}\), not associated with masses, applied to the various blocks;
* \(\delta_{x,i}\) (resp. \(\delta_{y,i}\)) is the virtual horizontal (resp. vertical) displacement of the point of application of the \(i\)-th weight \(P_{i}\), assuming the positive direction as the seismic action activating the mechanism acts (resp. upward);
* \(\delta_{k}\) is the virtual displacement of the point where the \(h\)-th external force is applied parallel to and with the same sign as the force;
* \(L_{fi}\) is the work of the internal forces. To perform the safety checks, the horizontal multiplier \(\alpha_{0}\) activating the mechanism is transformed into spectral acceleration \(a_{0}\)* according to Equation (12), in order to have consistency with the demand measure. \[a_{0}^{*}=\frac{\alpha_{0}\cdot S}{e^{*}Fc} \tag{12}\]  
In Equation (12), \(e^{*}\) is the portion of the participating mass in the considered mechanism, which is estimated according to Equation (13). \[e^{*}=\frac{\left(\sum_{i=1}^{n+m}P_{i}\cdot\delta_{x,i}\right)^{2}}{\sum_{i=1 }^{n+m}p_{i}\cdot\sum_{i=1}^{n+m}\left(P_{i}\cdot\delta_{x,i}^{2}\right)} \tag{13}\]  
If the considered collapse mechanism includes an isolated element or a portion of the construction supported on the ground, the safety check (ULS) is fulfilled if the spectral acceleration \(a_{0}\)* satisfies the following condition:  
\[q\cdot a_{0}^{*}\geq a_{g,ULS}\cdot S \tag{14}\]  
where \(S\) is the amplification due to the soil conditions (1.5 for the case under consideration) and the behavior factor is assumed as \(q=2.0\)[25]. Conversely, if the local mechanism affects a portion of the building located at a certain height, it is necessary to account for amplifying effects according to Equation (15):  
\[q\cdot a_{0}^{*}\geq S_{e}(T_{1})\cdot\psi(Z)\cdot\gamma \tag{15}\]  
where the following apply:  
* \(\psi(Z)\) is the first mode of vibration in the considered direction, normalized to one at the top of the building (in the absence of more accurate assessments, \(\psi(Z)=Z/H\), with \(H\) the height of the structure relative to the foundation);
* \(Z\) is the height, relative to the foundation of the building, of the centroid of the boundary lines between the blocks affected by the mechanism and the rest of the structure;
* \(\gamma\) is the corresponding modal participation factor, equal to \(3N/(2N+1)\).