## 2 Simplified Vulnerability Assessment Methods for Masonry and RC Building Structures
### Methods Belonging to Group B
#### 2.2.3 RE.SIS.TO(r)  
Method  
The method RE.SIS.TO(r) [5] is based on the assessment of the collapse acceleration of the building, PGA\({}_{\text{c}}\), evaluated through a simplified evaluation of the shear strength at each floor of the building. The spectral acceleration \(S_{a,c}\), corresponding to the collapse of the building, is converted in a capacity in terms of PGA, i.e., PGA\({}_{\text{c}}\), through the following relationship [30]:  
\[\text{PGA}_{\text{c}}=\frac{S_{a,c}}{\alpha_{PM}\cdot\alpha_{AD}\cdot\alpha_{ DT}\left(\frac{1}{\alpha_{DUC}}\right)} \tag{7}\]  
being \(\alpha_{PM}\) a modal participation factor, equal to 1.0 for one-floor buildings and 0.8 for multi-story buildings, \(\alpha_{AD}\) a spectral amplification factor equal to 2.50, \(\alpha_{DT}\) a factor taking into account dissipative phenomena and equal to 0.8 if the contribution of the infill walls is neglected or to 1.0 if it is significant, and \(\alpha_{DUC}\) is the structure factor, which can be assumed equal to 2.0 for masonry buildings.  
The building capacity in terms of spectral acceleration, \(S_{a,c}\), is defined as the minimum ratio between the resistant shear, \(V_{r,i,rid}\), at the i-th floor and the corresponding acting shear, \(V_{s,i}\):  
\[S_{a,c}=\text{min}\left(\frac{V_{r,i,rid}}{V_{S,i}}\right) \tag{8}\]  
The ratio of the resistant shear, \(V_{r,i,}\), to the acting shear, \(V_{s,i}\), at each floor represents, indeed, the structural performance of each floor in terms of acceleration. For masonry buildings, the resistant shear at the \(i\)-th floor is calculated with the Turnsek and Caccovic [31] formulation:  
\[V_{R,i}=(A_{min,i}\tau_{0})\cdot\sqrt{1+\frac{\sigma_{0,i}}{1.5\cdot\tau_{0}}} \tag{9}\]  
Being \(A_{min,i}\) the minimum area of resistant walls along with the main directions of the building at the \(i\)-th floor, \(\tau_{0}\) the shear strength of masonry in the absence of normal stress, and \(\sigma_{0,i}\) the average normal stress at the \(i\)-th floor.  
The resistant shear is further reduced by a factor, \(C_{rid}\), aimed to take into account the current conditions of the building by means of the GNDT form [11]:  
\[V_{r,i,ri} =C_{rid}\cdot V_{r,i} \tag{10}\] \[C_{rid}=\prod_{i=1}^{10}\left(1+\alpha\cdot\frac{V_{i}}{V_{pgg}}\right) \tag{11}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Masonry Class** & **Stories** & **Description** \\ \hline M1 & 1–3 & Confined masonry school with horizontal and vertical RC ties and RC slab (rigid) \\ \hline M2 & 1–2 & URM school with horizontal RC ties on the top of the main walls and jack-arch slab (flexible) \\ \hline M3 & 1 & URM school without any confinement and jack-arch slab (flexible) \\ \hline \hline \end{tabular}
\end{table}
Table 9: Masonry typologies proposed by [8].  
where \(V_{i}\) is the value assigned to each parameter provided in the GNDT form [11] without considering the third parameter related to the conventional strength, \(\alpha\) is a coefficient estimated by the method calibration (here considered equal to 1), and \(V_{Pogg}\) is the sum of the values of all the parameters \(V_{i}\) evaluated in class D.  
Based on the values of the ratio PGA\({}_{c}\)/PGA\({}_{d}\), being PGA\({}_{d}\) the design acceleration expected at the site, five classes of strength and vulnerability are defined (Table 10).