## 2 Simplified Vulnerability Assessment Methods for Masonry and RC Building Structures
### Methods Belonging to Group B
#### 2.2.4 DPCM 9/2/11 Method  
The Italian guidelines (DPCM 9/2/11, [16]) define the following expeditious safety index, \(I_{s}\), aimed to assess the seismic vulnerability according to a territorial scale:  
\[I_{s}=\frac{a_{SLV}}{a_{g,SLV}} \tag{12}\]  
being \(a_{g,SLV}\) the expected acceleration on rigid soil at the life safety limit state and \(a_{SLV}\) the collapse acceleration of the building, i.e., the previously defined PGA\({}_{c}\), with reference to the lower strength direction and defined as follows:  
\[a_{SLV}=\left\{\begin{array}{c}\frac{S_{e,SLV}(T_{1})}{S_{F0}}T_{B}\leq T_{1 }\leq T_{C}\\ \frac{S_{e,SLV}(T_{1})}{S_{F0}}\frac{T_{1}}{T_{C}}T_{C}\leq T_{1}\leq T_{D} \end{array}\right. \tag{13}\]  
with  
\[S_{e,SLV}=\frac{q\text{-}F_{SLV}}{e^{*}M} \tag{14}\]  
\[T_{1}=C_{1}H^{3/4} \tag{15}\]  
where \(S_{e,SLV}\) is the value of the elastic spectrum at the life safety limit state, \(q\) is the structure factor, \(M\) is the total seismic mass of the building, \(e^{*}\) is the ratio of participating mass in the considered failure mode, \(T_{1}\) is the main vibration period of the structure, i.e., \(C_{1}=0.05\) for masonry buildings, \(H\) is the building height, \(T_{B}\), \(T_{C}\), and \(T_{D}\) are the characteristic periods of the response spectrum, \(S=S_{S}\cdot S_{T}\) is the factor taking into account the subsoil typology and the topographic conditions (EC8 [32]), and \(F_{0}\) is the maximum value of the amplification factor.  
The shear strength of the building, \(F_{SLV}\), is calculated as the minimum value between those related to two main directions and for each floor of the building. As an example, for the \(i\)-th floor along the X direction, the shear strength is:  
\[F_{SLV,xi}=\frac{\mu_{xi}\xi_{xi}\cdot\zeta_{xi}\cdot A_{xi}\cdot\tau_{di}}{ \beta_{xi}} \tag{16}\]  
where \(\tau_{di}\) is the design shear strength of masonry of the \(i\)-th floor, defined according to the Turnsek and Caccovic [31] formulation as follows:  
\[\tau_{di}=\tau_{0d}\sqrt{1+\frac{\sigma_{0j}}{1.5\cdot\tau_{0d}}} \tag{17}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**PGA\({}_{d}\)/PGA\({}_{d}\)**} & \multicolumn{1}{c}{**Strength Class**} & \multicolumn{1}{c}{**Vulnerability Class**} \\ \hline
0–25\% & V & High \\ \hline
25–50\% & IV & Medium-High \\ \hline
50–75\% & III & Medium \\ \hline
75–100\% & II & Medium-Low \\ \hline \(>\)100\% & I & Low \\ \hline \hline \end{tabular}
\end{table}
Table 10: Vulnerability classification according to the RE.SIS.TO® method [5].  
being \(\tau_{0d}\) the design shear strength of masonry in the absence of normal stresses and \(\sigma_{0,i}\) the average normal stress at the \(i\)-th floor due to the dead and variable loads under the seismic load combination.  
Moreover, \(A_{xi}\) is the area of resistant walls along the X direction of the building at the \(i\)-th floor, \(\zeta_{xi}\) is a coefficient related to the strength of masonry spandrels (equal to 1.0 in the case of strong spandrels or to lower values, minimum 0.8, for weak spandrels), \(\beta_{xi}\) depends on the plan regularity of the i-th floor and varies from 1.00 to 1.25 (for safety 1.25 can be assumed), \(\xi_{xi}\) is a coefficient equal to 1.0 for shear or 0.8 for flexural failure of the masonry walls, \(\mu_{xi}\) is a coefficient related to strength and stiffness homogeneity of the masonry piers at the i-th floor and can be assumed to be equal to 0.8 for safety.