## 3 Numerical Analysis  
### Current Configuration  
To evaluate the seismic behavior of the building, linear and non-linear analyses have been performed. In particular, non-linear static analysis has been carried out using the equivalent frame method [18; 27] which represents a suitable approach for buildings characterized by a regular distribution of openings such as the one here analyzed (Figure 5b). The piers and the spandrels have been implemented through deformable beam elements interconnected by rigid offsets at their intersection. This modeling hypothesis is derived from the damages observed in this type of construction subjected to seismic event which show that, usually, cracks and failure modes are localized in piers and spandrels, while little damage is observed at their intersection, thus justifying the adoption of rigid offsets in the modelling process. Bending moments between orthogonal walls have been released to represent the unperfect toothing and avoid out-of-plane response of piers [28]. According to [26], the following collapse mechanisms have been considered for the structural elements:  
Figure 4: Roof (unit of measure is cm). Figure 3: First and second floor slab (unit of measure is cm). (i) diagonal shear cracking, (ii) sliding failure and (iii) rocking failure due to bending moment. Considering that the damage is assumed to be concentrated in localized sections of the building (in particular, in the mid and at the end of the monitored elements), non-linear beams with lumped inelasticity are used for piers and spandrels. The lumped inelasticity was modeled in the EFM with non-linear flexural or shear springs obtained considering, respectively, moment-rotation or shear-displacement laws. Hinges are characterized by an elastic perfectly plastic behavior where the ultimate displacement capacity, obtained considering the limit drift calculated as reported in [25], is taken equal to 0.6% for the flexural hinge and 0.4% for the shear hinge. The sectional limit strength was calculated by the finite element software considering the masonry mechanical characteristics listed in Table 1 and the actual stress distribution. Two horizontal load profiles have been considered to perform the pushover analysis, representing the inertia forces related to the occurrence of a seismic event: (i) acceleration distribution proportional to the first mode and (ii) uniform acceleration distribution. The different horizontal load profiles have been applied in both X and Y direction, considering both positive and negative signs. Figure 6 shows the capacity curves for X (red curves) and Y (green curves) directions obtained from the execution of the pushover analysis. Figure 5: (**a**) 3D Finite Element model of the building; (**b**) Equivalent frame model. Figure 6: Capacity curves (current configuration). To evaluate the seismic vulnerability of the structure, N2 method [29] has been used considering the seismic action defined according to [25], where the fundamental parameters are listed in Table 2.  
The seismic capacity of the structure was defined considering two different types of seismic vulnerability risk indices (RI) obtained as the ratio between the peak ground acceleration which leads to the collapse of the structure (PGA\({}_{\rm C}\)), the design peak ground acceleration reported in Table 3 (PGA\({}_{\rm D}\)), and the related return periods (RP\({}_{\rm C}\) and RP\({}_{\rm D}\)):  
\[\rm RI_{PGA}=PGA_{C}/PGA_{D}, \tag{1}\]  
\[\rm RI_{RP}=RP_{C}/RP_{D}. \tag{2}\]  
Table 3 shows the minimum risk index values obtained considering the capacity curves calculated using the above-mentioned different horizontal load profiles, both in terms of peak ground acceleration and related return period. It is possible to notice that the minimum value of risk index calculated is less than one, which characterizes a structure with a high vulnerability to the seismic action. Furthermore, with the N2 method, the behavior factor (q) that characterized the structure has been calculated and is equal to q = 2.