## 8 Vulnerability Curves and Damage Index Distribution  
The vulnerability curve allows to correlate the vulnerability index, damage index, and peak ground acceleration. Two limit-levels of acceleration are key to the analysis of damage: the acceleration associated to the beginning of the damage and the acceleration associated to the structural collapse. We recall that the damage value varies in the (0, 1) interval. The present investigation relies upon the study devised by [PERSON] and [PERSON] [4], who obtained a relation between vulnerability index, acceleration, and damage, by observing the damage levels of masonry buildings subjected to real earthquakes. Corresponding acceleration/damage relation can be modelled with a smooth vulnerability curve (Figure 15). For simplicity purposes, [PERSON] and [PERSON], instead of using a vulnerability curve, proposed to exploit a tri-linear law parametrized in terms of the values that the peak ground acceleration takes at early damage, \(\mathrm{PGA_{i}}\), and at the collapse, \(\mathrm{PGA_{c}}\). In this study, instead of a post-earthquake damage observation, the acceleration for the yield and collapse states were calculated by the pushover analysis [27; 33]. Then, using the vulnerability indices and yield and collapse accelerations, a new damage-vulnerability-peak ground acceleration relationship was derived. The damage index is expressed in the (0-1) interval via a tri-linear law, analogously to [4] though defined through two parameters: yield acceleration PGA\({}_{\rm y}\), which corresponds the beginning of the damage (d = 0), and collapse acceleration PGA\({}_{\rm c}\) (d = 1).  
The developed procedure has been applied to the observed test site. The basis for defining vulnerability curves are vulnerability indices, yield peak ground acceleration PGA\({}_{\rm y}\), and collapse peak ground acceleration PGA\({}_{\rm c}\), obtained by the pushover analysis for 18 analyzed buildings, as shown in Table 2. Yield acceleration PGA\({}_{\rm y}\) is assigned to PGA\({}_{\rm DL}\) and collapse acceleration PGA\({}_{\rm c}\) to PGA\({}_{\rm NC}\) limit states, respectively. As PGA\({}_{\rm y}\) and PGA\({}_{\rm c}\) depend on the vulnerability index I\({}_{\rm v}\), the values of PGA\({}_{\rm y}\), associated with damage d = 0, and PGA\({}_{\rm c}\), associated with damage d = 1, can be calculated for each value of I\({}_{\rm v}\).  
Figure 16 shows the vulnerability curves used for the estimation of the damage index of the buildings at the investigated area. These vulnerability curves are partially changed in comparison to those derived from the authors for the sole historical center [31], as here the study was extended to the entire test site. In fact, the influence of the buildings outside of the historical center with vulnerability indices mainly up to 30% changed the relationships between vulnerability index and peak ground accelerations, as well as the vulnerability curves for the low vulnerability buildings.  
The spatial distribution of the damage is represented by the damage index maps of the investigated area for the given intensity of the earthquake. Three seismic scenarios corresponding to return periods of 95, 225, and 475 years and demand peak ground accelerations of 0.11 g, 0.17 g, and 0.22 g, respectively, have been chosen. The damage to the buildings for different scenarios is presented in Figure 17.  
Figure 16: Vulnerability curves for the test site.  
Figure 15: Vulnerability curve and its idealization.  
Figure 17: Damage index distribution at the test site: (**a**) T = 95 years; (**b**) T = 225 years; (**c**) T = 475 years.