## 7 Vulnerability Index--PGA Relations  
The emphasis of this part of the paper is to investigate whether the seismic behavior of the building can be estimated from the vulnerability index parameters. Therefore, 18 buildings, which represent 16% of total 111 buildings with calculated vulnerability indices, were analyzed by the non-linear static (pushover) method. The buildings have been chosen considering different material and structural characteristics as well as the period of construction. Additionally, there are buildings among them with different number of the floors and regular and non-regular layout and elevation.  
The non-linear pushover analysis carried out on the stone masonry buildings in the old city center indicated a low global capacity in terms of the collapse peak ground acceleration, as well as low global accelerations of significant damage and damage limitation states. Numerical predictions of the acceleration achieved for structural collapse have indicated that no building meets the seismic demand of a\({}_{g}\) = 0.22 g for T = 475 yearsin both the x and y directions. Moreover, in the simulations, a conspicuous number of buildings reached the collapse at accelerations that are lower than the demand acceleration of a\({}_{g}\) = 0.11 g for a return period T = 95 years. The local mechanism failure induced by a lack of connection among perpendicular walls, and poor connections between floors/roofs and walls, was also analyzed for few stone masonry buildings in the old city center, where out-of-plane effect can be expected. The lowest acceleration was achieved for the global response of the buildings [33]. Buildings outside of the center are made of concrete or brick masonry with horizontal RC confining elements or both with horizontal and vertical confining elements  
Figure 12: Distribution of I\({}_{\
u}\) for 18 buildings analyzed by pushover analysis.  
Figure 13: Peak ground accelerations for NC, SD, and DL limit states calculated by pushover analysis.  
and rigid horizontal diaphragms. Therefore, the failure of the structure caused by local mechanisms is not expected and the behavior of the buildings represented by capacity accelerations will be analyzed assuming the global failure of the structure.  
The results for 18 analyzed buildings (10 in the historical center and 8 outside of the center) presented in Table 2 and Figures 12 and 13, are used to establish the vulnerability index--peak ground acceleration relation for the DL, SD, and NC limit states at the entire test site. Figure 14 shows a cloud of points representing the relationship between the vulnerability index calculated on the basis of 11 parameters \(\mathrm{I_{v}}\) and the critical peak ground accelerations associated with the DL, SD, and NC limits. The trend lines \(\mathrm{I_{v}}\)-\(\mathrm{PGA_{DL}}\), \(\mathrm{I_{v}}\)-\(\mathrm{PGA_{SD}}\) and \(\mathrm{I_{v}}\)-\(\mathrm{PGA_{NC}}\) for three limit states were obtained and are shown in Figure 14. The exponential functions were chosen as the most representative. They are used to approximate the yield, significant damage, and collapse peak ground accelerations for the entire test site. The values of yield and collapse accelerations are the basis for deriving vulnerability curves.  
The quality of the approximation of the results obtained by the pushover analysis and those represented by trend lines for the three limit states are validated by standard deviation. The comparison shows a significantly better quality of peak ground acceleration approximations for the NC and SD limit states than for the DL state. It is obvious that the most vulnerable buildings have a certain seismic load-bearing capacity expressed with peak acceleration. The derived trend lines are used to estimate peak ground accelerations for three limit states of the buildings using their vulnerability index. It should be noted that the results for near collapse and significant damage states approximate critical accelerations much better than the state of damage limitation.