# Seismic Vulnerability Assessment of Historical Masonry Buildings in Croatian Coastal Area
## 3 Development and Calibration of Vulnerability Model  
### Vulnerability Model  
The vulnerability index is not a relevant indicator of seismic risk because it does not give information about the behavior of the building under a specific seismic action. The seismic risk of buildings is often expressed by the damage caused by an earthquake of a certain intensity. Alternatively, it can be formulated in terms of a seismic safety index, defined as the ratio between the peak ground acceleration corresponding to the collapse of the structure and the design ground acceleration. In this paper, an approach based on the evaluation of the damage to the building was applied.  
Several studies established correlations between the vulnerability index, peak ground acceleration, or macro-seismic intensity, and the damage index. They present a cause-effect relation, where the earthquake is the cause and the damage is the effect. Therefore, vulnerability relates the ground acceleration to a certain level of damage. Two limit-levels of acceleration are important for the damage analysis: the acceleration corresponding to the beginning of the damage to a structure, and the acceleration corresponding to the collapse. The level of damage varies in the [0, 1] space.  
This study uses the approach developed by [PERSON] and [PERSON] [23], who derived a relation between damage, acceleration, and the vulnerability index by observing the damage to masonry buildings under real earthquakes. They studied a set of damaged masonry buildings in the historical city center of the towns of Venzone (Udine, Intensity IX MCS, May 1976 earthquake), Tarcento, and San Daniele (Udine, Intensity VIII MCS, May 1976 earthquake); they also considered some other buildings from the 1984 Parco d'Abruzzo earthquake (Intensity VII MCS). The level of damage to each building, as well as the level of the ground acceleration, were accounted for. Corresponding acceleration/damage laws can be represented with smooth vulnerability curves, such as the one shown in Figure 10.  
For simplicity, [PERSON] and [PERSON] substituted the vulnerability curve with a trilinear law defined by the values of the peak ground acceleration corresponding to early damage, PGAi, and to the collapse, PGAc, as follows:  
\[\text{d}=\left\{\begin{array}{cc}0&\text{PGA}<\text{PGAi}\\ (\text{PGA}-\text{PGAi})/(\text{PGAc}-\text{PGAi})&\text{PGAi}\leq\text{PGA} \leq\text{PGAc}\\ 1&\text{PGA})\text{PGAc}\end{array}\right., \tag{2}\]  
By means of this equation, the definition of the beginning of the damage and the collapse of buildings are related to the two values of acceleration.  
In this work, instead of a field observation of the damage caused by past earthquakes, a static non-linear pushover analysis was performed for the buildings at the test site; thus, the yield and collapse acceleration were determined. Then, using the previously computed vulnerability indexes and yield and collapse accelerations, new damage-vulnerability-peak ground acceleration relationships were derived. The damage index is expressed in the [0-1] space via a tri-linear law, shown in Figure 11, defined by two points: yield acceleration PGA\({}_{\text{y}}\), which represents the beginning of the damage (d = 0), and acceleration for the collapse of the building PGA\({}_{\text{c}}\) (d = 1).