# Seismic Vulnerability Assessment of Historical Masonry Buildings in Croatian Coastal Area
## 3 Development and Calibration of Vulnerability Model
### Static-Nonlinear Pushover Analysis of Representative Buildings
The yield displacement of the bilinear SDOF system \(d_{y}^{*}=2(d_{m}{}^{*}-E_{m}{}^{*}/F_{y}{}^{*})\) was obtained from the deformation energy \(E_{m}{}^{*}\) up to the formation of the plastic mechanism. The mass \(m^{*}\), the stiffness \(k^{*}\), and the period \(T^{*}\) of the equivalent SDOF system can be obtained as follows:  
\[m^{*}=\sum_{i=1}^{n}{\Phi_{i}}^{*}m_{i}\ ;\ k^{*}=\frac{F_{y}^{*}}{d_{y}^{*}}\ ;\ T^{*}=2\pi\sqrt{\frac{m^{*}}{k^{*}}}, \tag{3}\]  
The spectral yielding acceleration \(S_{ay}\) and the elastic spectral acceleration \(S_{ae}\) of an elastic SDOF with period \(T^{*}\) are calculated as:  
\[S_{ay}=\frac{F_{y}^{*}}{m^{*}}\ ;\ S_{ae}=S_{ae}(T^{*}) \tag{4}\]  
Reduction factor \(R_{\mu}\) is expressed as:  
\[R_{\mu}=\frac{S_{ae}}{S_{ay}} \tag{5}\]and used to calculate the displacement ductility factor:  
\[ \begin{array}{l}\mu_{r}=(\mathrm{R}_{\mu}-1)\frac{\mathrm{T}_{\mathrm{c}}}{ \mathrm{T}}+1\ \ \ \ ;\ \ \ \ \mathrm{T}^{*}<\mathrm{T}_{\mathrm{c}}\\ \mu_{r}=\mathrm{R}_{\mu}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ;\ \ \ \ \ \ \ \ \ \ \ \ \ \ ;\ \ \ \ \ \ \mathrm{T}^{*}\geq\mathrm{T}_{\mathrm{c}}\end{array}  \tag{6}\]  
Spectral inelastic demand acceleration \(\mathrm{S}_{\mathrm{ai}}\) and displacement \(\mathrm{S}_{\mathrm{di}}\) were derived as follows:  
\[\mathrm{S}_{\mathrm{ai}}=\frac{\mathrm{S}_{\mathrm{ae}}}{\mathrm{R}_{\mu}( \mu_{r}\mathrm{T})}\ \ ;\ \mathrm{S}_{\mathrm{di}}=\frac{\mu_{r}}{\mathrm{R}_{\mu}( \mu_{r}\mathrm{T})}\mathrm{S}_{\mathrm{de}}=\mu_{r}\frac{\mathrm{T}^{2}}{4 \pi^{2}}\mathrm{S}_{\mathrm{ai}}. \tag{7}\]  
The values of peak ground acceleration \(\mathrm{PGA}_{\mathrm{y}}\) and collapse acceleration \(\mathrm{PGA}_{\mathrm{c}}\) are calculated from the corresponding displacements according to the following procedure. The displacement demand \(\mathrm{d}_{r}\) was cast as a function of the spectral elastic displacement \(\mathrm{S}_{\mathrm{de}}(\mathrm{T}^{*})\) using the following analytical relationship:  
\[ \begin{array}{l}\mathrm{d}_{r}^{*}=\left[1+(\mathrm{R}_{\mu}-1)\frac{ \mathrm{T}_{\mathrm{c}}}{\mathrm{T}^{*}}\right]\frac{\mathrm{S}_{\mathrm{de} }(\mathrm{T}^{*})}{\mathrm{R}_{\mu}}\ \ \ \ ;\ \ \ \ \mathrm{T}^{*}<\mathrm{T}_{\mathrm{c}}\\ \mathrm{d}_{r}^{*}=\mathrm{S}_{\mathrm{de}}\left(\mathrm{T}^{*}\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ;\ \ \ \ \ \mathrm{T}^{*}\geq\mathrm{T}_{\mathrm{c}}\end{array}  \tag{8}\]  
The reduction factor can be calculated again as a function of the actual ductility \(\mu\) of SDOF system in the form:  
\[\begin{array}{l}\overline{\mathrm{R}}_{\mu}=1+(\mu-1)\frac{\mathrm{T}^{*}} {\mathrm{T}_{\mathrm{c}}}\ \ \ ;\ \ \ \mathrm{T}^{*}<\mathrm{T}_{\mathrm{c}}\\ \overline{\mathrm{R}}_{\mu}=\mu\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  
on the period T, soil factor S, and damping correction factor \(\eta\), and the characteristics periods T\({}_{\text{B}}\), T\({}_{\text{C}}\), and T\({}_{\text{D}}\) represent the lower and upper limits of each spectral acceleration branch. The peak ground accelerations PGA\({}_{\text{Y}}\) and PGA\({}_{\text{C}}\), corresponding to the yield displacement and to the ultimate displacement, respectively, can now be calculated in the form:  
\[\text{PGA}_{\text{Y}}=\frac{\text{S}_{\text{ae},\text{y}}\left(\text{T}^{*} \right)}{\text{f}_{\text{i}}(\text{T})}\ \ ;\ \text{PGA}_{\text{C}}=\frac{\text{S}_{\text{ae}, \text{c}}\left(\text{T}^{*}\right)}{\text{f}_{\text{i}}(\text{T})} \tag{15}\]  
The procedure of the evaluation of the yield and the collapse acceleration is shown for the building of the Public Library, a stone masonry structure built in the 19 th century, in Figure 12. It has three floors (ground floor, first floor, and attic) and is intended for public usage. It consists of a library and a hall for events on the ground floor and offices on the first floor. The structure is made of stone walls and wooden floors. The walls were built of natural carved stone, 50-cm thick, with medium-quality mortar.