# 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  
Peak ground accelerations at the yield and the collapse of the structure were evaluated through static non-linear pushover analysis, according to Eurocode 8 [41] and the corresponding Croatian standards [45; 48]. Eleven buildings in the town center, representative of the structural and material characteristics of the test area, were analyzed. The analysis was carried out with TREMURI software [49; 50], in which the building is modelled as a spatial structure where the walls resist to both vertical and horizontal loads. Walls are modeled by means of non-linear two-node macro-elements, representing whole masonry panels and piers. The macro-element considers both the shear-sliding damage failure mode and its evolution. It accounts for strength and stiffness degradation, and rocking  
Figure 11: Pushover curve and tri-linear acceleration/damage law. Figure 10: Vulnerability curve and its idealization. mechanisms, whereas the toe crushing effect is modeled by means of a phenomenological non-linear constitutive law with stiffness degradation under compression. The horizontal structures, such as floors, vaults and ceilings, transfer their vertical loads to the walls and transmit the horizontal actions to the walls. In this way, the structure is modeled by assembling the walls and the horizontal structures, both lacking bending rigidity outside of the level. Horizontal structures can have membrane stiffness or can be regarded as rigid. For each slab's typology, the connection of the structural components should be defined. A good connection to the masonry contributes to the resistance of the global system. In addition, the floor can divide its mass in a single direction or along two directions of the floor. If the floors possess bi-directional stiffness, it is necessary to indicate the vertical load percentage for the principal direction. The floors are modeled as orthotropic membrane three-node elements, with two degrees of freedom per node (displacements \(u_{x}\) and \(u_{y}\)), which are associated with a warping direction. This enables us to model both flexible and rigid floors. The same rules for floors also apply to roofs. A roof can be modeled as a part of a bearing system or just as a load-distributing frame. The response of the structure is investigated along the two geometrical orthogonal axes, in both the positive and negative directions. Non-regular distribution of the masses inside the structure is considered by means of the assumption of an eccentricity of the lateral loads, equal to \(\pm 5\%\) of the maximum floor dimension at each level. Three lateral load distributions--uniform, linear and modal distribution--and considering positive and negative eccentricities led to a total of 36 analyses. Each pushover analysis resulted in an MDOF capacity curve, which represents the relation between base shear force and the displacement of a control node placed at the top of the building. The pushover curve was scaled according to the N2 method described in Annex B of Eurocode 8 [46] using the transformation factor \(\Gamma=\Sigma m_{i}\Phi_{i}/\Sigma m_{i}{\Phi_{i}}^{2}\), where \(\Phi_{i}\) is the i-th component of the eigenvector and \(m_{i}\) is the mass of the node \(i\). For the actual base shear force F and the corresponding top displacement of the structure d of the MDOF system, the values \(F^{*}=F/\Gamma\) and \(d^{*}=d/\Gamma\) represent the base shear force and the displacement of the equivalent SDOF system, respectively. After the transformation of the MDOF curve in the SDOF one, a bilinear force-displacement diagram was obtained. The yield force \(F_{y}^{*}\), representing the actual strength of an idealized system, is equal to the base-shear force at the formation of the plastic mechanism. The initial stiffness was determined assuming an area equivalence between the equivalent and the bilinear system.