## 4 Capacity Curves  
In the proposed approach, the vulnerability assessment is based on capacity curves that are computed as described in the EC8 and which are used for NSA. A capacity curve is the nonlinear relation between the displacement of the control node (\(d_{n}\)), which is normally the centre of mass of the roof of the building, and the base shear force (\(F_{b}\)). The first step of any NSA method consists in the idealization of a single degree of freedom (SDOF), with stiffness \(k^{*}\) and mass \(m^{*}\) (Figure 7), which is equivalent to the initial multiple degrees of freedom (MDOF) dynamic system. The transformation process of the initial dynamic system with \(N\) degrees of freedom (DOF) is done throughout the adoption of a transformation factor (\(\Gamma\)) given by:  
\[\Gamma=\frac{m^{*}}{\sum_{i=1}^{N}m_{i}\cdot\phi_{i}^{2}}, \tag{6}\]  
\[m^{*}=\sum\
olimits_{i=1}^{N}m_{i}\cdot\phi_{i}, \tag{7}\]  
with \(m_{i}\) being the mass associated to each DOF in the MDOF dynamic system, and \(\phi_{i}\) being the configuration of the deformed shape that is adopted in the transformation process (normalized such as \(\phi_{n}\) = 1 in the control node \(n\)). Next, it is necessary to obtain the capacity curve of the MDOF system (Figure 7d), which can be computed by using any structural analysis software with that option. To do so, a set of forces (\(F_{0i}\)) are applied to the structure in each DOF. It is advisable to normalise these forces, so that the sum equals the unity (\(\sum_{i=1}^{N}F_{0i}=1\)), that being:  
\[F_{0i}=\frac{m_{i}\cdot\phi_{i}}{m^{*}}. \tag{8}\]  
In consequence, the base shear force will be equal to the load parameter (\(\lambda\)) that is normally computed by the pushover analysis software:  
\[F_{b}=\lambda\cdot\sum_{i=1}^{N}F_{0i}=\lambda. \tag{9}\]  
Figure 7: Scheme for obtaining the buildings capacity curves. Force patterns: (**a**) Uniform; (**b**) proportional to the high of the building; (**c**) based on a given mode of vibration. (**d**) Capacity curve of the multiple degrees of freedom (MDOF) dynamic system. (**e**) Capacity curve of the equivalent single degree of freedom (SDOF) dynamic system. The EC8 indicates that two force patterns must be considered to compute the capacity curves, which are a function of the adopted deformed shape configuration:  
* A uniform pattern, proportional to the mass of each degree of freedom, that being \(\phi_{n}\) = 1 (Figure 7a);
* a modal-like pattern, which can be proportional to the distance between the base (Figure 7b) and the degree of freedom (DOF), the configuration of the simplified Rayleigh method or corresponding to the configuration of a given computed vibration mode, normally the one with the highest mass participation in the direction where the forces are applied (Figure 7c). These forces can be obtained automatically by using the developed computer object, and instantly exported to MS Excel, to be then introduced in any structural nonlinear analysis software. According to the EC8, the two lateral vertical load patterns must be applied in both the positive and negative plan directions of the building (X and Y), and an accidental eccentricity of the centre of mass must also be considered (three mass centres for each direction). Accounting for all these rules, it is necessary to consider at least 24 capacity curves (\(2\times 2\times 2\times 3=24\)) for each building.