## 4 Capacity Curves
After obtaining the capacity curves of the MDOF structural system, it is possible to compute the capacity curves for the equivalent SDOF system (Figure 7e) just by using the transformation factor:  
\[d^{*}=\frac{d_{n}}{\Gamma}, \tag{10}\]  
\[F^{*}=\frac{F_{b}}{\Gamma}. \tag{11}\]  
In Figure 8, the user interface of the developed capacity curve computer object is presented, showing 12 capacity curves for the X direction as an example  
Damage is evaluated based on the performance point (the EC8 target displacement) and considering the limit states (LS) that are currently presented in Part 3 of Eurocode 8 (EC8-3), which are the LS of damage limitation (DL), the LS of significant damage (SD) and the LS of near collapse (NC), and also considering the LS for operationality (OP), as it is already presented in the Italian code NTC2018 [49; 50], and as it will probably be in the future generation of the Eurocodes [51], as presented in Figure 10 (where \(d_{u}\) is the ultimate displacement). The performance point may be computed using different NSA methods, namely the CSM and the N2 method, which are the ones that are implemented in the present version of the developed software. The main difficulty dealing with the determination of the damage level related to the LS presented in the EC8-3 is that each capacity curve presents different LS displacement values. This means that a capacity curve with higher resistance may also present a higher damage level, or even collapse, if it exhibits much less ductility. So, it is not very evident which is the capacity curve that presents the worst results in terms of damage, namely in accordance with the EC8-3. To overcome this problem, a new concept is proposed: The performance curve (Figure 11). Figure 10: Example of the localization of the limit states (LS) in one capacity curve. Figure 9: Connections between the damage assessment computer object that computes the performance point and the other independent objects. The performance curve represents the relation between a displacement of the control node and the percentage of a given response spectrum (used as input), which is necessary to obtain a target displacement (performance point) corresponding to the predefined displacement level. Basically, it represents all the performance points that are obtained when different percentages of a given response spectrum are considered. In this way, it becomes quite evident which capacity curves are the most problematic ones for a given damage LS, as can observed in Figure 11.  
At a first glance, this proposal seems to be quite inefficient, computationally speaking, because the analysis should be carried out many times for each percentage of the seismic action, which might be very time consuming, particularly when adopting iterative procedures. However, this is not the case for the implemented approach because the procedures of both the CSM and N2 methods are inverted. Instead of obtaining a target displacement (the performance point) associated to a given spectral acceleration, as usual, in the proposed approach the percentage of the spectral acceleration associated to a given predefined displacement is computed. This process is described in detail in the following sections.