## 5 Numerical Analyses
### Response Parameters  
To evaluate the fulfilment of the SD and NC limit states, the response of the dissipative members (braces and braced beams) was expressed in terms of ductility demand-to-capacity ratios. To this end, the damage index DI of the braces is defined in [19] as follows:  
\[\text{DI}=\frac{\mu_{\text{d}}-1}{\mu_{\text{f}}-1} \tag{7}\]  
where \(\mu_{\text{d}}\) and \(\mu_{\text{f}}\) are the ductility demand and capacity of the relevant brace, respectively. Specifically, the ductility demand was calculated as the ratio of the sum of the absolute values of maximum elongation and shortening of the brace over the elongation corresponding to yielding of the brace. In keeping with [19], with regard to the Near Collapse (NC) limit state, a limit value of the DI equal to 1.0 was considered, whereas a limit value of DI equal to 0.75 was considered to be representative of the achievement of the Significant Damage (SD) limit state.  
The damage index of the beams was calculated as the ratio of the demanded plastic rotation to the plastic rotation capacity, defined as reported in Eurocode 8 part 3 [34]. Specifically, at the achievement of the NC limit state, for axial load ratios not higher than 0.3, the rotation capacity was equal to 8 and 3 times the chord rotation at yield (\(\theta_{\text{y}}\)) for Class 1 and Class 2 cross-sections, respectively. At the achievement of the SD limit state, instead, the rotation capacity was equal to 6 \(\theta_{\text{y}}\) and 2 \(\theta_{\text{y}}\) for the Class 1 and Class 2 cross-sections, respectively.  
Regarding the fragile members (beams and columns), the verifications of yielding or instability were conducted in terms of the resistance index \(RI\), stability index \(SI\) and lateral torsional buckling index \(TI\). These indexes were calculated following the prescriptions  
Fragility curves were finally used to evaluate the probability of exceedance of the considered limit state for a peak ground acceleration equal to the target value; i.e., 0.15 g at the SD limit state and 0.189 g at the NC limit state. This latter value was determined by the following equation given in Eurocode 8 [16], assuming that the NC limit state should be fulfilled for seismic events with a probability of exceedance \(P_{\text{VR}}\) equal to 5% in 50 years:  
\[a_{\text{g}}=0.15g\left(10/P_{\text{VR}}\right)^{1/3} \tag{14}\]  
As an example, Figure 3 shows the fragility curves obtained for the 4-storey cases Q1.0 and Q2.5 at the SD limit state. The red curve is obtained starting from the \(a_{\text{g,SD}}\) values corresponding to the achievement of the deformation capacity of the dissipative members (first option discussed in this Section), whereas the black curve is obtained based on the \(a_{\text{g,SD}}\) values considering both ductile and fragile members (second option discussed in this Section). With regard to system Q1.0 (Figure 3a), if only ductile members were considered, the probability of exceedance of the SD limit state for \(a_{g}=0.15\) g (see the red arrow) would be equal to 8.1%, whereas it increases to 18.8% (see the black arrow) because of the earlier fragile collapse of some members. Instead, regarding system Q2.5, the analysis of Figure 3b shows that case Q2.5 is considerably oversized and the probability of exceedance of the SD limit state for \(a_{\text{g}}=0.15\) g is equal to 0% if only ductile members were considered. This is due to the restrictive conditions introduced (1) on the size of the brace of the upper floor due to the normalized slenderness \(\overline{\lambda}\) that has to be lower than 2.0; and (2) on the size of the braces of the other storeys because of the limits imposed to the maximum scattering in the distribution of the overstrength \(\Omega_{max}/\Omega_{min}\). However, the probability of exceedance of the SD limit state increases to 67.3% because of the fragile collapse of some members and thus the response of this case study is deemed not adequate because the abovementioned probability of exceedance is higher than 50%.