## 5 Numerical Analyses
### Response at the SD Limit State  
The probabilities of exceedance of the SD limit state (i.e., for \(a_{\text{g}}=0.15\) g) obtained for all the considered case studies are represented by the histograms of Figure 4. In particular, Figure 4a,b refer to the 4-storey and 8-storey structures, respectively. Plots in the first line show the results related to ductile members only, while those in the second line show the results obtained considering both the verifications of ductile and fragile members.  
As a general trend, the probabilities of exceedance of the SD limit state of the considered systems determined considering only ductile members are always lower than 10%. Thus, all the cases provide a satisfactory response. If the probability of exceedance of the SD limit state is evaluated considering the requirements of both ductile and fragile elements, cases k02 and k06 provide a probability of exceedance larger than 50% in the case of 4-storey structures, whereas case Q2.5 provides no suitable performance independently of the number of storeys of the buildings.  
Figure 3: Fragility curves at the SD limit state of a 4-storey case study: (a) Q = 1.0; (b) Q = 2.5.  
To investigate on this result, Figure 5 shows, for each designed structure, the percentage of cases in which the SD limit state is attained because of the achievement of the capacity in a brace, a column or a beam. For a fixed case study, when the majority of collapses are caused by failure of columns or failure of fragile beams, it means that the considered structure has a poor dissipative capacity and is characterized by a fragile seismic response. On the contrary, a ductile seismic response is provided by those structures that tend to collapse mainly due to the failure of the braces.  
The figure shows that in case Q2.5, the SD limit state is always achieved because of the fragile collapse of columns. In the case of 4-storey structures, this also occurs for cases k02 and k06, whereas in the cases k10 to k18 the column failure occurs for about 50% of the accelerograms. It is interesting to remark that the column cross-sections used for the 4-storey frames are the same as in the case studies k02 to k18 and that the column at the base has a minimum overstrength. Thus, if the column cross-sections are not oversized, a braced beam with moderate flexural stiffness and strength partially prevents column failure.  
The performance of the designed case study has also been evaluated in terms of the residual interstorey drifts. The limit value corresponding to the attainment of the SD  
Figure 4: Probability of exceedance of the SD limit state for \(a_{\mathrm{g}}=0.15\) g: (**a**) 4-storey structures; (**b**) 8-storey structures.  
Figure 5: Percentage number of accelerograms leading to DI = 0.75 in braces or \(RI=1\), \(SI=1\), \(TI=1\) in beams and columns: (**a**) 4-storey structures; (**b**) 8-storey structures.  
limit state was equal to 0.5% \(h\), \(h\) being the interstorey height, in accordance with the provision given by FEMA [37]. Values of residual interstorey drifts lower than 0.5% \(h\) are deemed acceptable and the structure is worthy to be repaired. As the interstorey height of the building is equal to 4.0 m, the limit value of the residual interstorey drift is equal to \(\Delta d_{lim}=20.0\) mm. The probabilities of exceedance of the maximum value of the residual interstorey drifts for a peak ground acceleration equal to 0.15 g are very low for all the considered structures and thus are not reported in any figure.