# IMPA versus Cloud Analysis and IDA: Different Methods to Evaluate Structural Seismic Fragility
## 4 Results  
### Nonlinear Static Analysis Results  
The pushover analysis has been performed in displacement control to reach a target displacement of 350 mm. The number of steps to reach the target displacement is 350 calculation steps with an increment of 1 mm each step. Figure 7 shows the capacity curves obtained applying two load distributions proportional to the first and second modal shapes, respectively. The capacity curve for the first load distribution reaches a maximum base shear of 492 kN corresponding to a top displacement of about 220 mm. The sequence of the plastic hinges activation with the relative calculation step is shown in Figure 8. A plastic hinge is activated when the reinforcement of the section reaches and exceeds the yield point: the first plastic hinges activated in the columns rather than in the beams, particularly in the upper stories which, according to Eurocode 8, have weak column-strong beam connections. Figure 7: (a) First and second modal shapes and (b) capacity curves of the frame. The two subsets of records studied in-depth (Sel. 1 NF and Sel. 7 FF (Table 1)) comprise two different records of the real seismic event to which the case study was exposed in 2016, the 6.5 magnitude Central Italy earthquake, which, as mentioned, caused severe damage to structural and especially non-structural component of the school. Figures 10 and 11 shows IDA curves and IMPA curves: the curves of IDA referring to the Norcia 2016 earthquake exceed the threshold of \(DCR_{LS}\) for the intensities of S\({}_{\text{a}}\) equal to 0.5 g and 0.57 g for NF and FF selection, respectively. The results of D-CA show that, as expected, the \(DCR_{LS}\) threshold for the SLV limit state is exceeded for both the near-fault record and far-field record, with \(DCR_{LS}\) values of 2.57 and 1.01, respectively (Figure 12). Similarly, even higher \(DCR_{LS}\) are obtained from the cloud analysis based on nonlinear static analysis (Figure 13). To check the consistency of modal and multimodal IMPA and pushover-based cloud, the developed fragility curves (Figure 14) are compared with those of IDA. The accuracy of the prediction of the different fragility models with respect to IDA is quantified by normalized root-mean-square deviation (_RMSD_). It is evaluated according to the following Equation (15), assuming that the values derived by Equation (12) (IDA) are the reference ones:  
\[RMSD(\%)=\sum_{i=1}^{n}\sqrt{\frac{{(\hat{y}_{i}-y_{i})}^{2}}{{\hat{y}_{i}}^{2 }}} \tag{15}\]  
where \(n\) is the number of points, and \(y_{i}\). and \(\hat{y}_{i}\) are the predicted and reference probability of exceeding the considered limit state (_LS_), respectively. Figure 11: IMPA: (**a**) comparison among IMPA curves for near-fault record selection and (**b**) far-field record selection; (**c**) comparison between the two fragility curves obtained considering only the first mode; (**d**) comparison among IMPA curves for near-fault record selection and (**e**) far-field record selection; (**f**) comparison between the two fragility curves obtained considering the first mode and second mode. Figure 12: D-CA: (**a**) comparison among dynamic analysis cloud data regressions for near-fault record selection and (**b**) far-field record selection; (**c**) comparison between the two fragility curves obtained. The following Table 4 shows the comparison in terms of the percentage variation of the median, 16% and 84% fractiles of the fragility curves with respect to IDA and the absolute values of the standard deviation of each method. Regarding the selection of NF records, the cloud method appears solid in estimating the 50%, 16% and 84% fractiles compared to IDA with the smallest normalized root-mean-square deviation of 8%. However, when it comes to far-field records, IMPA appears to be the most accurate methodology for estimating fragility. The inclusion of two or more vibration modes in the assessment of the maximum multimodal \(DCR_{LS}\) does not seem to be essential, as it leads to very conservative results.