## 3 Assessing the Effectiveness of the Proposed IDA Procedure
### Comparison of the Results  
Figures 7-11 show that \(DM\) is a non-monotonic function of \(IM\); the curves often display a twisting pattern with both softening and (counterintuitively) hardening behaviors. Such a trend is also typical for the IDA curves when other choices of \(DM\) and \(IM\) are made, as reported in [19]. Moreover, the curves obtained for different records tend to provide rather dispersed results. Comparing the graphs in Figures 7-11 almost always reveals more marked scatter in the \(x\)-direction with respect to the \(y\)-direction regardless of the value of the period (elastic or inelastic, approximated or rigorously calculated). If a reference to the elongated period is made, however, the scatter generally reduces, even if referring to the values of the elongated period does not significantly improve the effectiveness of the IDA procedure. In contrast, when more rigorous values of the elongated inelastic period are accounted for, namely, \(T_{inP}\) and \(T_{inSP}\), the IDA curves exhibit less dispersion (see Figures 10 and 11). A remarkable reduction in dispersion in both the \(x\)-direction and the \(y\)-direction is found when the \(IM\) parameter is calculated  
Figure 11: IDA curves obtained using the elongated smooth peak period \(T_{inSP}\) according to Equation (4): (a) \(x\)-direction; (b) \(y\)-direction. Figure 10: IDA curves obtained using the elongated peak period \(T_{inP}\) according to Equation (3): (a) \(x\)-direction; (b) \(y\)-direction. through the smooth peak period \(T_{inSP}\). A comparison between the graphs in Figure 11 and those in Figures 7-10 shows that the _IM_ parameter provided by Equation (4) is, in fact, the best choice among those considered in the present investigation, at least in reference to the building in Figure 3. To assess whether and to what extent Equation (4) may provide a good _IM_ indicator, the effectiveness of the different IDA procedures will be evaluated in terms of the dispersion of the _DM_ results at given _IM_ values for the considered building. According to the conventional procedure presented in [19], the collapse is assumed to be reached at the point from which the slope of the curve falls to a value less than 20% of the elastic slope. The values of _IM_ and _DM_ at the collapse point are referred to as _Sacollapse_ and _IDcollapse_. The collapse point enables to determine the response reduction factors [7] and to assess the seismic performance of existing structures [6]. To this purpose, the median of the collapse points of the IDA curves is used as global collapse criterion. The 16 th, 50 th, and 84 th percentiles of _Sacollapse_ and _IDcollapse_ achieved in this study are provided in Figures 12 and 13, for the two analysis directions and relevant to the five values of period considered. The histograms in Figure 12 show that, for any percentile, the values of _Sacollapse_ relevant to the elastic period are the highest, reaching almost 6 g for the 84 th percentile in both directions. Still, rather high values are reached for the approximate elongated period given by [PERSON] and [PERSON] [33]. On the contrary, the _Sacollapse_ values obtained with the approximate period of [PERSON] et al. [24] and with the periods \(T_{inP}\) and \(T_{inSP}\) increase slightly from the 16 th to the 84 th percentile, reaching in general values lower than 3 g, which are compatible with the expected capacity of the type of structure considered. The histograms in Figure 13 show that \(ID_{collapse}\) corresponding to the 16 th percentile has moderate values for any kind of period considered in this study. When the 50 th and 84 th percentile are concerned, the \(ID_{collapse}\) values calculated with the elastic period and with the [PERSON] and [PERSON] [33] elongated period, increase notably instead, particularly in the \(y\)-direction, where 12% is exceeded. In contrast, the values obtained with the \(T_{inp}\) and \(T_{inSP}\) periods show a moderate increase, reaching values below 4%.