## 5 Validity of PSDM's Assumptions
### Lognormality assumption  
The validity of the lognormality assumption is examined according to the Anderson-Darling (AD) test. In general, the Anderson-Darling test measures how well the data follow a particular distribution. The smaller the AD value, the better the distribution fits to the data. Through the AD test, the \(p\)-value is calculated in order to determine which distribution fits better to data set [63]. The AD test is implemented for each level of PGA considering a confidence level equal to 95%. For each level of PGA, the implemented hypothesis test is as follows: For \(\mathrm{H_{o}}\) (_null hypothesis_), the data follow lognormal distribution; and for \(\mathrm{H_{1}}\), the data do not follow the lognormal distribution. If the \(p\)-value for the AD test is lower than the significance level of 0.05, data do not follow lognormal distribution. On the other hand, the null hypothesis  
Figure 7: MLE, MM, IM percentiles, PSDM methodologies on the fragility assessment of the eight-story RC frame in terms of maximum curvature ductility \(\mu_{\varphi,\mathrm{max}}\) as a function of the PGA. Examined cases: (**a**) without the bounding effect, (**b**) \(\mathrm{d_{g}}=0.0\) cm, (**c**) \(\mathrm{d_{g}}=4.5\) cm, and (**d**) \(\mathrm{d_{g}}=9.0\) cm. cannot be rejected. In Figure 8, indicative results of the probability plots at the level of PGA = 0.355 g are presented, while Table 3 shows the \(p\)-values of all the examined cases. It can be observed that, although the assumption of the lognormal distribution cannot be rejected in the case of considering the \(\text{IDR}_{\text{max}}\) and \(\text{TDR}_{\text{max}}\) seismic demands, in the case of \(\mu_{\varphi,\text{max}}\), the distribution of the data does not match the lognormal distribution for all the examined levels of PGA (see Table 3). The second assumption of the PSDM concerns the relationship between the median structural demand and the intensity measure (IM). It is assumed that the mathematical relationship between EDP and IM can be approximated by a power law model and, thus, the representation of EDP-IM pairs in log-log space (lognormality assumption) implies that the structural response variates linearly (Equation (10)) with the intensity measure. Therefore, the median structural demand lies on a line at any level of IM [64]. However, linear representation of the PSDM along the range of IM is not always accurate and the adoption of a bilinear model is required to describe the variation of the structural demand especially when the components exhibit significant nonlinear behavior. In Figures 9 and 10, the accuracy of using the power law model to describe the demands of \(\text{IDR}_{\text{max}}\) and \(\text{TDR}_{\text{max}}\) as a function of the PGA, is examined. In these figures, the blue dots show the structural demands retrieved based on IDAs, the black rhombus dots represent the median value of EDPs' stripes at each level of IM, and the red lines depict the produced PSDM. Figure 9 shows that the probabilistic seismic demand model in terms of \(\text{IDR}_{\text{max}}\) is in good agreement with the median demand at each level of IM for all the examined cases. The same results are also yielded in the case of \(\text{TDR}_{\text{max}}\). As it can be observed in Figure 10, the median \(\text{TDR}_{\text{max}}\) demand at each level of PGA is very close with the probabilistic seismic demand model for all the examined cases. Therefore, the linear PSDM can be used to generate the seismic demands of the eight-story frame in terms of \(\text{IDR}_{\text{max}}\) and \(\text{TDR}_{\text{max}}\) in the context of the probabilistic assessment procedure. Figure 9: Comparative results in terms of \(\text{IDR}_{\text{max}}\) (%hst) at each level of PGA as deduced based on IDAs, PSDMs, and median values of demand. Examined cases: (**a**) without the bounding effect, (**b**) \(\text{d}_{\text{g}}\) = 0.0 cm, (**c**) \(\text{d}_{\text{g}}\) = 4.5 cm, and (**d**) \(\text{d}_{\text{g}}\) = 9.0 cm.