## 5 Validity of PSDM's Assumptions
### Lognormality assumption
For this purpose, the following three different hypothesis cases are considered:  
* Case 1 (_lognormality assumption_) In this case, only the lognormality assumption is considered for developing the fragility curves. So, the value of the probability is defined accounting the mean and the standard deviation of each distribution at a particular level of IM. * Case 2 (_lognormality assumption and power law model_) The lognormality assumption is considered in combination with the power law model. The median of the structural demand at a particular level of IM is based on the PSDM, while the dispersion is calculated for each level of IM through Equation (11). * Case 3 (_lognormality assumption, power law model, and homoscedasticity assumption_) The three basic assumptions of PSDM are considered for the development of the fragility curves. The available separation gap distances \(\mathrm{d}_{\mathrm{g}}\) between the adjacent structures and the case of without bounding effect are also incorporated. Furthermore, the discrete probability data points (empirical CDF method) and the MLE-based fragility curve (as a reliable continuous fragility curve) are used for the quantification of the bias induced through assumptions. In Figures 12 and 13, the fragility curves in terms of \(\mathrm{IDR}_{\mathrm{max}}\,|\,\mathrm{PGA}\) and \(\mathrm{TDR}_{\mathrm{max}}\,|\,\mathrm{PGA}\) based on the examined hypothesis cases (1-3) are presented. As observed, the examined fragility curves are in a good agreement with the assessment based on the MLE method. Therefore, the violation of the homoscedasticity assumption for both EDPs does not influence the fragility assessment of the bounding effect. So, PSDMs for both EDPs are sufficient estimators of the structural demand on the fragility assessment of the eight-story RC frame structure subjected to bounding. In Figure 14, the corresponding fragility curves that are produced considering the three different hypothesis cases in terms of \(\mu_{\varphi,\mathrm{max}}\,|\,\mathrm{PGA}\) at the performance level of DL, are presented. It is observed that the fragility curves are significantly influenced by the assumptions made on the formulation of PSDMs when the seismic performance of eight-story frame is evaluated without considering the bounding between the adjacent structure (Figure 14a). The error induced due to the power law model and the homoscedasticity assumption in the case of \(\mu_{\varphi,\max}\) seems to be considerably high. However, this outcome is not so intense when the effect of structural bounding is assessed. This is attributed to the use of bilinear PSDM to describe the critical inelastic behavior of the column at low levels of IM (Figure 14b-d). Figure 12: Influence of the PSDM assumptions on the fragility assessment of the eight-story RC frame in terms of \(\text{IDR}_{\max}\) (\(\%\text{h}_{\text{st}}\)) as a function of the PGA. Examined assumptions: Case 1—lognormality, Case 2—lognormality and power law model, and Case 3—SDM’s assumptions. Fragility curves (**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. Figure 13: _Cont_.