## 5 Validity of PSDM's Assumptions
### Lognormality assumption
In Figure 11, comparative results between linear and bilinear PSDMs of the column C20 in terms of \(\mu_{\varphi,\max}\) | PGA are presented. These results are compared with the corresponding median demands at each level of PGA. Figure 11: _Cont._  
Figure 10: Comparative results in terms of TDR\({}_{\max}\) (\(\%\)H\({}_{\text{tot}}\)) 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. It can be observed that the median data are differentiated from the PSDMs in the case of using a linear regression model. These differences are more pronounced in the case where the bounding effect is evaluated, and especially when the structural members start to exhibit significant nonlinear behavior due to the interaction with the adjacent shorter and stiffer structure. **Homoscedasticity Assumption**  
The last assumption for the formulation of the PSDM concerns the logarithm standard deviation (\(\beta_{\text{EDP[IM]}}\) of the demand. In order to evaluate the validity of homoscedasticity assumption, the area limited between the mean plus and minus one standard deviation of the associated normal distribution at each level of IM is defined. So, in Figures 9-11, the  
Figure 11: Comparative results in terms of \(\mu_{\varphi,\text{max}}\) (column C20) at each level of PGA as deduced based on IDAs, PSDMs, and median values of demand. Regression models: (**a**) linear, and (**b**) bilinear. Examined cases: (**1**) without the bounding effect, (**2**) \(\text{d}_{\text{g}}=0.0\) cm, (**3**) \(\text{d}_{\text{g}}=4.5\) cm, and (**4**) \(\text{d}_{\text{g}}=9.0\) cm. green shaded area shows the variation of the demands along the examined range of PGA. Figure 9 shows that the variation of demands in terms of the \(\mathrm{IDR}_{\mathrm{max}}\) is not constant but is slightly reduced for values of PGA greater than 0.255 g. Similar results are also deduced in the case of \(\mathrm{TDR}_{\mathrm{max}}\), as it can be observed in Figure 10.  
Therefore, the homoscedasticity assumption of producing the PSDM of \(\mathrm{IDR}_{\mathrm{max}}\) and \(\mathrm{TDR}_{\mathrm{max}}\) is not satisfied within the overall range of PGA. Furthermore, the interaction between the adjacent structures has not altered the outcomes related to the homoscedasticity assumption of the PSDM in terms of displacement EDPs. On the contrary, the case of structural bounding between adjacent structures has altered the outcomes related to the homoscedasticity of the PSDM in terms of \(\mu_{\varphi,\mathrm{max}}\,|\,\mathrm{PGA}\). As it is discussed below, this is attributed to the effect of bounding on the local inelastic demands of the column. Results of Figure 11 clearly indicate that the linear PSDM cannot adequately describe the maximum local curvature ductility demands, i.e., \(\mu_{\varphi,\mathrm{max}}\) of the critical column, due to the bounding effect (Figure 11a). For the bilinear PSDMs, the validity of the homoscedasticity assumption is evaluated at each linear branch of the regression model. Small fluctuations in the dispersion along the first branch of the regression model are observed, and the response of the column is still an elastic one. On the other hand, the nonlinear behavior of the structural member results in a large variation of the demands that differs significantly for each ground motion. These variations are depicted at the second branch of the PSDM. Thus, it could be pointed out that the nonlinear local demands of the structural member are not sufficiently reflected on the homoscedasticity assumption when only linear PSDM is adopted. **Interaction of Assumptions on Developing Fragility Curves**  
In this part of the study, the influence of the PSDM assumptions on the fragility curves is examined in order to provide a better insight of the bias induced.