## 2 Probabilistic Seismic Demand Analysis  
The structural bounding risk can be defined as the mean annual frequency (MAF) of exceedance of a given engineering demand parameter (edp) and is calculated as follows [13]:  
\[\lambda_{\mathrm{EDP}}(\mathrm{edp})=\int\mathrm{G}_{\mathrm{EDP}|\mathrm{IM} }(\mathrm{edp}\Big{|}\mathrm{im})|\mathrm{d}\lambda_{\mathrm{IM}}(\mathrm{im})| \tag{1}\]where \(\mathrm{G}_{\mathrm{EDP}|\mathrm{IM}}(\mathrm{edp}|\mathrm{im})\) denotes the complementary cumulative distribution function (CCDF) of \(\mathrm{EDP}=\mathrm{edp}\) conditional to \(\mathrm{IM}=\mathrm{im}\), \(\lambda_{\mathrm{IM}}(\mathrm{im})\) is the MAF of exceeding a certain value of \(\mathrm{im}\) at a given seismic site of interest, and \(\mathrm{d}(\cdot)\) is the differentiation operator. The seismic hazard \(\lambda_{\mathrm{IM}}(\mathrm{im})\) is defined through probabilistic seismic hazard analysis and can be approximated using the following expression proposed by [PERSON] and [PERSON] [39]:  
\[\lambda_{\mathrm{IM}}(\mathrm{im})=\mathrm{k}_{\mathrm{o}}\,\mathrm{im}^{- \mathrm{k}} \tag{2}\]  
where \(\mathrm{k}\) and \(\mathrm{k}_{\mathrm{o}}\) are the slope and the intercept of the seismic hazard curve. On the other hand, \(\mathrm{G}_{\mathrm{EDP}|\mathrm{IM}}(\mathrm{edp}|\mathrm{im})\) is the task of a probabilistic seismic demand analysis (PSDA). In this direction, a series of time history analyses are required to define the probabilistic seismic demand model (PSDM) that defines the relation between the median engineering demand parameter \(\mathrm{EDP}\) and the \(\mathrm{IM}\) based on the following power law model:  
\[\mathrm{EDP}\big{|}\mathrm{IM}=\mathrm{a}\mathrm{IM}^{\mathrm{b}} \tag{3}\]  
Linear regression analysis in log-log space is performed to calculate the coefficients a and b. Therefore, the Equation (3) is rewritten as follows:  
\[\ln\mathrm{EDP}|\mathrm{IM}=\mathrm{bln}\,\mathrm{IM}+\ln\mathrm{a}+\varepsilon |\mathrm{IM} \tag{4}\]  
where \(\varepsilon\,\mathrm{IM}\) is the random error with mean zero and standard deviation \(\beta\). Thereafter, the CCDF is evaluated based on the closed-form solution of the PSDM as follows:  
\[\mathrm{G}_{\mathrm{EDP}|\mathrm{IM}}(\mathrm{edp}|\mathrm{im})=\mathrm{P}[ \mathrm{EDP}|\,\mathrm{IM}\geq\mathrm{edp}|\mathrm{im}]=\Phi\Bigg{(}\frac{ \ln\mathrm{a}+\mathrm{bln}\,\mathrm{im}-\ln\hat{C}}{\beta_{\mathrm{EDP}| \mathrm{IM}}}\Bigg{)} \tag{5}\]  
where \(\beta_{\mathrm{EDP}|\mathrm{IM}}\) is the logarithm standard deviation and \(\hat{C}\) is the capacity of the \(\mathrm{EDP}\). It is noted that the results of the fragility analysis are influenced by uncertainties associated with the ground motions, the structural system properties, and the lack of knowledge. Furthermore, [PERSON] et al. [25] highlight the importance of considering the uncertainty and the dependence between different demand parameters in the fragility analysis of engineering structures. To deal with this issue, a multivariate PSDM was proposed. In this study, the uncertainties affecting the structural response due to variability of the selected ground motions are considered by assuming that the structural demands follow a lognormal distribution with the logarithm standard deviation \(\beta_{\mathrm{EDP}|\mathrm{IM}}\) calculated as follows:  
\[\beta_{\mathrm{EDP}|\mathrm{IM}}=\sqrt{\frac{\sum_{i=1}^{n}\bigl{(}\ln\, \mathrm{EDP}_{i}|\mathrm{IM}-\ln\,\mathrm{EDP}\big{|}\mathrm{IM}\bigr{)}^{2}} {n-2}} \tag{6}\]  
In the case of considering the nonlinear local demands of the structure, the logarithm standard deviation \(\beta_{\mathrm{EDP}|\mathrm{IM}}\) does not remain constant within the region of the examined IMs, and a bilinear regression model is required to conform with the homoscedasticity assumption. Details regarding the accurate definition of a PSDM in structural bounding conditions can be found in the work of [PERSON] and [PERSON] [15; 38].