## 5 Fragility Analysis of CFST with BRBs under Earthquake Sequences
### Probabilistic Seismic Demand Model (PSDM)  
The core of establishing PSDM is to develop a relationship between the seismic response and ground motion intensity based on nonlinear time history analysis. There are two main types of methods known as the IDA method and the cloud method. In this study, the cloud method is used to statistically analyze the results of the dynamic response of the structural underground motions and present the processed data in a cloud shape in the coordinate system. The cloud method usually assumes that the seismic demand and the ground motion intensity satisfy a logarithmic linear regression relationship. Compared with the IDA method, which requires scaling, the cloud method has high computational efficiency. In the present study, the employed ground motion inputs were selected considering the earthquake magnitudes and epicentral distances using the bin approach. The five ground motion bins developed by [PERSON] and [PERSON] [53] are applied, including (1) small magnitude and small distance (SMSR, \(5.8<\mathrm{Mw}<6.5\), \(13<\mathrm{R}<30\) km), (2) large magnitude and small distance (LMSR, \(6.5<\mathrm{Mw}<7.0\), \(13<\mathrm{R}<30\) km), (3) small magnitude and large distance (SMLR, \(5.8<\mathrm{Mw}<6.5\), \(30<\mathrm{R}<60\) km), (4) large magnitude and large distance (LMLR, \(6.5<\mathrm{Mw}<7.0\), \(30<\mathrm{R}<60\) km), and (5) near-fault ground motions (NF, \(\mathrm{R}<13\) km). In total, 80 ground motions (16 in each bin) are employed to take into account the ground motion randomness in assessing the seismic fragility.  
[PERSON] et al. [54] indicated that the structural seismic demand (\(\mu_{D}\)) and ground motion intensity measure (_IM_) satisfy the following logarithmic linear relationship.  
\[\ln(\mu_{D})=\ln a+b\ln(IM) \tag{19}\]  
where \(a\) and \(b\) are coefficients that can be obtained by linear regression analysis based on the analysis results of the cloud method.  
The PSDM of CFST-B is calculated using the above-described cloud method. Figure 14 shows the PSDM of the structure subjected to a single mainshock and different earthquake sequences with \(\delta=0.4\), 0.6, and 0.8. The details of regression results for each case are shown in Table 5. As shown in Figure 14, the seismic demand parameter (ISDRmax) and the ground motion intensity parameter (PGA) follow a logarithmic linear relationship well, with coefficients of determination R\({}^{2}\) of 0.76618, 0.75449, 0.77191, and 0.76804, respectively. All R\({}^{2}\) values obtained based on the results of the nonlinear time history analysis of CFST-B are greater than 0.75, which indicates that the PSDM is highly credible in describing the probabilistic characteristics of the seismic demand of the structure. This lays a foundation for the subsequent seismic fragility analysis based on the PSDM.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Cases** & **Linear Regression Model** & **R\({}^{2}\)** & \(\sigma_{D|IM}\) \\ \hline Mainshock ES(\(\delta=0.4\)) & \(\ln(\mathrm{ISDRx})=-4.03073+0.98723\ln(\mathrm{PGA})\) & 0.75449 & 0.28215 \\ ES(\(\delta=0.6\)) & \(\ln(\mathrm{ISDRx})=-3.86207+1.01474\ln(\mathrm{PGA})\) & 0.77191 & 0.29486 \\ ES(\(\delta=0.8\)) & \(\ln(\mathrm{ISDRx})=-3.79405+1.03846\ln(\mathrm{PGA})\) & 0.76804 & 0.31308 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Regression parameters for the PSDMs.