## 5 Fragility Analysis of CFST with BRBs under Earthquake Sequences
### Results and Analysis  
Assuming that the structural seismic demand follows a logarithmic linear distribution at different IMs, the seismic fragility function of a structure can be expressed by the equation below.  
\[P(D\geq C|IM)=\Phi\left[\frac{\ln(\mu_{D})-\ln(\mu_{C})}{\sqrt{\sigma_{D|IM}^{2}+ \sigma_{C}^{2}}}\right] \tag{20}\]  
where \(D\) and \(C\) are the seismic demand and structural capacity, respectively. \(\mu_{D}\) and \(\mu_{C}\) are the median values of \(D\) and \(C\). \(\sigma_{D|IM}\) and \(\sigma_{C}\) are the corresponding standard deviations.  
In this study, four structural damage limit states, known as slight damage, moderate damage, severe damage, and collapse, are considered. The division of structural damage states and the definition of the structural limit states used in this study are consistent with those specified in the Chinese Code for Seismic Design of Buildings [55]. Currently, because the CFST frame structure with BRBs is a new type of high-performance structural system, the deformation limits that characterize the seismic capacity of this structure have not been clearly specified in the code. To determine the damage limit states of CFST structures, reference is made to the \"Specification for Design of Steel-Concrete Mixed Structure of Tall Buildings\" [56], which recommends damage limit values for mixed frame structures under earthquakes. As a result, ISDRs of 1/400, 1/200, 1/100, and 1/50 corresponding to four damage limit states, known as slight damage (LS-1), moderate damage (LS-2), severe damage (LS-3), and collapse (LS-4), are employed to derive the seismic fragility curves herein.  
Figure 15 shows the fragility curves for the damage limit states LS-1, LS-2, LS-3, and LS-4. It can be observed that all seismic fragility curves under the earthquake sequence are located above those under the single mainshock, which indicates that all damage probabilities of different damage limit states for  
Figure 14: PSDM under main shock and earthquake sequences.  
the structures under earthquake sequences increased. An increasing \(\delta\), which means higher impact of the earthquake sequence on the cumulative structural damage, leads to a higher damage probability.  
The PGA corresponding to an exceedance probability of 50% in the seismic fragility curve is defined as the median PGA of the seismic fragility curve. The median PGAs corresponding to the structure exceeding LS-1 under earthquake sequences with \(\delta\) = 0.4, 0.6, and 0.8 decrease by 5%, 7%, and 12%, respectively, when compared with those under the mainshock. The median PGAs corresponding to the structure exceeding LS-2 under earthquake sequences with \(\delta\) = 0.4, 0.6, and 0.8 decrease by 9%, 11%, and 16%, respectively, when compared with those under the mainshock. The median PGAs corresponding to the structure exceeding LS-3 under earthquake sequences with \(\delta\) = 0.4, 0.6, and 0.8 decrease by 14%, 16%, and 19%, respectively, when compared with those under the mainshock. The median PGAs corresponding to the structure exceeding LS-4 under earthquake sequences with \(\delta\) = 0.4, 0.6, and 0.8 decrease by 15%, 25%, and 29%, respectively, when compared with those under the mainshock. These results show that the PGA corresponding to the structure exceeding each limit state under the earthquake sequence decreases when compared with that under the mainshock. Therefore, the structure is more likely to be damaged under an earthquake sequence than under a single mainshock when the other conditions remain the same. Therefore, earthquake sequences should be considered in the seismic design of structures.