## 5 Seismic Risk Assessment of Novel SCPC Frame
### Incremental Dynamic Analysis (IDA)
The vulnerability analysis usually uses fragility curves to evaluate the seismic performance. The vulnerability function are defined as follows:  
\[F(x)=\sum_{x}P[LS]SI=x]. \tag{5}\]  
\(F(\)x\) refers to the seismic vulnerability of the structure, value x refers to the value of certain seismic intensity corresponding to the \(SI\). The structural fragility function can be described by a lognormal model corresponding the limit state shown as follows [46]:  
\[F(x)=\Phi\left[\frac{\ln x-\ln m_{R}}{\sqrt{\beta_{d}^{2}+\beta_{c}^{2}}}\right], \tag{6}\]  
Figure 19: The IDA curves of novel SCPC frame. where \(\Phi\) is a standard normal distribution function, \(m_{R}\) is median value of the fragility function, \(\beta_{d}\), \(\beta_{c}\) are assigned to logarithmic standard deviation of the structural demand, capacity respectively. \(\beta_{d}\) can carried out by linear regression of the structural response under different intensities. \(\beta_{c}\) means 0.3 to construct the fragility curves in this paper [47]. The seismic hazard probability model \(H(\text{x})\) represents the annual exceeding probability of the certain seismic intensity \(SI\) at the certain site, which is related to the magnitude, epicenter distance and the attenuation regulation of ground motion [48]. If the seismic hazard probability is small, the simplified seismic hazard model can be used, the model is expressed as:  
\[H(x)=P[SI>x]=k_{0}x^{-k}, \tag{7}\]  
where \(k\) and \(k_{0}\) are shape coefficient of seismic hazard curves, the values of \(k\) and \(k_{0}\) are subjected to four parameters--1 annual exceeding probability with 10% exceeding probability in 50 years; 2 annual exceeding probability with 2% exceeding probability in 50 years; 3\(S_{a}\) (\(T_{1}\)) with 10% exceeding probability in 50 years; 4\(S_{a}\) (\(T_{1}\)) with 2% exceeding probability in 50 years. Compared with the seismic hazard probability model, the seismic hazard curves of the novel SCPC frame under near-fault ground motions is shown in Figure 20. From seismic hazard curves and Chinese Standard Code [29], \(S_{a}\) (\(T_{1}\)) and annual exceeding probability corresponding to the DBE level are 0.1176 and 1/475, respectively, \(S_{a}\) (\(T_{1}\)) and annual exceeding probability corresponding to the MCE level are 0.235 and 1/2475, respectively. Based on the fragility curves and the seismic hazard curves, the seismic risk assessment of a structure can be expressed by annual and 50-year exceeding probability of a certain limit state. The annual probability of exceeding a certain limit state can be calculated by the formula:  
Figure 17:0  
combined with Formulas (6) and (7), the annual probability can be rewritten as follows:  
Figure 17:1  
the 50-year exceeding probability corresponding to a certain limit state can be written as follows:  
Figure 17:2  
Figure 17:3 Seismic hazard curves of the novel SCPC frame. Based on the theory of fragility analysis and the results of IDA analysis under near-fault earthquake wave, the fragility curves of the novel SCPC structure are plotted in Figure 17:4 The black and red curves represent the limit state of IO and RE for the engineering demand parameters including \(\theta_{max}\) and \(\theta_{r,max}\), respectively. From the fragility curves, the vertical lines represent the corresponding spectral acceleration \(S_{a}\) (\(T_{1}\)) under the DBE and MCE levels, which are 0.117 g and 0.234 g calculated with 3% damping ratio of the novel SCPC frame. The vulnerability curves for the seismic limit state from IO to RE become smooth and move significantly toward the right, which means under the same \(S_{a}\) (\(T_{1}\)), the exceeding probabilities of the corresponding \(\theta_{max}\) and \(\theta_{r,max}\) drop obviously.