## 3 Examples and Discussion
### Calculation of Structural Earthquake Vulnerability Function  
The structure was analyzed with a non-linear model. The beam-column fiber section model is used to simulate the non-linear behavior of the structure under earthquake. The Kent-Scott-Park model is assigned to the concrete elements [22]. The Kent-Scott-Park model (KSP model) is a constitutive model used to describe the behavior of reinforced concrete materials. The KSP model is primarily used for predicting and analyzing the mechanical response of reinforced concrete structures under bending and shear loading [23]. The Kent-Scott-Park model was used as the control point to determine the constitutive relationship of stirrup-constrained concrete, and the constitutive relationship was established according to the beam-column fiber section model proposed by [PERSON] and [PERSON] [21]. [PERSON] obtained probabilistic seismic demand parameters through linear fitting of 1000 maximum  
Figure 5: (**a**) Frame structure plan; (**b**) Framework structure facade. interstory displacement angles \(\theta_{max}\) and spectral acceleration \(S_{a}\)[21]; the unit of \(S_{a}\) is g (9.8 m/s\({}^{2}\)), the interstory angle is dimensionless, and the fitting results are shown in Table 1:  
Based on the seismic vulnerability theory, the failure probability of the structure has been calculated at four different levels. The analytical equation that describes the relationship between the maximum interstory displacement angle of the frame structure subjected to longitudinal horizontal earthquake excitation and the spectral acceleration of seismic motion under a single earthquake is presented as follows [21]:  
\[lnm_{D|S_{a}}=-3.4408+0.8661 lnS_{a} \tag{22}\]  
The maximum interstory displacement angle could be calculated by the methods in [24; 25]. Substituting the aforementioned equation into the seismic vulnerability function equation, the expression of the single seismic vulnerability curve of the frame structure at different state levels can be obtained after simplification:  
\[\left\{ \begin{array}{l}P_{\text{slight damage}}=\Phi(1.925 lnS_{a}+6.376) \\ P_{\text{moderate damage}}=\Phi(1.925 lnS_{a}+2.588)\\ P_{\text{severe damage}}=\Phi(1.925 lnS_{a}+1.047)\\ P_{\text{complete damage}}=\Phi(1.925 lnS_{a}-0.493)\end{array} \right. \tag{23}\]  
The difference between the structural demand of the mainshock-aftershock and that of the single mainshock is 10% [26]. Therefore, the relation expression of seismic motion spectral acceleration under the action of the mainshock-aftershock is  
\[lnm^{\prime}_{D|S_{a}}=ln\Big{(}1.1m_{D|S_{a}}\Big{)}=ln1.1+lnm_{D|S_{a}} \tag{24}\]  
By substituting Equation (22) into Equation (24), the following expression can be derived [21]:  
\[lnm^{\prime}_{D|S_{a}}=-3.3455+0.8661 lnS_{a} \tag{25}\]  
Under the action of the mainshock-aftershock, the vulnerability curve expression of the frame structure at different state levels can be obtained according to Equation (23):  
\[\left\{ \begin{array}{l}P^{\prime}_{\text{slight damage}}=\Phi(1.925 lnS_{a}+6.58 8)\\ P^{\prime}_{\text{moderate damage}}=\Phi(1.925 lnS_{a}+2.799)\\ P^{\prime}_{\text{severe damage}}=\Phi(1.925 lnS_{a}+1.259)\\ P^{\prime}_{\text{complete damage}}=\Phi(1.925 lnS_{a}-0.281)\end{array} \right. \tag{26}\]  
The spectral acceleration of ground motion is the \(X\)-axis and the overpassing probability of structure at different levels is the \(Y\)-axis. The spectral acceleration follows a lognormal distribution. According to the equation of failure probability, the relationship between failure probability \(P_{f}\) and spectral acceleration \(S_{a}\) is obtained by linear interpolation. The longitudinal seismic vulnerability curve of eight-story reinforced concrete frame under a single seismic event is established. Figure 6 presents the vulnerability curves under the single shock and the mainshock-aftershock.