## 3 Examples and Discussion
### Analysis of the Seismic Resilience of the Mainshock-Aftershock  
The seismic resilience of the engineering structure can be evaluated based on its seismic performance, including the recovery time, cost required, and recoverability of the structure. Three functional recovery models are employed in this study to introduce the theory and methodology for assessing seismic engineering losses, and the seismic resilience of reinforced concrete frames was analyzed under different seismic intensities, including multiple occurrence earthquakes, fortification-level earthquakes, and rare occurrence earthquakes. By analyzing the seismic resilience of three distinct recovery models, an evaluation of the overall seismic performance of the engineering structure was conducted. Based on the seismic vulnerability curve equation, the seismic vulnerability of the structure was calculated separately for multiple occurrence earthquakes (the seismic impact coefficient is 0.076), fortification-level earthquakes (the seismic impact coefficient is 0.215), and rare occurrence earthquakes (the seismic impact coefficient is 0.429). (1) Multiple occurrence earthquakes  
Under the action of a single seismic event, the exceedance probability \(P\) for the loss of the structure under the four failure states can be calculated as follows:  
\[\left\{ \begin{array}{l}P_{\text{slight damage}}=\Phi[1.925 ln(0.076)+6.376] =\Phi(1.415)\\ P_{\text{moderate damage}}=\Phi[1.925 ln(0.076)+2.586]=\Phi(-2.375)\\ P_{\text{severe damage}}=\Phi[1.925 ln(0.076)+1.047]=\Phi(-3.914)\\ P_{\text{complete damage}}=\Phi[1.925 ln(0.076)-0.493]=\Phi(-5.454)\end{array} \right. \tag{27}\]  
Figure 6: Vulnerability curve of eight-story reinforced concrete frame. Under the action of the mainshock-aftershock, the exceedance probability \(P^{\prime}\) for the loss of the structure under the four failure states can be calculated as follows:  
\[\left\{ \begin{array}{l}P^{\prime}_{\text{slight damage}}=\Phi[1.925 ln(0.076)+6.588]=\Phi(1.627)\\ P^{\prime}_{\text{moderate damage}}=\Phi[1.925 ln(0.076)+2.800]=\Phi(-2.161)\\ P^{\prime}_{\text{severe damage}}=\Phi[1.925 ln(0.076)+1.259]=\Phi(-3.702)\\ P^{\prime}_{\text{complete damage}}=\Phi[1.925 ln(0.076)-0.281]=\Phi(-5.242) \end{array} \right. \tag{28}\]  
Fortification-level earthquakes  
Under the action of a single seismic event, the exceedance probability \(P\) for the loss of the structure under the four failure states can be calculated as follows:  
\[\left\{ \begin{array}{l}P_{\text{slight damage}}=\Phi[1.925 ln(0.215)+6.37 6]=\Phi(3.417)\\ P_{\text{moderate damage}}=\Phi[1.925 ln(0.215)+2.586]=\Phi(-0.373)\\ P_{\text{severe damage}}=\Phi[1.925 ln(0.215)+1.047]=\Phi(-1.912)\\ P_{\text{complete damage}}=\Phi[1.925 ln(0.215)-0.493]=\Phi(-3.452)\end{array} \right. \tag{29}\]  
Under the action of the mainshock-aftershock, the exceedance probability \(P^{\prime}\) for the loss of the structure under the four failure states can be calculated as follows:  
\[\left\{ \begin{array}{l}P^{\prime}_{\text{slight damage}}=\Phi[1.925 ln(0.215 )+6.588]=\Phi(3.629)\\ P^{\prime}_{\text{moderate damage}}=\Phi[1.925 ln(0.215)+2.800]=\Phi(-0.159)\\ P^{\prime}_{\text{severe damage}}=\Phi[1.925 ln(0.215)+1.259]=\Phi(-1.700)\\ P^{\prime}_{\text{complete damage}}=\Phi[1.925 ln(0.215)-0.281]=\Phi(-3.240) \end{array} \right. \tag{30}\]  
Rare occurrence earthquakes  
Under the action of a single seismic event, the exceedance probability \(P\) for the loss of the structure under the four failure states can be calculated as follows:  
\[\left\{ \begin{array}{l}P_{\text{slight damage}}=\Phi[1.925 ln(0.215)+6.37 6]=\Phi(4.747)\\ P_{\text{moderate damage}}=\Phi[1.925 ln(0.215)+2.586]=\Phi(0.957)\\ P_{\text{severe damage}}=\Phi[1.925 ln(0.215)+1.047]=\Phi(-0.582)\\ P_{\text{complete damage}}=\Phi[1.925 ln(0.215)-0.493]=\Phi(-2.122)\end{array} \right. \tag{31}\]  
Under the action of the mainshock-aftershock, the exceedance probability \(P^{\prime}\) for the loss of the structure under the four failure states can be calculated as follows:  
\[\left\{