## 3 Examples and Discussion
### Effect of the Mainshock-Aftershocks on the Seismic Resilience
#### 3.4.2 Function Curves under Mainshock and Mainshock-Aftershock  
In the case of multiple earthquakes, fortified earthquakes, and rare earthquakes, the loss function is calculated according to the calculation results of the loss index in Table 3. When the structure is damaged by the earthquake, the residual function index of the structure is displayed in Table 6.  
1. [label=(0)]
2. Frequently occurred earthquake Based on recovery time and the definition of resilience (1), a simulation is conducted on the trigonometric function recovery model under multiple earthquake intensities. Figure 7 illustrates the functional function curves of the structure solely subjected to the mainshock. As depicted in Figure 7, the structure exhibits a functional loss index of 0.0478 after being subjected to earthquake-induced damage, with a time duration of 24 days required for repair completion from the occurrence of the damage. The loss-of-function function of the structures under the mainshock-aftershock is simulated using the Monte Carlo method. The simulation results are presented in Table 7. It can be observed that, under the action of multiple earthquakes, the resilience index tends to converge to a value of 0.9981 when a random number of 100 is selected. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Earthquake**} & \multicolumn{2}{c}{**Frequently Occurred Earthquake**} & \multicolumn{2}{c}{**Fortified Earthquake**} & \multicolumn{2}{c}{**Rarely Occurred Earthquake**} \\ \cline{2-7}  & **Principal** & **Mainshock-** & **Principal** & **Mainshock-** & **Principal** & **Mainshock-** \\  & **Earthquake** & **Aftershock** & **Earthquake** & **Aftershock** & **Earthquake** & **Aftershock** \\ \hline Loss function index Residual function index & 0.0478 & 0.0505 & 0.1350 & 0.1632 & 0.3728 & 0.4903 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Calculation results of normalized functional and economic losses of structures under different seismic actions. Figure 7: Functional function curves of the structure under mainshock alone. As illustrated in Figure 8, this study focused on analyzing the functional function curves of the structure under six typical sets of mainshock-aftershocks, each characterized by different time intervals. In Figure 8a, it can be observed that the mainshock and aftershocks occur nearly simultaneously, resulting in a functional loss index of 0.0505 for the structure following seismic damage. Figure 8b,c depict the structure was damaged by the mainshock, and was followed by a period of repair over time. The shorter the time interval between the mainshock and aftershocks, the greater the slope of the recovery curve of aftershocks, and the faster the restoration speed. Figure 8d-f demonstrate a high degree of consistency. (2) Fortified earthquake  
As depicted in Figure 9, the structure exhibits a functional loss index of 0.1350 after being subjected to earthquake-induced damage, with a time duration of 74 days required for repair completion from the occurrence of the damage. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline Simulation times & \multicolumn{2}{c}{Principal earthquake} & \multicolumn{3}{c}{Mainshock-aftershock} \\ \cline{2-6}  & & 100 & 1000 & 10,000 & 100,000 \\ \hline Resilience index & 0.9984 & 0.9981 & 0.9981 & 0.9981 & 0.9981 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Seismic resilience index under multiple earthquakes. Figure 8: Functional function curves of the mainshock-aftershock structure at different recovery times. The loss-of-function function of the structures under the mainshock-aftershock is simulated using the Monte Carlo method. The simulation results are presented in Table 8. It can be observed that, under the action of fortified earthquakes, the resilience index tends to converge to a value of 0.9819 when a random number of 10,000 is selected.