## 3 Examples and Discussion
### Effect of the Mainshock-Aftershocks on the Seismic Resilience
#### 3.4.2 Function Curves under Mainshock and Mainshock-Aftershock
As illustrated in Figure 10, 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 10a, it can be observed that the mainshock and aftershocks occur nearly simultaneously; at this time, it is permissible to approximately superimpose the damage caused by aftershocks onto the damage caused by the mainshock. The structure exhibits a functional loss index of 0.1632 after seismic damage. Figure 10b-d 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 10e,f demonstrate a high degree of consistency. (3) Rarely occurred earthquake  
As depicted in Figure 11, the structure exhibits a functional loss index of 0.3728 after being subjected to earthquake-induced damage, with a time duration of 274 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 9. It can be observed that, under the action of rare earthquakes, the resilience index tends to converge to a value of 0.8016 when a random number of 10,000 is selected. \begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{Simulation times} & Principal earthquake & \multicolumn{3}{c}{Mainshock-aftershock} \\ \cline{2-6}  & & 100 & 1000 & 10,000 & 100,000 \\ \hline Resilience index & 0.9863 & 0.9819 & 0.9818 & 0.9819 & 0.9819 \\ \hline \end{tabular}
\end{table} 
Table 8: Seismic resilience index under fortified earthquake. \begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{Simulation times} & Principal earthquake & \multicolumn{3}{c}{Mainshock-aftershock} \\ \cline{2-6}  & & 100 & 1000 & 10,000 & 100,000 \\ \hline Resilience index & 0.8600 & 0.8023 & 0.8029 & 0.8016 & 0.8016 \\ \hline \end{tabular}
\end{table} 
Table 9: Resilience index of mainshock-aftershocks under rare earthquake. Figure 9: Functional function curve of the structure under mainshock alone. As illustrated in Figure 12, 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 12a, it can be observed that the mainshock and aftershocks occur nearly simultaneously; at this time, it is permissible to approximately superimpose the damage caused by aftershocks onto the damage caused by the mainshock. The structure exhibits a functional loss index of 0.4903 after seismic damage. Figure 12b-f 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 11: Functional function curve of the structure under mainshock alone. Figure 10: Functional curves of the mainshock-aftershock structure at different recovery times. In conclusion, taking into account economic factors and socio-cultural factors, the comparison of the area enclosed by the functional function and the coordinate axis allows for the following conclusions to be drawn:  
The function curve of the main earthquake and mainshock-aftershocks reveals a relationship between seismic intensity and various factors. As the seismic intensity increases, the slope of the curve decreases, resulting in a smaller enclosed area. This reduction signifies a decrease in the resilience of the system or structure, subsequently weakening its capacity to withstand the earthquake. Conversely, when the structure exhibits greater resilience, it indicates a stronger capacity for recovery, ultimately leading to a reduction in the economic loss incurred due to the earthquake.