## 3 Examples and Discussion
### Effect of the Mainshock-Aftershocks on the Seismic Resilience  
#### 3.4.1 Recovery Curve Parameter Calculation  
The recovery curve parameters in this section are calculated based on the assumption that the mainshock-aftershock time interval is 0. Based on the three recovery function Equations (12)-(14), the recovery curves for the three recovery models were calculated considering multiple earthquakes, fortified earthquakes, and rare earthquakes.  
Taking the recovery model of the mainshock and mainshock-aftershocks as an example, when the time interval between the mainshock and mainshock-aftershocks occurs, the structure experiences damage and subsequent restoration to restore its integrity to 99.5% of its pre-failure state in order to maintain its basic functionality. According to Equation (3) and Table 1, the functional function can be expressed as:  
\[\begin{cases}Q(t)=Q(t_{\text{OE}})=1-L_{\text{DE}}&t=t_{\text{OE}}\\ Q(t)=Q(t_{\text{OE}}+T_{\text{RE}})=0.995&t=t_{\text{OE}}+T_{\text{RE}}\end{cases} \tag{48}\]  
Substitute Equation (48) into Equation (13).  
\[\left\{\begin{array}{ll}1-L_{\text{DE}}=a\\ 0.995=\frac{a}{b}\times(1+csb)\end{array}\right. \tag{49}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Normalized Economic Loss index**} & \multirow{2}{*}{**Frequently Occurred Earthquake**} & \multicolumn{3}{c}{**Fortified Earthquake**} & \multicolumn{3}{c}{**Rarely Occurred**} \\ \cline{2-7}  & & **Principal** & **Mainshock-** & **Principal** & **Mainshock-** & **Earthquake** & **Mainshock-** \\  & **Earthquake** & **Aftershock** & **Earthquake** & **Aftershock** & **Earthquake** & **Aftershock** \\ \hline Direct economic losses & 0.0478 & 0.0505 & 0.1350 & 0.1632 & 0.3728 & 0.4903 \\ Indirect economic loss & 0.0335 & 0.0353 & 0.0945 & 0.1143 & 0.2610 & 0.3432 \\ Overall loss & 0.0813 & 0.0858 & 0.2295 & 0.2775 & 0.6338 & 0.8335 \\ Mainshock and mainshock-aftershock loss difference & 0.0045 & & 0.0480 & & 0.1997 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Results of normalized economic loss index calculation for structures under different earthquakes.  
In Equation (49), considering Table 3, the total economic loss undergoes normalization, and the parameters \(a=1\) and \(b=0.8405\) are calculated under the sole action of the main earthquake. The trigonometric recovery function of the structure under the condition of multiple earthquakes can be expressed as:  
\[f_{rec}(t)=\frac{1}{2}\times\left[1+cos\frac{2.6404t}{T_{\rm RE}}\right] \tag{50}\]  
Based on the normalized economic loss index of the mainshock-aftershock in Table 3, parameters \(a=1\) and \(b=2.6539\) are calculated using Equation (50). The recovery function of the structure under the action of the mainshock-aftershock can be expressed as:  
\[f^{\prime}_{rec}(t)=\frac{1}{2}\times\left[1+cos\frac{2.6539t}{T_{\rm RE}}\right] \tag{51}\]  
\[f^{\prime\prime}_{rec}(t)=\frac{1}{2}\times\left[1+cos\frac{2.4839t}{T_{\rm RE }}  
Notably, larger single earthquakes are associated with smaller areas enclosed by the functional curve and the \(X\)-axis within the recovery time range of the recovery curve. Conversely, a smaller angular velocity circle frequency corresponds to a larger enclosed area;
3. In the exponential function recovery model, the parameter indicates an exponential, which decreases as the earthquake intensity increases; the slope of the aftershock recovery function decreases slightly compared to that of a single earthquake event. Moreover, a higher index corresponds to a larger area enclosed by the functional curve and the \(X\)-axis within the recovery time range, while a lower index indicates a smaller enclosed area.