## 3 Examples and Discussion
### Structural Normalization and Direct Economic Loss  
The calculation methods for normalizing the direct economic loss of structures under three scenarios, multiple earthquakes, fortified earthquakes, and rare earthquakes, can be described as follows. (1) Normalized economic losses under multiple earthquakes  
Under the action of a single earthquake event and mainshock-aftershocks, the direct economic loss of the structure is shown in Equations (33) and (34):  
\[L_{\text{DE,L}}=5\%\times 0.9215+20\%\times 0.0088+50\%\times 0+97.5\%\times 0 =0.0478 \tag{33}\]  
\[L_{\text{DE,L}}^{\prime}=5\%\times 0.9481+20\%\times 0.0153+50\%\times 0+97.5\% \times 0 =0.0505 \tag{34}\]  
This paper incorporates an indirect economic loss ratio of \(\gamma=0.7\) and computes the summation of direct and indirect economic losses to determine the total normalized economic loss. Under the action of a single earthquake event and mainshock-aftershocks, the total normalized economic loss is  
\[L_{\text{DE}}=(1+\gamma)L_{\text{DE,L}}=1.7\times 0.0478=0.0813 \tag{35}\]  
\[L_{\text{DE}}^{\prime}=(1+\gamma)L_{\text{DE,L}}^{\prime}=1.7\times 0.0505=0.0858 \tag{36}\]  
The normalized economic loss of the structure after the earthquake, under the mainshock-aftershock, is slightly higher than the normalized economic loss under a single earthquake event. The difference \(\Delta L_{\text{DE}}\) between the two is calculated as follows:  
\[\Delta L_{\text{DE}}=L_{\text{DE}}-L_{\text{DE}}^{\prime}=0.0045 \tag{37}\]  
(2) Normalized economic losses under fortification-level earthquakes  
Under the action of a single earthquake event and mainshock-aftershocks, the direct economic loss of the normalized post-earthquake structure is shown in Equations (38) and (39):  
\[L_{\text{DE,L}}=5\%\times 0.9997+20\%\times 0.3552+50\%\times 0.0280+97.5\% \times 0=0.1350 \tag{38}\]  
\[L_{\text{DE,L}}^{\prime}=5\%\times 0.9999+20\%\times 0.4366+50\%\times 0.0446+97.5 \%\times 0.0037=1.6322 \tag{39}\]  
Under the action of a single earthquake event and mainshock-aftershocks, the total normalized economic loss is  
\[L_{\text{DE}}=(1+\gamma)L_{\text{DE,L}}=1.7\times 0.0478=0.2295 \tag{40}\]  
\[L_{\text{DE}}^{\prime}=(1+\gamma)L_{\text{DE,L}}^{\prime}=1.7\times 0.0505=0.2775 \tag{41}\]  
The normalized economic loss of the structure after the earthquake, under the mainshock-aftershock, is slightly higher than the normalized economic loss under a single earthquake event. The difference \(\Delta L_{\text{DE}}\) between the two is calculated as follows:  
\[\Delta L_{\text{DE}}=L_{\text{DE}}-L_{\text{DE}}^{\prime}=0.0480 \tag{42}\]  
(3) Normalized economic loss under rare earthquakes \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Earthquake**} & \multicolumn{2}{c}{**Slight Damage**} & \multicolumn{2}{c}{**Moderate Damage**} & \multicolumn{2}{c}{**Severe Damage**} & \multicolumn{2}{c}{**Complete Damage**} \\ \cline{2-9}  & **Principal** & **Aainshock-** & **Principal** & **Aainshock-** & **Principal** & **Mainshock-** & **Principal** & **Mainshock-** \\  & **Earthquake** & **Aftershock** & **Aftershock** & **Earthquake** & **Aftershock** & **Earthquake** & **Aftershock** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Probabilities of structural disruption at different state levels.