## 4 Structural Integrity of the Shaft under Liquefaction  
The allowable bending moment of the shaft was calculated using a sectional analysis in Response2000 [30]. The input geometric properties are summarized in Table 6. Based on the sectional analysis results, the cracking moment and maximum allowable moment were determined as 6438 kN\(\cdot\)m and 10,467 kN\(\cdot\)m, respectively. To evaluate the earthquake risk assessment for a tunnel shaft, the maximum moment value obtained from the numerical analysis was compared with the calculated cracking moment. The damage state of the underground structure could be classified into three states: minor/slight, moderate, and extensive. [PERSON] and [PERSON] [31] suggested a damage index applying the failure moment (Md) and measuring moment. In addition, a damage state using the ratio of the measuring moment to the yielding moment (My) was developed by [PERSON] et al. [32]. When the damage state was determined using the Md value, the predicted damage state was significantly more conservative than the actual seismic performance. Therefore, in this study, My was applied to determine the damage state and consider the performance uncertainty of a reinforced concrete structure [32]. The My value was referred to in the cracking moment from the sectional analysis. The damage states defined in previous research are summarized in Table 7. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Gross Conc.** & **Trans (n = 7.54)** \\ \hline Area (mm\({}^{2}\)) & \(3.07\times 10^{6}\) & \(3.12\times 10^{6}\) \\ Inertia (mm\({}^{4}\)) & \(8.9\times 10^{12}\) & \(9.1\times 10^{12}\) \\ yt (mm) & 2500 & 2500 \\ yb (mm) & 2500 & 2500 \\ st (mm\({}^{3}\)) & \(3.57\times 10^{12}\) & \(3.63\times 10^{12}\) \\ Sb (mm\({}^{3}\)) & \(3.57\times 10^{12}\) & \(3.63\times 10^{12}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Material properties of the model concrete shaft. Figure 9: Lateral residual displacement. The maximum moment values from the numerical analysis are summarized in Table 8. In addition, based on the numerical analysis results, damage index values were calculated for each case using the damage states developed by [PERSON] et al. [29]. Figure 10 shows the damage index results for each case. The maximum observed moment value was lower than the cracking moment for liquefiable deposit depths of 5 m and 10 m, and no damage states were expected. However, the shaft structure with a liquefiable layer of 15 m could experience slight damage under earthquakes with return periods of 500 and 1000 y. In addition, a moderate damage state could be produced under an earthquake having a return period of 2400 y. In the case of the shaft with a 21 m liquefiable layer, the damage index values for all cases were greater than 1.4, indicating that at least moderate damage could occur in the shaft structure with deep liquefiable deposits. In particular, the damage index of Cases 11 and 12 exceeded values of 2.3, which implies that extensive damage could occur under an earthquake with a return period of 1000 y. To evaluate the seismic risk of the shaft structure, a risk assessment framework was constructed based on the damage index analysis results for the bending moment. The rows are the return periods of the earthquake, which denotes the hazard intensity, and the columns are the case numbers, which indicate the thickness of the liquefiable layer. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Damage State** & \
\begin{tabular}{c} **Damage Index, M/Md** \\ **([PERSON] and [PERSON] (2012))** \\ \end{tabular} &
\begin{tabular}{c} **Damage Index M/My** \\ **([PERSON] et al. (2016))** \\ \end{tabular} \\ \hline None & DI \(<\) 1.0 & DI \(<\) 1.0 \\ Minor/slight & 1.0 \(<\) DI \(<\) 1.5 & 1.0 \(<\) DI \(<\) 1.4 \\ Moderate & 1.5 \(<\) DI \(<\) 2.5 & 1.4 \(<\) DI \(<\) 2.3 \\ Extensive & 2.5 \(<\) DI \(<\)3.5 & 2.3 \(<\) DI \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Damage states. Figure 10: Maximum bending moments in the numerical analysis.