## 2 Modeling Methodology and Conditions  
In this study, the seismic responses of a tunnel shaft during liquefaction were simulated using FLAC3D (Fast Lagrangian Analysis of Continua in 3 Dimensions) [22], which is a three-dimensional numerical simulation tool. The model system was virtually defined based on typical construction conditions of urban railway tunnels in Seoul, South Korea. The model tunnel of an urban railway (prototype) was located in the rock layer. A vertical shaft structure fabricated from reinforced concrete was rigidly installed at the top side of the tunnel body. A liquefiable loose sand layer was located above the rock layer, and the model shaft penetrated both layers vertically from the top side of the tunnel body. The primary dimensions and material properties of the tunnel shaft structure are presented in Table 1.  
The 12 analysis cases were composed of four depths of the base rock (5 m, 10 m, 15 m, and 21 m), i.e., thicknesses of the liquefiable layer, and three amplitudes of the input earthquake motion (500 y, 1000 y, and 2400 y), as summarized in Table 2 and Figure 1.  
A soil constitutive model for liquefiable soil should be able to simulate important dynamic behaviors such as pore water pressure generation and reduction in effective strength during liquefaction. In the present study, the Finn liquefaction model was adopted, which has been verified through comparison with various field observations and model tests in several previous studies [23; 24; 25; 26]. The Finn model correlates the increment of the volumetric strain per loading cycle \(\Delta\varepsilon_{vd}\) with increase in the shear strain amplitude \(\gamma\) during that particular cycle. The Finn model was incorporated into the Mohr-Coulomb plasticity model. For practical purposes, [PERSON] [27] proposed simple equations to determine these parameters as functions of the relative density of sand, as shown in Equations (1) and (2):  
\[\frac{\Delta\varepsilon_{vd}}{\gamma}=C_{1}\exp\left(-C_{2}\left(\frac{ \varepsilon_{vd}}{\gamma}\right)\right) \tag{1}\]  
\[C_{1}=7600(D_{r})^{-2.5},\,C_{2}=\frac{0.4}{C_{1}} \tag{2}\]  
where \(\varepsilon_{vd}\) is the accumulated irrecoverable volumetric strain to the \(N\)th cycle, \(\gamma\) is the shear strain at the \((N+1)\)th cycle, \(\Delta\varepsilon_{vd}\) is the increment of the volume decrease at the \((N+1)\)th cycle, and \(C_{1}\) and \(C_{2}\) are volume change constants. \begin{table}
\begin{tabular}{c c} \hline \hline
**Material Properties** & **Value** \\ \hline Outer diameter (m) & 5 \\ Concrete stiffness (MPa) & 24 \\ Elastic modulus of concrete(MPa) & 26,950 \\ Steel stiffness (MPa) & 300 \\ Inner diameter (m) & 4.6 \\ Elastic modulus of steel(MPa) & 200,000 \\ Unit weight of reinforced concrete (kN/m\({}^{3}\)) & 25 \\ Poisson’s ratio & 0.33 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of the tunnel shaft. \begin{table}
\begin{tabular}{c c} \hline \hline  & **Case Number** \\ \hline  & Depth of the base rock (m) & Return period of the input earthquake motion (y) \\ \hline
1 & 500 \\ \hline
2 & 5 m & 1000 \\ \hline
3 & 2400 \\ \hline
4 & 500 \\ \hline
5 & 10 m & 1000 \\ \hline
6 & 2400 \\ \hline
7 & 500 \\ \hline
8 & 15 m & 1000 \\ \hline
9 & 2400 \\ \hline
10 & 500 \\ \hline
11 & 1000 \\ \hline
12 & 2400 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: List of analysis cases considered in this study. The nonlinearity of soil significantly influences the dynamic behavior of the shaft. In this study, the hysteretic damping model was applied to consider both the nonlinear variations in the shear modulus and damping ratio and the energy dissipation. This model presents the \(G/G_{max}\) curve versus the logarithm of the cyclic strain and the tangent modulus as shown in Equations (3)-(5).