## 2 Modeling Methodology and Conditions
\[M_{t}=s^{2}(3-2s)-\frac{6s(1-s)}{L_{2}-L_{1}}\log_{10}\exp \tag{3}\]  
\[\mathrm{L}=\log_{10}(\gamma) \tag{4}\]  
\[\mathrm{s}=\frac{L_{2}-L}{L_{2}-L_{1}} \tag{5}\]  
where \(M_{t}\) is the tangent modulus, \(L\) is the logarithmic strain, and \(L_{1}\) and \(L_{2}\) and are the extreme values of \(L\). \(L_{1}\) and \(L_{2}\) represent the degradation rate and starting point of the degradation of \(G/G_{max}\) curve, respectively. \(L_{1}\) and \(L_{2}\) for the model soil were set as \(-3.65\) and \(0.5\), respectively, as proposed by [PERSON] and [PERSON] [28]. The maximum shear modulus of the soil, \(G_{max}\), was calculated using Equation (6), as proposed by [PERSON] and [PERSON] [29]. \[G_{max}=AF(e)(OCR)^{k}P_{a}^{1-n}\big{(}\sigma^{\prime}{}_{m}\big{)}^{n} \tag{6}\]  
where \(F(e)=1/(0.3+0.7e^{2})\), in which \(e\) is the void ratio; \(\sigma^{\prime}{}_{m}\) is the mean principal effective stress; \(P_{a}\) is the atmospheric pressure; \(k\) is the overconsolidation ratio exponent (which is equal to 0 in sand); and \(A\) and \(n\) are empirical coefficients, which were determined to have values of 247.73 and 0.567, respectively, for model soil [28]. An interface element that allows both slippage and separation between the soil and the shaft structure under strong earthquake loading was applied to simulate the dynamic  
Figure 1: Schematic views of the analysis cases. soil-structure interaction. The interface model attaches normal and shear springs with equivalent spring stiffness between the soil and shaft, as shown in Equation (7) [1]. In Equation (7), \(G\) reflects the nonlinear reduction in the soil shear stiffness; therefore, the nonlinear behavior at the interface can also be captured in the proposed model. \[K_{E}=max\bigg{[}\frac{K+(4/3)G}{\Delta z_{min}}\bigg{]} \tag{7}\]  
where \(K_{E}\) is the equivalent stiffness of the interface spring; \(K\) and \(G\) are the bulk and shear moduli of the soil, respectively; and \(\Delta z_{min}\) is the smallest width of an adjoining zone in the normal direction. One of the important points for modeling soil-structure systems under dynamic conditions is the proper simulation of the free-field boundary. Improper boundary conditions in dynamic simulations can cause the reflection of outward propagating waves back into the model and do not allow for the necessary energy radiation. Therefore, in this study, the free-field boundary model implemented in FLAC3D was used along the edge of the model. The free-field boundary is used to enforce the free-field motion such that the boundaries retain their non-reflecting properties and outward waves originating from the structure are properly absorbed. The input properties of the model soil used in the numerical simulation were determined based on typical values for Jumunjin sand, which is a Korean standard sand, as summarized in Tables 3 and 4. The model soil was defined as liquefiable loose sand and was completely saturated in the model. Figure 2 shows a representative modeling mesh of the target system (Cases 10, 11, and 12) used to calculate seismic responses of the tunnel shaft in this study. In this figure, only half of the system was modeled for efficiency of the analysis; the entire system can be obtained by reflection in the simulation tool. The geometry of the tunnel and rock layer was omitted in the analysis model and the input earthquake motion was directly induced at the base of the model because the shaft and tunnel in the rock layer were regarded as fully fixed. The shaft structure was modeled using a cylinder-type solid element with an elastic model. Dynamic analysis was then carried out based on a three-dimensional finite difference approach. The analysis proceeded via the following steps according to the construction and loading phases: (1) static equilibrium of the soil media, (2) construction of the tunnel shaft and attainment of static equilibrium of the overall system, and (3) dynamic analysis.