# Influence Analysis of Liquefiable Interlayer on Seismic Response of Underground Station Structure
## 2 Numerical Calculation Method
### Material Damping Selection  
Considering the calculation accuracy and time cost, we used local damping to reflect the nonlinear characteristics of soil materials and the energy loss of the soil-station structure interaction model in the dynamic analysis process. According to our engineering experience and previous research [25; 26], the critical damping ratio of rock and soil materials was taken as 10%, and the critical damping ratio of reinforced concrete materials was taken as 5%.  
For the sandy soil interlayer, the hysteresis effect due to soil liquefaction needs to be considered further in the dynamic calculation process. Since hysteresis damping can be employed simultaneously with other types of damping and does not affect the calculation time step, hysteresis damping was further applied to the sandy soil interlayer to characterize its hysteresis properties. Among the various models, the Hardin-Dmevich model requires only one parameter for the reference strain, and the fitted equation for the normalized cut-line modulus (\(M_{\mathrm{s}}\)) is shown in Formula (6):  
\[M_{\mathrm{s}}=\frac{G}{G_{\mathrm{max}}}=\frac{1}{1+\gamma/\gamma_{\mathrm{ref}}} \tag{6}\]  
For sandy soil materials, the reference strain \(\gamma_{\mathrm{ref}}\) is generally taken as 0.06. The modulus decay curves derived from the hysteresis damping model selected for this paper were compared with the experimental data of [PERSON] et al. [27], and the results are shown in  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{5}{c}{**Soil Material Parameters**} & \multicolumn{5}{c}{**Structure and Interface Parameters**} \\ \hline \multicolumn{1}{c}{\multirow{2}{*}{**Clay**}} & \multicolumn{2}{c}{**Sand**} & \multicolumn{2}{c}{**Wall and**} & \multicolumn{1}{c}{**Column**} \\ \hline \(D_{\mathrm{r}}\) (\%) & & 30 & 50 & 70 & material grade & C40 & C40 \\ \(G\) (Pa) & \(1.65\times 10^{7}\) & \(4.30\times 10^{6}\) & \(5.92\times 10^{6}\) & \(7.55\times 10^{6}\) & \(G\) (Pa) & \(1.35\times 10^{10}\) & \(1.36\times 10^{9}\) \\ \(K\) (Pa) & \(5.36\times 10^{7}\) & \(1.51\times 10^{7}\) & \(1.65\times 10^{7}\) & \(2.73\times 10^{7}\) & \(K\) (Pa) & \(1.81\times 10^{10}\) & \(2.18\times 10^{9}\) \\ \(\varphi\) (\({}^{\circ}\)) & \(15.4\) & 29 & 32 & 35 & \(\rho\) (kg/m\({}^{3}\)) & \(2500\) & \(302\) \\ \(\rho_{\mathrm{d}}\) (kg/m\({}^{3}\)) & \(1450\) & 1400 & 1460 & 1520 & \(k_{\mathrm{s}}\) (Pa/m) & \(1\times 10^{9}\) & \\ \(n\) & \(0.45\) & 0.46 & 0.42 & 0.38 & \(k_{\mathrm{n}}\) (Pa/m) & \(1\times 10^{9}\) & \\ \(k\) (cm/s) & \(1\times 10^{-6}\) & \(1\times 10^{-2}\) & \(7\times 10^{-3}\) & \(5.5\times 10^{-3}\) & \(\delta\) (\({}^{\circ}\)) & \(15\) & \\ \(c\) (Pa) & \(3\times 10^{4}\) & 0 & 0 & 0 & & & \\ \(C_{1}\) & & 0.77 & 0.21 & 0.09 & & & \\ \(C_{2}\) & & 0.26 & 0.93 & 2.16 & & & \\ \hline \hline \end{tabular}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{5}{c}{\(D_{\mathrm{r}}\): relative density of sandy soil; \(G\): shear modulus; \(K\): bulk modulus; \(\varphi\): friction angle of the soil; \(\rho_{\mathrm{d}}\): dry density; \(n\): porosity; \(k\): permeability; \(c\): cohesion; \(C_{1}\) and \(C_{2}\): liquefaction parameters; \(\rho\):density; \(k_{\mathrm{s}}\) and \(k_{\mathrm{n}}\): shear and normal interface stiffness; \(\delta\): friction angle of the interface surface.} \\ \end{tabular}
\end{table}
Table 2: Material parameters used in numerical analysis.  
Figure 3: Schematic diagram of numerical model meshing.  
Figure 4. It can be seen that the modulus decay curves fitted using the Hardin-Drnevich model are closer to the measured data.