# Influence Analysis of Liquefiable Interlayer on Seismic Response of Underground Station Structure
## 2 Numerical Calculation Method
### Numerical Model Establishment and Parameter Setting  
The Finn model was proposed by [PERSON] et al. [19] to solve the problem of volume strain and the pressure change pattern of soil under cyclic loading based on experiments. This model introduces the pore pressure rise model into the Mohr-Coulomb model, which can simulate the process of accumulation and development of super-pore water pressure in soil under dynamic loading until the point of liquefaction. [PERSON] [20] summarized and analyzed the test data of [PERSON] et al. to simplify the Finn model, and the derived strain increment calculation equation is shown in Formula (1):  
\[\frac{\Delta\varepsilon_{\text{vd}}}{\gamma}=C_{1}\text{exp}(\,-\,C_{2}\frac{ \varepsilon_{\text{vd}}}{\gamma}) \tag{1}\]  
where \(\gamma\) is the shear strain, \(\varepsilon_{\text{vd}}\) is the volume strain, and \(\Delta\varepsilon_{\text{vd}}\) is the volume strain increment. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Case**} & **Top Buried** & \multirow{2}{*}{**Thickness (H)**} & **Relative** & **Position Relative to** \\  & **Depth (\(D\))** & & **Density (\(D_{\text{r}}\))** & **Station Structure** \\ \hline a & 0 m & 5 m & 50\% & top \\ b & 9 m & 5 m & 50\% & middle \\ c & 19.89 m & 5 m & 50\% & bottom \\ d & 9 m & 5 m & 30\% & middle \\ e & 9 m & 5 m & 70\% & middle \\ f & 6 m & 12 m & 50\% & middle \\ g & 2.5 m & 20 m & 50\% & middle \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Distribution of the liquefiable interlayer in each case. Figure 2: Soil layer distribution and monitoring point layout of the site. In Formula (1), \(C_{1}\) and \(C_{2}\) are the constant related to the relative density of the sandy soil and the corrected slamming number, which is calculated empirically using the following formula:  
\[C_{1}=8.7(N_{1})_{60}^{-1.25} \tag{2}\]  
\[C_{2}=\frac{0.4}{C_{1}} \tag{3}\]  
\[D_{\mathrm{r}}=15(N_{1})_{60}^{\frac{1}{2}} \tag{4}\]  
Many scholars [21; 22; 23] have chosen the Finn model to describe the liquefaction behavior of soil and obtained satisfactory data results, which shows that the Finn model is an appropriate tool for investigating the liquefaction principal of soil in this paper. In order to improve computational accuracy in numerical analysis, interface elements are usually added between the soil and the structure in order to more realistically simulate the nonlinear contact phenomena between the structure and the soil. Empirically, the normal stiffness \(k_{\mathrm{n}}\) and shear stiffness \(k_{\mathrm{s}}\), the key parameters of the contact surface interface, can be taken as 10 times the equivalent stiffness of the surrounding \"hardest\" adjacent area [24], as shown in Formula (5). The internal friction angle can be approximated as roughly 0.5 times that of the surrounding soil layer:  
\[k_{\mathrm{n}}=k_{\mathrm{s}}=10\mathrm{max}\bigg{[}\bigg{(}K+\frac{4}{3}G \bigg{)}/\Delta z_{\mathrm{min}}\bigg{]} \tag{5}\]  
where \(K\) is the bulk modulus, \(G\) is the shear modulus, and \(\Delta z_{\mathrm{min}}\) is the minimum size of the connection area in the normal direction of the contact surface. Based on the finite difference software FLAC\({}^{\mathrm{3D}}\), a numerical model of the interaction between the soil layer and the station structure was established. In order to eliminate the influence of the model boundary effect as much as possible, the calculation range of the foundation is taken as 7 times the length of the station structure, being 150 m in the horizontal direction with a 50 m depth below the ground surface in the vertical direction. The elastic constitutive model is used for concrete, and the Mohr-Coulomb strength criterion is followed for site soils. During the seepage analysis, the soil elements were set up as isotropic models, and the station structure elements were set up as impermeable models. The Byrne-simplified Finn pore pressure model is used to simulate the liquefaction behavior of liquifiable interbedded site soils under seismic action due to the rise in pore water pressure.