# Deformation and Stress Analysis of Pile-Supported Immersed Tunnels under Seismic Loads
## 2 Methodology
### Numerical Model
#### 2.2.3 Viscoelastic Boundary and Seismic Loads  
As shown in Figure 5, the viscoelastic boundary is actually a continuous distribution of springs and dampers around the perimeter and bottom of the model, except for the free top surface. The spring stiffness and damping coefficients of the nodes are the product of the linear stiffness and damping and the control area of the nodes. The mechanical parameters of the boundary springs and dampers are determined by the mechanical properties of the surrounding rock. The stiffness \(K\) and damping coefficient \(C\) of the spring are shown in Equations (1) and (2). \[K_{\rm bT}=A_{\rm b}\alpha_{\rm T}\frac{G_{\rm s}}{R}\quad C_{\rm bT}=A_{\rm b} \rho C_{\rm S} \tag{1}\]  
\[K_{\rm bN}=A_{\rm b}\alpha_{\rm N}\frac{G_{\rm s}}{R}\quad C_{\rm bT}=A_{\rm b} \rho C_{\rm S} \tag{2}\]  
where \(G_{\rm s}\) and \(\rho\) are the shear modulus and density of the medium, respectively; \(R\) is the distance from the wave source to the artificial boundary (straight distance from the boundary point of the finite element model to the source of the earthquake. ); \(A_{\rm b}\) is the control area of the node \(b\) on the boundary; \(\alpha_{\rm N}\) and \(\alpha_{\rm T}\) are the correction coefficients for the normal and tangential directions, respectively; and \(C_{\rm S}\) and \(C_{\rm P}\) are the shear and longitudinal wave velocities of the medium and the _Lamey_ constants \(\beta\) and \(\delta\), respectively [40]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & **Marine Clay** & **SCP** & **Immersed** & **Gravel Cushion** & **Backfill** \\ \hline Consolidation index, \(\lambda\) & 0.113 & 0.023 & & & \\ \hline Swelling index, \(K\)r & 0.05058 & 0.0103 & & & \\ \hline Critical state parameter, \(M\) & 0.668 & 1.2 & & & \\ \hline Poisson’s ratio, \(\
u\) & 0.36 & 0.3 & 0.17 & 0.45 & 0.37 \\ \hline Density, \(\rho\) (g/m\({}^{3}\)) & 1.851 & 1.765 & 2.45 & 0.9 & 2 \\ \hline Buoyant density, \(\rho_{\rm w}\) (g/m\({}^{3}\)) & 0.801 & 0.715 & 1.4 & 0.85 & 0.95 \\ \hline Cohesion, \(C\) (KPa) & 7.4 & 0.5 & & & \\ \hline Friction angle \(\varphi\) (\({}^{\circ}\)) & 15 & 30 & & & \\ \hline Young’s modulus, \(E\) (KPa) & 2821.83 & \(E^{*}\) & 30,000 & 20,000 & 150,000 \\ \hline Permeability, \(\kappa\) (m/s) & \(1.8\times 10^{-8}\) & \(3\times 10^{-5}\) & & & \\ \hline Coefficient of lateral pressure & 0.577 & 0.5 & & & \\ \hline Initial void ratio, \(e_{0}\) & 1.059 & 0.72 & & & \\ \hline Degree of saturation & 1 & 1 & & & \\ \hline \hline \end{tabular} Note: \(E^{*}=0.6\sigma_{\rm at}(\sigma_{\rm m}/\sigma_{\rm a})^{\lambda}\), where “\(\sigma_{\rm a}\)” is the stress of atmospheric and “\(\sigma_{\rm m}\)” is the current average principal stress in the soil unit under consideration [42].  
\end{table} 
Table 2: Material constants for modeling. Neural networks and artificial intelligence effectively analyze the sensitivity of various factors in complex engineering [43; 44; 45; 46; 47; 48]. The effect of seismic loads on piled foundations is assumed as the dynamic load transfer process in Euler-Bernoulli beams with Winkler foundations in [PERSON] et al. [49; 50] and [PERSON] and [PERSON] [51]. In this study, the seismic wave input is realized by transforming the earthquake into a wave source problem and then acting on the boundary as a concentrated force. There is an equilibrium relationship between the equivalent force \(F_{\rm b}\), the normal stress \(\sigma_{\rm b}\)\({}^{\prime\prime}\)n and the force \(f(t)\) generated by the spring damper on any boundary point \(b(x_{\rm b},\,y_{\rm b},\,z_{\rm b})\) of the model [PERSON] et al. [52]. The conversion of seismic waves into equivalent seismic loads within the control area (\(A_{\rm b}\)) of point \(b\) on an artificial viscoelastic boundary as in [PERSON] et al. [39]. The applied equivalent load is shown in Equation (3).