## 3 Finite-Element Model
### Three-Dimensional FE Model  
The initial stress field, pore pressure field, and structural internal force of the 3D calculation model were generated using the initialization method. The final field and structural internal forces of the above plane model were considered as the initial states of the 3D calculation model. In dynamic analysis, both the numerical model and mesh size can affect the calculation speed and accuracy. Specifically, the representative holes 4-6 were \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Steel Corrugated** & **Material Density** & **Equivalent Density** & **Equivalent Thickness** & **Elastic Modulus (GP)** & **Poisson Ratio** \\
**Pipe Type** & \(\rho\) (g/cm\({}^{3}\)) & \(\rho_{1}\) (g/cm\({}^{3}\)) & \(t\) (mm) & & \(v\) \\ \hline
4–9 holes & 7.85 & 1.411 & 70 & 206 & 0.3 \\ Other holes & 7.85 & 1.171 & 65 & 206 & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameters of steel corrugated pipe and liner used in simulation. Figure 6: Distribution of soil stress (unit: Pa). Figure 7: Distribution of pore pressure (unit: Pa). considered along the driving direction, and the width of the model was 1/2 of the road width. The plane grid was expanded into a 3D shape along the road width, as shown in Figure 8. The road is not symmetrically arranged, the front is the sidewalk, and the rear is the concentration of the load. [PERSON] and [PERSON] [27] demonstrated that for accurate wave propagation, the mesh size must be less than 1/8-1/10 of the minimum wavelength \(\lambda_{\text{min}}\) and equal to the formation shear wave velocity \(u\) divided by the maximum input wave frequency \(f\). The vibration frequency measured under vehicle load is primarily concentrated in the range of 3-30 Hz. The maximum mesh size of the model is approximately in the range of 0.17-1.67 m when the shear wave velocity of the foundation is 50-100 m/s. The road surface area was refined by dividing the model grid. The 1.8 m wide carriageway was divided into two 0.9 m wide grids, and seventy-one 0.55 m wide grids were divided in the driving direction; the remaining grids were approximately 1.8 m wide. The displacement in the three directions at the bottom of the model was constrained, the four sides of the model were set as free-field boundaries, and vehicle loads that varied with time were applied in the carriageway. The range of the critical damping ratio (\(\xi\)) of geotechnical materials is typically 2-5%. In this study, a small damping ratio of 0.5% was used to satisfy the calculation requirements. As shown in Figure 9, the design load of the road is a first-class highway, and the vehicle load was considered as on the 10-wheeled vehicle type. The ground area of the front and the rear wheel of the vehicle is 0.2 m \(\times\) 0.3 m and 0.2 m \(\times\) 0.6 m [20], respectively. According to the specifications, the oval wheel ground area in this study was treated as a rectangle. Only a single-row load of 140 kN by the rear wheel was considered. The sinusoidal variation law was selected as the vehicle load model, which can reflect the vehicle load cycle and considers the influence of similar excitation forms with geometric irregularities, as shown in Formula (3). To facilitate the model in applying the vehicle load, the concentrated load _F_(_t_) should be equivalent to an evenly distributed load in areas (1.8 m \(\times\) 0.5 m), and the load should be applied in a moving manner [28; 29]. When the speed is 60 km/h, which is the maximum design speed, 0.033 s are required by the load to move from the current node to the next node, which is a 0.55 m distance between the adjacent two points. The time of load contact with the road surface is 0.012 s (t\({}_{0}\) = L\({}_{0}\)/v and L\({}_{0}\) is 0.2 m of wheel grounding length); thus, a complete load action and calculation period of 0.045 s can be obtained. The time-history curve of a single vehicle load is shown in Figure 10.  
Figure 8: Three-dimensional finite-element model.