## 4 Case Validation  
### Finite Element Simulation  
Finite element simulation software was used to verify the feasibility of the proposed method. The size of the model in Figure 8 is 200 (length) \(\times\) 100 (width) \(\times\) 50 (height), which eliminates the boundary effect. In this paper, the existing tunnel has a diameter of 6.2 m and a tunnel axis that is 16.9 m away from the ground. The mechanical properties of the simulated stratum and soil are shown in Table 1, and the existing tunnel's structural parameters are shown in Table 2. In the uniformly distributed load model, the ground load with a size of 40 m \(\times\) 20 m is 40 MPa. The soil in the finite element model is a layer of common soft soil. The constitutive model of soil strengthened with small strain stiffness is used to simulate the mechanical behavior of soft clay. The failure of the soil is defined by the Moor-Coulomb failure criterion. The equivalent bending stiffness of the tunnel is 7.8 \(\times\) 10\({}^{4}\) MN m\({}^{2}\).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Unit Weight \(\gamma_{s}\)** & **Elastic Modulus** & **Poisson’s Ratio** & **Cohesion \(c\)** & **Friction Angle** \\ (**kN/m\({}^{3}\)**) & \(E_{s}\) (MPa)** & \(\
u\) & **(KPa)** & **\(\varphi\) (\({}^{\circ}\))** \\ \hline
18 & 12 & 0.3 & 5 & 25 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Physical–mechanical parameters of soil.  
Figure 9 compares the results of the proposed analytical method with those of the finite element simulation of the longitudinal displacement of the existing tunnel subjected to surface surcharge. As can be seen from the figure, the analytical solution obtained by using the revised foundation bed coefficient is slightly larger than the finite element simulation result, which may be because the soil is continuous in the Pasternak foundation; this deviation is acceptable in practical engineering. Moreover, the existing tunnel settlement curve obtained from the analytical solution is in good agreement with the simulation result.  
The results of the other three simplified analytical elastic methods show that the prediction results for the method based on the Vesic foundation bed coefficient are clearly  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Extremal** & **Lining** & **Lining** & **Segment** & **Number of** & **Diameter of** & **Length of** & **Bol Elastic** \\
**Diameter** \(D\)** & **Thickness** \(t\) & **Width** \(I_{s}\) & **Hodulus** \(E_{s}\) & **Modulus** \(E_{s}\) & **Longitudinal** & **Joint** \(D_{b}\) & **Joint** \(I_{b}\) (mm) & **Modulus** \(E_{b}\) \\
**(m)** & **(m)** & **(m)** & **(m)** & **(m)** & **(mm)** & **(m)** & **(m)** \\ \hline
6.2 & 0.35 & 1.2 & 3.45 \(\times 10^{4}\) & 17 & 30 & 400 & 20.6 \(\times 10^{5}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Tunnel parameters of Shanghai Metro Line 9.  
Figure 8: Finite element model of the existing tunnel.  
Figure 9: Comparison of the analytical solution and finite element simulation.  
higher than the finite element simulation results. [PERSON]'s and Yu's foundation bed coefficients underestimate the tunnel settlement, which indicates that the tunnel settlement is inversely proportional to the soil foundation bed coefficient. Although the tunnel settlement calculated by the Yu foundation bed coefficient is close to the simulation results, this method is bound to underestimate the prediction results, which poses a risk to the safety of the tunnel structure; therefore, its use is not recommended.