## 3 Three-Dimensional Finite Element Analysis Model
### Material Properties  
The modified Cam-clay model was proposed by [PERSON] (1968) at the University of Cambridge, UK, based on the triaxial test of normally consolidated clay and weakly overconsolidated clay. This model can consider the hydrostatic yield characteristics, compressive stiffness, and shear shrinkage of geotechnical materials, which is suitable for normally consolidated soils or weakly overconsolidated soils, as well as strongly overconsolidated soils and other viscous soils. Considering that the soft soil involved in this project is normally consolidated soil, it is appropriate to adopt the modified Cam-clay model to simulate the stress-strain relationship of the soils. The main parameters of the modified Cam-clay model can be obtained by the geological survey report, and the specific conversion formulas are as follows. \[\lambda=\frac{C_{c}}{\ln 10}(\text{INCL}) \tag{2}\]  
Figure 4: Schematic diagram of the convergent displacement method. Figure 5: Schematic diagram of shield tunnel excavation. \[\kappa=\frac{C_{s}}{\ln 10}(\text{INCL}) \tag{3}\] \[M=\frac{6\sin\varphi^{\prime}}{3-\sin\varphi^{\prime}}(\text{CSL}) \tag{4}\]  
where \(\lambda\) is the slope of the initial isotropic normal consolidation curve INCL in the semi-logarithmic coordinate system; \(\kappa\) is the slope of the initial compression rebound curve in the semi-logarithmic coordinate system; \(M\) is the spatial slope of the critical state line CSL; \(C_{c}\) is the compression factor; \(C_{s}\) is the rebound index; \(\varphi^{\prime}\) is the effective internal friction angle of the soil. According to the modified Cam-clay model theory, its yield surface can be written as:  
\[\frac{M^{2}{p^{\prime}}^{2}}{q^{2}+M^{2}{p^{\prime}}^{2}}=\frac{{p^{\prime}}^{ 2}}{{p_{0}}^{2}} \tag{5}\]  
Furthermore, it can be rewritten as:  
\[{p_{0}}^{\prime}=\frac{q^{2}+M^{2}{p^{\prime}}^{2}}{M^{2}{p^{\prime}}}=\frac{ q^{2}}{M^{2}{p^{\prime}}}+{p^{\prime}} \tag{6}\]  
where \({p^{\prime}}\) and \(q\) are the average effective stress and deviator stress, respectively; \({p_{0}}^{\prime}\) is the initial value of \({p^{\prime}}\). According to the modified Cam-clay model theory, the void ratio \(e_{0}\) of the soil in the consolidated state is:  
\[ \begin{split} e_{0}&=e_{1}-\lambda\ln{p_{0}}^{ \prime}+\kappa\ln{\frac{p_{0}}{p^{\prime}}}\\ &=\,e_{1}-\lambda\ln\Bigl{(}\frac{q^{2}}{M^{2}{p^{\prime}}}+{p^{ \prime}}\Bigr{)}+\kappa\ln\Bigl{(}\frac{q^{2}}{M^{2}{p^{\prime}}}+1\Bigr{)} \end{split}  \tag{7}\]  
where \(e_{1}\) is the intercept of the normal consolidation curve INCL. In the ABAQUS/standard, if the soil uses the Gravity type to add gravity load, the material density should use the dry density of the soil. Therefore, when the saturated density of soil is known, the dry material density \(\rho_{d}\) is:  
\[\rho_{d}=\rho_{sat}-\rho_{w}\frac{e_{0}}{1+e_{0}} \tag{8}\]  
where \(\rho_{sat}\) is the saturated density of the soil; \(\rho_{w}\) is the density of water. The soil properties of at monitoring section shown in Table 1 were mainly obtained from triaxial tests and penetration testing, and the corresponding parameters of the modified Cam-clay model are shown in Table 2: \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Stratum** & \(\rho_{sat}\) **(g/cm\({}^{3}\))** & \(\varphi^{{}^{\prime}}\) **(\({}^{\prime}\))** & \(\
u\) & \(k\) **(m/s)** & \(C_{c}\) & \(C_{s}\) \\ \hline Silty clay\(\textcircled{1}_{1}\) & 1.72 & 20 & 0.35 & 5.5\(\times 10^{-7}\) & 0.245 & 0.038 \\ Mucky-silty clay\(\textcircled{2}_{\textcircled{3}}\) & 1.82 & 22.3 & 0.45 & 6.5\(\times 10^{-7}\) & 0.299 & 0.042 \\ Silty clay\(\textcircled{1}_{1}\) & 1.82 & 26.1 & 0.3 & 7.5\(\times 10^{-7}\) & 0.286 & 0.037 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the soil layer.