## 2 Engineering Background
### Establishment of Numerical Model
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Model Parameters** & **Plain Fill** & **Clay** & **Medium Sand** & **Gravel Sand** & **Moderately Weathered** & **Slightly Weathered** \\ \hline \begin{tabular}{c} Secant modulus corresponding to \\ partial stress of 0.5 times shear \\ strength under reference stress \\ \end{tabular} & \(E_{50}^{ref}\) (MPa) & 8 & 6 & 17 & 24 & 100 & 120 \\ \hline \begin{tabular}{c} Unloading and reloading \\ modulus under \\ \end{tabular} & \(E_{ur}^{ref}\) (MPa) & 24 & 18 & 51 & 72 & 300 & 360 \\ \hline \begin{tabular}{c} Gravity of unsaturated soil \\ \end{tabular} & \begin{tabular}{c} \(\gamma_{\text{unsat}}\) \\ (\(\text{kN}/\text{m}^{3}\)) \\ \end{tabular} & 18.4 & 17.2 & 19 & 20 & 19.7 & 20.9 \\ \hline \begin{tabular}{c} Gravity of saturated soil \\ \end{tabular} & \(\gamma_{\text{sat}}\) (\(\text{kN}/\text{m}^{3}\)) & 20 & 20.3 & 20 & 20 & 23.3 & 24.5 \\ \hline \
\begin{tabular}{c} Poisson’s ratio \\ \end{tabular} & \(\
u\) & 0.3 & 0.46 & 0.28 & 0.25 & 0.22 & 0.20 \\ \hline \begin{tabular}{c} Cohesion \\ \end{tabular} & \(\text{c}\) (kPa) & 28 & 25 & 0 & 0 & 35 & 100 \\ \hline \begin{tabular}{c} Friction angle \\ \end{tabular} & \(\varphi\) (\({}^{\circ}\)) & 19 & 21 & 28 & 30 & 33 & 35 \\ \hline \begin{tabular}{c} Power exponent of stiffness stress \\ level correlation \\ \end{tabular} & m & 0.6 & 0.8 & 0.5 & 0.5 & 0.5 & 0.5 \\ \hline \begin{tabular}{c} Shear strain corresponding to the \\ secant shear modulus attenuation of \\ 0.7 times of the initial shear modulus \\ \end{tabular} & \(\gamma_{0.7}\) (\(10^{-4}\)) & 2.1 & 2.5 & 2.7 & 1.5 & 1.2 & 1.0 \\ \hline \
\begin{tabular}{c} Reference initial shear modulus of \\ small strain \\ \end{tabular} & \(G_{0}^{ref}\) (MPa) & 65 & 74 & 85 & 120 & 500 & 750 \\ \hline
\begin{tabular}{c} Damage ratio \\ \end{tabular} & \(\text{R}_{\text{f}}\) & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Physical and mechanical parameters of the soil layer. Figure 7: PLAXIS 3D finite element model. The tunnel face support stress P is subject to the stable earth pressure of the heading face; its empirical formula [36] is shown in Equation (1). \[P\ =\ K_{0}\gamma^{\prime}H+p_{w}\pm 20 \tag{1}\]  
where \(K_{0}\) is the coefficient of static earth pressure, \(\gamma^{\prime}\) is the effective weight of soil, \(H\) is the overburden thickness of tunnel vault, \(p_{w}\) is water pressure, and 20 kPa is the fluctuating pressure. The calculation shows that the support force at the vault face is 148 kPa, and the pressure gradient is determined according to the gradient setting in the actual shield tunnelling process, which is 7 kPa/m. Through relevant literature research and field data analysis [37], the grouting pressure at the shield tail was found to be 1.2 times the supporting force at the vault face. Therefore, the grouting pressure P\({}_{0}\) at the vault is 177.6 kPa, and the pressure gradient is 16.7 kPa/m. The simulation process is as follows: (1) generation of initial in-situ stress field; (2) loading piles, including activated pile caps and Embedded Pile structural elements; (3) displacement of the previous stage of the model is cleared to 0, and the grouting reinforcement area is activated at the same time; (4) activation the installed lining, shield machine and segments; (5) for the first ring of shield excavation, activate the tunnelling parameters, including heading face force, jack thrust and grouting pressure; (6) in the second ring of shield excavation, freeze the tunnelling parameters of the previous stage and activation the tunnelling parameters of the new stage and segments of the previous stage; (7) by analogy, freeze the tunnelling parameters of the previous stage and activate the tunnelling parameters of the new stage. In the numerical simulation process, the cutter head of shield machine at the first ring of excavation is 22 m away from the pile foundation to ensure that the excavation can be carried out before the shield starts to affect the pile foundation.