## 4 Refined Numerical Simulation of Shield Tunneling
### Simulation of Interaction between Shield Skin and Soil  
Due to the need for shield rectification and resistance reduction, the diameter of the cutter head is usually larger than the diameter of the shield tail, and the shield shell is a conical surface with a certain slope [20]. The soil around the shield shell will create  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Layer**} & **Modulus of** & **Poisson’s** & **Cohesion** & **Internal** & **Bulk** & **Void** & **Permeability** \\  & **Elasticity (MPa)** & **Ratio** & **(kPa)** & **Argle (\({}^{\circ}\))** & **Density** & **Ratio** & **Coefficient** \\  & & & & & **Angle (\({}^{\circ}\))** & **(kN/m\({}^{3}\))** & **Ratio** & **(cm\(\cdot\)s\({}^{-1}\))** \\ \hline Upper silty clay & 5.5 & 0.30 & 15.0 & 21.0 & 19.6 & 0.85 & \(2.0\times 10^{-6}\) \\ Salt & 11.0 & 0.25 & 7.8 & 33.0 & 19.9 & 0.65 & \(4.0\times 10^{-5}\) \\ Lower clay silty & 6.0 & 0.30 & 22.0 & 21.0 & 19.8 & 0.73 & \(2.0\times 10^{-6}\) \\  & & & & & & & \(1.0\times 10^{-6}\) \\ Equivalent zone & 50.0 (early stage) & 0.30 & / & / & 21.0 & 0.85 & (early stage) \\  & 500.0 (later stage) & & & & & & \(1.0\times 10^{-9}\) \\  & & & & & & & (later stage) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Calculation parameters of soil and equivalent zone.  
Figure 6: Numerical model of EPB shield. (**a**) Grid division diagram. (**b**) Local features of the model.  
radial displacement pointing to the tunnel axis as the shield advances. In this paper, the node radial displacement was applied to the shield shell elements, and the surrounding soil elements were moved by the interaction between the lining elements and grid units to simulate the ground deformation caused by the overcut, as shown in Figure 7. The calculation correlation of radial displacement \(\Delta r\) was as follows:  
\[\Delta r=\frac{L_{r}\times(D_{e}-d_{e})}{2L_{e}} \tag{2}\]  
where \(D_{e}\) is the diameter of the cutter head; \(d_{e}\) is the diameter of the shield tail; \(L_{r}\) is the length of the segment; \(L_{e}\) is the length of the shield shell. This paper set \(L_{r}\) = 1.5 m, \(L_{e}\) = 5L\({}_{r}\) = 7.5 m and \(\Delta r\) = 0.001 m.  
The soil around the shield shell will create horizontal displacement pointing in the tunneling direction as the shield advances [29; 30]. In the current study, the constant node velocity along the tunneling direction was applied to the shield shell elements with the interfacial shear coupling spring activated to simulate the friction between the shield skin and the soil. According to the interface characteristics of the lining elements (Figure 8), the simulation method could create friction related to the normal compressive stress of the interface and control the shield posture. As shown in Figure 9, the soil around the shield shell generated horizontal displacement under the action of friction, resulting in ground uplift in front of the excavation face and ground settlement behind the excavation face. The friction between the shield skin and soil could be controlled by changing the friction angle of the interface shear coupling spring.  
Figure 8: Interface behavior of liner structural elements.  
Figure 7: Conical shield shell (180-fold magnified).