## 2 Numerical Approach
### The Numerical Model
Each degree of freedom has independent one-dimensional constitutive relationships and mechanical properties. Previous studies showed that the flexural stiffness \(K_{R}\), axial stiffness \(K_{C}\), and shear stiffness \(K_{T}\) of the link element exert a more significant influence on the mechanical properties of the shield tunnel. So, the effects of the three degrees of freedom related to the aforementioned stiffness parameters were mainly considered in the present research, which were set as elastic connections and represented by springs. The remaining three degrees of freedom were set as rigid connections. The specific connection form is shown in Figure 7a, which is a schematic representation of the \begin{table}
\begin{tabular}{c c c c c} \hline
**Lining Type** & \
\begin{tabular}{c} **Concete** \\ **Strength Grade** \\ \end{tabular} & **Thickness/mm** &
\begin{tabular}{c} **Elastic** \\ **Modulus/GPa** \\ \end{tabular} & **Poisson’s Ratio** \\ \hline Segment lining & C50 & 60 & 34.5 & 0.25 \\ \hline Secondary lining & C30 & 30 & 30 & 0.25 \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters of lining. composition of three elastic springs for any link element. The linear yield ideal elastoplastic constitutive relation illustrated in Figure 7b was adopted for springs on the three degrees of freedom [43], and it can be expressed as:  
\[\sigma_{b}=\left\{ \begin{array}{ll}K\text{-}\varepsilon_{b}&(0\leq\varepsilon_ {b}<\varepsilon_{b0})\\ \sigma_{bf}&(\varepsilon_{b0}\leq_{sb}<\varepsilon_{bu})\end{array} \right. \tag{3}\]  
where \(\sigma_{b}\) and \(\varepsilon_{b}\) represent the stress and the strain of the spring, respectively; \(\sigma_{bf}\) is the yield stress of spring; \(\varepsilon_{b0}\) is the strain corresponding to \(\sigma_{bf}\); \(\varepsilon_{bu}\) is the ultimate strain of spring; \(K\) denotes the stiffness of any spring from the three degrees of freedom with elastic connection. The spring stiffness was determined by the empirical values from the study [44], and the yield strength was a macroscopic strength parameter in the actual project, which was determined in conjunction with the findings of the present research and previous experimental studies [21], as shown in Table 3.  
Figure 6: Simulation method of bolts. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Rotational Stiffness** & **Axial Stiffness** & **Shear Stiffness** & **Yield Strength of** \\ \(\text{K}_{R}/\text{(MN}\cdot\text{m}\cdot\text{rad}^{-1})\) & \(\text{K}_{C}/\text{(MN}\cdot\text{m}^{-1})\) & \(\text{K}_{T}/\text{(MN}\cdot\text{m}^{-1})\) & **Spring \(\sigma_{bf}/\text{(MPa)}\)** \\ \hline \(100\) & \(1050\) & \(500\) & \(640\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of bolts between segments. Figure 7: Bolt characteristics. The contact characteristics of the contact surface between the segment lining and the secondary lining and their parameter value exert an important influence on the participation extent of the secondary lining in the structural bearing process. Previous studies have shown [23] that the contact characteristics should include contact and separation in the normal direction as well as adhesion and friction in the tangential direction. Except for the tension and pressure in the normal direction, the contact surface can also transmit a certain shear force in the tangential direction. The contact characteristics in the refined numerical calculation model consist of a linear spring with tensile strength in the normal direction and two linear yield springs in the tangential direction, which were implemented using the interface element, as shown in Figure 8a. Figure 8b shows the shear strength criterion for the interface element, where the compressive strength of the interface element too high so its compressive failure was ignored. The maximum shear stress \(\tau_{max}\) changes with the normal stress \(\sigma_{n}\), as shown in Equation (4):  
\[\tau_{max}= \begin{cases}c_{r}&(\sigma_{n}<-f_{t})\\ c&(-f_{t}\leq\sigma_{n}<0)\\ c+\sigma_{n}\tan\varphi&(0\leq\sigma_{n})\end{cases}