## 3 FEA Model and Validation
### Model Validation  
The lining stress could be affected significantly by adjacent TBM-driven processes. In order to verify the FEA model in which the lining stress variation is a major concern, the computational results are compared with the field monitoring data. The on-site bending moment can be calculated by rebar stress which has been recorded by the field monitoring program in this study.  
The stress variation of rebar has been expressed by Equation (1), and the rebar elastic modulus is \(2\times 10^{5}\) MPa. Thus, rebar strain \(\varepsilon_{s}\) can be obtained via Equation (14):  
\[\varepsilon_{s}=\frac{\sigma_{s}}{E_{s}} \tag{16}\]  
The concrete stress can be expressed as:  
\[\sigma_{c}=\frac{E_{c}}{E_{s}}\times\sigma_{s} \tag{17}\]  
Assuming that each lining can be regarded as a pure bending beam and the concrete strain is consistent with that of rebar, the bending moment of each lining can be obtained as follows.  
\[M=\frac{E_{c}\cdot\sigma_{s}\cdot I_{z}}{E_{s}\cdot y} \tag{18}\]  
\[\sigma_{c}=\frac{M\cdot y}{I_{z}} \tag{19}\]  
\[I_{z}=\frac{b\cdot h^{3}}{12} \tag{20}\]  
where \(M\), \(y\), \(I_{z}\). are the section bending moment, maximum distance from lining edge to neutral axis, and the moment of inertia, respectively.  
The FEA model parameters are shown in Table 2[34].  
Available monitoring results and FEA modelling results of WR39 and WR40 (see Figure 2) are shown in Figure 7 in which L denotes the distance between the TBM tail and studied lining.  
Figure 7 shows the comparison of the bending moment between the field monitoring and the FEA results. In Figure 7, positive and negative values indicate tension and compression stress state at the lining external surface, respectively. As shown in Figure 7a,b, the field measurement and FEA results show acceptable agreement with each other at WR39. As illustrated in Figure 7, negative and positive bending moment can be observed, respectively, at the tunnel crown/bottom and sidewall, indicating a \"horizontal oval\" deformation pattern of the lining during the WB tunneling process. However, for the bending moment at WR40 after 2.4 m of the tail passing-by (Figure 7d), the monitoring data at G3, G7, and G6 show a difference with the corresponding FEA results. This is probably caused by insufficient grouting pressure which might result in excessive outward displacement of segment. However, this study focuses on the ideal situation under which the tail-void is completely filled with grout. Therefore, the FEA model proposed could be employed to provide reliable prediction for the variation of the tunnel bending moment during the TBM-driven process in this study.  
\begin{table}
\begin{tabular}{c c c c c} \hline \(\mathbf{K_{0}}\) & **Grouting Pressure at Tunnel Crown** & **Grouting Pressure Gradient (kN/m)** & **Ground Contraction** & \(\delta\)**(mm)** \\ \hline
0.4 & Earth pressure & 7 & Non-uniform & 6 \\ \hline \end{tabular} Note: \(K_{0}\) denotes the lateral earth pressure coefficient used for determining the supporting pressure at the tunnel face.  
\end{table}
Table 2: Parameters used in presented FEA model [34].  
Figure 7: _Cont._  
pressure G and supporting pressure S in different working conditions, where \(G=K_{g}\times G_{0}\), \(S=K_{s}\times S_{0}\) (\(G_{0}\) and \(S_{0}\) are the values of Case 0).