## 5 Design Strategy of the Flexible Joint
### Parameter Analysis  
The length of the lining segments and the width and the strength of the flexible joints have a great influence on the stress and deformation characteristics of the lining. In previous studies, however, the main design parameters of flexible joint have been determined based on engineering experience, \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Name** & **Density/(kg-m-3)** & \begin{tabular}{c} **Young’s** \\ **Modulus/(GPa)** \\ \end{tabular} & \begin{tabular}{c} **Poisson’s Ratio** \\ **Poisson’s Ratio** \\ \end{tabular} & \
\begin{tabular}{c} **Internal Friction** \\ **Angle/(v)** \\ \end{tabular} &
\begin{tabular}{c} **Cohesion/(MPa)** \\ \end{tabular} \\ \hline Surrounding rock mass & 2200 & 6.7 & 0.25 & 40 & 0.8 \\ Lining segment & 2500 & 30.0 & 0.25 & - & - \\ Flexible material & 2000 & Pending & 0.25 & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of materials. Figure 11: Stress response. (**a**) Shear stress; (**b**) axial stress. Figure 10: Displacement response (**a**) Comparison of lining with and without flexible joints; (**b**) displacement contour. and the calculation method is not yet mature. Therefore, it is particularly important to determine the length of the lining segment, the width and the material strength of the flexible joint. Based on previous research results, the length of the concrete lining trolleys used in tunnel construction is generally 6-15 m. In this paper, the lengths of the lining segments are taken as 6 m, 10 m, and 15 m, respectively. To investigate the mechanism of the influence of the width of flexible joint, the widths of the flexible joint are taken as 0.5 m, 1.0 m, and 1.5 m, respectively. In addition, the material strength of the flexible joint is determined according to the ratio of the strength of the flexible joint to the strength of the lining segment, which are taken as 1/10, 1/50, and 1/100, respectively. The main design parameters and corresponding levels of flexible joints are shown in Table 5. The parameter study is carried out, keeping the magnitude of the fault dislocation at 1.0 m.  
Segment length: The deformation and stress response of the lining are studied in the cases where the width of the flexible joint is 1.0 m, the ratio of the strength of flexible joints to the strength of lining segments is 1/100, and the lengths of the lining segments are 6 m, 10 m and 15 m, respectively. Figure 12a shows the vertical displacement curve of tunnel with different segment lengths. Figure 12a,b show the stress response of the tunnel lining. It is obvious that the displacement of the lining increases with the decrease of the segment length. The lining structure will be more flexible to accommodate the displacement caused by fault dislocation with the smaller segment length. In addition, the stress of the tunnel lining is concentrated within a limited range of 30 m along the tunnel axis on both sides of the fault plane. The peak stress decreases with the decrease of the length of the lining segments. Therefore, the following conclusion is verified: The smaller segment length of the tunnel lining is beneficial to absorb the displacement generated under the fault dislocation, and the internal stress of the lining can be reduced to ensure the stability of the lining structure. Width of flexible joint: The deformation and stress response of the lining are studied in cases where the length of the lining segment is 1.0 m, the ratio of the strength of the flexible joints to the strength of the lining segments is 1/100, and the widths of the flexible joints are 0.5 m, 1.0 m and 1.5 m, respectively. The vertical displacement curves with different widths of flexible joint are shown in Figure 13a. Figure 13b,c show the shear stress and the axial stress of the tunnel with different widths of flexible joints. It is found that the vertical displacement of the lining increases with the increase of the width of the flexible joint.