## 5 Design Strategy of the Flexible Joint
### Orthogonal Array Test  
The orthogonal array test technique was utilized to investigate the magnitude of influence of the main design parameters of the flexible joint on the lining stress. Furthermore, the optimal combination for flexible joint design was presented by range analysis. Referring to the design parameters and the corresponding levels in Table 5, the orthogonal array test was carried out with an L9(3\({}^{4}\)) orthogonal array table, and the last one was listed as a blank column [33; 34]. A total of nine tests were carried out. The distribution curves of the shear stress and axial stress are shown in Figure 15, which can be compared with the stress response of the lining without flexible joints in Figure 11. It can be seen that under the fault dislocation of 1.0 m, the shear stress and axial stress distribution of the lining with flexible joints are basically the same as the distribution of the lining without flexible joints. Additionally, the application of the flexible joint can greatly reduce the internal stress of the tunnel lining. Table 6 shows the reduction of the maximum shear stress and maximum axial stress in the nine tests. It can be seen that different combinations of the main parameters have different effects on the reduction of the internal stress. In this paper, \(K_{i}\) (\(I=1\), 2, 3) denotes the test index of each parameter at each level; here, the test index refers to the reduction of the shear stress and axial stress, expressed in a percentage. \(\overline{K_{i}}\) is the average value of \(K_{i}\), and it can judge the optimal combination of the design parameters and the corresponding levels. R stands for the difference between each level of the same parameter, which is calculated by Equation (1). It can reflect the primary and secondary relationship of the parameters. \[\mathrm{R}=\mathrm{max}\overline{K_{1}},\overline{K_{2}},\overline{K_{3}}- \mathrm{min}\overline{K_{1}},\overline{K_{2}},\overline{K_{3}} \tag{1}\]  
Table 7 lists the sum value \(K_{Ei}\), the average value \(\overline{K_{Ei}}\), and the range value \(R_{E}\) for the shear stress and the axial stress at different parameters and corresponding levels. In the table, \(A_{i}\), \(B_{i}\) and  
Figure 14: Curves of the lining with flexible joints of different joint strengths. (**a**) Vertical displacement; (**b**) shear stress; (**c**) axial stress. (\(i=1,2,3\)) represent the segment length, the flexible joint width, and the material strength of flexible joint, respectively. For example, the influence of the different segment lengths on the test index is recorded by \(\overline{K}_{1}\), \(\overline{K}_{2}\) and \(\overline{K}_{3}\) in line A. It can be seen from the table that \(\overline{K}_{1}\), \(\overline{K}_{2}\) and \(\overline{K}_{3}\) are obviously different, which shows that the average value reflects the influence of the same parameter and different levels on the test index. Similarly, the influence of the flexible joint width and the flexible joint strength on the test index will also be manifested by \(\overline{K}_{1}\), \(\overline{K}_{2}\) and \(\overline{K}_{3}\) in lines B and C. In the range analysis, the larger the test index value is, the greater the reduction of the lining stress will be, and the safer the lining structure will be. The tendency chart, which reflects the influence level of each parameter, is shown in Figure 16 based on Table 7. With it, the optimal combination of the design parameters for the flexible joint is