5.1. Flexible Lining
Referring to the collected construction tunnels, some engineering cases adopt this approach [
29,
30]. Due to the lack of systematic design theory and calculation methods, the design parameters of flexible joint are usually determined empirically so that the application and development of flexible joint design are restricted.
In this paper, the design strategy of flexible joint for the cross-fault tunnel was investigated with the present numerical model. The tunnel lining is considered to be continuous zones and is separated into several independent sections along the longitudinal direction; materials with low strength are used to fill the gaps between the lining sections [
31]. The lining structure with flexible joint is shown in
Figure 9. The main parameters of the tunnel and the surrounding rock mass are shown in
Table 4.
Meanwhile, in order to present the influence of design parameters of the flexible joint more clearly, and to better show the stress response and deformation characteristics of the lining under different design parameters, in this model, the fault displacement is taken as 1.0 m. The longitudinal displacement profile and stress response under the fault displacement of 1.0 m are shown in
Figure 10 and
Figure 11, respectively. It is obvious that the application of the flexible joint makes the lining structure more flexible. The vertical displacement curve of the lining with flexible joints is not very smooth but slightly stepped. This is because the deformation of the material in the gaps between the lining segments is smaller than the deformation of the lining segments. Thus, the application of the flexible joint is an effective measure to accommodate the displacement of lining caused by fault dislocation. In addition, since the flexible connecting material with low strength can only transmit a small internal stress, the stress value at the flexible joint is small so that the stress loading on the lining structure is reduced [
32].
5.2. 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, 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. At the same time, increasing the width of the flexible joint can reduce the stress loading on the lining. However, it can be seen that there is small difference between these curves, indicating that the width of the flexible joint has little influence on the stress and deformation of the lining.
Strength of flexible joint: This section studies the deformation and stress response of the lining in cases where the length of the lining segment is 1.0 m, the width of the flexible joints is 1.0 m, and the ratios of the strength of the flexible joints to the strength of the lining segments are 1/10, 1/50 and 1/100.
Figure 14a shows the vertical displacement curve with different material strengths of the flexible joint.
Figure 14b,c show the shear stress and the axial stress of the tunnel with different strengths of the flexible joints, respectively. From the comparison of these curves, we can see that the weaker material strength of the flexible joints is beneficial to accommodate the displacement generated under the fault dislocation, reduce the internal stress of the tunnel lining, and increase the tunnel flexibility.
5.3. 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,
Ki (
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.
is the average value of
Ki, 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.
Table 7 lists the sum value
KEi, the average value
, and the range value
RE for the shear stress and the axial stress at different parameters and corresponding levels. In the table,
Ai,
Bi and
Ci (
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
,
and
in line A. It can be seen from the table that
,
and
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
,
and
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 obtained: The optimal combination concerning shear stress reduction is A
1B
3C
3, and the optimal combination for axial stress reduction is A
1B
2C
3.
Since the two optimal combinations are not consistent, the optimal level should be determined according to the primary and secondary relationship of the parameters. In this test, level B3 is the optimal level according to the influence of the shear reduction. While considering the axial stress reduction, the B2 level is the best choice. However, referring to the above results, the width of the flexible joint has less influence on the test index. When the width of the flexible joint is taken as B2, the reduction of the shear stress is 1.82%, which is lower than that of B3, and the reduction of the axial stress is 5.81%, which is higher than that of B3. Moreover, gaps between the lining segments that are too wide are not conducive to the safety of the lining structure.
Therefore, considering the current lining molding technology, the optimal combination of flexible joint is A1B2C3. It is suggested that a reasonable segment length is 6 m, a reasonable width of the flexible joint is 1.0 m, and a reasonable material strength of the flexible joint is 10% of the strength of the lining segments.