## 3 Results Analysis
### Dynamic Response of Tunnel Lining  
Figures 9-11 show the maximum vertical displacement, tensile stress, and compressive stress distributions of tunnel lining at each group of measuring points on the 15 th working condition. Numerical simulation results under other conditions are not displayed in this paper. The results show out that displacement of both the upper and lower tunnels increases significantly due to the action of train loads, the displacement response of these two tunnels decreases as the rock thickness increases, and the dynamic response becomes unfavorable for the tunnel lining at the tridimensional cross region as the rock thickness decreases. For example, the maximum displacement increment of the upper tunnel vault is \(-2.28\), \(-1.76\), \(-1.54\), and \(-1.46\) mm when rock thickness values are 1, 3, 5, and 10 m, respectively. The displacement at the interaction is also larger than that at other places in the longitudinal direction for both tunnels. When the rock thickness is larger than 3 m, the displacement increment of tunnel lining becomes relatively small about 20 m away from the intersection, which is approximately 1.5 times the tunnel width. However, the influence zone would be significantly increased to 50 m away from the intersection if the rock thickness between the two tunnels is 1 m.  
The most affected positions are different for the upper tunnel and the lower one under the action of dynamic train loads. The maximum displacement of the upper tunnel locates at the inverted arch, followed by the side wall and tunnel vault, while for the lower tunnel, the location order by degree of affected extent is the opposite--the maximum displacement of the lower tunnel locates at the arch, followed by the side wall and inverted arch. Figure 8: Initial displacement of tunnel lining. Figure 7: Initial stress distribution: (**a**) Maximum tensile stress, \(\sigma 1\); (**b**) maximum compressive stress, \(\sigma 3\). Figure 10: The maximum tensile stress, \(\sigma\)1; distribution of secondary lining: (**a**) vault of upper tunnel, (**b**) vault of lower tunnel, (**c**) left side wall of upper tunnel, (**d**) left side wall of lower tunnel, (**e**) inverted arch of upper tunnel, (**f**) inverted arch of lower tunnel. Figure 9: Vertical displacement of measuring points under different rock thickness: (**a**) arch of upper tunnel, (**b**) arch of lower tunnel, (**c**) left side wall of upper tunnel, (**d**) left side wall of lower tunnel, (**e**) invert of upper tunnel, (**f**) invert of lower tunnel. Similarly, the stress response of the two tunnels also indicates that thinner thickness of the rock column between the upper tunnel and the lower one is not good for the safety of the tunnel lining. For example, the maximum values of the lower tunnel inverted arch were 619.63, 580.96, 430.74, and 287.27 kPa when the thickness of rock was 1, 3, 5, and 10 m, respectively. It is worth noting that the maximum vertical displacement, tensile stress, and compressive stress of the lining are all at the tunnel intersection, whether the upper tunnel or the lower tunnel. It also shows that the farther the distance from the tunnel intersection, the smaller the corresponding dynamic index. It shows that the dynamic response of the tunnel intersection is the strongest and the weakest link of the three-dimensional intersection tunnel as shown in Table 7.  
The maximum tensile stress increment of the tunnel lining under different working conditions are shown in Table 7. It can be found out that the dynamic response of the adjacent tunnel is more significant when the surrounding rock condition is worse, the train speed is higher, or the intersection angle is smaller. Among the three conditions of a train passing through the cross tunnels, the dynamic response of the lining structure is the largest when two trains pass through the upper and the lower tunnels at the same time, which is about 0.789 MPa.