## 4 Conclusions  
In order to analyze the response characteristic of high-speed railway cross tunnel lining under dynamic train loads, this paper analyzed the displacement and stress response of tunnel lining under static strata loads and dynamic train loads using numerical analysis. We further analyzed the damage  
Figure 17: Strain evolutionary characteristic: (**a**) Strain increment; (**b**) Strain ratio.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Stress Level** & **0.6** & **0.65** & **0.7** & **0.75** \\ \hline Damage value & 0.27 & 0.32 & 0.33 & 0.37 \\ Strain ratio & 0.30 & 0.35 & 0.37 & 0.40 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Damage value and strain ratio of the inverted arch under different stress levels.  
evolutionary characteristics of the tunnel inverted arch under different dynamic load conditions and different initial static stress conditions with the indoor test method. The main conclusions are as follows:  
1. The displacement and stress levels of the tunnel lining are very significant at the cross position than at other positions. The stress increment of the tunnel lining due to dynamic train loads is more likely to induce a break in the tunnel lining at this position. To avoid the damage of the tunnel lining at the cross position, some countermeasures should be introduced, like ensuring the height of the rock column between the upper tunnel and the lower one is greater than 3 m, improving the strength of the rock column to make it superior to the surrounding rock of level IV, and avoiding the condition of two trains passing through the cross position at the same time.
2. The change of structural strain increment amplitude and strain ratio is obvious when the dynamic load stress level is higher, and the tunnel lining would be more easily damaged in the long-term service period. The tunnel inverted arch will be more likely to be intact when the dynamic stress level is 0.6. If the stress level of the dynamic load is higher than this threshold, the strain amplitude increment and strain ratio of the tunnel inverted arch will suffer from three stages during the dynamic loading process--the gradually increasing stage in the early loading period, the stable stage in the middle period, and finally, the breaking stage in the final period. From the perspective of the long-term service of the tunnel, the dynamic stress levels of the tunnel inverted arch should not be higher than 0.6.
3. The initial static load has an influence on both the tensile and compressive strain of the tunnel inverted arch, and the tensile strain is the controlling failure factor when the ultimate compressive strain is much small than its ultimate tensile strain. The change of strain amplitude and strain ratio would be more significant if the static stress level is higher, and the tunnel inverted arch would be easily damaged. When the stress level of static load is 0.65, the change of strain amplitude and strain ratio tends to be stable after growth in the early stage. Once the static load is higher than this value, the strain amplitude and strain ratio are significant in the initial stage and the failure stage. In order to ensure that the tunnel has long service capacity, the initial static stress level should be less than 0.65.  
It should be noted that the deformation and stress states of a tridimensional tunnel are influenced by many factors such as geological heterogeneity and construction disturbance. In the design of tridimensional cross tunnels, the influence of other factors should also be paid attention to, besides the analysis results in this paper.  
Conceptualization, A.W.; formal analysis, C.Z.; funding acquisition, C.S.; investigation, C.S.; methodology, A.W., C.S., and E.D.; supervision, H.H.; writing--original draft, E.D.; writing--review and editing, A.W., C.S., and W.Y. All authors have read and agreed to the published version of the manuscript.  
This research was funded by National Natural Science Foundation of China, grant number 51978670.  
The authors declare no conflict of interest.