## 5 Conclusions  
This paper focused on the spatial-temporal distribution of extreme pressures in high-speed railway tunnels and its long-term effects on the safety of the transportation system in the post-train stage. Based on the evolution and superpositions of pressure waves, a one-dimensional model referred to as the spatial-temporal distribution of the pressure state (SDPS) model is developed, and equations are first proposed to describe the pressure state variation with time and space in the tunnel. Based on the SDPS model, a spatial-temporal prediction method for the extreme pressure in a tunnel and a calculation program are developed, which can greatly reduce the calculation cost and time in the tunnel design stage. Finally, the dimensionless spatial distribution of the extreme parameters (MPP, MNP, and MPPR) in the tunnel under different working conditions is analyzed using dimensionless operation parameters (train/wave velocity ratio \(M\) and train/tunnel length ratio \(\varepsilon\)). The main conclusions are summarized as follows:  
1. The pressure wave in the tunnel follows the principle of wave superposition, which can be calculated quickly with the SDPS model proposed in this study. The proposed model was verified by the field measurements, and the relative errors of the predicted and measured MPP and MNP are between 1.35% and 1.85%.
2. In the post-train stage, the MPP is determined by the superposition of PWS, and MNP is related to PWS and PP. When the MPP and MNP are only determined by the superposition of the PWS, they are symmetrical about the midpoint of the tunnel axis and appear alternately around it. Otherwise, it would result in asymmetry in the location of the MPP and MNP.
3. When \(M\in[0.1,\,0.8]\) and \(\varepsilon\in[0.1,\,0.8]\), the dimensionless positions of the MPP, MNP, and MPPR are concentrated in the middle region of the tunnel (\(x/L_{tu}\in[0.33,\,0.67]\)); meaning that the tunnel lining and facilities should be strengthened in the MPPR of the built tunnel to prevent fatigue failure. In the future, avoid placing pressure-bearing tunnel facilities (such as fire doors, control boxes, signal lights, etc.) in the MPPR to extend the life span of tunnel facilities.  
It should be noted that due to the three-dimensional effect, the SDPS model may need to be modified within 50 m of the tunnel entrance, which will be discussed in the future.  
The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app13031350/s1](https://www.mdpi.com/article/10.3390/app13031350/s1)]([https://www.mdpi.com/article/10.3390/app13031350/s1](https://www.mdpi.com/article/10.3390/app13031350/s1)).  
Conceptualization, W.L., A.L., H.C. and L.C.; methodology, W.L., A.L., H.C. and L.C.; software, W.L.; validation, W.L.; formal analysis, W.L.; investigation, W.L.; resources, A.L. and H.C.; funding acquisition, L.C.; supervision, A.L., H.C. and L.C.; writing--original draft, W.L.; writing--review & editing, A.L. and H.C. All authors have read and agreed to the published version of the manuscript.  
The author [PERSON] was funded by the Natural Science Foundation of Shaanxi Province, and the grant number is 2020 JM-479.  
Thanks are extended to [PERSON], [PERSON], [PERSON], [PERSON] for their constructive suggestions, [PERSON] for the inspire of the function related with \(n_{u}\), and [PERSON], [PERSON] for assistance in program development.  
The authors declare no conflict of interest.