## 5 Conclusions  
This study combines indoor experiments and numerical simulations to analyze the tunnel excavation process, considering the time-dependent behavior of the surrounding rock using the Burgers creep model. The research focuses on investigating the influence of the back-arch excavation progress and the back-arch closure length on the settlement of the initial support arch. The key findings of this research are as follows:  
1. Decreasing the closure distance of the inverted arch leads to a reduction in the maximum settlement of the vault. This indicates that a smaller closure distance results in stronger stability of the initial support, with a more prominent effect of the initial support force ring. However, the maximum settlement of the vault is found to be insensitive to the exposed distance of the inverted arch. This suggests that the distance of the inverted arch has a less significant impact on the stability of the initial support. For a V-grade surrounding rock inverted arch, the sealing distance can be controlled within a range of 55 m, while for an IV-grade surrounding rock inverted arch, it can be controlled within a range of 70 m. The sealing distance for an III-grade surrounding rock inverted arch depends on specific site conditions.  
Figure 10: Tunnel vault settlement curves caused by different levels of surrounding rock and different excavation methods.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Closing Distance (m)**} & **Maximum Settlement of** & **Maximum Settlement of** & **Maximum Settlement of** \\  & **Class V Surrounding Rock** & **Class IV Surrounding Rock** & **Class III Surrounding Rock** \\  & **Arch (mm)** & **Arch (mm)** & **Arch (mm)** \\ \hline
30 & 13.33 & 4.82 & 4.49 \\
35 & 14.98 & 4.98 & 4.63 \\
40 & 16.38 & 5.48 & 5.00 \\
45 & 17.93 & 5.61 & 5.08 \\
50 & 21.71 & 6.18 & 5.36 \\
55 & 23.9 & 6.5 & 5.6 \\
60 & 25.1 & 6.72 & 5.74 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Tunnel deformation for different rock levels and different closure lengths.  
2. Following mechanized operation, the excavation length of the V-grade surrounding rock inverted arch can be controlled within a range of 5 m at a time. 3. In the case of the same surrounding rock conditions, the stability of the initial support of the tunnel is mainly influenced by the distance between the closure of the elevated arch; the stability of the initial support is less influenced by the elevated arch excavation step; the surrounding rock grade is different; the initial support of the tunnel is influenced by the distance between the closure of the elevated arch; the surrounding rock conditions are better; the stability of the initial support is less influenced by the change in the distance between the elevated arch. 4. The step method is suitable for excavation in Grade III, IV, and V surrounding rocks, and the length of excavation steps can be determined based on the settlement of the vault caused by the excavation. The numerical model developed in this study is based on the Songgangshan Tunnel and Shimengang Tunnel. The proposed method can be applied to study other sections of the project following the same approach.  
H.F.: Conceptualization, Methodology, Supervision, Project administration. W.X.: Methodology, Software, Data processing, Writing, original draft. Y.W.: Validation, Formal analysis, Writing, review, and editing. All authors have read and agreed to the published version of the manuscript.  
Subject of National Railway Group Co., Ltd. (J2018Z503) and National Natural Science Foundation of China (51978668).  
Not applicable.  
The authors thank the Advanced Research Center, Central South University, for providing the experiment conditions. The authors also express special thanks to the editors and anonymous reviewers for their constructive comments.  
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.