## 4 Numerical Simulation of Repeated Supportability of Roadway Roof  
According to the stepping temporary support mechanism, the bracket repeatedly comes into contact with the roadway roof during the support process. Consequently, the lithology and thickness of the roadway roof play a crucial role in determining the repeatability of the support. Furthermore, the original rock stress at varying roadway depths also influences the repeatability. In order to assess the repeatability of the roadway roof support, the subsequent numerical simulation investigates the changes in fracturing and displacement of the roadway roof following the implementation of stepping support. It considers different lithology, thickness, and burial depths of the roadway roof. Numerical Simulation Results of Repetitive Support under Different Direct Roof Lithology and Thickness  
The physical and mechanical properties of the direct roof rock bodies in numerical simulation are displayed in Table 1, based on the geological conditions of China Guanban Coal Mine. The direct roof of the roadway in this mine primarily consists of sandstone, sandy mudstone, and mudstone, which are three types of rock formations. Here, the capacity is the dry capacity  
The three direct-roof lithologies mentioned above are individually assigned in the numerical model, and a series of numerical simulations is conducted for each lithology. Based on the analysis of the composition of the surrounding rock in Section 3.1, the direct roof should have a minimum thickness of 0.8 m using the temporary support process. Therefore, three thicknesses of direct roof (0.8 m, 1.6 m, and 2.4 m) are selected for each group of numerical simulations, labeled as (a), (b), and (c), to examine the impact of different thicknesses on the repeatable support of the roadway under each lithology. Table 2 displays the designs of the three sets of numerical simulations. The roadway is assumed to be buried at a depth of 500 m, subjected to a vertical stress of 10 Mpa, with a maximum horizontal principal stress of 15 Mpa and a minimum horizontal principal stress of 12 Mpa, thereby simulating the original stress conditions of the rock surrounding the roadway. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Direct Roof** & **Density** & **Tensile** & **Bulk** & **Shear** & **Cohesive** & **Friction** & **Dry Bulk** \\
**Lithology** & **(kg/m3)** & **Strength** & **Modulus** & **Modulus** & **Force** & **Angle** & **Density** \\
**Sandstone** & 2680 & 2.74 & 20.2 & 3.6 & 5 & 35 & 26.2 \\ Sandy & 2450 & 1.0 & 10.56 & 8.96 & 2.5 & 30 & 24 \\ mudstone & 2250 & 0.6 & 7.11 & 3.86 & 2 & 28 & 22 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The physical and mechanical properties of different direct roof rock bodies. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Numerical Simulation** & **Direct Roof** & **Direct Roof** & **Direct Roof Thickness** \\
**Group** & **Lithology** & & & \\ \hline Group I & Sandstone & & & \\ \hline Group II & Sandy mudstone & (a) & (b) & (c) \\
**Group III** & Mudstone & 0.8 m & 1.6 m & 2.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Numerical simulation group. Following the completion of step-by-step excavation and support, the calculations continue until stress equilibrium is achieved. Figure 13 presents a schematic diagram illustrating the distribution of the excavation roadway area after the temporary support is finalized. Area 1 represents the temporary support area, Area 2 refers to the region after repeated support, and Area 3 corresponds to the permanent support zone. The results of Area 2 crushing and displacement are used as the basis for whether the roof can be supported repeatedly.