## 4 Numerical Simulation of Asymmetric Failure of Surrounding Rock in Roadway
### Analysis of Numerical Simulation Results  
The plastic zone of the surrounding rock of the roadway obtained by the numerical simulation is shown in Figure 15.  
The following can be derived from the analysis of Figure 15:  
Under different vertical distances, the shape and maximum range of the plastic zone of the roadway's surrounding rock change along with the variation in the number of cycle steps (namely, the horizontal distance between the roadway and the working face).  
1. When the vertical distance is 5 m and 12.5 m, with the increase in the number of cycling steps, the plastic zone shape of the surrounding rock of the roadway changes from circular to rectangular, and finally forms an X-shaped distribution, and the maximum range of the plastic zone increases significantly;
2. When the vertical distance is 20 m and 27.5 m, the expansion of the plastic zone of the roadway surrounding rock is restrained, and its shape gradually transforms from circular to butterfly shaped. Moreover, when the number of cycling steps is 35-40, the plastic zone on the upper right side of the roadway surrounding rock develops significantly towards the working face;
3. When the vertical distance is 35 m, as the distance between the roadway and the working face keeps increasing, the influence degree of the working face mining on  
Figure 14: The plastic zones of the surrounding rock of the roadway under different cycles.  
Figure 13: Hydrostatic pressure condition.  
the surrounding rock of the roadway reduces. The overall expansion trend of the plastic zone of the surrounding rock is restricted, and the final development form of the plastic zone is approximately rectangular. The maximum depth and orientation of the plastic zone of the roadway surrounding rock under different cycles are presented in detail in Table 4.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \begin{tabular}{c} **Vertical** \\ **Distance/m** \\ \end{tabular} & **Cycle Number** & \begin{tabular}{c} **Shape of the Plastic** \\ **Zone** \\ \end{tabular} & \
\begin{tabular}{c} **Direction of the** \\ **Roadway** \\ \end{tabular} &
\begin{tabular}{c} **Maximum Development Point of the Plastic Zone** \\ **and the Angle with the Horizontal Line/\({}^{\circ}\)** \\ \end{tabular} \\ \hline \multirow{4}{*}{5.0} & 1–17 & Circular & Right upper corner & 46 \\  & 18–30 & Rectangular & Left upper corner & 127 \\  & 31–35 & Rectangular & Left lower corner & 249 \\  & 36–40 & X-shaped & Right lower corner & 324 \\ \hline \multirow{4}{*}{12.5} & 1–14 & Circular & Right upper corner & 47 \\  & 15–30 & Rectangular & Left upper corner & 131 \\  & 31–35 & Rectangular & Left lower corner & 230 \\  & 36–40 & Butterfly-shaped & Right lower corner & 316 \\ \hline \multirow{4}{*}{20.0} & 1–21 & Circular & Right upper corner & 51 \\  & 22–30 & Rectangular & Left upper corner & 135 \\  & 31–35 & Rectangular & Left lower corner & 225 \\  & 36–40 & Butterfly-shaped & Right lower corner & 322 \\ \hline \multirow{4}{*}{27.5} & 1–21 & Circular & Right upper corner & 53 \\  & 22–30 & Rectangular & Left upper corner & 140 \\  & 31–35 & Rectangular & Left lower corner & 224 \\  & 36–40 & Butterfly-shaped & Right lower corner & 325 \\ \hline \multirow{4}{*}{35.0} & 1–23 & Circular & Right upper corner & \\  & 24–30 & & Left upper corner & \\  & 31–35 & Rectangular & Left lower corner & / \\  & 36–40 & & Right lower corner & \\ \hline \hline \end{tabular}
\end{table}
Table 4: The maximum depth and orientation of the plastic zone of the roadway’s surrounding rock under different cycles.  
Figure 15: The distribution of the plastic zone of the roadway’s surrounding rock.  
Based on the above analysis, it is evident that as the working face progresses gradually, the shape of the plastic zone of the roadway surrounding rock undergoes a continuous transformation from \"circular\" to \"butterfly-shaped\". During this process, the range of the plastic zone keeps increasing and develops in the direction of the working face.