## 4 Numerical Simulation of Asymmetric Failure of Surrounding Rock in Roadway
### Analysis of Error Between Numerical Simulation and Theoretical Calculation  
The comparison between the numerical simulation and the theoretical results is depicted in Figure 16, and the comparison of numerical simulation results with theoretical calculation data is included in Table 5. (As the plastic zone shape of the surrounding rock of the roadway at a vertical distance of 35 m did not exhibit a distinct \"butterfly\" development, it will not be discussed further). It can be derived from the comparative analysis of Figure 16 and Table 5 that the theoretical calculation results have the following situations compared with the numerical simulation results:  
1. When the surrounding rock reaches the ultimate strength, the surrounding rock mass transforms from an elastic body to a plastic one. Since the bearing capacity of the plastic zone is lower than that of the elastic zone, the stress of the stope further transfers to the deep elastic zone, thereby further increasing the error in the theoretical calculation. Moreover, the influence of the plastic zone of the surrounding rock of the stope was not considered in the theoretical calculation of this paper, resulting in the overall theoretical calculation results being smaller than the numerical simulation results;  
Figure 16: The comparison between the numerical simulation and the theoretical results. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & & \multicolumn{2}{c}{**Maximum**} & & & & \\
**Vertical** & **Direction of the** & **Deevelopment Point** & **Theoretical** & **The Difference** & **Between Numerical** & **Average** & **Relative Error** & **Combined** \\
**Distance/m** & **Roadway** & **Plastic Zone with the Horizontal Line\({}^{\prime}\)** & **the Angle\({}^{\prime}\)** & **Simulation and Theory** & **Error** & **Relative Error** & **Relative Error** \\ \hline  & Rightupper corner & 46 & 40–28 & 6–18 & 12 & 13.3\% & \\  & Left upper corner & 127 & 130–118 & –3–9 & 3 & 3.3\% & \\  & Left lower corner & 249 & 220–208 & 29–41 & 35 & 38.9\% & \\  & Right lower corner & 324 & 310–298 & 14–26 & 20 & 22.2\% & \\ \hline  & Rightupper corner & 47 & 21–78 & 26–392 & 32.6 & 36.2\% & \\  & Left upper corner & 131 & 111–97.8 & 20–33.2 & 26.6 & 29.5\% & \\  & Left lower corner & 230 & 201–187.8 & 29–42.1 & 35.55 & 39.5\% & \\  & Right lower corner & 316 & 291–277.8 & 25–382 & 31.6 & 35.1\% & \\ \hline  & Rightupper corner & 51 & 79.0–92 & 43.1–50.21 & 46.66 & 51.8\% & \\
20.0 & Left upper corner & 135 & 97.9–90.92 & 37.1–44.08 & 40.59 & 45.1\% & \\  & Left lower corner & 225 & 187.9–180.92 & 40.1–47.08 & 43.59 & 48.4\% & \\  & Right lower corner & 322 & 277–92.092 & 44.1–51.08 & 47.59 & 52.8\% & \\ \hline  & Rightupper corner & 53 & 2-0.01 & 51–52.99 & 52 & 57.8\% & \\
27.5 & Left upper corner & 140 & 92–90.01 & 48–49.99 & 49 & 54.4\% & \\  & Left lower corner & 224 & 182–180.01 & 42–43.99 & 43 & 47.7\% & \\  & Rightlower corner & 325 & 272–270.01 & 53–54.99 & 54 & 60\% & \\ \hline \end{tabular}
\end{table} 
Table 5: Comparison of numerical simulation results with theoretical calculation data. With the increase in the vertical distance between the roadway and the working face, the \"butterfly-shaped\" development of the plastic zone of the roadway surrounding rock becomes less sufficient. Moreover, as this paper only considers the disturbance of the working face mining to the underlying roadway and does not take into account the influence of the initial stress of the surrounding rock, the error between theory and simulation becomes larger as the vertical distance between the roadway and the working face increases. For instance, when the vertical distance between the roadway and the working face is 5 m, the combined relative error of theory and simulation is 19.4%, while when the vertical distance is 27.5 m, the combined relative error rises to 54.9%;
3. Under the same vertical distance, the calculation error of the plastic zone of the roadway roof is evidently lower than that of the plastic zone of the floor.