## 3 Methodology  
### Theoretical Calculation  
The rock mass in which the mining roadway is located can be categorized as a heterogeneous, anisotropic medium. Currently, it is difficult to accurately analyze the stress in the roadway surrounding rock in this type of rock mass using mechanics and mathematics. Therefore, the calculations need to be simplified. For several decades, the plastic zone radius of a circular roadway has been typically determined using the Kastner formula as follows:  
\[R_{P}=R_{0}\bigg{[}\frac{(p_{0}+\text{cot}\varphi)(1-\text{sin}\varphi)}{p_{1}+ \text{cot}\varphi}\bigg{]}^{\frac{1-\text{sin}\varphi}{2\text{sin}\varphi}} \tag{1}\]  
However, roadways are often subjected to bidirectional, unequal pressure fields. Based on the actual stress environment of the surrounding rock of a roadway, the theory of elastic mechanics, and the Mohr-Coulomb criterion, [PERSON] et al. (2016, 2017) derived the boundary equation for the plastic zone of the surrounding rock of a circular roadway under the condition of bidirectional, unequal pressure.  
\[\begin{split} 9(1-\lambda)^{2}(\frac{r}{R_{0}})^{8}&+[-12(1 -\lambda)^{2}+6(1-\lambda)^{2}\text{cos}2\theta]\times(\frac{r}{R_{0}})^{6}+[( 1-\lambda)^{2}-\text{sin}^{2}\varphi(1+\lambda^{2}+\frac{2\text{cos}\varphi}{ P_{\text{Ssin}\varphi}})^{2}]\\ &+[10(1-\lambda)^{2}\text{cos}^{2}2\theta-4(1-\lambda)^{2}\text{ sin}^{2}\varphi\text{cos}^{2}2\theta-2(1-\lambda)^{2}\text{sin}^{2}2\theta-4(1- \lambda^{2})\text{cos}2\theta\\ &+(1+\lambda)^{2}](\frac{r}{R_{0}})^{4}+[-4(1-\lambda)^{2}\text{ cos}4\theta+2(1-\lambda^{2})\text{cos}2\theta-4(1-\lambda^{2})\text{sin}^{2}2\varphi\text{ cos}2\theta\\ &-\frac{4\text{C}(1-\lambda)\text{sin}\text{cos}2\theta}{P_{3}} ](\frac{r}{R_{0}})^{2}=0\\ \end{split} \tag{2}\]  
where \(\lambda\) is the lateral pressure coefficient; \(r\) is the radius of the roadway; \(R_{0}\) is the distance from the boundary of the plastic zone to the center of the roadway; \(\theta\) represents  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Lithology**} & **Tensile** & **Compressive** & **Elastic** & **Poisson** & **Cohesion** & **Internal Friction** \\  & **Strength** & **Strength** & **Modulus** & **Ratio** & **(MPa)** & **(MPa)** & **(\(\circ\))** \\ \hline Sandstone of basic roof & 3.48 & 37.9 & 9976 & 0.21 & 4.8 & 49 \\ Sandstone of immediate roof & 3.70 & 38.5 & 10,023 & 0.23 & 6.7 & 50 \\ No. 1–2 upper coal & 1.98 & 11.2 & 7087 & 0.35 & 2.85 & 43.2 \\ Sandstone of immediate floor & 4.9 & 47.2 & 11,430 & 0.22 & 7.1 & 52.78 \\ Sandstone of basic floor & 3.89 & 40.1 & 10,477 & 0.23 & 6.8 & 52.09 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Rock mechanical parameters.  
Figure 4: Rock mechanics parameter test: (**a**) rock samples; (**b**) RMT-150B electro-hydraulic servo testing machine.  
the polar coordinates of any point in the roadway; \(C\) and \(\varphi\) are the cohesion and internal friction angle of the surrounding rock, respectively; \(P\) is the vertical stress.  
In this study, the abovementioned two formulas were used to calculate the plastic zone of the roadway surrounding rock, and the calculation results were compared and analyzed.