## 3 Results  
### Stress Distribution Characteristics  
Based on Table 1, the mathematical calculation model of the plane containing a single opening shown in Figure 1b can be transformed into the general stress state shown in Figure 1a. By solving the equations, the stress functions of the plane containing a single opening can be determined, and through these formulas, the stress state at any point within the plane can be obtained. According to the mathematical model shown in Figure 1a, the study solved the stress functions for the plane containing the opening under different \(\lambda\) and \(\beta\) conditions. In practical engineering, the failure of the surrounding rock of an opening often begins at a certain area of the opening boundary. Therefore, the stress characteristics of the opening boundary were selected as the focus of this study. The distribution curves of the boundary hoop stress are shown in Figure 4, where the dashed lines represent the initial mapped shape. The portions of the stress curves inside the opening are negative, indicating tensile stress, while those outside are positive, indicating compressive stress. The stress solution data are provided in the Supplementary Materials. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Principal Stress State**} & \multicolumn{3}{c}{**Non-Principal Stress State**} \\ \hline \(\lambda\) & \(\beta\) & \(\sigma_{x}\) & \(\sigma_{y}\) & \(\tau_{xy}\) \\ \hline \multirow{6}{*}{0} & 0 & 1.000 & 0.000 & 0.000 \\  & 15 & 0.933 & 0.067 & 0.250 \\  & 30 & 0.750 & 0.250 & 0.433 \\  & 45 & 0.500 & 0.500 & 0.500 \\  & 60 & 0.250 & 0.750 & 0.433 \\  & 75 & 0.067 & 0.933 & 0.250 \\  & 90 & 0.000 & 1.000 & 0.000 \\ \hline \multirow{6}{*}{0.2} & 0 & 1.000 & 0.200 & 0.000 \\  & 15 & 0.946 & 0.254 & 0.200 \\  & 30 & 0.800 & 0.400 & 0.346 \\  & 45 & 0.600 & 0.600 & 0.400 \\  & 60 & 0.400 & 0.800 & 0.346 \\  & 75 & 0.254 & 0.946 & 0.200 \\  & 90 & 0.200 & 1.000 & 0.000 \\ \hline \multirow{6}{*}{0.4} & 0 & 1.000 & 0.400 & 0.000 \\  & 15 & 0.960 & 0.440 & 0.150 \\  & 30 & 0.850 & 0.550 & 0.260 \\  & 45 & 0.700 & 0.700 & 0.300 \\  & 60 & 0.550 & 0.850 & 0.260 \\  & 75 & 0.440 & 0.960 & 0.150 \\  & 90 & 0.400 & 1.000 & 0.000 \\ \hline \multirow{6}{*}{0.6} & 0 & 1.000 & 0.600 & 0.000 \\  & 15 & 0.973 & 0.627 & 0.100 \\  & 30 & 0.900 & 0.700 & 0.173 \\  & 45 & 0.800 & 0.800 & 0.200 \\  & 60 & 0.700 & 0.900 & 0.173 \\  & 75 & 0.627 & 0.973 & 0.100 \\  & 90 & 0.600 & 1.000 & 0.000 \\ \hline \multirow{6}{*}{0.8} & 0 & 1.000 & 0.800 & 0.000 \\  & 15 & 0.987 & 0.813 & 0.050 \\  & 30 & 0.950 & 0.850 & 0.087 \\  & 45 & 0.900 & 0.900 & 0.100 \\  & 60 & 0.850 & 0.950 & 0.087 \\  & 75 & 0.813 & 0.987 & 0.050 \\  & 90 & 0.800 & 1.000 & 0.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Calculation cases with various geo-stress states. It can be observed from Figure 4a that when \(\beta=0^{\circ}\) (maximum principal stress \(\sigma_{\beta,max}\) is vertical), the stress distribution is symmetrical. The stress curves show sharp changes around \(135^{\circ}\) and \(225^{\circ}\), corresponding to the corners of the opening's straight walls where the compressive stress reaches its maximum. As \(\lambda\) increases, the compressive stress concentration in these two regions also increases. Specifically, when \(\lambda\) reaches 1.0 (red curve), the compressive stress concentration in these regions is the most pronounced. On the other hand, when \(\lambda\) is below 0.4, tensile stress appears in the roof and floor regions of the opening, and the maximum tensile stress occurs in the middle regions at \(90^{\circ}\) and \(270^{\circ}\), indicating that without or with low lateral pressure, the middle regions of the roof and floor of the opening are prone to tensile failure. As \(\lambda\) increases, the tensile stress in these regions around the opening gradually decreases, transforming completely into compressive stress when \(\lambda\) reaches 0.4.