## 3 Results
### Initial Failure Mode
For example, when \(\lambda\) = 0, the regions of the opening boundary perpendicular to the maximum principal stress direction exhibit significant tensile stress concentration, leading to initial tensile cracks in these areas (Figure 7a-d). It is obvious that without lateral pressure, tensile stresses dominate the failure process, resulting in tensile crack initiation. When \(\lambda\) = 0.2, the tensile stress concentration around the opening is no longer evident, and the formation and development of tensile cracks are inhibited. Accordingly, tensile failures no longer occur in the tensile stress regions around the opening. This transition highlights the impact of increasing lateral pressure in mitigating tensile stress and preventing tensile failures. As \(\lambda\) continues to increase, the tensile stress around the opening significantly decreases and gradually transforms into compressive stress, with initial failures around the opening primarily resulting from compressive stress concentration. When the maximum principal stress is vertical or horizontal (\(\beta\) = 0\({}^{\circ}\) or \(\beta\) = 90\({}^{\circ}\)), the hoop stress exhibits a symmetrical distribution about the opening's axis of symmetry. In these cases, the initial cracks around the opening also display a roughly symmetrical distribution. As \(\lambda\) increases, the regions of compressive stress concentration around the opening expand, correspondingly increasing the extent of initial failures along the opening boundary. Changes in \(\beta\) cause asymmetry in the stress distribution around the opening, leading to a shift in the failure locations to one side. This asymmetry, induced by the principal stress orientation, complicates the prediction of failure zones and necessitates adaptive design measures to address the varying stress conditions. The comprehensive analysis combining analytical stress solutions and numerical simulations robustly validates the initial failure modes around the opening. Significant stress concentration areas revealed by the analytical stress solutions align well with the numerical simulation results, identifying these zones as potential sites for initial failures. For instance, as the lateral pressure coefficient (\(\lambda\)) increases, the analytical solutions indicate a reduction in tensile stress around the opening, transitioning these regions into compressive stress zones. The numerical simulations corroborate this transition, showing a corresponding inhibition of tensile crack formation and a shift towards compressive stress-induced failures. When the lateral pressure coefficient is low (\(\lambda\) \(\leq\) 0.4), due to the lack of or insufficient lateral constraints, the rock mass is more likely to fail under lower external loads (Figure 7a-d). This phenomenon particularly agrees with the analytical stress distribution in the corresponding models shown in Figure 4, where the low lateral pressure (\(\lambda\)\(\leq\) 0.4) results in significant tensile stress concentrations. Additionally, as the principal stress angle (\(\beta\)) changes, the numerical simulations reveal corresponding shifts in the failure locations, consistent with the analytical predictions. This correlation confirms that the analytical approach accurately captures the critical stress zones prone to failure, demonstrating the effectiveness of these methods in predicting opening failure patterns. Overall, the integration of analytical and numerical methods provides a comprehensive understanding of the stress-induced failure mechanisms around openings. This combined approach not only validates the predictive accuracy of the analytical models but also offers detailed insights into the stress evolution and failure progression, which is essential for designing effective reinforcement strategies and ensuring the stability of underground structures.