## 4 Conclusions  
This study systematically investigated the stress distribution and mechanical behavior of rock masses with a three-center arch opening under various complex deep underground stress conditions with varying lateral pressure coefficients (\(\lambda\)) and principal stress directions (\(\beta\)). Combined with DEM numerical modeling, the study investigated their impact on the initial failure characteristics at the opening boundary. The main conclusions of the study are as follows:The lateral pressure coefficient mainly affects the stability of the opening by influencing the stress concentration around the surrounding rock. When the lateral pressure coefficient is at a low level, tensile stress concentration is likely to occur in the boundary area of the opening perpendicular to the maximum principal stress. Within the scope of this study, as the lateral pressure coefficient increases, the tensile stress disappears, and the compressive stress region expands, enhancing the stability of the opening.
2. The direction of the principal stress has a minor effect on the degree of tensile stress concentration around the surrounding rock of the opening, but it significantly impacts the stress distribution at the boundary of the opening. When the angle between the maximum principal stress in the plane and the vertical direction is 45\({}^{\circ}\), the compressive stress at the corners of the opening's straight walls reaches its maximum. This region is prone to failure due to the high compressive stress. In cases where the lateral pressure coefficient is not equal to 1, the stress distribution asymmetry is most pronounced.
3. DEM numerical simulations confirmed that initial failures occur in areas of compressive or tensile stress concentration, validating the analytical stress solutions. As the lateral pressure coefficient increases, the tensile stress decreases and eventually disappears, while the compressive stress-induced failure range significantly expands.  
The following supporting information can be downloaded at: [[https://figshare.com/articles/dataset/Raw_Data_for_applsci-3077289/26789905?file=48673189](https://figshare.com/articles/dataset/Raw_Data_for_applsci-3077289/26789905?file=48673189)]([https://figshare.com/articles/dataset/Raw_Data_for_applsci-3077289/26789905?file=48673189](https://figshare.com/articles/dataset/Raw_Data_for_applsci-3077289/26789905?file=48673189)) (accessed on 12 August 2024).  
Data curation, M.C. and Z.L.; formal analysis, L.T. and M.C.; investigation, R.C., X.S. and X.Q.; methodology, X.S., R.C. and X.Q. All authors have read and agreed to the published version of the manuscript.  
This research was funded by projects of the National Natural Science Foundation of China (No. 52104110, 52374152), Hunan Provincial Natural Science Foundation of China (No. 2020 JJ5743), Key R&D projects of the 14 th Five Year Plan (No. 2022 YFC2904602); and the CRSRI Open Research Program (No. CKWV2019738/KY).  
Not applicable.  
The data used to support the findings of this study are available from the corresponding author upon request.  
Author [PERSON] was employed by the Guangdong Province Dabaoshan Mining Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.