## 1 Introduction
The study introduced an improved method for calculating the mapping function, which defines the relationship between the physical and mapped planes. Based on this method, [PERSON] et al. [15] further investigated the stress distribution and mechanical behavior of rock masses with openings of various shapes under different lateral pressure coefficients. These studies involve various research factors such as different opening shapes and lateral pressure coefficients. In addition to the theory of complex functions, they also include research methods like numerical simulation and mechanical testing. Overall, these studies mainly solve the mapping functions through polygonal methods or optimization algorithms. Their findings indicate that the stress distribution characteristics around the opening primarily depend on the shape of the opening and the lateral pressure coefficient. When the opening boundary has sharp corners or there is a significant difference between the maximum and minimum principal stresses, the stress distribution around the opening rock mass shows noticeable non-uniformity with localized stress concentration. Comparisons between stress analytical solutions and mechanical test results further reveal that the characteristics of the stress distribution are consistent with the initial instability modes of the surrounding rock mass, based on which the stability and initial failure modes of the opening can be determined. These studies provide a comprehensive analytical framework for understanding the stress distribution and mechanical behavior of rock masses containing openings, offering valuable insights into the design and stability assessment of underground structures. These studies contribute to the optimization of engineering designs and the mitigation of potential failure risks in underground rock engineering. However, almost all these studies assume that the maximum and minimum principal stresses are aligned with the vertical and horizontal directions, respectively. Due to the constraints of experimental conditions, many laboratory tests are conducted under uniaxial compression conditions. Although this assumption significantly simplifies the calculation process and reduces the complexity of the solution, it can lead to substantial errors when the principal stress directions do not align with the vertical or horizontal directions. Some studies have found that the stability of openings, such as tunnels and roadways, is optimal when the local stress direction is orthogonal to the opening, while the stability decreases and the failure characteristics are influenced when the local stress direction is inclined [16; 17; 18; 19]. This indicates that the orientation of the in situ stress significantly impacts the stability and failure modes of the openings (e.g., deformation patterns, crack initiation, and propagation). However, these studies are primarily based on numerical simulations or laboratory mechanical tests, focusing on the mechanical properties and failure characteristics of the specimens without explaining the influence mechanism from the perspective of the stress field. To address these limitations, this study integrates analytical solutions with numerical simulations to systematically investigate the stress distribution around three-center arch openings under different geo-stress states characterized by lateral pressure coefficient and principal geo-stress direction, and its impact on the failure characteristics of the surrounding rock. The aim is to bridge the gap between mechanical properties and stress-field influence mechanisms, offering a more comprehensive understanding of how stress orientations affect opening stability.