## 1 Introduction  
In recent years, the demand for efficient and safe underground excavations and mining operations has significantly increased. Ensuring the stability of these subterranean structures is crucial to preventing accidents and maintaining operational efficacy. One of the challenging aspects of underground engineering is understanding and managing the stress distribution within the surrounding rock masses, especially when there are natural or man-made openings. The complex function theory, particularly the concepts of conformal mapping and the functions of a complex variable, has become a critical tool in the analysis of stress distribution around excavations in rock masses. The theory allows for a more accurate analytical solution for rock stress, especially around structures like tunnels in a homogeneous, isotropic elastic rock mass. The complex function theory has been utilized in a more sophisticated manner to study the stability of tunnel-surrounding rock in underground rock engineering. [PERSON] et al. [1] presented an analytical stress solution for a circular tunnel in a half-plane subjected to a concentrated force. The complex variable function method is employed to derive the closed-form solutions for the stress field, displacement field the support pressure of the tunnel. The solutions are validated using numerical analysis and are found to be accurate. [PERSON] et al. [2] focused on deriving an analytical solution for stress around a shallow buried lined circular tunnel under the deformation of the surrounding rock's inner edge by combining the complex potential functions with the conformal mapping method, which can be useful in the design and analysis of tunnel structures in the rock engineering field. [PERSON] et al. [3] delved into the deformation and stress theory concerning the surrounding rock of a shallow circular tunnel, setting up a bipotential stress potential complex function to explore the relationship between stress components, strain components stress potential functions, employing the complex variable function method for the analysis. The existing research largely focuses on the stress issues of underground circular tunnels explores them through the lens of the complex variable theory. Through these works, expressions for relevant stress components around such tunnel structures are derived, contributing to the understanding and analysis of stress and deformation in circular tunnels under various conditions. Generally, circular tunnels, with their simplistic shape, can easily be analyzed by linear mapping functions, linking them to the corresponding unit circle on the plane, thus simplifying the solving process. Whether dealing with a single circular tunnel or double circular tunnels, the related research has matured significantly [4]. However, in reality, underground rock formations encompass a variety of tunnel shapes, not limited to simple forms like circles or ellipses but also including rectangles, trapezoids, straight-wall arch shapes various other geometric configurations of underground spatial structures. Although the basic principles of solving stress problems of circular tunnels using complex variables remain consistent, the complex shapes of these other forms make the stress equilibrium conditions at the tunnel boundaries significantly more complicated, thereby increasing the difficulty of solving such problems. A considerable amount of research has been conducted on rock engineering issues of a single opening, employing the conformal mapping method in conjunction with the complex variable theory for the analysis [5; 6]. [PERSON] et al. [7; 8] introduced the use of the Box composite method and triangular interpolation method for solving the mapping functions of planes with tunnels of complex shapes. [PERSON] et al. [9] conducted a tunnel construction and surrounding rock stability analysis by focusing on the rock fracture evolution during the construction and operation of straight-walled arched tunnels.