# Analytical Prediction of Tunnel Deformation Beneath an Inclined Plane: Complex Potential Analysis
## 1 Introduction  
Estimating the loads on the ground surface before and after tunnel excavation is an essential concept in current engineering. The prediction of the ground deformation of a tunnel under a slope can be interpreted both analytically and numerically. Using analytical solutions to determine the stresses and displacements around the tunnel is also practical [1]. Indeed, it is essential to recognize the substantial impact of excavation and assisted construction procedures and their technical details on the distribution of stresses/deformations around the opening and its digital support system. Methods for predicting ground motion fall primarily into three categories: empirical methods [2; 3; 4; 5; 6; 7; 8], numerical methods [9; 10; 11; 12; 13; 14; 15] and finally, analytical methods [16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31]. Furthermore, other approaches based on an approximation of the Navier-Stokes equations (for incompressible materials at the free surface) are proposed, taking into account the absence of hydrostatic equilibrium, assuming that the medium has a small temperature variation [32]. Thus, according to [PERSON] et al. [33], the unique mechanical properties of the contact surface can then have a large influence on the deformation of the soil, thus interacting with the soil-structure contact. The soil response can be described not only by the soil failure states but also by the material stiffness. The mode of displacement of the wall (all buried materials) greatly influences the intensity and distribution of the compression pressure. In execution, different theories may be applicable, justified and verified by the assumption that the soil properties, the roughness of the structure, and the state of ground deformation may be admitted in the plastic or non-plastic domain. Depending on the excavation depth, many studies have been conducted on shallow tunnels, but few on sloping tunnels have been presented. Until now, some researchhas focused on the effect of non-uniform deformation caused by ground loss, offering estimations of the motion imposed by various deformation models associated with the open tunnel [26; 28; 31; 34; 35; 36; 37]. Furthermore, [PERSON] et al. [15] propose a unified displacement function to express by Fourier series a more detailed description of the deformation mode of a shallow circular tunnel cross-section when the ground surface is sloping. These various methods are based on linear elasticity theory in order to simplify the fixed stress shapes at the lower limit of the tunnel. These approaches can also be utilized to simultaneously determine the tunnel stresses and displacements when the tunnel is sloping. Recently, many engineers have used analytical approaches to predict the deformation of the ground surrounding a tunnel. Their applicability to the stability of a tunnel under a slope is less developed. Therefore, this paper presents an approximate solution based on the complex variable method to predict the tunnel deformation beneath a slope. The method is based on two parameters called \"complex potential functions\" ( \(\varphi(z)\), \(\Psi(z)\)), also known as \"Goursat functions\". The main idea is to obtain solutions to the deformation of the tunnel wall using the complex potential function under an acceptable limit. More precisely, at the boundary conditions, the stresses applied to the tunnel wall are evaluated as a function of the variation of the surface load while measuring the boundary stress and the displacement of the cross-section by combining the stress-displacement terms. Thus, analytically, the left and right infinite boundaries define the two complex displacement functions when developing the Laurent series. The displacement model is then integrated with the complex potentials to analytically predict the deformation induced by the inclined plane. The performance and viability of this method are explored using a series of analyses introduced by the complex displacement.