Analytical Solutions of Elastic Complex Variables for Tunnels with Complicated Shapes Under Geostress Field

The elastic solutions of isotropic complex noncircular deep or shallow tunnels are obtained by the complex variable theory in this paper. In the solution procedure, the mapping function in the form of power series is adopted and Cauchy integral formulas are widely applied to simplify the calculation. The unified solutions expressions for the mapping function with any number of series terms (namely the solving accuracy) are presented. When the terms number of the mapping function is determined, the solutions are accurate for the tunnels considering the influence of local rock gravity, as they are derived by analysis without any approximation and numerical method. Compared to previous analytical method that maps tunnels and surface boundary to rings, the surface boundary conditions of this method are not fully satisfied but the coefficients of the mapping function used are easier to solve, requiring fewer series, and the solution process is simpler and faster. Therefore, it is more suitable for deep and shallow rock tunnels that are not near the surface. Semi-circle arch, three-centered arch, horseshoe shape, asymmetric trapezoid and 'C' shape tunnels are analyzed as examples, and the validity of the results is verified by comparing the analytical solution obtained with the FEM numerical solution calculated in ANSYS software. Elastic analytical algorithm for shallow rock tunnels with complex shapes is proposed.Solutions proposed are valid for the mapping function in power series form with any number of terms.The impact of tunnel asymmetry is considered in this study.High-accuracy solutions for tunnels with complex shapes are obtained by controlling the number of mapping function terms