Elastic analysis of stress–displacement field for a lined circular tunnel at great depth due to ground loads and internal pressure

An elastic plane strain solution for stresses and displacements around a lined circular tunnel in an isotropic geomaterial due to uniform ground loads and internal pressure is presented. By applying the series expansion technique in the complex potential theory established by Muskhelishvili, the complex stress and displacement functions are presented in an appropriate form. Then, considering the continuity of the stress and displacement along the boundary, in conjunction with the comparisons of the same power exponent on both sides of the equation, the problem is simplified to solve a set of linear algebraic equations. Subsequently, the complex potentials in the liner and the surrounding geomaterial are explicitly derived, respectively. Sensitivity analyses indicate that if the relative thickness of the stiff liner is too high, tensile stresses likely occur along the tunnel opening and the interface between the liner and the surrounding geomaterial. In addition, it is found that the variations of stresses in the geomaterial intensively rely on the relative rigidity and thickness of the liner when the ratio between the distance of these points under investigation to the tunnel axis and the outer radius of liner is in the range from 1 to 2. Either too high or too low values for the relative rigidity and thickness of the liner are unfavorable for the structural stability. The derived solutions are verified by comparing the results for some special conditions with Kirsch’s solution.