Ground displacements and stresses around a non-uniformly deforming circular tunnel in elastic media using complex variable method

A closed-form analytical solution using complex variables has been determined to predict ground displacements and stresses around a non-uniformly deforming single circular tunnel in a linearly elastic weightless and ponderable (weighty) medium. For the weightless medium, the solution has been derived with the satisfaction of (1) stress-free conditions on the boundary of the elastic half-space and (2) the displacements’ boundary condition on the tunnel periphery. While for the ponderable medium, the solution has been obtained with the satisfaction of (1) geostatic stress conditions, (2) stress-free ground surface, and (3) non-uniformly deforming tunnel periphery but with an inclusion of the resultant buoyant force. The ground volume loss (GVL) is prescribed as an input in the form of a Gaussian distribution of the non-uniform radial displacements around the tunnel using the Fourier series. The displacements and stresses have been both evaluated. An elastic plane strain finite elements (FE) analysis has also been performed. The ground displacements from the proposed analytical formulation and the FE analysis were compared with each other and a close comparison between the two solutions has been clearly noted. The comparisons of the obtained solutions have also been provided with those observed in field from the literature. The proposed analytical formulation will form an accurate benchmark solution for determining the stresses and displacements in a linear elastic medium for a non-uniform deforming circular tunnel with a specified magnitude of GVL.