Analytic solution for an underground tunnel of arbitrary shape at a moderate depth

This paper focuses on the analytic determination of the stress field around a horizontal tunnel buried in an elastic medium at a moderate depth. The cross section of the tunnel is assumed to be arbitrary, and the medium is subjected to a vertical gravity and a lateral pressure in the horizontal direction. In contrast to the cases of deep-buried tunnels in which a constant initial stress field induced by the gravity and lateral pressure before tunnel excavation is used to derive corresponding classical solutions, the current case of a moderate-depth tunnel necessitates the consideration of a non-constant initial stress field varying linearly with the depth coordinate. In this setting, we employ several analytic techniques, in the context of the complex variable formalism for plane elasticity, to derive a modified solution for the full stress field in the medium after tunnel excavation and obtain a closed-form formula for evaluating the hoop stress around the tunnel. Comparisons with the finite element results, for a rectangle-semicircle-shaped tunnel at a depth approximately equal to two times the diameter of the tunnel, are made to validate the modified solution. Numerical examples are presented to illustrate the stress concentration around moderate-depth tunnels of equilaterally triangular shape, square shape and rectangle-semicircle shape. It is found that the modified solution deviates significantly from the classical counterpart in determining the stress concentration at the corners of the above-mentioned tunnels, and the differences between the modified and classical solutions depend highly on the ratio of the lateral pressure to the vertical pressure (caused by gravity).