An Analytical Solution for Tunneling via Virtual Cylinder Model

Deriving rigorous elastoplastic analytical solutions for shallow tunnels subject to a gravity-induced stress gradient presents significant mathematical challenges. This paper introduces a virtual cylindrical structure model to derive a closed-form elastoplastic solution for tunnel excavation. By evaluating the static equilibrium of infinitesimal elements, the methodology explicitly determines the plastic zone boundary via the Lambert W function and yields the elastoplastic distributions of stress and displacement fields under the Mohr–Coulomb criterion. The reliability of the derivations is verified by degenerating the equations under specific boundary conditions and comparing them with classical Lamé solutions, showing agreement at low friction angles (⌀ = 5° − 10°). A case study of a 14.5 m-diameter shield tunnel in the Yangtze River Delta is conducted to demonstrate its practical application. The analytical results show that the maximum convergence displacement is controlled within 15 mm, and a ground loss rate of 1.82% corresponds to an unloading ratio of 40%. The proposed method provides a theoretical tool for preliminary estimating excavation-induced disturbances in shallow homogeneous strata.