Closed-Form Solutions for a Circular Tunnel in Elastic-Brittle-Plastic Ground with the Original and Generalized Hoek–Brown Failure Criteria

The paper presents a closed-form solution for the convergence curve of a circular tunnel in an elasto-brittle-plastic rock mass with both the Hoek–Brown and generalized Hoek–Brown failure criteria, and a linear flow rule, i.e., the ratio between the minor and major plastic strain increments is constant. The improvement over the original solution of Brown et al. (J Geotech Eng ASCE 109(1):15–39, 1983) consists of taking into account the elastic strain variation in the plastic annulus, which was assumed to be fixed in the original solution by Brown et al. The improvement over Carranza-Torres’ solution (Int J Rock Mech Min Sci 41(Suppl 1):629–639, 2004) consists of providing a closed-form solution, rather than resorting to numerical integration of an ordinary differential equation. The presented solution, by rigorously following the theory of plasticity, takes into account that the elastic strain components change with radial and circumferential stress changes within the plastic annulus. For the original Hoek–Brown failure criterion, disregarding the elastic strain change leads to underestimate the convergence by up to 55%. For a rock mass failing according to the generalized Hoek–Brown failure criterion, using the original failure criterion leads to a high probability (97%) of underestimating the convergence by up to 100%. As a consequence, the onset or degree of squeezing may be underestimated, and the loading on the support/reinforcement calculated with the convergence/confinement method may be largely underestimated.