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Fast methods to solve the unloading problem of a cylindrical cavity or tunnel excavated in elasto-perfectly plastic, elasto-brittle or strain-softening materials under a hydrostatic stress field can be derived based on the self-similarity of the solution. As a consequence, they only apply when the rock mass is homogeneous and so exclude many cases of practical interest. We describe a robust and fast numerical technique that solves the tunnel unloading problem and estimates the ground reaction curve for a cylindrical cavity excavated in a rock mass with properties depending on the radial coordinate, where the solution is no longer self-similar. The solution is based on a continuation-like approach (associated with the unloading and with the incremental formulation of the elasto-plastic behavior), finite element spatial discretization and a combination of explicit sub-stepping schemes and implicit techniques to integrate the constitutive law, so as to tackle the difficulties associated with both strong strain-softening and elasto-brittle behaviors. The developed algorithm is used for two practical ground reaction curve computation applications. The first application refers to a tunnel surrounded by an aureole of material damaged by blasting and the second to a tunnel surrounded by a ring-like zone of reinforced (rock-bolted) material.
1. Introduction
The convergence-confinement method (CCM) is a technique that quantifies the interplay between tunnel and installed support in terms of strains and stresses. The method relies on three components: the ground reaction curve (GRC), which describes the relationship between diminishing internal pressure and deformation on the tunnel spring-line; the longitudinal deformation profile, which relates the deformation on the spring-line to distance to the face; and the support characteristic curve, which represents the stress–strain relationship for the support system. These together enable the designer to estimate the performance of the support system. In geomechanical practice, the method is usually applied, for design purposes, to tunnels where stresses are expected to surpassrock mass strength, that is, deep tunnels excavated in average to poor quality rock masses.
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