Roof stability in deep rock tunnels

A method is presented addressing quantitative assessment of tunnel roof stability, based on the kinematic approach of limit analysis. Long tunnels with both rectangular (flat-ceiling) and circular cross-sections are considered. The rock is governed by the Hoek-Brown strength envelope and the normality flow rule, and it is assumed to provide enough ductility at failure, making plasticity theorems applicable. A failing block in the collapse mechanism is separated from the stationary rock by a deformation band with a large gradient of velocity across its width. The shape of the block in the critical mechanism is found from the requirement of the mechanism's kinematic admissibility and an optimization procedure consistent with respective measures of stability. The stability number and the supporting pressure needed for tunnel stability are calculated first. Although less commonly used in rock engineering, a procedure is developed for estimating the factor of safety, defined as the ratio of the rock shear strength determined from the Hoek-Brown criterion to the demand on the strength. Curiously, for flat-ceiling tunnels, such definition of the factor of safety yields results equivalent to the ratio of a dimensionless group dependent on the uniaxial compressive strength and the size of the tunnel to the stability number. Such an equivalency does not hold for tunnels with ceilings of finite curvature. Not surprisingly, all measures of tunnel roof stability are strongly dependent on the Geological Strength Index that describes the quality of the rock.