Three-dimensional roof collapse analysis in circular tunnels in rock

Collapse of a roof in circular cross-section tunnels is analyzed. The kinematic approach of limit analysis is utilized, with strength of the rock described by the Hoek-Brown failure criterion. The parametric form of the Hoek-Brown function is used to avoid introduction of an alternative explicit form of the shear strength criterion. Three measures of safety are considered: stability number, the factor of safety, and the support pressure needed to assure roof stability. The shape of the rock block in the failure mechanism consists of a right elliptic cone with a piece-wise linear generatrix and a prismatic section inserted between the two halves of the cone. The complexity of the cross-section of the block with the tunnel makes for an intricate integration of the rates of the dissipated work and the work of external forces. All three measures of safety are strongly dependent on the quality of rock described in the Geological Strength Index, whereas the dependence on the rock type captured in coefficient m_i is less distinct. The length of the roof failure mechanism is subject to a constraint dependent on the spacing of the supporting ribs in the tunnel. All measures of safety (or stability) are distinctly dependent on the length constraint; the shorter the spacing between the ribs, the safer the tunnel against roof collapse. The 2D analysis yields the most conservative outcome.