A study has been made of the stability of blocks and wedges of rock subjected to gravity forces in the roofs of underground excavations. Rock wedges may become wholly or partly self supporting with the mobilization of shear resistance of discontinuities bounding such wedges. The influence of in-situ stresses and the relative stiffnesses of the intact rock, and shear and normal stiffnesses of the discontinuities has been investigated. A two dimensional, plane strain model of symmetric rock wedges in a horizontal stress field has been utilized. A comparison between the results of an extension of a closed form solution proposed by Bray(1981), and numerical analyses with Displacement Discontinuity elements (Crouch, 1976), are presented. These analyses indicate that the stability of the wedge and the stress redistribution around the wedge upon excavation of an opening is markedly influenced by the ratio of intact rock stiffness to seam normal and shear stiffness. It is shown that the analytical solution provides a reasonable measure of the lower and upper bounds of the loads necessary to cause failure of the wedge.
The problem considered (Fig. 1) is a two dimensional plane strain model containing two discontinuities. The discontinuities are symmetrically disposed with respect to a horizontal excavation which forms a wedge of rock bounded by these discontinuities. The angle of the intersecting discontinuities at the apex of the wedge is 2a. Mechanical properties of the two discontinuities are identical, being represented by linear normal and shear joint stiffnesses Kn and Ks and angle of friction ø. The rock wedge is loaded by an in-situ horizontal stress field oh and a surface traction t which represents the self weight of the rock wedge or an external force applied to the wedge. Only cases in which the half wedge apex angle (a) is less than the discontinuity friction angle ø have been considered. Thus, prior to excavation of the opening, the discontinuity is able to transmit the horizontal stress field without the discontinuity being in a state of limiting equilibrium. With excavation of the opening, the wedge is stabilized by the horizontal stress field. A downward force T (=2at) must be applied to cause the wedge to fail.
The Displacement Discontinuity (DD) method is an indirect boundary element formulation based on a solution which expresses displacements and stresses resulting from constant normal and shear displacement discontinuities over a finite line segment in an elastic body. The method comprises placing displacement discontinuities along the boundary region being analyzed. Seam or discontinuity elements may be represented as displacement discontinuity elements, the opposite faces of which are connected by elements representing the shear and normal stiffnesses (Ks and Kn) of the seam. In the formulation used here the normal and shear stiffnesses of the discontinuities have been modelled by linear elastic elements, but in general, non-linear discontinuity elements may be modelled with the programme. Seam failure is incorporated by imposing a constraint on the maximum shear stress that can be developed in a seam.