A methodology is presented for the calculation of the load carrying capacity of a 2D symmetric roof rock wedge within a confining surrounding stress field, taking into account the progressive nature of yield on the wedge joints. A two stage joint relaxation procedure is employed that allows for an initial elastic stress distribution. For any vertical displacement that strains the wedge joints, formulae for the calculation of the associated vertical resistance are provided. The carrying capacity of the wedge is found as the maximum calculated vertical resistance. Validation of the procedure is obtained with a distinct element code. A parametric investigation highlights the effect of joints strength and deformability on the pull out resistance of the rock wedge.
Potentially unstable rock wedges in the roof of underground excavations may be identified on the basis of the orientation and strength of the joints forming the wedge and the orientation of the excavation surfaces. Kinematical methods [1, 2, 3, 4] provide a convenient means for the assessment of the movement feasibility of polyhedral blocks around an excavation, but do not normally take into account the stress field around the block. However, as noticed by Goodman , a rock wedge with joint faces deeply inclined to the excavation walls may be considerably reinforced by friction mobilisation due to the confining stresses acting on the joint faces, as long as confinement is maintained despite weathering, relaxation, blasting , or any mining activity. The analysis of the stability of such a confined rock block has been originally proposed by Bray  who provided an analytical solution by assuming a two-stage relaxation procedure. In the first stage the joints are assumed infinitely stiff and the excavation is performed in a homogeneous, isotropic, linearly elastic, weightless medium; at this stage the confining force acting on the wedge is evaluated. In the second stage, the joints are assumed flexible and the rock mass is assumed rigid. During this stage, loading is due to the weight of the wedge, any supporting force and the confining forces calculated in the first stage. Based on the concepts of Bray , Goodman  calculated the support requirements of a polyhedral block under the influence of gravity, support and the in-situ stress field. For the latter, simplifying assumptions were employed to allow for a closed form solution to be derived. Crawford and Bray  further investigated the extent of the joint shear resistance mobilization, which occurs as the wedge deforms. Brady and Brown  gave analytic expressions for the factor of safety of long, uniform triangular prisms against falling from an underground roof. Sofianos  calculated the carrying capacity of symmetric and asymmetric roof wedges with symmetric loading on the joint faces and further analyzed the effect of a deformable wedge body. Elsworth  combined the solution of Bray  with an analytic solution for the forces acting on a 2D rock wedge adjacent to a circular tunnel in a plane hydrostatic stress field.