Vector Analysis of Keyblock Rotations

The mechanical response of rock slopes and excavations depends largely on the geometry of discontinuities and associated rock blocks. Commonly used rigid block models for determining stability, such as the sliding wedge model or the keyblock analysis, assume that initial displacements are pure translations. This paper examines kinematic and kinetic constraints on block rotations. The kinematic constraints include conditions that are independent of the free surface and therefore belong uniquely to the joint pyramid, as well as conditions that depend on the orientation of the free surface. The relevant criteria, expressed in terms of vector inequalities, are represented on the stereographic projection, allowing graphical solution of the kinematical inequalities. Additionally, rotational equilibrium of tetrahedral blocks is considered, the failure modes of block theory are generalized to include rotational modes and procedures for evaluating rotational stability are discussed. The results show that blocks found to be stable with respect to the translational modes of sliding and falling may yet fail by rotation.