Kinematic Analysis of Block Rotations in a Jointed Rock Mass using Graphical and Vector Methods

"The kinematic analysis of rock blocks is one part of a block stability assessment. It provides information about a block’s movability, i.e. if a block can detach from the adjacent rock mass and move towards the free space. This paper introduces an analytical method for the analysis of pure rotations of arbitrary finite polyhedral blocks. It enables the engineer to judge a block’s rotatability about a corner or an edge. The method is presented using a graphical and computational method. The application of the method is illustrated with an example. INTRODUCTIONThe kinematic analysis of rock blocks deals with the determination of the movability of blocks. Established methods typically cover: (a) translational motion (John 1968, Markland 1972, Londe et al. 1969, 1970, Goodman & Shi 1985, Warburton 1981) and (b) the specific rotational cases Toppling (Goodman & Bray 1977) and Slumping (Kieffer 1998). Mauldon & Goodman (1990, 1996) and Tonon (1998) introduced rotational kinematic analyses for three-joint pyramids and tetrahedral blocks extending Goodman & Shi’s block theory.All mentioned methods are subject to assumptions that limit the applicability to engineering problems. For example, the kinematic test of daylighting joints or intersections inherently assumes a finite block, which only can be proved by the finiteness theorem (Goodman & Shi 1985). Daylighting of an intersection or plane is not a sufficient condition for removability. Goodman & Shi’s and Warburton’s methods are limited to translational block motion. The methods for block rotations so far address tetrahedral blocks only. They also address only pure rotational modes and neglect the general motion of a rigid body. This paper introduces an analytical method for the analysis of the kinematic rotatability of finite rock blocks. It extends Mauldon's and Tonon's theory of rotatability of tetrahedra to arbitrary polyhedral blocks. A graphical method using the stereographic projection and a computational method are presented."