Block theory is an approach to stability analysis in fractured rock which examines sequentially the kinematics, mode and stability of rock blocks. This paper describes an alternative kinematic analysis based on the joint pole pyramid (dual to the joint pyramid) and the set of space pyramid poles. The proposed model works directly with the raw data of keyblock analysis, namely joint set and free surface poles, and allows explicit treatment of variable orientation data. A useful distinction between local and global removability, revealed by the proposed model but invisible to conventional block theory, is introduced and removability redefined to include local and global removability as sub-cases. The model has simple graphical interpretations, has considerable practical utility and is easy to apply in practice.
Block theory is a geometric approach to rock mass mechanics which examines in sequence kinematic constraints on block displacements, keyblock failure modes and keyblock stability. At the heart of block theory is the concept of removability. A block is removable if it is kinematically free to translate towards free space, and non-removable if the translation is blocked by the adjacent rock mass. For removability, the corresponding joint pyramid must be contained within the space pyramid, or JPSP. This useful result, which originated with Shi (1976) and Goodman and Shi (1985) has been applied successfully to a number of rock engineering projects (e.g., Scott and Kottenstette, 1993; Hatzor 1995; Vogt & Boyle 1995; Yeung 1995). This paper offers a broader interpretation, and a new definition, of removability. The basic notion - kinematic feasibility - remains unchanged, but the concepts of global and local removability are introduced and the definition of removability expanded to include local and global removability as sub-cases. The paper is limited to blocks exposed at a convex free space. The conditions for global removability, local removability and removability are expressed in terms of the joint pole pyramid JPP (dual to the joint pyramid), and the set S of space pyramid poles. Equivalent to Shi's condition JPCSP, for example, is the reciprocal statement S CJPP, which, together with JP ^ 0, is a necessary and sufficient condition for global removability (to use the new terminology) at a convex free space. The JPP-S formulation has a number of advantages over the JP-SP (or JP-EP) formulation of Goodman and Shi (1995), including the following. (1) Local and global removability, which are identifiable by the methods described in this paper but are invisible to the standard block theory interpretation of removability, are powerful concepts with real engineering utility. (2) The JPP-S formulation allows one to deal explicitly with variable discontinuity orientations (Mauldon, 1993). (3) The JPP-S formulation allows one to define the free space once and for all in terms of the set S of free face poles. Global and local removability are then determined by examining the intersection of the set S with JPP (S Ð /PP), which gives the precise range of removability of the object under study.