Consideration of secondary blocks in key-block analysis

Abstract

Abstract The theorem on the removability of a finite convex block states that a convex block is not removable (tapered) if its block pyramid is empty (BP=e) and its joint pyramid is also empty (JP=e), (Goodman, Shi 1985). In stereographic projection, this non-removable finite block is indicated by the missing projection plane. When the primary key-block is removed, the adjacent block which was originally non-removable could become a removable block. Such new removable blocks are designated as secondary key-blocks. Design engineers have a critical need to know both the primary and secondary key-blocks which may exist, but may not be evident, in the initial analysis. For example, to design the length of a rock anchor, the design engineer has to be aware that the key-block support tied to a secondary key-block may not be safe. The behavior of these secondary key-blocks was investigated by evaluating the originally non-removable blocks surrounding the primary key-block. When a primary key-block is removed, the contact surface between it and the next block becomes a free surface. The number of joints decrease while the number of free surfaces increase. This study was initiated by solving the two dimensional problem. After the formulations were developed, the approaches were extended to three dimensional problems. The study was done on non-repeated and repeated rock joint systems. The results of this study show that the secondary key-block analysis is a powerful tool for analyzing a progressive failure.