INTRODUCTION
ABSTRACT: The paper applies a new method based on block theory, stress analysis and linear programming to study the stability of a statically indeterminate 2-d block under arbitrary loads. The stability of keyblocks subjected to a range of directions of a fictitious loading or support force is analyzed using linear programming. Based on this optimization procedure, a keyblock can be classified as stable, unstable or potentially unstable. The range of loading force where a keyblock may be potentially unstable (uncertainty in stability) is affected by the indeterminacy of the force system and the angle of applied load. For a wide range of applied load angles, the uncertainty is small. Hence, the technique presented here would provide useful results for design.
Block theory is a useful analysis tool for investigating the stability of a rock mass (Goodman & Shi 1985). The method of block theory is based on a geometric analysis of blocks formed by the intersection of a particular set of half-spaces. First, all infinite and non- removable blocks are eliminated by kinematic analysis. Second, block theory uses kinetic analysis to differentiate between stable and unstable blocks without regard to surface metions arising from in-situ stresses. Finally, based on the physical and mechanical properties of the joint planes, the stability of keyblocks is assessed. If the keyblock is unstable, the required support force is calculated, and the support scheme is designed accordingly (Hatzor & Goodman 1993, Warburton 1993, Windsor 1997).
The standard block theory approach, however, can lead to excessively over-conservative design because it ignores the stabilizing effect of surface forces arising from field stresses (Mauldon et al. 1997a,b). Roof blocks bounded by steeply inclined joints in the crown of a tunnel, for example, may be stable because of frictional shear stresses on the joint plane(s). Block theory, however, would predict such roof blocks to be unstable under gravitational loading.
Not only does block theory lead to over- conservative design, but when the action of additional loading or support forces is considered, the problem becomes statically indeterminate. Current approaches to handling the indeterminacy include relaxation techniques (Brady & Brown 1993) and numerical methodsuch as DDA (Shi 1988, Yeung 1991, Lin et al. 1996), and UDEC (Cundall & Hart 1993). Geologic uncertainties in tunneling have been discussed by Einstein et al. (1996).
Using linear programming, limits on the stability of 2-d keyblocks under the combined influence of field stresses, self weight, and additional loading or support forces can be evaluated. The paper uses this optimization technique to examine the effects of applying loading forces at various angles relative to the free face of a keyblock in the crown of a tunnel (see Table 3 and Fig. 4).