Mining - Roof Slope at Deflected Supports

Analysis of a mine roof can be based on fixed-end beam behavior. The author here analyzes the effects of zero restraint at deflecting beam supports. Formulae are given for determining permissible support deflections. Analysis of a mine roof in a bedded deposit usually follows the assumption that the behavior of the roof is analogous to a beam with both ends fixed. Yet it is recognized that the end restraint itself varies with the thickness of cover, so that at shallow depth the restraint is minimum, and at great depth, maximum.' Consequently, an analysis on the basis of fixed ends or full restraint represents only one extreme in the possible range of conditions. Since the actual amount of restraint is difficult to ascertain, a more rational approach is to consider the two extreme cases of end restraint. Such situations regarding uncertainty of the restraining effects of supports are not uncommon in structural analysis, and are handled in a like manner.' Since previous studies9)4 have dealt with full restraint, this paper will analyze the effects of zero restraint at deflecting beam supports. In such a case a mine roof with several intermediate supports becomes analogous to a uniformly loaded continuous beam on simple supports, as shown in Fig. 1A. Such a structure is statically indeterminate to a degree equal to the number of intermediate supports, n. Therefore, with the three equilibrium equations, n additional equations are needed. A number of methods exist to analyze this problem, but The Theorem of Three Moments is Simplest. The continuous beam is divided at each support into a sequence of simply supported single span beams, with the unknown end moments, as appears in Fig. 1B. Continuity requires that the common ends of adjacent beams have the same slope (Fig. 1C). eiB= B(i+l)~= ei(i+l)and MiB= Mti+I)A= Mi(i+i). [1I A Glossary of all terms is given in Table I. Furthermore, if we first consider that the supports do not deflect, the slope at a support is the sum of the slopes due to each of the individual loads shown in Fig. 2, yielding: