The response of flat square rock roofs is examined numerically using a three dimensional explicit finite difference code. Roofs are modeled as plates, loaded by their own weight, vertically cracked with persistent joints along their abutments and their diagonals. Numerical results are presented in graphs for the normal stress distribution at the abutments and the diagonal cracks, the contact length within the joints at these sections and the deflection occurring at the centre of the plate. Conclusions for the critical plate parameters are derived.


Flat roofs are common in underground exploitation within stratified rock. Vertical deflection of such roofs detaches the lower strata from the above ones, creating thus a plate structure that bears its own weight only. If such plates are long enough in one direction compared to the other, they may be considered as rock beams and analyzed accordingly. However, when such plates have side lengths of similar magnitude, the assistance given to the structure by all edges may not be ignored, and the analysis should account for this. A simple case may thus pertain to a square plate rock roof bearing on all sides on stiff walls. Such a case may arise in the roof of an underground quarry that consists of hard rock thick strata, intact or jointed. Their stability may be governed by the ability of the roof rock structure to bear the imposed internal forces. These forces may cause initially cracking of the plate roof, although the cracked structure may continue to sustain load similarly with the case of a voussoir beam. Therefore it is worth examining the response of such platy roofs in the cracked stage, and to evaluate their limiting state. The mechanical response of such square rock roofs is endeavored, by employing threedimensional simulation in the cracked horizontal plate models of various shape factors, which are loaded by their own weight. Cracking of these plates may be assumed as persistent vertical planes along their abutments and diagonals. Such a mode of cracking corresponds to the yield line assumption employed for the analysis of reinforced concrete plates, already proposed by Beer and Meek [1]. Thus, for each roof rock plate 4 triangular solid rock prisms are formed that interact with each other.


Let us consider a thin elastic square rock plate, with side length s and thickness t, clamped along all its edges (Figure 1). distributed loading, tensile failure may be assumed at the abutments when the maximum tensile stress, acting in the upper part of the plate, exceeds the tensile strength of the rock. If we further assume that vertical tensile fractures form at the abutments, then, provided that slip is prevented, the plate may be considered as simply supported. For a simply supported square plate, the maximum stresses are shown to be near its centre, and further fracturing of the plate there may be assumed.

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