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International Journal of Differential Equations
Volume 2012 (2012), Article ID 806078, 21 pages
http://dx.doi.org/10.1155/2012/806078
Research Article

Controlled Roof Collapse during Secondary Mining in Coal Mines

School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa

Received 26 December 2011; Accepted 10 April 2012

Academic Editor: Jean Charpin

Copyright © 2012 Ashleigh Hutchinson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The problem considered is an investigation of the possible collapse of the roof between the pillar next to be mined in secondary coal mining and the first line of pillar remnants called snooks. The roof rock between the pillar, which is the working face, and the snook is modelled as an Euler-Bernoulli beam acted on at each end by a horizontal force and by its weight per unit length. The beam is clamped at the pillar and simply supported (hinged) at the snook. The dimensionless differential equation for the beam and the boundary conditions depend on one dimensionless number 𝐵 . We consider the range of values of 𝐵 before the displacement and curvature first become singular at 𝐵 = 𝐵 1 . The model predicts that for all practical purposes, the beam will break at the clamped end at the pillar. The failure of the beam for values of 𝐵 greater than 𝐵 1 is investigated computationally.