Controlled roof collapse during secondary mining in coal mines (Q446357)

Summary: 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.