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Controlled roof collapse during secondary mining in coal mines. (English) Zbl 1248.34028

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.

MSC:

34B60 Applications of boundary value problems involving ordinary differential equations
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
34C23 Bifurcation theory for ordinary differential equations
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References:

[1] J. N. van der Merwe, “Fundamental analysis of the interaction between overburden behaviour and snook stability in coal mines,” Journal of the South African Institute of Mining and Metallurgy, vol. 105, pp. 63-73, 2005.
[2] C. Please, D. P. Mason, C. M. Khalique, J. M. T. Ngnotchouye, J. N. van der Merwe, and H. Yilmaz, “Coal mine pillar extraction,” Submitted to International Journal of Rock Mechanics and Mining Sciences.
[3] N. van der Merwe, “Private communication,” 2012.
[4] C. Kimball and T.-W. Tsai, “Modeling of exural beams subjected to arbitrary end loads,” Journal of Mechanical Design, vol. 124, pp. 223-235, 2002.
[5] B. H. G. Brady and E. T. Brown, Rock Mechanics for Underground Mining, Chapman & Hall, London, UK, 2nd edition, 1993.
[6] L. Obert and W. L. Duvall, Rock Mechanics and the Design of Structures in Rock, Wiley & Sons, New York, NY, USA, 1967.
[7] E. P. Popov, Mechanics of Materials, Prentice Hall International, Englewood Cliffs, NJ, USA, 1978. · Zbl 0435.73019
[8] J. C. Jaeger and N. G. W. Cook, Fundamentals of Rock Mechanics, Methuen, London, UK, 1969.
[9] J. C. Jaegar, N. G. W. Cook, and R. W. Zimmerman, Fundamentals of Rock Mechanics, Blackwell, Oxford, UK, 4th edition, 2007.
[10] L. A. Segal and G. H. Handelman, Mathematics Applied to Continuum Mechanics, Macmillan, New York, NY, USA, 1977. · Zbl 0371.73006
[11] J. N. van der Merwe and B. J. Madden, in Rock Engineering for Underground Coal Mining, Special Publications, Series 8, p. 135, Southern African Institute of Mining and Metallurgy, Johannesburg, South Africa, 2nd edition, 2011.
[12] J. N. van der Merwe and B. J. Madden, in Rock Engineering for Underground Coal Mining, Special Publications, Series 8, pp. 208-218, Southern African Institute of Mining and Metallurgy, Johannesburg, South Africa, 2nd edition, 2011.
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