Instability of Initially Anisotropic Chain Pillars in Vertical Compression
A. I. Chanyshev, I. M. Abdulin, O. E. Belousova, O. A. Luk’ashko
Chinakal Institute of Mining, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia
Keywords: Initial anisotropy, chain pillar, characteristic equation, simple roots, multiple roots, critical load, height, preset load, allowable width
Abstract
A chain pillar is analyzed in plane deformation under vertical compression. It is assumed that the pillar is initially anisotropic, with variation of all six constants of elasticity. The influence of the initial anisotropic on the critical load in case of the pillar instability is estimated using the Leibenzon-Ishlinsky approach. For different multiplicities of roots of a characteristic equation, depending on constants of elastic stiffness, the values of critical loads when the pillar loses stability are found. In case of the simple roots, the critical load is found from the equality of the forth order determinant to zero. For the multiple roots, an analytical formula is obtained, which connects the critical load with the elastic constants and geometrical parameters of the pillar, generalizing the known relation for the initial isotropic medium by Ishlinsky. The influence of the ratio of all elastic constants to the Young modulus of material under vertical compression on the peak load values is analyzed. The cases when the critical width of a pillar, such that it can lose its stability, is determined at the known height of the pillar and its preset load are discussed. The substantial role of different moduli of the test material in the compression direction and across it is revealed, and the quantitative estimates of the influence exerted by the parameters of the material anisotropy on the critical load values are obtained.
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