Modeling cleaned excavations by plane mathematical solutions

Abstract

Laws governing the distribution of stresses and displacements around mine excavations are taken as the basis of practical recommendations for the mining of mineral resources.Familiar methods of computing the three-dimensional stress-strain state around cleaned excavations are based on either very rigorous assumptions that reduce to consideration of only one component ozz (see, for example, [i]), or numerical methods of implementing equations of elasticity theory, which also employ simplifying assumptions (symmetry, form of approximation, etc.).It is essential that even incomplete data on the deformation process of a mass with excavations (for example, calculation of only the normal stress component azz, displacements w, etc.) assume major significance in practice since they make it possible to answer a number of important specific questions over a broad range of mining-engineering situations that are encountered in conducting mining operations [1][2][3][4].A method of solving three-dimensional problems, which encompasses a class that permits the modeling of recesses by plane mathematical solutions, is developed below. Examples are cited for the computation of the stress-strain state of the rock mass around cleaned excavations for which one of the characteristic dimensions is considerably smaller than the other two. The simultaneous investigation of all three basic problems of rock mechanics within the framework of a single approach defining the nine components of the stress-strain state at a point opens new possibilities for the practical use of the method of modeling by plane mathematical solutions for the geomechanical basis of technological schemes for working mineral deposits. The proposed method essentially utilizes a scheme based not on the desire to obtain an immediate solution for a space with excavations, but on partitioning of the initial region into subregions for which a solution is known.For the case in question, this subregion is a half space. The most complex case when boundary conditions are formulated in stresses, i.e., when (i.i) can be represented, for example in the form o,, = P, o= = g = av, = r = O. is noted here.For this case, the system of equations as determined from (1.2) for condition (1.3) will be [5] Mining Institute, Siberian Branch, Academy of Sciences of the USSR, Novosibirsk.Translated from No. 5, pp. 22