Relation between stresses and shifts at the periphery of a working

Abstract

Extensive use is made of the methods of mechanics of continua to solve problems of rock mechanics.The complexity of problems of rock mechanics is due not only to the fact that the shape of individual workings is other than round (their number may be arbitrary) but also to the diverse mining-geological conditions and properties of rocks. This leads to considerable difficulties when evaluating underground structures for strength and stability.Regardless of the model of the medium, the theory of fracture and flow, an elastic solution is assumed as a first approximation.Existing schemes for evaluating mine workings represent the real solid rock as a homogeneous isotropic medium [i, 2]. They reflect to a reasonable degree the basic physical and kinematic characteristics of the behavior of rocks. The use of conformal mapping has widened the range of solvable problems [4, 5]. Boundary-value problems for a plane with a circular hole, and a half-plane, have been the most thoroughly examined. Touching on these are problems for which the function representing the real periphery on a circle or half-plane has been found.But if the representing function is irrational, to find ~(z) we obtain a singular integral equation which must be solved.Even with the most simplifying assumptions concerning contact, the modeling of a rock mass by a set of uniform particles in contact markedly complicates the problem.Obtaining a practical solution is problematic.Taking account of the overall complexity of description of real rocks, the most important trend is the experimental-analytical method of determining the stress--strain state in the vicinity of a working [6, 7]. It employs (and this is highly significant) field measurements of shifts along lines which are formed as the boundary conditions for a half-plane; as in the case of analytical solutions, we must find the functions ~(z) and ~(z), and we then describe the components of the stress and deformation tensor over the whole region, including the boundary.In practice, e.g., it is necessary to find the state of stress of the periphery of the working directly from the measured components of the boundary shifts directly, without actual determination of the complex potentials ~(z) and ~(z), or vice versa.However, kndwing the stresses and shifts at the periphery we can readily find ~(z) and ~(z), i.e., expand the solution to the region.Many methods of stress measurement in rocks are based on conversion of the measured deformations of a borehole periphery to stresses [8].Below we examine the problem of determination of the stress--strain state at the periphery of holes for the basic boundary problems without actual determination of the functions ~ (z) and ~(z).Let us consider a body S-bounded by a contour r (the complement of S-+ F with respect to the whole plane is called S+), the same stress--strain state of which is defined by various combinations of boundary conditions.We will keep to the notation in [3] throughout the remainder of this article.The boundary problems of the first and second...

Stress-Strain State of a Rock Mass With a Crack in the Roof

“…В данной работе затронут вопрос определения напряженно-деформированного состояния массива пород, ослабленного прямоугольной выработкой с трещиной, расположенной в кровле [1,2]. Исследование строится на основе системы сингулярных интегральных уравнений, связывающих все граничные значения компонент напряжений и смещений [3].…”

“…Система сингулярных интегральных уравнений, связывающая граничные значения компонент напряжений и смещений для произвольной односвязной области, приведена в [3], где…”

“…На основании [3] и условий (2) можно записать соотношения, связывающие граничные значения компонент напряжений и смещений на контуре выработки и линиях контакта. Они аналогичны тем, что получены в работах [4 -6], и из-за громоздкости записи не приводятся.…”

Stress-strain state of enclosing rock mass around an arbitrary cross-section excavation by measurement of displacements of the excavation walls

“…As an illustration, under analysis is the problem on an arbitrary-shaped excavation, with overspecified (in terms of the classical statement of the problem) conditions [1][2][3][4][5][6]. …”