Page 185 - Rock Mechanics For Underground Mining
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PRINCIPLES OF CLASSICAL STRESS ANALYSIS

























              Figure 6.1  An opening in a medium  analytical solutions, such as those collated by Poulos and Davis (1974), which can
              subject to initial stresses, for which  be used in excavation design. Use of any solution in a design exercise could not be
              is required the distribution of total  justified unless suitable tests were applied to establish its validity.
              stresses and excavation-induced dis-
                                          The following discussion considers as an example a long, horizontal opening of reg-
              placements.
                                        ular cross section excavated in an elastic medium. A representative section of the prob-
                                        lem geometry is shown in Figure 6.1. The far-field stresses are p yy (= p), p xx (= Kp),
                                        and p zz , and other field stress components are zero, i.e. the long axis of the excavation
                                        is parallel to a pre-mining principal stress axis. The problem is thus one of simple
                                        plane strain. It should be noted that in dealing with excavations in a stressed medium,
                                        it is possible to consider two approaches in the analysis. In the first case, analysis
                                        proceeds in terms of displacements, strains and stresses induced by excavation in
                                        a stressed medium, and the final state of stress is obtained by superposition of the
                                        field stresses. Alternatively, the analysis proceeds by determining the displacements,
                                        strains and stresses obtained by applying the field stresses to a medium containing
                                        the excavation. Clearly, in the two cases, the equilibrium states of stress are identi-
                                        cal, but the displacements are not. In this discussion, the first method of analysis is
                                        used.
                                          The conditions to be satisfied in any solution for the stress and displacement dis-
                                        tributions for a particular problem geometry and loading conditions are:

                                        (a) the boundary conditions for the problem;
                                        (b) the differential equations of equilibrium;
                                        (c) the constitutive equations for the material;
                                        (d) the strain compatibility equations.
                                        For the types of problem considered here, the boundary conditions are defined by
                                        the imposed state of traction or displacement at the excavation surface and the far-
                                        field stresses. For example, an excavation surface is typically traction free, so that,
                                        in Figures 6.1b and c, t x and t y ,or t l and t m , are zero over the complete surface of
                                        the opening. The other conditions are generally combined analytically to establish
                                        a governing equation, or field equation, for the medium under consideration. The
                                        objective then is to find the particular function which satisfies both the field equation
                                        for the system and the boundary conditions for the problem.
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