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EXCAVATION SHAPE AND BOUNDARY STRESSES
























              Figure 7.11  Definition of nomen-
              clature for an elliptical excavation
              with axes parallel to the field stresses.


                                        7.4 Excavation shape and boundary stresses

                                        The previous discussion has indicated that useful information on boundary stresses
                                        around a mine opening can be established from the elastic solution for the particular
                                        problem geometry even in the presence of discontinuities. It is now shown that simple,
                                        closed form solutions have greater engineering value than might be apparent from a
                                        first inspection.
                                          Figure 7.11 illustrates a long opening of elliptical cross section, with axes parallel
                                        to the pre-mining stresses. For the particular cases of   = 0,   = 0, and   = 0,   =
                                         /2, equation 6.21 reduces to


                                                                                       2W

                                                       A  = p(1 − K + 2q) = p 1 − K +                  (7.6)
                                                                                         A

                                                                    2K                   2H

                                                       B = p K − 1 +     = p K − 1 + K                 (7.7)
                                                                     q                      B
                                        where   A and   B are boundary circumferential stresses in the sidewall (A) and crown
                                        (B) of the excavation, and   A and   B are the radii of curvature at points A and
                                        B. Equation 7.6 indicates that if   A is small,   A is large. Equation 7.7 defines a
                                        similar relation between   B and   B . A generalisation drawn from these results is
                                        that high boundary curvature (i.e. 1/  ) leads to high boundary stresses, and that
                                        boundary curvature can be used in a semi-quantitative way to predict boundary
                                        stresses.
                                          Figure 7.12 shows an ovaloidal opening oriented with its major axis perpendicular
                                        to the pre-mining principal stress. The width/height ratio for the opening is three, and
                                        the radius of curvature for the side wall is H/2. For a ratio of 0.5 of the horizontal and
                                        vertical field principal stresses, the sidewall boundary stress is given, by substitution
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