Page 225 - Rock Mechanics For Underground Mining
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EFFECT OF PLANES OF WEAKNESS ON ELASTIC STRESS DISTRIBUTION






















              Figure 7.9  An inclined, radially ori-
              ented plane of weakness intersecting a
              circular excavation.
                                                                     ◦
                                        a feature inclined at an angle of 45 , the normal and shear stress components are
                                        obtained by substitution in the Kirsch equations (equations 6.18), and are given by
                                                                    p          a 2
                                                            n =   

 =  ∗ 1.5 1 +
                                                                    2          r 2
                                                                    p          2a 2  3a 4
                                                            =   r
 =  ∗ 0.5 1 +    −
                                                                    2           r 2  r 4
                                          The variation of the ratio  /  n is plotted in Figure 7.9b. The maximum value of the
                                        ratio, 0.357 at r/a = 2.5, corresponds to a mobilised angle of friction of 19.6 . The
                                                                                                      ◦
                                        far-field value of the ratio of the stresses corresponds to a mobilised angle of friction of
                                        18.5 . If the rock mass were in a state of limiting equilibrium under the field stresses,
                                            ◦
                                        the analysis indicates that mining the excavation could develop an extensive zone of
                                        slip along the plane of weakness. On the other hand, an angle of friction for the plane
                                        of weakness exceeding 19.6 would be sufficient to preclude slip anywhere in the
                                                               ◦
                                        medium.
                                        Case 5.  (Figure 7.10) The design problem shown in Figure 7.10a involves a circular
                                        opening to be excavated close to, but not intersecting, a plane of weakness. For
                                        purposes of illustration, the stress field is taken as hydrostatic. From the geometry
                                        given in Figure 7.10a, equations 7.2 for the stress distribution around a circular hole
                                        in a hydrostatic stress field, and the transformation equations, the normal and shear
                                        stresses on the plane are given by

                                                                          1
                                                              1
                                                           n = (  rr +   

 ) + (  rr −   

 ) cos 2
                                                              2           2
                                                                    a
                                                                     2
                                                            = p 1 −    cos 2
                                                                    r 2
                                                                         1
                                                            =   r
 cos 2  − (  rr −   

 ) sin 2
                                                                         2
                                                               a 2
                                                            = p   sin 2
                                                               r 2
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