Page 188 - Rock Mechanics For Underground Mining
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METHODS OF STRESS ANALYSIS

































              Figure 6.2 A thick-walled cylinder
              of elastic material, subject to interior
              pressure, p i , and exterior pressure, p o .

                                        requirement for uniqueness of displacements and the pressure boundary conditions
                                        for the problem. It can be shown that uniqueness of displacements requires B = 0,
                                        and that the stress components are then given by
                                                                        A
                                                                     rr =  + 2C
                                                                        r 2
                                                                          A
                                                                    

 =−   + 2C
                                                                          r 2
                                                                     r
 = 0                            (6.8)

                                        where
                                                                        2 2
                                                                      a b (p i − p o )
                                                                  A =     2   2
                                                                        (b − a )
                                                                          2
                                                                       p o b − p i a 2
                                                                 2C =
                                                                              2
                                                                         2
                                                                        (b − a )
                                          In these expressions, a and b are the inner and outer radii of the cylinder, and p i
                                        and p o are the pressures applied to its inner and outer surfaces.
                                          For problems involving two-dimensional geometry and biaxial stress, the most
                                        elegant solution of the biharmonic equation is obtained in terms of complex variable
                                        theory. The topic is discussed in some detail by Jaeger and Cook (1979). Briefly, it is
                                        shown that the Airy stress function may be expressed as the real part of two analytic
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