On the elastic distortion of a cylindrical hole by a localised hydrostatic pressure

When a hydrostatic pressure is applied over only a small part of the length of a cylindrical hole extending through an infinite elastic solid, the stresses and displacements differ considerably from those caused by the application of this pressure over the entire length of the hole. This problem has been discussed by H. M. Westergaard1 using an approximate method but it is not easy to assess the accuracy of his numerical results. It is the purpose of the present note to give an exact solution and to compare numerical results with those given by Westergaard. The analysis used here is a simple adaptation of that given by A. W. Rankin2 for the similar problem of a band of uniform pressure applied to a long cylindrical shaft. The numerical calculations are not so formidable as would appear at first sight and a method given by L. N. G. Filon3 for evaluating trigonometric integrals has proved very valuable in this connection. The results for the maximum radial displacement show that the approximation used by Westergaard is rather crude. 1. The analytical solution. We use cylindrical coordinates and consider the pressure loading as being given by c on the surface of the cylindrical hole r — a. With the usual notation4 we therefore require to find a stress function 4> satisfying V4<£ = 0, r > a, — oo < z < oo, (1)