The axisymmetric mixed problem of elasticity theory for a cone clamped along its side surface with an attached spherical segment

Abstract

The axisymmetric mixed problem of the stress state of an elastic cone, with a spherical segment attached to the base, is considered. The side surface of the cone is rigidly clamped, while the surface of the spherical segment is under a load. By using a new integral transformation over the meridial angle the problem is reduced in transformant space to a vector boundary value problem, the solution of which is constructed using the solution of a matrix boundary value problem. The unknown function (the derivative of the displacements), which occurs in the solution, is determined from the approximate solution of a singular integral equation, for which a preliminary investigation is carried out of the nature of the singularity of the function at the ends of the integration interval. Subsequent use of inverse integral transformations leads to the final solution of the initial problem. The values of the stresses obtained are compared with those that arise in the cone for a similar load, when sliding clamping conditions are specified on the side surface of the cone (for this case an exact solution of this problem is constructed, based on the known result).