The non-axisymmetric end loading of a truncated cone

Abstract

The problem considered is the equilibrium of a semi-infinite cone, truncated a fixed distance from its apparent apex. The cone is assumed to be loaded on its end surface of truncation, with the ruled sides being free from stress. The present formulation does not require that this loading be axisymmetric.Total-stress end loading problems are formulated in terms of the three stress variables prescriptible on the truncated end and three auxiliary variables, of the order of stresses with respect to differentiation. The three auxiliary variables carry the displacement information on the end surface and permit the integration of three of the Beltrami equations of compatibility.The six-vector satisfies a matrix partial differential equation whose constituent equations are obtained from the three integrated Beltrami equations, the defining equations for the auxiliary variables, and the equation of equilibrium containing the variables prescriptible on the truncated surface. The remaining equations of equilibrium are used to determine the stress variables not in the six vector.A separation of variables of the matrix equation yields a non-selfadjoint matrix differential equation, hence the eigenfunctions are non-orthogonal. A biorthogonality relation is derived from consideration of the adjoint problem to permit the numerical solution of particular boundary value problems.The decoupling of the axisymmetric problem in the case of axisymmetry is discussed, including the decoupling of the non-axisymmetric biorthogonality into biorthogonalities for the axisymmetric torsion problem and the axisymmetric torsionless problem.