The exact solution for the general bending problems of conical shells on the elastic foundation

Abstract

The general bending problem of conical shells on the elastic foundation (Winkler Medium) is not solved. In this paper, the displacement solution method for this problem is presented. From the governing differential equations in displacement form of conical shell and by introducing a displacement function U(s, θ), the differential equations are changed into an eight-order soluble partial differential equation about the displacement function U(s, θ) in which the coefficients are variable. At the same time, the expressions of the displacement and internal force components of the shell are also given by the diplacement function U(s θ). As special cases of this paper, the displacement function introduced by V.S.Vlasov in circular cylindrical shell^[5], the basic equation of the cylindrical shell on the elastic foundation and that of the circular plates on the elastic foundation are directly derived.

Under the arbitrary loads and boundary conditions, the general bending problem of the conical shell on the elastic foundation is reduced to find the displacement function U(s, θ). The general solution of the eight-order differential equation is obtained in series form. For the symmetric bending deformation of the conical shell on the elastic foundation, which has been widely used in practice, the detailed numerical results and boundary influence coefficients for edge loads have been obtained. These results have important meaning in analysis of conical shell combination construction on the elastic foundation and provide a valuable judgement for the numerical solution accuracy of some of the same type of the existing problem.