Generalized expressions for boundary conditions of shells of revolution

Abstract

For the boundary conditions of shells of revolution, traditionally, four out of the eight quantities which are the four displacements on the middle surface u, v, w and ϕ together with the four corresponding forces, are given. When the generalized displacements on the nodal circles are used as basic unknowns, the number of unknowns on a nodal circle is more than four^[1][2][3][4]. In this case, how to deal with the boundary conditions is still a problem that has not been solved satisfactorily yet. In this paper, the relations between the generalized and nongeneralized quantities of a shell's edge are derived according to the principle of virtual work. Seven types of common edges are studied and their expressions of boundary conditions in the form of generalized displacements or forces are given. The number of expressions for each type of edge may correspond with the number of unknowns used on a nodal circle. With these expressions, boundary conditions can be put directly into equations of motion of generalized displacement method so as to solve the generalized displacements. By so doing, the process of transformation and inverse transformation about unknowns in [2] is avoided. Not only is the argument simple and clear, but the calculation work is reduced. Having the set of generalized expressions of boundary conditions, the generalized displacement method of the shell of revolution may be more perfect in theory.