On an Accurate Theory for Circular Cylindrical Shells

Abstract

The accuracy of the characteristic roots obtained from solving a pair of complex conjugate fourth-order differential equations which govern the deformation of circular cylindrical shells is studied. Roots of the characteristic equations of the pair of fourth-order equations can be readily obtained in simple closed forms, unlike the Flu¨gge and other unreduced equations for circular cylindrical shells for which roots of the characteristic equations can only be found approximately by complicated numerical processes. In the present paper, roots from the pair of equations are computed for a range of significant parameters and comparisons are made with solutions based on other known equations. These results show that the pair of complex conjugate fourth-order equations has at least the same accuracy as the Flu¨gge equation and is as accurate as an equation can be within the scope of the Kirchhoff assumptions. The pair of complex conjugate governing equations for the homogeneous solutions of circular cylindrical shells is as follows: Lw=0andLw¯=0, in which L=(∇2+1)∇2+i1k∂2∂α2+k(1−ν)∂2∂β2−∂2∂α2∇2+∂2∂β2,L¯=thelinearcomplexconjugateoperatorofL,k=h2a3(1−ν2),h=thickness,a=radiusofcylinder,ν=Poisson’sratio,w=radialdisplacement. The particular solutions can be found from nonhomogeneous equation LLw¯=a4D∇4Z. It may be shown that the simplified equations, such as Donnell, Morley, Novozhilov, and other equations can be readily obtained from the present equation.