A Theory of Ring-Stiffened Cylinders

Abstract

5»6 The method of Hoff is to express the stress in the cylin­ der and in the ring as Fourier series in the angle. Then the total strain energy is calculated as a function of the Fourier coefficients. According to the variational principle of Castigliano the equilibrium state is deter­ mined by minimizing the strain energy. This mini­ mization process leads Hoff to an infinite set of simul­ taneous linear equations for the Fourier coefficients. Fortunately it proves possible to solve these equations by a step-by-step process. The stresses are then found by summing a sufficient number of terms of the Fourier series. The Hoff analysis is found to give good agree­ ment with experiment.5 In a previous note the writer followed the basic as- sumptions of the Hoff treatment, but Fourier series were not employed.7 Instead, classical methods of the calculus of variations were used to obtain the Euler differential equation. The Euler equation is an eighth order ordinary differential equation with constant coefficients.** The independent variable is the angle, and the dependent variable is the bending moment in the ring. The Fourier series method is then seen to be merely a special way of solving the Euler differential equation. An advantage of the differential equation approach to the problem is greater flexibility; thus there is no difficulty in taking into account a stiffening ring having variable cross section. The only difference in this case is that one of the coefficients of the differential equation becomes variable rather than constant. The solution of the differential equation can then be obtained by use of digital or analog computers. In HofFs treatment it is supposed that the applied forces lie in the plane of the ring. In a recent report, J. Adachi was concerned with the stress problem of the ring-stiffened cylinder when the applied forces are perpendicular to the plane of the ring.8 Adachi ideal­ ized the problem by supposing that it was sufficient to consider an end-stiffened flat sheet. He indicates the desirability of having a theory avoiding this idealiza­ tion. In this paper a general approach is developed which permits the applied force to have an arbitrary direction. Also the assumptions on the properties of the ring are relaxed. Thus in the Hoff treatment only flexure in the plane of the ring was taken into account. It was assumed that the ring was perfectly flexible perpendicu­ lar to its plane. In the present theory account is taken of flexure perpendicular to the plane and torsion as well as flexure in the plane. The theory is again based on Castigliano's principle. The strain energy is taken to consist of five separate terms resulting from shear stress r in the cylinder, nor­ mal stress a parallel to the axis of the cylinder, bending moment L perpendicular to the plane of the ring, torsional moment M of the ring, and bending moment N in the plane of the ring. The Euler equations re­ sulting may be written as two ordinary differential equations of the seventh order relating the moments MzndN.