Analysis Of Asymmetric Problems Of Bellows

Bellows is a flexible element, which first came into use as a pressure meter in steam locomotive in 1844 and is now widely applied as elastic-sensitive element, seal and expansion joint in modern equipment. Common bellows are shells of revolution with wrinkled meridian. Much analytical work has been done on its symmetric problems including the axial elastic spring rate, the stress distribution under pressure and axial forces, the nonlinear behavior between the axial load and the related displacement, etc. For the last two or three decades, the asymmetric problems such as the pure bending and the bending by lateral forces, the asymmetric buckling under internal or external pressure, the natural frequencies and modes of lateral or axial vibration have been under analytical and experimental studies. However, to the best of the author's knowledge, few solution of the asymmetric problem is commonly agreeable, and no paper is concerned with the nonlinear behavior of bellows under bending loads. Further more, all the studies are confined to shells with a single material ply leaving out shells with multipliers. This paper presents a systematic study on the following asymmetric problems:Firstly, a linear governing equation expressed in complex variable for the toroidal shell with moderately slender ratio bending by lateral forces in a meridional plane is derived from E.L. Axelrad's theory of flexible shells. And a general solution for theproblem is presented, which is constituted of a particular solution and a homogeneous solution with the four integral constants to be solved by boundary conditions. The present equation is simplified but coincided with the fact that as the shell is subjected to bending loads the flexure and the effects of the transverse force mainly occur in the meridional plane. And the present general solution gets rid of the difficulties that all known particular solutions expressed in series of trigonometric functions have the limitation of convergence and the disagreement to boundary conditions. It is the key to successfully solving a bellows made of by torus.Secondly, the general solution is applied to analysis of the asymmetric deformations of bellows. The stress distributions and the stiffness coefficients are calculated by the general solution for the pure bending of -shaped bellows, C-shaped bellows and U-shaped bellows. The same calculations are made in the bending by lateral forces. In addition, an initial parameter formulation for toroidal shells without the limitation of the slender ratio is presented on Axelrad's flexible shell theory in order to test the general solution. The results of the general solution are compared with the initial parameter formulation, the standards of EJMA, and the available tests. It is shown that the present general solution is effective for solving the asymmetric deformations of bellows.Thirdly, an improvement on the finite-element displacement method to analysis of the nonlinear behavior of shells of revolution under pure bending or the bending by lateral force is presented. A systematic procedure for the formulation is based upon the idea of perturbation that the vector of nodal displacement components and the vector of nodal force components for each element are directlyexpanded by taking weighed root-mean-square value of circumferential strain of the shell as a perturbation parameter. For solution, the shell is discretized into a series of conical frusta for the convenience of practice, Sander's nonlinear geometric equations of moderate small rotation are used, and the shell with multipliers is considered. After that, the present formulation is applied to solve a number of U-shaped bellows, and the results of the linear part are compared with the former general solution, and the results including the nonlinear effects are compared with the available experimental results.Lastly, both the free-vibration and the stability of bellows are discussed by use of solving a linear eigenvalue problem, in which the shell with multipliers is considered. T...