Analysis of Circumferential Buckling of U-Shaped Bellows and Related Structures Part I: Formulations, Buckling of Annular Plates
Zhu Weiping
(Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University 149 Yanchang Road, Shanghai 200072, China)

Abstract£º U-Shaped bellows is composed of annular plates and the semi-toroidal shells with positive and negative Gaussian curvatures, which is commonly used as a displacement compensator in modern pipeline system. It may be buckling when the pressure of medium transmitted in the pipeline exceeds a critical value. In the stability problems of bellows, the circumferential buckling or so-called in-plane squirm is too complex to be solved by analytical methods. Even the finite element methods for it are also seldom used yet. In this study, the circumferential buckling of U-shaped bellows as well as the related structures (annular plates, toroidal shells and semi-toroidal shells) is systematically evaluated by using the finite element method. The segments of the toroidal shells (a special shell of revolution) are used as elements to idealize the structures. If necessary the segments can be reduced into annular plates automatically. The present method is confined to the elastic material and to the mineralized eigenvalue problem, but the finite prebuckling rotations and the follower force effect of the pressure are considered, so the obtained stressª²stiffness matrix is asymmetric.
    The study is divided into three parts, i.e. Part I£º Formulations, Buckling of Annular Plates; Part II£º Buckling of Toroidal Shells and Semi-Toroidal Shells; Part III: Mechanism of Inª²Plane Squirm of Bellows. This paper is the first part. It presents that the finite element method is formulated and the circumferential buckling problems of the annular plates with different ratios of the inside radius to the outside radius and with different boundary conditions under uniform radial pressure are calculated. In which, the axial symmetric buckling treated as a special case. The prebuckling stress distributions, the critical loads and the corresponding modes are displayed. It turns out that the present critical loads are almost as same as the exact values based upon the von Karman's equatio ns of large deflections of plates provided by other authors.
Keywords: Stability of bellows, finite element method, buck ling of annular plates, prebuckling stress distributions, critical loads, buckling modes.