U型波纹管及相关结构环向屈曲的有限元分析(Ⅰ)--基本方程及环板的屈曲

U型波纹管是现代管道系统中最常见的一种位移补偿器 ,它由环板和具有正、负Gauss曲率的半圆环壳组成 ,在管道所传输的介质的压力作用下会发生屈曲。其中环向屈曲最为复杂 ,精确的理论分析非常困难 ,有限元分析也不多见。作者在分析前人工作的基础上 ,以圆环壳段为单元 (特定的旋转壳段单元 ,能自动退化成环板单元 ) ,限于弹性范围和线性化特征值问题 ,对介质压力作用下U型波纹管及其相关结构 (圆环板、圆环壳、半圆环壳 )的环向屈曲问题进行了分析。考虑了结构屈曲前的弯曲 ,计及压力的二次势能 ,导出的应力刚度矩阵和载荷刚度矩阵是非对称的。全部工作分为三部分 :(Ⅰ )基本方程 ,环板的屈曲 ;(Ⅱ )圆环壳、半圆环壳的屈曲 ;(Ⅲ )波纹管平面失稳的机理。本文为第一部分 ,除推导公式外 ,对不同边界和不同内外径之比的环板在径向均匀压力作用下的环向屈曲进行了计算 (轴对称的径向屈曲作为特例得到 ) ,给出了前屈曲应力分布、临界载荷及相应的屈曲模态 ,并将临界压力的值与前人基于vonK偄ｒｍ偄ｎ大挠度板的精确解进行了比较 ,吻合良好。

Analysis ofCircumferential Buckling of U-Shaped Bellows and Related Structures Part Ⅰ: Formulations, Buckling of Annular Plates

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 stress stiffness matrix is asymmetric. The study is divided into three parts, i.e. Part I: Formulations, Buckling of Annular Plates; Part II: Buckling of Toroidal Shells and Semi Toroidal Shells; Part III:Mechanism of In Plane 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 equations of large deflections of plates provided by other authors.