Hamilton体系下矩形薄板受抛物线压力载荷的屈曲分析

BUCKLING ANALYSIS OF RECTANGULAR THIN PLATES UNDER PARABOLIC COMPRESSIONS AT EDGES IN HAMILTON SYSTEM

针对四边简支矩形薄板在两对边相向的非线性分布压力下的面内应力分布以及屈曲问题,应用弹性力学的Hamilton体系和Galerkin法进行了研究.基于弹性力学的平面矩形域Hamilton体系,根据辛本征向量展开解法,得到了对应于零本征值和非零本征值的含待定常数的实数型面内应力分布通解.依据必须满足的应力边界条件,导出了矩形薄板在抛物线分布载荷下的面内应力分布.考虑到应力分布表达式的复杂性,用完全的解析方法得到屈曲载荷是不可能的.因此,运用基于虚功原理的Galerkin法,根据四边简支矩形薄板弯曲的位移边界条件,给出了不同长宽比矩形薄板受抛物线分布载荷的屈曲临界载荷.通过与已有文献中DQ法给出的数值计算结果比较,表明了本文求解方法的有效性和正确性.基于所给出的结果,可望为解决矩形薄板在非线性分布载荷下的面内应力分布以及屈曲问题提供一种新的研究方法.

The distribution of in-plane stresses and buckling of rectangular elastic thin plates with four edges simply supported, subjected to in-plane pressures along any two opposite edges, are studied by using the Hamilton system of elasticity and Galerkin method. The general solutions of the in-plane stress distribution are obtained by using the symplectic eigen-solution expansion method at first. Then, the Galerkin method is employed for obtaining the buckling loads of rectangular plates with various aspect ratios. The numerical results agree very well with the existing DQ （differential quadrature） data, which confirms the validity of the proposed method. Obviously, the symplectic eigen-solution expansion method provides a new way for solving the bending of rectangular thin plates.