矩形薄板面内非线性分布载荷下的辛弹性力学解

SYMPLECTIC ELASTICITY SOLUTIONS FOR THIN RECTANGULAR PLATES SUBJECTED TO NON-LINEAR DISTRIBUTED IN-PLANE LOADINGS

用辛弹性力学理论,根据平面矩形域本征向量展开解法,得到了对应于零本征值和非零本征值的本征向量解,以及含待定常数的面内应力分布通解,依据必须满足的应力边界条件,利用符号运算软件Maple,导出了矩形薄板在半余弦分布载荷作用下的面内应力表达式。为了验证方法的有效性和所得到的公式的正确性,具体分析了正方形薄板在两种非线性形式载荷——半余弦和抛物线分布载荷作用下的例子。算例结果与微分求积法及其有限元法得到的数值结果极其相近。基于所给出的结果,可望为工程应用中的屈曲分析提供合理的前期准备。

The eigenvector solutions corresponding to the zero and nonzero eigenvalues are carried out according to the symplectic eingen-solution expansion method in rectangular domains. Including the nonzero eigenvalues in the eigenvector solution yields the general solution of the in-plane stress with undetermined constants. After applying the boundary conditions, one gets a set of coupled equations to determine the unknown constants. These equations are solved by the software Maple. The formula determining the stress distribution of a thin rectangular elastic plate subjected to in-plane compressive loads varying half-cosine along two opposite edges are derived. The examples of the square plates under half-cosine and parabolic load distributions are analyzed to verify the efficiency and accuracy of the proposed method. The results are agreed well with the numerical results of differential quadrature （DQ） method and FEM. The results reported herein could provide reasonable preparations for the buckling analysis in engineering applications.