含对边自由边界矩形板屈曲失稳的有限差分法求解

Finite difference method for the solution of buckling instability for rectangular plates with opposite free boundary edges

为了精确求解含对边自由边界矩形板的屈曲失稳问题.首先,利用有限差分法将忽略剪切效应的屈曲控制微分方程及边界条件进行离散化;将屈曲失稳问题转化为求解挠度系数矩阵广义特征值的问题.同时,以加载对边简支、非加载对边自由矩形板为例,通过与文献解、有限元仿真解进行比较,验证了有限差分法求解程序及有限元仿真解的精确性.由于厚板较薄板在厚度方向存在明显的剪切效应,且板件几何参数变化对剪切效应的影响较难度量.因此,参照精确的有限元仿真解,拟合出了在给定误差范围内,有限差分法精确求解屈曲失稳问题几何参数适用范围的计算公式.最后,经实例参数分析验证表明,在拟合公式限定的几何参数范围内有限差分法可精确求解含对边自由边界矩形板的屈曲失稳问题,为考虑剪切效应矩形板的屈曲失稳问题提供了新的解决方法.

To accurately solve the buckling instability problem of a rectangular plate with free boundaries on the opposite edges,the finite difference method,FDM,was introduced.The buckling differential equations and boundary conditions equations ignoring the shear effect were discretized by the FDM;the buckling instability problem was transformed into a problem of solving the generalized eigenvalues of the deflection coefficient matrix.At the same time,the opposite simply supported edges subjected to unidirectional pressure were chosen as the case study,the accuracy of the FDM results and the finite element method results were verified by comparing them with the previous results.The effect of the plate thickness on critical buckling load is difficult to measure for the moderately thick plates.Therefore,a fitting formula was obtained to give the applicable range of geometric parameters for the FDM within a given relative error range.Finally,the effect of geometrical parameters on the critical buckling load was analyzed.And the results show that the FDM used in this paper can solve the buckling instability problem of the rectangular plate with two opposite free edges accurately within the geometrical parameters range given by the fitting formula,which provides a new solution to the buckling instability problem both for the thin plates and moderately thick plates.