弹性薄板弯曲问题的边界轮廓法  

A BOUNDARY CONTOUR METHOD FOR THE NUMERICAL SOLUTION OF ELASTIC THIN PLATE BENDING PROBLEMS

作　　者：周慎杰[1] 孙树勋 曹志远[2] 王威强[1] 

机构地区：[1]山东工业大学环境与化工学院,济南250061 [2]同济大学工程力学系固体力学教育部重点实验室,上海200092

出　　处：《力学学报》2000年第6期717-726,共10页Chinese Journal of Theoretical and Applied Mechanics

基　　金：山东省自然科学基金(Q99A01)

摘　　要：导出了弹性薄板弯曲问题边界积分方程的另一种形式，基于这种方程，提出了平板弯曲问题的边界轮廓法，讨论了三次边界单元边界轮廓法的计算列式，并给出了计算内力的边界轮廓法方程．该法无需进行数值积分计算，完全避免了角点问题和奇异积分计算．给出的算例，与解析解相比较，证实该方法的有效性．A variant of the usual boundary element method, called the boundary contour method, has been presented for elasticity in the literature in recent years. The new method requires no numerical integrals at all for two-dimensional problems and numerical evaluation of the line integrals only for three-dimensional problems. A boundary contour method for elastic thin plate bending problems has been presented first in this paper. By defining new boundary variables a new type of the boundary integral equation from the Betti's formulation has been derived. It is shown that the integrand vector of the boundary integral equation has the property of divergence free everywhere except at the point of singularity if the field variables correspond to a force free elastic plate with the same elastic constants as the fundamental solution. Therefore, the line integral on the usual boundary elements is transformed into the evaluation of potential functions at points on the boundary of a plate. The cubic shape function is chosen according to the complex expression of deflection solutions and the configuration of correspondent boundary elements is given. The numerical implementation with cubic boundary elements is carried out. This approach avoids completely numerical integration and the modeling of corners. The formulation of the boundary contour method is presented for evaluating moments. This formulation is regular at the points of the boundary except the ends of boundary elements such that the effect of the boundary layer is removed. To verify the effectiveness of the present method, some plate bending examples are solved by the present method. Comparison between the results of the boundary contour method and the analytical results shows well agreement.

关 键 词：边界轮廓法 边界元法 弹性薄板 弯曲 

分 类 号：O343.1[理学—固体力学] TU330.1[理学—力学]