平面单元及平面问题的一种缩放不变性

Scaling Invariance of Plane Elements and Plane Problems

证明了有限元中二维单元的一个特性,即厚度不变时,将单元按比例缩放,单元的刚度矩阵不变。由此指出二维问题的一个奇特现象:当厚度不变时,将模型按比例缩放,则在相同的约束和(集中)载荷下的结构变形不变。对分布载荷和力偶,若把它们折算成集中力时保持力的大小不变,则结论也成立。

A characteristic of the 2D finite element is proved that when the thickness remains constant,if the element is scaled proportionally,the stiffness matrix of the element remains unchanged.Therefore,a peculiar phenomenon of 2D problem is pointed out:when the thickness is kept constant,if the model is scaled proportionally,the deformation of the structure will remain unchanged under the same constraint and concentrated load.For distributed loads and couples,if the magnitude of the force remains unchanged when they are converted into equivalent concentrated forces,the conclusion is also valid.