浅曲梁大挠度分析的余能有限元方法

Finite element method based on complementary energy principle for large deflection of shallow beam

以高玉臣提出的弹性大变形余能原理为基础,利用Lagrange乘子,放松平衡方程和力边界条件对余能泛函的约束,推导出广义的余能原理。根据极分解定理,将变形分为刚性转动和纯变形两部分,则余能也包含相应的两部分,一部分与刚性转动有关,而另一部分与纯变形有关。使用线弹性本构关系,建立了可用于几何非线性计算的有限元模型。应用更新的Lagrange列式法,给出了增量形式的有限元公式。数值计算结果表明,该方法可用于浅曲梁的几何大变形计算。

Based on the complementary energy principle for large elasticity proposed by Gao Yuchen, the generalized complementary energy principle （GCEP） was deduced when the constraints of equilibrium equations and force boundary conditions were released by the Lagrange multiplier method. According to the polar decomposition theorem, the deformation could be decomposed into two parts of rigid rotation part and pure deformation part, and then the complementary energy also included two parts in which one part was related to rigid rotation while the other was related to pure deformation. Using the linear elastic constitutive relation, the finite element model that could be used to geometrical nonlinear problem was established. With the update Lagrange method, the incremental finite element formulas were obtained. The numerical results show that the FEM based on GCEP can be used to geometrical nonlinear computation for shallow beam.