基于材料力学弯曲理论求解非线性梁的探讨

基于材料力学弯曲理论,利用广义变分原理研究了非线性本构关系静不定梁支承反力的求解,指出了有关文献求解非线性本构关系静不定梁支承反力存在的错误．研究结果表明：本构关系为的静不定梁弯曲时,会同时存在曲率为正、曲率为负的梁段．当n为奇数时,曲率为正梁段的余能表达形式与曲率为负的余能表达式相同．当n为偶数时,曲率为正梁段的余能表达式与曲率为负梁段的余能表达式则不相同,存在正负号的差异．

Discussion on Solving Nonlinear Beam Based on Bending Theory of Material Mechanics

Based on the bending theory of material mechanics and the generalized variational principle, the solution of the support reaction of statically indeterminate beam with nonlinear constitutive relationship was studied. The errors in solving the support reaction of statically indeterminate beam with nonlinear constitutive relationship in the relevant literature were pointed out. The results show that when the statically indeterminate beam with constitutive relation  is bent, there will be beam segments with positive curvature and negative curvature at the same time. When n is an odd number, the expression of complementary energy of the beam segment with positive curvature is the same as that of the beam segment with negative curvature. When n is an even number, the expression of complementary energy of the beam segment with positive curvature is different from that of the beam segment with negative curvature, in terms of the sign.