大位移小转角空间曲梁的弹性力学方程

ELASTICITY EQUATIONS FOR SPATIAL CURVED BEAMS WITH LARGE DISPLACEMENT AND SMALL ROTATION

  • 摘要: 为了提高曲线结构的分析效率建立空间曲梁几何非线性单元,在分析和总结前人关于曲梁研究成果的基础上,该文就曲梁相关方程做了更具普遍性的分析,给出了数学上更严密的结果:在几何方程方面,在作者前期工作的基础上,按小转角原则对已有结果进行简化,以矩阵形式给出了该类曲梁的几何关系;在平衡方程方面,用微分几何的思想推导了真正意义上空间双向弯、扭的曲梁平衡微分方程,并分别以矢量和矩阵的形式给出相关结果;从加权余量法和变形功相等的原则出发,推导了以截面内力和位移表达的曲梁非线性虚功方程;最后,从广义Hooke本构关系出发,推导了截面内力和截面位移之间关系表达式,为下一步建立几何非线性曲梁单元进行了必要的准备。

     

    Abstract: Based on previous researches on curved beams, the paper presents improved equations of elasticity to raise the efficiency of curve structure analysis, and then develops the geometrical non-linear spatial curved beam element. In the aspect of geometry, the geometrical relation between displacement and deformation is expressed by a series of matrix formulas. In the aspect of equilibrium relation, the differential equation for the equilibrium of curved beam is derived using differential geometry theory. The relevant formulas are given in forms of vector and matrix. Then the internal force-displacement virtual work equation is established based on the weighted residual method and the principle of equivalent deformation work. At last, to develop the geometrical non-linear curved beam element, the relationship between section internal force and displacement is proposed from the view of generalized Hooke constitutive relation.

     

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