一类指定应力问题的变分原理与应用

A Variational Principle and Applications for a Class of Specified Stress Problems

针对有限元分析中对应力或内力有指定条件的问题,引入非弹性应变作为实现指定应力条件的附加未知量,在小变形条件下描述了指定应力条件应当满足的弹性力学控制方程;以位移和未知非弹性应变作为独立变量,建立了具有指定应力条件问题的势能变分原理和虚功方程;以位移、弹性应变、未知非弹性应变和应力为独立变量,建立了一个含四类变量的广义变分原理.在基于变分原理得到的桁架单元和梁单元平衡方程中,指定轴力和需要的调整量以对偶形式出现,可实现调整量已知情况下的常规受力分析,又可在轴力指定条件下获得需要的调整量;同时考虑了材料刚度和内力对结构的影响,改进了目前预应力筋模拟的等效荷载法和实体力筋法,还可用于拉索结构的索力优化和调整算法.通过拉索结构位移优化和索力调整的数值算例,验证了该文理论与算法的可行性及精度.

To solve some problems in the finite element analysis with specified conditions for stresses or inter⁃nal forces,the inelastic strain as an additional unknown to meet the specified stress condition was introduced.The elastic mechanics governing equations meeting the specified stress conditions were described under the small deformation assumption.The potential variational principle and the virtual work equation were established with the displacements and unknown inelastic strains as independent variables.Then a generalized variational principle with 4 types of independent variables of the displacement,the elastic strain,the unknown inelastic strain and the stress,was presented.Based on the variational principle,the specified axial forces and the re⁃quired adjustment quantities were arranged in a dual form in the equations for truss and beam elements.The or⁃dinary stress analysis was realized under the condition with known adjustment quantities,and the required ad⁃justment quantities were obtained with specified axial force conditions.The effects of material stiffness and in⁃ternal force of a structure were considered in the new method with the prestressing reinforcement simulation improved based on the equivalent load method or the real physical reinforcement method.The method applies to the optimization and adjustment of cable forces in related structures.Numerical simulations of displacement optimization and cable tension adjustment for a cable⁃stayed structure demonstrate the feasibility and accuracy of the presented theory and algorithm.