柔性梁几何非线性/后屈曲分析的改进势能列式方法研究

针对基于Updated-Lagrangian列式的能量方法存在:1) 由于位移模型的近似性而带来虚假节点力;2) 在分析节点空间转动效应上存在争议;3) 势能高阶项由于物理概念不明确给简化列式带来困难等问题,提出描述柔性梁构件有限位移过程受力状态变化的势能列式方法。根据连续介质力学极分解定理,将典型增量步内单元内力势能分解为刚体变位下初始节点力势能和自然变形中积累的初始节点力势能和应变能,推导了满足刚体运动检验和变形后节点受力平衡的空间梁单元几何刚度矩阵。建立全面反映构件非线性大位移行为的增量割线刚度矩阵显式列式。数值分析结果表明,势能列式能准确预测任意荷载作用下结构非线性平衡路径,物理概念清晰,适应工程实践对一般杆系结构非线性分析需求。

IMPROVED POTENTIAL ENERGY FORMULATION FOR GEOMETRICALLY NONLINEAR/ POST-BUCKLING ANALYSIS OF FLEXIBLE BEAM STRUCTURES

A novel potential energy approach accounting for the finite displacement effects of beam members have been put forward to address the following issues resulting from applying the well-recognized energy method under the framework of Updated-Lagrangian formulation, which primarily refer to 1) the artificial nodal forces caused by the approximate displacement field model, 2) the theoretical ambiguity regarding the treatment of joint equilibrium during spatial rotations, and 3) the lack of rational physical interpretations about some high order components of total potential energy may pose difficulties as whether they can be ruled out to simplify formulation. Based on the well-known polar decomposition theorem, the potential energy of member forces absorbed during a typical incremental step can be subdivided into three parts, potential energy due to initial member forces developed in the rigid body motion, potential energy caused by natural deformations of the member, and the strain energy, respectively. Using the derived potential energy, the geometric stiffness matrix of a solid spatial beam element that satisfies the rigid body motion test and joint equilibrium requirement in the deformed configuration have been obtained. In this regard, an explicitly formulated incremental secant stiffness matrix has been established to describe member large displacement behavior. Numerical studies revealed that the proposed energy method is capable of reliably tracing nonlinear equilibrium path with a clear physical meaning in analyzing general flexible beam structures.