转动矩阵在弯扭屈曲和几何非线性分析中的研究

Study of rotation matrix in flexural-torsional stability and geometrically nonlinear analysis

薄壁钢梁在利用有限单元模型进行弯扭屈曲分析和三维非线性分析中,都可以通过转动矩阵来得到位移-应变的非线性关系,因此转动矩阵的精度决定了应变的准确性.在弯扭屈曲中,利用小转动理论可以得到较新势能方程,但分析的结果会高于实际值;而大转动理论可得到传统势能方程,会更接近实际值.在几何非线性分析中,由小转动理论得到的几何刚度矩阵,当结构发生空间有限转动时,边角结点弯矩会产生不平衡的附加力矩,导致结点不能保持平衡和转动连续性,需要对相应的连带力矩矩阵进行修正.而由大转动理论得到的几何刚度矩阵能保持结点平衡和转动连续性,但该单元不能通过刚体检验.

In formulating a finite element model for the flexural-torsional stability and three dimensional nonlinear analyses of thin-walled beams, a rotation matrix is usually used to obtain a nonlinear strain-displacement relationship. Hence, the accuracy of the nonlinear strains is related to the accuracy of the rotation matrix. It is founded that a finite element based on a small rotation matrix may derive the alternative equations and predict incorrect elastic flexural-torsional load bucklings of beams, while a large rotation matrix may derive the classical equations and predict incorrect elastic flexural-torsional buckling loads. In geometrically nonlinear analysis, the geometrical nonlinear stiffness matrix based on a small rotation matrix will yield the induced moment effects which can be attributed to the rotational characteristic feature of nodal moments. The equilibrium is not satisfied and it also leads to rotational discontinuities at the joints of deformed space frame structures, which needs a correct matrix. The geometrical nonlinear stiffness matrix based on a large rotation matrix makes equilibrium and rotational continuities at nodal points, but the derived beam element does not pass spatial rigid body motion tests.