基于混合近似法的纤维梁单元非线性求解方法#br#

环境荷载作用下土木工程结构往往产生较大的非线性变形，包括材料非线性和几何大变形两部分，而两者非线性问题的数值求解往往需要花费高昂的计算代价。隔离非线性有限元法是将材料应变分解为弹性与塑性两部分，在控制方程中实现刚度矩阵的弹塑性分离，使用Woodbury公式进行非线性求解。目前，该方法仅用于求解局部材料非线性问题。文章基于隔离非线性有限元法，推导同时考虑材料和几何非线性（简称混合非线性）的纤维梁单元控制方程；依据Woodbury公式和组合近似法的基本理论，提出混合近似法的高效非线性求解方法，并利用时间复杂度理论，对该方法与传统有限元法进行计算效率对比分析；将提出的方法应用于某钢框架结构的地震非线性反应分析，计算结果表明：文中方法可以在求解材料非线性时考虑几何大变形问题，在保证计算精度的同时使计算效率得到明显提升。

Nonlinear solution method for fiber beam element based on the hybrid approximations method

Nonlinearity problems in civil engineering structures mainly involve the material nonlinearity and geometric nonlinearity (i.e., hybrid nonlinearities) under environmental loads. Because the computational cost is high, great attentions have been paid to the numerical solution of hybrid nonlinearity problem. The inelasticity-separated finite element method (IS FEM) is an efficient nonlinear solution method, and based on the concept, the strain of a nonlinear material can be decomposed into the linear-elastic and inelastic components. Consequently, the elastic and plastic components in the stiffness matrices from the governing equation can be separated, such that the Woodbury formula can be used to solve local material nonlinearity problems. In this study, the governing equation of the fiber beam element considering hybrid nonlinear behaviors is derived using the IS FEM. In order to efficiently solve the governing equation, the efficient hybrid approximations method is proposed based on the Woodbury formula and the combined approximations (CA) method. In addition, the time complexity theory is used to evaluate the calculation efficiency of the present method, and the results show that it is obviously superior to the conventional finite element method. Finally, based on a numerical example, it is proved that the proposed method can produce accurate results efficiently, and thus, appear to be appealing in solving hybrid nonlinear problems of the fiber beam element.