Abstract The nonlinear response of straight, constant cross section beams is investigated by means of a two-level finite element solution procedure. The higher level model is built starting from the Hellinger-Reissner principle, with the normal stress resultant and moment resultant as additional unknowns field beside the beam reference line position and orientation field. A lower-level nonlinear cross-section problem is defined for each integration point. The two-level models are linked together by the normal stress resultant and moment resultant, in one direction, and by the variation of the complementary strain energy in the other: the cross-section level do deform in such a way that the the normal stress resultant and moment resultant are equal to those of the beam model, while the higher level beam model receives the gradient and Hessian of the complementary strain energy with respect to the resultants. The complementary strain energy gradient and Hessian are computed by defining suitable first and second order adjoint problems.

中文翻译：

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一种两级非线性梁分析方法

摘要 通过两级有限元求解程序研究了直截面梁的非线性响应。更高层次的模型是从 Hellinger-Reissner 原理开始构建的，法向应力合力和力矩合力作为梁参考线位置和方向场旁边的附加未知场。为每个积分点定义了一个较低级别的非线性横截面问题。两级模型通过一个方向上的法向应力合力和力矩合力以及另一个方向上互补应变能的变化连接在一起：横截面水平确实以这样的方式变形，使得法向应力合力和力矩合力等于梁模型的合力和力矩，而更高级别的梁模型接收相对于结果的互补应变能的梯度和 Hessian。互补应变能梯度和 Hessian 是通过定义合适的一阶和二阶伴随问题来计算的。