This paper deals with the buckling phenomenon of periodic Vierendeel beams. Closed-form solutions for critical loads and deformed shapes are presented. They are built by exploiting several auxiliary solutions obtained for the discrete periodic girder and for a geometrically nonlinear micro-polar equivalent model. In particular, the girder when subjected to sinusoidal self-equilibrated systems of inner bending moments (self-moments) is analysed. The corresponding results are used for solving the large-deflection equilibrium problem of the continuous equivalent model by means of the eigenfunction expansion technique. Girder buckling conditions are then defined in terms of kinematics of the micro-polar model: more precisely, they are attained when special distributions of self-moments, able to bend the continuous system without violating compatibility of shear strains, act in the girder. It is shown that these systems, neglected in the theories presented so far, have a significant stiffening effect on the buckling girder behaviour. Moreover, they are governed by the continuity equation for micro-rotations that is solved in closed form by the Galerkin method, with the micro-polar model eigenfunctions as basis functions. The accuracy of the proposed solutions is verified by comparing them with those achieved by a series of finite element girder models.

本文讨论了周期性Vierendeel梁的屈曲现象。提出了用于临界载荷和变形形状的封闭形式的解决方案。通过利用为离散周期梁和几何非线性微极性等效模型获得的几种辅助解决方案来构建它们。尤其是，分析了大梁承受内部弯矩（自矩）的正弦自平衡系统时的情况。利用特征函数展开技术，将相应的结果用于求解连续等效模型的大挠度平衡问题。然后根据微极模型的运动学定义梁屈曲条件：更精确地说，当自矩的特殊分布，能够弯曲连续系统而不会破坏剪切应变的相容性，作用于大梁。结果表明，这些系统在迄今为止提出的理论中都被忽略，但对屈曲梁的​​行为具有明显的加强作用。此外，它们由微旋转的连续性方程控制，该方程由Galerkin方法以闭合形式求解，以微极性模型的本征函数为基础函数。通过将它们与一系列有限元梁模型所获得的结果进行比较，可以验证所提出解决方案的准确性。它们由微旋转的连续性方程控制，该方程通过伽勒金方法以闭合形式求解，并以微极模型的本征函数为基础。通过将它们与一系列有限元梁模型所获得的结果进行比较，可以验证所提出解决方案的准确性。它们由微旋转的连续性方程控制，该方程通过伽勒金方法以闭合形式求解，并以微极模型的本征函数为基础。通过将它们与一系列有限元梁模型所获得的结果进行比较，可以验证所提出解决方案的准确性。