Buckling analysis of periodic Vierendeel beams by a micro-polar homogenized model

This paper investigates the buckling analysis of periodic beams by a geometrically nonlinear equivalent micro-polar model that is built through the results of the unit cell transfer matrix eigenanalysis. The periodic system considered is the Vierendeel girder under compressive axial loads. The stiffness properties of the equivalent model are evaluated by an averaging process of unit cell strain energies associated with the inner force transmission modes, without any assumption on the real beam kinematics. Hence, equilibrium equations are achieved by the virtual power principle. Closed-form solutions are obtained for the girder critical loads and deformed shapes. They are of great accuracy in a wide range of conditions. However, model longitudinal shear strains are geometrically not compatible when the shear force is not uniform. Therefore, in cases where the girder response is dominated by these strains, the accuracy of buckling load estimates may be poor. To overcome this limitation, using a particular solution of the model equilibrium equations, the search for the critical load is carried out re-stating the buckling problem in an alternative form: conditions are investigated under which a system of self-equilibrated inner bending moments, able to bend the equivalent beam without violating geometrical compatibility, will exist. It is shown that this system is defined by an integral equation that, when solved by the Galerkin method, leads to buckling load estimates and deformed shapes that are in very close agreement with the ones obtained from classical finite element models.