Comparison of the Von Kármán and Kirchhoff models for the post-buckling and vibrations of elastic beams

Sébastien Neukirch (1) , Morteza Yavari (2) , Noël Challamel (3) , Olivier Thomas (4)

1 DALEMBERT - Institut Jean Le Rond d'Alembert
2 Islamic Azad University
3 IRDL - Institut de Recherche Dupuy de Lôme
4 LISPEN - Laboratoire d’Ingénierie des Systèmes Physiques et Numériques

We compare different models describing the buckling, post-buckling and vibrations of elastic beams in the plane. Focus is put on the first buckled equilibrium solution and the first two vibration modes around it. In the incipient post-buckling regime, the classic Woinowsky-Krieger model is known to grasp the behavior of the system. It is based on the von Kármán approximation, a 2nd order expansion in the strains of the buckled beam. But as the curvature of the beam becomes larger, the Woinowsky-Krieger model starts to show limitations and we introduce a 3rd order model, derived from the geometrically-exact Kirchhoff model. We discuss and quantify the shortcomings of the Woinowsky-Krieger model and the contributions of the 3rd order terms in the new model, and we compare them both to the Kirchhoff model. Different ways to nondi-mensionalize the models are compared and we believe that, although this study is performed for specific boundary conditions, the present results have a general scope and can be used as abacuses to estimate the validity range of the simplified models.

Mots clés: natural frequencies; postbuckling; nonlinearities

Domaines: Mécanique des solides [physics.class-ph]; Mécanique des structures [physics.class-ph]

Sébastien Neukirch, Morteza Yavari, Noël Challamel, Olivier Thomas. Comparison of the Von Kármán and Kirchhoff models for the post-buckling and vibrations of elastic beams. Journal of Theoretical, Computational and Applied Mechanics, 2021, Issue 2 | 2022, pp.1-18. ⟨10.46298/jtcam.6828⟩. ⟨hal-02957425v3⟩