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

Article dans une revue avec comité de lecture
Author
NEUKIRCH, Sébastien
541882 Institut Jean Le Rond d'Alembert [DALEMBERT]
YAVARI, Morteza
555032 Islamic Azad University
CHALLAMEL, Noël
172265 Université de Bretagne Sud [UBS]
457614 Institut de Recherche Dupuy de Lôme [IRDL]
ccTHOMAS, Olivier
543315 Laboratoire d’Ingénierie des Systèmes Physiques et Numériques [LISPEN]

URI
http://hdl.handle.net/10985/22704
DOI
10.46298/jtcam.6828
Date
2021-05
Journal
Journal of Theoretical, Computational and Applied Mechanics

Abstract

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.

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