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

jtcam:6828 - Journal of Theoretical, Computational and Applied Mechanics, May 21, 2021 - https://doi.org/10.46298/jtcam.6828
Comparison of the Von Kármán and Kirchhoff models for the post-buckling and vibrations of elastic beams

Authors: Sébastien Neukirch ; Morteza Yavari ; Noël Challamel ; Olivier Thomas

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.


Published on: May 21, 2021
Submitted on: October 7, 2020
Keywords: postbuckling,natural frequencies,nonlinearities,[PHYS.MECA.SOLID]Physics [physics]/Mechanics [physics]/Solid mechanics [physics.class-ph],[PHYS.MECA.STRU]Physics [physics]/Mechanics [physics]/Structural mechanics [physics.class-ph]


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