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

Sébastien Neukirch; Morteza Yavari; Noël Challamel; Olivier Thomas

Pré-Publication, Document De Travail Année : 2021

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

(1) , , ,

Sébastien Neukirch

Fonction : Auteur
PersonId : 5879
IdHAL : sebastien-neukirch
ORCID : 0000-0003-0665-9634
IdRef : 161508219

Institut Jean Le Rond d'Alembert

Morteza Yavari

Fonction : Auteur

Noël Challamel

Fonction : Auteur

Olivier Thomas

Fonction : Auteur
PersonId : 171248
IdHAL : olivier-thomas
ORCID : 0000-0001-7240-5259
IdRef : 061423793

Résumé

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 nonlinearties

Domaines

Mécanique des structures [physics.class-ph] Mécanique des solides [physics.class-ph]

Fichier principal

vk3.pdf (1.41 Mo)

data/fig2_marron.txt (28.58 Ko)

data/fig2_noir.txt (18.98 Ko)

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data/fig3_marron.txt (29.39 Ko)

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data/fig4_vert.txt (46.53 Ko)

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data/fig5_shapeA.txt (14.82 Ko)

data/fig5_shapeB.txt (14.84 Ko)

data/fig5_shapeC.txt (14.83 Ko)

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data/fig6a_noir_1200.txt (56.63 Ko)

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data/fig7a_noir_19200.txt (80.61 Ko)

data/fig7a_noir_4800.txt (19.77 Ko)

data/fig7a_noir_76800.txt (58.59 Ko)

data/fig7a_vert_1200.txt (30.55 Ko)

data/fig7a_vert_19200.txt (31.69 Ko)

data/fig7a_vert_4800.txt (48.21 Ko)

data/fig7a_vert_76800.txt (48.2 Ko)

data/fig7b_noir_1200.txt (56.63 Ko)

data/fig7b_noir_19200.txt (77.83 Ko)

data/fig7b_noir_4800.txt (19.05 Ko)

data/fig7b_noir_76800.txt (56.33 Ko)

data/fig7b_vert_1200.txt (29.7 Ko)

data/fig7b_vert_19200.txt (30.55 Ko)

data/fig7b_vert_4800.txt (46.53 Ko)

data/fig7b_vert_76800.txt (46.29 Ko)

data/fig7c_noir_1200.txt (46.59 Ko)

data/fig7c_noir_19200.txt (80.24 Ko)

data/fig7c_noir_4800.txt (13.91 Ko)

data/fig7c_noir_76800.txt (58.08 Ko)

data/fig7d_noir_1200.txt (47.3 Ko)

data/fig7d_noir_19200.txt (80.98 Ko)

data/fig7d_noir_4800.txt (14.09 Ko)

data/fig7d_noir_76800.txt (58.73 Ko)

data/fig8a_marron.txt (28.58 Ko)

data/fig8a_noir.txt (18.98 Ko)

data/fig8a_rouge.txt (43.08 Ko)

data/fig8a_vert.txt (46.28 Ko)

data/fig8b_marron.txt (28.54 Ko)

data/fig8b_noir.txt (19.08 Ko)

data/fig8b_rouge.txt (42.87 Ko)

data/fig8b_vert.txt (46.09 Ko)

data/fig8c_marron.txt (28.16 Ko)

data/fig8c_noir.txt (18.36 Ko)

data/fig8c_rouge.txt (41.31 Ko)

data/fig8c_vert.txt (44.41 Ko)

data/fig8d_noir.txt (24.99 Ko)

data/fig8d_rouge.txt (31.31 Ko)

data/fig8d_vert.txt (9.69 Ko)

data/fig9a_noir_1.txt (27.52 Ko)

data/fig9a_noir_2.txt (15.77 Ko)

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data/fig9b_noir_1.txt (14.24 Ko)

data/fig9b_noir_2.txt (15.59 Ko)

data/fig9b_noir_3.txt (13.9 Ko)

vibration clamped ext norm19.pdf (203.65 Ko)

vk3 all figs.pdf (436.32 Ko)

vk3 clamped norm19.pdf (184.61 Ko)

Origine : Fichiers produits par l'(les) auteur(s)

Sébastien Neukirch : Connectez-vous pour contacter le contributeur

https://hal.science/hal-02957425

Soumis le : vendredi 2 avril 2021-08:55:38

Dernière modification le : samedi 7 octobre 2023-21:36:22

Dates et versions

hal-02957425 , version 1 (05-10-2020)

hal-02957425 , version 2 (02-04-2021)

hal-02957425 , version 3 (19-05-2021)

Identifiants

HAL Id : hal-02957425 , version 2

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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. 2021. ⟨hal-02957425v2⟩

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

Résumé

Mots clés

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