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Explicit analysis of large transformation of a Timoshenko beam: post-buckling solution, bifurcation, and catastrophes

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Abstract

This paper exposes full analytical solutions of a plane, quasi-static but large transformation of a Timoshenko beam. The problem is first re-formulated in the form of a Cauchy initial value problem where load (force and moment) is prescribed at one end and kinematics (translation, rotation) at the other one. With such formalism, solutions are explicit for any load and existence, and unicity and regularity of the solution of the problem are proven. Therefore, analytical post-buckling solutions were found with different regimes driven explicitly by two invariants of the problem. The paper presents how these solutions of a Cauchy initial value problem may help tackle (i) boundary problems, where physical quantities (of load, position or section orientation) are prescribed at both ends and (ii) problems of quasi-static instabilities. In particular, several problems of bifurcation are explicitly formulated in case of buckling or catastrophe.

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Acknowledgements

This work was supported in part by a grant from the Walid Joumblatt Foundation For University Studies and the Henri Lebesgue Center. The authors gratefully acknowledge the support of IRMAR - CNRS UMR 6625 for providing financial support. The authors are very grateful to Professor Lalaonirina Rakotomanana for the careful and thoughtful comments on the work.

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Authors and Affiliations

  1. CNRS, IRMAR - UMR 6625, Univ. Rennes, 35000, Rennes, France

    Marwan Hariz & Loïc Le Marrec

  2. CNRS, Univ. Evry, Laboratoire de Mathématiques et Modélisation d’Evry, Université Paris-Saclay, 91037, Evry-Courcouronnes, France

    Jean Lerbet

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  1. Marwan Hariz
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  2. Loïc Le Marrec
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Correspondence to Marwan Hariz.

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Hariz, M., Le Marrec, L. & Lerbet, J. Explicit analysis of large transformation of a Timoshenko beam: post-buckling solution, bifurcation, and catastrophes. Acta Mech 232, 3565–3589 (2021). https://doi.org/10.1007/s00707-021-02993-8

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  • DOI: https://doi.org/10.1007/s00707-021-02993-8

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