Sébastien Neukirch
Institut Jean le Rond d'Alembert
Centre National de la Recherche Scientifique
Sorbonne Université, Campus Pierre et Marie Curie
Paris, Francetel: +33 1 44 27 72 61
e-mail: sebastien.neukirch (-atat-) upmc.fr
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Comparison of the Von Karman and Kirchhoff models for the post-buckling and vibrations of elastic beams
Sebastien Neukirch, Morteza Yavari, Noel Challamel, Olivier Thomas
Journal of Theoretical, Computational and Applied Mechanics, May 21 (2021) p 1-18
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 non-linear model is known to grasp the behavior of the system. It is based on the von Karman approximation, a 2nd order expansion in the rotation and vertical displacement of the buckled beam. But as the rotation and the vertical displacement in the beam become 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. Furthermore, we show that the limit in the validity range of the Woinowsky-Krieger model is only marginally affected by the slenderness ratio of the beam. Different ways to nondimensionalize 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.
Key words : nonlinearities, postbuckling, natural frequencies
DOI: 10.46298/jtcam.6828
Diamond Open Access: journal version hal-02957425v3 and supplementary data hal-02957425v2
Submitted on oct 7th, 2020
Reports received on dec. 2020
2nd version submitted on march, 2021
Accepted on March 11th, 2021
Published May 17th, 2021
Reviewer 1
The manuscript “Comparison of the Von Karman and Kirchhoff models for the post-buckling and vibrations of elastic beams” presents a thorough investigation of the (differences in the) performance of the two well-established beam models of Woinowsky-Krieger and Kirchhoff for nonlinear post-buckling behavior. In my opinion, the manuscript is well written and presents relevant new insights into the formulations of the beam models and their applicability. Thus, I recommend acceptance for publication in Journal of Theoretical, Computational and Applied Mechanics provide that below minor comments are taken into account.Reviewer 2
The present article studies the buckling and post-buckling behavior of a plane, geometrically nonlinear beam problem in terms of nonlinear equilibirum configurations and small vibrations w.r.t. to these configurations. The geometrically exact Kirchhoff beam model is considered as reference solution in this context. Based on this reference solution, the accuracy and scope of applicability of the well-known (approximate) Woinowsky-Krieger (WK) model is critically studied. It is demonstrated that some of the shortcomings of this model can be traced back to the fact that it does not represent a consistent 2nd order model. Based on this finding, the authors propose a consistent 3rd order model derived from Kirchhoff’s beam theory, which is capable of representing vibration modes in the post-buckling regime more accurately.All in all, the article is well written, mathematical derivations are sound and the presented findings as well as the proposed 3rd order model are interesting and novel. Therefore, the article can be considered as suitable for publication in JTCAM. Still, before publication the following minor issues should be addressed.
1) The study with the consistently derived third order model is very interesting. However, as the authors stated, in contrast to the WK model it’s difficult to find analytical solutions for the third order model. Do the authors still see a significant practical benefit when using this third order model as compared to the full Kirchhoff model or is the motivation for the third order model of a pure didactic nature? A short comment on this issue in the revised manuscript would be appreciated.
2)What discretization scheme is used to solve the Kirchhoff problem numerically? What is the computational effort to solve this problem? For comparison, what is the effort of solving the 3rd order model (related to comment 1) above)? Please provide this information in the revised manuscript.
3) The authors took the geometrically exact Kirchhoff beam model as reference solution for the considered 2D buckling problem of a straight, clamped beam. In recent years significant research efforts have been spent on the derivation of finite element formulations according to the Kirchhoff beam theory allowing to solve 3D large deformation beam problems in the most general sense, e.g. w.r.t. to general curved initial geometries, cross-section shapes and boundary conditions, see e.g. [1-4]. In my opinion, it would be beneficial for potential readers to provide this information in the introduction.
4)In the range of reasonably high slenderness ratios, i.e. the range of applicability of the Kirchhoff model, I would expect that axial strains have only minor influence on the buckling solution. Just out of curiosity: In the authors’ opinion, could the process of expanding the differential equations e.g. up to third-order terms and of deriving analytical solutions be simplified, when assuming inextensibility, i.e. e=0 or X’^2 + Y’^2=1?
5)In Figure 2, would it be reasonable to mention/discuss also the Euler solution for the critical buckling load?
Reviewer 3
The authors carefully consider the implications of the conventional approach to treat moderate geometrically nonlinear behavior of straight beams. This model, which is frequently used in practice, is attributed by the authors to S. Woinowsky-Krieger. Comparing solutions, predicted by this model to the geometrically exact solutions for Kirchhoff beams, the authors demonstrate the limitations of the WK model and accurately estimate its range of applicability. The direct simulation results are supported by the asymptotic analysis of the exact nonlinear equations. Moreover, the authors suggest another 3rd order model, which is simpler than the fully nonlinear one, but which is capable of describing effects, not captured by the WK model.