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Acta Mechanica
Published in: Acta Mechanica 9/2021

29-06-2021 | Original Paper

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

Authors: Marwan Hariz, Loïc Le Marrec, Jean Lerbet

Published in: Acta Mechanica | Issue 9/2021

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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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Literature
1.
Han, S.M., Benaroya, H., Wei, T.: Dynamics of transversely vibrating beams using four engineering theories. J. Sound Vib. 225(5), 935–988 (1999)CrossRef
2.
Dell’Isola, F., Corte, A Della, Greco, L., Luongo, A.: Plane bias extension test for a continuum with two inextensible families of fibers: a variational treatment with Lagrange multipliers and a perturbation solution. Int. J. Solids Struct. 81, 1–12 (2016)CrossRef
3.
Timoshenko, S.P.: On the transverse vibrations of bars of uniform cross-section. Lond. Edinb. Dublin Philos. Mag. J. Sci. 43(253), 125–131 (1922)CrossRef
4.
Timoshenko, S.P., Gere, J.M.: Theory of Elastic Stability. Courier Corporation, Chelmsford (2009)
5.
Elishakoff, I.E.: Handbook on Timoshenko-Ehrenfest beam and Uflyand-Mindlin Plate Theories. World Scientific, Singapore (2019)CrossRef
6.
Mohyeddin, A., Fereidoon, A.: An analytical solution for the large deflection problem of Timoshenko beams under three-point bending. Int. J. Mech. Sci. 78, 135–139 (2014)CrossRef
7.
Li, D.-K., Li, X.-F.: Large deflection and rotation of Timoshenko beams with frictional end supports under three-point bending. C. R. Méc. 344(8), 556–568 (2016)CrossRef
8.
Cosserat, E., Cosserat, F.: Théorie des corps déformables, A. Hermann et fils (1909)
9.
Eugster, S.R., Harsch, J.: A variational formulation of classical nonlinear beam theories. In: Abali, B.E., Giorgio, I. (eds.) Developments and Novel Approaches in Nonlinear Solid Body Mechanics, pp. 95–121. Springer International Publishing, Berlin (2020)CrossRef
10.
Antman, S.: Nonlinear Problems of Elasticity, Volume 107 of Applied Mathematical Sciences, 2nd edn, p. 1. Springer, New York (2005)
11.
Reissner, E.: On one-dimensional finite-strain beam theory: the plane problem. Zeitschrift für angewandte Mathematik und Physik ZAMP 23(5), 795–804 (1972)CrossRef
12.
Simo, J.C.: A finite strain beam formulation. The three-dimensional dynamic problem. I. Comput. Methods Appl. Mech. Eng. 49(1), 55–70 (1985)CrossRef
13.
Rakotomanana, L.: Eléments de dynamique des solides et structures déformables, PPUR Presses polytechniques (2009)
14.
Le Marrec, L., Lerbet, J., Rakotomanana, L.R.: Vibration of a Timoshenko beam supporting arbitrary large pre-deformation. Acta Mech. 229(1), 109–132 (2018)MathSciNetCrossRef
15.
Bigoni, D.: Nonlinear Solid Mechanics: Bifurcation Theory and Material Instability. Cambridge University Press, Cambridge (2012)CrossRef
16.
Reissner, E.: Some remarks on the problem of column buckling. Ingenieur-Archiv 52(1–2), 115–119 (1982)CrossRef
17.
Cedolin, L., et al.: Stability of Structures: Elastic, Inelastic, Fracture and Damage Theories. World Scientific, Singapore (2010)MATH
18.
Bažant, Z.P.: A correlation study of formulations of incremental deformation and stability of continuous bodies. J. Appl. Mech. 38(4), 919–928 (1971)CrossRef
19.
Dell’Isola, F., Corte, A. Della, Battista, A., Barchiesi, E.: Extensible beam models in large deformation under distributed loading: A numerical study on multiplicity of solutions. In: Higher Gradient Materials and Related Generalized Continua, pp. 19–41. Springer (2019)
20.
Corte, A.D., Battista, A., Dell’Isola, F., Seppecher, P.: Large deformations of Timoshenko and Euler beams under distributed load. Zeitschrift für Angewandte Mathematik und Physik ZAMP 70(2), 52 (2019)MathSciNetCrossRef
21.
Meyer, K.R.: Jacobi elliptic functions from a dynamical systems point of view. The American Mathematical Monthly 108(8), 729–737 (2001)MathSciNetCrossRef
22.
Baker, T.E., Bill, A.: Jacobi elliptic functions and the complete solution to the bead on the hoop problem. Am. J. Phys. 80(6), 506–514 (2012)CrossRef
23.
Olver, F.W., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Hardback and CD-ROM. Cambridge University Press, Cambridge (2010)MATH
24.
Ohtsuki, A.: An analysis of large deflection in a symmetrical three-point bending of beam. Bull. JSME 29(253), 1988–1995 (1986)CrossRef
25.
Chucheepsakul, S., Wang, C.M., He, X.Q., Monprapussorn, T.: Double curvature bending of variable-arc-length elasticas. J. Appl. Mech. 66(1), 87–94 (1999)CrossRef
26.
Magnusson, A., Ristinmaa, M., Ljung, C.: Behaviour of the extensible elastica solution. Int. J. Solids Struct. 38(46–47), 8441–8457 (2001)CrossRef
27.
Humer, A.: Exact solutions for the buckling and postbuckling of shear-deformable beams. Acta Mech. 224(7), 1493–1525 (2013)MathSciNetCrossRef
28.
Humer, A., Pechstein, A.S.: Exact solutions for the buckling and postbuckling of a shear-deformable cantilever subjected to a follower force. Acta Mech. 230(11), 3889–3907 (2019)MathSciNetCrossRef
29.
Batista, M.: Large deflections of shear-deformable cantilever beam subject to a tip follower force. Int. J. Mech. Sci. 75, 388–395 (2013)CrossRef
30.
Batista, M.: Analytical treatment of equilibrium configurations of cantilever under terminal loads using Jacobi elliptical functions. Int. J. Solids Struct. 51(13), 2308–2326 (2014)CrossRef
31.
Batista, M.: A closed-form solution for Reissner planar finite-strain beam using Jacobi elliptic functions. Int. J. Solids Struct. 87, 153–166 (2016)CrossRef
32.
Timoshenko, S.P.: Lxvi. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Lond. Edinb. Dublin Philos. Mag. J. Sci. 41(245), 744–746 (1921)CrossRef
33.
Le Marrec, L., Zhang, D., Ostoja-Starzewski, M.: Three-dimensional vibrations of a helically wound cable modeled as a Timoshenko rod. Acta Mech. 229(2), 677–695 (2018)MathSciNetCrossRef
34.
Forgit, C., Lemoine, B., Le Marrec, L., Rakotomanana, L.: A Timoshenko-like model for the study of three-dimensional vibrations of an elastic ring of general cross-section. Acta Mech. 227(9), 2543–2575 (2016)MathSciNetCrossRef
35.
Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. Tata McGraw-Hill Education, New York (1955)MATH
36.
Thom, R.: Structural Stability and Morphogenesis. CRC Press, Boca Raton (2018)
37.
Poston, T., Stewart, I.: Catastrophe Theory and Its Applications. Courier Corporation, Chelmsford (2014)MATH
38.
Metadata
Title
Explicit analysis of large transformation of a Timoshenko beam: post-buckling solution, bifurcation, and catastrophes
Authors
Marwan Hariz
Loïc Le Marrec
Jean Lerbet
Publication date
29-06-2021
Publisher
Springer Vienna
Published in
Acta Mechanica / Issue 9/2021
Print ISSN: 0001-5970
Electronic ISSN: 1619-6937
DOI
https://doi.org/10.1007/s00707-021-02993-8

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