Timoshenko beam under finite and dynamic transformations: Lagrangian coordinates and Hamiltonian structures

Oscar Cosserat; Loïc Le Marrec; Oscar Cosserat; Loïc Le Marrec

doi:10.3934/cam.2026002

Communications in Analysis and Mechanics

2026, Volume 18, Issue 1: 37-69. doi: 10.3934/cam.2026002

Previous Article Next Article

Research article

Timoshenko beam under finite and dynamic transformations: Lagrangian coordinates and Hamiltonian structures

Oscar Cosserat ^{1
,
,},
Loïc Le Marrec ²

1.
Georg-August-University Göttingen, Bunsenstraße 3-5, 37073, Göttingen, Germany
2.
Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

Received: 03 April 2025 Revised: 25 September 2025 Accepted: 26 September 2025 Published: 12 January 2026
35F20, 74K10, 37K58

In the framework of Timoshenko beam, the material parameters were inherently prescribed on the material moving frame. In this regard, we derived the strong and weak formulations of the dynamics under finite transformation in Lagrangian coordinates. Accordingly, analytical mechanics tools were used to deduce a new Hamiltonian formulation of the model which proved to be remarkably simple and synthetic.
- Timoshenko beam,
- geometrically exact formulation,
- large transformation,
- Lagrangian coordinates,
- Hamiltonian mechanics,
- Poisson brackets
Citation: Oscar Cosserat, Loïc Le Marrec. Timoshenko beam under finite and dynamic transformations: Lagrangian coordinates and Hamiltonian structures[J]. Communications in Analysis and Mechanics, 2026, 18(1): 37-69. doi: 10.3934/cam.2026002

Related Papers:

Abstract

In the framework of Timoshenko beam, the material parameters were inherently prescribed on the material moving frame. In this regard, we derived the strong and weak formulations of the dynamics under finite transformation in Lagrangian coordinates. Accordingly, analytical mechanics tools were used to deduce a new Hamiltonian formulation of the model which proved to be remarkably simple and synthetic.

References

[1]	L. Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, apud Marcum-Michaelem Bousquet, 1744.
[2]	R. Levien, The elastica: a mathematical history, Tech. Rep. UCB/EECS-2008-103, 2008. http://www2.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-103.html
[3]	I. Elishakoff, Who developed the so-called Timoshenko beam theory? Math. Mech. Solids, 25 (2020), 97–116. https://doi.org/10.1177/1081286519856931
[4]	S. P. Timoshenko, LXVI. on the correction for shear of the differential equation for transverse vibrations of prismatic bars, Lond. Edinb. Dubl. Phil. Mag., 41 (1921), 744–746. https://doi.org/10.1080/14786442108636264 doi: 10.1080/14786442108636264
[5]	P. D. Simão, Influence of shear deformations on the buckling of columns using the Generalized Beam Theory and energy principles, Eur. J. Mech. A Solids, 61 (2017), 216–234. https://doi.org/10.1016/j.euromechsol.2016.09.015 doi: 10.1016/j.euromechsol.2016.09.015
[6]	B. V. Sankar, An elasticity solution for functionally graded beams, Compos. Sci. Technol., 61 (2001), 689–696. https://doi.org/10.1016/S0266-3538(01)00007-0 doi: 10.1016/S0266-3538(01)00007-0
[7]	L. Rakotomanana, Eléments de dynamique des solides et structures déformables, PPUR Presses polytechniques, 2009.
[8]	L. Le Marrec, J. Lerbet, L. R. Rakotomanana, Vibration of a Timoshenko beam supporting arbitrary large pre-deformation, Acta Mech, 229 (2018), 109–132. https://doi.org/10.1007/s00707-017-1953-x doi: 10.1007/s00707-017-1953-x
[9]	Z. P. Bažant, M. Jirásek, Nonlocal integral formulations of plasticity and damage: Survey of progress, J. Eng. Mech, 128 (2002), 1119–1149. https://doi.org/10.1061/(ASCE)0733-9399(2002)128:11(1119) doi: 10.1061/(ASCE)0733-9399(2002)128:11(1119)
[10]	C. M. Wang, Y. Y. Zhang, S. S. Ramesh, S. Kitipornchai, Buckling analysis of micro- and nano-rods/tubes based on nonlocal Timoshenko beam theory, J. Phys. D: Appl. Phys., 39 (2006), 3904. https://doi.org/10.1088/0022-3727/39/17/029 doi: 10.1088/0022-3727/39/17/029
[11]	M. Spagnuolo, U. Andreaus, A targeted review on large deformations of planar elastic beams: extensibility, distributed loads, buckling and post-buckling, Math. Mech. Solids, 24 (2019), 258–280. https://doi.org/10.1177/1081286517737000 doi: 10.1177/1081286517737000
[12]	M. Hariz, L. Le Marrec, J. Lerbet, Explicit analysis of large transformation of a Timoshenko beam: post-buckling solution, bifurcation, and catastrophes, Acta Mech, 232 (2021), 3565–3589. https://doi.org/10.1007/s00707-021-02993-8 doi: 10.1007/s00707-021-02993-8
[13]	P. Koutsogiannakis, D. Bigoni, F. Dal Corso, Double restabilization and design of force–displacement response of the extensible elastica with movable constraints, Eur. J. Mech. A Solids, 100 (2023), 104745. https://doi.org/10.1016/j.euromechsol.2022.104745 doi: 10.1016/j.euromechsol.2022.104745
[14]	M. Hariz, L. Le Marrec, J. Lerbet, Buckling of Timoshenko beam under two-parameter elastic foundations, Int. J. Solids Struct., 244–245 (2022), 111–583. https://doi.org/10.1016/j.ijsolstr.2022.111583 doi: 10.1016/j.ijsolstr.2022.111583
[15]	E. Reissner, On one-dimensional large-displacement ﬁnite-strain beam theory, Stud. Appl. Math., 52 (1973), 87–95. https://doi.org/10.1002/sapm197352287 doi: 10.1002/sapm197352287
[16]	E. Reissner, On one-dimensional finite-strain beam theory: the plane problem, Z. Angew. Math. Phys., 23 (1972), 795–804.
[17]	A. B. Whitman, C. N. DeSilva, An exact solution in a nonlinear theory of rods, J Elasticity, 4 (1974), 265–-280. https://doi.org/10.1007/BF00048610 doi: 10.1007/BF00048610
[18]	J. C. Simo, A finite strain beam formulation. The three-dimensional dynamic problem. Part I, Comput. Methods Appl. Mech. Eng., 49 (1985), 55–70. https://doi.org/10.1016/0045-7825(85)90050-7 doi: 10.1016/0045-7825(85)90050-7
[19]	P. Mata, A. Barbat, S. Oller, R. Boroschek, Inelastic Analysis of Geometrically Exact Rods, Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), 2007. https://doi.org/10.13140/2.1.1068.5764
[20]	M. M. Attard, Finite strain-beam theory, Int. J. Solids Struct., 40 (2003), 4563–4584. https://doi.org/10.1016/S0020-7683(03)00216-6 doi: 10.1016/S0020-7683(03)00216-6
[21]	E. Cosserat, F. Cosserat, Théorie des Corps Déformables, Nature, 81 (1909), 67. https://doi.org/10.1038/081067a0 doi: 10.1038/081067a0
[22]	R. K. Kapania, J. Li, On a geometrically exact curved/twisted beam theory under rigid cross-section assumption, Comput Mech, 30 (2023), 428–443. https://doi.org/10.1007/s00466-003-0421-8 doi: 10.1007/s00466-003-0421-8
[23]	R. K. Kapania, J. Li, A formulation and implementation of geometrically exact curved beam elements incorporating finite strains and finite rotations, Comput Mech, 30 (2003), 444–459. https://doi.org/10.1007/s00466-003-0422-7 doi: 10.1007/s00466-003-0422-7
[24]	F. Auricchio, P. Carotenuto, On the geometrically exact beam model: A consistent, effective and simple derivation from three-dimensional finite-elasticity, Int. J. Solids Struct., 45 (2008), 4766–4781. https://doi.org/10.1016/j.ijsolstr.2008.04.015 doi: 10.1016/j.ijsolstr.2008.04.015
[25]	M. Yilmaz, M. Omurtag, Large deflection of 3D curved rods: An objective formulation with principal axes transformations, Comput. Struct., 163 (2016), 71–82. https://doi.org/10.1016/j.compstruc.2015.10.010 doi: 10.1016/j.compstruc.2015.10.010
[26]	F. Gruttmann, R. Sauer, W. Wagner, Theory and numerics of three-dimensional beams with elastoplastic material behaviour, Internat. J. Numer. Methods Engrg., 48 (2000), 1675–1702. 10.1002/1097-0207(20000830)48:12 < 1675::AID-NME957>3.0.CO; 2-6 doi: 10.1002/1097-0207(20000830)48:12 < 1675::AID-NME957>3.0.CO; 2-6
[27]	S. Levyakov, Formulation of a geometrically nonlinear 3D beam finite element based on kinematic-group approach, Appl. Math. Model., 39 (2015), 6207–6222. https://doi.org/10.1016/j.apm.2015.01.064 doi: 10.1016/j.apm.2015.01.064
[28]	P. de Mattos Pimenta, S. Maassen, C. da Costa e Silva, J. Schröder, A Fully Nonlinear Beam Model of Bernoulli–Euler Type, Springer International Publishing, Cham, 2020. https://doi.org/10.1007/978-3-030-33520-5_5
[29]	C. da Costa e Silva, S. F. Maassen, P. M. Pimenta, J. Schröder, A simple finite element for the geometrically exact analysis of Bernoulli–Euler rods, Comput Mech, 65 (2020), 905–923. https://doi.org/10.1007/s00466-019-01800-5 doi: 10.1007/s00466-019-01800-5
[30]	C. da Costa e Silva, S. F. Maassen, P. M. Pimenta, J. Schröder, On the simultaneous use of simple geometrically exact shear-rigid rod and shell ﬁnite elements, Comput Mech, 67 (2021), 867–881. https://doi.org/10.1007/s00466-020-01967-2 doi: 10.1007/s00466-020-01967-2
[31]	H. Murakami, Development of an active curved beam model - part i: Kinematics and integrability conditions, J. Appl. Mech., 86 (2017), 061002. https://doi.org/10.1115/1.4036308 doi: 10.1115/1.4036308
[32]	H. Murakami, Development of an active curved beam model - part ii: Kinetics and internal activation, J. Appl. Mech., 86 (2017), 061003. https://doi.org/10.1115/1.4036317 doi: 10.1115/1.4036317
[33]	C. Lanczos, The variational principles of mechanics, in: Mathematical Expositions, University of Toronto Press, 1949. https://doi.org/10.3138/9781487583057
[34]	K. Satō, On the governing equations for vibration and stability of a Timoshenko beam: Hamilton's principle, J Sound Vib, 145 (1991), 338–340. https://api.semanticscholar.org/CorpusID:119965926 doi: https://api.semanticscholar.org/CorpusID:119965926
[35]	J. E. Marsden, T. S. Ratiu, Introduction to mechanics and symmetry: A basic exposition of classical mechanical systems, in: Texts in Applied Mathematics, Springer New York, NY, 1999. https://doi.org/10.1007/978-0-387-21792-5
[36]	M. Chadha, M. D. Todd, An introductory treatise on reduced balance laws of Cosserat beams, Int. J. Solids Struct., 126–127 (2017), 54–73. https://doi.org/10.1016/j.ijsolstr.2017.07.028 doi: 10.1016/j.ijsolstr.2017.07.028
[37]	A. Ibrahimbegovic, R. L. Taylor, On the role of frame-invariance in structural mechanics models at finite rotations, Comput. Methods Appl. Mech. Eng., 191 (2002), 5159–5176. https://doi.org/10.1016/S0045-7825(02)00442-5 doi: 10.1016/S0045-7825(02)00442-5
[38]	J. C. Simo, L. Vu-Quoc, A Geometrically-exact rod model incorporating shear and torsion-warping deformation, Int. J. Solids Struct., 27 (1991), 371–393. https://doi.org/10.1016/0020-7683(91)90089-X doi: 10.1016/0020-7683(91)90089-X
[39]	J. C. Simo, J. E. Marsden, P. S. Krishnaprasad, The Hamiltonian structure of nonlinear elasticity: The material and convective representations of solids, rods, and plates, Arch. Rational Mech. Anal., 104 (1988), 125–183. https://doi.org/10.1007/BF00251673 doi: 10.1007/BF00251673
[40]	Z. P. Bažant, L. Cedolin, Stability of Structures, WORLD SCIENTIFIC, 2010. https://www.worldscientific.com/doi/pdf/10.1142/7828
[41]	M. Chadha, M. D. Todd, Poisson bracket formulation of a higher-order, geometrically-exact beam, Appl. Math. Lett., 113 (2021), 106842. https://doi.org/10.1016/j.aml.2020.106842 doi: 10.1016/j.aml.2020.106842
[42]	J. C. Simo, N. Tarnow, M. Doblare, nonlinear dynamics of three-dimensional rods: Exact energy and momentum conserving algorithms, Internat. J. Numer. Methods Engrg., 38 (1995), 1431–1473. https://doi.org/10.1002/nme.1620380903 doi: 10.1002/nme.1620380903
[43]	M. A. Crisﬁeld, G. Jelenić, Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite-element implementation, Proc. R. Soc. Lond. A., 455 (1999), 1125–1147. https://doi.org/10.1098/rspa.1999.0352 doi: 10.1098/rspa.1999.0352
[44]	S. Ghosh, D. Roy, Consistent quaternion interpolation for objective finite element approximation of geometrically exact beam, Comput. Methods Appl. Mech. Eng., 198 (2008), 555–571. https://doi.org/10.1016/j.cma.2008.09.004 doi: 10.1016/j.cma.2008.09.004
[45]	S. Ghosh, D. Roy, Consistent quaternion interpolation for objective finite element approximation of geometrically exact beam, Comput Mech, 44 (2009), 103–118. https://doi.org/10.1007/s00466-008-0358-z doi: 10.1007/s00466-008-0358-z
[46]	F. Demoures, F. Gay-Balmaz, S. Leyendecker, S. Ober-Blöbaum, T. S. Ratiu, Y. Weinand, Discrete variational Lie group formulation of geometrically exact beam dynamics, Numer. Math., 130 (2015), 73–123. https://doi.org/10.1007/s00211-014-0659-4 doi: 10.1007/s00211-014-0659-4
[47]	S. Hante, D. Tumiotto, M. Arnold, A Lie group variational integration approach to the full discretization of a constrained geometrically exact Cosserat beam model, Multibody Syst Dyn, 54 (2002), 97–123. https://doi.org/10.1007/s11044-021-09807-8 doi: 10.1007/s11044-021-09807-8
[48]	M. Herrmann, P. Kotyczka, Relative-kinematic formulation of geometrically exact beam dynamics based on lie-group variational integrators, 2024. https://doi.org/10.21203/rs.3.rs-4163759/v1
[49]	P. Zhong, G. Huang, G. Yang, Frame-invariance in finite element formulations of geometrically exact rods, Appl. Math. Mech., 37 (2016), 1669–1688. https://doi.org/10.1007/s10483-016-2147-8 doi: 10.1007/s10483-016-2147-8
[50]	M. Hariz, L. Le Marrec, J. Lerbet, Analysis of one-dimensional structure using Lie groups approach, Math. Mech. Complex Syst., 12 (2024), 333–358. https://doi.org/10.2140/memocs.2024.12.333 doi: 10.2140/memocs.2024.12.333
[51]	J. Lerbet, Intrinsic formulation of dynamics of curvilinear systems, Volume 6: 5th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B, and C, USA, 2005. https://doi.org/10.1115/DETC2005-84127
[52]	J. Lerbet, General equations of curvilinear systems: A coordinate-free approach by using lie group theory, Mech. Res. Commun., 30 (2003), 505–512. https://doi.org/10.1016/S0093-6413(03)00028-4 doi: 10.1016/S0093-6413(03)00028-4
[53]	P. A. Djondjorov, Invariant properties of Timoshenko beam equations, Int. J. Eng. Sci., 33 (1995), 2103–2114. https://doi.org/10.1016/0020-7225(95)00056-4 doi: 10.1016/0020-7225(95)00056-4
[54]	J. C. Simo, L. Vu-Quoc, A three-dimensional finite-strain rod model. part II: Computational aspects, Comput. Methods Appl. Mech. Eng., 58 (1986), 79–116. https://doi.org/10.1016/0045-7825(86)90079-4 doi: 10.1016/0045-7825(86)90079-4
[55]	A. Brugnoli, D. Alazard, V. Pommier-Budinger, D. Matignon, Port-Hamiltonian formulation and symplectic discretization of plate models Part I: Mindlin model for thick plates, Appl. Math. Model., 75 (2019), 940–960. https://doi.org/10.1016/j.apm.2019.04.035 doi: 10.1016/j.apm.2019.04.035
[56]	A. Bendimerad-Hohl, D. Matignon, G. Haine, L. Lefèvre, On Stokes-Lagrange and Stokes-Dirac representations for 1D distributed port-Hamiltonian systems, IFAC-PapersOnLine, 58 (2024), 238–243. https://doi.org/10.1016/j.ifacol.2024.10.174 doi: 10.1016/j.ifacol.2024.10.174
[57]	H. Poincaré, Sur une forme nouvelle des équations de la mécanique, CR Acad. Sci, 132 (1901), 369–371.
[58]	J. E. Marsden, T. S. Ratiu, A. Weinstein, Semidirect products and reduction in mechanics, Trans. Amer. Math. Soc., 281 (1984), 147–177. https://doi.org/10.2307/1999527 doi: 10.2307/1999527
[59]	D. D. Holm, J. E. Marsden, T. S. Ratiu, The Euler–Poincaré equations and semidirect products with applications to continuum theoriesm, Adv. Math., 137 (1998), 1–81. https://doi.org/10.1006/aima.1998.1721 doi: 10.1006/aima.1998.1721
[60]	J. E. Marsden, T. S. Ratiu, A. Weinstein, Reduction and Hamiltonian structures on duals of semi-direct product Lie algebras, Cont. Math., 28 (1984), 55–100. https://doi.org/10.1090/conm/028 doi: 10.1090/conm/028
[61]	J. C. Simo, The (symmetric) Hessian for geometrically nonlinear models in solid mechanics: Intrinsic definition and geometric interpretation, Comput. Methods Appl. Mech. Engrg., 96 (1992), 189–200. https://doi.org/10.1016/0045-7825(92)90131-3 doi: 10.1016/0045-7825(92)90131-3
[62]	P. Libermann, C.-M. Marle, Symplectic geometry and analytical mechanics, in: Mathematics and Its Applications, Springer Dordrecht, 1987. https://doi.org/10.1007/978-94-009-3807-6
[63]	J. N. Reddy, On the dynamic behaviour of the timoshenko beam ﬁnite elements, Sadhana, 24 (1999), 175-–198. https://doi.org/10.1007/BF02745800 doi: 10.1007/BF02745800
[64]	W. Hu, X. Xi, Z. Song, C. Zhang, Z. Deng, Coupling dynamic behaviors of axially moving cracked cantilevered beam subjected to transverse harmonic load, Mech. Syst. Sig. Process., 204 (2023), 110757. https://doi.org/10.1016/j.ymssp.2023.110757 doi: 10.1016/j.ymssp.2023.110757
[65]	W. Hu, P. Cui, Z. Han, J. Yan, C. Zhang, Z. Deng, Coupling dynamic problem of a completely free weightless thick plate in geostationary orbit, Appl. Math. Model., 137 (2025), 115628. https://doi.org/10.1016/j.apm.2024.07.035 doi: 10.1016/j.apm.2024.07.035
[66]	M. Xu, W. Hu, Z. Han, H. Bai, Z. Deng, C. Zhang, Symmetry-breaking dynamics of a flexible hub-beam system rotating around an eccentric axis, Mech. Syst. Sig. Process., 222 (2025), 111757. https://doi.org/10.1016/j.ymssp.2024.111757 doi: 10.1016/j.ymssp.2024.111757

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)