, Volume 51, Issue 11, pp 2589–2606 | Cite as

A comprehensive analysis of hardening/softening behaviour of shearable planar beams with whatever axial boundary constraint

  • S. LenciEmail author
  • F. Clementi
  • G. Rega
Nonlinear Dynamics, Identification and Monitoring of Structures


The free nonlinear oscillations of a planar elastic beam are investigated based on a comprehensive asymptotic treatment of the exact equations of motion. With the aim of investigating the behaviour also for low slenderness, shear deformations and rotational inertia are taken into account. Attention is payed to the influence of the geometrical and mechanical parameters, and of the boundary conditions in changing the nonlinear behaviour from softening to hardening. An axial linear spring is added to one end of the beam, and it is shown how the behaviour changes qualitatively on passing from the hinged-hinged (commonly hardening) to the hinged-supported (commonly softening) case. Some interesting, and partially unexpected, results are obtained also for values of the slenderness moderately low but still in the realm of practical applications.


Geometrically exact beam model Asymptotic analysis  Nonlinear free vibrations Axially restrained/unrestrained beams High/low slenderness Hardening/softening behaviour 



This work has been partially supported by the Italian Ministry of Education, University and Research (MIUR) by the PRIN funded program 2010/11 N.2010MBJK5B “Dynamics, stability and control of flexible structures”.


  1. 1.
    Atluri S (1973) Nonlinear vibrations of hinged beam including nonlinear inertia effects. ASME J Appl Mech 40:121–126zbMATHGoogle Scholar
  2. 2.
    Luongo A, Rega G, Vestroni F (1986) On nonlinear dynamics of planar shear indeformable beams. ASME J Appl Mech 53:619–624zbMATHGoogle Scholar
  3. 3.
    Kauderer H (1958) Nichtlinear mechanik. Springer, Berlin. ISBN: 978-3-642-92734-8zbMATHGoogle Scholar
  4. 4.
    Crespo da Silva MRM (1988) Nonlinear flexural-flexural-torsional-extensional dynamics of beams. II. Response analysis. Int J Solids Struct 24:1235–1242zbMATHGoogle Scholar
  5. 5.
    Mettler E (1962) Dynamic buckling. In: Flugge (ed) Handbook of engineering mechanics. McGraw-Hill, New York. ISBN: 0070213925Google Scholar
  6. 6.
    Lacarbonara W, Yabuno H (2006) Refined models of elastic beams undergoing large in-plane motions: theory and experiments. Int J Solids Struct 43:5066–5084zbMATHGoogle Scholar
  7. 7.
    Lacarbonara W (2013) Nonlinear structural mechanics. Springer, New York. ISBN: 978-1-4419-1276-3zbMATHGoogle Scholar
  8. 8.
    Simo JC, Vu-Quoc L (1991) A geometrically-exact beam model incorporating shear and torsion warping deformation. Int J Solids Struct 27:371–393zbMATHGoogle Scholar
  9. 9.
    Nayfeh AH, Pai PF (2004) Linear and nonlinear structural mechanics. Wiley, New York. ISBN: 978-0-471-59356-0zbMATHGoogle Scholar
  10. 10.
    Cao DQ, Tucker RW (2008) Nonlinear dynamics of elastic rods using the Cosserat theory: modelling and simulation. Int J Solids Struct 45:460–477zbMATHGoogle Scholar
  11. 11.
    Stoykov S, Ribeiro P (2010) Nonlinear forced vibrations and static deformations of 3D beams with rectangular cross section: the influence of warping, shear deformation and longitudinal displacements. Int J Mech Sci 52:1505–1521Google Scholar
  12. 12.
    Luongo A, Zulli D (2013) Mathematical models of beams and cables. Wiley-ISTE, New York. ISBN: 978-1-84821-421-7zbMATHGoogle Scholar
  13. 13.
    Formica G, Arena A, Lacarbonara W, Dankowicz H (2013) Coupling FEM with parameter continuation for analysis and bifurcations of periodic responses in nonlinear structures. ASME J Comput Nonlinear Dyn 8:021013Google Scholar
  14. 14.
    Lenci S, Rega G (2015) Nonlinear free vibrations of planar elastic beams: a unified treatment of geometrical and mechanical effects, IUTAM Procedia (in press)Google Scholar
  15. 15.
    Nayfeh A (2004) Introduction to perturbation techniques. Wiley-VCH, Weinheim. ISBN: 0-978-471-31013-6zbMATHGoogle Scholar
  16. 16.
    Kovacic I, Rand R (2013) About a class of nonlinear oscillators with amplitude-independent frequency. Nonlinear Dyn 74:455–465MathSciNetzbMATHGoogle Scholar
  17. 17.
    Timoshenko S (1955) Vibrations problems in engineering. Wolfenden Press, New York. ISBN: 1406774650Google Scholar
  18. 18.
    Huang TC (1961) The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions. ASME J Appl Mech 28:579–584MathSciNetzbMATHGoogle Scholar
  19. 19.
    Lenci S, Rega G (2015) Asymptotic analysis of axial-transversal coupling in the free nonlinear vibrations of Timoshenko beams with arbitrary slenderness and axial boundary conditions (submitted)Google Scholar
  20. 20.
    Clementi F, Demeio L, Mazzilli CEN, Lenci S (2015) Nonlinearvibrations of non-uniform beams by the MTS asymptotic expansionmethod, Contin Mech Thermodyn. doi: 10.1007/s00161-014-0368-3 zbMATHGoogle Scholar
  21. 21.
    Srinil N, Rega G (2007) The effects of kinematic condensation on internally resonant forced vibrations of shallow horizontal cables. Int J Non-Linear Mech 42:180–195zbMATHGoogle Scholar
  22. 22.
    Lagomarsino S, Penna A, Galasco A, Cattari S (2013) TREMURI program: an equivalent frame model for the nonlinear seismic analysis of masonry buildings. Eng Struct 56:1787–1799Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Civil and Buildings Engineering, and ArchitecturePolytechnic University of MarcheAnconaItaly
  2. 2.Department of Structural and Geotechnical EngineeringSapienza University of RomeRomeItaly

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