Advertisement

Acta Mechanica

, Volume 225, Issue 7, pp 1967–1984 | Cite as

A general method for analyzing moderately large deflections of a non-uniform beam: an infinite Bernoulli–Euler–von Kármán beam on a nonlinear elastic foundation

  • T. S. JangEmail author
Article

Abstract

The present paper concerns a semi-analytical procedure for moderately large deflections of an infinite non-uniform static beam resting on a nonlinear elastic foundation. To construct the procedure, we first derive a nonlinear differential equation of a Bernoulli–Euler–von Kármán “non-uniform” beam on a “nonlinear” elastic foundation, where geometrical nonlinearities due to moderately large deflection and beam non-uniformity are effectively taken into account. The nonlinear differential equation is transformed into an equivalent system of nonlinear integral equations by a canonical representation. Based on the equivalent system, we propose a method for the moderately large deflection analysis as a general approach to an infinite non-uniform beam having a variable flexural rigidity and a variable axial rigidity. The method proposed here is a functional iterative procedure, not only fairly simple but straightforward to apply. Here, a parameter, called a base point of the method, is also newly introduced, which controls its convergence rate. An illustrative example is presented to investigate the validity of the method, which shows that just a few iterations are only demanded for a reasonable solution.

Keywords

Base Point Nonlinear Differential Equation Elastic Foundation Geometrical Nonlinearity Steady State Error 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Lee S.Y., Ke H.Y., Kuo Y.H.: Exact static deflection of a non-uniform Bernoulli–Euler beam with general elastic end restraints. Comput. Struct. 36, 91–97 (1990)CrossRefzbMATHGoogle Scholar
  2. 2.
    Kuo Y.H., Lee S.Y.: Deflection of non-uniform beams resting on a nonlinear elastic foundation. Comput. Struct. 51, 513–519 (1994)CrossRefzbMATHGoogle Scholar
  3. 3.
    Jang T.S., Sung H.G.: A new semi-analytical method for the non-linear static analysis of an infinite beam on a non-linear elastic foundation: a general approach to a variable beam cross-section. Int. J. Non-linear Mech. 47, 132–139 (2012)CrossRefGoogle Scholar
  4. 4.
    Parker D.F.: An asymptotic analysis of large deflections and rotations of elastic rods. Int. J. Solids Struct. 15, 361–377 (1979)CrossRefzbMATHGoogle Scholar
  5. 5.
    Pleus P., Sayir M.: A second order theory for large deflections of slender beams. J. Appl. Math. Phys. 34, 192–217 (1983)CrossRefzbMATHGoogle Scholar
  6. 6.
    Jang T.S.: A new semi-analytical approach to large deflections of Bernoulli–Euler– von Kármán beams on a linear elastic foundation: nonlinear analysis of infinite beams. Int. J. Mech. Sci. 66, 22–32 (2013)CrossRefGoogle Scholar
  7. 7.
    Katsikadelis J.T., Tsiatas G.C.: Large deflection analysis of beams with variable stiffness. Acta Mech. 164, 1–13 (2003)CrossRefzbMATHGoogle Scholar
  8. 8.
    Tsiatas G.C.: Nonlinear analysis of non-uniform beams on nonlinear elastic foundation. Acta Mech. 209, 141–152 (2010)CrossRefzbMATHGoogle Scholar
  9. 9.
    Brunner W., Irschik H.: Vibrations of multi-layered composite elasto-viscoplastic beams using second-order theory. Non-linear Dyn. 6, 37–48 (1994)Google Scholar
  10. 10.
    Adam C., Ziegler F.: Moderately large forced oblique vibrations of elastic-viscoplastic deteriorating slightly curved beams. Arch. Appl. Mech. 67, 375–392 (1997)CrossRefzbMATHGoogle Scholar
  11. 11.
    Li X.-F., Tang G.-J., Shen Z.-B., Lee K.Y.: Vibration of nonclassical shear beams with Winkler–Pasternak-type restraint. Acta Mech. 223, 953–966 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Fotiu P., Irschik H., Ziegler F.: Forced vibrations of an elasto-plastic and deteriorating beam. Acta Mech. 69, 193–203 (1987)CrossRefzbMATHGoogle Scholar
  13. 13.
    Irschik H., Ziegler F.: Dynamics of linear elastic structures with selfstress: A unified treatment for linear and nonlinear problems. Z. Angew. Math. Mech. 68, 199–205 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Fotiu P.A.: Elastodynamics of thin plates with internal dissipative processes, part I: theoretical foundations. Acta Mech. 95, 29–50 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Fotiu P.A.: Elastodynamics of thin plates with internal dissipative processes, part II: computational aspects. Acta Mech. 98, 187–212 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Timoshenko, S.: Method of analysis of statistical and dynamical stress in rail. Proc. Int. Congr. Appl. Mech. Zürich 407–418 (1926)Google Scholar
  17. 17.
    Kenney J.T.: Steady-state vibrations of beam on elastic foundation for moving load. J. Appl. Mech. 21, 359–364 (1954)zbMATHGoogle Scholar
  18. 18.
    Saito H., Murakami T.: Vibrations of an infinite beam on an elastic foundation with consideration of mass of a foundation. Jpn. Soc. Mech. Eng. 12, 200–205 (1969)CrossRefGoogle Scholar
  19. 19.
    Fryba L.: Infinite beam on an elastic foundation subjected to a moving load. Aplikace Matematiky 2, 105–132 (1957)MathSciNetGoogle Scholar
  20. 20.
    Greenberg M.D.: Foundations of Applied Mathematics. Prentice-Hall Inc., New Jersey (1978)zbMATHGoogle Scholar
  21. 21.
    Choi, S.W., Jang, T.S.: Existence and uniqueness of non-linear deflections of an infinite beam resting on a non-uniform non-linear elastic foundation. Bound. Value Probl. 2012(5): doi: 10.1186/1687-2770-2012-5 (2012)

Copyright information

© Springer-Verlag Wien 2014

Authors and Affiliations

  1. 1.Naval Architecture and Ocean EngineeringPusan National UniversityBusanRepublic of Korea

Personalised recommendations