Skip to main content
SpringerLink
Log in

Explicit solution to the large deformation of a cantilever beam under point load at the free tip using the variational iteration method-II

  • Published:
Journal of Mechanical Science and Technology Aims and scope Submit manuscript

Abstract

This study focuses on a new analytical method called the variational iteration method-II (VIM-II) for the differential equation of the large deformation of a cantilever beam under point load at the free tip. The rotation angles as well as the horizontal and vertical displacements of a cantilever beam with large deformation are calculated in an explicit analytical form. A comparison of the results with those of some numerical and analytical methods shows the simplicity and effectiveness of VIM-II. VIM-II is proven to be a powerful technique that can be used to obtain accurate solutions that cannot be provided otherwise by perturbation and other methods. The accuracy and convergence of the method are also investigated and compared with those of other methods. The results showed good agreement between VIM-II and other methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. D. Landau and E. M. Lifshitz, Course of theoretical physics, Vol. 7: Theory of Elasticity (Oxford: Pergamon Press) Chap., 17 (1986).

    Google Scholar 

  2. R. Feynman, R. B. Leighton and M. Sands, The feynman lectures of physics, Volume II: Mainly Electromagnetism and Matter (Massaxhusets: Addison-Wesley Publishing) Chap., 38 (1989).

    Google Scholar 

  3. B. N. Rao and G. V. Rao, On the large deflection of cantilever beams with end rotational load, Zeitschrift für Angewandte Mathematik und Mechanik, 66(10) (1986) 507–509.

    Article  MATH  Google Scholar 

  4. B. N. Rao and G. V. Rao, Large deflections of a cantilever beam subjected to a rotational distributed loading, Forschung im Ingenieurwesen, 55(4) (1989) 116–120.

    Article  Google Scholar 

  5. T. Beléndez, C. Neipp and A. Beléndez, Large and small deflections of a cantilever beam, European Journal of Physics, 23(3) (2002) 371–379.

    Article  Google Scholar 

  6. B. N. Rao and G. V. Rao, Large deflections of a nonuniform cantilever beam with end rotational load, Forschung im Ingenieurwesen, 54(1) (1988) 24–26.

    Article  Google Scholar 

  7. R. Schmidt and D. A. Dadeppo, Approximate analysis of large deflections of beams, Zeitschrift für Angewandte Mathematik und Mechanik, 51 (1971) 233–234.

    Article  MATH  Google Scholar 

  8. B. N. Rao, G. L. N. Babu and G. V. Rao, Large deflection analysis of a spring hinged cantilever beam subjected to a tip concentrated rational load, Zeitschrift für Angewandte Mathematik und Mechanik, 67(10) (1987) 519–520.

    Article  Google Scholar 

  9. J. H. He, Variational iteration method for autonomous ordinary differential systems, Applied Mathematics and Computation, 114 (2000) 115–123.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. H. He, Variational Approach for nonlinear oscillators, Chaos. Soliton. Fractals, 34(5) (2007) 1430–1439.

    Article  MATH  Google Scholar 

  11. J. H. He, Max-min approach to nonlinear oscillators, Int. J. Nonlinear Sci. Numer. Simul., 9(2) (2008) 207–210.

    Google Scholar 

  12. S. Bagheri, A. Nikkar and H. Ghaffarzadeh, Study of nonlinear vibration of Euler-Bernoulli beams by using analytical approximate techniques, Latin American Journal of Solids and Structures, 11(1) (2014) 157–168.

    Article  Google Scholar 

  13. S. S. Ganji, A. Barari and D. D. Ganji, Approximate analysis of two-mass-spring systems and buckling of a column, Computers & Mathematics with Applications, 61(4) (2011) 1088–1095.

    Article  MathSciNet  MATH  Google Scholar 

  14. H. Yaghoobi and M. Torabi, Novel solution for acceleration motion of a vertically falling non-spherical particle by VIM-Padé approximant, Powder Technology, 215–216 (2012) 206–209.

    Article  Google Scholar 

  15. J. Wang, J. K. Chen and S. Liao, An explicit solution of the large deformation of a cantilever beam under point load at the free tip, Journal of Computational and Applied Mathematics, 212 (2008) 320–330.

    Article  MathSciNet  MATH  Google Scholar 

  16. N. Tolou and J. L. Herder, A semi-analytical approach to large deflections in compliant beams under point load, Mathematical Problems in Engineering, Article ID 910896, 13 pages (2009) doi:10.1155/2009/910896.

    Google Scholar 

  17. P. Salehi, H. Yaghoobi and M. Torabi, Application of the differential transformation method and variational iteration method to large deformation of cantilever beams under point load, Journal of Mechanical Science and Technology, 26(9) (2012) 2879–2887.

    Article  Google Scholar 

  18. D. D. Ganji, H. B. Rokni, M. Hadi Rafiee, A. A. Imani, M. Esfandyaripour and M. Sheikholeslami, Reconstruction of variational iteration method for boundary value problems in structural engineering and fluid mechanics, Int. J. of Nonlinear Dynamics in Eng. and Sci., 3 (2011) 1–10.

    Google Scholar 

  19. A. A. Imani, D. D. Ganji, H. B. Rokni, H. Latifizadeh, E. Hesameddini and M. H. Rafiee, Approximate traveling wave solution for shallow water wave equation, Applied Mathematical Modelling, 36(4) (2012) 1550–1557.

    Article  MathSciNet  MATH  Google Scholar 

  20. J. M. Gere and S. P. Timoshenko, Mechanics of materials, PWS, Boston, Mass, USA (1997).

    Google Scholar 

  21. J. H. He, Asymptotic methods for solitary solutions a compactons, Abstract and Applied Analysis, Vol. 2012 (2012) Article ID 916793, doi:10.1155/2012/916793.

    Google Scholar 

  22. J. H. He, A short remark on fractional variational iteration method, Physics Letters A, 375(38) (2011) 3362–3364.

    Article  MathSciNet  MATH  Google Scholar 

  23. E. Hesameddin and H. Latifizadeh, Reconstruction of variational iteration algorithms using laplace transform, Internat. J. Nonlinear Sci. Numer. Simulation, 10(10) (2009) 1365–1370.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Civil Engineering, University of Tabriz, Tabriz, Iran

    Hosein Ghaffarzadeh & Ali Nikkar

Authors
  1. Hosein Ghaffarzadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ali Nikkar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hosein Ghaffarzadeh.

Additional information

Recommended by Associate Editor Jun-Sik Kim

Hosein Ghaffarzadeh received his Ph.D. in Structural Engineering in 2005 from Shiraz University. His research interests are structural dynamics, structural control, and damage detection. He is currently an assistant professor in the Faculty of Civil Engineering and Faculty Deputy at the University of Tabriz, Tabriz, Iran.

Ali Nikkar received his Bachelor’s degree in Civil Engineering from Shomal University, Amol, Iran and his M. Sc. Degree in Earthquake Engineering from the University of Tabriz. His research interests include formulation and analysis of problems in solid and structural mechanics, nonlinear vibration of structural systems, and asymptotic methods.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ghaffarzadeh, H., Nikkar, A. Explicit solution to the large deformation of a cantilever beam under point load at the free tip using the variational iteration method-II. J Mech Sci Technol 27, 3433–3438 (2013). https://doi.org/10.1007/s12206-013-0866-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12206-013-0866-4

Keywords

Search

Navigation