Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2013, Article ID 341063, 8 pages
http://dx.doi.org/10.1155/2013/341063
Research Article

Buckling of Euler Columns with a Continuous Elastic Restraint via Homotopy Analysis Method

1Department of Mathematics, Nevşehir University, 50300 Nevşehir, Turkey
2Department of Mathematics, Niğde University, 51100 Niğde, Turkey
3Department of Civil Engineering, Kocaeli University, 41380 Kocaeli, Turkey

Received 12 October 2012; Revised 10 December 2012; Accepted 10 December 2012

Academic Editor: Fazlollah Soleymani

Copyright © 2013 Aytekin Eryılmaz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. L. Euler, “Die altitudine colomnarum sub proprio pondere corruentium,” Acta Academiae Scientiarum Imperialis Petropolitanae, 1778, in Latin. View at Google Scholar
  2. A. G. Greenhill, “Determination of the greatest height consistent with stability that a vertical pole on mast can be made and of the greatest height to which a tree of given proportions can grow,” in Proceedings of the Cambridge Philosophical Society IV, 1883.
  3. A. N. Dinnik, “Design of columns of varying cross-section,” Transactions of the ASME, vol. 51, 1929. View at Google Scholar
  4. T. R. Karman and M. A. Biot, Mathematical Methods in Engineering, McGraw Hill, New York, NY, USA, 1940.
  5. S. P. Timoshenko and J. M. Gere, Theory of Elastic Stability, McGraw Hill, New York, NY, USA, 1961.
  6. C. M. Wang, C. Y. Wang, and J. N. Reddy, Exact Solutions for Buckling of Structural Members, CRC Press LLC, Florida, Fla, USA, 2005.
  7. H. Bleich, Buckling Strength of Metal Structures, McGraw-Hill, New York, NY, USA, 1952.
  8. A. N. Dinnik, “Design of columns of varying cross-section,” Transactions of the ASME, Applied Mechanics, vol. 51, 1932. View at Google Scholar
  9. I. Elishakoff and F. Pellegrini, “Exact and effective approximate solution solutions of some divergent type non-conservative problems,” Journal of Sound and Vibration, vol. 114, pp. 144–148, 1987. View at Google Scholar
  10. I. Elishakoff and F. Pellegrini, “Application of bessel and lommel functions, and the undetermined multiplier galerkin method version, for instability of non-uniform column,” Journal of Sound and Vibration, vol. 115, no. 1, pp. 182–186, 1987. View at Google Scholar · View at Scopus
  11. I. Elishakoff and F. Pellegrini, “Exact solutions for buckling of some divergence-type nonconservative systems in terms of bessel and lommel functions,” Computer Methods in Applied Mechanics and Engineering, vol. 66, no. 1, pp. 107–119, 1988. View at Google Scholar · View at Scopus
  12. M. Eisenberger, “Buckling loads for variable cross-section members with variable axial forces,” International Journal of Solids and Structures, vol. 27, no. 2, pp. 135–143, 1991. View at Google Scholar · View at Scopus
  13. J. M. Gere and W. O. Carter, “Critical buckling loads for tapered columns,” Journal of Structural Engineering ASCE, vol. 88, no. 1, pp. 1–11, 1962. View at Google Scholar
  14. F. Arbabi and F. Li, “Buckling of variable cross-section columns integral equation approach,” Journal of Structural Engineering ASCE, vol. 117, no. 8, 1991. View at Google Scholar
  15. A. Siginer, “Buckling of columns of variable flexural rigidity,” Journal of Engineering Mechanics, ASCE, vol. 118, no. 3, pp. 543–640, 1992. View at Google Scholar
  16. Q. Li, H. Cao, and G. Li, “Stability analysis of a bar with multi-segments of varying cross-section,” Computers and Structures, vol. 53, no. 5, pp. 1085–1089, 1994. View at Google Scholar · View at Scopus
  17. L. Qiusheng, C. Hong, and L. Guiqing, “Stability analysis of bars with varying cross-section,” International Journal of Solids and Structures, vol. 32, no. 21, pp. 3217–3228, 1995. View at Google Scholar · View at Scopus
  18. Q. Li, H. Cao, and G. Li, “Static and dynamic analysis of straight bars with variable cross-section,” Computers and Structures, vol. 59, no. 6, pp. 1185–1191, 1996. View at Publisher · View at Google Scholar · View at Scopus
  19. J. H. B. Sampaio and J. R. Hundhausen, “A mathematical model and analytical solution for buckling of inclined beam-columns,” Applied Mathematical Modelling, vol. 22, no. 6, pp. 405–421, 1998. View at Google Scholar · View at Scopus
  20. J. B. Keller, “The shape of the strongest column,” Archive for Rational Mechanics and Analysis, vol. 5, no. 1, pp. 275–285, 1960. View at Publisher · View at Google Scholar · View at Scopus
  21. I. Tadjbakhsh and J. B. Keller, “Strongest columns and isoperimetric inequalities for eigenvalues,” Journal of Applied Mechanics ASME, vol. 29, pp. 159–164, 1962. View at Google Scholar
  22. J. E. Taylor, “The strongest column—an energy approach,” Journal of Applied Mechanics ASME, vol. 34, pp. 486–487, 1967. View at Google Scholar
  23. M. T. Atay and S. B. Coşkun, “Elastic stability of Euler columns with a continuous elastic restraint using variational iteration method,” Computers and Mathematics with Applications, vol. 58, no. 11-12, pp. 2528–2534, 2009. View at Publisher · View at Google Scholar · View at Scopus
  24. S. B. Coşkun and M. T. Atay, “Determination of critical buckling load for elastic columns of constant and variable cross-sections using variational iteration method,” Computers and Mathematics with Applications, vol. 58, no. 11-12, pp. 2260–2266, 2009. View at Publisher · View at Google Scholar · View at Scopus
  25. M. T. Atay, “Determination of buckling loads of tilted buckled column with varying flexural rigidity using variational iteration method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, no. 2, pp. 97–103, 2010. View at Google Scholar · View at Scopus
  26. F. Okay, M. T. Atay, and S. B. Coçkun, “Determination of buckling loads and mode shapes of a heavy vertical column under its own weight using the variational iteration method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, no. 10, pp. 851–857, 2010. View at Google Scholar · View at Scopus
  27. S. B. Coşkun, “Determination of critical buckling loads for euler columns of variable flexural stiffness with a continuous elastic restraint using homotopy perturbation method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 2, pp. 191–197, 2009. View at Google Scholar · View at Scopus
  28. M. T. Atay, “Determination of critical buckling loads for variable stiffness euler columns using homotopy perturbation method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 2, pp. 199–206, 2009. View at Google Scholar · View at Scopus
  29. S. B. Coşkun, “Analysis of tilt-buckling of euler columns with varying flexural stiffness using homotopy perturbation method,” Mathematical Modelling and Analysis, vol. 15, no. 3, pp. 275–286, 2010. View at Publisher · View at Google Scholar · View at Scopus
  30. S. Pinarbasi, “Stability analysis of non-uniform rectangular beams using homotopy perturbation method,” Mathematical Problems in Engineering, vol. 2012, Article ID 197483, 18 pages, 2012. View at Google Scholar
  31. Y. Huang and Q. Z. Luo, “A simple method to determine the critical buckling loads for axially inhomogeneous beams with elastic restraint,” Computers and Mathematics with Applications, vol. 61, no. 9, pp. 2510–2517, 2011. View at Publisher · View at Google Scholar · View at Scopus
  32. Z. X. Yuan and X. W. Wang, “Buckling and post-buckling analysis of extensible beam-columns by using the differential quadrature method,” Computers & Mathematics with Applications, vol. 62, no. 12, pp. 4499–4513, 2011. View at Google Scholar
  33. S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, CRC Press, 2004.
  34. S. J. Liao, “Homotopy analysis method: a new analytic method for nonlinear problems,” Applied Mathematics and Mechanics, vol. 19, no. 10, pp. 957–962, 1998. View at Google Scholar · View at Scopus
  35. S. J. Liao, “A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat plate,” Journal of Fluid Mechanics, vol. 385, pp. 101–128, 1999. View at Google Scholar · View at Scopus
  36. S. J. Liao, “An explicit analytic solution to the Thomas-Fermi equation,” Applied Mathematics and Computation, vol. 144, no. 2-3, pp. 495–506, 2003. View at Publisher · View at Google Scholar · View at Scopus
  37. T. Hayat, T. Javed, and M. Sajid, “Analytic solution for rotating flow and heat transfer analysis of a third-grade fluid,” Acta Mechanica, vol. 191, no. 3-4, pp. 219–229, 2007. View at Publisher · View at Google Scholar · View at Scopus
  38. T. Hayat, S. B. Khan, M. Sajid, and S. Asghar, “Rotating flow of a third grade fluid in a porous space with Hall current,” Nonlinear Dynamics, vol. 49, no. 1-2, pp. 83–91, 2007. View at Publisher · View at Google Scholar · View at Scopus
  39. S. Abbasbandy, “Homotopy analysis method for heat radiation equations,” International Communications in Heat and Mass Transfer, vol. 34, no. 3, pp. 380–387, 2007. View at Publisher · View at Google Scholar · View at Scopus
  40. S. Abbasbandy and E. Shivanian, “A new analytical technique to solve Fredholm's integral equations,” Numerical Algorithms, vol. 56, no. 1, pp. 27–43, 2011. View at Publisher · View at Google Scholar · View at Scopus
  41. S. Abbasbandy and F. S. Zakaria, “Soliton solutions for the fifth-order KdV equation with the homotopy analysis method,” Nonlinear Dynamics, vol. 51, no. 1-2, pp. 83–87, 2008. View at Publisher · View at Google Scholar · View at Scopus
  42. Z. Wang, L. Zou, and H. Zhang, “Applying homotopy analysis method for solving differential-difference equation,” Physics Letters A, vol. 369, no. 1-2, pp. 77–84, 2007. View at Publisher · View at Google Scholar · View at Scopus
  43. A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “Solutions of time-dependent Emden-Fowler type equations by homotopy analysis method,” Physics Letters A, vol. 371, no. 1-2, pp. 72–82, 2007. View at Publisher · View at Google Scholar · View at Scopus
  44. M. Inc, “On exact solution of Laplace equation with Dirichlet and Neumann boundary conditions by the homotopy analysis method,” Physics Letters A, vol. 365, no. 5-6, pp. 412–415, 2007. View at Publisher · View at Google Scholar · View at Scopus
  45. F. Awawdeh, A. Adawi, and S. Al-Shara, “Analytic solution of multi pantograph equation,” Journal of Applied Mathematics and Decision Sciences, vol. 2008, Article ID 605064, 11 pages, 2008. View at Publisher · View at Google Scholar