Acta Mechanica Solida Sinica

, Volume 26, Issue 2, pp 197–204 | Cite as

Post-Buckling Behavior of a Double-Hinged Rod under Self-Weight

Article

Abstract

Post-buckling phenomena of slender rods have attracted great attention for both theoretical and engineering aspects. In this study, we explored the post-buckling behavior of a slender rod with two hinged ends under its self-weight. We first established the potential energy functional of the system, and then derived the governing differential equations according to the principle of least potential energy. We further addressed the physical meaning of the Lagrange multiplier by analyzing the force equilibrium. A computer code of shooting method was developed by using the commercial software MathCAD, which has proved efficient in computing the post-buckling configurations of the rod. We finally discussed the buckling of an oil sucker rod adopting our numerical results, which will be beneficial to the engineering design.

Key words

slender rod energy functional variation governing equation post-buckling 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Love, A.E.H., A Treatise on the Mathematical Theory of Elasticity (4th ed.). Dover Publications, New York, 1944.MATHGoogle Scholar
  2. [2]
    Bishopp, K.E. and Drucker, D.C., Large deflections of cantilever beams. Quarterly Journal of Mechanics and Applied Mathematics, 1945, 3: 272–275.MathSciNetMATHGoogle Scholar
  3. [3]
    Liu, J.L. and Feng, X.Q., Capillary adhesion of microbeams: finite deformation analyses. Chinese Physics Letters, 2007, 24: 2349–2352.CrossRefGoogle Scholar
  4. [4]
    Glassmaker, N.J. and Hui, C.Y., Elastica solution for a nanotube formed by self-adhesion of a folded thin film. Journal of Applied Physics, 2004, 96: 3429–3444.CrossRefGoogle Scholar
  5. [5]
    Wang, C.Y., A critical review of the heavy elastica. International Journal of Mechanical Sciences, 1986, 28: 549–559.CrossRefGoogle Scholar
  6. [6]
    Gurfinkerl, G., Buckling of elastically restrained columns. ASCE Journal of the Structure Division, 1965, 91: 159–183.Google Scholar
  7. [7]
    Haftka, R. and Nachbar, W., Post-buckling analysis of an elastically-restrained column. International Journal of Solids and Structures, 1970, 6: 1433–1449.CrossRefGoogle Scholar
  8. [8]
    Wang, C.Y., Post-buckling of a clamped-simply supported elastica. International Journal of Non-Linear Mechanics, 1997, 32: 115–1122.MATHGoogle Scholar
  9. [9]
    Huang, T. and Dareing, D.W., Buckling and lateral vibration of drill pipe. Journal of Engineering for Industry, 1968, 90: 613–619.CrossRefGoogle Scholar
  10. [10]
    Huang, T. and Dareing, D.W., Buckling and frequencies of long vertical pipes. Journal of the Engineering Mechanics Division, 1969, 95: 167–182.Google Scholar
  11. [11]
    Wang, C.M. and Ang, K.K., Buckling capacities of braced heavy columns under an axial load. Computers and Structures, 1988, 28: 563–571.CrossRefGoogle Scholar
  12. [12]
    Vaz, M.A. and Silva, D.F.C., Post-buckling analysis of slender elastic rods subjected to terminal force. International Journal of Non-Linear Mechanics, 2002, 34: 483–492.MATHGoogle Scholar
  13. [13]
    Tan, Z. and Witz, J.A., On the delfected configuration of a slender elastic rod subject to parallel terminal forces and moments. Proceedings of the Royal Society A: Mathematical, Physical & Engineering Sciences, 1995, 449: 337–349.CrossRefGoogle Scholar
  14. [14]
    Lee, B.K. and Oh, S.J., Elastica and buckling load of simple tapered columns with constant volume. International Journal of Non-Linear Mechanics, 2000, 37: 2507–2518.MATHGoogle Scholar
  15. [15]
    Greenhill, M.A., Determination of the greatest height consistent with stability that a vertical pole or mast can be made, and of the greatest height to which a tree of given proportions can grow. Proceedings of the Cambridge Philosophical Society, 1881, 4: 65–73.MATHGoogle Scholar
  16. [16]
    Lubinski, A., A study of the buckling of rotary drilling stings. API Drilling and Production Practice, 1950: 178–214.Google Scholar
  17. [17]
    Frisch-Fay, R., The analysis of a vertical and a horizontal cantilever under a uniformly distributed load. Journal of the Franklin Institute, 1961, 271: 192–199.MathSciNetCrossRefGoogle Scholar
  18. [18]
    Vaz, M.A. and Patel, M.H., Initial post-buckling of submerged slender vertical structures subjected to distributed axial tension. Applied Ocean Research, 1998, 20: 325–335.CrossRefGoogle Scholar
  19. [19]
    Wang, C.Y. and Waston, L.T., Overhang of a heavy elastic sheet. Journal of Applied Mathematics and Physics, 1982, 33: 17–23.CrossRefGoogle Scholar
  20. [20]
    Vaz, M.A. and Patel, M. H., Analysis of drill strings in vertical and deviated holes using the Galerkin technique. Engineering Structures, 1995, 17: 437–442.CrossRefGoogle Scholar
  21. [21]
    Patel, M.H. and Vaz, M.A., On the mechanics of submerged vertical slender structures subjected to varying axial tension. Proceedings of the Royal Society A: Mathematical, Physical & Engineering Sciences, 1996, 354: 609–648.MATHGoogle Scholar
  22. [22]
    Kokkinis, T. and Bernitsas, M.M., Post-buckling analysis of heavy columns with application to marine risers. SNAME Journal of ship Research, 1985, 29: 162–169.Google Scholar
  23. [23]
    Duan, W.H. and Wang, C.M., Exact solution for buckling of columns including self-weight. Journal of Engineering Mechanics, 2008, 134: 16–119.CrossRefGoogle Scholar
  24. [24]
    Wang, C.Y., Vibration of a standing heavy column with intermediate support. Journal of Vibration and Acoustics, 2010, 132: 044502.CrossRefGoogle Scholar
  25. [25]
    Vaz, M.A. and Mascaro, G.H.W., Post-buckling analysis of slender elastic vertical rods subjected to terminal forces and self-weight. International Journal of Non-linear Mechanics, 2005, 40: 1049–1056.CrossRefGoogle Scholar
  26. [26]
    Mahadevan, L. and Keller, J.B., Periodic folding of thin sheets. SIAM Journal on Applied Mathematics, 1999, 41: 115–131.MathSciNetMATHGoogle Scholar
  27. [27]
    Lloyd, D.W., Shanahan, W.J., and Konopasek, M., The folding of heavy fabric sheets. International Journal of Mechanical Sciences, 1978, 20: 521–527.CrossRefGoogle Scholar
  28. [28]
    MathCad, Mathcad 2000 Professional for PC. Mashsoft Inc., 2000.Google Scholar
  29. [29]
    Ghatak, A. and Das, A.L., Kink instability of a highly deformable elastic cylinder. Physical Review Letters, 2007, 99: 076101.CrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2013

Authors and Affiliations

  1. 1.Department of Engineering MechanicsChina University of PetroleumQingdaoChina
  2. 2.School of Civil Engineering and TransportationSouth China University of TechnologyGuangzhouChina
  3. 3.College of Architecture and Civil EngineeringTaiyuan University of TechnologyTaiyuanChina

Personalised recommendations