Acta Mechanica

, Volume 229, Issue 12, pp 4753–4770 | Cite as

Critical points of the clamped–pinned elastica

  • P. SinghEmail author
  • V. G. A. Goss
Open Access
Original Paper


We investigate equilibrium configurations of the clamped–pinned elastica where the pinned end can be displaced towards, and past, the clamped end. Solving the nonlinear ordinary differential equation for the clamped–pinned elastica for any mode in terms of elliptic integrals, we find sets of equations which govern the equilibrium configurations for given displacements. Equilibrium configurations for various displacements of the pinned end and any mode are obtained by numerically solving those sets of equations. A physical quantity, such as the force that arises in the elastica due to displacement of the pinned end, is taken to be a function of displacement. Although it is generally not possible to define a physical quantity as a function of displacement explicitly, an equation for the rate of change of this physical quantity with respect to displacement can be found by partial differentiation of the sets of equations which govern the equilibrium configurations. Setting that rate of change to zero provides a constraint equation for locating the critical points of that physical quantity. That constraint equation and the sets of equations which govern the equilibrium configurations are solved numerically to obtain the critical points of the physical quantity. Our procedure is demonstrated by finding local extrema on force–displacement plots (useful when analysing the stability of equilibrium configurations) and the maximum deflection of the elastica. Finally, we suggest how our procedure has scope for wider application.



This research was funded by London South Bank University. We would like to thank the reviewers for valuable comments and suggestions.


  1. 1.
    Euler, L.: Methodus inveniendi lineas curvas maximi minimivi proprietate gaudentes. Appendix 1: De curvis elasticis, Bousquet, Lausanne, and Geneva (1744)Google Scholar
  2. 2.
    Levien, R.: The elastica: a mathematical history. Technical report UCB/EECS-2008-103, EECS Department, University of California, Berkeley, August 2008.
  3. 3.
    Goss, V.G.A.: The history of the planar elastica: insights into mechanics and scientific method. Sci. Educ. 18(8), 1057–1082 (2009). CrossRefGoogle Scholar
  4. 4.
    Tongyun, W.: A numerical study of elastica using constrained optimisation method, diploma thesis. University of Singapore, Department of Civil Engineering (2004)Google Scholar
  5. 5.
    Wang, C.: Post-buckling of a clamped-simply supported elastica. Int. J. Non Linear Mech. 32(6), 1115–1122 (1997). CrossRefGoogle Scholar
  6. 6.
    Humer, A.: Exact solutions for the buckling and postbuckling of shear-deformable beams. Acta Mech. 224(7), 1493–1525 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Maddocks, J.H.: Stability of nonlinearly elastic rods. Arch. Ration. Mech. Anal. 85(4), 311–354 (1984). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Thompson, J.M.T.: Stability of elastic structures and their loading devices. J. Mech. Eng. Sci. 3(2), 153–162 (1961)CrossRefGoogle Scholar
  9. 9.
    Bigoni, D., Bosi, F., Misseroni, D., Dal Corso, F., Noselli, G.: New phenomena in nonlinear elastic structures: from tensile buckling to configurational forces. In: Bigoni, D. (ed.) Extremely Deformable Structures, pp. 55–135. Springer, Vienna (2015)CrossRefGoogle Scholar
  10. 10.
    Mikata, Y.: Complete solution of elastica for a clamped-hinged beam, and its applications to a carbon nanotube. Acta Mech. 190(1), 133–150 (2007). CrossRefzbMATHGoogle Scholar
  11. 11.
    Banu, S., Saha, G., Saha, S.: Multisegment integration technique for post-buckling analysis of a pinned-fixed slender elastic rod. BRAC Univ. J. 5(2), 1–7 (2008)Google Scholar
  12. 12.
    Kuznetsov, V., Levyakov, S.: Complete solution of the stability problem for elastica of Euler’s column. Int. J. Non Linear Mech. 37(6), 1003–1009 (2002). CrossRefGoogle Scholar
  13. 13.
    Batista, M.: A simplified method to investigate the stability of cantilever rod equilibrium forms. Mech. Res. Commun. 67, 13–17 (2015). CrossRefGoogle Scholar
  14. 14.
    Batista, M.: On stability of elastic rod planar equilibrium configurations. Int. J. Solids Struct. 72, 144–152 (2015). CrossRefGoogle Scholar
  15. 15.
    Timoshenko, S., Gere, J.M.: Theory of Elastic Stability, 2nd edn. McGraw-Hill Book Co, New York (1961)Google Scholar
  16. 16.
    Goss, V.G.A., van der Heijden, G.H.M., Thompson, J.M.T., Neukirch, S.: Experiments on snap buckling, hysteresis and loop formation in twisted rods. Exp. Mech. 45(2), 101–111 (2005). CrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.School of EngineeringLondon South Bank UniversityLondonUK

Personalised recommendations