Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Euler, L.: Methodus inveniendi lineas curvas maximi minimivi proprietate gaudentes. Appendix 1: De curvis elasticis, Bousquet, Lausanne, and Geneva (1744)
Levien, R.: The elastica: a mathematical history. Technical report UCB/EECS-2008-103, EECS Department, University of California, Berkeley, August 2008. http://www2.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-103.html
Goss, V.G.A.: The history of the planar elastica: insights into mechanics and scientific method. Sci. Educ. 18(8), 1057–1082 (2009). https://doi.org/10.1007/s11191-008-9166-2
Tongyun, W.: A numerical study of elastica using constrained optimisation method, diploma thesis. University of Singapore, Department of Civil Engineering (2004)
Wang, C.: Post-buckling of a clamped-simply supported elastica. Int. J. Non Linear Mech. 32(6), 1115–1122 (1997). https://doi.org/10.1016/S0020-7462(96)00125-4. http://www.sciencedirect.com/science/article/pii/S0020746296001254
Humer, A.: Exact solutions for the buckling and postbuckling of shear-deformable beams. Acta Mech. 224(7), 1493–1525 (2013). https://doi.org/10.1007/s00707-013-0818-1
Maddocks, J.H.: Stability of nonlinearly elastic rods. Arch. Ration. Mech. Anal. 85(4), 311–354 (1984). https://doi.org/10.1007/BF00275737
Thompson, J.M.T.: Stability of elastic structures and their loading devices. J. Mech. Eng. Sci. 3(2), 153–162 (1961)
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)
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). https://doi.org/10.1007/s00707-006-0402-z
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)
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). https://doi.org/10.1016/S0020-7462(00)00114-1. http://www.sciencedirect.com/science/article/pii/S0020746200001141
Batista, M.: A simplified method to investigate the stability of cantilever rod equilibrium forms. Mech. Res. Commun. 67, 13–17 (2015). https://doi.org/10.1016/j.mechrescom.2015.04.009. http://www.sciencedirect.com/science/article/pii/S0093641315000877
Batista, M.: On stability of elastic rod planar equilibrium configurations. Int. J. Solids Struct. 72, 144–152 (2015). https://doi.org/10.1016/j.ijsolstr.2015.07.024. http://www.sciencedirect.com/science/article/pii/S0020768315003303
Timoshenko, S., Gere, J.M.: Theory of Elastic Stability, 2nd edn. McGraw-Hill Book Co, New York (1961)
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). https://doi.org/10.1007/BF02428182
Acknowledgements
This research was funded by London South Bank University. We would like to thank the reviewers for valuable comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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.
About this article
Cite this article
Singh, P., Goss, V.G.A. Critical points of the clamped–pinned elastica. Acta Mech 229, 4753–4770 (2018). https://doi.org/10.1007/s00707-018-2259-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-018-2259-3