Critical points of the clamped–pinned elastica
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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.Euler, L.: Methodus inveniendi lineas curvas maximi minimivi proprietate gaudentes. Appendix 1: De curvis elasticis, Bousquet, Lausanne, and Geneva (1744)Google Scholar
- 2.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
- 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.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 CrossRefGoogle Scholar
- 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.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 CrossRefGoogle Scholar
- 13.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 CrossRefGoogle Scholar
- 14.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 CrossRefGoogle Scholar
- 15.Timoshenko, S., Gere, J.M.: Theory of Elastic Stability, 2nd edn. McGraw-Hill Book Co, New York (1961)Google Scholar
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