Journal of Elasticity

, Volume 105, Issue 1–2, pp 117–136 | Cite as

On Stability Analyses of Three Classical Buckling Problems for the Elastic Strut

Open Access


It is common practice in analyses of the configurations of an elastica to use Jacobi’s necessary condition to establish conditions for stability. Analyses of this type date to Born’s seminal work on the elastica in 1906 and continue to the present day. Legendre developed a treatment of the second variation which predates Jacobi’s. The purpose of this paper is to explore Legendre’s treatment with the aid of three classical buckling problems for elastic struts. Central to this treatment is the issue of existence of solutions to a Riccati differential equation. We present two different variational formulations for the buckling problems, both of which lead to the same Riccati equation, and we demonstrate that the conclusions from Legendre and Jacobi’s treatments are equivalent for some sets of boundary conditions. In addition, the failure of both treatments to classify stable configurations of a free-free strut are contrasted.


Elastica Variational Methods Elastic Stability Buckling 

Mathematics Subject Classification (2000)

34B15 49B10 74G60 74K10 93C10 


  1. 1.
    Bell, D.J., Jacobson, D.H.: Singular Optimal Control Problems. Academic Press, London (1975) MATHGoogle Scholar
  2. 2.
    Bolza, O.: Lectures on the Calculus of Variations. University of Chicago Press, Chicago (1907) Google Scholar
  3. 3.
    Born, M.: Untersuchungen über die Stabilität der elastischen Linie in Ebene und Raum, unter verschiedenen Grenzbedingungen. Dieterichsche Universitäts-Buchdruckerei, Göttingen (1906) Google Scholar
  4. 4.
    Bryson, A.E. Jr., Ho, Y.C.: Applied Optimal Control: Optimization, Estimation, and Control. Hemisphere Publishing Corporation, Washington (1975). Revised printing Google Scholar
  5. 5.
    Faruk Senan, N.A., O’Reilly, O.M., Tresierras, T.N.: Modeling the growth and branching of plants: a simple rod-based model. J. Mech. Phys. Solids 56(10), 3021–3036 (2008). doi:10.1016/j.jmps.2008.06.005 MathSciNetADSMATHCrossRefGoogle Scholar
  6. 6.
    Gelfand, I.M., Fomin, S.V.: Calculus of Variations. Prentice Hall, Englewood Cliffs (1964) Google Scholar
  7. 7.
    Jacobson, D.H.: A new necessary condition of optimality for singular control problems. SIAM J. Control 7, 578–595 (1969). doi:10.1137/0307042 MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Jin, M., Bao, Z.B.: Sufficient conditions for stability of Euler elasticas. Mech. Res. Commun. 35(3), 193–200 (2008). doi:10.1016/j.mechrescom.2007.09.001 MathSciNetCrossRefGoogle Scholar
  9. 9.
    Johnson, C.D., Gibson, J.E.: Singular solutions in problems of optimal control. IEEE Trans. Autom. Control AC-8, 4–15 (1963) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kuznetsov, V.V., Levyakov, S.V.: Complete solution of the stability problem for elastica of Euler’s column. Int. J. Non-Linear Mech. 37(6), 1003–1009 (2002). doi:10.1016/S0020-7462(00)00114-1 MATHCrossRefGoogle Scholar
  11. 11.
    Leitmann, G.: The Calculus of Variations and Optimal Control. Plenum, New York (1981) MATHGoogle Scholar
  12. 12.
    Levyakov, S.V.: Stability analysis of curvilinear configurations of an inextensible elastic rod with clamped ends. Mech. Res. Commun. 36(5), 612–617 (2009). doi:10.1016/j.mechrescom.2009.01.005 MathSciNetCrossRefGoogle Scholar
  13. 13.
    Levyakov, S.V., Kuznetsov, V.V.: Stability analysis of planar equilibrium configurations of elastic rods subjected to end loads. Acta Mech. 211(1–2), 73–87 (2010). doi:10.1007/s00707-009-0213-0 MATHCrossRefGoogle Scholar
  14. 14.
    Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity, 4th edn. Cambridge University Press, Cambridge (1927) MATHGoogle Scholar
  15. 15.
    Maddocks, J.H.: Stability of nonlinearly elastic rods. Arch. Ration. Mech. Anal. 85(4), 311–354 (1984). doi:10.1007/BF00275737 MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Manning, R.S.: Conjugate points revisited and Neumann-Neumann problems. SIAM Rev. 51(1), 193–212 (2009). doi:10.1137/060668547 MathSciNetADSMATHCrossRefGoogle Scholar
  17. 17.
    Manning, R.S., Rogers, K.A., Maddocks, J.H.: Isoperimetric conjugate points with application to the stability of DNA minicircles. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 454(1980), 3047–3074 (1998). doi:10.1098/rspa.1998.0291 MathSciNetADSMATHCrossRefGoogle Scholar
  18. 18.
    McDanell, J.P., Powers, W.F.: New Jacobi-type necessary and sufficient conditions for singular optimization problems. AIAA J. 8, 1416–1420 (1970). doi:10.2514/3.5917 MathSciNetADSMATHCrossRefGoogle Scholar
  19. 19.
    O’Reilly, O.M., Tresierras, T.N.: On the static equilibria of branched elastic rods. Int. J. Eng. Sci. 49(2), 212–227 (2011). doi:10.1016/j.ijengsci.2010.11.008 CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Pedregal, P.: Introduction to Optimization. Springer, New York (2004) MATHGoogle Scholar
  21. 21.
    Rosenblueth, J.F.: Conjugate journey in optimal control. Int. Math. Forum 2(13–16), 633–674 (2007). MathSciNetMATHGoogle Scholar
  22. 22.
    Sachkov, Y.L.: Optimality of Euler elasticae. Dokl. Akad. Nauk 417(1), 23–25 (2007). doi:10.1134/S106456240706004X Google Scholar
  23. 23.
    Sachkov, Y.L.: Conjugate points in the Euler elastic problem. J. Dyn. Control Syst. 14(3), 409–439 (2008). doi:10.1007/s10883-008-9044-x MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Zeidan, V.: Sufficient conditions for variational problems with variable endpoints: Coupled points. Appl. Math. Optim. 27(2), 191–209 (1993). doi:10.1007/BF01195982 MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of California at BerkeleyBerkeleyUSA

Personalised recommendations