Skip to main content
SpringerLink
Log in

Stability analysis of planar equilibrium configurations of elastic rods subjected to end loads

  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

A semi-analytical method is proposed for investigating the stability of planar equilibrium configurations of an inextensible elastic rod under end-loading conditions. The method is based on representing the second variation of the constrained strain energy of the rod as a diagonal quadratic form using the eigensolutions of an auxiliary Sturm–Liouville problem. The coefficients of the resulting form which determine the sign of the second variation are analyzed by numerically solving an initial-value problem. Examples of curvilinear configurations of rods and a circular ring under point loads are considered and their stability is analyzed using the proposed method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Love A.E.H. (1944) A Treatise on the Mathematical Theory of Elasticity. Dover, New York

    MATH  Google Scholar 

  2. Timoshenko S.P. (1953) History of Strength of Materials. McGraw-Hill, New York

    Google Scholar 

  3. Basoco M.A. (1941) On the inflexional elastica. Am. Math. Mon. 48: 303–309

    Article  MathSciNet  Google Scholar 

  4. Goss, V.G.A.: The history of the planar elastica: insights into mechanics and scientific method. Sci Educ. doi:10.1007/s11191-008-9166-2

  5. Magnusson A., Ristinmaa M., Ljung C. (2001) Behavior of the extensible elastica solution. Int. J. Solids Struct. 38: 8441–8457

    Article  MATH  Google Scholar 

  6. Sotiropoulou A.B., Panayotounakos D.E. (2004) Exact parametric analytic solutions of the elastica ODEs for bars including effects of the transverse deformation. Int. J. NonLinear Mech. 39: 1555–1570

    Article  MATH  MathSciNet  Google Scholar 

  7. Filipich C.P., Rosales M.B. (2000) A further study on the postbuckling of extensible elastic rods. Int. J. NonLinear Mech. 35: 997–1022

    Article  MATH  Google Scholar 

  8. Wang C.M., Lam K.Y., He X.Q. (1998) Instability of variable arc-length elastica under follower force. Mech. Res. Commun. 25: 189–194

    Article  MATH  Google Scholar 

  9. Chucheepsakul S., Phungpaigram B. (2004) Elliptic integral solutions of variable-arc-length elastica under an inclined follower force. ZAMM. Z. Angew. Math. Mech. 84: 29–38

    Article  MATH  MathSciNet  Google Scholar 

  10. Plaut R.H., Taylor R.P., Dillard D.A. (2004) Postbuckling and vibration of a flexible strip clamped at its ends to a hinged substrate. Int. J. Solids Struct. 41: 859–870

    Article  MATH  Google Scholar 

  11. van der Heijden G.H.M., Neukirch S., Goss V.G.A., Thompson J.M.T. (2003) Instability and self-contact phenomena in the writhing of clamped rods. Int. J. Mech. Sci. 45: 161–196

    Article  MATH  Google Scholar 

  12. Mikata Y. (2007) Complete solution of elastica for a clamped-hinged beam, and its applications to a carbon nanotube. Acta Mechanica 190: 133–150

    Article  MATH  Google Scholar 

  13. Jerome J.W. (1975) Smooth interpolating curves of prescribed length and minimum curvature. Proc. Am. Math. Soc. 51: 62–66

    Article  MATH  MathSciNet  Google Scholar 

  14. Horn B.K.P. (1983) The curve of least energy. ACM Trans. Math. Softw. 9: 441–460

    Article  MATH  MathSciNet  Google Scholar 

  15. Jurdjevic V. (1993) The geometry of the ball-plate problem. Arch. Rat. Mech. Anal. 124: 305–328

    Article  MATH  MathSciNet  Google Scholar 

  16. Maddocks J.H. (1984) Stability of nonlinearly elastic rods. Arch. Rat. Mech. Anal. 85: 311–354

    Article  MATH  MathSciNet  Google Scholar 

  17. Domokos G. (1994) Global description of elastic bars. Z. Angew. Math. Mech. 74: 289–291

    Article  MathSciNet  Google Scholar 

  18. Gaspar Z., Domokos G., Szeberenyi I. (1997) A parallel algorithm for the global computation of elastic bar structures. Comput. Assist. Mech. Eng. Sci. 4: 55–68

    MATH  Google Scholar 

  19. Domokos G. (2002) The odd stability of the Euler beam. In: Seyranian A., Elishakoff I. (eds) Modern Problems of Structural Stability, vol 436. Springer, Wien, pp. 58–71

  20. Levyakov S.V. (2001) States of equilibrium and secondary loss of stability of a straight rod loaded by an axial force. J. Appl. Mech. Technical Phys. 42: 153–160

    MATH  Google Scholar 

  21. Born, M.: Stabilität der elastischen Linie in Ebene und Raum. Preisschrift und Dissertation, Dieterichsche Universitäts-Buchdruckerei Göttingen (1906). Reprinted in: Ausgewählte Abhandlungen, vol. 1, pp 5–101. Vanderhoeck & Ruppert, Göttingen (1963)

  22. Raboud D.W., Lipsett A.W., Faulkner M.G., Diep J. (2001) Stability evaluation of very flexible cantilever beams. Int. J. NonLinear Mech. 36: 1109–1122

    Article  MATH  Google Scholar 

  23. Sachkov Y.L. (2008) Conjugate points in the Euler elastic problem. J. Dyn. Control. Syst. 14: 409–439

    Article  MathSciNet  Google Scholar 

  24. Jin M., Bao Z.B. (2008) Sufficient conditions for stability of Euler Elasticas. Mech. Res. Commun. 35: 193–200

    Article  MathSciNet  Google Scholar 

  25. Hoffman K.A., Manning R.S., Paffenroth R.C. (2002) Calculation of the stability index in parameter-dependent calculus of variations problems: buckling of a twisted elastic strut. SIAM J. Appl. Dyn. Syst. 1: 115–145

    Article  MATH  MathSciNet  Google Scholar 

  26. Kuznetsov V.V., Levyakov S.V. (2002) Complete solution of the stability problem for elastica of Euler’s column. Int. J. NonLinear Mech. 37: 1003–1009

    Article  MATH  Google Scholar 

  27. Jairazbhoy V.A, Petukhov P., Qu J.M. (2008) Large deflection of thin plates in cylindrical bending—non-unique solutions. Int. J. Solids Struct. 45: 3203–3218

    Article  MATH  Google Scholar 

  28. Polyanin A.D., Zaitsev V.F. (1995) Handbook of Exact Solutions for Ordinary Differential Equations. CRC Press, Boca Raton

    MATH  Google Scholar 

  29. Freid I. (1981) Stability and equilibrium of the straight and curved elastica—finite element computation. Comput. Methods Appl. Mech. Eng. 28: 49–61

    Article  Google Scholar 

  30. Krylov, A.N.: Selected Works. Izd. Akad. Nauk SSSR, Moscow (1958) (in Russian)

  31. Seide P. (1973) Postbuckling behavior of circular rings with two or four concentrated loads. Int. J. NonLinear Mech. 8: 169–178

    Article  MATH  Google Scholar 

  32. Seide P., Albano E.D. (1973) Bifurcation of circular rings under normal concentrated loads. Trans. ASME. Ser. E J. Appl. Mech. 40: 233–238

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Engineering Mathematics, Novosibirsk State Technical University, 630092, Novosibirsk, Russia

    S. V. Levyakov & V. V. Kuznetsov

Authors
  1. S. V. Levyakov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. V. Kuznetsov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. V. Levyakov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Levyakov, S.V., Kuznetsov, V.V. Stability analysis of planar equilibrium configurations of elastic rods subjected to end loads. Acta Mech 211, 73–87 (2010). https://doi.org/10.1007/s00707-009-0213-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-009-0213-0

Keywords

Search

Navigation