An Introduction to Compound and Coupled Buckling and Dynamic Bifurcations

  • G. Augusti
  • V. Sepe
  • A. Paolone
Conference paper
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 379)


This Part I is a general introduction to the subject matter of the Volume, spanning over some of the main « theoretical aspects ».

The first Chapter presents the simple strut models and equations that allowed to discover and describe for the first time, about 30 years ago, the phenomenon of compound buckling in a static framework. In the second Chapter, the most significant of these models is used to illustrate the probabilistic characteristics of the load capacity of an imperfection-sensitive structure.

In Chapter 3, instability is tackled in a dynamic context. Mechanical systems subject to non-conservative actions and containing two real control parameters are considered. Two simple models are used to show double Hopf and Hopf -divergence bifurcation. Post-critical solutions are discussed in terms of non-linear coupling between respectively two dynamic modes or one dynamic and one buckling mode.


Hopf Bifurcation Bifurcation Diagram Bifurcation Point Initial Imperfection Collapse Load 
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  1. [1]
    Augusti G.: “Some Problems in Structural Instability, with special reference to beam-columns of I-section”, Part I: Investigations on the basic types of elastic buckling and post-buckling by means of semi-rigid models; Ph.D. Thesis; University of Cambridge, Department of Engineering, 1964.Google Scholar
  2. [2]
    Augusti G.: “Stabilità di strutture elastiche in presenza di grandi spostamenti”; Atti dell’ Accademia delle Scienze Fisiche e Matematiche di Napoli, Vol.IV, Serie 3’, N. 5; 1964.Google Scholar
  3. [3]
    Thompson J.M.T., Hunt G.W.: A General Theory of Elastic Stability; John Wiley & Sons, 1973.MATHGoogle Scholar
  4. [4]
    Cederbaum G., Arbocz J.: “On the reliability of imperfection-sensitive long isotropic cylindrical shells”; Structural Safety, 1996, Vol. 18 (No.1), pp. 1–10.CrossRefGoogle Scholar
  5. [5]
    Augusti G.: “Il carico di collasso delle aste caricate di punta”; Giornale del Genio Civile, 1966 (No.1).Google Scholar
  6. [6]
    Augusti G., Baratta A.: “Teoria probabilistica delle aste compresse”; Costruzioni Metalliche, 1971 (No.1).Google Scholar
  7. [7]
    Elishakoff I., Marcus S., Starnes J.H. Jr: “On Vibrational Imperfection Sensitivity of Augusti’s Model Structure in the vicinity of a Non-Linear Static State”; Int. J. Non-Linear Mechanics, 1996, Vol. 31 (No.2), pp. 229–236.CrossRefMATHGoogle Scholar
  8. [8]
    Pignataro M., Rizzi N., Luongo A.: Stability, Bifurcation, and Postcritical Behaviour of Elastic Systems, Elsevier, Amsterdam, 1991. ( Italian original, Rome, 1983 ).Google Scholar
  9. [9]
    Bolotin V.V.: Non-Conservative Problems of the Theory of Elastic Stability, Pergamon Press, Oxford 1963, ( Russian original, Moscow, 1961 ).Google Scholar
  10. [10]
    Ziegler H.: Principles of Structural Stability, Blaisdell Publishing Company, London 1968.Google Scholar
  11. [11]
    Paolone A.: Metodi Asintotici per l’analisi di stabilità e biforcazione di sistemi meccanici non lineari non conservativi, Phd. Thesis, Rome 1995 (in Italian).Google Scholar
  12. [12]
    Arnold V.I.: Ordinary Differential Equations, M.I.T. Press: Cambrige 1973, ( Russian original, Moscow, 1971 ).Google Scholar
  13. [13]
    Arnold V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations, Springer Verlag, New York, Heidelberg, Berlin 1982, ( Russian original, Moscow, 1977 ).Google Scholar
  14. [14]
    Guckenheiner J. and Holmes P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of vector Fields, Springer-Verlag, New-York, 1983.CrossRefGoogle Scholar
  15. [15]
    Troger H., Steindl A.: Nonlinear Stability and Bifurcation Theory, Springer Verlag, Wien, New York 1991.Google Scholar
  16. [16]
    Nayfeh A.H.: Method of Normal Forms, Wiley-Interscience, New York 1993.MATHGoogle Scholar
  17. [17]
    Nayfeh A.H. and Mook D.T.: Nonlinear Oscillations, Wiley, New-York 1979.MATHGoogle Scholar
  18. [18]
    Nayfeh A.H.: Introduction to Perturbation Techniques, Wiley-Interscience, New York 1991.Google Scholar
  19. [19]
    Sethna P.R.: On Averaged and Normal Form Equations, Nonlinear Dynamics, 7 (1995), 1–10.MathSciNetGoogle Scholar
  20. [20]
    Huseyin K.: Multiple Parameter Stability Theory and its Applications, Clarendon Press: Oxford 1986.MATHGoogle Scholar
  21. [21]
    Nayfeh A.H., and Balachandran B.: Nonlinear Dynamics: Concepts and Applications, Wiley-Interscience, New York 1993.Google Scholar
  22. [22]
    Luongo A. and Paolone A.: Perturbation Methods for Bifurcation Analysis from Multiple Nonresonant Complex Eigenvalues, 1996, ( Submitted).Google Scholar
  23. [23]
    Luongo A. and Paolone A.: Perturbation Methods for Bifurcation Analysis from Bifurcation from One Zero and a Purely Imaginary Pair Eigenvalues, 1996, ( Submitted).Google Scholar
  24. [24]
    Iooss G. and Joseph D.D.: Elementary Stability and Bifurcation Theory, Springer-Verlag, New-York 1980.CrossRefMATHGoogle Scholar
  25. [25]
    Cohen D. S.: Bifurcation from Multiple Complex Eigenvalues, Journal of Mathematical Analysis and Applications, 57 (1977), 505–521.CrossRefMATHMathSciNetGoogle Scholar
  26. [26]
    Piccardo G.: A Methodology for the Study of Coupled Aeroelastic Phenomena, J. Wind Eng. Aerodyn, 48 (1993), 241–252.CrossRefGoogle Scholar
  27. [27]
    Novak M.: Aeroelastic Galloping of Prismatic Bodies, Eng. Mech. Division, ASCE, 96 (1969), 115–142, N. EM1.Google Scholar
  28. [28]
    Langford W.F.: Periodic and steady-state mode interactions lead to tori, SIAM J. Appl. Math. 37 (1979), 22–48.MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Wien 1998

Authors and Affiliations

  • G. Augusti
    • 1
  • V. Sepe
  • A. Paolone
  1. 1.University of Roma “La Sapienza”RomaItaly

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