Nonlinear Dynamic Buckling and Stability of Autonomous Dissipative Discrete Structural Systems

Potential Systems
  • A. N. Kounadis
Part of the International Centre for Mechanical Sciences book series (CISM, volume 342)


The Nonlinear dynamic buckling and stability of nonlinearly elastic autonomous discrete systems under conservative step loading (of infinite or finite duration), impact and impulsive loading is examined in detail. Discrete structural systems which under the same loading applied statically exhibit snap-through buckling are mainly considered. Emphasis is focused on the coupling effect of nonlinearities (geometric and/or material) and structural damping. A qualitative discussion of the dynamic buckling mechanism is comprehensively presented on the basis of energy and topological concepts in the light of recent progress of nonlinear dynamics and chaos. This leads to useful criteria for establishing exact, approximate and lower-upper bound dynamic buckling estimates without integrating the highly nonlinear equations of motion. It is shown that dynamic buckling can be defined as an escaped motion through a saddle of the (static) unstable postbuckling equilibrium path which leads either to an “unbounded” motion or to a point attractor associated with a remote stable equilibrium point. As byproducts of this analysis various phenomena such as restabilization (metastability), loading discontinuity, sensitivity to initial-conditions and to damping as well as chaoticlike Phenomena, are revealed. The theory is illustrated with analyses of single, two and three degrees of freedom models.


Saddle Point Total Potential Energy Equilibrium Path Potential System Stable Equilibrium Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Hoff, N.J. and Bruce, V.G.: Dynamic Analysis of the Buckling of Laterally Loaded Flat Arches, Journal of Mathematics and Physics, 32 (4) (1954), 276–288.zbMATHMathSciNetGoogle Scholar
  2. 2.
    Grigolyuk, E.I.: Nonlinear Vibration and Stability of Shallow Shells, Izvestiya Akademii Nauk.,SSSR, 3(33),1955 translated in Applied Mechanics Series of Institute of Engineering Research, Univ. of California, 131, 1960.Google Scholar
  3. 3.
    Humphreys, J.S. and Bodner, S.R.: Dynamic Buckling of Shallow Shells Under Impulsive Loading, Proceedings of the ASCE, Journal of Engineering Mechanics Division, 88, EM2, (1962), 17–36.Google Scholar
  4. 4.
    Simitses, G.J.: Dynamic Snap-Through Buckling of Low Arches and Spherical Shells, Ph.D. Dissertation, Dept. of Aeronautics and Astronautics, Stanford Univ. Stanford, CA, 1965.Google Scholar
  5. 5.
    Humphreys, J.S.: On Dynamic Snap-Buckling of Shallow Arches, AIAA Journal, 4, No. 5, (1966), 878–886.CrossRefGoogle Scholar
  6. 6.
    Humphreys, J.S.: On the Adequacy of Energy Criteria for Dynamic Buckling of Arches, AIAA Journal, 4, (1966), 921–926.CrossRefGoogle Scholar
  7. 7.
    Hutchinson, J.W. and Budiansky, B.: Dynamic Buckling Estimates, AIAA Journal, 4, (1966), 525–530.CrossRefGoogle Scholar
  8. 8.
    Budianksy, B.: Dynamic Buckling of Elastic Structures, Criteria and Estimates, Proceedings of the International Conference on Dynamic Stability of Structures, Pergamon, New York, (1967), 83–106.Google Scholar
  9. 9.
    Thompson, J.M.T., Dynamic Buckling Under Step Loading, Dynamic Stability of Structures, Pergamon, New York, (1967), 215–236.Google Scholar
  10. 10.
    Handelman, G.H., A Simple Model of Stability of Structures, SIAM Review, 17, (1975), 593–604.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kounadis, A.N., Dynamic Snap-through Buckling of a Timoshenko Two-Bar Frame Under Suddenly Applied Load, ZAMM, 59, (1979), 523–531.zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Simitses, G.J., Kounadis, A.N. and Giri, J., Dynamic Buckling of Simple Frames Under Step Load, Journal of Engineering Mechanics Division, 105, EMS, (1979), 896–900.Google Scholar
  13. 13.
    Holasut, S., and Ruiz, C., Effect of an Impulsive Disturbing Load on the Stability of a Statically Loaded Structure, International Journal of Impact Engineering, 3, No. 1, (1985), 57–73.CrossRefGoogle Scholar
  14. 14.
    Bolotin, V.V.: Dynamic Stability of Elastic Systems, (Trans. from the Russian), Holden-Day, San Fransisco, 1964.Google Scholar
  15. 15.
    Kounadis, A.N.: On the Parametric Resonance of Columns Carrying Concentrated Masses, J. Struct. Mech., 5 (4), (1977), 383–394.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Simitses, G.J.: Instability of Dynamically Loaded Structures, Applied Mechanics Reviews, Vol. 44, No. 10, (1987), 1403–1408.CrossRefGoogle Scholar
  17. 17.
    Kounadis, A.N. and Mallis, J.: Dynamic Stability of Initially Crooked Columns Under a Time-Dependent Axial Displacement of Their Support, Q.J.Mech. Appl. Math. 41 (1), (1988), 580–596.MathSciNetGoogle Scholar
  18. 18.
    Raftoyiannis, J. and Kounadis, A.N.: Dynamic Buckling of Limit Point Systems Under Step Loading, Dynamics and Stability of Systems, 3, No. 3,4, (1988), 219–234.zbMATHMathSciNetGoogle Scholar
  19. 19.
    Kounadis, A.N., Mahrenholtz, O., and Bogacz, R.,: Nonlinear Dynamic Stability of a Simple Floating Bridge Model, Ingenieur Archiv, 60, No. 4, (1990), 262–273.zbMATHCrossRefGoogle Scholar
  20. 20.
    Kounadis, A.N. and Raftoyiannis, J.: Dynamic Stability Criteria of Nonlinear Elastic Damped/Undamped Systems Under Step Loading, AIAA J. 28, No. 7, (1990), 1217–1223.CrossRefGoogle Scholar
  21. 21.
    Kounadis, A.N.: Nonlinear Dynamic Buckling of Discrete Dissipative or Nondissipative Systems Under Step Loading, AIAA J. 29, No. 2, (1991), 280–289.CrossRefGoogle Scholar
  22. 22.
    Perko, L.: Differential Equations and Dynamical Systems, Springer-Verlag, New York, 1991.zbMATHCrossRefGoogle Scholar
  23. 23.
    Meirovitch, L.: Methods of Analytical Dynamics, Mc Graw-Hill, New York, 1970.Google Scholar
  24. 24.
    Gantmacher, F.: Lectures in Analytical Mechanics, MIR Publishers, Moscow, 1970.Google Scholar
  25. 25.
    Green, A.E. and Zerna, W.: Theoretical Elasticity, Clarendon Press, Oxford, 1968.zbMATHGoogle Scholar
  26. 26.
    Koiter, W.T.: Over de Stabilitat van het Elastisch Evenwicht, Dissert., Delft, Holland, H.J. Paris, Amsterdam, 1945.Google Scholar
  27. 27.
    Thompson, J.M.T. and Hunt, G.W.: A General Theory of Elastic Stability, John Wiley and Sons, London, 1973.zbMATHGoogle Scholar
  28. 28.
    Simitses, G.J.: Dynamic Stability of Suddenly Loaded Structures, Springer-Verlag, New York, 1990.zbMATHCrossRefGoogle Scholar
  29. 29.
    Kounadis, A.N.: Nonlinear Dynamic Buckling of Discrete Structural Systems Under Impact Loading, Intern. J. of Solids and Structures, 30, No 21, (1993), 2895–2909.zbMATHCrossRefGoogle Scholar
  30. 30.
    Kounadis, A.N.: Criteria and Estimates in the Nonlinear Dynamic Buckling of Discrete Systems Under Step Loading,XVII International Congress on Mechanics, IUTAM, Grenoble, France, Aug. 21–27, 1988.Google Scholar
  31. 31.
    Kounadis, A.N., Mallis, J. and Raftoyiannis, J.: Dynamic Buckling Estimates for Discrete Systems Under Step Loading, ZAMM Z, angew, Math. Mech. 71, No 10, (1991), 391–402.zbMATHCrossRefGoogle Scholar
  32. Kounadis, A.N.: Chaoslike Phenomena in the Nonlinear Dynamic Stability of Discrete Damped or Undamped Systems Under Step Loading, Int. J. Nonlinear Mechanics, 26, No 3/4, (1991), 301–311.zbMATHCrossRefGoogle Scholar
  33. 33.
    Kounadis, A.N.: Nonlinear Dynamic Buckling and Stability of Autonomous Structural Systems, Int. J. Mech. Sciences, 35 (8), (1993), 643–656.zbMATHCrossRefGoogle Scholar
  34. 34.
    Kounadis, A.N.: An Efficient and Simple Approximate Technique for Solving Nonlinear Initial and Boundary-Value Problems, Computational Mechanics, 9, (1992), 221–231.zbMATHMathSciNetGoogle Scholar
  35. 35.
    Marsden, J.E. and Mc Cracken, M., “The Hopf bifurcation and its applications”, Springer-Verlag, New York, 1976.zbMATHCrossRefGoogle Scholar
  36. 36.
    Sethna, P.R. and Schapiro, S.M., “Nonlinear behavior of flutter unstable dynamical systems with gyroscopic and circulatory forces”, J. Appl. Mech., 44, (1977), 755–762.zbMATHCrossRefGoogle Scholar
  37. 37.
    Huseyin, K., “On the Stability of Equilibrium paths associated with autonomous systems”, J. Appl. Mech., 48, (1981), 183–187.zbMATHCrossRefGoogle Scholar
  38. 38.
    Holmes, P.J. and Moon, F.C., “Strange Attractors and Chaos in Nonlinear Mechanics”, J. Appl. Mech., 50, (1983), 1021–1032.CrossRefGoogle Scholar
  39. 39.
    Thompson, J.M.T. and Stewart, H.B., “Nonlinear Dynamics and Chaos”, Wiley, Chichester,1986.Google Scholar
  40. 40.
    Huseyin, K., “Multiple-parameter Stability Theory and its Applications”, Oxford University Press, London, 1986.Google Scholar
  41. 41.
    Kounadis, A.N., “Global Bifurcation with Chaotic and other Stability Phenomena in Simple Structural Systems”, in Proceedings of Int.Congress in Honour of Professor P.S.Theocaris, A.hens, Sept. 25–26, (1989), 277–299.Google Scholar
  42. 42.
    Kalathas, N. and Kounadis, A.N., “Metastability and Chaoslike phenomena in nonlinear dynamic buckling of a simple two-mass system under step load”, Ingenieur-Archiv, 61, (1991), 162–173.zbMATHGoogle Scholar
  43. 43.
    Coddington, E.A. and Levinson, N., “Theory of Ordinary differential equations”, Mc Graw-Hill Book Co., New York, 1955.Google Scholar
  44. 44.
    Struble, R.A., “Nonlinear Differential Equations”, Mc Graw-Hill Book Co., New York, 1962.Google Scholar
  45. 45.
    Kounadis, A.N., “Static and Dynamic, Local and Global Bifurcations in Nonlinear Autonomous Structural Systems”, AIAA J., 31 (8), (1993), 1468–1477.zbMATHMathSciNetCrossRefGoogle Scholar
  46. 46.
    Gantmacher, F.R., “The Theory of Matrices”, Chelsea Publishing Co., Vol.II, New York, 1964.Google Scholar
  47. 47.
    Atadan, A.S. and Huseyin, K., “On the Oscillatory Instability of Multiple-Parameter Systems”,Int.J.Engng.Sci. 23,No 8, (1985), 857–873.zbMATHMathSciNetCrossRefGoogle Scholar
  48. 48.
    Car, J., “Applications of Centre Manifold Theory”, Springer-Verlag, New York, 1981.Google Scholar
  49. 49.
    Kounadis, A.N., “Some New Instability Phenomena in Non-linear Discrete Systems”, Eurodyn 90, European Conference on Structural Dynamics Bochum,June 5–7,1991, edited by Balkema,A.A., 1990, 103–111.Google Scholar
  50. 50.
    Kounadis, A.N., “A Qualitative Analysis for the Local and Global Dynamic Buckling and Stability of Autonomous Discrete Systems”, QJMAM,(1993), to be published.Google Scholar
  51. 51.
    Kounadis, A.N. Gantes, Ch. and Kandakis, G. “Numerical Solutions and Theoretical Predictions Based on Energy Criteria for Establishing the Dynamic Response of Autonomous Dissipative/Nondissipative Systems”, 1st National Congress of Comput. Mechanics, Athens, Sept. 3–4, (1992), 601–609.Google Scholar
  52. 52.
    Andronov, A.A. and Pontryagin, L., “Systemes Grossiers”, Dokl. Acad. Nauk. SSSR, 14, (1937), 247–251.Google Scholar
  53. 53.
    Goldsmith, W., “Impact, the Theory and Physical Behavior of Colliding Solids”, Edward Arnold LTD, London, 1960.Google Scholar
  54. 54.
    Kounadis, A.N. and Gantes, C. and Simitses, G.J., “Nonlinear Instability of Structures Subjected to Impact Loading”, Proceed. 2nd Europ. Conf. Struct. Dynamics, Eurodyn 93, Trondheim, Norway, 21–23 June, (1993), 641–649.Google Scholar

Copyright information

© Springer-Verlag Wien 1995

Authors and Affiliations

  • A. N. Kounadis
    • 1
  1. 1.National Technical University of AthensAthensGreece

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