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Computational concepts in structural stability

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Summary

The present essay contains a general structural stability theory for discretized structural systems. Instabilities are essential constituents of nonlinear structural responses, the computational assessment of which in a modern treatment is exclusively based on incremental-iterative (step-wise) numerical techniques, applied to the tangential equation of motion. The paper derives this fundamental equation as first variation of the nonlinear equation of motion in its standard form and its phase projection. Further, it transforms the principle of virtual work for arbitrary nonlinear (Kelvin-Voigt) continua into its incremental variant and finally into the consistent tangential equation of motion. Its various applications then are demonstrated to classes of time-independent and time-dependent, unstable structural responses, for which suitable numerical instruments are outlined. The derived algorithms are based on the concepts of Lyapunow exponents and Poincaré multipliers which are introduced as universal stability measures. Qualitative and quantitative convergence properties of perturbations in the phase space enable the proper establishment of stability definitions. The validity of the received concepts is illustrated by several examples.

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References

  1. Ramm, E. (ed.) (1982), “Buckling of Shells”, Springer-Verlag, Berlin.

    MATH  Google Scholar 

  2. Bushnell, D. (1981), “Buckling of Shells-Pitfall for Designers”,JAIAA,9, pp. 1183–1226.

    Google Scholar 

  3. Willems, J.L. (1970), “Stability Theory of Dynamical Systems”, Th. Nelson & Sons Ltd., London.

    MATH  Google Scholar 

  4. Müller, P.C. (1977), “Stabilität und Matrizen”, Springer-Verlag, Berlin.

    MATH  Google Scholar 

  5. Bellman, R. (1969) (1st ed. 1953), “Stability Theory of Differential Equations”, Dover Publications Inc., New York.

    Google Scholar 

  6. Ljapunow, A.M. (1949), “Problème général de la stabilité du mouvement”, Translation of 2nd doctoral thesis from 1893,Ann. Math. Studies, Princeton.

  7. Ramm, E. (1981), “Strategies for Tracing the Nonlinear Response Near Limit Points”, In: W. Wunderlichet al. (eds.), “Nonlinear Finite element Analysis in Structural Mechanics”, Springer-Verlag, Berlin, pp. 63–89.

    Google Scholar 

  8. Crisfield, M.A. (1981), “A Fast Incremental/Iterative Solution Procedure that Handles Snap-Through”,Comp. & Struct.,13, pp. 55–62.

    Article  MATH  Google Scholar 

  9. Riks, E. (1979), “An Incremental Approach to the Solution of Snapping and Buckling Problems”,Int. J. Solids Struct.,15, pp. 529–551.

    Article  MATH  MathSciNet  Google Scholar 

  10. Nawrotzki, P., Krätzig, W.B. and Montag, U. (1994), “On Kinetic Instability Phenomena of Elasto-Plastic Shell Structures”,Proc. 2nd Int. Conf. Comp. Struct. Technologies, Adv. Comp. Mech., Civil-Comp. Ltd., Edinburgh, pp. 83–97.

  11. Basar, Y. and Krätzig, W.B. (1989), “A Consistent Shell Theory for Finite Deformations”Acta Mechanica,76, pp. 73–87.

    Article  MathSciNet  Google Scholar 

  12. Zienkiewicz, O.C. (1989), “The Finite Element Method”, 3rd edition, McGraw-Hill Book Company Ltd., London.

    Google Scholar 

  13. Argyris, J.H. and Mlejnek, H.P. (1986), “Die Methode der Finiten Elemente”, Band 1. Friedw. Vieweg & Sohn, Braunschweig.

    MATH  Google Scholar 

  14. Krätzig, W.B. (1989), “Eine einheitliche statische und dynamische Stabilitätstheorie für Pfadverfolgungsalgorithmen in der numerischen Festkörpermechanik”,ZAMM,69, 7, pp. 203–213.

    Article  MATH  Google Scholar 

  15. Krätzig, W.B. (1990), “Fundamentals of Numerical Algorithms for Static and Dynamic Instability Phenomena of Thin Shells” In: W.B. Krätzig, E. Oñate (eds.),Computational Mechanics of Nonlinear Response of Shells, Springer-Verlag Berlin, pp. 101–124.

    Google Scholar 

  16. Zurmühl, R. and Falk, S. (1984/1986, “Matrizen und ihre Anwendungen”, Vol. 1/2, Springer-Verlag, Berlin/Heidelberg/New York/Tokyo.

    Google Scholar 

  17. Eckstein, U. (1983), “Nichtlineare Stabilitätsberechnung elastischer Schalentragwerke”,Techn. Report,83, 3, Inst. for Struct. Engng., Ruhr-University, Bochum.

    Google Scholar 

  18. Stein, E., Wagner W. and Wriggers, P. (1989), “Grundlagen nichtlinearer Berechnungsverfahren in der Strukturmechanik”, In: E. Stein (ed.),Nichtlineare Berechnungen im Konstruktiven Ingenieurbau, Springer-Verlag, Berlin, pp. 1–53.

    Google Scholar 

  19. Thompson, J.M.T. and Hunt, G.W. (1973), “A General Theory of Elastic Stability”, John Wiley & Sons Ltd., London.

    MATH  Google Scholar 

  20. Leipholz, H. (1968), “Stabilitätstheorie”, Teubner, Stuttgart.

    MATH  Google Scholar 

  21. Ziegler, H. (1968), “Principles of Structural Stability”, Blaisdell, Massachusetts/Toronto/London.

    Google Scholar 

  22. Abbot, J.P. (1978), “An Efficient Algorithm for the Determination of Certain Bifurcation Points”,J. Comp. Appl. Math.,4, pp. 19–27.

    Article  Google Scholar 

  23. Brendel, B. and Ramm, E. (1982), “Nichtlineare Stabilitätsuntersuchungen mit der Methode der finiten Elemente”,Ing. Archiv,51, pp. 337–362.

    Article  MATH  Google Scholar 

  24. Werner, B. and Spence, A. (1984), “The Computation of Symmetry-Breaking Bifurcation Points”,SIAM J. Num. Anal.,21, pp. 388–399.

    Article  MATH  MathSciNet  Google Scholar 

  25. Weinitschke, H.J. (1985), “On the Calculation of Limit and Bifurcation Points in Stability Problems of Elastic Shells”,Int. J. Sol. Struct.,21, pp. 79–95.

    Article  MATH  MathSciNet  Google Scholar 

  26. Wriggers, P., Wagner, W. and Miehe, C. (1988), “A Quadradically Convergent Procedure for the Calculation of Stability Points in Finite Element Analysis”,Comp. Meth. Appl. Mech. Engng.,70, pp. 329–347.

    Article  MATH  Google Scholar 

  27. Wriggers, P. (1993), “Fundamentals of Nonlinear Response Analysis of Discretized Systems”, Lecture Notes, CISM-Course on Nonlinear Stability of Structures, Udine.

  28. Choong, K.K. and Hangai, Y. (1993), “Review on Methods of Bifucation Analysis for Geometrically Nonlinear Structures”,IASS-Bulletin,34, pp. 133–149.

    Google Scholar 

  29. Koiter, W.T. (1945), “On the Stability of Elastic Equilibrium” (in Dutch), H.J. Paris, Amsterdam.

    Google Scholar 

  30. Krätzig, W.B. and Li, L.Y. (1992), “On rigorous stability conditions for dynamic quasi- bifucations”,Int. J. Sol. Struct.,29, 1, pp. 97–104.

    Article  MATH  Google Scholar 

  31. Kreuzer, E. (1987), “Numerische Untersuchung nichtlinearer dynamischer Systeme”, Springer-Verlag, Berlin.

    MATH  Google Scholar 

  32. Eller, C. (1988) “Lineare und nichtlineare Stabilitätsanalyse periodisch erregter, visko-elastischer Strukturen”,Mitt.,88, 2, Inst. für Konstruktiven Ingenieurbau, Ruhr-Universität, Bochum.

    Google Scholar 

  33. Müller, P.C., Schiehlen, W.O. (1976), “Lineare Schwingungen”, Akad. Verlagsges., Wiesbaden.

    MATH  Google Scholar 

  34. Thompson, J.M.T. and Stewart, H.B. (1986), “Nonlinear Dynamics and Chaos”, J. Wiley and Sons, Chichester.

    MATH  Google Scholar 

  35. Schuster, H.G. (1989), “Deterministic Chaos”, VCH Verlagsges., Weinheim.

    MATH  Google Scholar 

  36. Krätzig, W.B. and Eller, C. (1992), “Numerical Algorithms for Unstable Dynamic Shell Responses”,Computers & Structures,44, 1/2, pp. 263–271.

    Article  Google Scholar 

  37. Kunick, A. and Steeb, W.H. (1986), “Chaos in dynamischen Systemen”, BI Wissenschaftsverlag, Mannheim/Wien/Zürich.

    MATH  Google Scholar 

  38. Nawrotzki, P. (1994), “Numerical Stability Analysis of Arbitrary Structural Responses”,Techn. Report,94, 2, Inst. for Struct. Engng., Ruhr-University, Bochum.

    Google Scholar 

  39. Moon, F.C. (1987), “Chaotic Vibrations”, J. Wiley & Sons, Inc., New York.

    MATH  Google Scholar 

  40. Argyris, J., Faust, G. and Haase, M. (1994), “Die Erforschung des Chaos”, Vieweg Verlag, Braunschweig/Wiesbaden.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institute for Statics and Dynamics, Ruhr-University Bochum, 44780, Bochum, Germany

    W. B. Krätzig

  2. Hochtief, 45128, Essen, Germany

    P. Nawrotzki

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  1. W. B. Krätzig
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  2. P. Nawrotzki
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Krätzig, W.B., Nawrotzki, P. Computational concepts in structural stability. ARCO 3, 81–119 (1996). https://doi.org/10.1007/BF02736131

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