Large Deflection Finite Element Analysis of Pre- and Postcritical Response of Thin Elastic Frames

  • D. Karamanlidis
  • A. Honecker
  • K. Knothe

Abstract

A finite element variational approach and a computational algorithm are presented for predicting the nonlinear static, pre- and postcritical response of elastic framed structures. An updated Lagrangian formulation is used and the structure is discretized by using curved, mixed-hybrid beam elements. Throughout the development correction terms are maintained in the incremental energy functional to reduce the drifting of the approximate solution from the true solution. These correction terms correspond to checks on the stress resultants equilibrium and compatibility in the reference state. To tracing the unstable postcritical load deflection path in snap-through and bifurcation problems a generalized incrementation procedure has been developed and implemented. The high accuracy and effectiveness of the proposed approach is demonstrated by means of well-selected numerical examples.

Keywords

Librium Nite Mast Poss Cula 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Dupuis, G.A.; Hibitt, H.D.; McNamara, S.F.; Marcal, P.V.: Nonlinear material and geometric behavior of shell structures. Comp. & Struct. 1 (1971) 223–239.CrossRefGoogle Scholar
  2. 2.
    Bathe, K.-J.; Bolourchi, S.: Large displacement analysis of three -dimensional beam structures. Int. J.Num. Meth. Engng. 14 (1979) 961–986.MATHCrossRefGoogle Scholar
  3. 3.
    Ramm, E.: Geometrisch nichtlineare Elastostatik und finite Elemente. Habilitationsschrift, Universität Stuttgart, 1976.Google Scholar
  4. 4.
    Frey, F.: L’analyse statique non lineaire des structures par la methode des elements finis et son application a la construction metallique. Ph. D. Thesis, Universite de Liege, 1978.Google Scholar
  5. 5.
    Argyris, J.H.; Dunne, P.C.: On the application of the natural mode technique to small strain large displacement problems. Proc. World Congress on Finite Element Methods in Structural Mechanics, Bournemouth/England, 1975.Google Scholar
  6. 6.
    Pirotin, S.D.: Incremental large deflection analyses of elastic structures. Ph. D. Thesis, M.I.T., 1971.Google Scholar
  7. 7.
    Pian, T.H.H.: Derivation of element stiffness matrices by assumed stress distributions, AIAA J.2 (1964)1333–1336.Google Scholar
  8. 8.
    Atluri, S.: On the hybrid stress finite element model for incremental analysis of large deflection problems. Int. J. Solids Struct. 9 (1973) 1177–1191.MATHCrossRefGoogle Scholar
  9. 9.
    Boland, P.L.: Large deflection analysis of thin elastic structures by the assumed stress hybrid finite element method. Ph. D. Thesis, M.I.T., 1975.Google Scholar
  10. 10.
    Bäcklund, J.: Finite element analysis of nonlinear structures. Ph. D. Thesis, Göteborg, 1973.Google Scholar
  11. 11.
    Noor, A.K.; Greene, W.H.; Hartley, S.J.: Nonlinear finite element analysis of curved beams. Comp. Meth.Appl.Mech. Engng. 12 (1977) 289–307.ADSMATHCrossRefGoogle Scholar
  12. 12.
    Wunderlich, W.; Beverungen, G.: Geometrisch nichtlineare Theorie eben gekrümmter Stäbe. Bauingenieur 52 (1977) 225–237.Google Scholar
  13. 13.
    Karamanlidis, D.; Tsuzuki, O.; Knothe, K.: A study of the geometrically nonlinear behaviour of plane curved beam structures using a mixed hybrid finite element procedure. ILR Mitt. 54, Berlin, 1978.Google Scholar
  14. 14.
    Karamanlidis, D.: Finite Elementmodelle zur numerischen Berechnung des geometrisch nichtlinearen Verhaltens ebener Rahmentragwerke im unter-und überkritischen Bereich. Fortschr. - Ber. VDI - Z 1/63, Düsseldorf, 1980.Google Scholar
  15. 15.
    Mast, St.; Karamanlidis, D.: Elastoplastische Berechnungen von ebenen Rahmentragwerken nach einem gemischt-hybriden Finite-Element-Verfahren. DER STAHLBAU (in press).Google Scholar
  16. 16.
    Karamanlidis, D.; Knothe, K.: Geometrisch nichtlineare Berechnung von ebenen Stabwerken auf der Grundlage eines gemischt-hybriden Finite-Element-Verfahrens. To appear in: Ingenieur Archiv.Google Scholar
  17. 17.
    Washizu, K.: Variational methods in elasticity and plasticity, 2nd Edition. Pergamon Press, 1975.Google Scholar
  18. 18.
    Pian, T.H.H.: Variational principles for incremental finite element methods. J. Franklin Inst. 302 (1976) 473–488.MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Stricklin, J.A.; Haisler, W.E.; von Riesemann, W.A.; Evaluation of solution procedures for material and/or geometric nonlinear structural analysis. AIAA J. 11 (1973) 292–299.ADSMATHCrossRefGoogle Scholar
  20. 20.
    Honecker, A.: Entwicklung, Implementierung und Austestung von Lösungsalgorithmen für Gleichungssysteme mit singulär werdender Funktionalmatrix. Diploma Thesis, Berlin, 1980.Google Scholar
  21. 21.
    Bergan, P.G.; Horrigmoe, G.; Krakeland, B.; Söreide, T.: Solution techniques for nonlinear finite element problems. Int. J. num. Meth. Engng. 12 (1978) 1677–1696.MATHCrossRefGoogle Scholar
  22. 22.
    Bergan, P.G.; Roland, I.; Söreide, T.H.: Use of the current stiffness parameter in solution of nonlinear problems. Energy Methods in Finite Element Analysis,edited by R. Glowinski, E.Y. Rodin and O.C. Zienkiewicz. John Wiley & Sons Ltd., 265–282, 1979.Google Scholar
  23. 23.
    Bergan, P.G.: Solution algorithm s for nonlinear structural problems. Proc. Int. Conf. Engineering Application of the finite Element Method, 13.1 — 13.38, Hövik/Norway, 1979Google Scholar
  24. 24.
    Bergan, P.G.; Soreide, T.H.: Solution of Large displacement and instability problems using the current stiffness parameter. Proc. Int. Conf. Finite Elements in Nonlinear Solid and Structural Mechanics, Geilo/Norway, 1977.Google Scholar
  25. 25.
    Remseth, S.N.; Holthe, K.; Bergan, P.G.; Roland, S.: Tube buckling analysis by the finite element method. Ibidem.Google Scholar
  26. 26.
    Krakeland, B.: Nonlinear analysis of shells using degenerate isoparametric element. Ibidem.Google Scholar
  27. 27.
    Wood, R.D.: Zienkiewicz, O.C.: Geometrically nonlinear finite element analysis of beams, frames, arches and axissymmetric shells. Comp. & Struct. 7 (1977) 725–735.MATHCrossRefGoogle Scholar
  28. 28.
    Noor, A.K.; Peters, J.M.: Reduced basis technique for nonlinear analysis of structures. AIAA J. 18 (1980) 455–462.ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1981

Authors and Affiliations

  • D. Karamanlidis
    • 1
  • A. Honecker
    • 1
  • K. Knothe
    • 1
  1. 1.Technische Universitat BerlinGermany

Personalised recommendations