Nonlinear Stability Analysis of Shells with the Finite Element Method

  • W. Wagner
Part of the International Centre for Mechanical Sciences book series (CISM, volume 328)


The investigation of the nonlinear response of shell structures requires besides knowledge about geometrical and material nonlinear behaviour the insight in the stability response. Here three main aspects arise. These are associated with the detection of singular points (e.g. limit or bifurcation points), the path-following in the pre- and postcritical range and a branch-switching between different paths. These problems are treated in this paper using the finite element method. For this purpose we summarize the necessary finite element formulations, where we emphasize the higher order derivatives. In a next section we present a brief overview on path-following methods. A main aspect of stability analysis is the detection of singular points. Thus, we introduce a definition of singular points, derive methods to detect the type of singular point and report possibilities to treat the stability considerations in an accompanying way. Furthermore we discuss modern concepts to calculate singular points directly using so called extended systems. Remarks on branch-switching procedures terminate the theoretical considerations. At the end of the paper some numerical examples are given to illustrate the derived methods and algorithms.


Singular Point Eigenvalue Problem Constraint Equation Bifurcation Point Extended System 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Riks, E.: The Application of Newtons Method to the Problem of Elastic Stability, J. Appl. Mech. 39 (1972) 1060–1066.CrossRefMATHGoogle Scholar
  2. 2.
    Riks, E.: An Incremental Approach to the Solution of Snapping and Buckling Problems, Int. J. Solids & Struct. 15 (1979) 529–551.MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Crisfield, M. A.: A Fast Incremental/Iterative Solution Procedure that Handles Snap Through, Comp. amp; Struct. 13 (1981) 55–62.CrossRefMATHGoogle Scholar
  4. 4.
    Ramm, E.: Strategies for Tracing the Nonlinear Response Near Limit Points. In: W. Wunderlich, E. Stein, K.-J. Bathe (eds.): Nonlinear Finite Element Analysis in Structural Mechanics, Springer, Berlin, Heidelberg, New-York (1981) 63–89.CrossRefGoogle Scholar
  5. 5.
    Bergan, P. G., Horrigmoe, G., Krakeland, B., Soreide, T. H.: Solution Techniques for Non-linear Finite Element Problems, Int. J. Num. Meth. Engng. 12 (1978), 1677–1696.CrossRefMATHGoogle Scholar
  6. 6.
    Decker, D. W., Keller, H. B.: Solution Branching—A Constructive Technique, in: Holmes, P.(ed): New Approaches to Nonlinear Problems in Dynamics, SIAM (1980), 53–69.Google Scholar
  7. 7.
    Wagner, W., Wriggers, P.: A Simple Method for the Calculation of Postcritical Branches, Engineering Computations, 5 (1988), 103–109.CrossRefGoogle Scholar
  8. 8.
    Mittelmann, H.-D., Weber, H.: Numerical Methods for Bifurcation Problems–a Survey and Classification. In: Mittelmann, Weber (eds.): Bifurcation Problems and their Numerical Solution, ISNM 54, Birkhâuser, Basel, Boston, Stuttgart (1980) 1–45.CrossRefGoogle Scholar
  9. 9.
    Moore, G., Spence, A.: The Calculation of Turning Points of Nonlinear Equations, SIAM, J. Numer. Anal. Comput. 17 (1980), 567–575.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Wriggers, P., Wagner, W., Miehe, C.: A Quadratically Convergent Procedure for the Calculation of Stability Points in Finite Element Analysis, Comp. Meth. Appl. Mech. Engng. 70 (1988), 329–347.CrossRefMATHGoogle Scholar
  11. 11.
    Wriggers, P., Simo, J.C.: A General Procedure for the Direct Computation of Turning and Bifurcation Points, Int. J. Num. Meth. Engng. 30 (1990), 155–176.CrossRefMATHGoogle Scholar
  12. 12.
    Schweizerhof, K. H., Wriggers, P.: Consistent Linearization for Path Following Methods in Nonlinear FE Analysis, Comp. Meth. Appl. Mech. Engng. 59 (1986), 261–279.CrossRefMATHGoogle Scholar
  13. 13.
    Keller, H. B.: Numerical Solution of Bifurcation and Nonlinear Eigenvalue Problems. In: Rabinowitz, P. (ed.): Application of Bifurcation Theory, Academic Press, New York (1977), 359–384.Google Scholar
  14. 14.
    Rheinboldt, W. C.: Numerical Analysis of Continuation Methods for Nonlinear Structural Problems, Comp. & Struct. 13 (1981) 103–113.MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Riks, E.: Some Computational Aspects of Stability Analysis of Nonlinear Structures, Comp. Meth. Appl. Mech. Engng. 47 (1984), 219–260.CrossRefMATHGoogle Scholar
  16. 16.
    Batoz, J. L., Dhatt, G.: Incremental Displacement Algorithms for Non-Linear Problems, Int. J. Num. Meth. Engng., 14 (1979), 1262–1267.MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Fried, I.: Orthogonal Trajectory Accession to the nonlinear Equilibrium Curve, Comp. Meth. Appl. Mech. Engng. 47 (1984) 283–297.MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Wagner, W.: Zur Behandlung von Stabilitätsproblemen der Elastostatik mit der Methode der Finiten Elemente, Forschungs-und Seminarberichte aus dem Bereich der Mechanik der Universität Hannover, F91 /1 (1991).Google Scholar
  19. 19.
    Zienkiewicz O. C., Taylor, R. L.: The Finite Element Method, Vol.1–2, 4. Edn., Mc Graw-Hill, London, 1989/1991.Google Scholar
  20. 20.
    Simo, J. C., Wriggers, P., Schweizerhof, K., Taylor, R. L.: Finite Deformation Postbuckling Analysis Involving Inelasticity and Contact Constraints, Int. J. Num: Meth. Engng., 23 (1986), 779–800.CrossRefMATHGoogle Scholar
  21. 21.
    Wagner, W.: A simple Finite Element Model for Beams with Finite Rotations, in preparation.Google Scholar
  22. 22.
    Wagner, W.: A Finite Element Model for Nonlinear Shells of Revolution with Finite Rotations, Int. J. Num. Meth. Engng. 29 (1990), 1455–1471.CrossRefMATHGoogle Scholar
  23. 23.
    Brendel, B., Ramm, E.: Nichtlineare Stabilitätsuntersuchungen mit der Methode der finiten Elemente, Ing. Archiv 51 (1982), 337–362.MATHGoogle Scholar
  24. 24.
    Hildebrandt, T. IL, Graves, L. M.: Implicit Functions and their Differentials in General Analysis. A.M.S. Transactions 29 (1927), 127–153.MathSciNetMATHGoogle Scholar
  25. 25.
    Chan, T. F.: Deflation Techniques and Block-Elimination Algorithms for Solving Bordered Singular Systems, SIAM, J. Sci. Stat. Comput. 5 (1984), 121–134.CrossRefMATHGoogle Scholar
  26. 26.
    Koiter W. T.: On the Stability of Elastic Equilibrium, Translation of ‘Over de Stabiliteit von het Elastisch Evenwicht’, Polytechnic Institute Delft, H. J. Paris Publisher Amsterdam 1945, NASA TT F-10, 833, 1967.Google Scholar
  27. 27.
    Jepson A. D., Spence, A.: Folds in Solutions of two Parameter Systems and their Calculation, SIAM, J. Numer. Anal. 22 (1985), 347–369.MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Dennis, J. E., Schnabel, R. B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall Inc., Englewood Cliffs, New Jersey (1983).MATHGoogle Scholar
  29. 29.
    Abbott, J. P.: An Efficient Algorithm for the Determination of certain Bifurcation Points, J. Comp. Appl. Math. 4 (1978), 19–27.MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Werner, B., Spence, A.: The Computation of Symmetry-Breaking Bifurcation Points, SIAM J. Num. Anal. 21 (1984), 388–399.MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Weinitschke, H. J.: On the Calculation of Limit and Bifurcation Points in stability Problems of Elastic Shells, Int. J. Solids Struct. 21 (1985), 79–95.MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Stein, E., Wagner, W., Wriggers, P.: Concepts of Modeling and Discretization of Elastic Shells for Nonlinear Finite Element Analysis, in: Proceedings of the Mathematics of Finite Elements and Applications VI MAFELAP 1987 Conference, ed. J. R. Whiteman, Academic Press, London (1988), 205–232.Google Scholar
  33. 33.
    Horrigmoe, G.: Finite Element Analysis of Free-Form Shells, Rep. No. 77–2, Inst. for Statikk, Division of Structural Mechanics, The Norwegian Instiute of Technology, University of Trondheim, Norway.Google Scholar
  34. 34.
    Noor, A. K.: Recent Advances in Reduction Methods for Nonlinear Problems, Comp. & Struct. 13 (1981) 31–44.CrossRefMATHGoogle Scholar
  35. 35.
    Belytschko, T., Tsay, C. S.: A Stabilization Procedure for the Quadrilateral Plate Element with One—Point Quadrature. Int. J. Num. Meth. Engng., 19 (1983), 405–419.CrossRefMATHGoogle Scholar
  36. 36.
    Wagner, W.: Zur Formulierung eines Zylinderschalenelementes mit vollständig reduzierter Integration, ZAMM, 68 (1988), T430–433.Google Scholar

Copyright information

© Springer-Verlag Wien 1992

Authors and Affiliations

  • W. Wagner
    • 1
  1. 1.University of HannoverHannoverGermany

Personalised recommendations