# Nonlinear Stability Analysis of Shells with the Finite Element Method

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

## Abstract

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.

## Keywords

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.

## References

1. 1.
Riks, E.: The Application of Newtons Method to the Problem of Elastic Stability, J. Appl. Mech. 39 (1972) 1060–1066.
2. 2.
Riks, E.: An Incremental Approach to the Solution of Snapping and Buckling Problems, Int. J. Solids & Struct. 15 (1979) 529–551.
3. 3.
Crisfield, M. A.: A Fast Incremental/Iterative Solution Procedure that Handles Snap Through, Comp. amp; Struct. 13 (1981) 55–62.
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.
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.
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.
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.
9. 9.
Moore, G., Spence, A.: The Calculation of Turning Points of Nonlinear Equations, SIAM, J. Numer. Anal. Comput. 17 (1980), 567–575.
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.
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.
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.
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.
15. 15.
Riks, E.: Some Computational Aspects of Stability Analysis of Nonlinear Structures, Comp. Meth. Appl. Mech. Engng. 47 (1984), 219–260.
16. 16.
Batoz, J. L., Dhatt, G.: Incremental Displacement Algorithms for Non-Linear Problems, Int. J. Num. Meth. Engng., 14 (1979), 1262–1267.
17. 17.
Fried, I.: Orthogonal Trajectory Accession to the nonlinear Equilibrium Curve, Comp. Meth. Appl. Mech. Engng. 47 (1984) 283–297.
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.
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.
23. 23.
Brendel, B., Ramm, E.: Nichtlineare Stabilitätsuntersuchungen mit der Methode der finiten Elemente, Ing. Archiv 51 (1982), 337–362.
24. 24.
Hildebrandt, T. IL, Graves, L. M.: Implicit Functions and their Differentials in General Analysis. A.M.S. Transactions 29 (1927), 127–153.
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.
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.
28. 28.
Dennis, J. E., Schnabel, R. B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall Inc., Englewood Cliffs, New Jersey (1983).
29. 29.
Abbott, J. P.: An Efficient Algorithm for the Determination of certain Bifurcation Points, J. Comp. Appl. Math. 4 (1978), 19–27.
30. 30.
Werner, B., Spence, A.: The Computation of Symmetry-Breaking Bifurcation Points, SIAM J. Num. Anal. 21 (1984), 388–399.
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.
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.
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.
36. 36.
Wagner, W.: Zur Formulierung eines Zylinderschalenelementes mit vollständig reduzierter Integration, ZAMM, 68 (1988), T430–433.Google Scholar