Skip to main content
SpringerLink
Log in

Theory and numerics of thin elastic shells with finite rotations

Theorie und numerische Analysis dünner elastischer Schalen mit endlichen Drehungen

  • Originals
  • Published:
Ingenieur-Archiv Aims and scope Submit manuscript

Summary

A bending theory for thin shells undergoing finite rotations is presented, and its associated finite element model is described. The kinematic assumption is based on a Reissner-Mindlin theory. The starting point for the derivation of the strain measures is the polar decomposition of the material deformation gradient. The work-conjugate stress resultants and stress couples are integrals of the Biot stress tensor. This tensor is invariant with respect to rigid body motions and therefore appropriate for the formulation of constitutive equations. The rotations are described by using Eulerian angles. The finite element discretization of arbitrary shells is performed using isoparametric elements. The advantage of the proposed shell formulation and its numerical model is shown by application to different non-linear plate and shell problems. Finite rotations can be calculated within one load increment. Thus the step size of the load increment is only imited by the local convergence behaviour of Newton's method or the appearance of stability phenomena.

Übersicht

Es wird eine Biegetheorie elastischer Schalen mit endlichen Drehungen dargestellt und die numerische Behandlung mit der Methode der finiten Elemente beschrieben. Dazu müssen die Verzerrungen und die schwache Form des Gleichgewichts angegeben werden. Es wird eine Reissner-Mindlin-Kinematik zugrunde gelegt. Die Verwendung des Greenschen Verzerrungstensors führt im Schalenraum bei endlichen Drehungen auf sehr komplizierte Ausdrücke. Daher wird von der polaren Zerlegung des materiallen Deformationsgradienten ausgegangen. In diesem Fall sind die Kräfte und Momente Spannungsresultierende des Biot-Spannungstensors, der invariant gegenüber Starrkörperbewegungen und somit für ein Stoffgesetz geeignet ist. Als Drehkinematen werden Eulerwinkel verwandt. Um allgemeine Schalengeometrien beschreiben zu können, wird eine isoparametrische Elementformulierung gewählt. Endliche Drehungen können in den berechneten Platten- und Schalenproblemen in einem Lastschritt aufgebracht werden. Die Lastschrittweite wird nur durch das Lösungsverfahren und das Auftreten von Stabilitätspunkten beschränkt.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Atluri, S. N.: Alternative stress and conjugate strain measures, and mixed variational formulations involving rigid rotations, for computational analysis of finitely deformed solids, with application to plates and shells. Comput. Struct. 18 (1984) 93–116

    Google Scholar 

  2. Argyris, J. H.; Balmer, H.; Doltsinis, J. St.; Dunne, P. C.; Haase, M.; Kleiber, M.; Malejannakis, G. A.; Mlejnek, H.-P.; Müller, M.; Scharpf, D. W.: Finite element method — the natural approach. Comput. Methods. Appl. Mech. Eng. 17/18 (1979) 1–106

    Google Scholar 

  3. Badur, J.; Pietraszkiewicz, W.: On geometrically non-linear theory of elastic shells derived from pseudocosserat continuum with constrained microrotations. In: Pietraszkiewicz, W. (ed.) Finite rotations in structural mechanics, Proc. of the Euromech Colloquium 197, Jabłonna, Poland, 1985, pp. 19–32. Berlin, Heidelberg, New York: Springer 1986

    Google Scholar 

  4. Belytschko, T.; Bachrach, W. E.: Efficient implementation of quadrilaterals with high coarse mesh accuracy. Methods. Appl. Mech. Eng. 54 (1986) 279–301

    Google Scholar 

  5. Bufler, H.: The Biot stresses in nonlinear elasticity and the associated generalized variational principles. Ing. Arch. 55 (1985) 450–462

    Google Scholar 

  6. Gruttmann, F.: Theorie und Numerik schubelastischer Schalen mit endlichen Drehungen unter Verwendung der Biot-Spannungen. Forschungs- und Seminarberichte aus dem Bereich der Mechanik der Universität Hannover, Bericht Nr. F88/1, Hannover 1988

  7. Harte, R.: Doppelt gekrümmte finite Dreieckelemente für die lineare und geometrisch nichtlineare Berechnung allgemeiner Flächentragwerke. Institut für Konstruktiven Ingenieurbau, Ruhr-Universität in Bochum, Mitteilung Nr. 82-10, Nov. 1982

  8. Hughes, T. J. R.; Pister, K. S.: Consistent linearization in mechanics of solids and structures. Comput. Struct. 8 (1978) 391–397

    Google Scholar 

  9. Kayuk, Ya. F.; Sakhatskiy, V. G.: On the nonlinear theory of shells based on the notion of a finite rotation. Sov. Appl. Mech. 21 (1985) 65–73

    Google Scholar 

  10. Libai, A.; Simmonds, J. G.: Nonlinear elastic shell theory. New York: Academic Press 1983

    Google Scholar 

  11. Makowski, J.; Stumpf, H.: Finite strains and rotations in shells. In: Pietrazkiewicz, W. (ed.): Finite rotations in structural mechanics, Proc. of the Euromech Colloquium 197, Jabłonna, Poland, 1985, pp. 175–194. Berlin, Heidelberg, New York: Springer 1986

    Google Scholar 

  12. Malkus, D. S.; Hughes, T. R. J.: Mixed finite element methods — reduced and selective integration techniques: a unification of concepts. Comput. Methods. Appl. Mech. Eng. 15 (1978) 63–81

    Google Scholar 

  13. Marsden, J. E.; Hughes, T. J. R.: Mathematical foundations of elasticity. Englewood Cliffs: Prentice-Hall 1983

    Google Scholar 

  14. Nolte, L. P.: Beitrag zur Herleitung und vergleichende Untersuchung geometrisch nichtlinearer Schalentheorien unter Berücksichtigung großer Rotationen. Mitt. Inst. f. Mechanik, No. 39, Ruhr-Universität Bochum 1983

  15. Park, K. C.; Stanley, G. M.: A curved C 0 shell element based on assumed natural-coordinate strains. J. Appl. Mech. 53 (1986) 278–290

    Google Scholar 

  16. Pietraszkiewicz, W.: Finite rotations in the non-linear theory of thin shells. In: Olszak, W. (ed.) Thin shells theory, new trends and applications, pp. 155–208. Wien, New York: Springer 1980

    Google Scholar 

  17. Ramm, E.: Geometrisch nicht lineare Elastostatik und finite Elemente. Bericht Nr. 76-2, Institut für Baustatik der Universität Stuttgart 1976

  18. Recke, L.; Wunderlich, W.: Rotations as primary unknowns in the nonlinear theory of shells and corresponding finite element models. In: Pietraszkiewicz, W. (ed.) Finite rotations in structural mechanics (Proc. of the Euromech Colloquium 197, Jabłonna, Poland, 1985), pp. 239–258. Berlin, Heidelberg, New York: Springer 1986

    Google Scholar 

  19. Reissner, E.: A note on two-dimensional, finite-deformation theories of shells. Int. J. Non-Linear Mech. 17 (1982) 217–221

    Google Scholar 

  20. Reissner, E.: On one-dimensional finite strain beam theory, the plane problem. J. Appl. Math. Phys. 23 (1972) 795–804

    Google Scholar 

  21. Schweizerhof, K.; Wriggers, P.: Consistent linearizations for path following methods in nonlinear FE analysis. Comput. Methods. Appl. Mech. Eng. 59 (1986) 261–279

    Google Scholar 

  22. Simmonds, J. G.; Danielson, D. A.: Non-linear shell theory with finite rotations and stress-function vectors. J. Appl. Mech. 39 (1972) 1085–1090

    Google Scholar 

  23. Simo, J. C.: A finite strain beam formulation. The three-dimensional dynamic problem, Part I. Comput. Methods. Appl. Mech. Eng. 49 (1985) 55–70

    Google Scholar 

  24. Simo, J. C.; Vu-Quoc, L.: Three-dimensional finite-strain rod model, Part II: Computational aspects. Comput. Methods Appl. Mech. Eng. 58 (1986) 79–116

    Google Scholar 

  25. Stein, E.; Wagner, W.; Wriggers, P.: Concepts of modeling and discretization of elastic shells for nonlinear finite element analysis. In: Whiteman, J. R. (ed.) Proc. of MAFELAP 1987 Conf. London: Academic Press (to appear)

  26. Wempner, G.: Finite elements, finite rotations and small strains; Int. J. Sol. Struct. 5 (1969) 117–153

    Google Scholar 

  27. Zienkiewicz, O. C.: The finite element method. London: Mc Graw Hill 1977

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Baumechanik und Numerische Mechanik, Universität Hannover, Appelstraße 9 A, D-3000, Hannover 1, Federal Republic of Germany

    F. Gruttmann,  E. Stein &  P. Wriggers

Authors
  1. F. Gruttmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Stein
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. Wriggers
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Dedicated to the memory of Dr. Dieter Withum who died on June 25, 1979. At the time of his death, Dr. Withum was Professor of Engineering Mechanics at the University of the German Federal Armed Forces in Munich. He would have been 60 years old on September 29, 1988

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gruttmann, F., Stein, E. & Wriggers, P. Theory and numerics of thin elastic shells with finite rotations. Ing. arch 59, 54–67 (1989). https://doi.org/10.1007/BF00536631

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00536631

Keywords

Search

Navigation