A Variational Formulation of Shallow Shells

  • A. Ibrahimbegovic
  • F. Frey
  • G. Fonder
  • C. Massonnet


A sound variational framework applicable to shallow shell analysis is proposed in this work. Addition of so-called drilling degrees of freedom, usually absent in the shell analysis, is facilitated within the proposed framework at the continuum level. Therefore, it provides a solid theoretical basis for development of thick and thin curved shell elements with six degrees of freedom per node.


Variational Principle Variational Formulation Shell Element Deformation Gradient Polar Decomposition 
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  1. 1.
    Ahmad S., B.M. Irons and O.C. Zienkiewicz, Analysis of Thick and Thin Shell Structures by Curved Elements, Int. J. Numer. Methods Eng., 2, 419–451, 1970CrossRefGoogle Scholar
  2. 2.
    Allman D.J., A Compatible Triangular Element Including Vertex Rotations for Plane Elasticity Problems, Comput. Struct., 19, 1–8, 1984MATHCrossRefGoogle Scholar
  3. 3.
    Argyris J., An Excursion into Large Rotations, Comput. Methods Appl. Mech. Eng., 32, 85–155, 1982MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bergan P.G. and C.A. Felippa, A Triangular Membrane Element with Rotational Degrees of Freedom, Comput. Methods Appl. Mech., 50, 25–60, 1985MATHCrossRefGoogle Scholar
  5. 5.
    Fonder G. and R.W. Clough, Explicit Addition of Rigid Body Motion in Curved Finite Elements, AIAA J., 11, 305–315, 1973CrossRefGoogle Scholar
  6. 6.
    Fraeijs de Veubeke B., A New Variational Principle for Finite Elastic Displacements, Int. J. Engng. Sci., 10, 745–763, 1972MATHCrossRefGoogle Scholar
  7. 7.
    Frey F., Shell Finite Elements with Six Degrees of Freedom per Node, in Analytical and Computational Models for Shells (eds. A.K. Noor, T. Be-lytschko, J.C. Simo), ASME, 291–317, 1989Google Scholar
  8. 8.
    Gurtin M., An Introduction to Continuum Mechanics, Academic Press, 1981MATHGoogle Scholar
  9. 9.
    Hughes T.J.R. and F. Brezzi, On Drilling Degrees of Freedom, Comput. Methods Appl. Mech. Eng., 72, 105–121, 1989MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Ibrahimbegovic A., R.L. Taylor and E.L. Wilson, A Robust Membrane Quadrilateral Element With Drilling Degrees of Freedom, Int. J. Numer. Methods Eng., 30, 445–457, 1990MATHCrossRefGoogle Scholar
  11. 11.
    Ibrahimbegovic A. and F. Frey, Quadrilateral Membrane Elements with Rotational Degrees of Freedom. Comput. Struct., submitted, 1991Google Scholar
  12. 12.
    Idelsohn S., On the Use of Deep, Shallow or Flat Shell Finite Elements for the Analysis of Thin Shell Structures, Comput. Methods Appl. Mech. Eng., 26, 321–330, 1981MATHCrossRefGoogle Scholar
  13. 13.
    Jetteur Ph. and F. Frey, A Four Node Marguerre Element for Nonlinear Shell Analysis, Eng. Comput., 3, 276–282, 1986CrossRefGoogle Scholar
  14. 14.
    Marsden J.E. and T.J.R. Hughes, Mathematical Foundations of Elasticity, Prentice Hall, Englewood Cliffs, New Jersey, 1983MATHGoogle Scholar
  15. 15.
    Naghdi P.M., On a Variational Theorem in Elasticity and Its Application to Shell Theory, J. Appl. Mech., 12, 647–653, 1964MathSciNetCrossRefGoogle Scholar
  16. 16.
    Ogden R.W., Nonlinear Elastic Deformations, John Wiley, London, 1984MATHGoogle Scholar
  17. 17.
    Reissner E., On the Form of Variationally Derived Shell Equations, J. Appl. Mech., 86, 233–238, 1964MathSciNetCrossRefGoogle Scholar
  18. 18.
    Reissner E., A Note on Variational Principles in Elasticity, Int. J. Solids Struct., 1, 93–95, 1965CrossRefGoogle Scholar
  19. 19.
    Simo J.C. and D.D. Fox, On a Stress Resultants Geometrically Exact Shell Model. Part I: Formulation and Optimal Parameterization, Comp. Methods Appl. Mech. Eng., 72, 267–304, 1989MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Washizu K., Variational Methods in Elasticity and Plasticity, Pergamon Press, Oxford, 1982MATHGoogle Scholar
  21. 21.
    Zienkiewicz O.C. and R.L. Taylor, The Finite Element Method: Basic Formulation and Linear Problems, vol I, McGraw-Hill, London, 1989Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • A. Ibrahimbegovic
    • 1
  • F. Frey
    • 1
  • G. Fonder
    • 2
  • C. Massonnet
    • 2
  1. 1.Department of Civil EngineeringSwiss Federal Institute of TechnologyLausanneSwitzerland
  2. 2.Department of Civil EngineeringUniversity of LiègeLiègeBelgium

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