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Finite element analysis of shell structures

Archives of Computational Methods in Engineering volume 4, pages 3–61 (1997)Cite this article

Summary

A survey of effective finite element formulations for the analysis of shell structures is presented. First, the basic requirements for shell elements are discussed, in which it is emphasized that generality and reliability are most important items. A general displacement-based formulation is then briefly reviewed. This formulation is not effective, but it is used as a starting point for developing a general and effective approach using the mixed interpolation of the tensorial components. The formulation of various MITC elements (that is, elements based on Mixed Interpolation of Tensorial Components) are presented. Theoretical results (applicable to plate analysis) and various numerical results of analyses of plates and shells are summarized. These illustrate some current capabilities and the potential for further finite element developments.

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References

  1. Noor, A.et al. (eds.) (1989), “Analytical and Computational Models of Shells”,ASME Special Publication. CED-3.

  2. Bathe, K.J. and Dvorkin, E. (1986), “A Formulation of General Shell Elements—the Use of Mixed Interpolation of Tensorial Components”,International Journal for Numerical Methods in Engineering,22, pp. 697–722.

    Article  MATH  Google Scholar 

  3. Love, A.E.H. (1888), “The Small Free Vibrations and Deformation of a Thin Elastic Shells”,Phil. Trans. Roy. Soc. London, (Ser. A) 179), pp. 491–546.

    Article  Google Scholar 

  4. Koiter, W.T. (1959), “A Consistent First Approximation in General Theory of Thin Elastic Shells”, InProc. IUTAM Symposium on the Theory of Thin Elastic Shells. Delft, pp. 12–33.

  5. Sanders, J.L. Jr. (1959), “An Improved First Approximation Theory for Thin Shells”,Tech. Report R-24, NASA.

  6. Kardestuncer, H. (ed.) (1987), “Finite Element Handbook”, MacGraw-Hill.

  7. Ahmad, S., Irons, B.M. and Zienkiewicz, O.C. (1970), “Analysis of Thick and Thin Shell Structures by Curved Finite Elements”,International Journal for Numerical Methods in Engineering,2, pp. 419–451.

    Article  Google Scholar 

  8. Ramm. E. (1977), “A Plate/Shell Element for Large Deflections and Rotations”, In Bathe, K.J. (ed.),Formulations and Computational Algorithms in Finite Element Analysis, M.I.T. Press.

  9. Bathe, K.J. and Bolourchi, S. (1979), “A Geometric and Material Nonlinear Plate and Shell Element”,Computers & Structures,11, 23–48.

    Article  Google Scholar 

  10. Zienkiewicz, O.C., Taylor, R.L. and Too, J.M. (1971), “Reduced Integration Techniques in General Analysis of Plates and Shells”,International Journal for Numerical Methods in Engineering,3, pp. 275–290.

    Article  MATH  Google Scholar 

  11. Oñate, E., Hinton, E. and Glover, N. (1978), “Techniques for Improving the Performance of the Ahmad Shell Elements”,Proc. Int. Conf. on Applied Numerical Modelling, Pentech Press, Madrid, pp. 30–53.

    Google Scholar 

  12. Hughes, T.J.R., Cohen, M. and Haroun, M. (1978), “Reduced and Selective Integration Techniques in the Finite Element Analysis of Plates”,Nuclear Engineering Design,46, pp. 203–222.

    Article  Google Scholar 

  13. Malkus, D.S. and Hughes, T.J.R. (1978), “Mixed Finite Element Methods-Reduced and Selective Integration Techniques: A Unification of Concepts”,Computer Methods in Applied Mechanics and Engineering,15, pp. 63–81.

    Article  MATH  Google Scholar 

  14. Bercovier, M. (1978), “Perturbation of a Mixed Variational Problems. Applications to Mixed Finite Element Methods”,R.A.I.R.O. Anal. Numer.,12, pp. 211–236.

    MathSciNet  MATH  Google Scholar 

  15. Belytschko, T., Liu, W.K., Ong, S.J. and Lam, D. (1985), “Implementation and Application of a 9-node Lagrange Shell Element with Spurious Mode Control”,Computers & Structures,20, (1–3), pp. 121–128.

    Article  MATH  Google Scholar 

  16. Belytschko, T., Ong, S.J. and Liu, W.K. (1985), “A Consistent Control of Spurious Singular Modes in the 9-Node Lagrangian Element for the Laplace and Mindlin Plate Equations”,Computer Methods in Applied Mechanics and Engineering,44, pp. 269–295.

    Article  MathSciNet  Google Scholar 

  17. Bathe, K.J. and Brezzi, F. (1985), “On the Convergence of a Four-Node Plate Bending Element Based on Mindlin/Reissner Plate Theory and a Mixed Interpolation”. In Whiteman, J.R. (ed.),Proceedings of the Conference on Mathematics of Finite Elements and Applications V, Academic Press, pp. 491–503.

  18. Bathe, K.J. and Brezzi, F. (1987), “A Simplified Analysis of Two-Plate Bending Elements—the MITC4 and MITC9 Elements”, In Pande, G.N. and Middleton, J. (eds.),Proceedings of the International Conference on Numerical Methods in Engineering (NUMETA 87), University College of Swansca, Wales, p. D46.

  19. Brezzi, F., Bathe, K.J. and Fortin, M. (1989), “Mixed-Interpolated Elements for Reissner-Mindlin Plates”,International Journal for Numerical Methods in Engineering,28, pp. 1787–1801.

    Article  MathSciNet  MATH  Google Scholar 

  20. Brezzi, F., Fortin, M. and Stenberg, R. (1991), “Error Analysis of Mixed-Interpolated Elements for Reissner-Mindlin Plates”,Mathematical Models and Methods in Applied Sciences,1, pp. 125–151.

    Article  MathSciNet  MATH  Google Scholar 

  21. Dvorkin, E. and Bathe, K.J. (1984), “A Continuum Mechanics Based Four-Node Shell Element for General Nonlinear Analysis”,Engineering Computations,1, pp. 77–88.

    Article  Google Scholar 

  22. Bathe, K.J. and Dvorkin, E. (1985), “A Four-Node Plate Bending Element Based on Mindlin-Reissner Plate Theory and a Mixed Interpolation”,International Journal for Numerical Methods in Engineering,21, pp. 367–383.

    Article  MATH  Google Scholar 

  23. Bathe, K.J., Cho, S.W., Bucalem, M.L. and Brezzi, F. (1989), “On Our MITC Plate Bending/Shell Elements”, In Noor, A.et al. (eds.),Analytical and Computational Models of Shells, CED-3, ASME Special Publication, pp. 261–278.

  24. Bathe, K.J., Brezzi, F. and Cho, S.W. (1989), “The MITC7 and MITC9 Plate Bending Elements”,Computers & Structures,32, (3/4), pp. 797–814.

    Article  MATH  Google Scholar 

  25. Bathe, K.J., Bucalem, M.L. and Brezzi, F. (1990), “Displacement and Stress Convergence of Our MITC Plate Bending Elements”,Engineering Computations,7, 4, pp. 291–302.

    Article  Google Scholar 

  26. Pitkäranta, J. (1991), “The Problem of Membrane Locking in Finite Element Analysis of Cylindrical Shells”, Report, Helsinki University of Technology, Otakaari 1, SF-02150 Espoo, Finland.

    Google Scholar 

  27. Hughes, T.J.R. and Tezduyar, T.E. (1981), “Finite Elements Based upon Mindlin Plate Theory with Particular Reference to the Four-Node Bilinear Isoparametric Element”,Journal of Applied Mechanics, Transactions of ASME,48, pp. 587–596.

    Article  MATH  Google Scholar 

  28. MacNeal, R.H. (1982), “Derivation of Element Stiffness Matrices by Assumed Strain Distributions”,Nuclear Engineering Design,70, pp. 3–12.

    Article  Google Scholar 

  29. Bathe, K.J. (1996), “Finite Element Procedures”, Prentice Hall, Inc., Englewood Cliffs.

    Google Scholar 

  30. Schweizerhof, K. and Ramm, E. (1984), “Displacement Dependent Pressure Loads in Nonlinear Finite Element Analysis”,Computers & Structures,28, pp. 1099–1114.

    Article  Google Scholar 

  31. Argyris, J. (1982), “An Excursion with Large Rotations”,Computer Methods in Applied Mechanics and Engineering,32, pp. 85–155.

    Article  MathSciNet  MATH  Google Scholar 

  32. Bathe, K.J. (1986), “Finite Elements in CAD and ADINA”,Journal of Nuclear Engineering and Design,98, pp. 57–67.

    Article  Google Scholar 

  33. Bathe, K.J. and Ho, L. (1981), “Finite Elements in CAD and ADINA”,Nonlinear Finite Element Analysis in Structural Mechanics, Wunderlich, W.et al. (eds.), Springer Verlag, Berlin.

    Google Scholar 

  34. Hughes, T.J.R., Taylor, R.L. and Kanoknukulchai (1977), “A Simple and Efficient Finite Element for Plate Bending”,International Journal for Numerical Methods in Engineering,11, pp. 1529–1543.

    Article  MATH  Google Scholar 

  35. Huang, H.C. and Hinton, E. (1986), “A New Nine-Node Degenerated Shell Element with Enhanced Membrane and Shear Interpolation”,International Journal for Numerical Methods in Engineering,22, pp. 73–92.

    Article  MathSciNet  MATH  Google Scholar 

  36. Park, K.C. and Stanley, G.M. (1986), “A curvedC o Shell Element Based on Assumed Natural-Coordinate Strains”,Journal of Applied Mechanics, Transactions of ASME, 53, pp. 278–290.

    Article  MATH  Google Scholar 

  37. Jang, J. and Pinsky, P.M. (1987), “An Assumed Covariant Strain Based 9-Node Shell Element,”International Journal for Numerical Methods in Engineering,24, pp. 2389–2411.

    Article  MathSciNet  MATH  Google Scholar 

  38. Bucalem, M.L. and Bathe, K.J. (1993), “Higher-Order MITC General Shell Elements”,International Journal for Numerical Methods in Engineering,36, pp. 3729–3754.

    Article  MathSciNet  MATH  Google Scholar 

  39. Sussman, T. and Bathe, K.J. (1987), “A Finite Element Formulation for Nonlinear Incompressible Elastic and Inelastic Analysis”,Computers & Structures,26, pp. 357–409.

    Article  MATH  Google Scholar 

  40. Häggblad, B. and Bathe, K.J. (1990), “Specifications of Boundary Conditions for Reissner-Mindlin Plate Bending Finite Elements”,International Journal for Numerical Methods in Engineering,30, pp. 981–1011.

    Article  MATH  Google Scholar 

  41. Arnold, D.N. and Falk, R.S. (1989), “Edge Effects in Reissner-Mindlin Plate Theory”, In Noor, A.et al., (eds.)Analytical and Computational Models of Shells, CED-3, ASME Special Publication.

  42. ADINA R&D (1990), “ADINA—A Finite Element Program for Automatic Dynamic Incremental Nonlinear Analysis”, Report 90-1, ADINA R&D, Watertown, MA.

    Google Scholar 

  43. Lindberg, G.M., Olson, M.D. and Cowper, E.R. (1969), “New Developments in the Finite Element Analysis of Shells”,National Research Council of Canada, Quarterly Bulletin of the Division of Mechanical Engineering and the National Aeronautical Establishment,4, pp. 1–38.

    Google Scholar 

  44. Sussman, T. and Bathe, K.J. (1986), “Studies of Finite Element Procedures. Stress Band Plots and the Evaluation of Finite Element Meshes”,Engineering Computations,3, pp. 178–191.

    Article  Google Scholar 

  45. Morley, L.S.D. and Morris, A.J. (1978), “Conflict Between Finite Elements and Shell Theory”, Technical report, Royal Aicraft Establishment Report, London.

  46. Bathe, K.J. and Dvorkin, E.N. (1983), “On the Automatic Solution of Nonlinear Finite Element Equations”,Computers & Structures,17, 5–6, pp. 871–879.

    Article  Google Scholar 

  47. Leicester, R.H. (1968), “Finite Deformations of Shallow Shells”, InProceedings of American Society of Civil Engineers,94, (EM6), pp. 1409–1423.

  48. Sabir, A.B. and Lock, A.C. (1973), “The Application of Finite Elements in the Large Deflection Geometrically Nonlinear Behavior of Cylindrical Shells”, In Brebbia, A. and Tottcham, H. (eds.),Variational Methods in Engineering, pp. 7/66–67/75, Southampton University Press.

  49. Chapelle, D. and Bathe, K.J. (1993), “The Inf-Sup Test”,Computer and Structures,47, pp. 537–545.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Laboratório de Mecânica Computacional Departamento de Engenharia de Estruturas e Fundações, Escola Politécnica da Universidade de São Paulo, 05508-900, São Paulo, SP, Brasil

    M. L. Bucalem

  2. Department of Mechanical Engineering, Massachusetts Institute of Technology, 02139, Cambridge, MA, USA

    K. J. Bathe

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  1. M. L. Bucalem
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  2. K. J. Bathe
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Bucalem, M.L., Bathe, K.J. Finite element analysis of shell structures. ARCO 4, 3–61 (1997). https://doi.org/10.1007/BF02818930

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Keywords

  • Shell Element
  • Nite Element
  • Director Vector
  • Transverse Shear Strain
  • Hemispherical Shell