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Formulation and evaluation of incompatible but convergent rational quadrilateral membrane elements

  • Batoz, J.L. (Universite de Technologie de Compiegne, Laboratoire de Genie Mecanique pour les Materiaux et les Structures) ;
  • Hammadi, F. (Universite de Technologie de Compiegne, Laboratoire de Genie Mecanique pour les Materiaux et les Structures) ;
  • Zheng, C. (Universite de Technologie de Compiegne, Laboratoire de Genie Mecanique pour les Materiaux et les Structures) ;
  • Zhong, W. (Dalian University of Technology)
  • Published : 2000.02.25

Abstract

This paper presents four incompatible but convergent Rational quadrilateral elements, two four-node elements (RQ4Z and RQ4B) and two five-node elements (RQ5Z and RQ5B). The difference between the so-called Rational Finite Element (Zhong and Zeng 1996) and the Free Formulation (Bergan and Nygard 1984) are discussed and compared. The importance of the mode completeness in these formulations is emphasized. Numerical results for several benchmark problems show the good performance of these elements. The two five-nodes elements RQ5Z and RQ5B, which can be viewed as complete quadratic mode elements (with seven stress modes), always give better results than the four nodes elements RQ4Z and RQ4B.

References

  1. Batoz, J.L. and Dhatt. G. (1992), Modelisation des Structures par Elements Finis. Poutres et Plaques, Hermes, Paris.
  2. Batoz, J.L., Hammadi, F. , Zheng, C.L. and Zhong, W.X. (1998), "On the linear analysis of plates and shells using a new sixteen dof flat shell element", Advances in Finite Element Procedures and Techniques, ed by B.H.V. Topping, Civil-Comp press, Edinburgh, Scotland, UK, 31-41., also to be published in Computers and Structures.
  3. Bergan, P.G. and Hanssen, L. (1975), "A new approach for deriving good finite elements", MAFELAP II Conference, Brunel University (1976), The Mathematics of Finite Elements and Applications, II, ed by J.R. Whiteman, Academic Press, London, 483-497.
  4. Bergan, P.G. and Nygard, M.K. (1984), "Finite elements with increased freedom in closing shape functions", International Journal for Numerical Methods in Engineering, 20, 643-664. https://doi.org/10.1002/nme.1620200405
  5. Chen, W. and Cheung, Y.K. (1987), "A new approach for the hybrid element method", International Journal for Numerical Methods in Engineering, 24, 1697-1709. https://doi.org/10.1002/nme.1620240907
  6. Irons, B.M. and Razzaque, A. (1972), "Experiences with the patch test for convergence of finite elements", The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Edition A.K. Aziz, Academic press, NewYork, 551-587.
  7. Jirousek, J. (1978), "Basis for development of large finite element locally satisfying all field equations", Computers Methods in Applied Mechanics and Engineering, 14, 65-92. https://doi.org/10.1016/0045-7825(78)90013-0
  8. Jirousek, J. and Teodorescu, P. (1982), "Large finite elements method for the solution of problems in the theory of elasticity", Computers and Structures, 15, 575-587. https://doi.org/10.1016/0045-7949(82)90009-8
  9. Park, K.C. and Stanley, G.M. (1986), "A curved $C^{\circ}$ shell element based on assumed natural-coordinate strains", Journal of Applied Mecanics, 53, 51-54.
  10. Pian, T.H.H. (1964), "Derivation of element stiffness matrices by assumed stress distributions", AIAA J-2, 1333- 1376. https://doi.org/10.2514/3.2546
  11. Pian, T.H.H. and Chen, D.P. (1982), "Alternative ways for formulation of hybrid stress elements", International Journal for Numerical Methods in Engineering, 18, 1679-1684. https://doi.org/10.1002/nme.1620181107
  12. Pian, T.H.H. and Sumihara, K. (1984), "Rational approach for assumed stress finite element", International Journal for Numerical Methods in Engineering, 20, 1685-1695. https://doi.org/10.1002/nme.1620200911
  13. Piltner, R. and Taylor, R. (1995), "A quadrilateral mixed finite element with two enhanced strain modes", International Journal for Numerical Methods in Engineering, 38, 1783-1808. https://doi.org/10.1002/nme.1620381102
  14. Piltner, R. and Taylor, R. (1999), "A systematic construction of B-bar functions for linear and non-linear mixedenhanced finite elements for plane elasticity problems", International Journal for Numerical Methods in Engineering, 44, 615-639. https://doi.org/10.1002/(SICI)1097-0207(19990220)44:5<615::AID-NME518>3.0.CO;2-U
  15. Simo, J.C. and Rifai, M.S. (1990), "A class of mixed assumed strain methods and the method of incompatible modes", International Journal for Numerical Methods in Engineering, 29, 1595-1638. https://doi.org/10.1002/nme.1620290802
  16. Taig, I.C. and Kerr, R. (1964), "Some problems in the discrete element representation of aircraft structure", Matrix Method of the Structural Analysis, Pergamon Press, London, 267-315.
  17. Tang, L.M., Chen, W.J. and Liu, Y.X. (1984), "Formulation of quasi-conforming element and Hu-Washizu principle", Computers and Structures, 19, 247-250. https://doi.org/10.1016/0045-7949(84)90224-4
  18. Taylor, R.L., Beresford, P.J. and E.L. Wilson, (1976), "A non-conforming element for stress analysis", International Journal for Numerical Methods in Engineering, 10, 1211-1219. https://doi.org/10.1002/nme.1620100602
  19. Turner, M., Clough, R., Martin, H. and Topp, L. (1956), "Stiffness and deflection analysis of complex structures", J. Aeronaut. Sci., 23, 1805-1823.
  20. Wilson, E.L., Taylor, R.L., Doherty, W.P. and Ghaboussi, J. (1973), "Incompatible displacement models", Numerical and Computer Methods in Structural Mechanics, Academic Press, New York, 43-57.
  21. Zhao, P., Pian, T.H.H. and Sheng, T. (1997), "A new formulation of isoparametric finite elements and the relationship between hybrid stress element and incompatible element", International Journal for Numerical Methods in Engineering, 40, 15-27. https://doi.org/10.1002/(SICI)1097-0207(19970115)40:1<15::AID-NME46>3.0.CO;2-K
  22. Zhong, W.X. and Zeng. J. (1996), "Rational finite elements", Journal of Computational Structural Mechanics and Application, in Chinese, 13, 1-8.

Cited by

  1. Finite element linear and nonlinear, static and dynamic analysis of structural elements, an addendum vol.19, pp.5, 2002, https://doi.org/10.1108/02644400210435843