Plate Finite Element with Physical Shape Functions: Correctness of the Formulation

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Abstract

The formulation of a plate finite element with so called ‘physical’ shape functions is revisited. The derivation of the ‘physical’ shape functions is based on Hencky-Bolle theory of moderately thick plates. The considered finite element was assessed in the past, and the tests showed that the solution convergence was achieved in a wide range of thickness to in-plane dimensions ratios. In this paper a holistic correctness assessment is presented, which covers three criteria: the ellipticity, the consistency and the inf-sup conditions. Fulfilment of these criteria assures the existence of a unique solution, and a stable and optimal convergence to the correct solution. The algorithms of the numerical tests for each test case are presented and the tests are performed for the considered formulation. In result it is concluded that the finite element formulation passes every test and therefore is a good choice for modeling plate structural elements regardless of their thickness.

1. Gilewski W., On the criteria for evaluation of finite elements - from Timoshenko beam to Hencky-Bolle plate. (in Polish) Warsaw University of Technology Publishing House (OWPW): Warsaw, 2005.

2. Gilewski W., Radwańska M., A survey of finite element models for the analysis of moderately thick shells. Finite Element Analysis and Design 1991; 9:1-21.

3. Radwańska M., An overview of selected plate and shell finite element models with graphic presentation of geometric equation. Comp. Ass. Mech. Eng. Sc. 2007; 14:431-456.

4. Witkowski W., 4-node combined shell element with semi-EAS-ANS strain interpolations in 6-parameter shell theories with drilling degrees of freedom. Comp. Mech. 2009; 43(2):307-319.

5. Kączkowski Z., Plates-static calculations. (in Polish) Arkady: Warsaw, 1980.

6. Gilewski W., Gomuliński A., Physical shape functions in finite element analysis of moderately thick plates. Int. Journ. Num. Meth. Engrg. 1991; 32:1115-1136.

7. Bathe K.J., Dvorkin E., A formulation of general shell elements - The use of mixed interpolation of tensorial components. Int. J. Num. Meth. Eng. 1986; 22:697-722.

8. Bathe K.J., The inf-sup condition and its evaluation for mixed finite element methods. Comp. Struct. 2001; 79:243-252.

9. Gilewski W., A simple method for the analysis of correctness of FEM formulations. Computational Mechanics. Theory and Applications, 1995; 1:678-683.

10. Brezzi F., On existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. RAIRO, Anal. Numer. 1974; 8:129-151.

11. Babuska I., The finite element method with Lagrange multipliers. Num. Math. 1977; 20: 322-333.

12. Chapelle D., Bathe K.J., The inf-sup test. Computers and Structures. 1993; 47:537-545.

13. Bathe K.J., Finite element procedures. Prentice Hall: Engelwood Cliffs, NJ, 1996.

14. Bathe K.J., Iosilevich A., Chapelle D., An evaluation of the MITC shell elements. Comp. Struct. 2000; 75: 1-30.

15. Berkovic M., Mijuca D., Grozdanovic I., Some continuous stress mixed formulations and inf-sup test. Comp. Ass. Mech. Eng. Sci. 2001; 8:141-153.

16. Kirmse A., Bending-dominated deformations of thin spherical shells: Analysis and finite-element approximation. SIAM J. Numer. Anal. 1993; 30:1015-1040.

17. Pitkaranta J., Sanchez-Palencia E., On the asymptotic behavior of sensitive shells with small thickness. C. R. Acad. Sci. Paris 1997; 325:127-134.

18. Chapelle D., Bathe K.J., Fundamental considerations for the finite element analysis of shell structures. Comp. Struct. 1998; 66:19-36.

19. Gilewski W., Sitek M., The inf-sup condition tests for shell/plate finite elements, Archives of Civil Engrg. 2011; 57(4):425-447.

20. Iosilevich A., Bathe K.J., Brezzi F., On evaluating the inf-sup condition for plate bending elements. Int. Journ. Num. Meth. Engrg. 1997; 40:3639-3663.

21. Hencky H., Uber die berűcksichtigung der schubverzerrung in ebenen platen. Ingenieur-Archiv. 1947; 16(1):72-76.

22. Bolle L., Contribution un probleme lineaire de flexion d’un plaque elastique. Bulletin Technique de la Suisse Romande 1947; 73(21-22):281-285&293-298.

23. Reissner E., Reflections on the theory of elastic plates. Applied Mechanics Review 1985; 38(11):1453-1464.

24. Gilewski W., Sitek M., Evaluation of shell finite elements: ellipticity, consistency and inf-sup condition. Some practical examples. CMM-2007, Łodź-Spała 2007.

25. Sitek M., Correctness assessment of beam, plate and shell finite elements, PhD Thesis (in Polish) Warsaw University of Technology Publishing House (OWPW): Warsaw, 2010.

Archives of Civil Engineering

The Journal of Polish Academy of Sciences

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