Novel quadrilateral elements based on explicit Hermite polynomials for bending of Kirchhoff–Love plates

  • Alireza Beheshti
Original Paper


The contribution addresses the finite element analysis of bending of plates given the Kirchhoff–Love model. To analyze the static deformation of plates with different loadings and geometries, the principle of virtual work is used to extract the weak form. Following deriving the strain field, stresses and resultants may be obtained. For constructing four-node quadrilateral plate elements, the Hermite polynomials defined with respect to the variables in the parent space are applied explicitly. Based on the approximated field of displacement, the stiffness matrix and the load vector in the finite element method are obtained. To demonstrate the performance of the subparametric 4-node plate elements, some known, classical examples in structural mechanics are solved and there are comparisons with the analytical solutions available in the literature.


Plate bending FEM Kirchhoff–Love model Hermite polynomials 


  1. 1.
    Zienkiewicz OC, Too J, Taylor RL (1971) Reduced integration technique in general analysis of plates and shells. Int J Numer Methods Eng 3:275–290CrossRefzbMATHGoogle Scholar
  2. 2.
    Hughes TJR, Cohen M, Harou M (1978) Reduced and selective integration techniques in the finite element analysis of plates. Nucl Eng Des 46:203–222CrossRefGoogle Scholar
  3. 3.
    Malkus DS, Hughes TJR (1978) Mixed finite element methods in reduced and selective integration techniques: a unification of concepts. Comput Methods Appl Mech Eng 15:63–81CrossRefzbMATHGoogle Scholar
  4. 4.
    Zienkiewicz OC, Qu S, Taylor RL, Nakazawa S (1986) The patch test for mixed formulations. Int J Numer Methods Eng 23:1873–1883MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Soh AK, Long ZF, Cen S (1999) A new nine DOF triangular element for analysis of thick and thin plates. Comput Mech 24:408–417CrossRefzbMATHGoogle Scholar
  6. 6.
    Piltner R, Joseph DS (2001) An accurate low order plate bending element with thickness change and enhanced strains. Comput Mech 27:353–359CrossRefzbMATHGoogle Scholar
  7. 7.
    Wu F, Liu GR, Li GY, Cheng AG, He ZC (2014) A new hybrid smoothed FEM for static and free vibration analyses of Reissner–Mindlin plates. Comput Mech 54:865–890CrossRefzbMATHGoogle Scholar
  8. 8.
    Li T, Ma X, Xili J, Chen W (2016) Higher-order hybrid stress triangular Mindlin plate element. Comput Mech 58:911–928MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Zienkiewicz OC, Cheung YK (1964) The finite element method for analysis of elastic isotropic and orthotropic slabs. Proc Inst Civ Eng 28:471–488Google Scholar
  10. 10.
    Henshell RD, Walters D, Warburton GB (1972) A new family of curvilinear plate bending elements for vibration and stability. J Sound Vibr 20:327–343CrossRefzbMATHGoogle Scholar
  11. 11.
    Bazeley GP, Cheung YK, Irons BM, Zienkiewicz OC (1966) Triangular elements in bending—conforming and non-conforming solutions. In: Proceedings of 1st conference on matrix methods in structural mechanics AFFDL-TR-66-80, pp 547–576Google Scholar
  12. 12.
    Morley LSD (1971) On the constant moment plate bending element. J Strain Anal 6:20–24CrossRefGoogle Scholar
  13. 13.
    Zienkiewicz OC, Taylor RL (2005) The finite element method for solid and structural mechanics, 6th edn. Elsevier Butterworth-Heinemann, OxfordzbMATHGoogle Scholar
  14. 14.
    Clough RW, Tocher JL (1966) Finite element stiffness matrices for analysis of plate bending. In: Proceedings of 1st conference on matrix methods in structural mechanics AFFDL-TR-66-80, pp 515–545Google Scholar
  15. 15.
    Irons BM (1969) A conforming quartic triangular element for plate bending. Int J Numer Methods Eng 1:29–46CrossRefzbMATHGoogle Scholar
  16. 16.
    Clough RW, Felippa CA (1968) A refined quadrilateral element for analysis of plate bending. In: Proceedings of 2nd conference on matrix methods in structural mechanics, AFFDL TR-68-150, pp 399–440Google Scholar
  17. 17.
    de Veubeke BF (1968) A conforming finite element for plate bending. Int J Solids Struct 4:95–108CrossRefzbMATHGoogle Scholar
  18. 18.
    Argyris JH, Fried I, Scharpf DW (1968) The tuba family of plate elements for the matrix displacement method. Aeronaut J 72:701–709Google Scholar
  19. 19.
    Bell K (1969) A refined triangular plate bending element. Int J Numer Meth Eng 1:101–122CrossRefGoogle Scholar
  20. 20.
    Pian THH (1964) Derivation of element stiffness matrices by assumed stress distribution. J AIAA 2:1332–1336CrossRefGoogle Scholar
  21. 21.
    Herrmann LR (1968) Finite element bending analysis of plates. J Eng Mech ASCE 94:13–25Google Scholar
  22. 22.
    Stricklin JA, Haisler W, Tisdale P, Gunderson K (1969) A rapidly converging triangle plate element. J AIAA 7:180–181CrossRefzbMATHGoogle Scholar
  23. 23.
    Razzaque A (1973) Program for triangular bending element with derivative smoothing. Int J Numer Methods Eng 5:588–589CrossRefGoogle Scholar
  24. 24.
    Dhatt G, Marcotte L, Matte Y (1986) A new triangular discrete Kirchhoff plate–shell element. Int J Numer Methods Eng 23:453–470CrossRefzbMATHGoogle Scholar
  25. 25.
    Felippa CA, Militello C (1999) Construction of optimal 3-node plate bending triangles by templates. Comput Mech 24:1–13CrossRefzbMATHGoogle Scholar
  26. 26.
    Bogner FK, Fox RL, Schmit LA (1966) The generation of interelement-compatible stiffness and mass matrices by the use of interpolation formulae. In: Proceedings of 1st conference on matrix methods in structural mechanics AFFDL-TR-66-80, pp 397–443Google Scholar
  27. 27.
    Petera J, Pittman J (1994) Isoparametric hermite elements. Int J Numer Methods Eng 37:3489–3519MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ugural AC (1981) Stresses in plates and shells. McGraw-Hill, New YorkzbMATHGoogle Scholar
  29. 29.
    Hughes TRJ (1987) The finite element method. Prentice Hall, Upper Saddle RiverzbMATHGoogle Scholar
  30. 30.
    Wriggers P (2008) Nonlinear finite element methods. Springer, BerlinzbMATHGoogle Scholar
  31. 31.
    Chernuka MW, Cowper GR, Lindberg GM, Olson MD (1972) Finite element analysis of plates with curved edges. Int J Numer Methods Eng 4:49–65CrossRefzbMATHGoogle Scholar
  32. 32.
    Timoshenko S, Woinowsky-Krieger S (1959) Theory of plates and shells. McGraw-Hill, New YorkzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of GuilanRashtIran

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