Studies of 4-node Membrane Element with Analytical Stiffness-Matrix Based on the Quadrilateral Area Coordinates

  • Yu Du
  • Song Cen
Conference paper


The new Quadrilateral Area Coordinate (QAC) method is a powerful tool to construct 2D finite element models [1]. Compared with the traditional models using isoparametric coordinates, these new models are less sensitive to mesh distortion. Various elements based on QAC method have been successfully developed, among which the 4-node quadrilateral membrane element AGQ6-I is a typical one [2].

Theoretically, by using the area integral formulae [3] the analytical stiffness-matrix could be obtained, which may greatly benefit the computation procedure. However, as the derivation of the analytical expression is relatively complicated, all the papers by far still adopt the numerical integration method for computer coding, which indeed impedes the advantages of the QAC method. So in this paper, by introducing the basic QAC formulae into two famous symbolic computation softwares, Maple and Mathematica, the analytical expression of the stiffness-matrix of AGQ6-I is obtained for the first time. Then a corresponding FORTRAN subroutine is compiled. Numerical examples show that the present scheme exhibits excellent performances in computation efficiency when compared with the QAC element using numerical integration and the isoparametric element. Furthermore, some general remarks for simplification are also concluded from the derivation process, which may provide significant reference for other researches.


  1. 1.
    Long YQ, Li JX, Long ZF, Cen S. Area coordinates used in quadrilateral elements. Commun. Numer. Meth. Engng., 1999; 15(8): 533–545.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Chen XM, Cen S, Long YQ, Yao ZH. Membrane elements insensitive to distortion using the quadrilateral area coordinate method. Computers & Structures, 2004; 82(1): 35–54.CrossRefGoogle Scholar
  3. 3.
    Long ZF, Li JX, Cen S, Long YQ. Some basic formulae for area coordinates used in quadrilateral elements. Commun. Numer. Meth. Engng., 1999; 15(12): 841–852.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Tsinghua University Press & Springer 2006

Authors and Affiliations

  • Yu Du
    • 1
  • Song Cen
    • 1
    • 2
  1. 1.School of AerospaceTsinghua UniversityBeijingChina
  2. 2.Failure Mechanics LaboratoryTsinghua UniversityBeijingChina

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