An improved incompatible dsq element using free formulation approach

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A formulation of a quadratic incompatible quadrilateral element, called DSQK, using a combination of free formulation approach, independent transverse shear strains, and discrete shear constraints, is described in this paper. This new element, which includes transverse shear effects and is valid for thin and thick plates, has 4 nodes and 3 DOFs per node (transverse displacement w and rotations βx and βy at the corner nodes). The couple between lower order and higher order bending energy is assumed to be zero in the DSQK element to fulfill the constant bending patch test. The independent transverse shear strain is expressed only by second derivatives of the rotations obtained from a unified and integrated kinematic relationship, constitutive law, and equilibrium equations. The validation based on individual element tests, patch tests, and convergence tests deliver good results for thin to thick plates of various geometries.

Original languageEnglish
Pages (from-to)665-679
Number of pages15
JournalStructural Engineering and Mechanics
Issue number6
Publication statusPublished - 25 Jun 2021


  • Discrete shear
  • DSQK
  • Free formulation
  • Incompatible finite element
  • Independent transverse shear strain


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