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A formulation for the 4‐node quadrilateral element
Author(s) -
Hueck Ulrich,
Wriggers Peter
Publication year - 1995
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620381802
Subject(s) - quadrilateral , stiffness matrix , direct stiffness method , mathematics , finite element method , mathematical analysis , linear elasticity , virtual work , tangent stiffness matrix , displacement field , geometry , elasticity (physics) , matrix (chemical analysis) , mixed finite element method , structural engineering , physics , engineering , materials science , thermodynamics , composite material
A formulation for the plane 4‐node quadrilateral finite element is developed based on the principle of virtual displacements for a deformable body. Incompatible modes are added to the standard displacement field. Then expressions for gradient operators are obtained from an expansion of the basis functions into a second‐order Taylor series in the physical co‐ordinates. The internal degrees of freedom of the incompatible modes are eliminated on the element level. A modified change of variables is used to integrate the element matrices. For a linear elastic material, the element stiffness matrix can be separated into two parts. These are equivalent to a stiffness matrix obtained from underintegration and a stabilization matrix. The formulation includes the cases of plane stress and plane strain as well as the analysis of incompressible materials. Further, the approach is suitable for non‐linear analysis. There, an application is given for the calculation of inelastic problems in physically non‐linear elasticity. The element is efficient to implement and it is frame invariant. Locking effects and zero‐energy modes are avoided as well as singularities of the stiffness matrix due to geometric distortion. A high accuracy is obtained for numerical solutions in displacements and stresses.