An isostatic quadrilateral membrane finite element with drilling rotations and no spurious modes

A new quadrilateral four node membrane finite element based on a mixed HellingerReissner variational formulation is proposed. Displacement and stress interpolations are defined by 12 kinematical DOFs (two displacements and one drilling rotation per node) and 9 stress parameters. The displacement interpolation is obtained as a sum of three contributions. The first two correspond to compatible modes that assume a linear and quadratic (Allman-like) shape along the sides. The latter corresponds to a cubic incompatible mode depending on the average nodal rotations of the element. The stress interpolation is obtained from a complete quadratic polynomial by enforcing the internal bulk equilibrium and three further uλ Pian equilibrium conditions, so obtaining an equilibrated and non-redundant field. The compliance and compatibility matrices are derived analytically, using an efficient boundary integration scheme. Numerical comparisons show that the proposed element performs better and is less sensitive to mesh distortion than similar elements in the literature. The constant stress states are recovered exactly and a very accurate recovery, for both stress and rotation fields, is also obtained in bending as well as in shear contexts. As shown by some numerical tests in buckling problems, the element is suitable for extension to nonlinear analysis. © 2011 Elsevier B.V. All rights reserved.