A new isostatic quadrilateral membrane finite element and its use in geometrically nonlinear analysis

In a previous work [1], a new quadrilateral four node membrane finite element, called HQ4-9β, based on a mixed Hellinger–Reissner variational formulation was proposed. Displacement and stress interpolations are defined by 12 kinematical DOFs (two displacements and one drilling rotation for each node) and 9 stress parameters, respectively. 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 [2]. 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 equilibrium equations and three further constraining conditions à la Pian involving incompatible displacements, so obtaining an equilibrated and isostatic approximation. The features and the behavior of the element in the linear context make it very suitable for the geometrically nonlinear analysis of slender folded plate structures when coupled with an appropriate flexural description and a corotational solution strategy. In the present paper, after a description of HQ4-9β, a Koiter asymptotic analysis of folded plate structures, is proposed. The analysis is developed following the general framework based on the corotational approach proposed in [3] and using the plate element proposed in [4] for the out-plane behavior. Some benchmarks are presented and discussed to show the accuracy and the efficiency of the proposed geometrically nonlinear analysis of the element.