A new isostatic quadrilateral membrane finite element and its use in geometrically nonlinear analysis R. Casciaro 1 , S. de Miranda 2 , A. Madeo 1 , F. Ubertini 2 , G. Zagari 1 1 MODELING, Università della Calabria, Italy, rcasciaro@unical.it, antoniomadeo@labmec.unical.it, gzagari@unical.it 2 DICAM, Università di Bologna, Italy E-mail: stefano.demiranda@unibo.it, francesco.ubertini@unibo.it Keywords: Mixed finite element, isostatic and equilibrated stress, drilling rotations, corotational for- mulation, 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 con- straining 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, af- ter 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. 1 INTRODUCTION The general framework based on corotational formulation discussed in [3] allows the geometri- cally nonlinear FEM analysis of beam or plate assemblages, either using path–following or asymp- totic numerical strategies, simply starting from finite elements defined for linear analysis. In the same paper, the proposed approach was used for 3D beam assemblages with very good results. A preliminary implementation for plate assemblages [5] provided quite satisfactory results but also suggested improvements by a better tuning of the plate element used, particularly with reference to its in-plane (membrane) behavior. Whit this in mind, in the paper [1] a linear membrane finite element called HQ4-9β, has been developed. This element when coupled with an appropriate flexural description and a corotational solution strategy, is suitable for use in the geometrical nonlinear analysis of slender folded plates and flat shells. Really, this element satisfies the following fundamental requirements: - a good linear response, free from locking and spurious zero-energy modes, that is the basic prerequisite for reliability and good performance also in the nonlinear context; 1