A rotation-free thin shell quadrilateral

In this paper a four-node quadrilateral finite element for the analysis of smooth thin shells is presented. The main feature of the element, an extension of previous developments in triangles, is that the translational displacements of the middle surface are the only degrees of freedom. The membrane behavior results from a standard bilinear interpolation of the geometry within the element. With the aim of an efficient element in codes with explicit time integration, one point quadrature is used in the element area. To avoid spurious deformation modes (hourglass modes) membrane forces resulting from a perturbation stabilization technique are included. For the computation of the curvature tensor a patch of five elements (the element and the four adjacent elements) is defined. The curvature field, assumed constant within the element, is expressed in terms of the deformation gradient at the element boundary and is dependent on the position of the twelve nodes included in the patch. In some problems a bending deformed configuration may occur without associated energy. A cost-effective perturbation stabilization scheme is used to control it. General boundary conditions are shown to be easily implemented. The element denoted BSQ (for Basic Shell Quadrilateral) is based on a Total Lagrangian Formulation and has been implemented in codes with implicit and explicit integration. To assess the element performance and convergence properties a set of numerical examples are presented, including geometrically linear and non-linear problems with large strain plasticity. The results obtained show good convergence properties. For several examples different values of the stabilization coefficients have been considered to study the sensitivity of the results to such coefficients. In general this sensitivity appears to be low as the mesh is refined and the results are obtained with a fixed set of coefficients.