Analysis of a rotation-free 4-node shell element

In this paper, a shell element for small and large deformations is presented based on the extension of the methodology to derive triangular shell element without rotational degrees of freedom (so-called rotation-free). As in our original triangular S3 element, the curvatures are computed resorting to the surrounding elements. However, the extension to a quadrilateral element requires internal curvatures in order to avoid singular bending stiffness. The quadrilateral area co-ordinates interpolation is used to establish the required expressions between the rigid-body modes of normal nodal translations and the normal through thickness bending strains at mid-side. In order to propose an attractive low-cost shell element, the one-point quadrature is achieved at the centre for the membrane strains, which are superposed to the bending strains in the centred co-rotational local frame. The membrane hourglass control is obtained by the perturbation stabilization procedure. Free, simply supported and clamped edges are considered without introducing virtual nodes or elements. Several numerical examples with regular and irregular meshes are performed to show the convergence, accuracy and the reasonable little sensitivity to geometric distortion. Based on an updated Lagrangian formulation and Newton iterations, the large displacements of the pinched hemispherical shell show the effectiveness of the proposed simplified element (S4). Finally, the deep drawing of a square box including large plastic strains with contact and friction completes the ability of the rotation-free quadrilateral element for sheet-metal-forming simulatio