DIRECT FORMULATION OF A HYBRID 4-NODE SHELL ELEMENT WITH DRILLING DEGREES OF FREEDOM

A simple 4-node assumed-stress hybrid quadrilateral shell element with rotational or 'drilling' degrees of freedom is formulated. The element formulation is based directly on a 4-node element. This direct formulation requires fewer computations than a similar element that is derived from an 'internal' 8-node isoparametric element in which the midside degrees of freedom are eliminated by expressing them in terms of displacements and rotations at corner nodes. The formulation is based on the principle of minimum complementary energy. The membrane part of the element has 12 degrees of freedom, including the drilling degrees of freedom. The bending part of the element also has 12 degrees of freedom. The bending part of the element uses the Reissner Mindlin plate theory which takes into account the transverse shear effects. Quadratic variations for both in-plane and out-of-plane displacement fields and linear variations for both in-plane and out-of-plane rotation fields are assumed along the edges of the element. The element Cartesian-co-ordinate system is selected such as to make the stress field invariant with respect to node numbering. The membrane part of the stress field is based on a 9-parameter equilibrating stress field, while the bending part is based on a 13-parameter equilibrating stress field. The element passes the patch test, is nearly insensitive to mesh distortion, alleviates the 'locking' phenomenon, is invariant with respect to node numbering, has no hidden spurious modes, and produces accurate and reliable results.