A nine-node corotational curved quadrilateral shell element for smooth, folded, and multishell structures

Abstract

A nine-node corotational curved quadrilateral shell element with novel treatment for rotation at intersection of folded and multishell structures is presented. The element's corotational reference frame is defined by the two bisectors of the diagonal vectors generated using the four corner nodes and their cross product. In reference frame, the element rigid-body rotations are excluded in calculating the local nodal variables from the global nodal variables. Rotations are not represented by axial (pseudo) vectors but by components of polar (proper) vectors, of which additivity and commutativity lead to symmetry of the tangent stiffness matrix. In the global coordinate system, the two smallest components of the midsurface normal vector at each node of a smooth shell or at nodes away from the intersection of nonsmooth shells are defined as rotational variables. In addition, of the two nodal orientation vectors at intersections of folded and multishell structures, two smallest components of one vector, together with one smaller component of another vector, are employed as rotational variables, leading to the desired additive property for all nodal variables in a nonlinear incremental solution procedure. In the local coordinate system, the two smallest components of the midsurface normal vector(s) at any node of a smooth shell or in each smooth patch of nonsmooth shell are defined as rotational variables. Different from other existing corotational finite-element formulations, the resulting element tangent stiffness matrix is symmetric owing to the commutativity of the local nodal variables in calculating the second derivative of strain energy with respect to these nodal variables. To alleviate membrane and shear locking phenomena, the membrane strains and the out-of-plane shear strains are replaced with assumed strains, using the Mixed Interpolation of Tensorial Components approach, for obtaining the element tangent stiffness matrices and the internal force vector. Finally, a series of benchmark, challenging smooth, folded, and multishell structures undergoing large displacements and large rotations are analyzed to demonstrate the reliability and computational accuracy of the proposed formulation.