Keywords: co-rotational finite elements, quadrilateral shell elements, vectorial rotational variables, additives, commutative, symmetric tangent stiffness matrix, membrane locking, shear locking, assumed strain, Hellinger-Reissner functional, elastic shells, large displacements, finite element method, FEM
A mixed co-rotational curved quadrilateral shell element
This paper presents a 9-node isoparametric formulation of degenerated curved shell element, which is based on a co-rotational framework. Within this framework, the local reference axes are easily defined by the two diagonal vectors generated by the four corner nodes in the current deformed configuration. Furthermore, vectorial rotational variables are utilised, which consist of the two smaller components of each nodal orientation vector, enabling additive and commutative rotational operations. The element strain energy is evaluated in the local co-rotational system using the Green strain, where the Hellinger-Reissner functional is employed with assumed strain tensors for the membrane and shear strain fields to alleviate the membrane/shear locking phenomena. By defining the equilibrium conditions in terms of the out-of-balance between the work-conjugate internal and external forces, symmetric element tangent stiffness matrices are achieved in both the local and global systems, thus leading to considerable computational benefits. Finally, several examples of elastic shells subject to large displacements are considered, which demonstrate favourable performance of the proposed finite element formulation.