An Advanced Co-rotational Curved Quadrilateral Shell Element

An advanced 9-node curved quadrilateral shell element for large displacement and large rotation analysis is developed, where the local coordinate system is a co-rotational framework, it translates and rotates rigidly with the element, but does not deform with the element. Thus, the contribution of rigid-body motion to nodal displacements and rotations can be excluded in advance, making it simpler to obtain the element tangent stiffness matrix. There are 3 translational degrees of freedom and 2 rotational degrees of freedom at each node, and the latters are the two smallest components of the mid-surface normal vector at each node, and additive in an incremental solution procedure, taking advantage in updating the tangent stiffness matrix. Furthermore, the Green-Lagrange strains specialized for the shallow curved shell are employed, and the internal force vector and the element tangent stiffness matrix are calculated respectively as the first derivative and the second derivative of the strain energy with respect to local nodal variables. Considering the commutativity of all nodal variables, the achieved element tangent stiffness matrix is symmetric.The uniformly reduced integration method can eliminate/alleviate locking problems, but sometimes it may lead to the occurrence of element rank deficiency and spurious singular modes. To overcome these problems, the Hellinger-Reissner functional is adopted in which part of conforming strains are replaced by assumed strains. The assumed strains consist of the lower-order strain and the higher-order strain, and the former is interpolated linearly by using the corresponding displacement-based strain, while the latter playing the role of stabilizing strain is formulated based on an assumed strain procedure. The element tangent stiffness matrix derived from Hellinger-Reissner variational principle is still symmetric.Finally, 8 elastic flat plate and curved shell problems are solved to evaluate the performance of the proposed element. The generalized displacement control method is employed in solving these problems. It is shown that locking phenomena has been eliminated/alleviated in the present element, and the reliability, computational efficiency and convergence of the element are satisfying.