Nine-node shell elements with 6 dofs/node based on two-level approximations. Part I: Theory and linear tests

The paper concerns 9-node quadrilateral shell elements derived for Reissner's kinematics. They are based on the Green strain and potential energy, and are applicable to large (unrestricted) rotations. The characteristic features of the developed elements are as follows:1. Drilling rotation is included via the drill rotation constraint (RC) imposed by the penalty method. Hence, the elements have 6 dofs per node, i.e. three displacements and three rotational parameters, including drilling rotation.2. Transverse shear and membrane locking as well as the in-plane shear over-stiffening are avoided using the two-level approximation applied to the strain (assumed strain method). This method does not affect the drilling RC.3. A modification of the two-level approximation method is proposed, consisting in treating the sampling and the numerical integration together, which results in six sampling points being replaced by two sampling lines. The two-level approximation is applied to components in the ortho-normal basis at the element center, which differs our element from the MITC family of elements, which uses the covariant strain components.4. Selective reduced integration (SRI) approach is revised. The total functional is split into several parts, and a suitable integration rule is found for each part, yielding an efficient element which shows very good mesh convergence.Two 9-node shell elements are developed and subjected to a range of benchmark tests, to establish the sensitivity to mesh distortion, the coarse mesh accuracy, and to confirm the lack of locking. Our results are compared with results obtained by the MITC9 element of ADINA and the S9R5 element of ABAQUS.