Based on a co-rotational framework, a 3-noded iso-parametric element formulation of 3D beam was presented, which was used for accurate modelling of frame structures with large displacements and large rotations. Firstly, a co-rotational framework was fixed at the internal node of the element, it translates and rotates with the node rigidly; then, vectorial rotational variables were defined, they are three smaller components of the cross-sectional principal vectors at each node, sometimes they represent different components of the cross-sectional principal vectors in incremental solution procedure so as to avoid the occurrence of ill-conditioned tangent stiffness matrix; thereafter, the internal force vector and tangent stiffness matrix in local system was derived from the strain energy of the element as its first partial derivative and second partial derivative with respect to local variables, respectively, and a symmetric tangent stiffness matrix was achieved; finally, several examples were analysed to illustrate the reliability and accuracy of this procedure.

A new 9-node co-rotational curved quadrilateral shell element formulation is presented in this paper. Different from other existing co-rotational element formulations: (1) Additive rotational nodal variables are utilized in the present formulation, they are two well-chosen components of the mid-surface normal vector at each node, and are additive in an incremental solution procedure; (2) the internal force vector and the element tangent stiffness matrix are respectively the first derivative and the second derivative of the element strain energy with respect to the nodal variables, furthermore, all nodal variables are commutative in calculating the second derivatives, resulting in symmetric element tangent stiffness matrices in the local and global coordinate systems;(3) the element tangent stiffness matrix is updated using the total values of the nodal variables in an incremental solution procedure, making it advantageous for solving dynamic problems. Finally, several examples are solved to verify the reliability and computational efficiency of the proposed element formulation.

Summary 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.

A new 3-node co-rotational element formulation for 3D beam is presented. The present formulation differs from existing co-rotational formulations as follows: 1) vectorial rotational variables are used to replace traditional angular rotational variables, thus all nodal variables are additive in incremental solution procedure; 2) the Hellinger-Reissner functional is introduced to eliminate membrane and shear locking phenomena, with assumed membrane strains and shear strains employed to replace part of conforming strains; 3) all nodal variables are commutative in differentiating Hellinger-Reissner functional with respect to these variables, resulting in a symmetric element tangent stiffness matrix; 4) the total values of nodal variables are used to update the element tangent stiffness matrix, making it advantageous in solving dynamic problems. Several examples of elastic beams with large displacements and large rotations are analysed to verify the computational efficiency and reliability of the present beam element formulation.

SUMMARYA new curved quadrilateral composite shell element using vectorial rotational variables is presented. An advanced co-rotational framework defined by the two vectors generated by the four corner nodes is employed to extract pure element deformation from large displacement/rotation problems, and thus an element-independent formulation is obtained. The present line of formulation differs from other corotational formulations in that (i) all nodal variables are additive in an incremental solution procedure, (ii) the resulting element tangent stiffness is symmetric, and (iii) is updated using the total values of the nodal variables, making solving dynamic problems highly efficient. To overcome locking problems, uniformly reduced integration is used to compute the internal force vector and the element tangent stiffness matrix. A stabilized assumed strain procedure is employed to avoid spurious zero-energy modes. Several examples involving composite plates and shells with large displacements and large rotations are presented to testify to the reliability, computational efficiency, and accuracy of the present formulation.