A 6-node Co-rotational Curved Triangular Shell Element For Smooth,Folded And Multi-shell Structures

A 6-node corotational curved triangular shell element with novel treatment for rotation at intersection of folded and multishell structures is presented.The local coordinate system of the element is defined by the vectors directing from one vertex to the other two vertexes 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,which simplifying the formulation of element in the local coordinate system.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.Compared with other existing co-rotational curved triangular shell element,the element developed in this paper has the following advantages:1) With the vectorial variables,rotations are not represented by axial (pseudo) vectors but by components of polar (proper) vectors,leading to the desired additive property for rotational variables.2) When employing the traditional rotational variables,the element tangent stiffness matrix is asymmetrical.However,with the vectorial variables,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,which can save computer storage resources and improve computational efficiency of the element formulation.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 complicate dynamic problems.To alleviate membrane and shear locking phenomena,the membrane strains and the out-of-plane shear strains are replaced with assumed strains.The line integration method proposed by MacNeal and the Discrete Shear Gap (DSG) method are used for obtaining the element tangent stiffness matrices and the internal force vector.The reliability and computational accuracy of the presented element formulation are verified through two smooth shell problems and six different types of non-smooth shell problems undergoing large displacements and large rotations.