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A six-node co-rotational curved triangular shell finite element with a novel rotation treatment for folded and multi-shell structures is presented. Different from other co-rotational triangular element formulations, 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 matrices in both local and global coordinate systems. In the co-rotational local coordinate system, the two smallest components of the shell director are defined as the nodal rotational variables. Similarly, the two smallest components of each director in the global coordinate system are adopted as the global rotational variables for nodes located either on smooth shells or away from non-smooth shell intersections. At intersections of folded and multi-shells, global rotational variables are defined as three selected components of an orthogonal triad initially oriented along the global coordinate system axes. As such, the vectorial rotational variables enable simple additive update of all nodal variables in an incremental-iterative procedure, resulting in significant enhancement in computational efficiency for large deformation analysis. To alleviate membrane and shear locking phenomena, an assumed strain method is employed in obtaining the element tangent stiffness matrices and the internal force vector. The effectiveness of the presented co-rotational triangular shell element formulation is verified by analyzing several benchmark problems of smooth, folded and multi-shell structures undergoing large displacements and large rotations. [ABSTRACT FROM AUTHOR]

A 6-node co-rotational triangular elasto-plastic shell element is developed. The local coordinate system of the element is defined by the vectors directing from one vertex to the other two vertices and their cross product. Based on such a co-rotational framework, the element rigid-body rotations are excluded in calculating the local nodal variables from the global nodal variables. The two smallest components of each nodal orientation vector are defined as rotational variables, resulting in the desired additive property for all nodal variables in a nonlinear incremental solution procedure. Different from other existing co-rotational finite element formulations, both the element tangent stiffness matrices in the local and in the global coordinate systems are symmetric owing to the commutativity of the nodal variables in calculating the second derivatives of the strain energy with respect to the local nodal variables and, through chain differentiation, with respect to the global nodal variables. For elasto-plastic analysis, the Maxwell-Huber-Hencky-von Mises yield criterion is employed together with the backward-Euler return-mapping method for the evaluation of the elasto-plastic stress state, where a consistent tangent modulus matrix is employed. To overcome locking problems, the assumed linear membrane strains and shear strains are obtained by using the line integration method proposed by MacNeal, and the assumed higher-order membrane strains are obtained by enforcing the stationarity of the mixed displacement-strain canonical functional, these assumed strains are then employed to replace the corresponding conforming strains. The reliability and convergence of the present 6-node triangular shell element formulation are verified through two elastic plate patch tests as well as two elastic and five elasto-plastic plate/shell problems undergoing large displacements and large rotations. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a novel and efficient 3-D Hybrid-EAS solid element formulation to accurately predict interlaminar stresses in thick laminated beams, plates, and shells. The element formulation is based on the mixed three-field Fraeijs de Veubeke–Hu–Washizu (FHW) functional principle. The element is designed to have quadratic C 0 transverse shear stress field through the thickness direction, while the displacement field remains linear. The continuity of the transverse shear stress at the layer interfaces, together with the vanishing transverse shear stresses at the outer surfaces of the composite structure, are satisfied exactly by using the transverse-shear-stress degrees of freedom (dofs). This method is more elegant than adding the above stress constraints on the FHW functional via a penalty or Lagrange multipliers, and is amenable for implementation in general purpose finite-element codes. Numerical examples show the excellent agreement with Pagano’s exact solution for composite beam/plate problems. The present element satisfies the interlaminar stress continuity, and accurately captures the quadratic variation of the transverse shear stresses in the thickness when compared to other element formulations, in particular the well-known hybrid formulation by Mau, Tong, and Pian (1972) [23]. In addition, the present approach is readily generalizable to the case with large deformation and nonlinear materials. [Copyright &y& Elsevier]