A 4-node Co-rotational Quadrilateral Shell Element For Smooth And Non-smooth Shell Structures

A four-node co-rotational quadrilateral shell element for smooth and non-smooth shell structures is presented.Each node of the element has three translational degrees of freedom and two or three vectorial rotational degrees of freedom.For the nodes of a smooth shell or nodes away from the intersection of non-smooth shells,the two smallest components of the mid-surface normal vector are defined as the nodal rotational variables.For the nodes at intersections of non-smooth shells,two smallest components of one orientation vector,together with one smaller or the smallest component of another nodal orientation vector,are employed as rotational variables.In a nonlinear incremental solution procedure,the vectorial rotational variables are additive and symmetric tangent stiffness matrices are obtained in both global and local coordinate systems,thus,one-dimensional linear storage scheme can be adopted,saving computer storage and computing time effectively.In reference frame,the local coordinate system translates and rotates rigidly with the element,but does not deform with the element,thus,the element rigid-body rotations can be excluded in calculating the local nodal variables from the global nodal variables,which simplifying the formulation of element in the local coordinate system.To alleviate membrane and shear locking phenomena,one-point quadrature is adopted in calculating the element tangent stiffness matrices and the internal force vector.The strategies used in the mixed interpolation of tensorial components approach are employed in defining the assumed shear strains in the natural coordinate system.An equivalent form of the standard isoparametric function and the physical stabilized method are employed to avoid the occurrence of spurious zero energy modes.Six generalized strains are defined for obtaining stabilized element formulation.The generalized displacement control method is employed in nonlinear incremental solution procedures.The reliability,computational accuracy and convergence are verified through three smooth shell problems and six non-smooth shell problems undergoing large displacements and large rotations.