A new four-node degenerated shell element with drilling degrees of freedom(DOF) is proposed. It is shown that the Allman-type displacement fields can be extended to three dimensional displacement fields based on the beam theory. Using these approximated displacement fields, new displacement fields of a four-node degenerated shell element with drilling DOF are derived from those of an eight-node degenerated shell element.
The assumed strain method is used for the transverse shear strains to circumvent the transverse shear locking. Conventional steps of calculating stiffness matrices are followed and 2x2 Gaussian quadratures are used. Through-thickness numerical integration is included in the formulation for the analysis of composite laminated shell problems. Two kinds of penalty energy functions are used to restrain two spurious modes. It is shown through numerical examples that the penalty energies do not deteriorate the element performance.
Many numerical problems including the patch tests and benchmark tests are solved for investigating the performance of the proposed element. The element is passed for both the membrane patch test and the bending patch test. It is concluded that the element shows good accuracy for most of benchmark problems and is less sensitive to element distortions and can be applied reliably to thin or moderately thick shell problems. And the proposed element shows good accuracy for not only isotropic but also composite laminated structures. It should be noted that true drilling rotations can be calculated accurately by introducing the penalty function comprising rotations and the differentiated terms of displacements. Stiffened or folded structures are handled more conveniently by the proposed element which has six degrees of freedom per node.
