Most structural analyses are concerned with the behavior of shell structures that are the effective form of relatively thin and flexible structures. Deformation of the shell structures is generally calculated by the finite element analysis with shell elements. The demand of shell models is rapidly growing in a wide range of finite element analyses of the shell structures in order to perform large-scale computations in complicated three-dimensional problems of structural analyses and forming processes. Among various types of shell elements, the four-node bilinear degenerated shell element is widely used for numerical analysis Many recent efforts have been done to include the sixth degree of freedom of the drilling rotation compatible to the physical meaning.
In this paper, a four-node degenerated shell element with the drilling DOF is developed to solve nonlinear problems with the implicit numerical scheme based on the Belytschko-Leviathan's shell element and the Zhu-Zacharia's bilinear quadrilateral shell element. The drilling DOF has been interpolated by a cubic polynomial function of the beam deflection mechanism and the assumed strain method for the purpose of eliminating the in-plane shear locking phenomenon induced by the third order polynomial interpolation. A suitable assumed strain method for the drilling DOF is obtained by the Rayleigh-Ritz method and is applied to portion of the drilling DOF among the in-plane strain components of a shell element. The transverse shear locking phenomenon is prevented by the Dvorkin and Bathe's assumed strain method. The one point integration rule is adopted for the computational efficiency. Since the reduced integration results in spurious energy modes so called hourglass modes, the physical stabilization method is applied in order to control these spurious modes. The shell element developed is extended to deal with the geometric and the material nonlinearities. The material nonlinearity is considered by applying the return mapping method for a plane stress problem by Simo and Taylor in order to deal with the elasto-plastic constitutive model.
Several benchmark problems and example problems were solved to investigate the performance of the element developed. The eigenvalue analysis confirmed the validity of the developed shell element and the analysis of linear problems demonstrated the accuracy and convergence of the element developed. Results of linear problems demonstrated that the element developed was free from the in-plane shear locking phenomenon and had good convergence and accuracy under the in-plane deformation and the warped geometry even with the coarse mesh system. The result form the full hemispherical shell problem showed that the membrane locking phenomenon still occurred with the coarse mesh system since it was influenced by the term with ξη.
Nonlinear problems including the geometrical and the material nonlinearities were chosen in order to confirm the versatility of the drilling degree of freedom and the robustness in the large rotational problem. The nonlinear problems of the bending, the torsion, the plate with a hole, collapse of a square box and collapse of a S-rail have been solved with the element developed and the results fully demonstrate the versatility and performance of the developed shell element. Results showed that the element developed was useful not only in linear problems but also in nonlinear problems including the geometrical and the material nonlinearities.