The main aim of this paper is to propose the new formulation for a four node degenerated shell element based on the assumed covariant strain method.
The original Mindlin-type plate and shell elements perform reasonably well for moderately thick shell situations. However, for thin shells when full integration is used to evaluate the stiffness matrix, overstiff solutions are often produced due to shear and membrane locking. To alleviate these deficiencies, several methods have been proposed by many investigators in the past such as the selective and reduced integration schemes, heterosis elements, stabilization methods, and discrete Kirchhoff elements, etc. These approaches, however, provide a number of distinct limitations. The selective or reduced integration scheme is not always successful in overcoming the locking problem and thus the resulting solutions may be overstiff for problems with highly constrained boundaries when coarse meshes are used in particular. Furthermore, for the problems with lightly constrained boundaries, these schemes may engender the rank deficiency. Heterosis elements exhibit the overstiffening effect for the problem with irregular mesh. The stabilization methods involve certain parameters which still lack appropriate physical interpretations. Discrete Kirchhoff elements do not include the transverse shear deformation effect by incorporating the Kirchhoff hypothesis and therefore still circumvent the membrane locking. There is still no general consensus in favor of a particular approach due to its aforementioned inherent limitations.
Lately, Dvorkin and Bathe, Park and Stanley, Huang and Hinton, Jang and Pinsky have developed various elements based on the assumed strain methods. The first two elements seem to be among the best available 4 node shell elements. The four node element proposed by Park and Stanley uses the assumed strains for all the strain components, i.e. membrane, bending and transverse shear strains. But because the assumed bending strain cannot satisfy the rigid body motion condition, this element locks the solution for a pinched hemispherical shell problem unless the reduced one-point integration is invoked. And this element has a restriction on transverse shear deformations and the application to the laminated composite shell because of the thickness pre-integration. Another four node element developed by Dvorkin and Bathe uses assumed strains only for the transverse shear components, but not for the membrane components because the membrane strain cannot be decoupled from the in-plane strain in their element. Therefore, this element is still overstiff for the membrane behavior.
In this paper, a simple and efficient four node degenerated shell element which overcomes aforementioned drawbacks is proposed. This element uses the assumed strain concept in terms of covariant strains referred to the element natural coordinate system. In formulation of the element, the assumed shear strains are used to avoid the shear locking. The covariant membrane strains are separated from the covariant in-plane strains which consist of the combination of the membrane and bending strains by mid-surface interpolation. With these separated strains, it is now possible to use the assumed membrane strains to alleviate the membrane locking by eliminating the violation in the rigid body motion condition caused by applying assumed bending strains.
Since this element is used under the three dimensional stress-strain condition, this element is applicable to both thin and thick shells and can be easily implemented into the existing finite element analysis program by minimum modification. Numerical examples are presented to evaluate the performance of the new element developed. The proposed element has no hourglass modes and the numerical results indicate that this element has a rapid convergence, provides a reasonable accuracy for the stresses, and also improves the membrane bending behavior of the shell.
Subsequently the new element is also extended to the geometrically or materially nonlinear analysis of shell structures undergoing large deformation in conjunction with the total Lagrangian scheme. The nonlinear equilibrium equations are solved by using the Newton-Raphson method combined with the arc-length control. Numerical tests show that this new degenerated shell element provides robustness and accuracy for nonlinear problems.