### Tech Briefs

Remarks on the Shell Elements in ADINA

The shell elements available in ADINA are based on using Mixed Interpolated Tensorial Components and are therefore referred to as MITC elements. The elements have been developed over the last two decades and have been analysed extensively in numerical experiments and mathematically, see references below (and the references therein).

Considering overall reliability and effectiveness, the elements represent the best, most powerful, shell elements available in the field. The elements in ADINA v. 8.3 are:

- The MITC4 element is the 4-node shell element. An option is available, whereby this element can also be used with incompatible modes to improve the in-plane bending behavior. This is the most-used element.

- The MITC3 element is the 3-node shell element.This is a spatially isotropic element and is generally used with the MITC4 element when triangular elements are needed in the mesh.

- The MITC9 element is the 9-node shell element; the 9th node is the center node and does not couple into adjacent elements, hence the element can also be accessed by pre- and post-processors as an 8-node element.

- The MITC16 element is the 16-node shell element. It is computationally the most expensive element, but not very expensive, and can be useful when high accuracy needs to be achieved with coarse meshes.

The ADINA mesh generator is able to generate shell element meshes that contain almost only quadrilateral elements, even for very complex shell structures. Mostly, the quadrilateral MITC4 element is used but the quadrilateral MITC9 and MITC16 elements can also be very useful.

All elements of course do not contain any spurious zero energy mode and show good convergence behavior.

All these elements can be used in linear analysis, and in large displacement and large strain analyses, e.g. in the simulations of collapse of shells and metal forming problems.

The proper numerical and mathematical evaluations of shell elements have been addressed in the references below, where it is also shown that the simple test problems frequently used in the literature are not sufficient to identify whether a shell element is really reliable and effective.

Well-chosen additional test problems have been proposed and solved in the references. While these additional test problems should be solved in order to fully see the reliability and effectiveness of a shell element, we present here the results obtained in a common test problem, the pre-twisted cantilever beam problem, frequently used in the literature — merely to demonstrate the predictive capabilities of the elements in a well-known linear analysis test.

The figures below show the pre-twisted cantilever beam problem considered and the calculated tip displacement (normalized to the solution obtained with a very fine MITC9 element mesh) using the various shell elements in ADINA. Excellent predictions even with coarse meshes are obtained. It is remarkable that the model of only two MITC9 shell elements gives an error of only 2 percent, and so does the model of only one single MITC16 shell element!

Three different views of the 1x2 MITC9 element mesh of the undeformed pre-twisted cantilever beam,

L = 12 m, b = 1.1 m, thickness = 0.0032 m, in-plane shear load in z-direction at the tip

K. J. Bathe, *Finite Element Procedures*, Prentice Hall, 1996.

D. Chapelle and K. J. Bathe, *The Finite Element Analysis of Shells — Fundamentals*, Springer, 2003.

E. Dvorkin and K. J. Bathe, “A Continuum Mechanics Based Four-Node Shell Element for General Nonlinear Analysis”, *Engineering Computations*, __1__, 77-88, 1984.

M. L. Bucalem and K. J. Bathe, “Higher-Order MITC General Shell Elements”, *Int. J. for Numerical Methods in Engineering*, __36__, 3729-3754, 1993.

D. Chapelle and K. J. Bathe, “Fundamental Considerations for the Finite Element Analysis of Shell Structures”, *Computers & Structures*, __66__, no. 1, 19-36, 711-712, 1998.

P. S. Lee and K. J. Bathe, “On the Asymptotic Behavior of Shell Structures and the Evaluation in Finite Element Solutions”, *Computers & Structures*, __80__, 235-255, 2002.

J. F. Hiller and K. J. Bathe, “Measuring Convergence of Mixed Finite Element Discretizations: An Application to Shell Structures”, *Computers & Structures*, __81__, 639-654, 2003.

P. S. Lee and K. J. Bathe, “Development of MITC Isotropic Triangular Shell Finite Elements”, *Computers & Structures*, __82__, 945-962, 2004.