Keywords: finite element, object-oriented, plate, patch test.

The classical Heterosis plate finite element [1], with different interpolation functions for transversal displacements and for rotations, that uses selective reduced integration, is an isoparametric element with excellent performance. However, it satisfies only the "weak" form of the patch test. This paper presents the formulation of a modified Heterosis element that due to the object-oriented finite element architecture has an easy implementation despite the need for selective integration and modified interpolation functions. The modified element passes the "strong" form (for arbitrary shape) of the patch test. Its performance is compared to the original Heterosis plate element. The element as a part of a shell element was successfully applied to model various structures, including layered wood-concrete partial composite floors and steel beams.

The use of the 2x2 uniform reduced integration Gaussian quadrature has been a standard technique [1] to suppress locking for membrane and transverse shear strains of the eight node shell elements, however two non-communicating spurious modes (one for membrane, one for bending) are introduced. When 2x2 uniform reduced integration is used with nine node shell elements seven spurious modes (five are communicating) are introduced representing a serious disadvantage compared to the eight node shell elements. However, an advantage of the nine node element over the eight node element is that it can correctly interpolate quadratic displacement states.

An improved 9-node Mindlin plate element [2], named Heterosis, involving Lagrange interpolation functions for bending and serendipity interpolation functions for transverse shear was introduced. Later, it was used [3] as a component of a shell element. A selective integration technique was adopted, using 3x3 full integration quadrature for bending and 2x2 reduced integration quadrature for membrane and transverse shear. As a result, the advantages of the nine-node Lagrange and of the eight-node serendipity elements were incorporated in the Heterosis element. The one non-communicative spurious mode induced is not a significant deficiency. While the Heterosis element with bilinear element geometry proved to be an element with excellent performance [3], the element does not pass the constant curvature patch test for arbitrary element shape. However, it passes the patch test for parallelogram element shape that is regarded as the weak form of the test.

In an effort to develop an eight node shell element with bilinear geometry that can correctly represent quadratic displacement states the use of hybrid, metric and parametric, interpolation was suggested in Reference [4], and was applied to this study.

In this paper the hybrid interpolation is applied to the transverse displacement of the Heterosis plate element in the development of the "modified Heterosis" element that correctly interpolates any quadratic transverse displacement imposed at the external nodes, and inherits the advantages [3] of the original Heterosis element. The modified Heterosis plate element was implemented into an object-oriented system for structural analysis and design developed by the author, as a descendant of the original Heterosis plate element's class.

1

Zienkiewicz, O.C., "The Finite Element Method in Engineering Science", McGraw-Hill, London, 1971.

2

Hughes, T.J.R., Cohen, M., "The "Heterosis" Finite Element for Plate Bending", Computer and Structures, Vol. 9, pp. 445-450, Pergamon Press Ltd., 1978. doi:10.1016/0045-7949(78)90041-X

3

Hughes, T.J.R., "The Finite Element Method - Linear Static and Dynamic Finite Element Analysis", Dover Publications, Mineola, NY, 2000.

4

MacNeal, R.H., Harder, R.L., "Eight Nodes or Nine?", International Journal Numerical Methods Eng., No. 33, pp. 1049-1058, 1992. doi:10.1002/nme.1620330510

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