Keywords: plate elements, shear flexibility, Reissner-Mindlin plate theory, shear locking, balanced interpolation.

Plate bending elements have been and still are the subject of many papers. Zienkiewicz [1] has given a very good overview of the historical development dating back to around 1965. In this paper focus will be entirely on plate elements taking the shear flexibility into account i.e. using Reissner-Mindlin plate theory. Two main approaches have been used either a displacement based approach or a hybrid interpolation approach. In this paper the focus is entirely on the displacement based method. In Reissner-Mindlin plate theory the shear forces are described independently of the bending moments in opposition to the Kirchoff-Love plate theory. That means for a displacement based method both the transverse displacement w and the rotations of cross-sections of the plate theta_x and theta_y have to be interpolated separately.

A very important issue in shear flexible elements is the risk of shear locking which can lead to erroneous results. Shear locking will typically show up when the plate becomes thin and the results will not be similar to the results for an equivalent analysis using Kirchoff-Love theory. Another important issue in the displacement based method is either to have fully compatible elements or at least full-filling the patch-test i.e. that the incompatibilities at the element boundaries are sufficiently small.

In the paper a new triangular plate bending element with linear bending stress is formulated. The trend in finite elements have been towards relatively simple elements, but to the authors believe many applications will benefit from a more accurate element. The shear strains are defined by one of the cross-sectional rotations theta_x and theta_y and the derivative of the transverse displacement w with respect to either x or y. In order to have a fully consistent interpolation providing a balanced interpolation of the shear strains we choose a cubic interpolation which will give ten degrees of freedom. The nodal degrees of freedom could be chosen as a transverse displacement at each corner, two at each side placed equidistant and one in the center of the element. However, we have chosen a slightly different approach by replacing the two midside displacements by a transverse displacement and a rotation perpendicular to the side in the mid-side node.

From the interpolation of the twenty-two degrees of freedom the stiffness matrix can be calculated using standard techniques. The transverse displacement in the center can be eliminated at the element level using standard procedures. At the mid-side nodes there will be four degrees of freedom namely a transverse displacement, two rotations of the cross-sections and finally a rotation of the mid-surface. The last degree of freedom is not very practical to implement in commercial systems, but if inter-element continuity is enforced the element would be fully compatible. It can be shown that if the inter-element continuity is not enforced the element will still pass the patch test. By neglecting the inter-element continuity the three mid-side rotations can be eliminated at the element level leading to an element with eighteen degrees of freedom.

1

O.C. Zienkiewicz, R.C. Taylor, "The Finite Element Method", Vol. 2. McGraw-Hill, fourth edition, 1992.

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