Keywords: plate bending, Reissner-Mindlin hypothesis, finite element analysis.

The application of the finite element method to the problem of plate bending has been the subject of intensive research over the past few decades. In particular, a great variety of finite elements have been developed on the basis of the Reissner-Mindlin plate bending theory e.g. [1,2,3,4], which subsumes Kirchhoff's bending theory for very thin plates where transverse shear strains are ignored.

The starting point for this work is based on the fact that, for an elastic plate with uniform thickness and material, the exact solution of the Reissner-Mindlin differential equations includes shear strains if and only if the generalised curvature strains follow a linear or higher order variation over the plate domain. The lowest order exact solution which includes constant shear strains is one where the curvature strains vary linearly and, consequently, the rotations and transverse displacement vary according to quadratic and cubic functions, respectively.

This paper proposes a displacement-based Reissner-Mindlin triangular element for plate bending analysis, which employs cubic and quadratic shape functions for the w and fields, respectively. The tangential transverse shear strain is constrained to a constant value along each of the element edges, and is considered as a degree of freedom which is shared with adjacent elements.

In order to satisfy the constant curvature patch test, two alternative approaches are proposed: i) a conforming approach, where an internal element freedom associated with a bubble function for the w field is included, and ii) a non-conforming approach, where a substitute shear field based on the edge shear freedoms is employed. Significantly, the conforming formulation has the added benefit of representing linear curvature exact solutions, whereas the non-conforming formulation can be applied as a discrete Kirchhoff element by restraining all edge shear freedoms. Consideration is given in the paper to the applicability of the proposed triangular element to thin plate bending analysis, particularly in relation to its susceptibility to shear locking. A new expression is proposed for evaluating the shear locking performance of element meshes, which shows that the non-conforming formulation is completely lock free, whereas the conforming formulation locks only for heavily constrained coarse meshes.

The proposed triangular element has been implemented within ADAPTIC [5], which is used in two numerical examples to investigate the performance of both the conforming and non-conforming types. Both element types are shown to converge to the same solution, comparing favourably against theoretical predictions. However, the conforming type is clearly demonstrated to provide superior convergence characteristics to the non-conforming type, even for very thin plates. In this respect, the fully-integrated conforming element is confirmed to be relatively lock free, with shear locking occurring only for the coarsest of meshes having clamped supports all-round. With the non-conforming element becoming equivalent to a discrete Kirchhoff element at very small thickness, the obtained results illustrate that the conforming triangular element is more powerful than the corresponding discrete Kirchhoff element for thin plate analysis. Obviously, the proposed Reissner-Mindlin triangular element is also applicable to bending analysis of moderately thick plates, the conforming type displaying even better convergence characteristics as the plate thickness is increased.

1

Hughes, T.J.R., Tezduyar, T.E., "Finite Elements Based upon Mindlin Plate Theory with Particular Reference to the Four-Node Bilinear Isoparametric Element", Journal of Applied Mechanics, 48, 587-596, 1981.

2

Bathe, K.J., Dvorkin, E.N., "A Four-Node Plate Bending Element Based on Mindlin-Reissner Plate Theory and a Mixed Interpolation", International Journal for Numerical Methods in Engineering, 21, 367-383, 1985. doi:10.1002/nme.1620210213

3

Zienkiewicz, O.C., Xu, Z., Zeng, L.F., Samuelsson, A., and Wiberg, N., "Linked Interpolation for Reissner-Mindlin Elements: Part I - A Simple Quadrilateral", International Journal for Numerical Methods in Engineering, 36, 3043-3056, 1993. doi:10.1002/nme.1620361802

4

Bathe, K.J., Brezzi, F., Cho, S.W., "The MITC7 and MITC9 Plate Bending Elements", Computers & Structures, 32(3/4), 797-814, 1989. doi:10.1016/0045-7949(89)90365-9

5

Izzuddin, B.A., Nonlinear Dynamic Analysis of Framed Structures, PhD Thesis, Department of Civil Engineering, Imperial College, University of London, 1991.

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