Mindlin plate elements are chosen which can be formed as an extension of the solutions used for the Kirchoff plate theory. The transverse displacements for the Mindlin element correspond to those of the Kirchoff element, but with no rotations. Two adequate independent rotation fields must be included. This simple approach will be less effective then the modern mixed formulations [2], but it seems suitable for basic experiments, and it is possible that fully compatible solutions of controled precision could be produced and used for comparison in cases for which exact solutions do not exist.

The first element used in the comparison is a sixteen node rectangular Lagrangian element with a bicubic polynomial expansion which is incomplete. The second element is a ten node full cubic triangular element. The same nodal mesh can be used for both types of element. The elements satisfy only minimal continuity requirements. As all three kinematic fields are approximated with the same function type, the element can exactly reproduce the field of the basic Kirchoff plate element. The comparison is based on two cases for which exact solutions are possible.

For the numerical experiment a quadratic plate with the base 20 was chosen. Five values of the plate thickness were considered: = 4.0; 2.0; 1.0; 0.4; 0.2. The material constants were: E = 15.0, =0.25. The radius of the loaded centered circle was = 1.0, and the load intensity p = 2.0. The calculations were performed on a symmetric quarter of the plate with a regular quadratic meshes with up to 1600 nodal points.

The influence of the concentration of the transverse loading on a relatively small area is tested by using the solution for a centrally loaded circular plate from which a rectangular subdomain is interpreted as a clamped rectangular plate with prescribed boundary displacements and rotations [3].

The approximation for deflections and bending moments in the midpoint showed the typical characteristic of a strict kinematic formulation. The transversal forces on the boundary of the plate show a similar precision as the bending moments. Weaker results are obtained for the transverse force for points lying on the border of the loaded area; especially if the point is not on the border of a finite element. Also in this test the approximation with the incomplete polynomial has shown to give slightly better results.

The sensitivity of the approximations to the typical corner concentrations is tested on a plate loaded only on the boundaries. Two opposite sides have "hard hinge" boundary conditions, and the two remaining sides have "soft hinge" boundary conditions. On these sides the loading is cosinusoidally distributed torsional moments [4].

The results for three characteristic values for the corner points are presented; the rotation, the torsional moment calculated for the finite element, the transversal force in the hard hinge. All three values have their extremum in this point. The approximation of the rotation shows the smallest errors, and the errors are largest for the torsional moment. The concentration of the transversal forces is in the Kirchoff theory substituted with concentrated forces, and was often discussed and analysed. Here a comparison with exact values is given. For the plate with the hight/base = 1/100 and a numerical model with approximately 40000 degrees of freedom, an error for the peek of the transversal forces of under 1% was achieved.

1

Zienkiewicz, O.C., Taylor, R.L., The Finite Element Method, Butterworth-Heinemann, London 2000.

2

Taylor, R.L., Govidjee, S., A Quadratic Linked Plate Element With an Exact Thin Plate Limit, Technical Report: UCB/SEMM-2002/10.

3

Werner, H., Lazarevic, D., Fresl, K., "Comparison of the exact rotational-symmetric Mindlin plate solutions with FEM solutions defined on a rectangular domain", 12th ACME Annual Conference Cardiff 2004,pp 67-70.

4

Werner, H., Lazarevic, D., Fresl, K., "Boundary Torsion Benchmark for Mindlin Plate Numerical Solutions", XXV Iberian Latin American Congress on Computational Methods in Engineering, Recifee, November 2004. pp117-118.

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