Triangular plate-bending on three point supports

This problem illustrates triangular plate-bending on three point supports. The results are compared to the theoretical solution.

This page discusses:

Problem description
Results and discussion
Input files
References
Tables
Figures

ProductsAbaqus/Standard

Problem description

Two meshes, coarse and fine, are considered. The coarse mesh is discretized with seven nodes and either six 3-node elements or three 4-node elements. The fine mesh is discretized with nineteen nodes and either twenty-four 3-node elements or twelve 4-node elements. The material is linear elastic with a Young's modulus of 207 × 10⁹ and a Poisson's ratio of 0.25. The plate has a thickness, t, of 0.00254. c = 0.138564 and l = 0.24. There are three corner-point supports in the z-direction. A uniform distributed tangential moment of 300/length and a linear distributed twisting moment of 194.85/length are applied on each boundary.

Results and discussion

The equivalent nodal moments are calculated at nodes 1, 2, 3, 6, and 7. The vertical displacement of the centroid, node 7, is also calculated. The theoretical solution is given in Table 1, where the equivalent nodal moments have been calculated by applying the principle of virtual displacements with a linear function for the rotation corresponding to the tangential moment and a quadratic function for the rotation corresponding to the twisting moment. Results for the coarse meshes are given in Table 2 to Table 5, and results for the fine meshes are given in Table 6 to Table 9. For the mesh densities used and due to the extrapolation of integration point quantities, the nodal moments show sizable errors compared to the theoretical solution. The predicted centroidal displacements are larger than the theoretical value, approaching the theoretical values as the mesh density increases.

Input files

ese4sfsh.inp: S4 elements, fine mesh.
ese4smsh.inp: S4 elements, coarse mesh.
esf4sfsh.inp: S4R elements, fine mesh.
esf4smsh.inp: S4R elements, coarse mesh.
es54sfsh.inp: S4R5 elements, fine mesh.
es54smsh.inp: S4R5 elements, coarse mesh.
es63sfsh.inp: STRI3 elements, fine mesh.
es63smsh.inp: STRI3 elements, coarse mesh.

References

Robinson, J., “Triangular Plate-Bending on Three Point Supports,” Finite Element News, no. 1, 1992.

Tables

Table 1. Theoretical solution.
NODE	$M_{x}$	$M_{y}$	$M_{x y}$
1	300.0	75.0	194.86
2	−37.7	412.5	0.0
3	300.0	75.0	−194.86
6	300.0	75.0	0.0
7	187.5	187.5	0.0
Centroidal displacement = 2.1226 × 10⁻³

Table 2. S4 elements, coarse mesh.
NODE	$M_{x}$	$M_{y}$	$M_{x y}$
1	194.7	64.38	−112.9
2	−0.7924	259.9	0.0
3	194.7	64.38	112.9
6	273.0	73.07	0.0
7	303.4	303.4	0.0
Centroidal displacement = 3.6602 × 10⁻³

Table 3. S4R elements, coarse mesh.
NODE	$M_{x}$	$M_{y}$	$M_{x y}$
1	243.0	132.0	−96.16
2	76.47	298.5	0.0
3	243.0	132.0	96.16
6	243.0	132.0	0.0
7	187.5	187.5	0.0
Centroidal displacement = 3.2232 × 10⁻³

Table 4. S4R5 elements, coarse mesh.
NODE	$M_{x}$	$M_{y}$	$M_{x y}$
1	243.6	131.4	−97.11
2	75.38	299.6	0.0
3	243.6	131.4	97.11
6	243.6	131.4	0.0
7	187.5	187.5	0.0
Centroidal displacement = 3.1924 × 10⁻³

Table 5. STRI3 elements, coarse mesh.
NODE	$M_{x}$	$M_{y}$	$M_{x y}$
1	101.5	251.5	−129.9
2	326.4	26.56	0.0
3	101.5	251.5	129.9
6	50.33	355.5	0.0
7	183.1	183.1	0.0
Centroidal displacement = 2.7551 × 10⁻³

Table 6. S4 elements, fine mesh.
NODE	$M_{x}$	$M_{y}$	$M_{x y}$
1	233.3	59.27	−151.5
2	−9.773	332.3	0.0
3	233.3	59.27	151.5
6	275.7	71.22	0.0
7	240.1	247.5	0.0
Centroidal displacement = 2.5038 × 10⁻³

Table 7. S4R elements, fine mesh.
NODE	$M_{x}$	$M_{y}$	$M_{x y}$
1	260.9	102.0	−139.4
2	19.36	352.0	0.0
3	260.9	102.0	139.4
6	273.7	108.2	0.0
7	184.7	191.2	0.0
Centroidal displacement = 2.4042 × 10⁻³

Table 8. S4R5 elements, fine mesh.
NODE	$M_{x}$	$M_{y}$	$M_{x y}$
1	261.3	101.1	−140.6
2	18.91	353.9	0.0
3	261.3	101.1	140.6
6	273.6	108.7	0.0
7	184.6	191.3	0.0
Centroidal displacement = 2.4022 × 10⁻³

Table 9. STRI3 elements, fine mesh.
NODE	$M_{x}$	$M_{y}$	$M_{x y}$
1	83.59	272.1	−160.3
2	370.3	−8.347	0.0
3	83.59	272.1	160.3
6	62.72	333.9	0.0
7	183.6	187.4	0.0
Centroidal displacement = 2.3259 × 10⁻³

Figures