Results.
| Example | Reference | Records | Largest deviation |
|---|---|---|---|
| Straight cantilever, rectangular elements | Published source | 3 | 9.84% |
| Straight cantilever, trapezoidal elements | Published source | 3 | 19.70% |
| Straight cantilever, parallelogram elements | Published source | 3 | 13.40% |
| Straight cantilever, in-plane shear load, mesh refinement | Published source | 4 | 9.84% |
| Curved beam | Published source | 8 | 6.63% |
| Twisted beam | Published source | 8 | 0.373% |
Problem. The beam problems of MacNeal and Harder (1985) are standard tests proposed to measure how a shell element behaves on coarse and distorted meshes. The straight cantilever is solved with rectangular, trapezoidal and parallelogram elements, and the curved and twisted beams are loaded at their tips. Refined meshes of the same problems are given beside the standard mesh.
Reference document. The reference values are the beam theory values given by MacNeal and Harder (1985). As the paper could not be accessed directly, the values were confirmed from secondary sources that quote it and from the Abaqus benchmarks manual.
The tip displacement of the straight cantilever under a shear load is the sum of the bending and shear contributions.
Numerical model. The straight cantilever is 6.0 × 0.2 × 0.1 and is modelled on the standard mesh with 6 × 1 quadrilateral elements; in the distorted meshes the inner edges are inclined at 45°. The curved beam is a 90° arc with inner radius 4.12 and outer radius 4.32, and the twisted beam is 12 long and twisted through 90° end to end. Every beam is fixed at one end and carries a unit load at the other.
Straight cantilever, rectangular elements
Parameters. mesh = 6×1
Published source. MacNeal and Harder (1985), A proposed standard set of problems to test finite element accuracy
| Quantity | Reference | Computed | Deviation |
|---|---|---|---|
| Tip displacement, axial load | 0.00003 | 0.0000300136 | 0.0452% |
| Tip displacement, in-plane shear load | 0.1081 | 0.0974613 | 9.84% |
| Tip displacement, out-of-plane shear load | 0.4321 | 0.423486 | 1.99% |
Note. On the standard 6 × 1 mesh there is a single element through the depth, so the result under the in-plane load stays below the reference. The result approaches the reference as the mesh is refined, and the tolerances are a regression band that locks today's value. The 2 to 4% difference under the out-of-plane load does not come from the mesh. For a thick, narrow section shell theory gives a slightly different bending and torsional stiffness from the beam theory behind the reference, and the difference does not vanish as the mesh is refined.
Straight cantilever, trapezoidal elements
Parameters. mesh = 6×1
Published source. MacNeal and Harder (1985), A proposed standard set of problems to test finite element accuracy
| Quantity | Reference | Computed | Deviation |
|---|---|---|---|
| Tip displacement, axial load | 0.00003 | 0.0000300053 | 0.0178% |
| Tip displacement, in-plane shear load | 0.1081 | 0.0867999 | 19.70% |
| Tip displacement, out-of-plane shear load | 0.4321 | 0.416205 | 3.68% |
Note. On the standard 6 × 1 mesh there is a single element through the depth, so the result under the in-plane load stays below the reference. The result approaches the reference as the mesh is refined, and the tolerances are a regression band that locks today's value. The 2 to 4% difference under the out-of-plane load does not come from the mesh. For a thick, narrow section shell theory gives a slightly different bending and torsional stiffness from the beam theory behind the reference, and the difference does not vanish as the mesh is refined.
Straight cantilever, parallelogram elements
Parameters. mesh = 6×1
Published source. MacNeal and Harder (1985), A proposed standard set of problems to test finite element accuracy
| Quantity | Reference | Computed | Deviation |
|---|---|---|---|
| Tip displacement, axial load | 0.00003 | 0.0000300003 | 0.0010% |
| Tip displacement, in-plane shear load | 0.1081 | 0.0936433 | 13.40% |
| Tip displacement, out-of-plane shear load | 0.4321 | 0.422572 | 2.20% |
Note. On the standard 6 × 1 mesh there is a single element through the depth, so the result under the in-plane load stays below the reference. The result approaches the reference as the mesh is refined, and the tolerances are a regression band that locks today's value. The 2 to 4% difference under the out-of-plane load does not come from the mesh. For a thick, narrow section shell theory gives a slightly different bending and torsional stiffness from the beam theory behind the reference, and the difference does not vanish as the mesh is refined.
Straight cantilever, in-plane shear load, mesh refinement
Published source. MacNeal and Harder (1985), A proposed standard set of problems to test finite element accuracy
| Quantity | Condition | Reference | Computed | Deviation |
|---|---|---|---|---|
| Tip displacement, in-plane shear load | mesh = 6×1 | 0.1081 | 0.0974613 | 9.84% |
| Tip displacement, in-plane shear load | mesh = 12×2 | 0.1081 | 0.10497 | 2.90% |
| Tip displacement, in-plane shear load | mesh = 24×4 | 0.1081 | 0.107143 | 0.885% |
| Tip displacement, in-plane shear load | mesh = 48×8 | 0.1081 | 0.107751 | 0.323% |
Curved beam
Published source. MacNeal and Harder (1985), A proposed standard set of problems to test finite element accuracy
| Quantity | Condition | Reference | Computed | Deviation |
|---|---|---|---|---|
| Tip displacement, in-plane load | mesh = 6×1 | 0.08734 | 0.0815463 | 6.63% |
| Tip displacement, in-plane load | mesh = 12×1 | 0.08734 | 0.0828022 | 5.20% |
| Tip displacement, in-plane load | mesh = 24×1 | 0.08734 | 0.0829889 | 4.98% |
| Tip displacement, in-plane load | mesh = 48×1 | 0.08734 | 0.0830307 | 4.93% |
| Tip displacement, out-of-plane load | mesh = 6×1 | 0.5022 | 0.473197 | 5.78% |
| Tip displacement, out-of-plane load | mesh = 12×1 | 0.5022 | 0.47962 | 4.50% |
| Tip displacement, out-of-plane load | mesh = 24×1 | 0.5022 | 0.481879 | 4.05% |
| Tip displacement, out-of-plane load | mesh = 48×1 | 0.5022 | 0.482642 | 3.89% |
Note. The mesh is refined only along the beam axis, so a single row of elements remains through the depth. The result under the in-plane load therefore settles about 5% below the reference. The 4 to 6% difference under the out-of-plane load does not come from the mesh either. For a thick, narrow section shell theory gives a slightly different bending and torsional stiffness from the beam theory behind the reference. The tolerances are a regression band.
Twisted beam
Published source. MacNeal and Harder (1985), A proposed standard set of problems to test finite element accuracy
| Quantity | Condition | Reference | Computed | Deviation |
|---|---|---|---|---|
| Tip displacement, load in the plane of the tip section | mesh = 12×2 | 0.005424 | 0.00540378 | 0.373% |
| Tip displacement, load in the plane of the tip section | mesh = 24×4 | 0.005424 | 0.00541218 | 0.218% |
| Tip displacement, load in the plane of the tip section | mesh = 48×8 | 0.005424 | 0.00541603 | 0.147% |
| Tip displacement, load in the plane of the tip section | mesh = 96×16 | 0.005424 | 0.00541751 | 0.120% |
| Tip displacement, load normal to the tip section | mesh = 12×2 | 0.001754 | 0.00174974 | 0.243% |
| Tip displacement, load normal to the tip section | mesh = 24×4 | 0.001754 | 0.00175166 | 0.134% |
| Tip displacement, load normal to the tip section | mesh = 48×8 | 0.001754 | 0.00175269 | 0.0744% |
| Tip displacement, load normal to the tip section | mesh = 96×16 | 0.001754 | 0.00175316 | 0.0478% |