Shear-loaded Cantilever Beam
[ Problem Description ]
[ Finite Element Analyses ]
[ Input and Output Files ]
The purposes of this example are:
In our numerical experiment, a shear-loaded cantilever beam of
length L = 48", height h = 12", width w = 1", is loaded
with force P = 40,000 lb at the end.
See the upper-most diagram in Figure 1.
Figure 1 : Finite Element Mesh for a Short Cantilever Beam.
The cantilever beam is constructed from one material type --
Young's Modulus E = 30000 ksi and Poisson's Ratio = 0.25.
From elasticity, the analytical solution for the tip displacement is
w = 0.3553 (in)
Finite element solutions are computed for:
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A mesh of four square elements (as shown in Figure above);
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Finer meshes of rectangular finite elements constructed by bisection;
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Irregular meshes of four and sixteen quadrilateral
elements, as shown in the lower sections of Figure 1.
Table 1 summarizes the numerical results,
with the asterisk (*) denoting the irregular mesh.
Meshes | 4 X 1 | 8 X 2 | 16 X 4 | 4 X 1* |
8 X 2* |
ALADDIN's Shell Element |
0.3445 | 0.3504 | 0.3543 | 0.3066 | 0.3455 |
Error to Theoretical Solutions |
3.039% | 1.379% | 0.282% | 13.707% | 2.758% |
ANSYS-5.0 Shell Element |
0.2424 | 0.3162 | 0.3449 | 0.2126 | 0.2996 |
Sabir's Element |
0.3281 | 0.3454 | 0.3527 | --- | --- |
Allman's Element |
0.3026 | 0.3394 | 0.3512 | --- | --- |
Bilinear Element |
0.2424 | 0.3162 | 0.3447 | --- | --- |
Table 1 : Summary of Tip Displacements for various Finite Element Meshs.
The Sabir finite element [2] is a rectangular element with the drilling degree of freedom.
The Allman finite element [1] is a rectangular element with the vertex rotation.
The bilinear element is a rectangular constant strain element without any
nodal rotational degree of freedom.
Conclusions
The numerical results from this experiment suggest that:
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With the same regular meshes, the Shell Finite Element with Drilling Degree of Freedom
gives more accurate results than other shell finite elements in the literature.
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For the same irregularly shaped meshes,
this shell element provides much greater accuracy than shell element of ANSYS-5.0.
Readers should note that the latter is a four node flat shell
element having six degrees of freedom per node in
which a drilling degree of freedom based on an approach
suggested by Kanok-Nukulchai is included.
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The numerical results also suggest that this shell element gives reasonably
accurate and rapidly convergent results with distorted meshes.
References
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Allman D.J., "A Quadrilateral Finite Element Including Vertex Rotations
for Plane Elasticity Analysis,"
International Journal for Numerical Methods in Engineering,
Vol. 26, 1988, pp. 2645-2655.
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Sabir A.B., "A Rectangular and a Triangular Plane Elasticity Element
with Drilling Degrees of Freedom,"
Proceedings of the Second International Conference on Variational Methods in Engineering,
Brebbia C.A. (ed.), Southhampton University, July 1985, Springer-Verlag,
Berlin, 1985. pp. 17-25.
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Click here to visit the complete input file
for the 8x2 irregularly shaped finite element mesh.
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Click here to visit the complete output
file.
Developed in July 1996 by Lanheng Jin & Mark Austin
Last Modified September 27, 1996
Copyright © 1996, Lanheng Jin and Mark Austin, Department of Civil
Engineering, University of Maryland