Shearloaded Cantilever Beam
[ Problem Description ]
[ Finite Element Analyses ]
[ Input and Output Files ]
The purposes of this example are:
In our numerical experiment, a shearloaded 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 uppermost 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:

A mesh of four square elements (as shown in Figure above);

Finer meshes of rectangular finite elements constructed by bisection;

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% 
ANSYS5.0 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:

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.

For the same irregularly shaped meshes,
this shell element provides much greater accuracy than shell element of ANSYS5.0.
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 KanokNukulchai is included.

The numerical results also suggest that this shell element gives reasonably
accurate and rapidly convergent results with distorted meshes.
References

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. 26452655.

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, SpringerVerlag,
Berlin, 1985. pp. 1725.

Click here to visit the complete input file
for the 8x2 irregularly shaped finite element mesh.

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