# Shear-loaded Cantilever Beam

### PROBLEM DESCRIPTION

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 ANALYSES

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.

Meshes4 X 18 X 216 X 44 X 1* 8 X 2*
ALADDIN's Shell Element 0.34450.35040.35430.30660.3455
Error to Theoretical Solutions 3.039%1.379%0.282%13.707%2.758%
ANSYS-5.0 Shell Element 0.24240.31620.34490.21260.2996
Sabir's Element 0.32810.34540.3527------
Allman's Element 0.30260.33940.3512------
Bilinear Element 0.24240.31620.3447------

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

References

1. 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.
2. 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.

### INPUT AND OUTPUT FILES

• 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