Shear-loaded Cantilever Beam

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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:

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.


The numerical results from this experiment suggest that:


  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.


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