Civil Engineering Reference
In-Depth Information
and the total angle of twist would be
8(10
−
6
)(1000 )
8(10
−
3
) rad
=
L
=
1
.
=
1
.
or about 0.1 degree. The exact solution [5] for this problem shows the angle of twist per
unit length to be
=
1
.
42(10
−
6
) rad/mm
.
Hence, our very simple model is in error by
about 27 percent.
9.10 SUMMARY
In this chapter, we present the development of the most basic finite elements used
in stress analysis in solid mechanics. As the finite element method was originally
developed for stress analysis, the range of element and problem types that can be
analyzed by the method are very large. Our description of the basic concepts is
intended to give the reader insight on the general procedures used to develop
element equations and understand the ramifications on element, hence, model,
formulation for various states of stress. As mentioned in the context of plate
bending in Section 9.8, finite element analysis involves many advanced topics in
engineering not generally covered in an undergraduate program. The interested
reader is referred to the many advanced-level texts on the finite element method
for further study. The intent here is to introduce the basic concepts and generate
interest in learning more of the subject.
REFERENCES
1.
Kreyszig, E.
Advanced Engineering Mathematics,
5th ed. New York: John Wiley
and Sons, 1983.
2.
Shigley, J. E.
Mechanical Engineering Design,
6th ed. New York: McGraw-Hill,
1998.
3.
Cowper, G. R. “Gaussian Quadrature Formulas for Triangles.”
International
Journal for Numerical Methods in Engineering
7 (1973).
4.
Huebner, K. H., and E. A. Thornton.
The Finite Element Method for Engineers,
2nd ed. New York: John Wiley and Sons, 1982.
5.
Timoshenko, S. P., and J. N. Goodier.
Theory of Elasticity,
New York: McGraw-
Hill, 1970.
PROBLEMS
9.1
Use the general stress-strain relations from Appendix B and the assumptions of
plane stress to derive Equations 9.2.
z
N
]
T
9.2
Let
{
z
}
be the
N
×
1 column matrix [
z
1
z
2
z
3
···
and let
[
A
]
be an
N
×
T
[
A
]
{
z
}
always results in
N
real-valued matrix. Show that the matrix product
{
z
}
a scalar, quadratic function of the components
z
i
.
