Civil Engineering Reference
In-Depth Information
and the final stiffness matrix for a general 3-D beam element is observed to be a
12
12
symmetric matrix composed of the individual stiffness matrices repre-
senting axial loading, two-plane bending, and torsion.
The general beam element can be utilized in finite element analyses of three-
dimensional frame structures. As with most finite elements, it is often necessary
to transform the element matrices from the element coordinate system to the
global coordinates. The transformation procedure is quite similar to that dis-
cussed for the bar and two-dimensional beam elements, except, of course, for the
added algebraic complexity arising from the size of the stiffness matrix and
certain orientation details required.
×
4.9 CLOSING REMARKS
In this chapter, finite elements for beam bending are formulated using elastic
flexure theory from elementary strength of materials. The resulting elements are
very useful in modeling frame structures in two or three dimensions. A general
three-dimensional beam element including axial, bending, and torsional effects
is developed by, in effect, superposition of a spar element, two flexure elements,
and a torsional element.
In development of the beam elements, stiffening of the elements owing to
tensile loading, the possibility of buckling under compressive axial loading, and
transverse shear effects have not been included. In most commercial finite
element software packages, each of these concerns is an option that can be taken
into account at the user's discretion.
REFERENCES
1.
Beer, F. P., E. R. Johnston, and J. T. DeWolf.
Mechanics of Materials,
3rd ed.
New York: McGraw-Hill, 2002.
2.
Budynas, R.
Advanced Strength and Applied Stress Analysis,
2nd ed. New York:
McGraw-Hill, 1999.
PROBLEMS
4.1
Two identical beam elements are connected at a common node as shown in
Figure P4.1. Assuming that the nodal displacements
v
i
,
i
are known, use
Equation 4.32 to show that the normal stress
x
is, in general, discontinuous
at the common element boundary (i.e., at node 2). Under what condition(s)
would the stress be continuous?
1
2
3
Figure P4.1
