Civil Engineering Reference
In-Depth Information
We now reexamine the energy equation of the Example 2.7 to develop a more-
general form, which will be of significant value in more complicated systems to
be discussed in later chapters. The system or global displacement vector is
U
1
U
2
U
3
U
4
{
U
} =
(2.56)
and, as derived, the global stiffness matrix is
k
1
−
k
1
0
0
−
k
1
k
1
+
2
k
2
−
2
k
2
0
[
K
]
=
(2.57)
0
−
2
k
2
2
k
2
+
k
3
−
k
3
0
0
−
k
3
k
3
If we form the matrix triple product
1
2
{
1
2
[
U
1
T
[
K
]
U
}
{
U
}=
U
2
U
3
U
4
]
k
1
−
k
1
0
0
U
1
U
2
U
3
U
4
−
k
1
k
1
+
2
k
2
−
2
k
2
0
×
(2.58)
0
−
2
k
2
2
k
2
+
k
3
−
k
3
0
0
−
k
3
k
3
and carry out the matrix operations, we find that the expression is identical to the
strain energy of the system. As will be shown, the matrix triple product of Equa-
tion 2.58 represents the strain energy of any elastic system. If the strain energy
can be expressed in the form of this triple product, the stiffness matrix will have
been obtained, since the displacements are readily identifiable.
2.6 SUMMARY
Two linear mechanical elements, the idealized elastic spring and an elastic tension-
compression member (bar) have been used to introduce the basic concepts involved in
formulating the equations governing a finite element. The element equations are obtained
by both a straightforward equilibrium approach and a strain energy method using the first
theorem of Castigliano. The principle of minimum potential also is introduced. The next
chapter shows how the one-dimensional bar element can be used to demonstrate the finite
element model assembly procedures in the context of some simple two- and three-
dimensional structures.
REFERENCES
1.
Budynas, R.
Advanced Strength and Applied Stress Analysis.
2d ed. New York:
McGraw-Hill, 1998.
2.
Love, A. E. H.
A Treatise on the Mathematical Theory of Elasticity.
New York:
Dover Publications, 1944.
