Page 315 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

302 Introduction to the Finite Element Method Chap. 10

I u I
k n
-v w w ----- ^ ^F^ku

EA/i n
■F={FA/l)u
Figure 10.1-1.
10.1 ELEMENT STIFFNESS AND MASS

Axial element. An element with pinned ends can support only axial forces
and hence will act like a spring. Figure 10.1-1 shows a spring and a uniform rod
pinned to a stationary wall and subjected to a force F. The force-displacement
relationships for the two cases are simply
Spring f = ku
( 10.1-1)
Uniform rod

In general, these axial elements can be a part of a pin-connected structure
that allows displacement of both ends. In the finite element method, the displace­
ment and force at each end of the element must be accounted for with proper sign.
Figure 10.1-2 shows an axial element labeled with displacements Wj, U2, and forces
Fj, F2, all in the positive sense. If we write the force-displacement relationship in
terms of the stiffness matrix, the equation is

( 10.1-2)
^22
The elements of the first column of the stiffness matrix represent the forces at the
two ends when = \ and U2 = 0, as shown in Fig. 10.1-3. Thus, = ku^ and
F2 = —kuy
Similarly, by letting U2 = I and = 0, we obtain, as in Fig. 10.1-4, F^ =
-ku2 and F2 = ku2. Thus, Eq. (10.1-2) becomes

(10.1-2')

1 k 2
r -^2
F F Figure 10.1-2.

pr k
—- -.... i
i/,=t
"2' Figure 10.1-3.