Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

P
B
751
METHOd OF VIRTuAL wORk
P 2
A
C
δ 1
δ2
P 1
P 1
E
O
P δ
2
1
δ2
(a)
FIGURE 17.20 External work done by two loads on an axial rod.
(b)
D
δ
Now, suppose that load P 1 has already been applied to the rod and a second load P 2 is
gradually added, as shown in Figure 17.20a. The load P 2 causes the rod to elongate by an
additional amount δ 2 . The work done initially by the gradual application of the first load P 1 is
W
= Pδ
(a)
1 2 1 1
which corresponds to area OAE shown in Figure 17.20b. The work done by the gradual
application of the second load P 2 is
W
= Pδ
(b)
1 2 2 2
which corresponds to area ABC. Area ACDE, the remaining area under the load–deformation
diagram, represents the work performed by load P 1 as the rod deforms by the amount δ 2 :
W = P 1 δ 2
(c)
Note that in this case load P 1 does not change its magnitude, because it was fully acting
on the rod before load P 2 was applied.
To summarize, when a load is gradually applied, the expression for work contains the
factor ½, as seen in Equations (a) and (b). Since the loads P 1 and P 2 increase from 0 to their
maximum values, the terms ½P 1 and ½P 2 can be thought of as average loads. If a load is
constant, however, the expression for work does not contain the factor ½, as seen in Equation
(c). These two types of expressions—one with the factor ½ and the other without that
factor—will be used to develop different methods for computing deflections.
The expressions for the work of concentrated moments are similar in form to those of
concentrated forces. A concentrated moment does work when it rotates through an angle.
The work dW that a concentrated moment M performs as it rotates through an incremental
angle dθ is given by
dW
= Mdθ
The total work of a gradually applied concentrated moment M through the rotation angle θ
can be expressed by
W
=
∫
θ
0
Mdθ