Section 5.5
Solid Mechanics Part I
Kelly
200
5.5 Virtual Work
Consider a mass attached to a spring and pulled by an applied force
apl
F
, Fig. 5.5.1a.
When the mass is in equilibrium,
0
=
+
apl
spr
F
F
, where
kx
F
spr
−
=
is the spring force
with
x
the distance from the spring reference position.
Figure 5.5.1: a force extending an elastic spring; (a) block in equilibrium, (b) block
not at its equilibrium position
In order to develop a number of powerful techniques based on a concept known as
virtual work
, imagine that the mass is not in fact at its equilibrium position but at an
(incorrect) non-equilibrium position
x
x
δ
+
, Fig. 5.5.1b.
The imaginary displacement
x
δ
is called a
virtual displacement
.
Define the
virtual work
W
δ
done by a force to be the
equilibrium force times this small imaginary displacement
x
δ
.
It should be emphasized
that virtual work is not real work – no work has been performed since
x
δ
is not a real
displacement which has taken place; this is more like a “thought experiment”.
The virtual
work of the spring force is then
x
kx
x
F
W
spr
spr
δ
δ
δ
−
=
=
.
The virtual work of the applied
force is
x
F
W
apl
apl
δ
δ
=
.
The total virtual work is
(
)
x
F
kx
W
W
W
apl
apl
spr
δ
δ
δ
δ
+
−
=
+
=
(5.5.1)
There are two ways of viewing this expression.
First, if the system is in equilibrium
(
0
=
+
−
apl
F
kx
) then the virtual work is zero,
0
=
W
δ
.
Alternatively, if the virtual work
is zero then, since
x
δ
is arbitrary, the system must be in equilibrium.
Thus the virtual
work idea gives one an alternative means of determining whether a system is in
equilibrium.
The symbol
δ
is called a
variation
so that, for example,
x
δ
is a
variation in the
displacement
(from equilibrium).
Virtual work is explored further in the following section.
spr
F
apl
F
spr
F
apl
F
e
x
x
δ
)
a
(
)
b
(

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