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3.3
Reciprocal Theorems
Consider a linear elastic body su
bje
cted to two different sets of forces.
S
uppose the body
is first subjected to surface forces
p
1
and then, a
t
the same locations as
p
1
and in the same
directions, is subjected to applied surface forces
p
2
. For the sake of
br
evity, body forces will
be ignored. From Chapter 2, Eq. (2.27), the external work done by
p
1
is
1
2
u
1
p
1
dS
=
W
e
11
(3.19)
S
p
where
u
1
are the displacements which result from the application of
p
1
.
The notation
W
e
11
is used to indicate the
w
ork of the force set 1 moving through the displacement s
e
t 1. Now
let additional forces
p
2
be applied, causing displacements
u
2
,
while force set
p
1
is held
constant. The work of
p
2
moving through
u
2
is
1
2
u
2
W
e
22
=
p
2
dS
(3.20)
S
p
An additional increment of work, say
W
e
12
,
will be done by the first forces
p
1
moving through
the displacements
u
2
of the second set of forces. Since the first forces
p
1
remain constant
during these further displacements
u
2
,
we have
u
2
W
e
12
=
p
1
dS
(3.21)
S
p
Thus, the total external work performed by the two sets of forces is
W
e
=
W
e
11
+
W
e
22
+
W
e
12
(3.22)
Now, remove the loads and apply them again, but in reverse order. For this second case,
the total work done will be
W
e
=
W
e
22
+
W
e
11
+
W
e
21
(3.23)
where
W
e
22
is again given by Eq. (3.20),
W
e
11
by Eq. (3.19), and
W
e
21
is now
u
1
W
e
21
=
p
2
dS
(3.24)
S
p
That is,
W
e
21
is the work performed by force set 2 moving through the incremental displace-
ments associated with force set 1.
The principle of superposition is applicable for this linearly elastic body. Since in this case
the total work of the applied forces must be independent of the order of application of the
loading, it follows by equating
W
e
of Eqs. (3.22) and (3.23) that
W
e
12
=
W
e
21
(3.25)
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