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to rewrite this expression as
V
σ
ij
p
Vi
u
i
dV
p
j
u
j
dS
−
ij
dV
+
=−
(3.34)
V
S
It follows from Gauss' integral theorem in the form of Eq. (3.32) (switch the superscript
∗
to the adjoining variables) that Eq. (3.34) can be written as
p
∗
Vi
p
Vi
u
i
p
i
u
i
p
i
−
u
i
dV
+
dV
=−
dS
+
u
i
dS
(3.35)
V
V
S
S
or
u
i
p
i
u
i
p
∗
Vi
p
i
p
Vi
dV
+
dS
=
u
i
dV
+
u
i
dS
(3.36)
V
S
V
S
Typically, this expression is referred to as Betti's theorem for linearly elastic problems. If
desired
,
the surface
i
ntegrals can be written as
S
=
S
p
+
S
u
so that a distinction between
p
i
and
p
i
or
u
i
and
u
i
can be made.
We proceed to consider a special case of Betti's theorem. Equation (3.25) appears
as
u
2
u
1
p
1
dS
=
p
2
dS
(3.37)
S
p
S
p
Suppose that for our linearly elastic
bo
dy, e
ac
h force
sy
stem contains only a single non-
zero force. Desi
gn
ate these forces as
P
1
and
P
2
. First,
P
1
is
a
pplied at point 1 and causes
a displacement
P
1
f
21
at point 2 (in the direction in whi
ch
P
2
is to be applied), where
f
ij
is the displacem
e
nt at
i
due to a unit force at
j
. Next,
P
2
is
applied at point 2, causing
a displace
ment
P
2
f
12
a
t poin
t 1 (in the direction in which
P
1
was applied). According to
Eq. (3.37),
P
1
(
)
=
(
)
=
.
P
2
f
12
P
2
P
1
f
21
or
f
12
f
21
In general,
f
ij
=
f
ji
(3.38)
This represents
Maxwell's reciprocal theorem
which can be expressed as the following:
For a linearly elastic body subjected to two unit (or equal in magnitude) forces, the
displacement at the location of (and in the direction of ) the first force caused by the second
force is equal to the displacement at the location of (and in the direction of ) the second
force which is due to the first force.
By definition,
f
ij
are the
influence
or
flexibility coefficients
. They form the elements of what
later will be called the flexibility matrix. Maxwell's theorem shows that flexibility matrices
must be symmetric for linear structures.
The forces and displacements in Maxwell's theorem may be the usual forces and dis-
placements, as well as moments and corresponding rotations, or combinations of for
ces
and moments
a
nd respective displacements and rotations. For example, suppose that
P
1
is a force and
P
2
is a moment (of the same magnitude, say unity, but with different units)
applied to a beam. Then
f
21
is the rotation at point 2 due to a unit transverse force at 1. And
f
12
is the deflection at point 1 due to a unit moment at point 2.
EXAMPLE 3.16 Betti's Reciprocal Theorem
If the deflection is known along a cantilevered beam with a force
P
at the free end (Fig. 3.14a),
find the free end deflection of the beams of Figs. 3.14b, c, and d.
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