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Axial force in each member
Axial force
N
in each
due to a unit load at
a
in the
L
EA
N
δ
N
member due to th
e
horizontal direction. All actual
Member (
J
)
actual loads (here
P
)
loads are removed
4
3
√
3
P
1
3
4
9
√
3
PL
EA
1
−
−
2
3
√
3
P
5
6
5
9
√
3
PL
EA
2
−
−
4
3
√
3
P
1
3
4
9
√
3
PL
EA
3
4
3
√
3
P
1
3
4
9
√
3
PL
EA
4
−
−
2
3
√
3
P
1
3
2
9
√
3
PL
EA
5
−
−
1
√
3
P
1
2
1
2
√
3
PL
EA
6
−
−
2
3
√
3
P
1
3
2
9
√
3
PL
EA
7
−
−
2
3
√
3
P
2
3
4
9
√
3
PL
EA
−
−
8
2
3
√
3
P
1
3
2
9
√
3
PL
EA
9
−
−
1
3
√
3
P
1
6
1
18
√
3
PL
EA
10
−
−
2
3
√
3
P
1
3
2
9
√
3
PL
EA
11
−
−
L
EA
N
δ
N
=−
2
9
√
3
PL
EA
11 bars
FIGURE 3.13
Tabular solution for Example 3.15.
of available conditions of equilibrium. The theorem of least work is obtained by setting the
right-hand side of Eq. (3.11) equal to zero, i.e.,
U
i
∂
∂
P
=
0
(3.18)
where the forces
P
are the redundant forces. Typically, the redundants are chosen to be
reactions, although they can also be internal forces. In the notation of the force method of
Chapter 5, Eq. (3.18) would appear as
U
i
/∂
0
.
For linearly elastic solids, Eq. (3.18) is sometimes referred to as
Castigliano's theorem of
least work
or
Castligliano's theorem of compatibility
. The first edition of this topic provides
applications of the theorem of least work.
∂
X
=
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