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FIGURE 3.3
Cantilevered beam.
beam is given by
L
M
2
2
EI
U
i
=
dx
(1)
0
Since Castigliano's theorem, part II is related to the principle of complementary virtual
work, the forces must be in equilibrium.
For
ou
r beam, t
he
conditions of equilibrium give (Fig. 3.3b)
M
=−
M
0
−
Px
.
In utilizing
M
0
with
U
i
defined by (1), it is generally simpler to differentiate the
integrand first and then to carry out the integration rather than the opposite. Thus,
U
i
/∂
U
i
/∂
∂
P
or
∂
L
U
i
∂
=
∂
1
EI
M
∂
M
w
P
=
dx
(2)
0
∂
P
0
With
M
=−
M
0
−
Px
and
∂
M
/∂
P
=−
x,
L
0
(
M
0
L
2
2
EI
PL
3
3
EI
1
EI
w
=
M
0
+
Px
)
xdx
=
+
(3)
0
For the end slope,
∂
M
/∂
M
0
=−
1 and
L
L
0
(
U
i
θ
0
=
∂
1
EI
M
∂
M
1
EI
M
0
=
dx
=
M
0
+
Px
)
dx
∂
∂
M
0
0
PL
2
2
EI
M
0
L
EI
=
+
(4)
Positive (negative) values of
0
in
dicate that th
ese
variables are in the same (opposite)
direction as the corresponding force
P
and moment
M
0
w
0
and
θ
.
EXAMPLE 3.7 Truss Analysis
Find the vertical displacement of point
b
of the simple truss of Fig. 3.4.
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