Civil Engineering Reference
In-Depth Information
w
A
B
L
P
w
x
A
B
L
Figure 4.21.
Example 4.10 Castigliano's second theorem.
A free-body diagram of the right-hand portion of the beam is shown in
Figure 4.22. The moment equation is found from equilibrium on the free-
body. Also, the partial derivative of the moment equation is taken with
respect to the particular force
P
. These are substituted into Equation 4.9
to find the deflection at point
B
.
wx
P
x/2
M
V
x
Figure 4.22.
Example 4.10 Castigliano's second theorem.
M wx
x
wx
2
Σ
M
==−− −∴=−
0
Px M
−
Px
B
2
2
∂
∂
M
P
=−
x
L
L
2
L
3
∂
∂
M
P
dx
EI
wx
dx
EI
wx
dx
EI
−
()
=
∫
∫
∫
∆
=
M
=−
−
Px
x
+
Px
2
B
2
2
0
0
0
4
3
wL
EI
PL
EI
∆
=+
B
8
3
Since
P
is a fictitious applied load placed on the beam only to find the
deflection, it can be removed and the deflection at
B
becomes the following:
wL
EI
4
∆
B
=
8
