sm12 46

________________________________________________________________________
PROBLEM (12.46) A cantilever beam is loaded as shown in Fig. P12.46. Apply Castigliano’s
theorem to determine the deflection of the free end.
A
v
SOLUTION
Segment AC
11
M
Qx M Q x=− ∂ ∂ =−
Segment BC
22
2()
2M
W
M
Qx x a x
Q
∂
=− − − =−
∂
Thus,
3
11 22
0
aa
Aa
MM MM
vdx
EI Q EI Q
∂∂
=+
∂∂
∫∫
dx
Let Q=0. Hence
33
232
() (2 )
22
aa
Aaa
ww
vxaxdxxaxa
EI EI
=−= −+
∫∫
2
xdx
3
4322
2
243 2
a
a
wx ax ax
EI
=−+
4
10
3wa EI
=↓
A
2
a
w
Q
C
a
B
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