sm12 50

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PROBLEM (12.50) A truss ABC of three steel members supports a vertical load P at C as
shown in Fig. P12.50. The cross-sectional areas of members are AAB , ABC , and AAC . What is the
vertical displacement v
δ
of joint C? Apply Castigliano’s theorem.
Given: AAB = 0.002 m, ABC = 0.003 m, AAC = 0.0026 m, P = 25 kN, E = 210 GPa
SOLUTION
1()
BC BC AC AC
AB AB
vAB BC AC
FF FF
FF
U
PE A P A P A P
δ
∂
∂
∂
∂
== + +
∂∂∂
∑∂
(1)
Joint C
We have
22
5.4 2.25 5.85
AC
Lm=+=
P
22
3 2.25 3.75
BC
Lm=+ =
2.4
AB
Lm
=
5.4 3
0: 0
5.85 3.75
xAC
FFF=− + =
∑BC
2.25 2.25
0: 0
5.85 3.75
yAC
FFF=− + =
∑BC
Solving,
3.25 3.75
AC AC
FPFP==
Joint B
3
0: 0
3.75
xAB BC
FF F
=
+=
∑
FP3
AB
=
Equation (1) is therefore
3
33 3.75 3.25
[( )(3) ( )(3.75) ( )(3.25)] 13.25(10 )
(10 ) 2 3 2.6
vPP
EE
δ
−
=++=
The value of displacement at C is
3
39
25(10 )
13.25(10 ) 1.577
210(10 )
vmm
δ ==↓
32.25
FAC
4
25 C
5.
2.
F
BC
F
BC
RB
B
3 25
2.
FAB
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