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FIGURE P3.11
Out-of-plane displacement of frame.
Hint:
Use
EI
∂
M
M
GJ
∂
M
t
M
t
∂
k
s
GA
∂
V
V
V
d
horizontal
=
dx
+
dx
+
dx
∂
P
P
∂
P
PL
3
EI
+
3
PL
3
4
PL
k
s
AG
3.12 Determine the horizontal displacement of point
a
in the
abc
plane of the bar of Fig.
P3.12. Consider bending and torsion of the members. Let
G
3
Answer:
V
d
horizontal
=
GJ
+
4
E
. The cross-sections
are constant and circular. Also, compute the rotation of point
c
in the
abc
plane.
=
0
.
Answer:
3500
/(
EI
)
+
2400
/(
EJ
)
,
2100
/(
EI
)
+
2400
/(
EJ
)
3.13 Find the relative displacement between points
a
and
b
of the truss of Fig. P3.13a. Also,
indicate how the rotation of member 1 can be determined.
Hint:
Apply a pair of opposing forces
Q
at
a
and
b
(Fig. P3.13b). Then Castigliano's
se
co
n
d theo
re
m gives
V
a
+
U
i
/∂
Q
in
U
i
V
b
=
∂
Q
|
Q
=
0
. To compute
∂
N
/∂
, remove
1at
a
and
b
.
The rotation of bar 1 is found by applying a couple
M
∗
as shown in Fig. P3.13c. The
rotation is then
P
e
,
P
d
, and
P
b
and calculate
N
in each bar due to
Q
=
M
∗
|
M
∗
=
0
.
3.14 Consider the curved bar of Fig. P3.14 with
P
acting perpendicular to the plane of the
bar. Find the horizontal, out-of-plane deflection and angle of twist of end
a
of the bar.
U
i
/∂
θ
=
∂
Hint:
If bending and twisting effects are taken into account,
π/
2
π/
2
M
EI
∂
M
GJ
∂
M
t
M
t
∂
V
a
horizontal
=
Rd
α
+
Rd
α
∂
P
P
0
0
with
M
=
PR
sin
α
,
M
t
=
PR
(
1
−
cos
α).
Add fictitious torque
M
t
at
a
, then
M
t
=
0
=
π/
2
π/
2
∂
U
M
EI
∂
M
GJ
∂
M
t
M
t
φ
a
=
Rd
α
+
Rd
α
∂
M
t
∂
M
t
∂
M
t
0
0
M
t
=
0
=
α
−
M
t
sin
α
=
(
−
α)
+
M
t
cos
α
with
M
PR
sin
,
M
t
PR
1
cos
.
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