399
*8–40. Use the method of virtual work and determine the
slope at point A of the beam made from steel.
E=29(103)
ksi,
I=245
in4.
5 ft
8 k
5 ft
AB
8 k
5 ft 5 ft
C
29(10
3
)(245)
Ans.
uA=4.05(10 –3) rad
SOLUTION
0M
dx
29(10
3
)(245)
C
8–41. Solve Prob. 8–40 using Castigliano’s theorem.
401
8–42. Determine the displacement at point D. Use the
principle of virtual work. EI is constant.
2 m2
m3
m
30 kN>m
60 kN
B
AC
D
SOLUTION
The virtual and real moment functions shown in Figs. a and b,
respectively.
1
kN #∆D=
L
L
0
mM
EI
dx =2
L
2 m
0
a1
2 x1
b
(30x1)dx1
EI
+
L
3 m
0
a
–
1
2 x2
b
[–(15x2
2+30x2)] dx2
EI
1
kN
#
∆D=
2935 kN2#m3
8 EI
∆D=
2935 kN #m3
8EI
T Ans.
Ans.
2935 kN #m3
8EI
SOLUTION
8EI
2 m2
m3
m
30 kN>m
60 kN
B
AC
D
8–43. Determine the displacement at point D. Use Castigliano’s
theorem. EI is constant.
Ans.
2935 kN #m3
*8–44. Determine the horizontal displacement at A. Take
E=200
GPa.
The moment of inertia of each segment of the
frame is indicated in the figure. Assume D is a pin support. Use
the method of virtual work. C
B
AD
8
mIAB 5 200(106) mm4
IBC 5 300(106) mm4
ICD 5 200(106) mm4
6 m
60 kN
>
m
SOLUTION
Referring to the virtual and moment functions shown in Figs. a and b,
SOLUTION
0M
1
C
B
AD
8
mIAB 5 200(106) mm4
IBC 5 300(106) mm4
ICD 5 200(106) mm4
6 m
60 kN
>
m
8–45. Solve Prob. 8–44 using Castigliano’s theorem.
405
8–46. The L-shaped frame is made from two fixed-connected
segments. Determine the vertical displacement of the end C.
Use the method of virtual work. EI is constant.
B
C
A
12 ft
9 ft
2 k
>
ft
SOLUTION
Referring to the virtual and real moment functions shown in
Figs. a and b, respectively,
1
k #(∆C)y=
L
L
0
mM
EI
dx =
L
9 ft
0
(–x1)
a
–
x
1
3
27
b
dx1
EI
+
L
12 ft
0
(–9)(–27)dx2
EI
1
k
#
(∆C)y=
16767 k2#ft3
5EI
(
∆C)y=
16767 k #ft3
5EI
T Ans.
Ans.
(
∆C)y=
16767 k #ft3
5EI
T
SOLUTION
Referring to the virtual and real moment functions shown in
Figs. a and b, respectively,
L
9 ft
(–1)
a
–
x
1
3
b
dx1
12 ft
4 EI
8–47. The L-shaped frame is made from two fixed-connected
segments. Determine the slope at point C. Use the method of
virtual work. EI is constant.
B
C
A
12 ft
9 ft
2 k
>
ft
407
B
C
A
12 ft
9 ft
2 k
>
ft
*8–48. Solve Prob. 8–46 using Castigliano’s theorem.
SOLUTION
0M
0M
1
5EI
SOLUTION
1#∆=
LL
0
mM
EI
d
x
1
2)dx1+
640 000 lb
#
ft
3
EI
0
L10
0
EI
8–49 Use the method of virtual work and determine the
horizontal and vertical displacements of point C. There is a
fixed support at A and fixed joint at B. EI is constant.
A
8 ft
10 ft
C
400 lb>ft
B
A
8 ft
10 ft
C
400 lb>ft
B
8–50 Solve Prob. 8–49 using Castigliano’s theorem.
Ans.
640 000 lb #ft3
SOLUTION
0M
1
0P
=0
0M
2
0P
=10 –x
2
Setting
P=0
,
M1=200x1
2
M2=12 800
∆
Ch
=
LL
0
M
a
0M
0Pb
dx
EI
=
1
EI
cL8
0
(200x1
2)(0)dx1+
L10
0
12 800(10 –x2)dx2
d
=
640 000 lb
#
ft
3
EI
Ans.
0M
1
0P
=x1
0M
2
0P
=
8
Setting
P=0
,
M1=200x1
2
M2=12 800
∆
Cv
=
LL
0
M
a
0M
0Pb
dx
EI
=
1
EI
cL8
0
(200x1
2)(x1)dx1+
L10
0
(12 800)(8)dx2
d
=
1 228 000 lb
#
ft
3
EI
Ans.
1 228 000 lb #ft3