588
CHAPTER 9
Deflections of Beams
Use the method of superposition.
(a) D
EFLECTION
d
B
AT THE FREE END
(1) Part
CB
of the beam:
(2) Part
AC
of the beam:
(
d
B
)
2
5
d
C
1
u
C
¢
L
2
≤
5
7
PL
3
24
EI
2
u
C
5
P
(
L
/
2)
2
2
EI
2
1
(
PL
/
2)(
L
/
EI
2
5
3
PL
2
8
EI
2
d
C
5
P
(
L
/
3
3
EI
2
1
(
PL
/
L
/
2
2
EI
2
5
5
PL
3
48
EI
2
(
d
B
)
1
5
P
3
EI
1
¢
L
2
≤
3
5
PL
3
24
EI
1
(3) Total deflection at point
B
(b) P
RISMATIC BEAM
Ratio:
(c) G
RAPH OF RATIO
r
5
d
B
d
1
5
1
8
¢
1
1
7
I
1
I
2
≤
d
1
5
PL
3
3
EI
1
d
B
5
(
d
B
)
1
1
(
d
B
)
2
5
PL
3
24
EI
1
¢
1
1
7
I
1
I
2
≤
B
C
I
1
P
L
2
—
1
0.5
0
12345
I
2
I
1
—
r
Problem 9.72
The cantilever beam
ACB
shown in the figure supports
a uniform load of intensity
q
throughout its length. The beam has moments
of inertia
I
2
and
I
1
in parts
AC
and
CB
, respectively.
(a) Using the method of superposition, determine the deflection
d
B
at the free end due to the uniform load.
(b) Determine the ratio
r
of the deflection
d
B
to the deflection
d
1
at the free end of a prismatic cantilever with moment of inertia
I
1
carrying
the same load.
(c) Plot a graph of the deflection ratio
r
versus the ratio
I
2
/
I
1
of the
moments of inertia. (Let
I
2
/
I
1
vary from 1 to 5.)
B
C
A
I
1
I
2
q
L
2
—
L
2
—
C
A
I
2
P
L
2
—
PL
2
—
Nonprismatic Beams
Problem 9.71
The cantilever beam
ACB
shown in the figure has moments
of inertia
I
2
and
I
1
in parts
AC
and
CB
, respectively.
(a) Using the method of superposition, determine the deflection
d
B
at the
free end due to the load
P
.
(b) Determine the ratio
r
of the deflection
d
B
to the deflection
d
1
at the free
end of a prismatic cantilever with moment of inertia
I
1
carrying the same load.
(c) Plot a graph of the deflection ratio
r
versus the ratio
I
2
/
I
1
of the moments
of inertia. (Let
I
2
/
I
1
vary from 1 to 5.)
Solution 9.71
Cantilever beam (nonprismatic)
B
C
A
I
1
I
2
P
L
2
—
L
2
—
r
1
1.00
2
0.56
3
0.42
4
0.34
5
0.30
I
2
I
1
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View Full DocumentSolution 9.72
Cantilever beam (nonprismatic)
SECTION 9.7
Nonprismatic Beams
589
Use the method of superposition
(a) D
EFLECTION
d
B
AT THE FREE END
(1) Part
CB
of the beam:
(2) Part
AC
of the beam:
(
d
B
)
2
5
d
C
1
u
C
¢
L
2
≤
5
15
qL
4
128
EI
2
5
7
qL
3
48
EI
2
u
C
5
q
(
L
/
2)
3
6
EI
2
1
(
qL
/
2)(
L
/
2
2
EI
2
1
(
qL
2
/
8)(
L
/
EI
2
d
C
5
q
(
L
/
4
8
EI
2
1
¢
qL
2
≤
(
L
/
3
3
EI
2
1
¢
qL
2
8
≤
¢
L
2
≤
2
2
EI
2
5
17
qL
4
384
EI
2
(
d
B
)
1
5
q
8
EI
1
¢
L
2
≤
4
5
qL
4
128
EI
1
(3) Total deflection at point
B
(b) P
RISMATIC BEAM
Ratio:
(c) G
RAPH OF RATIO
r
5
d
B
d
1
5
1
16
¢
1
1
15
I
1
I
2
≤
d
1
5
qL
4
8
EI
1
d
B
5
(
d
B
)
1
1
(
d
B
)
2
5
qL
4
128
EI
1
¢
1
1
15
I
1
I
2
≤
I
1
B
C
q
L
2
I
2
C
A
q
L
2
—
qL
2
—
qL
2
8
—
I
2
I
1
1
1
2
3
45
0.5
O
r
Problem 9.73
A simple beam
ABCD
has moment of inertia
I
near
the supports and moment of inertia 2
I
in the middle region, as shown
in the figure. A uniform load of intensity
q
acts over the entire length
of the beam.
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 Spring '08
 Korkolis
 Superposition

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