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SECTION 9.5
Method of Superposition
571
Problem 9.511
Determine the angle of rotation
u
B
and deflection
d
B
at
the free end of a cantilever beam
AB
supporting a parabolic load defined
by the equation
q
5
q
0
x
2
/
L
2
(see figure).
Solution 9.511
Cantilever beam (parabolic load)
A
B
q
0
x
y
L
L
OAD
:
qdx
5
element of load
q
5
q
0
x
2
L
2
T
ABLE
G1, C
ASE
5
(Set
a
equal to
x
)
5
q
0
6
EIL
2
#
L
0
(
x
4
)(3
L
2
x
)
dx
5
13
q
0
L
4
180
EI
5
1
6
EI
#
L
0
¢
q
0
x
2
L
2
≤
(
x
2
L
2
x
)
dx
d
B
5
#
L
0
(
qdx
)(
x
2
)
6
EI
(3
L
2
x
)
5
q
0
2
EIL
2
#
L
0
x
4
dx
5
q
0
L
3
10
EI
u
B
5
#
L
0
(
qdx
x
2
)
2
EI
5
1
2
EI
#
L
0
¢
q
0
x
2
L
2
≤
x
2
dx
A
B
qdx
a
L
Problem 9.512
A simple beam
AB
supports a uniform load of
intensity
q
acting over the middle region of the span (see figure).
Determine the angle of rotation
u
A
at the lefthand support
and the deflection
d
max
at the midpoint.
Solution 9.512
Simple beam (partial uniform load)
A
B
L
a
a
q
L
OAD
:
qdx
5
element of load
T
ABLE
G2, C
ASE
6
Replace
P
by
qdx
Replace
a
by
x
Integrate
x
from
a
to
L
/2
T
ABLE
G2, C
ASE
6
Replace
P
by
qdx
Replace
a
by
x
Integrate
x
from
a
to
L
/2
d
max
5
Pa
24
EI
L
2
2
4
a
2
)
5
q
24
EI
(
L
3
2
6
a
2
L
1
4
a
3
)
u
A
5
#
L
/
2
a
qdx
2
EI
(
x
L
2
x
)
5
q
2
EI
#
L
/
2
a
(
xL
2
x
2
)
dx
u
A
5
Pa
(
L
2
a
)
2
EI
A
B
L
/
2
L
/
2
a
a
x
x
qdx
qdx
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CHAPTER 9
Deflections of Beams
A
LTERNATE SOLUTION
(not recommended; algebra is
extremely lengthy)
Table G2, Case 3
5
q
24
EI
(
L
3
2
6
La
2
1
4
a
3
)
u
A
5
q
(
L
2
a
)
2
24
LEI
[2
L
2
(
L
2
a
)]
2
2
qa
2
24
LEI
(2
L
2
a
)
2
5
q
384
EI
(5
L
4
2
24
a
2
L
2
1
16
a
4
)
5
q
24
EI
#
L
/
2
a
(3
L
2
x
2
4
x
3
)
dx
d
max
5
#
L
/
2
a
qdx
24
EI
(
x
)(3
L
2
2
4
x
2
)
d
max
5
q
384
EI
L
4
2
24
L
3
a
2
1
16
a
4
)
2
6
L
¢
L
2
≤
2
1
2
¢
L
2
≤
3
R
5
qa
2
24
LEI
B
2
La
2
1
4
L
2
¢
L
2
≤
1
a
2
¢
L
2
≤
2
4
L
(
L
2
a
)
¢
L
2
≤
2
1
L
¢
L
2
≤
3
R
1
4
L
2
(
L
2
a
)
2
1
2(
L
2
a
)
2
¢
L
2
≤
2
d
max
5
q
(
L
/
2)
24
LEI
B
(
L
2
a
)
4
2
4
L
(
L
2
a
)
3
A
B
aa
q
q
La
q
A
a
=
Problem 9.513
The overhanging beam
ABCD
supports two
concentrated loads
P
and
Q
(see figure).
(a) For what ratio
P
/
Q
will the deflection at point
B
be zero?
(b) For what ratio will the deflection at point
D
be zero?
Solution 9.513
Overhanging beam
(a) D
EFLECTION AT POINT
B
Table G2, Cases 4 and 7
P
Q
5
3
a
L
d
B
5
PL
3
48
EI
2
Qa
¢
L
2
16
EI
≤
5
0
(b) D
EFLECTION AT POINT
D
Table G2, Case 4; Table G1, Case 4;
Table G2, Case 7
P
Q
5
16
a
(
L
1
a
)
3
L
2
d
D
52
PL
2
16
EI
(
a
)
1
Qa
3
3
EI
1
Qa
¢
L
3
EI
≤
(
a
)
5
0
A
D
C
B
a
P
Q
L
2
—
L
2
—
Problem 9.514
A thin metal strip of total weight
W
and length
L
is
placed across the top of a flat table of width
L
/3 as shown in the figure.
What is the clearance
d
between the strip and the middle of the
table? (The strip of metal has flexural rigidity
EI
.)
Solution 9.514
Thin metal strip
SECTION 9.5
Method of Superposition
573
d
L
3
—
L
3
—
L
6
—
L
6
—
W
5
total weight
EI
5
flexural rigidity
F
REE BODY DIAGRAM
(the part of the strip above the
table)
q
5
W
L
T
ABLE
G2, C
ASES
1
AND
10
But
:
∴
d
5
19
WL
3
31,104
EI
q
5
W
L
5
19
qL
4
31,104
EI
52
5
qL
4
31,104
EI
1
qL
4
1296
EI
d
5
q
384
EI
¢
L
3
≤
4
1
M
0
8
EI
¢
L
3
≤
2
L
/
6
L
/
6
q
M
o
M
o
qL
2
18
—
=
Problem 9.515
An overhanging beam
ABC
with flexural rigidity
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This note was uploaded on 03/30/2008 for the course CVEN 205 taught by Professor Muliana during the Spring '08 term at Texas A&M.
 Spring '08
 Muliana

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