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Problem 10.42
The propped cantilever beam shown in the figure
supports a uniform load of intensity
q
on the lefthand half of the beam.
Find the reactions
R
A
,
R
B
, and
M
A
, and then draw the shearforce and
bendingmoment diagrams, labeling all critical ordinates.
Solution 10.42
Propped cantilever beam
SECTION 10.4
Method of Superposition
643
Select
R
B
as redundant.
E
QUILIBRIUM
R
ELEASED STRUCTURE AND FORCE

DISPLACEMENT
RELATIONS
C
OMPATIBILITY
d
B
5
(
d
B
)
1
2
(
d
B
)
2
5
0
Substitute for (
d
B
)
1
and (
d
B
)
2
and solve for
R
B
:
O
THER REACTIONS
(
FROM EQUILIBRIUM
)
M
A
5
9
qL
2
128
R
A
5
57
qL
128
R
B
5
7
qL
128
M
A
5
qL
2
8
2
R
B
L
R
A
5
qL
2
2
R
B
S
HEAR

FORCE AND BENDING

MOMENT DIAGRAMS
M
max
5
945
qL
2
32,768
x
1
5
57
L
128
A
B
M
A
R
A
R
B
q
L
—
2
L
—
2
A
B
(
d
B
)
1
5
7
qL
4
384
EI
(
d
B
)
2
5
R
B
L
3
3
EI
AB
L
R
B
q
L
2
L
2
x
1
R
A
R
B
V
O
2
M
O
M
max
M
A
2
Problem 10.43
The figure shows a propped cantilever beam
ABC
having span length
L
and an overhang of length
a
. A concentrated load
P
acts at the end of the overhang.
Determine the reactions
R
A
,
R
B
, and
M
A
for this beam. Also, draw the
shearforce and bendingmoment diagrams, labeling all critical ordinates.
A
L
a
B
C
M
A
R
A
R
B
P
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View Full Document Solution 10.43
Beam with an overhang
644
CHAPTER 10
Statically Indeterminate Beams
Select
M
A
as redundant.
E
QUILIBRIUM
R
ELEASED STRUCTURE AND FORCE

DISPL
.
EQS
.
C
OMPATIBILITY
u
A
5
(
u
A
)
1
2
(
u
A
)
2
5
0
Substitute for (
u
A
)
1
and (
u
A
)
2
and solve for
M
A
:
M
A
5
Pa
2
R
B
5
1
L
(
M
A
1
PL
1
Pa
)
R
A
5
1
L
(
M
A
1
Pa
)
O
THER REACTIONS
(
FROM EQUILIBRIUM
)
S
HEAR

FORCE AND BENDING

MOMENT DIAGRAMS
R
B
5
P
2
L
(2
L
1
3
a
)
R
A
5
3
Pa
2
L
A
B
(
u
A
)
1
(
u
A
)
2
M
A
L
(
u
A
)
2
5
M
A
L
3
EI
(
u
A
)
1
5
PaL
6
EI
C
P
a
P
3
Pa
2
L
V
O
2
Pa
2
Pa
M
O
2
L
3
Problem 10.44
Two flat beams
AB
and
CD
, lying in horizontal planes,
cross at right angles and jointly support a vertical load
P
at their
midpoints (see figure). Before the load
P
is applied, the beams just touch
each other. Both beams are made of the same material and have the same
widths. Also, the ends of both beams are simply supported. The lengths
of beams
AB
and
CD
are
L
AB
and
L
CD
, respectively.
What should be the ratio
t
AB
/
t
CD
of the thicknesses of the beams
if all four reactions are to be the same?
Solution 10.44
Two beams supporting a load
P
P
B
D
C
A
t
AB
t
CD
For all four reactions to be the same, each beam
must support onehalf of the load
P
.
D
EFLECTIONS
d
CD
5
(
P
/
2)
L
3
CD
48
EI
CD
d
AB
5
(
P
/
L
3
AB
48
EI
AB
C
OMPATIBILITY
d
AB
5
d
CD
or
M
OMENT OF INERTIA
t
AB
t
CD
5
L
AB
L
CD
∴
L
3
AB
t
3
AB
5
L
3
CD
t
3
CD
I
CD
5
1
12
bt
3
CD
I
AB
5
1
12
bt
3
AB
L
3
AB
I
AB
5
L
3
CD
I
CD
Problem 10.45
Determine the fixedend moments (
M
A
and
M
B
) and
fixedend forces (
R
A
and
R
B
) for a beam of length
L
supporting a
triangular load of maximum intensity
q
0
(see figure). Then draw the
shearforce and bendingmoment diagrams, labeling all critical ordinates.
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This note was uploaded on 03/19/2008 for the course E M 316 taught by Professor Korkolis during the Spring '08 term at University of Texas at Austin.
 Spring '08
 Korkolis
 Superposition, Shear

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