This preview shows pages 1–3. Sign up to view the full content.
Nonprismatic Beams
Problem
5.71
A tapered cantilever beam
AB
of length
L
has square
cross sections and supports a concentrated load
P
at the free end (see
figure on the next page). The width and height of the beam vary linearly
from
h
A
at the free end to
h
B
at the fixed end.
Determine the distance
x
from the free end
A
to the cross section of
maximum bending stress if
h
B
5
3
h
A
. What is the magnitude
s
max
of the
maximum bending stress? What is the ratio of the maximum stress to the
largest stress
s
B
at the support?
Solution 5.71
Tapered cantilever beam
SECTION 5.7
Nonprismatic Beams
321
A
h
A
h
B
B
x
P
L
S
QUARE CROSS SECTIONS
h
A
5
height and width at smaller end
h
B
5
height and width at larger end
h
x
5
height and width at distance
x
S
TRESS AT DISTANCE
x
A
T END
A
:
x
5
0
s
A
5
0
A
T SUPPORT
B
:
x
5
L
s
B
5
2
PL
9(
h
A
)
3
s
1
5
M
x
S
x
5
6
Px
(
h
A
)
3
¢
1
1
2
x
L
≤
3
S
x
5
1
6
(
h
x
)
3
5
h
A
3
6
¢
1
1
2
x
L
≤
3
h
x
5
h
A
1
(
h
B
2
h
A
)
¢
x
L
≤
5
h
A
¢
1
1
2
x
L
≤
h
B
h
A
5
3
C
ROSS SECTION OF MAXIMUM STRESS
Set
Evaluate the derivative, set it equal
to zero, and solve for
x
.
M
AXIMUM BENDING STRESS
Ratio of
s
max
to
s
B
s
max
s
B
5
2
s
max
5
(
s
1
)
x
5
L
/
4
5
4
PL
h
A
)
3
x
5
L
4
d
s
1
dx
5
0
A
B
x
P
L
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentProblem
5.72
A tall signboard is supported by two vertical
beams consisting of thinwalled, tapered circular tubes (see
figure). For purposes of this analysis, each beam may be
represented as a cantilever
AB
of length
L
5
8.0 m subjected
to a lateral load
P
5
2.4 kN at the free end. The tubes have
constant thickness
t
5
10.0 mm and
average
diameters
d
A
5
90 mm and
d
B
5
270 mm at ends
A
and
B
, respectively.
Because the thickness is small compared to the diameters,
the moment of inertia at any cross section may be obtained
from the formula
I
5
p
d
3
t
/8 (see Case 22, Appendix D), and
therefore the section modulus may be obtained from the
formula
S
5
p
d
2
t
/4.
At what distance
x
from the free end does the maximum
bending stress occur? What is the magnitude
s
max
of the
maximum bending stress? What is the ratio of the maximum
stress to the largest stress
s
B
at the support?
Solution 5.72
Tapered circular tube
322
CHAPTER 5
Stresses in Beams (Basic Topics)
Wind
load
P
= 2.4 kN
A
B
x
t
= 10.0 mm
d
A
= 90 mm
d
B
= 270 mm
L
= 8.0 m
P
5
2.4 kN
L
5
8.0 m
t
5
10 mm
d
5
average diameter
At end
A
:
d
A
5
90 mm
At support
B
:
d
B
5
270 mm
A
T DISTANCE
x:
M
x
5
Px
5
2400
xx
5
meters,
M
x
5
N
?
m
s
1
5
M
x
S
x
5
2400
x
20.25
p
¢
1
1
2
x
L
≤
2
Ê
L
5
meters,
s
1
5
MPa
5
20,250
p
¢
1
1
2
x
L
≤
2
Ê
S
x
5
mm
3
S
x
5
p
4
(
d
x
)
2
(
t
)
5
p
4
(90)
2
¢
1
1
2
x
L
≤
2
(10)
d
x
5
d
A
1
(
d
B
2
d
A
)
¢
x
L
≤
5
90
1
180
x
L
5
90
¢
1
1
2
x
L
≤
A
T END
A
:
x
5
0
s
1
5
s
A
5
0
A
T SUPPORT
B
:
x
5
L
5
8.0 m
s
1
5
s
B
5
33.53 MPa
C
ROSS SECTION OF MAXIMUM STRESS
Set
Evaluate the derivative, set it equal to
zero, and solve for
x
.
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '08
 Muliana

Click to edit the document details