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Solution 2.38
Bar with a hole
80
CHAPTER 2
Axially Loaded Numbers
d
2
d
1
L
4
P
P
—
L
4
—
L
2
—
d
d
5
diameter of hole
S
HORTENING
d
OF THE BAR
(Eq. 1)
N
UMERICAL VALUES
(
DATA
):
d
5
maximum allowable shortening of the bar
5
8.0 mm
5
PL
p
E
¢
1
d
1
2
2
d
2
1
1
d
1
2
1
2
d
2
2
≤
5
P
E
C
L
/
4
p
4
(
d
1
2
2
d
2
)
1
L
/
4
p
4
d
1
2
1
L
/
2
p
4
d
2
2
S
d
5
a
N
i
L
i
E
i
A
i
5
P
E
a
L
i
A
i
P
5
110 kN
L
5
1.2 m
E
5
4.0 GPa
d
1
5
100 mm
d
max
5
maximum allowable diameter of the hole
d
2
5
60 mm
S
UBSTITUTE NUMERICAL VALUES INTO
E
Q
. (1)
FOR
d
AND SOLVE FOR
d
5
d
max
:
U
NITS
: Newtons and meters
3
d
max
5
23.9
mm
—
d
5
0.02387
m
d
2
5
569.81
3
10
2
6
m
2
5
761.598
2
100
2
555.556
5
106.042
1
0.01
2
d
2
761.598
5
1
0.01
2
d
2
1
1
0.01
1
2
0.0036
B
1
(0.1)
2
2
d
2
1
1
(0.1)
2
1
2
(0.06)
2
R
0.008
5
(110,000)(1.2)
p
(4.0
3
10
9
)
Problem 2.39
A wood pile, driven into the earth, supports a load
P
entirely
by friction along its sides (see figure). The friction force
f
per unit length of pile
is assumed to be uniformly distributed over the surface of the pile. The pile has
length
L
, crosssectional area
A
, and modulus of elasticity
E
.
(a) Derive a formula for the shortening
d
of the pile in terms of
P
,
L
,
E
,
and
A
.
(b) Draw a diagram showing how the compressive stress
s
c
varies throughout
the length of the pile.
L
P
f
Problem 2.38
A bar
ABC
of length
L
consists of two parts
of equal lengths but different diameters (see figure). Segment
AB
has diameter
d
1
5
100 mm and segment
BC
has diameter
d
2
5
60 mm. Both segments have length
L
/2
5
0.6 m. A
longitudinal hole of diameter
d
is drilled through segment
AB
for onehalf of its length (distance
L
/4
5
0.3 m). The bar is
made of plastic having modulus of elasticity
E
5
4.0 GPa.
Compressive loads
P
5
110 kN act at the ends of the bar.
If the shortening of the bar is limited to 8.0 mm, what
is the maximum allowable diameter
d
max
of the hole?
d
2
d
1
L
4
P
P
AB
C
—
L
4
—
L
2
—
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View Full DocumentSolution 2.39
Wood pile with friction
SECTION 2.3
Changes in Lengths under Nonuniform Conditions
81
F
ROM FREE

BODY DIAGRAM OF PILE
:
(Eq. 1)
(a) S
HORTENING
d
OF PILE
:
At distance
y
from the base:
©
F
vert
5
0
Ê
c
1
T
2
Ê
fL
2
P
5
0
Ê
f
5
P
L
N
(
y
)
5
axial force
N
(
y
)
5
fy
(Eq. 2)
(b) C
OMPRESSIVE STRESS
s
c
IN PILE
At the base (
y
5
0):
s
c
5
0
See the diagram above.
At
the
top(
y
5
L
):
s
c
5
P
A
s
c
5
N
(
y
)
A
5
fy
A
5
Py
AL
—
d
5
PL
2
EA
—
d
5
#
L
0
d
d
5
f
EA
#
L
0
ydy
5
fL
2
2
EA
5
PL
2
EA
d
d
5
N
(
y
)
dy
EA
5
fy
dy
EA
L
y
P
f
dy
f =
P
L
s
c
=
P
y
AL
P
A
0
Compressive stress
in pile
Friction force
per unit
length of pile
Problem 2.310
A prismatic bar
AB
of length
L
, crosssectional area
A
, modulus
of elasticity
E
, and weight
W
hangs vertically under its own weight (see figure).
(a) Derive a formula for the downward displacement
d
C
of point
C
, located
at distance
h
from the lower end of the bar.
(b) What is the elongation
d
B
of the entire bar?
(c) What is the ratio
b
of the elongation of the upper half of the bar to the
elongation of the lower half of the bar?
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 Spring '08
 Gardoni

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