Assignment # 03
CHAPTER 7
σ0
PROBLEM 7.81
100 MPa
σ0
The state of plane stress shown occurs in a machine component made of a steel with
325 MPa. Using the maximum-distortion-energy criterion, determine whether
Y
yield will occur when (a) 0 200 MPa, (b) 0 240 MPa, (c) 0 280 MPa. If
yield does not occur, determine the corresponding factor of safety.
SOLUTION
2
ave
(a)
0
200 MPa
ave
a
2
a
F . S.
(b)
0
240 MPa
ave
a
2
a
F . S.
(c)
0
280 MPa
ave
a
2
a
2
b
a
b
R
0
x
y
2
xy
2
100 MPa
200 MPa
ave
2
b
R
a
100 MPa,
b
b
ave
R
300 MPa
264.56 MPa < 325 MPa
(No yielding)
325
264.56
F . S . 1.228
240 MPa
ave
2
b
R
a
140 MPa,
b
b
ave
R
340 MPa
295.97 MPa < 325 MPa
(No yielding)
325
295.97
F . S . 1.098
280 MPa
ave
R
180 MPa,
329.24 MPa > 325 MPa
b
ave
R
380 MPa
(Yielding occurs)
PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use.
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1119
PROBLEM 7.82
σ0
100 MPa
σ0
Solve Prob. 7.81, using the maximum-shearing-stress criterion.
PROBLEM 7.81 The state of plane stress shown occurs in a machine component
made of a steel with Y 325 MPa. Using the maximum-distortion-energy criterion,
determine whether yield will occur when (a) 0 200 MPa, (b) 0 240 MPa,
(c) 0 280 MPa. If yield does not occur, determine the corresponding factor of
safety.
SOLUTION
2
ave
(a)
0
200 MPa:
ave
a
max
2
max
F . S.
(b)
0
240 MPa:
ave
a
max
2
(c)
0
280 MPa:
max
ave
a
max
2
max
R
0
x
y
2
xy
2
100 MPa
200 MPa
ave
0,
R
100 MPa
ave
R
300 MPa
300 MPa
min
max
b
min
300 MPa
325 MPa
(No yielding)
325
300
F . S . 1.083
240 MPa
ave
0,
R
140 MPa,
min
max
min
b
ave
R
340 MPa
340 MPa
340 MPa > 325 MPa
(Yielding occurs)
280 MPa
ave
0,
max
R
180 MPa,
min
min
b
ave
R
380 MPa
380 MPa
380 MPa
325 MPa
(Yielding occurs)
PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use.
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted
on a website, in whole or part.
1120
PROBLEM 7.83
21 ksi
τ xy
The state of plane stress shown occurs in a machine component made of a steel with
45 ksi. Using the maximum-distortion-energy criterion, determine whether
Y
yield will occur when (a) xy 9 ksi, (b) xy 18 ksi, (c) xy 20 ksi. If yield does
not occur, determine the corresponding factor of safety.
36 ksi
SOLUTION
36 ksi,
x
For stresses in xy plane,
1
(
2
ave
x
y
21 ksi,
y
y)
x
z
0
28.5 ksi
7.5 ksi
2
2
(a)
xy
9 ksi
a
ave
2
a
F .S .
2
b
x
R
R
a
y
2
xy
2
40.215 ksi,
ave
b
34.977 ksi
b
(7.5) 2
R
(9) 2
11.715 ksi
16.875 ksi
45 ksi
(No yielding)
45
39.977
F .S .
1.287
2
(b)
x
18 ksi R
xy
ave
a
2
a
F .S .
2
b
R
a
y
2
xy
2
48 ksi,
ave
b
44.193 ksi
b
(7.5)2
R
(18)2
19.5 ksi
9 ksi
45 ksi
(No yielding)
45
44.193
F .S .
1.018
2
(c)
xy
20 ksi
a
ave
2
a
2
b
R
R
a
x
2
xy
2
49.86 ksi,
b
y
b
46.732 ksi
ave
(7.5) 2
R
(20) 2
21.36 ksi
7.14 ksi
45 ksi
(Yielding occurs)
PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use.
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted
on a website, in whole or part.
1121
PROBLEM 7.84
21 ksi
Solve Prob. 7.83, using the maximum-shearing-stress criterion.
τ xy
PROBLEM 7.83 The state of plane stress shown occurs in a machine component
made of a steel with Y 45 ksi. Using the maximum-distortion-energy criterion,
determine whether yield will occur when (a) xy 9 ksi, (b) xy 18 ksi,
(c) xy 20 ksi. If yield does not occur, determine the corresponding factor of
safety.
36 ksi
SOLUTION
36 ksi,
x
21 ksi,
y
0
z
For stress in xy plane,
1
(
2
ave
y)
x
28.5 ksi
x
y
2
7.5 ksi
2
(a)
xy
9 ksi
a
ave
max
2
max
F .S.
x
R
R
2
40.215 ksi,
34.977 ksi,
max
y
b
11.715 ksi
ave
R
16.875 ksi
0
min
40.215 ksi
min
2
xy
(No yielding)
45 ksi
45
40.215
F .S .
1.119
2
(b)
xy
a
2
x
18 ksi R
ave
max
48 ksi
max
max
y
2
xy
2
R
48 ksi,
9 ksi
0
min
min
R
ave
b
19.5 ksi
48 ksi
(Yielding occurs)
45 ksi
2
(c)
xy
a
max
2
max
x
20 ksi R
ave
R
min
min
2
xy
2
49.86 ksi
49.86 ksi
max
y
b
ave
21.36 ksi
R
7.14 ksi
0
49.86 ksi
(Yielding occurs)
45 ksi
PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use.
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted
on a website, in whole or part.
1122
PROBLEM 7.85
The 38-mm-diameter shaft AB is made of a grade of steel for which the yield
strength is Y 250 MPa. Using the maximum-shearing-stress criterion,
determine the magnitude of the torque T for which yield occurs when
P 240 kN.
B
T
P
A
d = 38 mm
SOLUTION
P
A
x
y
ave
240 103 N
d2
4
4
1.13411 103 mm 2
240 103
1.13411 10
P
A
0
1
(
2
(38) 2
x
y)
x
y
3
1
2
211.62 106 Pa
2
2R
2
xy
2
Y
4
xy
From torsion:
xy
J
c
T
1
2
1
4
2
xy
2
max
2
x
2
xy
4
2
x
2
x
1
250 2
2
2
x
2
Y
211.62 2
66.553 106 Pa
Tc
J
J
c4
1
d
2
2
xy
y
66.553 MPa
2
211.62 MPa
x
2
R
1.13411 10 3 m 2
T
2
38
2
xy
c
4
204.71 103 mm 4
204.71 10 9 m 4
19 10 3 m
(204.71 10 9 )(66.553 106 )
19 10 3
717 N m
PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use.
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted
on a website, in whole or part.
1123
PROBLEM 7.86
Solve Prob. 7.85, using the maximum-distortion-energy criterion.
B
PROBLEM 7.85 The 38-mm-diameter shaft AB is made of a grade of steel
for which the yield strength is Y 250 MPa. Using the maximum-shearingstress criterion, determine the magnitude of the torque T for which yield
occurs when P 240 kN.
T
P
A
d = 38 mm
SOLUTION
P
A
x
y
ave
240 103 N
4
d2
(38) 2
4
1.13411 103 mm 2
240 103
1.13411 10
P
A
0
1
(
2
x
y)
x
y
1
2
211.62 106 Pa
3
2
a
2
b
1
4
2
xy
2
2
xy
2
x
a
ave
R
1
2
x
1
4
2
x
2
xy
b
ave
R
1
2
x
1
4
2
x
2
xy
a b
1
4
2
x
1
4
2
x
2
xy
xy
1
4
x
2
x
3
x
2
xy
1
4
2
xy
2
x
1
4
211.62 MPa
x
2
R
1.13411 10 3 m 2
2
x
2
xy
1
4
2
x
2
xy
2
x
2
xy
1
4
2
x
1
4
2
x
2
xy
2
Y
1 2
2
Y
x
3
1
2502 211.622
3
76.848 MPa
76.848 106 Pa
PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use.
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted
on a website, in whole or part.
1124
PROBLEM 7.86 (Continued)
From torsion,
xy
J
c
T
Tc
J
2
c4
1
d
2
T
38
2 2
J
xy
c
4
204.71 103 mm 4
204.71 10 9 m 4
19 10 3 m
(204.71 10 9 )(76.848 106 )
19 10 3
828 N m
PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use.
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted
on a website, in whole or part.
1125