208 3. THE DIMENSION OF A COMPONENT WITH REQUIRED RELIABILITY
3.28. Use the modified Monte Carlo method to do Problem 3.27.
3.29. e critical section of a machined rotating shaft is at the shoulder section, as shown in
Figure 3.20. e bending moment M on the shoulder section follows a normal distribu-
tion with a mean
M
D 1:5 (klb.in) and a standard deviation
M
D 0:25 (klb.in). e
shaft material’s ultimate strength is 61.5 (ksi). Its endurance limit S
0
e
follows a normal
distribution with a mean
S
0
e
D 24:7 (ksi) and a standard deviation
S
0
e
D 2:14 (ksi),
which are based on the fully reversed bending fatigue specimen tests. is shaft is de-
signed to have an infinite life. Determine the diameter d of the shaft with the required
reliability 0.99 when it has a dimension tolerance
˙
0:005
00
.
1
32
R "
Ø 2.000"
d
Figure 3.20: Schematic of a segment of a shaft.
3.30. Use a modified Monte Carlo method to do Problem 3.29.
3.31. A machined constant circular bar is subjected to cyclic axial loading spectrum. Accord-
ing to the design specification, the mean of the cyclic axial loading is F
m
D 14:21 (klb).
e amplitude F
a
of the cyclic axial loading follows a uniform distribution between
8.25 (klb) and 12.25 (klb). e number of cycles of this cyclic axial loading is n
L
D
450;000 (cycles). e ultimate material strength S
u
is 75 (ksi). e three distribution
parameters of material fatigue strength index K
0
for the standard specimen under fully-
reversed bending stress are m D 8:21;
ln K
0
D 41:738, and
ln K
0
D 0:357. For the ma-
terial fatigue strength index K
0
, the stress unit is ksi. Use the modified R-F method to
determine the diameter of the bar with the reliability 0.95 when its dimension tolerance
is ˙0:005
00
.
3.32. Use the modified Monte Carlo method to do Problem 3.31.
3.33. A machined stepped circular bar, as shown in Figure 3.20 is subjected to cyclic axial load-
ing spectrum. According to the design specification, the mean F
m
and amplitude F
a
of
the cyclic loading are F
m
D 10:68 (klb) and F
a
D 7:82 (klb). e number of cycles of this
cyclic axial loading follows a normal distribution with a mean
n
L
D 350;000 (cycles)
and a standard deviation
n
L
D 14;000 (cycles). e ultimate material strength S
u
is
3.7. EXERCISES 209
75 (ksi). e three distribution parameters of material fatigue strength index K
0
for the
standard specimen under fully-reversed bending stress are m D 8:21;
ln K
0
D 41:738,
and
ln K
0
D 0:357. For the material fatigue strength index K
0
, the stress unit is ksi.
Use the modified R-F method to determine the diameter of the bar with the required
reliability 0.99 when its dimension tolerance is ˙0:005
00
.
3.34. Use the modified Monte Carlo method to do Problem 3.33.
3.35. A machined single-shear pin is under a cyclic shearing loading spectrum. e mean
of the cyclic shearing loading is V
m
D 12:7 (klb). e amplitude of the cyclic shearing
loading follows a normal distribution with a mean
V
a
D 8:39 (klb) and a standard
deviation
V
a
D 1:35 (klb). e number of cycles n
L
of this cyclic shearing loading is a
constant n
L
D 500;000 (cycles). e ultimate material strength S
u
is 75 (ksi). e three
distribution parameters of material fatigue strength index K
0
for the standard specimen
under fully-reversed bending stress are m D 8:21;
ln K
0
D 41:738, and
ln K
0
D 0:357.
For the material fatigue strength index K
0
, the stress unit is ksi. Use the modified R-F
method to determine the diameter of the pin with the required reliability 0.95 when its
dimension tolerance is ˙0:005
00
.
3.36. Use the modified Monte Carlo method to do Problem 3.35.
3.37. A machined double-shear pin is subjected to a cyclic shear loading spectrum. According
to the design specification, the cyclic shear loading spectrum is listed in Table 3.72. e
ultimate material strength S
u
is 75 (ksi). e three distribution parameters of material
fatigue strength index K
0
for the standard specimen under fully reversed bending stress
are m D 8:21;
ln K
0
D 41:738, and
ln K
0
D 0:357. For the material fatigue strength
index K
0
, the stress unit is ksi. Determine the diameter of the pin with a reliability 0.99
when its dimension tolerance is ˙0:005
00
.
Table 3.72: e cyclic shear loading for Problem 3.37
Loading
Level #
Number of
Cycles n
L
Mean of the Cyclic
Shear Loading V
m
(klb)
Amplitude of the Cyclic
Shear Loading V
a
(klb)
1 4,000 3.422 6.251
2 500,000 3.422 4.815
3.38. Use the modified Monte Carlo method to do Problem 3.37.
3.39. e critical section of a machined shaft with a shoulder is at the shoulder section, as
shown in Figure 3.20. It is subjected to a cyclic torque loading spectrum. According to
the design specification, the cyclic torsion loading spectrum is listed in Table 3.73. e
210 3. THE DIMENSION OF A COMPONENT WITH REQUIRED RELIABILITY
ultimate material strength S
u
is 75 (ksi). e three distribution parameters of material
fatigue strength index K
0
for the standard specimen under fully-reversed bending stress
are m D 8:21;
ln K
0
D 41:738, and
ln K
0
D 0:357. For the material fatigue strength
index K
0
, the stress unit is ksi. Determine the diameter of the shaft with the required
reliability 0.99 when its dimension tolerance is ˙0:005
00
.
Table 3.73: e cyclic fatigue spectrum for example Problem 3.39
Loading
Level #
Number of Cycles n
Li
Mean T
mi
of the Cyclic
Torque (klb.in)
Amplitude T
ai
of the
Cyclic Torque (klb.in)
μ
n
Li
σ
n
Li
1 450,000 26,000 1.89 4.32
2 6,500 500 1.89 6.09
3.40. Use the modified Monte Carlo method to do Problem 3.39.
3.41. A machined constant circular shaft is subjected to a cyclic torque loading spectrum.
According to the design specification, the cyclic torsion loading spectrum is listed in
Table 3.74. e ultimate material strength S
u
is 75 (ksi). e three distribution pa-
rameters of material fatigue strength index K
0
for the standard specimen under fully
reversed bending stress are m D 8:21;
ln K
0
D 41:738, and
ln K
0
D 0:357. For the ma-
terial fatigue strength index K
0
, the stress unit is ksi. Use the modified R-F method to
determine the diameter of the shaft with a reliability 0.99 when its dimension tolerance
is ˙0:005
00
.
Table 3.74: e cyclic fatigue spectrum for example Problem 3.41
Loading
Level#
Number of Cycles n
Li
Torque Mean T
m
(klb.in)
Torque Amplitude T
a
(klb.in)
μ
T
a
σ
T
a
1 4,800,000 3.5 4.2 0.32
2 6,000 3.5 7.5 0.86
3.42. Use the modified Monte Carlo method to do Problem 3.41.
3.43. e constant solid rotating shaft is subjected to cyclic combined bending-torque load-
ing. According to the design specification, the combined cyclic loadings are listed in Ta-
ble 3.75. e number of cycles of this combined loading is n
L
D 520;000 (cycles). e
ultimate material strength S
u
is 75 (ksi). e three distribution parameters of material
fatigue strength index K
0
for the standard specimen under fully reversed bending stress
3.7. EXERCISES 211
are m D 8:21;
ln K
0
D 41:738, and
ln K
0
D 0:357. For the material fatigue strength
index K
0
, the stress unit is ksi. Determine the diameter of the shaft with a reliability
0.99 when its dimension tolerance is ˙0:005
00
.
Table 3.75: e cyclic combined loading spectrum for Example 3.43
Loading
Level #
Number of
Cycles n
Li
Torque T
i
(klb.in) Bending Moment M
i
(klb.in)
1 6,500 3.52 5.75
2 450,000 3.52 3.15
3.44. Use the modified Monte Carlo method to do Problem 3.43.
3.45. e critical section of a rotating shaft is at the shoulder section, as shown in Figure 3.20
and is subjected to cyclic combined bending-torque loading. According to the design
specification, the torque T is a constant 4:0 (klb.in). e bending moment M follows
a uniform distribution between 2.85 (klb.in) and 3.5 (klb.in). e number of cycles of
this combined loading is n
L
D 520;000 (cycles). e ultimate material strength S
u
is
75 (ksi). e three distribution parameters of material fatigue strength index K
0
for the
standard specimen under fully reversed bending stress are m D 8:21;
ln K
0
D 41:738,
and
ln K
0
D 0:357. For the material fatigue strength index K
0
, the stress unit is ksi.
Determine the diameter of the shaft with a reliability 0.99 when its dimension tolerance
is ˙0:005
00
.
3.46. Use the Monte Carlo method to do Problem 3.45.
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