112 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
[22] Le, Xiaobin, A probabilistic fatigue damage model for describing the entire set of fatigue
test data of the same material, ASME International Mechanical Engineering Congress and
Exposition, IMECE–10224, Salt Lake City, UT, November 8–14, 2019. 74, 75
[23] Zong, W. H. and Le, Xiaobin, Probabilistic Design Method of Mechanical Components,
Shanghai Jiao Tong University Publisher, September 1995, Shanghai, China.
[24] Le, Xiaobin and Peterson, M. L., A method for fatigue-based reliability when the loading
of the component is unknown, International Journal of Fatigue, vol. 21(6), pp. 603–610,
1999. DOI: 10.1016/S0142-1123(99)00016-X. 74, 75
2.12 EXERCISES
2.1. What is fatigue? Describe one fatigue failure example.
2.2. Explain the fatigue failure mechanism. For a component with the same type of material
and dimensions under the same type of cyclic loading, which one will be failure first
if one is machined with a lathe and another one is machined with a polished surface
finish? Explain the choice.
2.3. A group of fatigue test for a steel specimen is listed in Table 2.62. Calculate the slope
of the traditional S-N curve in both ln-axis scales.
Table 2.62: Steel specimen
Stress amplitude (ksi) Sample size Fatigue life (cycles) × 10
3
25.14 4 328.12, 315.87, 337.62
30.38 2 64.18, 71.62
35.81 5 16.53, 16.95, 17.39, 18.03, 17.27
42.85 3 3.32, 3.85, 3.62
2.4. A group of fatigue test for a metal specimen is listed in Table 2.63. Calculate the slope
of the traditional S-N curve in both ln-axis scales.
2.5. A machined bar with a diameter 2.150
00
is subjected to cyclic axial loading. Its ultimate
material strength is 61.5 ksi. If the fatigue strength is obtained from a standard pol-
ished specimen under a fully reversed bending stress, determine k
a
, k
b
, and k
c
for this
component.
2.6. A forged shaft with a diameter 1.750
00
is subjected to cyclic bending stress. Its ultimate
material strength is 91.7 ksi. If the fatigue strength is obtained from a standard pol-
ished specimen under a fully reversed bending stress, determine k
a
, k
b
, and k
c
for this
component.
2.12. EXERCISES 113
Table 2.63: Metal specimen
Stress amplitude (Mpa) Sample size Fatigue life (cycles) × 10
3
700 5 3.28, 3.81, 3.74, 3.69, 3.78
650 3 17.28, 18.09, 16.89
500 3 52.11, 49.83, 53.72
200 4 243.32, 239.97, 245.74, 244.29
2.7. A component is subjected to cyclic bending stress with a stress mean
m
D 18:38 (ksi)
and a stress amplitude
a
D 12:52 (ksi). Its ultimate material strength is 71.8 ksi. Cal-
culate the equivalent stress amplitude of a fully reversed cyclic stress.
2.8. A component is subjected to cyclic axial stress with a maximum stress
max
D 11:78 (ksi)
and a minimum stress
min
D 50:14 (ksi). Its ultimate material strength is 61.5 ksi.
Calculate the equivalent stress amplitude of a fully reversed cyclic stress.
2.9. A shaft shoulder with a fillet radius r D 0:032
00
is subjected cyclic bending stress. Its
theoretical stress concentration factor due to static bending stress is 1.69. e shaft
material ultimate strength is 61.5 ksi. Determine its fatigue stress concentration factor.
2.10. A plate with a center transverse hole is subjected to cyclic axial loading. e radius of
the hole is 0.25
00
. Its theoretical stress concentration factor due to static bending stress
is 2.58. e shaft material ultimate strength is 61.5 ksi. Determine its fatigue stress
concentration factor.
2.11. A machined constant circular beam with a diameter d D 1:250 ˙0:005
00
is subjected a
cyclic bending loading. e mean bending loading M
m
of the cyclic bending moment is
1.72 (klb.in). e bending moment amplitude M
a
of the cyclic bending moment follows
a normal distribution with a mean
M
a
D 2:26 (klb.in) and a standard deviation
M
a
D
0:93 (klb.in). e ultimate material strength is 61.5 (ksi). e component endurance
limit S
e
follows a normal distribution with a mean
S
e
D 22:4 (ksi) and a standard
deviation
S
e
D 1:44 (ksi). is bar is designed to have an infinite life. (1) Establish
the limit state function of this beam. (2) Calculate the reliability of the beam under the
cyclic bending loading.
2.12. A rectangular bar with a height h D 1:250 ˙ 0:005
00
and a width b D 2:250 ˙ 0:005
00
is
subjected to cyclic axial loading with a mean F
m
D 32:15 (klb). e axial loading ampli-
tude F
a
follows a uniform distribution between 7.15 (klb) and 9.25 (klb). e ultimate
material strength is 61.5 (ksi). e component endurance limit S
e
follows a normal
distribution with a mean
S
e
D 17:4 (ksi) and a standard deviation
S
e
D 2:84 (ksi).
is bar is designed to have an infinite life. (1) Establish the limit state function of this
problem. (2) Calculate the reliability of the bar under the cyclic bending loading.
114 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
2.13. A shaft with a diameter d D 1:500 ˙ 0:005
00
is subjected to a cyclic torsion loading.
e torque can be treated as a fully reversed cyclic torsion. e torque amplitude in the
unit of klb-in can be treated as lognormal distribution with a log mean
ln T
D 1:85
and log standard deviation
ln T
D 0:062. e component torsion endurance limit S
e
follows a normal distribution with a mean
S
e
D 12:4 (ksi) and a standard deviation
S
e
D 1:02 (ksi). is shaft is designed to have an infinite life. (1) Establish the limit
state function of this problem. (2) Calculate the reliability of the shaft under the cyclic
bending loading
2.14. What is the P-S-N curve approach? What are the two sets of curves that the P-S-N
curve can provide?
2.15. Conduct literature research and find an example where the P-S-N curves are presented.
2.16. A bar is subjected to cyclic axial stress with a mean stress
m
D 12:6 (ksi) and stress
amplitude
a
D 8:6 (ksi). According to the design specification, the bar has a design
life n
L
D 380;000 (cycles). e bar fatigue life N
C
at the stress level with a mean stress
m
D 12:6 (ksi) and stress amplitude
a
D 8:6 (ksi) follow a normal distribution with
a mean
N
C
D 440;000 (cycles) and a standard deviation
N
C
D 34;500. Calculate its
reliability.
2.17. A beam is subjected to a fully reversed cyclic bending stress with a constant num-
ber of cycles n
L
D 450;000 (cycles). e stress amplitude of this fully reversed cyclic
bending stress
a
follows a Weibull distribution with a scale parameter D 18:25 (ksi)
and a shape parameter ˇ D 1:5. e beam fatigue strength S
f
at the fatigue life N D
450;000 (cycles) follows a normal distribution with a mean
S
f
D 21:98 (ksi) and a
standard deviation
S
f
D 1:78 (ksi). Calculate its reliability
2.18. A square bar is subjected to cyclic fully reversed axial stress with a constant stress ampli-
tude
a
D 24:6 (ksi). Its number of cycles of this fully reversed axial stress can be treated
as a normal distribution with a mean
n
L
D 125;000 (cycles) and a standard deviation
n
L
D 5600 (cycles). e bar fatigue life N
C
at the fatigue strength level S
f
D 24:6 (ksi)
follows a lognormal distribution with a log mean
ln N
C
D 12:13 and a standard devi-
ation
ln N
C
D 0:249. Calculate its reliability.
2.19. A round bar is subjected to three constant cyclic stresses as listed in the 2nd and 3rd
columns of Table 2.64. e distributed fatigue life of this bar at the corresponding stress
levels are listed in the 4th and 5th columns of Table 2.64. Calculate its reliability.
2.20. A beam is subjected to three constant cyclic bending stresses with a distributed number
of cycles as listed in the 2nd, 3rd, and 4th columns of Table 2.65. e distributed fatigue
life of this bar at the corresponding stress levels are listed in the 5th and 6th columns of
Table 2.65. Calculate its reliability.
2.12. EXERCISES 115
Table 2.64: Cyclic stresses
Cyclic Loading Stress Spectrum Component P-N Distributions
Stress
Level #
σ
ai
(ksi) n
Li
Fatigue Life N
Ci
(normal distribution)
μ
N
Ci
σ
N
Ci
1 25 90,800 151,000 12,100
2 35 4,500 8,800 790
3 40 1,825 2,830 235
Table 2.65: Cyclic bending stresses
Cyclic Loading Stress Spectrum Component P-N Distributions
Stress
Level #
σ
ai
(ksi)
Number of Cycles n
Li
(normal distribution)
Fatigue Life N
Ci
(normal
distribution)
μ
n
Li
σ
n
Li
μ
N
Ci
σ
N
Ci
1 25 80,900
5,900
151,000 12,100
2 35 4,050
398
8,800 790
3 40 1,625
150
2,830 235
2.21. A shaft is subjected to model #6 cyclic bending stress. e numbers of cycles of each
stress level are constant and are listed in the 2nd column of Table 2.66. e fully re-
versed bending stress follows a normal distribution, and their distribution parameters
are listed in the 3rd and 4th columns of Table 2.66. e component fatigue strengths
S
f
at the corresponding fatigue life are normally distributed random variable, and their
distribution parameters are listed in the 5th and 6th columns of Table 2.66. Calculate
its reliability.
Table 2.66: Cyclic bending stresses
Cyclic Loading Stress Spectrum Component P-S
f i
Distributions
Stress
Level #
n
Li
(ksi)
Stress Amplitude σ
ai
(normal distribution)
Fatigue Strength S
f i
(normal distribution)
μ
σ
ai
σ
σ
ai
μ
S
fi
σ
S
fi
1 3,000 27.15
2.12
39.77 3.18
2 20,000 21.5
2.19
31.78 2.86
3 300,000 18.34
1.25
23.06 1.73
116 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
2.22. Conduct literature research to find a fatigue test under at least three different stress
level and with more than 30 tests. en use the K-D probabilistic model to determine
its three distribution parameters: m,
ln K
0
, and
ln K
0
.
2.23. A bar with a diameter 1:125 ˙ 0:005
00
is subjected to a constant cyclic axial loading,
which is listed in the first three columns of Table 2.67. e ultimate material strength
S
u
is 75 (ksi). e component fatigue strength index K on the critical section for the
axial cyclic loading is listed in the last three columns of Table 2.67. For the component
fatigue strength index K, the stress unit is ksi. Calculate the reliability of this bar.
Table 2.67: Cyclic axial loading
Cyclic Axial Loading
Component Fatigue Strength Index K
(lognormal distribution)
n
L
F
m
F
a
m μ
lnK
σ
lnK
480,000 20.74 (klb) 18.82 (klb) 8.21 41.738 0.357
2.24. A bar with a diameter 1:500 ˙ 0:005
00
is subjected to a fully reversed cyclic axial loading.
e axial loading amplitude follows a normal distribution. Information of this cyclic
axial loading is listed in the first three columns of Table 2.68. e component fatigue
strength index K on the critical section for the axial cyclic loading is listed in the last
three columns of Table 2.68. For the component fatigue strength index K, the stress
unit is ksi. Calculate the reliability of this bar.
Table 2.68: Cyclic axial loading
Cyclic Axial Loading
Component Fatigue Strength Index K
(lognormal distribution)
n
L
Normally Distributed F
a
m μ
lnK
σ
lnK
μ
F
a
σ
F
a
250,000 50.68 (klb) 4.82 (klb) 8.21 41.738 0.357
2.25. A round bar with a diameter 1:250 ˙ 0:005
00
is subjected to a fully reversed cyclic bend-
ing loading with a constant bending moment amplitude. e number of cycles of this
cyclic bending loading follows a normal distribution. Information of this cyclic bending
loading is listed in the first three columns of Table 2.69. e component fatigue strength
index K on the critical section for the cyclic bending loading is listed in the last three
columns of Table 2.69. For the component fatigue strength index K, the stress unit is
ksi. Calculate the reliability of this bar.
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