2.12. EXERCISES 117
Table 2.69: Reverse cyclic bending loading
Cyclic Bending Loading
Component Fatigue Strength Index K
(lognormal distribution)
Normally Distributed n
L
M
a
m μ
lnK
σ
lnK
μ
n
L
σ
n
L
200,000 8,900 8.86 (klb/in) 8.21 41.738 0.357
2.26. A square beam with a side height 1:75 ˙ 0:005
00
is subjected to several cyclic bending
loading. e number of cycles and the bending moment amplitudes of this cyclic bend-
ing loading are all constants. Information of this fully reversed bending loading is listed
in the first three columns of Table 2.70. e component fatigue strength index K on
the critical section for the cyclic bending loading is listed in the last three columns of
Table 2.70. For the component fatigue strength index K, the stress unit is ksi. Calculate
the reliability of this beam.
Table 2.70: Several cyclic bending loading
Cyclic Bending Loading
Component Fatigue Strength Index K
(lognormal distribution)
n
L
M
m
(klb.in) M
a
(klb.in) m μ
lnK
σ
lnK
3,500 10.64 18.24
8.21 41.738 0.35730,000 10.64 14.73
450,000 10.64 8.75
2.27. A square beam with a side height
1:500
˙
0:005
00
is subjected to several fully reversed
cyclic bending loading. e number of cycles in each loading level is constants. e
bending moment amplitudes follow normal distributions. Information of this fully re-
versed bending loading is listed in the first three columns of Table 2.71. e component
fatigue strength index K on the critical section for the cyclic bending loading is listed
in the last three columns of Table 2.71. For the component fatigue strength index K,
the stress unit is ksi. Calculate the reliability of this bar.
2.28. A shaft with a diameter 1:125 ˙0:005
00
is subjected to several fully reversed cyclic tor-
sion loading. e torsion amplitude in each loading level is constant. e number of
cycles in each torsion level follows a normal distribution. Information of this fully re-
versed torsion loading is listed in the first three columns of Table 2.72. e component
fatigue strength index K on the critical section for the cyclic torsion loading is listed in
118 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
Table 2.71: Fully reversed cyclic bending loading
Cyclic Bending Loading
Component Fatigue Strength Index K
(lognormal distribution)
n
L
Normally Distrubuted M
a
m μ
lnK
σ
lnK
μ
M
a
(klb.in) σ
M
a
(klb.in)
2,500 16.51 1.49
8.21 41.738 0.35740,000 11.08 0.89
350,000 7.86 0.53
Table 2.72: Fully reversed cyclic torsion loading
Cyclic Torsion Loading
Component Fatigue Strength Index K
(lognormal distribution)
Normally Distributed n
L
T
a
(klb.in)
m μ
lnK
σ
lnK
μ
n
L
σ
n
L
2,000 160 8.84
8.21 37.308 0.518
560,000 35,400 3.87
the last three columns of Table 2.72. For the component fatigue strength index K, the
stress unit is ksi. Calculate the reliability of this shaft.
2.29. e critical section of a solid rotating shaft with a diameter 1:250 ˙ 0:005
00
is subjected
to cyclic combined loading. According to the design specification, the cyclic loading
spectrum can be described by several stress levels and listed in the first three columns
of Table 2.73. e material ultimate tensile strength S
u
of the shaft is 75 (ksi). e
component fatigue strength index K on the critical section for the cyclic bending stress
is listed in the last three columns of Table 2.73. For the component fatigue strength
index K, the stress unit is ksi. Calculate the reliability of the shaft.
Table 2.73: Cyclic combined loading
Cyclic Axial Loading
Component Fatigue Strength Index K
(lognormal distribution)
n
L
T
i
(klb.in)
M
i
(klb.in) m μ
lnK
σ
lnK
3,000 4.78 7.32
8.21 41.738 0.357
200,000 4.78 4.16
2.12. EXERCISES 119
2.30. e critical section of a solid rotating shaft with a diameter 2:250 ˙ 0:005
00
is subjected
to cyclic combined loading. According to the design specification, the cyclic loading
spectrum can be described by several stress levels with distributed stress amplitudes and
are listed in the first four columns of Table 2.74. e material ultimate tensile strength
S
u
of the shaft is 75 (ksi). e component fatigue strength index K on the critical
section for the cyclic bending stress is listed in the last three columns of Table 2.74. For
the component fatigue strength index K, the stress unit is ksi. Calculate the reliability
of the rotating shaft.
Table 2.74: Cyclic combined loading
Cyclic Combined Loading
Component Fatigue Strength Index
K (lognormal distribution)
n
L
T
(klb.in)
Normally Distributed M
m μ
lnK
σ
lnK
μ
M
(klb.in) σ
M
(klb.in)
20,000 21.83
34.92 2.56
8.21 41.738 0.357
400,000 21.83
20.72 1.94
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