2.8. RELIABILITY OF A COMPONENT BY THE P-S-N CURVES APPROACH 55
For the stress level 2 to stress level 3, we have:
ˇ
2
D
ln N
C 2
ln
n
L2eq
ln N
C 2
D
11:01311 ln.27658:8/
0:191
D 3:986858: (d)
n
eq23
D exp
.
9:47966 3:986858 0:195
/
D 6016:27 (e)
n
3eq
D n
L3
C n
eq23
D 2800 C 6016:27 D 8816:27: (f)
e stress level 3 is the last. erefore, the reliability of the component in this example per
Equation (2.49) is
R D ˆ
"
ln N
C 3
ln
n
3eq
ln N
C 3
#
D ˆ
9:47966 ln.8816:27/
0:195
D ˆ.2:02721/ D 0:9787: (g)
In Example 2.14, if we convert the cyclic stress from the stress level 3 to the stress level 2,
and then from the stress level 2 to the stress level 1, we will get the reliability R D ˆ
.
2:07511
/
D
0:9810
. e results are slightly different because it is an approximate estimation with the assump-
tion of the equivalent fatigue damage concept.
2.8.7 RELIABILITY OF A COMPONENT UNDER MODEL #5 CYCLIC
LOADING SPECTRUM
Model #5 cyclic loading spectrum consists of multiple constant stress amplitudes of cyclic load-
ings with corresponding distributed cycle numbers at each cyclic stress level. is section will
discuss how to calculate the reliability of a component under model #5 cyclic loading spectrum.
It is very difficult to create the limit state function of a component under model #5 cyclic
loading spectrum. e author, in 2016, proposed an approach with a modified equivalent fatigue
damage concept to deal with this type of problem [16]. Let us use two stress loading levels with
a distributed number of cycles as an example to explain this approach. e cyclic loading and
corresponding component fatigue life at two different stress levels are listed in Table 2.22. n
Li
is a distributed number of cycles of the fully reversed cyclic loading with a stress amplitude
ai
in the stress level #i. N
Ci
is the distributed component fatigue life at the fully reversed fatigue
strength S
0
f
D
ai
in the stress level #i . e component fatigue life N
Ci
can be determined by
Equations (2.34), (2.35), and (2.36), which have been discussed in Section 2.8.2. Two assump-
tions [16] for this approach are as follows.
Assumption One: e reliability index of the component under cyclic loading is used as an
indirect index for measuring fatigue damage of a component. To transfer a distributed cyclic
number n
Li
at the stress level
ai
to the distributed cyclic number n
Lj
at the stress level
aj
, the
reliability index of the component due to n
Li
at the cyclic stress level
ai
should be equal to
56 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
Table 2.22: Two cyclic stress levels of model #5 cyclic loading spectrum
Stress
Level
#
Model #5 Cyclic Loading Spectrum
Component Fatigue Life N
C
at the
Given Fatigue Strength S
'
f
Cyclic Stress
Amplitude (constant)
Distributed
Number of Cycles
Fatigue Strength
(constant)
Distributed
Fatigue Life N
C
i σ
ai
n
Li
S
'
f
= σ
ai
N
Ci
j σ
aj
n
Lj
S
'
f
= σ
aj
N
Cj
the reliability index of the component due to an equivalent random variable n
ij
. is random
variable n
ij
has the same type of distribution and the same standard deviation as the cyclic
loading n
Lj
, but its mean is the equivalent cyclic number n
eqij
.
Assumption Two: e equivalent cyclic number n
eqij
is a deterministic cyclic number and
will only affect the mean value of the distributed cyclic number at the cyclic stress level
aj
. So the
new distributed cyclic number including the equivalent cyclic number n
eqij
at the cyclic stress
level
aj
will be n
eqij
C n
Lj
, which can be defined as n
j eq
. When compared with the original
random variable n
Lj
, the new random variable n
j eq
will have the same type of distribution and
the same standard deviation but have an increase in its mean value by n
eqij
.
We will discuss two cases in the following sections: (1) both the component fatigue life
N
C
and the cyclic number n
L
of the cyclic loading are the normal distribution; and (2) both
the component fatigue life N
C
and the cyclic number n
L
of the cyclic loading are log-normal
distribution.
Both Normal Distributions
When both the component fatigue life N
C
and the cyclic number n
L
of model #5 cyclic load-
ing at the cyclic stress levels are normal distributions, as shown in Table 2.23. We can use the
following procedure to determine the equivalent cyclic number n
eqij
and the new distributed
cyclic number n
j eq
at the stress level
aj
. en, we can calculate the reliability of a component
under such cyclic loading spectrum.
e limit state function of the component due to the cyclic loading at the stress level #i
is:
g
.
N
Ci
; n
Li
/
D N
Ci
n
Li
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(2.50)
Since both the component fatigue life N
Ci
(strength) and the number of cycles n
Li
of the cyclic
loading are normal distributions, the limit state function will be a normal distribution. e
2.8. RELIABILITY OF A COMPONENT BY THE P-S-N CURVES APPROACH 57
Table 2.23: Normal distributions for n
L
and N
C
Stress
Level
#
Model #5 Cyclic Loading Spectrum Component Fatigue Life N
C
Stress
Level
n
L
(normal distribution)
Fatigue
Strength
Distributed Fatigue Life N
C
Mean
Standard
Deviation
Mean
Standard
Deviation
i σ
ai
μ
n
Li
σ
n
Li
S
'
f
= σ
ai
μ
N
Ci
σ
N
Ci
j σ
aj
μ
n
Lj
σ
n
Lj
S
'
f
= σ
aj
μ
N
Cj
σ
N
Cj
reliability index of the limit state function (2.50) can be directly calculated by the following
equation:
ˇ
i
D
N
Ci
Li
q
.
N
Ci
/
2
C
.
Li
/
2
: (2.51)
Per Assumptions One and Two, the reliability index for the equivalent cyclic number n
eqij
of
the cyclic loading from the stress level #i to the stress level #j should have the same reliability
index ˇ
i
:
ˇ
i
D
N
Cj
n
eqij
q
N
Cj
2
C
Lj
2
: (2.52)
Rearrange Equation (2.52), the equivalent cyclic number n
eqij
from the stress level #i to the
stress level #j is:
n
eqij
D
N
Cj
ˇ
i
q
N
Cj
2
C
n
Lj
2
: (2.53)
Per Assumption Two, the new distributed cyclic number n
j eq
at the stress level
aj
will have
following mean and the same standard deviation:
j eq
D n
eqij
C
L
j
n
j eq
D
L
j
;
(2.54)
where
j eq
and
n
j eq
are the mean and the standard deviation of the normally distributed
random variable n
j eq
.
If there are more than two stress levels in the model #5 cyclic loading spectrum, we can
use above procedures to continuously convert cyclic loading from one level to next level per
Equations (2.51)–(2.54) until the equivalent cyclic loading in the last stress level has included
all transferred equivalent cycles. If the stress level #j is the last stress level, the reliability of the
component under model #5 cyclic loading spectrum is:
R D ˆ
0
B
@
N
Cj
n
j eq
q
N
Cj
2
C
n
Lj
2
1
C
A
; (2.55)
58 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
where ˆ
.
/
is the CDF of standard normal distribution.
Example 2.15
A component is subjected to model #5 cyclic loading spectrum with three stress levels as listed in
Table 2.24. Both the number of cycles n
L
and the component fatigue life N
C
follow normal dis-
tributions. eir distribution parameters at three stress levels are listed in Table 2.24. Calculate
the reliability of this component.
Table 2.24: Normal distributions for n
L
and N
C
for Example 2.15
Stress
Level
#
Cyclic
Stress
Amplitude
Number of cycles n
L
(normal distribution)
Component Fatigue Life N
C
(normal distribution)
Mean (μ
n
Li
)
Standard
Deviation (σ
n
Li
)
Mean (μ
N
Ci
)
Standard
Deviation (σ
N
Ci
)
1 45 (ksi) 11,000 1,200 45,000 3,600
2 40 (ksi) 32,000 5,400 118,800 11,000
3 35 (ksi) 112,000 9,800 356,200 26,000
Solution:
e equivalent cyclic number n
eq12
of the cyclic stress from the stress level 1 to the stress level
2 per Equations (2.51) and (2.53) is:
ˇ
1
D
N
C1
n
L1
q
N
C1
2
C
n
L1
2
D
45000 11000
q
.
3600
/
2
C
.
1200
/
2
D 8:959787 (a)
n
eq12
D
N
C 2
ˇ
1
q
N
C 2
2
C
n
L2
2
D 118800 8:959787
q
.
11000
/
2
C
.
5400
/
2
D 9000:96: (b)
erefore, the number of cycles n
2eq
in the stress level #2 including the transferred number of
cycles from the stress level #1 will have the mean and standard deviation per Equation (2.54):
2eq
D n
eq12
C
L
2
D 9000:96 C 32000 D 51006:96
2eq
D
L
2
D 5400:
(c)
Now, repeat above calculations to convert the cyclic stress at the stress level #2 with the distri-
bution parameters in Equation (c) to an equivalent number of cycles n
eq23
at the stress level
#3.
2.8. RELIABILITY OF A COMPONENT BY THE P-S-N CURVES APPROACH 59
Per Equation (2.51), we have:
ˇ
2
D
N
C 2
2eq
q
N
C 2
2
C
2eq
2
D
118800 51006:96
q
.
11000
/
2
C
.
5400
/
2
D 5:532329: (d)
Per Equation (2.53)
n
eq23
D
N
C 3
ˇ
2
q
N
C 3
2
C
.
L3
/
2
D 356200 5:532329
q
.
26000
/
2
C
.
9800
/
2
D 202480:9: (e)
Per Equation (2.54), the cyclic stress n
3eq
in the stress level #3, including all of the transferred
number of cycles from the stress level #1 and #2 will have a following mean and standard devi-
ation:
3eq
D n
eq23
C
L
3
D 202480:9 C 112000 D 314480:9
3eq
D
L
3
D 9800:
(f)
Since the stress level #3 is the last stress level, we can use Equation (2.55) to calculate the relia-
bility of the component by using the equivalent number of cycles n
3eq
, which includes all three
cyclic stresses.
R D ˆ
0
B
@
N
C 3
3eq
q
N
C 3
2
C
.
L3
/
2
1
C
A
D ˆ
356200 314480:9
p
.26000/
2
C .9800/
2
!
D ˆ
.
1:501465
/
D 0:9334: (g)
Both Lognormal Distributions
When both the component fatigue life N
C
and the cyclic number n
L
of model #5 cyclic loading
at the cyclic stress levels are lognormal distributions, as shown in Table 2.25. We can use the
following procedure to determine the equivalent cyclic number n
eqij
and the new distributed
cyclic number n
j eq
at the stress level
aj
. en, we can calculate the reliability of a component
under model #5 cyclic loading spectrum.
For both lognormal distribution, the event .N
Ci
> n
Li
/ is the same as the event .ln.N
C i
/ >
ln.n
Li
// because both N
Ci
and n
Li
are positive. e limit state function of the component due to
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