64 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
2.8.8 RELIABILITY OF A COMPONENT UNDER MODEL #6 CYCLIC
LOADING SPECTRUM
Model #6 cyclic loading spectrum is several distributed cyclic stress amplitudes at specified cycle
numbers, that is, .n
Li
;
ai
; i D 1; 2; : : : /. Here, n
Li
is a constant number of cycles in the cyclic
number level #i. Fully reversed cyclic stress level
ai
in the cyclic number level #i is a distributed
random variable. e corresponding component fatigue strength data will be the component
fatigue strength S
cfi
at the given fatigue life N D n
Li
. e component fatigue strength S
cfi
at
the given fatigue life N D n
Li
can be calculated per Equation (2.29). Its mean and standard
deviation can be calculated through Equations (2.30) and (2.31). is section will discuss how
to calculate the reliability of a component under model #6 cyclic loading spectrum.
It is difficult to establish the limit state function of a component under model #6 cyclic
loading spectrum. It is typically assumed that the influence of the sequence of cyclic loading on
fatigue life or fatigue damage can be negligible. erefore, each loading condition in the model
#6 can be treated as an independent event. e author proposed an approach [17] in 2017 to
estimate the reliability of a component under such cyclic loading spectrum. is approach has
the following two assumptions.
Assumption One: Based on the concepts of the widely accepted Miner rule [7, 10, 18], the
effect of the sequence of cyclic loading on the fatigue damage during the service life of the
component can be ignored, so that each cyclic loading stress condition .n
Li
;
ai
/ can be treated
as an independent random event.
Assumption Two: Since the fatigue damage of the component due to these independent cyclic
stress conditions is assumed to be independent, the estimation of the reliability R of the com-
ponent under Model #6 .n
Li
;
ai
/ is equal to the multiplication of the reliability R
i
of the com-
ponent under each cyclic loading condition .n
Li
;
ai
/.
Assumption One is mainly based on the widely accepted linear cumulative fatigue damage
theory. Assumption Two is a natural extension of Assumption One, but it is the expression of
the reliability computational method. So, according to Assumption Two, the reliability of a
component under the model #6 cyclic loading spectrum can be modeled as a series of reliability
block diagrams, each of which represents the component under each cyclic loading condition.
us, the reliability R of a component under model #6 cyclic loading spectrum is:
R D
L
Y
iD1
R
i
; (2.66)
2.8. RELIABILITY OF A COMPONENT BY THE P-S-N CURVES APPROACH 65
R
i
is the reliability of the component under cyclic stress .n
Li
;
ai
/ and can be calculated based
on the following limit state function of the component under cyclic stress .n
Li
;
ai
/:
g
S
cfi
;
ai
D S
cfi
ai
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure;
(2.67)
S
cfi
and
ai
are the component fatigue strength and the cyclic stress amplitude at the cyclic
number level #i.
When both the fatigue strength S
cfi
and the cyclic loading stress level
ai
at the given cycle
number n
Li
are normal distributions, the reliability R
i
can be directly calculated as the following:
R
i
D P
S
cfi
>
ai
D ˆ
2
6
4
S
cfi
ai
q
S
cfi
2
C
.
ai
/
2
3
7
5
; (2.68)
where
S
cfi
and
S
cfi
are the mean and the standard deviation of normally distributed S
cfi
.
ai
and
ai
are the mean and the standard deviation of normally distributed
ai
. ˆ
.
/
is the CDF
of standard normal distribution.
When both the fatigue strength S
cfi
and the cyclic loading stress level
ai
at the given
cycle number n
Li
are lognormal distributions, the reliability R
i
can be directly calculated as the
following:
R
i
D P
S
cfi
>
ai
D P
ln.S
cfi
/ > ln.
ai
D ˆ
2
6
4
ln S
cfi
ln
ai
q
ln S
cfi
2
C
.
ln
ai
/
2
3
7
5
; (2.69)
where
ln S
cfi
and
ln S
cfi
are the log-mean and the log standard deviation of log-normally dis-
tributed S
cfi
.
ln
ai
and
ln
ai
are the log-mean and the log standard deviation of log-normally
distributed
ai
. ˆ
.
/
is the CDF of standard normal distribution.
For a general case with any other type of distributions for S
cfi
and
ai
, the H-L, R-F
method, or Monte Carlo method discussed in Chapter 3 can be used to calculate the reliability
of the component per the limit state function Equation (2.67).
Example 2.17
A component is subjected to two distinguished fully reversed cyclic bending stresses due to two
designed functions. e fully reversed cyclic bending stresses can be described by two normal
distributed stress amplitudes at 8,000 cycles and 200,000 cycles, as shown in Table 2.27. e cor-
responding fatigue strength of the component at the given fatigue life 8,000 cycles and 200,000
cycles can be described as a normal distribution, as shown in Table 2.27. Calculate the reliability
of the component.
66 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
Table 2.27: Distribution parameters of S
cfi
and
ai
for Example 2.17
Level
#
Number of
Cycles n
Li
(constant)
Cyclic Stress Amplitude σ
ai
(ksi) (normal distribution)
Component Fatigue Strength S
cfi
(ksi) (normal distribution)
μ
σ
ai
σ
σ
ai
μ
S
cfi
σ
S
cfi
1 8,000 34.25 4.15 50.19 4.72
2 200,000 29.13 2.78 37.72 3.16
Solution:
Since both the component fatigue strength S
cfi
and the cyclic stress amplitude
ai
are normal
distributions, the reliability R
1
of the component under cyclic stress
a1
in the cyclic number
level #1 per Equation (2.68) is:
R
1
D P
S
cf 1
>
a1
D ˆ
2
6
4
S
cf 1
a1
q
S
cf 1
2
C
a1
2
3
7
5
D ˆ
2
6
4
50:19 34:25
q
.
4:72
/
2
C
.
4:15
/
2
3
7
5
D ˆ.2:536208/ D 0:994397: (a)
Repeat the same calculation in the cyclic number level #2 per Equation (2.68),
R
2
D P
S
cf 2
>
a2
D ˆ
2
6
4
S
cf 2
a2
q
S
cf 2
2
C
a2
2
3
7
5
D ˆ
"
37:72 29:13
p
.3:16/
2
C .2:78/
2
#
D ˆ.2:040962/ D 0:979373: (b)
Per Equation (2.66), the reliability of the component under this model #6 cyclic loading spec-
trum is:
R D
2
Y
iD1
R
i
D R
1
R
2
D 0:994397 0:979373 D 0:9739: (c)
2.8. RELIABILITY OF A COMPONENT BY THE P-S-N CURVES APPROACH 67
Example 2.18
A component is subjected to two distinguished fully reversed cyclic bending stresses due to
two designed functions. e fully reversed cyclic bending stresses can be described by two nor-
mally distributed stress amplitudes at 5,000 cycles and 300,000 cycles, as shown in Table 2.28.
e corresponding fatigue strength of the component at the given fatigue life 5,000 cycles and
300,000 cycles can be described as log-normal distribution, as shown in Table 2.28. Calculate
the reliability of the component.
Table 2.28: Distribution parameters of S
cfi
and
ai
for Example 2.18
Level
#
Number of
Cycles n
Li
(constant)
Cyclic Stress Amplitude σ
ai
(ksi) (normal distribution)
Component Fatigue Strength S
cfi
(ksi) (lognormal distribution)
μ
σ
ai
σ
σ
ai
μ
lnS
cfi
σ
lnS
cfi
1 5,000 54.2 6.775 4.3562 0.0321
2 300,000 45.2 5.3336 4.0507 0.0315
Solution:
Since the component fatigue strength S
cfi
are lognormal distributions and the cyclic stress am-
plitude
ai
are normal distributions, we need to use Equation (2.68) to establish the limit state
functions.
e limit state functions for the cyclic number level #1 and #2 are
g
S
cf 1
;
a1
D S
cf 1
a1
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure
(a)
g
S
cf 2
;
a3
D S
cf 2
a2
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(b)
Based on the limit state functions (a) and (b), the R-F method can be used to calculate their
reliabilities. We can follow the procedure and the flowchart of the R-F method presented in Ap-
pendix A.2 to compile a MATLAB program. e iterative results for the limit state function (a)
are listed in Table 2.29. From the iterative results, the reliability index ˇ
1
and corresponding
reliability R
1
of the component in this example are:
ˇ
1
D 3:296752I R
1
D ˆ.3:296752/ D 0:999511:
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