52 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
erefore, the range of the reliability of the component with a 95% confidence level will be:
R D 1 F D 0:9464 ˙ 0:0004:
2.8.6 RELIABILITY OF A COMPONENT UNDER MODEL #4 CYCLIC
LOADING SPECTRUM
Model #4 cyclic loading spectrum is multiple constant cyclic stress levels with multiple constant
numbers of cycles, which has been discussed in Section 1.2. When a component is under the
model #4 cyclic loading spectrum, no direct limit state function can be established. However,
the equivalent fatigue damage concept [11, 15] proposed by Dr. Dimitri B. Kececioglu in 1977
can be used to estimate the reliability of the component under such cyclic loading spectrum. e
assumption in the approach is that the cyclic number at a cyclic stress level could be transferred
to another stress level with an equivalent cyclic number under the condition that the probability
of a safe status of the component at the original stress level is the same as that at the transferred
stress level with the equivalent cyclic number. In this approach, the reliability index of the com-
ponent under cyclic loading is used as an indirect index for measuring the fatigue damage of a
component.
Let us use two levels of the model #4 cyclic loading spectrum as listed in Table 2.20 to
demonstrate the equivalent fatigue damage concept and procedure. e corresponding fatigue
life distributions are also listed in Table 2.20. In Table 2.20,
ai
and
aj
are the fully reversed
cyclic stress amplitudes in the cyclic stress levels i and j . n
Li
and n
Lj
are the numbers of cycles in
the cyclic stress levels i and j . ey are all constants for the model #4 cyclic loading spectrum,
that is, deterministic values.
S
0
f
is material fatigue strength, which is equal to the cyclic stress
amplitude of the corresponding stress level, as shown in Table 2.20. e component fatigue life
N
C
at the given fatigue strength S
0
f
D
ai
is typically a lognormal distribution, which has been
discussed in Section 2.8.2. It can be obtained per Equation (2.34). Its distribution parameters
can be calculated per Equations (2.35) and (2.36). In Table 2.20,
ln N
Ci
and
ln N
Ci
are the mean
and the standard deviation of the component fatigue life N
Ci
at the cyclic stress level i.
ln N
Cj
and
ln N
Cj
are the mean and the standard deviation of the component fatigue life N
Cj
at the
cyclic stress level j .
e general procedure for transferring cyclic stress level from stress level i to the stress
level j is described as the following.
Step 1: Calculate the index of the fatigue damage of the component due to the cyclic stress
.
ai
; n
Li
/ in the stress level i.
e index of the fatigue damage of the component due to the cyclic stress .
ai
; n
Li
/ in the stress
level #i can be indirectly represented by the probability P
.
N
C
> n
Li
/
D P
Œ
ln
.
N
C
/
> ln
.
n
Li
/
,
2.8. RELIABILITY OF A COMPONENT BY THE P-S-N CURVES APPROACH 53
Table 2.20: Two cyclic stress levels and corresponding component fatigue life
Model #4 Cyclic Loading Spectrum
Component Fatigue Life N
C
(lognormal distribution)
Stress level # Cyclic stress
Number of
cycles
Cyclic stress
μ
lnN
C
σ
lnN
C
i σ
ai
n
Li
S
'
f
= σ
ai
μ
lnN
Ci
σ
lnN
Ci
j σ
aj
n
Lj
S
'
f
= σ
aj
μ
lnN
Cj
σ
lnN
Cj
which is the reliability of the component under the cyclic stress level #i. e reliability can
be directly represented by the reliability index ˇ. erefore, we can use the reliability index to
represent fatigue damage due to the cyclic loading. e reliability index ˇ
i
of the component in
the cyclic stress level #i will be:
ˇ
i
D
ln N
Ci
ln
.
n
Li
/
ln N
Ci
: (2.45)
Step 2: Calculate the equivalent number of cycles from the cyclic stress in stress level #i to the
cyclic stress level #j .
Let n
eqij
represent the equivalent number of cycles to the cyclic stress level #j . According to
the equivalent fatigue damage concept, the reliability index of this n
eqij
in the stress level #j
will be
ˇ
i
. erefore, we have:
ˇ
i
D
ln N
Cj
ln
n
eqij
ln N
Cj
: (2.46)
By rearranging Equation (2.46), we have:
n
eqij
D exp
ln N
Cj
ˇ
i
ln N
Cj
: (2.47)
Step 3: Combine cyclic stresses of the two-stress levels.
e component fatigue damage due to the two cyclic stresses .
ai
; n
Li
/ and .
aj
; n
Lj
/ in two stress
levels #i and #j will be equal to the component fatigue damage due to cyclic stress .
aj
; n
Lj
C
n
eqij
/ in the stress level #j . So, the total equivalent number of cycles n
j eq
in the stress level
j is:
n
j eq
D n
Lj
C n
eqij
; (2.48)
where n
j eq
is the equivalent number of cycles, including the original number of cycles in current
stress level and the transferred equivalent number of cycles from another stress level.
54 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
Step 4: Calculate the reliability of the component due to model #4 cyclic loading spectrum.
If there are more than two stress levels in the model #4 cyclic loading spectrum, we can repeat
Step 1 to Step 3 to combine two stress levels into the next stress level, until we reach the last
stress level. Let us assume that the j stress level is the last stress level of model #4 cyclic loading
spectrum. e reliability of the component due to all stress levels will be:
R D ˆ
"
ln N
Cj
ln
n
Lj
C n
eqij
ln N
Cj
#
: (2.49)
If the component fatigue life N
C
follows another type of distribution, we can follow the
above steps to run a similar calculation by using the equivalent fatigue damage concept.
Example 2.14
A component is subjected to model #4 cyclic loading spectrum, as shown in Table 2.21. e
component fatigue life N
c
under the corresponding stress levels are lognormal distributions, as
shown in Table 2.21. Use the equivalent fatigue damage concept to calculate the reliability of
the component.
Table 2.21: Model #4 cyclic loading spectrum for Example 2.14
Model #4 Cyclic Loading Spectrum
Component Fatigue Life N
C
(lognormal distribution)
Level #
Cyclic stress
(psi)
Number of
cycles
Cyclic stress
(psi)
μ
lnN
C
σ
lnN
C
1 65,000 81,000 65,000 12.95987 0.198
2 85,000 16,000 85,000 11.01311 0.197
3 105,000 2,800 105,000 9.47966 0.195
Solution:
We can use Equations (2.45), (2.47), and (2.48) twice to convert the number of cycles in stress
level 1 to level 2, and then from stress level 2 to the last stress level 3.
For the stress level 1 to stress level 2, we have:
ˇ
1
D
ln N
C1
ln
.
n
L1
/
ln N
C1
D
12:95987 ln.81000/
0:198
D 8:437205 (a)
n
eq12
D exp
.
11:01311 8:437205 0:197
/
D 11658:8 (b)
n
2eq
D n
L2
C n
eq12
D 16000 C 11658:8 D 27658:8: (c)
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