60 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
Table 2.25: Log-normal distributions for n
L
and N
C
Stress
Level
#
Cyclic Stress
Amplitude
(constant)
Number of Cycles n
L
(lognormal distribution)
Component Fatigue Life N
C
(lognormal distribution)
Mean
Standard
Deviation
Standard
Deviation
Mean
i σ
ai
μ
lnn
Li
σ
lnn
Li
μ
lnN
Ci
σ
lnN
Ci
j σ
aj
μ
lnn
Lj
σ
lnn
Lj
μ
lnN
Cj
σ
lnN
Cj
the cyclic loading at the stress level #i is:
g
.
N
Ci
; n
Li
/
D ln
.
N
Ci
/
ln
.
n
Li
/
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(2.56)
Now, both ln.N
Ci
/ and ln.n
Li
/ are a normal distribution. We can repeat the same calculations as
those in Section 2.8.7. e reliability index of the limit state function (2.56) in the stress level
#i is:
ˇ
i
D
ln N
Ci
ln Li
q
.
ln N
Ci
/
2
C
.
ln Li
/
2
: (2.57)
e 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
ln N
Cj
ln
n
eqij
q
N
Cj
2
C
Lj
2
: (2.58)
From Equation (2.58), the equivalent cyclic number n
eqij
from the stress level i to the stress
level j is:
n
eqij
D exp
ln N
Cj
ˇ
i
q
ln N
Cj
2
C
ln n
Lj
2
: (2.59)
Per Assumption Two in Section 2.8.7, n
eqij
will be added to the mean of n
Lj
. e mean of the
lognormally distributed n
Lj
with the log mean
ln n
Lj
and log standard deviation
ln n
Lj
can be
calculated by the following Equation (2.60) [8]:
n
Lj
D exp
"
ln n
Lj
C
ln n
Lj
2
2
#
: (2.60)
2.8. RELIABILITY OF A COMPONENT BY THE P-S-N CURVES APPROACH 61
erefore, the new mean for the new distributed cyclic number n
j eq
at the stress level
aj
will
be:
j eq
D n
eqij
C
n
Lj
D n
eqij
C exp
"
ln n
Lj
C
ln n
Lj
2
2
#
: (2.61)
e new distributed cyclic number n
j eq
at the stress level
aj
will still be a lognormal distribu-
tion. e log standard deviation of the new distributed cyclic number n
j eq
is the same as that
of n
Lj
.
ln n
j eq
D
ln L
j
: (2.62)
For a lognormal distribution, the log mean of the new distributed cyclic number n
j eq
is:
j eq
D exp
"
ln n
j eq
C
ln n
j eq
2
2
#
: (2.63)
Rearranging Equation (2.63), the log mean of the new distributed cyclic number
n
j eq
is
ln n
j eq
D ln
j eq
ln n
j eq
2
2
; (2.64)
where
ln n
j
eq
and
ln n
j
eq
are the log mean and the log standard deviation of the lognormally
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 stress from one level to next level per Equa-
tions (2.57)–(2.64) until the equivalent cyclic stress in the last stress level has included all trans-
ferred 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
@
ln N
Cj
ln n
j eq
q
ln N
Cj
2
C
ln n
j eq
2
1
C
A
; (2.65)
where ˆ
.
/
is the CDF of standard normal distribution.
Example 2.16
e component made of steel is subjected to model #5 cyclic loading spectrum with three fully
reversed bending stress levels. Both the numbers of cycles of the fully reversed bending stress
n
L
and the component fatigue life N
C
at each stress level are lognormal distributions as listed
in Table 2.26. Calculate the reliability of the component under such cyclic loading spectrum.
62 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
Table 2.26: Lognormal distributions for n
L
and N
C
for Example 2.16
Stress
Level
#
Stress
Amplitude
(ksi)
Number of Cycles n
L
(lognormal distribution)
Component Fatigue Life N
C
(lognormal distribution)
μ
lnn
Li
σ
lnn
Li
μ
lnN
Ci
σ
ln
N
Ci
1 66 10.8 0.19 12.86454 0.24868
2 86 8.9 0.18 10.85669 0.15658
3 106 8.1 0.16 9.44520 0.16809
Solution:
e equivalent cyclic number n
eq12
of the cyclic stress from the stress level #1 to the stress
level #2 per Equations (2.57) and (2.59) is:
ˇ
1
D
ln N
C1
ln L1
q
ln N
C1
2
C
.
ln L1
/
2
D
12:86454 10:8
q
.
0:24868
/
2
C
.
0:19
/
2
D 6:596892 (a)
n
eq12
D exp
ln N
C 2
ˇ
1
q
ln N
C 2
2
C
ln n
L2
2
D exp
10:85699 6:596892
q
.
0:15658
/
2
C
.
0:18
/
2
D 10751:99; (b)
the new mean for the new distributed cyclic number n
2eq
at the stress level #2 per Equa-
tion (2.61) is:
2eq
D n
eq12
C exp
"
ln n
L2
C
ln n
L2
2
2
#
D 10751:99 C exp
"
8:9 C
.
0:18
/
2
2
#
D 18203:71: (c)
e log standard deviation of the new distributed cyclic number n
2eq
per Equation (2.62) is
ln n
2eq
D
ln L
2
D 0:18 (d)
the log mean of the new distributed cyclic number n
2eq
per Equation (2.64) is
ln n
2eq
D ln
2eq
ln n
2eq
2
2
D ln
.
18203;71
/
.
0:18
/
2
2
D 9:793181: (e)
Now, repeat above calculations to convert the cyclic stress at the stress level #2 with the distri-
bution parameters in Equations (d) and (e) 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 63
Per Equation (2.57), the reliability index ˇ
2
of the component under the new distributed
cyclic number n
2eq
in the stress level #2 is:
ˇ
2
D
ln N
C 2
ln n
2eq
q
ln N
C 2
2
C
.
ln L2
/
2
D
10:85699 9:793181
q
.0:15658/
2
C
.
0:18
/
2
D 4:457785: (f )
Per Equation (2.59), the equivalent cyclic number n
eq23
from the stress level #2 to the stress
level 3:
n
eq23
D exp
ln N
C 3
ˇ
2
q
ln N
C 3
2
C
ln n
L
3
2
D exp
9:4452 4:457785
q
.
0:16809
/
2
C
.
0:16
/
2
D 4494:921I (g)
the new mean for the new distributed cyclic number n
3eq
at the stress level #3 per Equa-
tion (2.61) is:
3eq
D n
eq23
C exp
"
ln n
L3
C
ln n
L3
2
2
#
D 4494:921 C exp
8:1 C
.0:16/
2
2
D 7831:83: (h)
e log standard deviation of the new distributed cyclic number n
3eq
per Equation (2.62) is
ln n
3eq
D
ln L
3
D 0:16I (i)
the log-mean of the new distributed cyclic number n
3eq
per Equation (2.64) is
ln n
3eq
D ln
3eq
ln n
3eq
2
2
D ln.7831:83/
.0:16/
2
2
D 8:953151: (j)
Since the stress level #3 is the last stress level, we can use Equation (2.65) 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
@
ln N
C 3
ln n
3eq
q
ln N
C 3
2
C
ln n
3eq
2
1
C
A
D ˆ
9:4452 8:953151
p
.0:16809/
2
C .0:16/
2
!
D ˆ.2:120303/ D 0:9830:
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