68 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
Table 2.29: e iterative results at the cyclic number level #1
Iterative #
S
c
*
f 1
σ
a
*
1
β* Δ|β*|
1 78.0005 78.0005 3.289598
2 75.1051 75.1051 3.296735 0.007137
3 75.24278 75.24278 3.296752 1.7E-05
e iterative results for the limit state function (b) is listed in Table 2.30. From the iterative
results, the reliability index ˇ
2
and corresponding reliability R
2
of the component in this example
are:
ˇ
2
D 2:175236I R
2
D ˆ.2:175236/ D 0:985194:
e reliability of the component under this cyclic loading spectrum per Equation (2.66)
is:
R D
2
Y
iD1
R
i
D R
1
R
2
D 0:999511 0:985194 D 0:9847:
Table 2.30: e iterative results at the cyclic number level #2
Iterative #
S
c
*
f 2
σ
a
*
2
β* Δ|β*|
1 57.46615 57.46615 2.172718
2 56.17362 56.17362 2.175234 0.002516
3 56.21165 56.21165 2.175236 2.25E-06
2.8.9 THE RELIABILITY OF A COMPONENT WITH P-S-N CURVES BY
THE MONTE CARLO METHOD
e limit state function must be established first before the Monte Carlo method can be used to
calculate the reliability of a component under cyclic loading Spectrum. When the P-S-N curves
are used as the fatigue strength data, the limit state function of a component under six possible
cyclic loading spectrums are listed here.
For model #1 cyclic loading spectrum .
a
; n
L
/, which is a constant number of cycle n
L
at a
constant fully reversed cyclic stress amplitude
a
, we can have two versions of limit state function
per Equation (2.38) when the component fatigue strength S
cf
is used or per Equation (2.40)
2.8. RELIABILITY OF A COMPONENT BY THE P-S-N CURVES APPROACH 69
when the component fatigue life N
C
:
g
k
a
; k
c
; K
f
; S
0
f
D
k
a
k
b
k
c
K
f
S
0
f
a
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure
(2.38)
g
k
a
; k
c
; K
f
; N
D N
k
a
k
b
k
c
K
f
m
n
L
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(2.40)
For model #2 cyclic loading spectrum .
a
; n
L
/, which is a distributed number of cycle n
L
at
a constant fully reversed cyclic stress amplitude
a
, the limit state function can be established
per Equation (2.41) when the component fatigue life N
C
at the given constant cyclic stress
amplitude
a
is provided. e limit state function will be constructed per Equation (2.42) when
the material fatigue life N at the given constant cyclic stress amplitude
a
is provided:
g
.
N
c
; n
L
/
D N
c
n
L
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure
(2.41)
g
k
a
; k
c
; K
f
; N; n
L
D N
k
a
k
b
k
c
K
f
m
n
L
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(2.42)
For model #3 cyclic loading spectrum .
a
; n
L
/, which is a distributed fully reversed cyclic stress
amplitude
a
, with a constant number of cycles n
L
, the limit state function can be established
per Equation (2.43) when the component fatigue strength S
Cf
at the given constant fatigue life
N D n
L
is provided. e limit state function will be constructed per Equation (2.44) when the
material fatigue strength S
0
f
at the given constant fatigue life N D n
L
is provided:
g
S
Cf
;
a
D S
Cf
a
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure
(2.43)
g
k
a
; k
c
; K
f
; S
0
f
;
a
D
k
a
k
b
k
c
K
f
S
0
f
a
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(2.44)
70 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
For model #6 cyclic loading spectrum .
ai
; n
Li
; i D 1; 2; : : : /, which are distributed fully re-
versed cyclic stress amplitude
ai
with a constant number of cycles n
Li
in the cyclic number level
#i, the limit state function of the component for each cyclic number level #i can be established
per Equation (2.67):
g
S
cfi
;
ai
D S
cfi
ai
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state i D 1; 2; : : :
< 0 Failure:
(2.67)
For all the above cases, we can directly use the Monte Carlo method to calculate the reliability
of a component per its limit state function. e Monte Carlo method procedure and program
flowchart is displayed in Appendix A.3.
For model #4 and model #5 cyclic loading spectrums .
ai
; n
Li
; i D 1; 2; : : : /, we could not
build their limit state functions, but the equivalent fatigue damage concepts could be used to
calculate the reliability.
e author presented an approach [19] in 2018 to use the Monte Carlo method to cal-
culate the reliability of a component under these two cyclic loading spectrums. e two key
concepts in the widely accepted Miner rule are that fatigue damage caused by cyclic loading
could be treated as independent random events and could be cumulated linearly. Based on these
two key concepts in the Miner rule, the following is the computational algorithm for imple-
menting the Monte Carlo method to calculate the reliability of a component under the model
#4 and #5 cyclic loading spectrum.
e accumulated fatigue damage F
j
in the j th trial of the Monte Carlo method is:
F
j
D
iDI
X
iD1
n
Lij
N
Cij
; (2.70)
where n
Lij
is a randomly generated sample of the distributed number of cycles n
Li
at the cyclic
stress level
ai
in the j th trial for the model #5 cyclic loading spectrum. n
Lij
will be equal to n
Li
for the model #4 cyclic loading spectrum. N
Cij
is a randomly generated sample of the distributed
component fatigue life N
Ci
at the cyclic stress level
ai
in the j th trial. e subscript i represents
the ith cyclic stress level. e symbol I represents the total number of different cyclic stress
levels. e trial result tn
j
of the j th Monte Carlo simulation will be determined per the following
equation:
tn
j
D
8
<
:
1; if F
j
< 1
0; if F
j
1:
(2.71)
In Equation (2.71), the trial result tn
j
D 1 represents that the component in the j th trail is
safe because the cumulative damage F
j
is less than 1. e trial result tn
j
D 0 indicates that the
2.8. RELIABILITY OF A COMPONENT BY THE P-S-N CURVES APPROACH 71
component in the j th trail fails because the cumulative damage F
j
is larger than 1. e sum of
all trial results
P
tn
j
will be the number of trails with a safe status. So, the component reliability
R under such cyclic loading spectrum is:
R D
0
@
N
t
X
j D1
tn
j
1
A
=N
t
; (2.72)
where N
t
is the total number of trials in the Monte Carlo simulation. Since the limit state
function of a component under cyclic loading spectrum is typically not very complicated, N
t
D
15;998;400 can be used.
Example 2.19
Use the Monte Carlo method to calculate the reliability of Example 2.14 in Section 2.8.6. A
component is subjected to cyclic loading at three different constant fully reversed cyclic stress
levels with three different constant numbers of cycles as listed in Table 2.31. e component
fatigue life N
c
under the corresponding stress levels are also listed in Table 2.31. Use the Monte
Carlo method to calculate the reliability of the component.
Table 2.31: Model #4 cyclic loading spectrum and corresponding component fatigue life for
Example 2.19
Stress
Level
i
Cyclic Stress
Amplitude
σ
ai
(psi)
Number of
Cycles n
Li
Component Fatigue Life N
Ci
at σ
ai
(psi)
(lognormal distribution)
μ
lnN
Ci
σ
lnN
Ci
1 65,000 81,000 12.95987 0.198
2 85,000 16,000 11.01311 0.197
3 105,000 2,800 9.47966 0.195
Solution:
We can use the above Equations (2.70), (2.71), and (2.72) to compile the Monte Carlo method
program, which is displayed in Appendix A.3. Based on the proposed computational algorithm,
the MATLAB program can be compiled, and the reliability of the component from the MAT-
LAB program is:
R D
0
@
N
t
X
j D1
tn
j
1
A
=N
t
D
15992388
15998400
D 0:9996:
e result from Example 2.14 by using the equivalent fatigue damage concept is R D
0:9877. e result of the Monte Carlo method is 0.9996. e results are not the same. However,
the relative difference is only 2.1%.
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