40 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
distribution parameters of the component fatigue life N
c
will be:
ln N
c
D
ln N
C m ln
k
a
k
b
k
c
K
f
!
(2.35)
ln N
c
D
v
u
u
u
t
.
ln N
/
2
C m
2
2
4
k
a
k
a
2
C
k
c
k
c
2
C
K
f
K
f
!
2
3
5
; (2.36)
where
ln N
and
ln N
are the log mean and the log standard deviation of the material fatigue life
at the given cyclic stress level.
ln N
c
and
ln N
c
are the log mean and the log standard deviation
of the component fatigue life at the given cyclic stress level.
Figure 2.13 schematically depicts the material S-N curve per Equation (2.1) and the com-
ponent S-N curve per Equation (2.32). e component fatigue life N
c
per Equation (2.34) is
schematically displayed by a horizontal line. At the given cyclic stress level S
0
f
, which is exactly
equal to the cyclic loading stress level, the material fatigue life is shrinking along the horizontal
line from N to N
C
. In the P-S-N curve approach, N
C
is a random variable and will be described
by a distribution function.
e component fatigue strength S
cf
per Equation (2.29) is schematically displayed by a
vertical line in Figure 2.13. At the given fatigue life N , which is exactly equal to the number of
cycles of the cyclic loading stress, the material fatigue strength S
0
f
is shrinking along the vertical
line from
S
0
f
to S
Cf
. In the P-S-N curve approach, S
Cf
is a random variable and will be described
by a distribution function.
logS
'
f
S
'
f
S
'
f
logS
f
logN
logN
c
S
cf
N
c
N
N
10
3
10
6
Component S-N curve
Material S-N curve
Fatigue Life
Fatigue Strength
Figure 2.13: Schematic of a material S-N curve and a component S-N curve.
e applications of the P N
C
curves at a given cyclic loading stress level or P S
Cf
curves at a given fatigue life will be demonstrated in the following sections.
2.8. RELIABILITY OF A COMPONENT BY THE P-S-N CURVES APPROACH 41
2.8.3 RELIABILITY OF A COMPONENT UNDER MODEL #1 CYCLIC
LOADING SPECTRUM
e general description of model #1 is .
a
;
m
; n
L
/, where
a
is a constant stress amplitude of
the cyclic stress,
m
is constant mean stress of the cyclic stress, and n
L
is the number of cycles of
the cyclic loading. Since provided P-S-N curves are typically obtained based on fully reversed
cyclic stress fatigue tests, the non-zero-mean cyclic stress will be converted into an equivalent
stress amplitude
aeq
of a fully reversed cyclic stress per Equation (2.21).
When the P S
cf
curve of component fatigue strength S
cf
at the given fatigue life N D
n
L
are provided, the reliability of the component can be directly calculated by the following
equation:
R D P
S
cf
>
aeq
D 1
Z
aeq
1
f
S
cf
.s/ds D 1 F
S
cf
aeq
; (2.37)
where f
S
cf
.s/ and F
S
cf
.s/ are the probability density function (PDF) and cumulative distribution
function (CDF) of the component fatigue strength S
cf
at the given fatigue life N which is equal
to the number of cycles n
L
of the model #1 cyclic stress.
aeq
is a constant equivalent stress
amplitude of the cyclic stress.
When the P S
cf
curve of component fatigue strength S
cf
at the given fatigue life N D
n
L
are obtained through the material P-S-N curves, that is per Equation (2.29), the component
fatigue strength at the fatigue life N D n
L
is:
S
cf
D
k
a
k
b
k
c
K
f
S
0
f
at the given fatigue life N : (2.29)
en the limit state function of the component under this situation is:
g
k
a
; k
c
; K
f
; S
0
f
D
k
a
k
b
k
c
K
f
S
0
f
aeq
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure;
(2.38)
where k
a
; k
b
, and k
c
are the surface finish modification factor, the size modification factor, and
the loading modification factor, respectively. K
f
is the fatigue stress concentration factor. S
0
f
is
the material fatigue strength at the fatigue life N D n
L
.
aeq
is a constant equivalent stress
amplitude of the cyclic stress, which can be calculated per Equation (2.21). In Equation (2.38),
k
b
and
aeq
will be treated as deterministic constants. e reliability of the component under
such a cyclic loading can be calculated by using the limit state function Equation (2.38) with
the H-L method, R-F method, or Monte Carlo method.
When the P N
c
curves of component fatigue life N
c
at the given fatigue cyclic stress
level S
0
f
D
aeq
are provided, the reliability of the component can be directly calculated by the
42 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
following equation:
R D P
.
N
c
> n
L
/
D 1
Z
n
L
1
f
N
c
.
n
/
d n D 1 F
N
c
.n
L
/; (2.39)
where f
N
c
.n/ and F
N
c
.n/ are the PDF and CDF of the component fatigue life N
c
at the given
cyclic stress level S
0
f
which is equal to
aeq
of model #1 cyclic stress. n
L
is the number of cycles
of the model #1 cyclic loading stress. n
L
is a constant number for the model #1 cyclic loading
stress
When the P N
c
curve of component fatigue life N
c
at the given fatigue cyclic stress
level S
0
f
D
aeq
are obtained through the material P-S-N curves, that is per Equation (
2.34),
the limit state function of this problem is:
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)
where k
a
; k
b
; k
c
, and K
f
are the same as those in Equation (2.38). N is the material fatigue
life at the cyclic stress level S
0
f
D
aeq
. n
L
is a constant number for the model #1 cyclic loading
stress. e reliability of the component under such a cyclic loading can be calculated by using
the limit state function Equation (2.40) with the H-L method, R-F method, or Monte Carlo
method.
For a component under model #1 cyclic loading spectrum, we could use Equations (2.37)
or (2.39) to directly calculate component’s reliability. Or, we can use the limit station functions
(2.38) and (2.40) to calculate component’s reliability. ree examples are presented to show how
to calculate the reliability of a component under model #1 cyclic loading spectrum.
Example 2.9
A beam at its critical section is subjected to a fully reversed cyclic bending stress with a constant
stress amplitude
a
D 35:5 (ksi) and a constant number of cycles n
L
D 3:90 10
5
(cycles). After
a series of calculation, the component fatigue life of the beam at its critical section under the
fully reversed cyclic bending stress S
cf
D 35:5 (ksi) follows a lognormal distribution with a log
mean
ln N
D 13:305 and a log standard deviation
ln N
D 0:187. Calculate the reliability of the
beam.
Solution:
For this problem, the component fatigue life P N
C
distribution at the given cyclic stress level
is given. We can directly use Equation (2.39) to calculate the reliability of the beam.
If the Microsoft Excel formula is used,
R D P
.
N
c
> n
L
/
D 1 F
N
c
.
n
L
/
D 1 LOGNORM:DIST
3:90 10
5
; 13:305; 0:187; t rue
D 0:9894:
2.8. RELIABILITY OF A COMPONENT BY THE P-S-N CURVES APPROACH 43
If the MATLAB function is used,
R D P
.
N
c
> n
L
/
D 1 F
N
c
.
n
L
/
D 1 logncdf
3:90 10
5
; 13:305; 0:187
D 0:9894:
Example 2.10
A constant cross-section bar is subject to a fully reversed cyclic axial stress with a constant stress
amplitude
a
D 23:5 (ksi) and a constant number of cycles n
L
D 1 10
4
(cycles). e material
fatigue life at the fully reversed rotating bending stress amplitude S
0
f
D 23:5 (ksi) follows a
lognormal distribution with a log mean
ln N
D 13:72 and a log standard deviation
ln N
D
0:124. e bar can be treated as a hot-rolled component. Its ultimate strength is 45.4 (ksi), and
its slope of the traditional S-N curve m is 8.30. (1) Determine the distribution parameters of
the component fatigue life at the given cyclic stress level. (2) Calculate the reliability of this bar.
Solution:
(1) Determine the distribution parameters of the component fatigue life
N
c
at the given cyclic
stress level.
Since the material fatigue life distribution at the given cyclic stress level is provided, Equa-
tion (2.34) will be used to get the component fatigue life.
Per Equations (2.14)–(2.16), for a hot-rolled component, the distribution parameters of
the surface finish modification factor k
a
are:
k
a
D 16:45
.
S
ut
/
0:7427
D 16:45
.
45:4
/
0:7427
D 0:9671 (a)
k
a
D
k
a
k
a
D 0:098 0:9671 D 0:09478: (b)
Per Equation (2.17), the size modification factor k
b
is:
k
b
D 1: (c)
Per Equations (2.18)–(2.20), the loading modification factor k
c
for cyclic axial loading are:
k
c
D 0:774: (d)
k
c
D
k
c
k
c
D 0:163 0:774 D 0:1262: (e)
In this example, K
f
will be 1 because of a constant cross-section bar.
e component fatigue life at the given cyclic stress level will be:
N
c
D N
.
k
a
k
c
/
8:30
: (f)
44 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
If it is assumed that the component fatigue life N
c
still follows a log-normal distribution, we
can calculate its log-mean and log-standard deviation per Equations (2.35) and (2.36):
ln N
c
D
ln N
C m ln
k
a
k
c
D 13:72 C 8:30 ln
.
0:9671 0:774
/
D 11:393 (g)
ln N
c
D
v
u
u
t
.
ln N
/
2
C m
2
"
k
a
k
a
2
C
k
c
k
c
2
#
D
v
u
u
t
.
0:124
/
2
C 8:30
2
"
0:09478
0:9671
2
C
0:1262
0:774
2
#
D 1:5838 (h)
(2) Calculate the reliability of this bar.
If we use the distribution parameters in Equations (g) and (h), that is, N
c
is a log-
normal distribution with
ln N
c
D 11:393 and
ln N
c
D 1:5838, the reliability of the bar per
Equation (2.40) will be:
R D P
.
N
c
> n
L
/
D 1 F
N
c
.
n
L
/
D 1 logncdf
.
1000; 11:393; 1:5838
/
D 0:9159:
Of course, this is only an approximate result. If the limit state function Equation (2.40) for
this example is used, we need to use the R-F or Monte Carlo method to compile a MATLAB
program to calculate the reliability of the bar and will get a more accurate result.
Example 2.11
e critical section of a machined rotating shaft is at its shoulder section. e schematic of the
shoulder section is shown in Figure 2.14. e critical section is subjected to fully reversed cyclic
bending stress with a constant stress amplitude
a
D 10:67 (ksi) and a constant number of cycles
n
L
D 3 10
5
(cycles). e material fatigue strength S
0
f
at the fatigue life N D 3:5 10
5
from
fully reversed rotating bending stress tests follows a normal distribution with a mean
S
0
f
D
26:52 (ksi) and standard deviation
S
0
f
D 1:98 (ksi). e ultimate strength of this material is
61.5 (ksi). (1) Express the component fatigue strength at the fatigue life N D 3:5 10
5
. (2)
Calculate the reliability of the shaft.
Solution:
(1) e component fatigue strength at the fatigue life N D 3:5 10
5
:
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