2.8. RELIABILITY OF A COMPONENT BY THE P-S-N CURVES APPROACH 37
Table 2.13: e P-N distribution at different fatigue strength levels
Axial Stress
Level #
σ
a
(ksi) σ
m
(ksi)
Equivalent Stress
σ
a˗eqi
(ksi)
Lognormal Distributed Fatigue Life
μ
lnN
σ
lnN
1 20.833 20.833 35.126 11.4736 0.238106
2 22.083 22.083 38.832 11.0410 0.227320
3 22.5 22.5 40.139 10.9551 0.242377
4 22.917 22.917 41.485 10.7831 0.225735
5 23.333 23.333 42.871 10.716 0.241955
ˇ is the shape parameter, and is the scale parameters. For the normal distribution of S
0
f
,
S
0
f
is a mean and
S
0
f
is a standard deviation. For steels, the P-S distribution at the fatigue life
N D 10
6
(number of cycles to failure) in Table 2.14 is the distribution function of the material
endurance limit.
2.8.2 THE COMPONENT P-S-N CURVES
ere are two sets of distribution functions of a material P-S-N curves, which are the P-N
distribution of the material fatigue life N at a given stress level and the P-S distribution of
the material fatigue strength S
0
f
at a given fatigue life N . After the differences between the
material fatigue test specimens and the component are considered by some modification factors,
accordingly, we will have two sets of distribution functions of a component P-S-N curves, which
will be the P N
c
distribution of the component fatigue life N
c
and the P S
f
distribution
of the component fatigue strength S
f
.
For the component fatigue strength S
f
at the given fatigue life N , we can provide two
approaches to modify the material fatigue strength S
0
f
. In the first approach (the traditional
approach), the surface finish modification factor k
a
, the size modification factor k
b
, and the
loading modification factor k
c
are used to modify the material fatigue strength S
0
f
to get the
component fatigue strength S
f
, as shown in Equation (2.13). While the fatigue strength con-
centration factor K
f
is used to modify the stress amplitude
aeq
. e limit state function of a
component at the given fatigue life N will be:
g
S
f
; K
f
;
aeq
D S
f
K
f
aeq
D k
a
k
b
k
c
S
0
f
K
f
aeq
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(2.27)
38 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
Table 2.14: e P-S distributions at different fatigue life of three materials
AISI 1045 Steel
Rotary Bending, WQ for 1520°F Tempered at 1210°F, K
t
= 1; S
ut
= 105 ksi; S
y
= 82 ksi
Fatigue Life (number
of cycles to failure)
3-Parameter Weibull Distribution of S
'
f
Normal Distribution of S
'
f
γ β η μ
S
'
f
σ
S
'
f
10
4
79.0 2.6 86.2 85.40 2.640
10
5
67.0 2.75 73.0 72.34 2.092
10
6
56.7 2.85 61.65 61.11 1.672
AISI 3140 Steel
Rotary Bending, OQ for 1520°F Tempered at 1300°F, K
t
= 1; S
ut
= 108 ksi; S
y
= 87 ksi
Fatigue Life (number
of cycles to failure)
3-Parameter Weibull Distribution Normal Distribution
γ β η μ
S
'
f
σ
S
'
f
10
3
89 3.7 100.4 99.29 3.063
10
4
74 5.2 87.7 86.61 2.740
10
5
66 5 76.7 75.82 2.213
10
6
57 5.5 67.2 66.42 1.942
A-286 Stainless steel
Fully Reversed Axial Load, K
t
= 1; S
ut
= 90 ksi; S
y
= 46 ksi
Fatigue Life (number
of cycles to failure)
3-Parameter Weibull Distribution Normal Distribution
γ β η μ
S
'
f
σ
S
'
f
10
4
40 1.84 54 52.44 7.072
10
5
31 2.1 43 41.63 5.347
10
6
24 2.2 34 32.856 4.267
For the reliability calculation, since K
f
is always larger than 1, the above limit state function can
be converted into the following equivalent limit state function,
g
S
f
; K
f
;
aeq
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:
(2.28)
We will use the symbol S
cf
to represent the component fatigue strength with the consideration
of all modification factors including the fatigue stress concentration factor K
f
:
S
cf
D
k
a
k
b
k
c
K
f
S
0
f
at the given fatigue life N : (2.29)
2.8. RELIABILITY OF A COMPONENT BY THE P-S-N CURVES APPROACH 39
Equation (2.29) can be used to build the limit state function Equation (2.28) of the component
under cyclic stress at the given fatigue life. is result is the second approach to obtain com-
ponent fatigue strength at the given fatigue life. We can assume that the modification factors
will not change the type of distributions, but only change the distribution parameters. So, we
can assume that S
cf
will have the same type of distribution as that of S
0
f
. If S
0
f
follows a normal
distribution with a mean
S
0
f
and a standard deviation
S
0
f
, the normally distributed S
cf
will
have the following mean
S
cf
and standard deviation
S
cf
:
S
cf
D
k
a
k
b
k
c
K
f
S
0
f
(2.30)
S
cf
D
k
a
k
b
k
c
K
f
S
0
f
v
u
u
t
k
a
k
a
2
C
k
c
k
c
2
C
K
f
K
f
!
2
C
S
0
f
S
0
f
!
2
; (2.31)
where
k
a
and
k
a
are the mean and the standard deviation of the surface finish modification
factor k
a
and determined per Equations (2.14), (2.15), and (2.16). k
b
is the size modification
factor and determined per Equation (2.17).
k
c
and
k
c
are the mean and the standard deviation
of the load modification factor k
c
and determined per Equations (2.18), (2.19), and (2.20).
K
f
and
K
f
are the mean and standard deviation of the fatigue stress concentration factor K
f
and
determined per Equations (2.22), (2.23), (2.24), and (2.25).
S
0
f
and
S
0
f
are the mean and the
standard deviation of material fatigue strength S
0
f
at the given fatigue life.
S
cf
and
S
cf
are the
mean and the standard deviation of the component fatigue strength at the given fatigue life.
For the component fatigue life N
c
at the given cyclic stress level, we can use the following
approach to modify the material fatigue life N . In a traditional S-N curve per Equation (2.1),
the component S-N curve will be:
N
k
a
k
b
k
c
K
f
S
0
f
m
D constant: (2.32)
We can rearrange Equation (2.32) as follows:
N
k
a
k
b
k
c
K
f
m
S
0
f
m
D constant: (2.33)
Based on Equation (2.33), we can use N
c
to represent the component fatigue life after the
consideration of all modification factors at the given cyclic stress level:
N
c
D N
k
a
k
b
k
c
K
f
m
: (2.34)
It is assumed that the component fatigue life N
c
has the same type of distribution as that of N .
If N is a lognormal distribution with a log-mean
ln N
and log-standard deviation
ln N
, the
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