2.9. THE PROBABILISTIC FATIGUE DAMAGE THEORY (THE K-D MODEL) 81
2.9.5 THE PROBABILISTIC FATIGUE DAMAGE THEORY (THE K-D
MODEL)
Based on the definitions of the component fatigue strength index K and the component fatigue
damage index D, the general limit state function of a component under cyclic loading is:
g
.
K; D
/
D K D D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(2.86)
If the material fatigue strength index K
0
is given, the limit state function of a component under
model #1, model #2, or model #3 cyclic loading spectrum is:
g
.
K; D
/
D K D D
.
k
a
k
b
k
c
/
m
K
0
n
L
K
f
a
m
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(2.87)
All variables in Equation (
2.87) have the same meanings as those in Equations (2.79) and (2.84).
If the material fatigue strength index K
0
is given, the limit state function of a component
under model #4, model #5, or model #6 cyclic loading spectrum is:
g
.
K; D
/
D K D D
.
k
a
k
b
k
c
/
m
K
0
L
X
iD1
n
Li
K
f
ai
m
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(2.88)
All variables in Equation (2.88) have the same meanings as those in Equations (2.79) and (2.85).
When this probabilistic fatigue damage model (the K-D model) is used, the reliability of a
component under any cyclic loading spectrum can be calculated per Equations (2.87) or (2.88).
2.9.6 RELIABILITY OF A COMPONENT UNDER CYCLIC AXIAL
LOADING
Per Equations (2.87) or (2.88), we can establish the limit state function of a component under
any type of cyclic axial loading spectrum. After the limit state function of a component under
cyclic axial loading is established, we can use the H-L method, R-F method, or Monte Carlo
method to calculate its reliability. In this section, we will use two examples to demonstrate how
to calculate the reliability of a component under cyclic axial loading spectrum.
Example 2.22
A constant round bar with a diameter 0:850 ˙ 0:005
00
is subjected to model #1 cyclic axial loading
spectrum as listed in Table 2.37. e ultimate material strength S
u
is 75 (ksi). ree parameters
82 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
of the component fatigue strength index K are m D 8:21,
ln K
D 41:738, and
ln K
D 0:357.
For the component fatigue strength index K, the stress unit is ksi. Calculate the reliability of
this bar.
Table 2.37: Model #2 cyclic axial loading for Example 2.22
Number of
Cycles n
L
The Mean of the Cyclic
Axial Loading F
m
(klb)
The Amplitude of Cyclic Axial Loading
F
a
(klb) (normal distribution)
μ
F
a
σ
F
a
103000 8.85 14.11 1.51
Solution:
(1) e cyclic axial stress and the component fatigue damage index.
e mean stress
m
and the stress amplitude
a
of the bar due to the cyclic axial loading
are:
m
D
F
m
d
2
=4
D
4F
m
d
2
(a)
a
D
F
a
d
2
=4
D
4F
a
d
2
: (b)
Since the cyclic axial stress is a non-zero mean cyclic stress, we need to convert it into a fully
reversed cyclic axial stress per Equation (2.83). e equivalent stress amplitude of this converted
fully reversed cyclic stress is:
aeq
D
a
S
u
.
S
u
m
/
D
4F
a
S
u
.
S
u
d
2
4F
m
/
: (c)
e component fatigue damage index D of the bar under model #2 cyclic loading spectrum per
Equation (2.84) is
D D n
L
K
f
a
8:21
D n
L
4F
a
S
u
.
S
u
d
2
4F
m
/
8:21
: (d)
(2) e limit state function of this bar.
2.9. THE PROBABILISTIC FATIGUE DAMAGE THEORY (THE K-D MODEL) 83
e limit state function of the bar per Equation (2.87) is
g
.
K; F
a
; d
/
D K n
L
4F
a
S
u
.
S
u
d
2
4F
m
/
8:21
D K 103000
300F
a
.
75d
2
35:4
/
8:21
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(e)
In this limit state function, we have three random variables. e diameter d will be treated as
a normal distribution. Its mean and standard deviation can be determined per Equation (1.1).
e distribution parameters of these three random variables are listed in Table 2.38.
Table 2.38: e distribution parameters of random variables in Equation (e)
K (log-normal) F
a
(klb) d (in)
μ
lnK
σ
lnK
μ
Fa
σ
Fa
μ
d
σ
d
41.738 0.357 14.11 1.51 0.85 0.00125
(3) Reliability of the bar.
e limit state function (e) contains two normal distributions and one log-normal dis-
tribution. We will use the R-F method to calculate its reliability, which is displayed in Ap-
pendix A.2. We can follow the procedure and the flowchart of the R-F method to create a
MATLAB program. e iterative results are listed in Table 2.39. From the iterative results, the
reliability index ˇ and the corresponding reliability R of the bar in this example are:
ˇ D 2:172022 R D ˆ
.
2:172022
/
D 0:9851:
Table 2.39: e iterative results of Example 2.22 by the R-F method
Iterative #
K
*
F
a
*
d
*
β
*
|∆β
*
|
1 1.43E+18 14.11 0.775131 2.161288
2 9.22E+17 17.13139 0.852768 2.17213 0.010842
3 9.5E+17 17.04937 0.84992 2.172022 0.000108
4 9.51E+17 17.05197 0.849898 2.172022 1.01E-07
84 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
Example 2.23
A bar with a diameter 0:820 ˙ 0:005
00
is subjected to model #6 cyclic axial loading spectrum
listed in Table 2.40. e ultimate material strength S
u
is 75 (ksi). ree parameters of the com-
ponent fatigue strength index K are m D 8:21,
ln K
D 41:738, and
ln K
D 0:357. For the com-
ponent fatigue strength index K, the stress unit is ksi. Calculate the reliability of this bar.
Table 2.40: Model #6 cyclic axial loading spectrum for Example 2.23
Level #
The Number of Cycles
n
Li
(constant)
The Fully Reversed Axial Loading
Amplitude F
ai
(klb) (normal distribution)
μ
F
ai
σ
F
ai
1 5,000 22.15 3.25
2 200,000 12.45 1.5
Solution:
(1) e cyclic axial stress and the component fatigue damage index.
Since the cyclic axial loadings are fully reversed cyclic axial loading, the fully reversed cyclic
axial stress amplitudes will be as follows.
In the level #1, the fully reversed axial stress amplitude
a1
is
a1
D
F
a1
d
2
=4
D
4F
a1
d
2
: (a)
In the level #2, the fully reversed axial stress amplitude
a2
is
a1
D
F
a2
d
2
=4
D
4F
a2
d
2
: (b)
e component fatigue damage index D of the bar under this model #6 cyclic loading per Equa-
tion (2.85) is:
D D n
L1
4F
a1
d
2
m
C n
L1
4F
a2
d
2
m
: (c)
(2) e limit state function of this bar.
e limit state function of a bar due to model #6 cyclic axial loading spectrum can be
established per Equation (2.88):
g
.
K; F
a1
; F
a2
; d
/
D K n
L1
4F
a1
d
2
m
n
L1
4F
a2
d
2
m
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(d)
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