2.9. THE PROBABILISTIC FATIGUE DAMAGE THEORY (THE K-D MODEL) 85
e diameter d will be treated as a normal distribution. Its mean and standard deviation can be
determined per Equation (1.1). ere are four random variables in the limit state function (d).
K is a log-normal distribution. F
a1
, F
a2
, and d are normal distributions. eir distribution
parameters in Equation (d) are listed in Table 2.41.
Table 2.41: e distribution parameters of random variables in Equation (d)
K (log-normal) F
a1
(klb) F
a2
(klb) d (in)
μ
lnK
σ
F
a
μ
Fa1
σ
Fa1
μ
Fa2
σ
Fa2
μ
d
σ
d
41.738 0.357 22.15 3.25 12.45 1.5 0.82 0.00125
(3) Reliability of the bar.
We will use the Monte Carlo method to calculate the reliability of this example. e
Monte Carlo method is displayed in Appendix A.3. We can follow the Monte Carlo method
and the program flowchart to create a MATLAB program.
Since the limit state function is not too complicated, we will use the trial number N D
1;598;400. e reliability of this component R by the Monte Carlo method is
R D
1;578;582
1;598;400
D 0:9876:
2.9.7 RELIABILITY OF A COMPONENT UNDER CYCLIC DIRECT
SHEARING LOADING
Per Equation (2.87) or Equation (2.88), we can establish the limit state function of a component
under any type of cyclic direct shearing loading spectrum and then calculate its reliability. In this
section, we will use two examples to demonstrate how to calculate the reliability of a component
under a cyclic direct-shearing loading spectrum.
Example 2.24
A single-shearing pin with a diameter 1:125 ˙ 0:005
00
is under a zero-to-maximum cyclic direct
shearing loading. e maximum shear loading V
max
of this cyclic shearing loading can be treated
as a constant V
max
D 26:75 (klb). e number of cycles n
L
of this cyclic shearing loading is also
treated as a constant n
L
D 500;000 (cycles). e ultimate material strength S
u
of the pin is
75 (ksi). ree parameters of the component fatigue strength index K on the critical section for
the cyclic shear loading are m D 8:21,
ln K
D 37:308, and
ln K
D 0:518. For the component
fatigue strength index K, the stress unit is ksi. Calculate the reliability of the pin.
86 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
Solution:
(1) e cyclic direct shearing stress and the component fatigue damage index.
e mean shear stress
m
and the shear stress amplitude
a
of the pin due to this zero-to-
maximum cyclic shearing loading are:
m
D
V
m
A
D
V
max
=2
d
2
=4
D
2V
max
d
2
(a)
a
D
V
a
A
D
V
max
=2
d
2
=4
D
2V
max
d
2
:
(b)
Since this is non-zero-mean cyclic shear stress, the equivalent stress amplitude of a fully reversed
cyclic shear stress is:
aeq
D
a
S
u
.
S
u
m
/
D
2V
max
d
2
S
u
.
S
u
2V
max
=d
2
/
D
2V
max
S
u
.
d
2
S
u
2V
max
/
: (c)
e component fatigue damage index of this pin under model #1 cyclic shear stress per Equa-
tion (2.84) is:
D D n
L
2V
max
S
u
.
d
2
S
u
2V
max
/
8:21
: (d)
(2) e limit state function.
e limit state function of the pin under model #1 cyclic loading spectrum per Equa-
tion (2.87) is:
g
.
K; d
/
D K n
L
2V
max
S
u
.
d
2
S
u
2V
max
/
8:21
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(e)
ere are two random variables in the limit state function (e). e dimension d can be treated as a
normal distribution, and its mean and its standard deviation can be calculated per Equation (1.1).
e distribution parameters in the limit state function (e) are listed in Table 2.42.
(3) e reliability of the single-shear pin.
We will use the Monte Carlo method to calculate the reliability of this example. We
can follow the Monte Carlo method and the program flowchart in Appendix A.3 to create a
MATLAB program. Since the limit state function is not too complicated, we will use the trial
number N D 1;598;400. e reliability of this component R by the Monte Carlo method is
R D
1;583;621
1;598;400
D 0:9908:
2.9. THE PROBABILISTIC FATIGUE DAMAGE THEORY (THE K-D MODEL) 87
Table 2.42: e distribution parameters of random variables in Equation (e)
K (lognormal) d (in)
μ
lnK
σ
lnK
μ
d
σ
d
37.308 0.518 1.125 0.00125
Example 2.25
A double-shearing pin with a diameter 0:500 ˙0:005
00
is subjected to model #3 cyclic shear load-
ing spectrum, as shown in Table 2.43. e ultimate material strength S
u
of the pin is 75 (ksi).
ree parameters of the component fatigue strength index K on the critical section for the
cyclic shear loading are m D 8:21,
ln K
D 37:308, and
ln K
D 0:518. For the component fa-
tigue strength index K, the stress unit is ksi. Calculate the reliability of the pin.
Table 2.43: Model #3 cyclic shearing loading spectrum for Example 2.25
Number of
Cycles n
L
Mean of the Cyclic
Shear Loading
V
m
(klb)
Amplitude of Cyclic Shear Loading
V
a
(klb) (normal distribution)
μ
V
a
σ
V
a
600,000 3.422 4.815 0.6
Solution:
(1) e cyclic shearing stress and the component fatigue damage index.
e mean shear stress
m
and the shear stress amplitude
a
of the pin due to this cyclic
shearing loading are:
m
D
V
m
=2
A
D
V
m
=2
d
2
=4
D
2V
m
d
2
(a)
a
D
V
a
=2
A
D
V
a
=2
d
2
=4
D
2V
a
d
2
: (b)
Since this is non-zero mean cyclic shear stress, the equivalent stress amplitude of a fully reversed
cyclic shear stress is:
aeq
D
a
S
u
.
S
u
m
/
D
2V
a
S
u
.
d
2
S
u
2V
m
/
: (c)
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