88 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
e component fatigue damage index of this pin under model #3 cyclic shear loading spectrum
per Equation (2.84) is:
D D n
L
2V
a
S
u
.
d
2
S
u
2V
m
/
8:21
: (d)
(2) e limit state function.
e limit state function of the pin under model #3 cyclic shearing loading spectrum per
Equation (2.87) is:
g
.
K; V
a
; d
/
D K n
L
2V
a
S
u
.
d
2
S
u
2V
m
/
8:21
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(e)
ere are three random variables in the limit state function (e). e dimension d can be treated as
a normal distribution, and its mean and standard deviation can be calculated per Equation (1.1).
e distribution parameters in the limit state function (e) are listed in Table 2.44.
(3) e reliability of the double-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;581;583
1;598;400
D 0:9895:
Table 2.44: e distribution parameters of random variables in Equation (e)
K (lognormal) V
a
(klb) d (in)
μ
lnK
σ
lnK
μ
Va
σ
Va
μ
d
σ
d
37.308 0.518 4.815 0.6 0.5 0.00125
2.9.8 RELIABILITY OF A SHAFT UNDER CYCLIC TORSION LOADING
Per Equation (2.87) or Equation (2.88), we can establish the limit state function of a shaft under
any type of cyclic torsion loading spectrum and then calculate its reliability by using the H-L
method, or the R-F method and the Monte Carlo method. In this section, we will use two
2.9. THE PROBABILISTIC FATIGUE DAMAGE THEORY (THE K-D MODEL) 89
examples to demonstrate how to calculate the reliability of a component under cyclic torsion
loading spectrum.
Example 2.26
A shaft with a diameter 1:250 ˙ 0:005
00
is subjected to model #4 cyclic torsion loading spectrum
as listed in Table 2.45. e ultimate material strength S
u
of the shaft is 75 (ksi). ree parameters
of the component fatigue strength index K on the critical section for the cyclic torsion 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 shaft.
Table 2.45: e model #4 cyclic torsion loading spectrum for Example 2.26
Loading
Level #
Number of Cycles n
Li
Mean T
mi
of the Cyclic
Torque (klb.in)
Amplitude T
ai
of the
Cyclic Torque (klb.in)
1 6,000 2.25 5.13
2 500,000 2.25 9.42
Solution:
(1) e cyclic torsion stress and the component fatigue damage index.
For the loading level #1, we have the mean shear stress
m1
, the shear stress amplitude
a1
and their corresponding equivalent shear stress amplitude
eq1
m1
D
T
m1
d=2
J
D
T
m1
d=2
d
4
=32
D
16T
m1
d
3
(a)
a1
D
T
a1
d=2
J
D
T
a1
d=2
d
4
=32
D
16T
a1
d
3
(b)
eq1
D
a1
S
u
S
u
m1
D
16
a1
S
u
d
3
S
u
16T
m1
: (c)
For the loading level #2, by repeating the above calculation, we have
eq2
D
a2
S
u
S
u
m2
D
16
a2
S
u
d
3
S
u
16T
m2
: (d)
e component fatigue damage index of this shaft under model #4 cyclic torsion stress per
Equation (2.85) is:
D D n
L1
16
a1
S
u
d
3
S
u
16T
m1
8:21
C n
L2
16
a2
S
u
d
3
S
u
16T
m2
8:21
: (e)
(2) e limit state function of the shaft.
90 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
e limit state function of the shaft under model #4 cyclic loading spectrum per Equa-
tion (2.88) is :
g
.
K; d
/
D K n
L1
16
a1
S
u
d
3
S
u
16T
m1
8:21
n
L2
16
a2
S
u
d
3
S
u
16T
m2
8:21
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(f)
ere are two random variables in the limit state function (f). e dimension d can be treated
as a normal distribution. Its mean and standard deviation can be calculated per Equation (1.1).
e distribution parameters in the limit state function (f) are listed in Table 2.46.
(3) e reliability of the shaft.
We will 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 shaft R by the Monte Carlo method is
R D
1;582;682
1;598;400
D 0:9902:
Table 2.46: e distribution parameters of random variables in Equation (f)
K (lognormal) d (in)
μ
lnK
σ
lnK
μ
d
σ
d
37.308 0.518 1.25 0.00125
Example 2.27
A shaft with a diameter
1:500
˙
0:005
00
is subjected to model #3 cyclic torsion loading spectrum
as listed in Table 2.47. e ultimate material strength S
u
of the shaft 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 shaft.
Solution:
(1) e cyclic torsion stress and the component fatigue damage index.
2.9. THE PROBABILISTIC FATIGUE DAMAGE THEORY (THE K-D MODEL) 91
Table 2.47: e model #3 cyclic loading spectrum for Example 2.27
Number of
Cycles n
L
Mean Torque
T
m
(klb.in)
Normally Distributed Torque Amplitude T
a
(klb.in)
The Mean μ
T
a
The Standard
Deviation σ
T
a
400,000 4.5 8.9 0.85
e mean shear stress
m
, the shear stress amplitude
a
and their corresponding equivalent
shear stress amplitude
eq
of the shaft due to the model #3 cyclic torque loading are:
m
D
T
m
d=2
J
D
T
m
d=2
d
4
=32
D
16T
m
d
3
(a)
a
D
T
a
d=2
J
D
T
a
d=2
d
4
=32
D
16T
a
d
3
(b)
eq
D
a
S
us
S
us
m
D
16T
a
S
us
d
3
S
us
16T
m
: (c)
e component fatigue damage index of this shaft under model #3 cyclic shear stress per Equa-
tion (2.84) is:
D D n
L
16
a
S
u
d
3
S
u
16T
m
8:21
: (d)
(2) e limit state function of this shaft.
e limit state function of the shaft under model #3 cyclic loading spectrum per Equa-
tion (2.87) is:
g
.
K; T
a
; d
/
D K n
L
16T
a
S
u
d
3
S
u
16T
m
8:21
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(e)
ere are three random variables in the limit state function (e). e dimension d can be treated as
a normal distribution, and its mean and standard deviation can be calculated per Equation (1.1).
e distribution parameters in the limit state function (e) are listed in Table 2.48.
(3) e reliability of the shaft.
e limit state function (e) contains two normal distributions and one log-normal dis-
tribution. We will follow the procedure of the R-F method in Section 3.2.4 and the flowchart
in Appendix A.2 to create a MATLAB program. e iterative results are listed in Table 2.49.
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