96 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
Table 2.53: e distribution parameters of random variables in Equation (e)
K (lognormal) M
a
(klb.in) d (in)
μ
lnK
σ
lnK
μ
M
a
σ
M
a
μ
d
σ
d
32.476 0.279 16.8 3.19 2.500 0.00125
Table 2.54: e iterative results of Example 2.29 by the R-F method
Iterative #
K
*
M
a
*
d
*
β
*
|∆β
*
|
1 1.32E+14 16.8 2.345547 1.557169
2 1.14E+14 21.63949 2.494604 1.624744 0.067575
3 1.12E+14 21.76716 2.499996 1.624707 3.69E-05
2.9.10 RELIABILITY OF A COMPONENT UNDER CYCLIC COMBINED
LOADING
Component under cyclic combined loading is very complicated because the frequencies of in-
dependent loadings might not be in phase. In this section, we will only discuss a rotating shaft
under a combined-torques-bending loading.
For a rotating shaft, the cyclic bending stress on a rotating shaft is mainly due to the
rotation of the shaft under a bending moment. For a combined stress fatigue issue, we can use
the Von Mises stress as the equivalent stress to run related fatigue calculation [2]. We will use
the following assumptions to study the fatigue issue of a rotating shaft.
(1) e cyclic stress due to the combined loading will be a cyclic Von Mises stress.
(2) e mean of the cyclic Von Mises stress is mainly induced by the acting torque on the
rotating shaft.
(3) e stress amplitude of the cyclic Von Mises stress is mainly induced by the acting bending
moment on the rotating shaft.
(4) e modified Goodman approach will be used to consider the effect of mean stress in
cyclic Von Mises stress.
Now, we will discuss the cyclic combined loading
.
n
Li
; T
i
; M
i
; i D 1; 2; : : : ; L
/
, where i
is the ith loading level. n
Li
, T
i
, and M
i
are the number of cycles, the torque and the bending
moment, respectively, in the i th loading level. L is the number of different combined loading
levels. n
Li
could be a constant number or a distributed number of cycles in a combined loading
level. M
i
can be a constant bending moment or a distributed bending moment in a combined
loading level. But T
i
will be a constant in each loading level. In the ith combined loading level
2.9. THE PROBABILISTIC FATIGUE DAMAGE THEORY (THE K-D MODEL) 97
with a bending moment M
i
and a torque T
i
, the mean stress
vonmi
, and the stress amplitude
vonai
of a cyclic Von Mises stress will be:
vonmi
D
p
3K
fs
T
i
(2.89)
vonai
D K
f
M
i
; (2.90)
where K
fs
and K
f
are the fatigue stress concentration factors for torsion stress and bending stress
on the critical section, respectively, which has been discussed in Section 2.6.
T
i
is the outer layer
nominal shear stress of the rotating shaft on the critical section due to the torque T
i
.
M
i
is the
maximum nominal bending stress on the rotating shaft due to a bending moment M
i
.
Since it is a non-zero-mean cyclic combined stress, we need to convert it into an equiv-
alent fully-reversed cyclic stress. e equivalent stress amplitude
aeqi
of this transferred fully
reversed cyclic stress is:
aeqi
D
vonai
S
u
S
u
vonmi
D
S
u
K
f
M
i
S
u
p
3K
fs
T
i
: (2.91)
e component fatigue damage index D will be:
D D
L
X
iD1
n
Li
S
u
K
f
M
i
S
u
p
3K
fs
T
i
!
m
: (2.92)
For a solid shaft with a diameter d , Equation (2.92) for the component fatigue damage index
D will become
D D
L
X
iD1
n
Li
32S
u
K
f
M
i
d
3
S
u
16
p
3K
fs
T
i
!
m
: (2.93)
For a hollow solid shaft with an inner diameter d
0
and an outer diameter d, Equa-
tion (2.92) for the component fatigue damage index D will become
D D
L
X
iD1
n
Li
"
32S
u
K
f
M
i
d
d
4
d
4
0
S
u
16
p
3K
fs
T
i
d
#
m
: (2.94)
e component fatigue strength K for a cyclic combined loading will be obtained based
on the cyclic bending stress because the cyclic Von Mises stress is mainly due to the bending
moment and the shaft rotation. erefore, the limit state function of a rotating shaft due to a
combined loading is:
g
.
K; D
/
D K D D
8
ˆ
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
ˆ
:
K
L
X
iD1
n
Li
"
32S
u
K
f
M
i
d
3
S
u
16
p
3K
fs
T
i
#
m
solid shaft
K
L
X
iD1
n
Li
"
32S
u
K
f
M
i
d
d
4
d
4
0
S
u
16
p
3K
fs
T
i
d
#
m
hollow shaft:
(2.95)
98 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
Equation (2.95) can be used to calculate the reliability of a rotating shaft under cyclic combined
loading. Now, we will use two examples to demonstrate how to calculate the reliability of a
rotating shaft under cyclic combined loading.
Example 2.30
e critical section of a solid rotating shaft with a diameter 1:750 ˙0:005
00
is subjected to cyclic
combined loading listed in Table 2.55. e ultimate material strength S
u
of the beam is 75 (ksi).
ree parameters of the component fatigue strength index K on the critical section for the cyclic
bending loading 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 the shaft.
Table 2.55: e cyclic combined loading spectrum for Example 2.30
Stress
Level # i
Number of Cycles n
Li
Torque T
i
(klb.in)
Bending Moment
M
i
(klb.in)
1 5,500 17.75 15.75
2 580,000 10.29 10.15
Solution:
(1) e cyclic Von Mises stress and the component fatigue damage index.
For the stress level #1, the mean Von Mises stress
vonm1
per Equation (2.49), the Von
Mises stress amplitude
vona1
per Equation (2.90) and its corresponding equivalent Von Mises
stress amplitude
aeq1
per Equation (2.91) are
vonm1
D
p
3K
fs
T
1
D
p
3
16T
1
d
3
(a)
vona1
D K
f
M
1
D
32M
1
d
3
(b)
aeq1
D
vona1
S
u
S
u
vonm
D
32M
1
S
u
d
3
S
u
16
p
3T
1
: (c)
For the stress level #2, we repeat the above calculations,
aeq2
D
32M
2
S
u
d
3
S
u
16
p
3T
2
: (d)
e component fatigue damage index of this shaft under the cyclic combined loading per Equa-
tion (2.94) is:
D D n
L1
32M
1
S
u
d
3
S
u
16
p
3T
1
8:21
C n
L2
32M
2
S
u
d
3
S
u
16
p
3T
2
8:21
: (e)
2.9. THE PROBABILISTIC FATIGUE DAMAGE THEORY (THE K-D MODEL) 99
(2) e limit state function of the shaft.
e limit state function of the rotating shaft in this example per Equation (2.95) is:
g
.
K; d
/
D K n
L1
32M
1
S
u
d
3
S
u
16
p
3T
1
8:21
n
L2
32M
2
S
u
d
3
S
u
16
p
3T
2
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 2.56.
(3) e reliability of the rotating 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;581;687
1;598;400
D 0:9895:
Table 2.56: e distribution parameters of random variables in Equation (e)
K (lognormal) d (in)
μ
lnK
σ
lnK
μ
d
σ
d
41.738 0.357 1.750 0.00125
Example 2.31
e critical section of a solid rotating shaft with a diameter 2:150 ˙ 0:005
00
is subjected to cyclic
combined loading listed in Table 2.57. e ultimate material strength S
u
of the beam is 75 (ksi).
ree parameters of the component fatigue strength index K on the critical section for the cyclic
bending loading 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 the rotating shaft.
Solution:
1. e cyclic Von Mises stress and the component fatigue damage index.
100 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
Table 2.57: e cyclic combined loading spectrum for Example 2.31
Number of
Cycles
Torque T (klb.in)
Normally Distributed Bending Moment
M (klb.in)
μ
M
σ
M
450,000 21.15 21.34 1.31
e mean Von Mises stress
vonm
per Equation (2.49), the Von Mises stress amplitude
vona
per Equation (2.90), and its corresponding equivalent Von Mises stress amplitude
aeq
per Equation (2.91) of the rotating shaft due to the cyclic combined loadings are:
vonm
D
p
3K
fs
T
D
p
3
16T
d
3
(a)
vona
D K
f
M
D
32M
d
3
(b)
aeq
D
vona
S
u
S
u
vonm
D
32MS
u
d
3
S
u
16
p
3T
: (c)
e component fatigue damage index of this shaft under the cyclic combined loading per Equa-
tion (2.94) is:
D D N
L
32MS
u
d
3
S
u
16
p
3T
8:21
: (d)
(2) e limit state function of the rotating shaft.
e limit state function of the rotating shaft under the cyclic combined loading in this
example per Equation (2.95) is:
g
.
K; M; d
/
D K N
L
32MS
u
d
3
S
u
16
p
3T
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. 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 2.58.
(3) e reliability of the rotating shaft.
e limit state function (e) contains two normal distributions and one log-normal distri-
bution. We will follow the procedure of the R-F method and the flowchart in Appendix A.2
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