198 3. THE DIMENSION OF A COMPONENT WITH REQUIRED RELIABILITY
Table 3.68: e iterative results of Example 3.27 by the modified R-F method
Iterative #
K
0
*
k
a
*
b
*
h
*
|∆h
*
|
1 1.32E+14 0.9053 2 2.703691
2 1.09E+14 0.868503 1.999953 2.777668 0.073977
3 1.11E+14 0.868067 1.999907 2.897338
0.119671
4 1.11E+14 0.867973 1.999954 2.902817
0.005478
5 1.11E+14 0.86797 1.999956 2.903057
0.00024
6 1.11E+14 0.86797 1.999956 2.903068
1.06E-05
3.4.7 COMPONENT UNDER CYCLIC COMBINED LOADING
SPECTRUM
e component fatigue issue due to the combined stresses is very complicated because the fre-
quencies of independent loadings might not be in phase. e limit state function of a rotating
shaft under cyclic combined bending-torsion loading is provided per Equation (2.95). Now, we
will only show one example of a rotating shaft under a cyclic combined bending-torsion loading
to determine the diameter with the required reliability.
Example 3.28
e critical section of a solid rotating shaft is at its shoulder section, as shown in Figure 3.14 and
is subjected to cyclic combined loading. According to the design specification, the cyclic fatigue
loading spectrum can be described by model #4 cyclic combining loads as listed in Table 3.69.
e ultimate material strength S
u
is 75 (ksi). e three distribution parameters of material fa-
tigue strength index K
0
for the standard specimen under the fully-reversed bending stress are
m D 8:21;
ln K
0
D 41:738, and
ln K
0
D 0:357. For the material fatigue strength index K
0
, the
stress unit is ksi. Determine the diameter of the shaft with a reliability 0.99 when its dimension
tolerance is ˙0:005
00
.
1
32
R "
Ø 2.000"
d
Figure 3.14: Schematic of the segment of a shaft with a shoulder.
3.4. DIMENSION OF A COMPONENT UNDER CYCLIC LOADING SPECTRUM 199
Table 3.69: e model #4 cyclic combined loading spectrum for Example 3.28
Level # i Number of Cycles n
Li
Torque T
i
(klb.in) Bending Moment M
i
(klb-in)
1 5,500 4.29 5.75
2 580,000 4.29 3.15
Solution:
(1) Preliminary design for determining the dimension-dependent parameters k
b
, K
t
, and K
f
.
For the component under cyclic combined loading with a stress concentration, there are
three dimension-dependent parameters k
b
, K
f
, and K
fs
in this fatigue problem. According
to the schematic of the shaft with a shoulder, we can assume that it has a sharp-fillet. e
preliminary static stress concentration factors per Table 3.2 will be
K
t
D 2:7 for bending, (a)
K
ts
D 2:2 for torsion: (b)
e preliminary size modification factor k
b
per Equation (3.3) will be
k
b
D 0:87: (c)
e size modification factor k
b
will be updated per Equation (2.17). e fatigue stress concen-
tration factor K
f
and K
fs
can be calculated and updated per Equations (2.22)–(2.25) when K
t
,
K
ts
and the fillet radius are known.
(2) e cyclic Von Mises stress and the component fatigue damage index.
e mean Von Mises stress
vonmi
per Equation (2.89), the Von Mises stress amplitude
vonai
per Equation (2.90) and its corresponding equivalent Von Mises stress amplitude
aeqi
per Equation (2.91) of the rotating shaft in this example are as follows.
For the stress level #1,
vonm1
D
p
3K
fs
T
1
D
p
3K
fs
16T
1
d
3
(d)
vona1
D K
f
M
1
D K
f
32M
1
d
3
(e)
aeq1
D
vona1
S
u
S
u
vonm
D
32K
f
M
1
S
u
d
3
S
u
16K
fs
p
3T
1
: (f )
For the stress level #2, we repeat the above calculations:
aeq2
D
32K
f
M
2
S
u
d
3
S
u
16K
fs
p
3T
2
: (g)
200 3. THE DIMENSION OF A COMPONENT WITH REQUIRED RELIABILITY
e component fatigue damage index of this shaft in this example per Equation (2.94) is:
D D n
L1
"
32K
f
M
1
S
u
d
3
S
u
16K
fs
p
3T
1
#
8:21
C n
L2
"
32K
f
M
2
S
u
d
3
S
u
16K
fs
p
3T
2
#
8:21
(h)
(3) e limit state function.
e component fatigue strength index K can be calculated per Equation (2.79).
K D
.
k
a
k
b
k
c
/
m
K
0
: (i)
e surface finish modification factor k
a
follows a normal distribution. Its mean and standard
deviation can be determined per Equations (2.14), (2.15), and (2.16). k
b
is treated as a deter-
ministic value and can be calculated per Equation (2.17). e mean and the standard deviation
of the load modification factor k
c
is assumed to be 1 because this combined cyclic loading is
mainly caused by cyclic bending stress.
e limit state function of the rotating shaft in this example per Equation (2.95) is:
g
K
0
; k
a
; K
f
; K
fs
; d
D
.
k
a
k
b
/
m
K
0
n
L1
"
32K
f
M
1
S
u
d
3
S
u
16K
fs
p
3T
1
#
8:21
n
L2
"
32K
f
M
2
S
u
d
3
S
u
16K
fs
p
3T
2
#
8:21
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(j)
ere are five random variables in the limit state function (j). K
0
is a lognormal distribution.
e rests are normal distributions. e dimension d can be treated as normal distributions. Its
mean and standard deviation can be calculated per Equation (1.1). e distribution parameters
in the limit state function (j) are listed in Table 3.70. In the table, K
f
and K
fs
will be updated
in each iterative step.
Table 3.70: e distribution parameters of random variables in Equation (j)
K
0
(lognormal) k
a
K
f
K
fs
d (in)
μ
lnK
0
σ
lnK
0
μ
k
a
σ
k
a
μ
K
f
σ
K
f
μ
K
fs
σ
K
fs
μ
d
σ
d
41.738 0.357 0.9053 0.05432 2.0337 0.1627 1.6552 0.1324
μ
d
0.00125
(4) Use the modified R-F method to determine the diameter d with reliability 0.99.
We will use the modified R-F to conduct this component dimension design, which has
been discussed in Section 3.2.4. We can follow the procedure discussed in Section 3.2.4 and the
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