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