3.4. DIMENSION OF A COMPONENT UNDER CYCLIC LOADING SPECTRUM 189
(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
: (h)
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 determin-
istic value and can be calculated per Equation (2.17). Its value will be updated in each iterative
step by using the newly available diameter of the shaft. e mean and the standard deviation of
the load modification factor k
c
can be calculated per Equations (2.18), (2.19), and (2.20).
e limit state function of the shaft for this example per Equation (2.88) is
g
K
0
; k
a
; k
c
; K
fs
; d
D
.
k
a
k
b
k
c
/
m
K
0
n
L1
K
fs
16
a1
S
u
d
3
S
u
16K
fs
T
m1
8:21
n
L2
K
fs
16
a2
S
u
d
3
S
u
16K
fs
T
m2
8:21
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(i)
ere are five random variables in the limit state function (i). e K
0
is a lognormal distribution.
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 (i)
are listed in Table 3.58.
Table 3.58: e distribution parameters of random variables in Equation (i)
K (lognormal) k
a
k
c
K
fs
d (in)
μ
lnK
σ
lnK
μ
k
a
σ
k
a
μ
k
c
σ
k
c
μ
K
fs
σ
K
fs
μ
d
σ
d
41.738 0.357 0.8588 0.05153 0.774 0.1262 1.6552 0.1324
1.25
0.00125
(4) Use the modified R-F method to determine the diameter d .
In the limit state function (i), there are one lognormal distribution and four normal distri-
bution. Now, 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 program flowchart shown in Figure 3.4 to compile a MATLAB program. e iterative
progress is listed in Table 3.59.
According to the result obtained from the program, the mean of the diameter with a
reliability 0.99 is
d
D 1:363
00
: (j)