3.2. DIMENSION DESIGN WITH REQUIRED RELIABILITY 141
Example 3.5
e critical section of a shaft is in the shoulder section, as shown in Figure 3.4. e shear yield
strength S
sy
(ksi) of a shaft follows a normal distribution with a mean
S
sy
D 31 (ksi) and the
standard deviation
S
sy
D
2:4
(ksi). e torque applied on the shaft
T
(klb.in) follows a two-
parameter Weibull distribution with the scale parameter D 20 and the shape parameter ˇ D 3.
Determine the diameter of the shaft with a reliability 0.99 when the dimension tolerance is
˙0:005.
1
16
R "
Ø 3.250"
d
Figure 3.4: Schematic of the segment of a shoulder of a shaft.
Solution:
(1) Preliminary design for determining the static shear stress concentration factor K
ts
.
is is a static design problem. According to the schematic of the stepped shaft, we can
assume that it has a well-rounded fillet. Per Table 3.2, we have the preliminary static stress
concentration factor for the shearing loading:
K
ts
D 1:6: (a)
(2) e limit state function.
e shear stress of the shaft caused by the torque T is
D K
ts
T
16
d
3
D K
ts
16T
d
3
: (b)
e limit state function of the shaft is
g
T; S
ys
; K
ts
; d
D S
ys
K
ts
16T
d
3
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(c)
142 3. THE DIMENSION OF A COMPONENT WITH REQUIRED RELIABILITY
In the limit state function, there are four random variables. S
ys
; K
ts
, and d follow normal dis-
tributions. T follows two-parameter Weibull distribution. e standard deviation of normally
distributed d can be determined per Equation (1.1). e mean and standard deviation of the
static shearing stress concentration factor K
ts
will be determined per Equation (1.3). eir dis-
tribution parameters in the limit state function (c) are listed in Table 3.11. e K
ts
in this table
will be updated in each iterative step by using the new available value
d
of the shaft diameter.
Table 3.11: Distribution parameters for Example 3.5
S
ys
(ksi) T (klb.in) Weibull Distribution K
ts
d (in)
μ
S
y
σ
S
y
η β μ
K
ts
σ
K
ts
μ
d
σ
d
31 2.4 20 3 1.6 0.08
μ
d
0.00125
(3) e reliability index ˇ of the shaft with a reliability 0.99.
e reliability index ˇ with the required reliability R D 0:99 per Equation (3.2) is
ˇ D ˆ
1
.
0:99
/
D 2:326348: (d)
(4) Use the modified R-F method to determine the dimension.
Following the modified R-F method procedure discussed above and the flowchart shown
in Figure 3.3, we can make a MATLAB program. e program is listed in Appendix B.5 as
“M-R-F program-Example 3.5.” e iterative results are listed in Table 3.12.
Table 3.12: e iterative results of Example 3.5 by the modified R-F method
Iterative #
T
*
S
y
*
K
t
*
s
d
*
|∆d
*
|
1 17.85959 31 1.6 1.674429
2 23.27808 30.65033 1.607528 1.835977 0.161549
3 23.28036 30.64817 1.918192 1.838956 0.002978
4 23.14829 30.51093 1.92268 1.949733 0.110778
5 23.1735 30.50655 1.93525 1.952054 0.002321
6 23.1677 30.50057 1.93568 1.956264 0.004209
7 23.16837 30.5003 1.936161 1.956433 0.000169
8 23.16816 30.50007 1.936187 1.956594 0.000161
9 23.16818 30.50005 1.936205 1.956604 9.47E-06
According to the result obtained from the program, the mean of the diameter with a
reliability 0.99 is
d
D 1:957
00
: (e)
3.2. DIMENSION DESIGN WITH REQUIRED RELIABILITY 143
erefore, the diameter of the shaft with the required reliability 0.99 under the specified loading
will be
d D 1:957 ˙ 0:005
00
:
Example 3.6
A machined double-shear pin is under a cyclic shearing loading spectrum. e mean shearing
loading can be treated as a constant V
m
D 10:125 (klb). e shearing loading amplitude V
a
can
be treated as a normal distribution with a mean
V
a
D 8:72 (klb) and a standard deviation
V
a
D
0:357 (klb). e number of cycles n
L
of this cyclic shearing loading is treated as a constant n
L
D
500000 (cycles). e ultimate material strength S
u
of the pin is 75 (ksi). e three parameters of
the material fatigue strength index K
0
on the critical section for a fully reversed bending loading
on the standard fatigue specimen 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 pin with
a reliability 0.99 when the dimension tolerance is ˙0:005.
Solution
(1) Preliminary design for determining k
b
for fatigue design.
is problem does not have any stress concentration. However, it is a fatigue issue; the
size modification factor is a dimension-dependent parameter. e preliminary size modification
factor per Equation (3.3) will be
k
b
D 0:87: (a)
(2) e cyclic stress and the component fatigue damage index.
e mean shear stress
m
and the shear stress amplitude
a
of the pin due to this cyclic
shearing loading are:
m
D
V
m
=2
A
D
V
m
=2
d
2
=4
D
2V
m
d
2
(b)
a
D
V
a
=2
A
D
V
a
=2
d
2
=4
D
2V
a
d
2
: (c)
Since this is non-zero-mean cyclic shear stress, the equivalent stress amplitude of a fully reversed
cyclic shear stress is:
aeq
D
a
S
u
.
S
u
m
/
D
2V
a
d
2
S
u
.
S
u
2V
m
=d
2
/
D
2V
a
S
u
.
d
2
S
u
2V
m
/
: (d)
e component fatigue damage index of this pin under this model #3 cyclic shear loading per
Equation (2.84) is:
D D n
L
2V
a
S
u
.
d
2
S
u
2V
m
/
8:21
: (e)
144 3. THE DIMENSION OF A COMPONENT WITH REQUIRED RELIABILITY
(3) e limit state function.
e limit state function of the pin under this cyclic shearing loading spectrum per Equa-
tion (2.87) is:
g
.
K
0
; k
a
; k
c
; V
a
; d
/
D .k
a
k
b
k
c
/
8:21
K
0
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:
(f)
ere are five random variables in the limit state function (f). K
0
is a lognormal distribution. All
others are normal distributions. e mean and standard deviation of the surface modification
factor k
a
can be determined per Equations (2.14), (2.15), and (2.16). e mean and standard
deviation of the load modification factor k
c
can be determined per Equations (2.18), (2.19),
and (2.20). e size modification factor k
b
is a deterministic and can be determined per Equa-
tion (2.17). It needs to be updated in each iterative step when the
d
is updated. e dimension
d is a normal distribution. Its standard deviation can be calculated per Equation (1.1). eir
distribution parameters in the limit state function (f) are listed in Table 3.13.
Table 3.13: e distribution parameters of random variables in Equation (f)
K
0
k
a
k
c
V
a
(klb) d (in)
μ
lnK
0
σ
lnK
0
μ
k
a
σ
k
a
μ
k
c
σ
k
c
μ
V
a
σ
V
a
μ
d
σ
d
41.738 0.357 0.8588 0.05153 0.583 0.07171 8.72 0.357
μ
d
0.00125
(4) e reliability index ˇ of the double-shear pin with the reliability 0.99.
e reliability index ˇ with the required reliability R D 0:99 per Equation (3.2) is
ˇ D ˆ
1
.
0:99
/
D 2:326348: (g)
(5) Use the modified R-F method to determine the dimension.
We can follow the modified R-F method procedure discussed above and the flowchart
shown in Figure 3.4 to compile a MATLAB program. e iterative results are listed in Ta-
ble 3.14.
According to the result obtained from the program, the mean of the diameter d of the
pin with the reliability 0.99 is
d
D 0:716
00
: (h)
erefore, the diameter of the pin with the required reliability 0.99 under the specified loading
will be
d D 0:716 ˙ 0:005
00
:
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