132 3. THE DIMENSION OF A COMPONENT WITH REQUIRED RELIABILITY
Step 7: Check convergence condition and the mean
d
of the dimension d .
d is the dimension in the unit of inch. erefore, the convergence condition for the dimension
d can be:
abs
d
1
d
0
< 0:0001
00
: (3.16)
If the convergence condition (3.16) is not satisfied, we need to update the design point by using
the following recurrence of Equation (3.17) and go back to Step 5:
X
0
i
D X
1
i
i D 1; : : : ; n
d
0
D d
1
:
(3.17)
If the convergence condition (3.16) is satisfied, the
d
in Equation (3.15) is the mean
d
of the
dimension d with the required reliability under the specified loading.
Since the modified H-L method is an iterative process, we should use the program for
calculation. e program flowchart of the modified H-L method is shown in Figure 3.2.
Example 3.3
Use the modified H-L method to do Example 3.1.
A circular stepped bar as shown in Figure 3.1 is subjected to axial loading F , which follows
a normal distribution with a mean
F
D 28:72 (klb) and a standard deviation
F
D 2:87 (klb).
e material of this bar is ductile. e yield strength S
y
of this bar’s material follows a normal
distribution a mean
S
y
D 32:2 (ksi) and a standard deviation
S
y
D 3:63 (ksi). Determine the
diameter d of the bar with a reliability 0.99 when its dimension tolerance is ˙0:005.
Solution:
(1) Preliminary design for determining K
t
.
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
K
t
D 1:9: (a)
(2) e limit state function.
e normal stress of the bar caused by the axial loading F is
D K
t
F
d
2
=4
D K
t
4F
d
2
: (b)
e limit state function of the bar is
g
S
y
; F; K
t
; d
D S
y
K
t
4F
d
2
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(c)