3.2. DIMENSION DESIGN WITH REQUIRED RELIABILITY 129
Per Equations (3.3) and (3.4), the mean and the standard deviation of the limit state
function are:
g
D g
S
y
;
M
;
b
;
h
D
S
y
6
M
b
2
h
D 32:2
6 50:25
2:000
2
h
D 32:2
150:75
2
h
(c)
g
D
v
u
u
t
S
y
2
C
6
M
b
2
h
!
2
C
6
M
b
2
b
2
h
!
2
C
12
M
h
b
3
h
!
2
D
s
13:1769 C
9:7344
4
h
C
0:03551
4
h
C
0:56814
6
h
D
s
13:1769 C
9:7699
4
h
C
0:56814
6
h
: (d)
(3) e diameter of the beam with a reliability 0.95.
e reliability index ˇ with the required reliability R D 0:95 per Equation (3.2) is
ˇ D ˆ
1
.
0:95
/
D 1:64485: (e)
Per Equation (3.5), we have the following equation:
1:64485 D
32:2
29:125
2
h
s
113:1769 C
9:7699
4
h
C
0:56814
6
h
D
32:2
3
h
150:75
h
q
13:1769
6
h
C 9:7699
2
h
C 0:56814
: (f)
By solving Equation (f), we have:
h
D 2:400
00
: (g)
erefore, the height of the beam with the required reliability 0.95 under the specified loading
will be
h D 2:400 ˙ 0:010
00
:
3.2.3 DIMENSION DESIGN BY THE MODIFIED H-L METHOD
When a limit state function of a component is a nonlinear function of all normal distributions,
we can use the modified H-L method to determine the component dimension with the required
reliability under specified loading [3–5]. e H-L method iteratively calculates the reliability
index ˇ and then uses converged reliability index ˇ to calculate the reliability, which has been
discussed in Section 3.6 of Volume 1 [1]. It is also displayed in Appendix A.1 of this book.
For a dimension design, we already know the reliability index ˇ, and we will use the following
modified H-L method to determine iteratively the mean
d
of the dimension.
130 3. THE DIMENSION OF A COMPONENT WITH REQUIRED RELIABILITY
e general procedure for the modified H-L method for dimension design is explained
and displayed as follows.
Step 1: Preliminary design for determining K
t
for static or K
f
and k
b
for fatigue design.
Per Section 3.2.1, we can determine the dimension-dependent parameters K
t
for static or K
f
and k
b
for fatigue issue if necessary.
Step 2: Establish the limit state function.
e general limit state function of a component for dimension design is listed in Equation (3.1)
and redisplayed here:
g
.
X
1
; : : : ; X
n
; d
/
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure;
(3.1)
where X
i
.i D 1; 2; : : : ; n/ is a random variable related to component strength or loading, which
could be any type of distributions. d is a normal distribution dimension with a mean
d
and a
standard deviation
d
. For component dimension design,
d
is the variable for solving and
d
is determined by the manufacturing process and is treated as a known variable.
Step 3: Calculate the reliability index ˇ.
According to the required reliability of the component, we can determine the reliability index ˇ
per Equation (3.2) and is redisplayed here:
ˇ D ˆ
1
.
R
/
D norminv
.
R
/
: (3.2)
Step 4: Pick an initial design point P
0
X
0
1
; : : : ; X
0
n
; : : : ; d
0
.
e initial design point must be on the surface of the limit state function as specified by Equa-
tion (3.1). We can use the mean values for the first n variables and then use the limit state
function (3.1) to determine d
0
:
X
0
i
D
X
i
i D 1; 2; : : : ; n
g
X
0
1
; : : : ; X
0
n
; d
0
D 0:
(3.9)
When the actual limit state function is provided, we can rearrange the second equation in Equa-
tion (3.9) and express d
0
by using X
0
1
; X
0
2
; : : : and X
0
n
. Let’s use the following equation to
represent this:
d
0
D g
1
X
0
1
; X
0
2
; : : : ; X
0
n
: (3.10)
Step 5: Calculate the new design point P
1
X
1
1
; : : : ; X
1
n
; d
1
.
3.2. DIMENSION DESIGN WITH REQUIRED RELIABILITY 131
Based on the H-L method that was discussed in Section 3.6 of Volume 1 [1] and also concisely
displayed in Appendix A.1 of this book, we can calculate the Taylor Series coefficients:
G
i
j
P
0
D
X
i
@g
.
X
1
; : : : ; X
n
; d
/
@X
i
ˇ
ˇ
ˇ
ˇ
P
0
i D 1; 2; : : : ; n
G
d
j
P
0
D
d
@g
.
X
1
; : : : ; X
n
; d
/
@d
ˇ
ˇ
ˇ
ˇ
P
0
:
(3.11)
Let us use variable G
0
to represent the following equation:
G
0
D
v
u
u
t
n
X
iD1
.
G
i
j
P
0
/
2
C
.
G
d
j
P
0
/
2
: (3.12)
Since the reliability index ˇ is a known value, we can use the same approach shown in the H-L
method to calculate the new design point P
1
X
1
1
; : : : ; X
1
n
; d
1
:
X
1
i
D
X
i
C
X
i
ˇ
G
i
j
P
0
G
0
i D 1; : : : ; n: (3.13)
d
1
will be determined by the limit state function per Equation (3.10) and can be displayed as
follows:
d
1
D g
1
X
1
1
; X
1
2
; : : : ; X
1
n
: (3.14)
Step 6: Update the dimension-dependent parameters.
If there are dimension-dependent parameters such as static stress concentration factor K
t
for
static loading, the fatigue stress concentration factor K
f
and the size factor k
b
for cyclic loadings,
we need to use the new dimensions to update these dimension-dependent parameters. Since
the value of any random variable at the design point in the H-L method is determined by this
equation:
d
1
D
d
C
d
ˇ
G
d
j
P
0
G
0
:
Rearranging the above equation, we can get the equation for
d
:
d
D d
1
d
ˇ
G
d
j
P
0
G
0
; (3.15)
where
G
d
j
P
0
G
0
will be the value calculated in Equations (3.11) and (3.12).
After this
d
is known, the geometric dimensions for the stress concentration areas are
all known. We can update the static stress concentration factor K
t
. en, we can use Equa-
tions (2.22), (2.23), and (2.24) to update the fatigue stress concentration factor K
f
and use
Equation (2.17) to update the size modification factor k
b
.
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)
3.2. DIMENSION DESIGN WITH REQUIRED RELIABILITY 133
Calculate Taylor Series coefficients
< 0.0001
Figure 3.2: e program flowchart for the modified H-L method.
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