136 3. THE DIMENSION OF A COMPONENT WITH REQUIRED RELIABILITY
Table 3.9: Distribution parameters for Example 3.4
S
y
(ksi) M (klb.in) b (in) h (in)
μ
S
y
σ
S
y
μ
M
σ
M
μ
b
σ
b
μ
h
μ
h
σ
h
32.2 3.63 50.25 1.04 2.000 0.0025 0.0025
Table 3.10: e iterative results of Example 3.1 by the modified H-L method
Iterative #
S
y
*
M
*
b
*
h
*
|∆d
*
|
1 32.2 50.25 2 2.163718
2 26.32887 50.55881 1.981349 2.411446 0.247727
3 26.29544 50.50239 1.984518 2.409705 0.001741
4 26.29542 50.50235 1.984563 2.409678 2.68E-05
(3) Use the modified H-L method to determine the dimension.
We can follow the procedure, and the flowchart in Figure 3.2 discussed above, we can
make the MATLAB program. e iterative results are listed in Table 3.10.
According to the result obtained from the program, the mean of the beam heigh with a
reliability 0.99 is
h
D 2:410
00
: (d)
erefore, the height of the beam with the required reliability 0.95 under the specified loading
will be
h D 2:410 ˙ 0:010
00
:
3.2.4 DIMENSION DESIGN BY THE MODIFIED R-F METHOD
When the limit state function (3.1) contains at least one non-normal distribution, we need to use
the modified R-F method to design the component dimension of a component with the required
reliability under specified loading [5]. e R-F method iteratively calculates the reliability index
ˇ and then uses converged reliability index ˇ to calculate the reliability, which has been discussed
in Section 3.7 of Volume 1 [1] and is also concisely displayed in Appendix A.2 of this book. Since
the reliability index ˇ is a known value for a dimension design, we will use the modified R-F
method to determine iteratively the mean
d
of the dimension d . e general procedure of the
modified R-F method for dimension design is explained and displayed as follows.
Step 1: Preliminary design for determining K
t
for static design or K
f
and k
b
for fatigue design.
3.2. DIMENSION DESIGN WITH REQUIRED RELIABILITY 137
Per Section 3.2.1, we can determine the dimension-dependent parameters K
t
for static design
or K
f
and k
b
for fatigue design if necessary.
Step 2: Establish the limit state function.
For a clear description of the modified R-F method procedure, we can rearrange the limit
state function (3.1) into the following form of the limit state function.
g
.
X
1
; : : : ; X
r
; X
rC1
; : : : ; X
n
; d
/
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure;
(3.18)
where the first r random variables are non-normal distributed random variables and the rest
.n r/ random variables are normally distributed random variables. e last one is normally
distributed dimension with a mean
d
and a standard deviation
d
.
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: Calculate the mean for non-normal distributed random variables.
For non-normally distributed random variables, we can calculate their means based on their type
of distributions, which was discussed in Chapter 3 of Volume 1 [1].
Step 5: Pick an initial design point P
0
X
0
1
; : : : ; X
0
r
; X
0
rC1
; : : : ; X
0
n
; d
0
.
e initial design point could be any point, but it must be on the surface of the limit state
function. Typically, we can use the means of the first n random variables as the values X
0
i
.i D
1; : : : ; n/ in the initial design point, the value d
0
will be determined by the limit state function:
X
0
i
D
X
i
i D 1; 2; : : : ; n
g
X
0
1
; : : : ; X
0
n
; d
0
D 0;
(3.19)
where
X
i
is the mean of every random variable X
i
.i D 1; : : : ; n/. For a non-normally dis-
tributed random variable X
i
.i D 1; : : : ; r/, we will use the mean values that have been calculated
in Step 4.
When the actual limit state function is provided, we can rearrange the second equation
in Equation (3.19) 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
: (3.20)
138 3. THE DIMENSION OF A COMPONENT WITH REQUIRED RELIABILITY
Step 6: e means and standard deviations of variables at the design point
P
0
X
0
1
; : : : ; X
0
n
; d
0
.
For non-normally distributed random variables, we convert them into equivalent normal dis-
tributed random variable [1] per Equation (3.21). For the first r non-normal random variables
in the limit state function described in Equation (3.18), we have the following equations:
z
0
X
i
D ˆ
1
F
X
i
X
0
i

D norminv
F
X
i
X
0
i

X
i
eq
D
1
f
X
i
X
0
i
z
0
X
i
D
1
f
X
i
X
0
i
normpdf
z
0
X
i
i D 1; 2; : : : ; r (3.21)
X
i
eq
D x
0
i
z
0
X
i
X
i
eq
;
where x
0
i
is the value of the non-normally distributed random variable X
i
at the design point
P
0
. f
X
i
x
0
i
and F
X
i
x
0
i
are the PDF and CDF of the non-normally distributed random
variable X
i
at the design point P
0
.
X
i
eq
and
X
i
eq
are the equivalent mean and the equivalent
standard deviation of the equivalent normally distributed random variable.
Now every random variable in the limit state function in Equation (3.18) at the design
point P
0
is a normally distributed random variable. e mean
eX
i
and standard deviation
eX
i
of these normal distributed random variable at the design point P
0
are:
eX
i
D
8
<
:
X
i
eq
i D 1; 2; : : : ; r
X
i
i D r C 1; : : : ; n
(3.22)
eX
i
D
(
X
i
eq
i D 1; 2; : : : ; r
X
i
i D r C 1; : : : ; n:
(3.23)
Step 7: Calculate the new design point P
1
X
1
1
; : : : ; X
1
n
; d
1
.
For the R-F method, we can calculate the Taylor Series coefficients:
G
i
j
P
0
D
eX
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
:
(3.24)
Let’s 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
: (3.25)
e new design point P
1
X
1
1
; : : : ; X
1
n
; d
1
can be calculated by the following equations:
X
1
i
D
X
i
C
X
i
ˇ
G
i
j
P
0
G
0
i D 1; : : : ; n (3.26)
3.2. DIMENSION DESIGN WITH REQUIRED RELIABILITY 139
d
1
will be determined by the limit state function per Equation (3.18) and can be displayed as
the following equation:
d
1
D g
1
X
1
1
; X
1
2
; : : : ; X
1
n
: (3.27)
Step 8: Update the dimension-dependent parameters.
If there are some 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
loading, 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
; (3.28)
where
G
d
j
P
0
G
0
will be the value calculated in Equations (3.24) and (3.25).
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
if necessary. en, 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
if necessary.
Step 9: Check the 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
: (3.29)
If the convergence condition (3.29) is not satisfied, we need to update the design point by using
the following recurrence of Equation (3.30) and go back to Step 6:
X
0
i
D X
1
i
i D 1; : : : ; n
d
0
D d
1
:
(3.30)
If the convergence condition (3.29) is satisfied, the
d
in Equation (3.28) is the mean of the
dimension with the required reliability under the specified loading.
Since the modified R-F method is an iterative process, we should use the program for
calculation. e program flowchart is shown in Figure 3.3.
140 3. THE DIMENSION OF A COMPONENT WITH REQUIRED RELIABILITY
Start
Calculate Taylor Series coefficients
, and (3-22)
Figure 3.3: e program flowchart for the modified R-F method.
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