3.2. DIMENSION DESIGN WITH REQUIRED RELIABILITY 123
Table 3.2: e preliminary stress concentration factors for a stepped bar/plate
Stepped Plate or Bar Bending Axial
Shoulder fi llet–sharp (r/W = 0.02) 2.7 3.0
Shoulder fi llet–well-rounded (r/W = 0.1) 1.7 1.9
Table 3.3: e preliminary stress concentration factors for a plate with a center hole
Plate with a Center Hole Bending Loading Axial Loading
Narrow plate (d/W = 2) 1 2.1
Plate (d/W = 3) 0.667 2.3
In fatigue design, the fatigue stress concentration factor is a dimension-dependent which
will be determined by dimension, fillet radius, and type of material. e fatigue stress concen-
tration factor can be calculated per Equations (2.22), (2.23), and (2.24) in Section 2.6 when the
static stress concentration factor, fillet radius, and ultimate strength are known. Tables 3.1, 3.2,
and 3.3 can help to determine the preliminary static stress concentration factor. As we mentioned
above, the fillet radius can be pre-determined by the type of fillets and its purpose according to
the sketched structure of the component. We can also pre-select the type of material. erefore,
we can determine a preliminary estimation of the fatigue stress concentration factor.
In fatigue design, the size modification factor is also a dimension-dependent parameter
and can be calculated per Equation (2.17) in Section 2.4. e size dimension factor for bending
or torsion has a value between 0.807–1.12. We can use 0.87 as its preliminary estimation:
k
b
D
8
<
:
0:87 For bending or torsion loading
1 For axial loading:
(3.3)
In the following sections, we will discuss how to determine the mean
d
of a normally dis-
tributed dimension with the required reliability R.
3.2.2 DIMENSION DESIGN BY THE FOSM METHOD
When all random variables are normally distributed random variables in the limit state func-
tion (1.1), we can use the FOSM method to estimate
d
with the required reliability R under
specified loading condition. e FOSM method for calculating the reliability of a component
has been discussed in Section 3.5 of Volume 1 [1]. Now, we will discuss how to use the FOSM
to conduct the dimension design with the required reliability. e procedure of the dimension
design by the FOSM method is as follows.
Step 1: Preliminary design for determining K
t
for static design or K
f
and k
b
for fatigue design.
124 3. THE DIMENSION OF A COMPONENT WITH REQUIRED RELIABILITY
For a component dimension design under static loading, we can determine the preliminary static
stress concentration factor, as discussed in Section 3.2.1 if the critical section is in the stress
concentration area. For a component dimension design under cyclic fatigue loading spectrum, we
can determine the preliminary fatigue stress concentration factor, as discussed in Section 3.2.1
if the critical section is in the stress concentration area. For the size modification factor, we will
use 0.87 for bending and torsion or 1 for axial loading per Equation (3.3).
Step 2: Use the FOSM method to calculate
0
d
.
According to the FOSM method, the mean and standard deviation of the limit state function [1]
will be:
g
D g
X
1
; : : : ;
X
n
;
0
d
(3.4)
g
D
v
u
u
t
n
X
iD1
@g
.
X
1
; : : : ; X
n
; d
/
@X
i
ˇ
ˇ
ˇ
ˇ
means
X
i
2
C
@g
.
X
1
; : : : ; X
n
; d
/
@d
ˇ
ˇ
ˇ
ˇ
means
d
2
; (3.5)
where
X
i
and
X
i
are the mean and the standard deviation of normally distributed X
i
.
g
and
g
are the mean and the standard deviation of the limit state function g
.
X
1
; : : : ; X
n
; d
/
.
d
is
the standard deviation of the dimension and is pre-determined by the dimension tolerance per
Equation (1.1).
For dimension design, the reliability of a component is given; that is, R D ˆ.ˇ/. ere-
fore, the design equation for a dimension with the required reliabilit R by using the FOSM
method [1] is
ˇ D ˆ
1
.
R
/
D
g
g
D
g
X
1
; : : : ;
X
n
;
0
d
v
u
u
t
n
X
iD1
@g
.
X
1
; : : : ; X
n
; d
/
@X
i
ˇ
ˇ
ˇ
ˇ
means
X
i
2
C
@g
.
X
1
; : : : ; X
n
; d
/
@d
ˇ
ˇ
ˇ
ˇ
means
d
2
: (3.6)
In Equation (3.6),
0
d
is the only one unknown and can be solved.
Step 3: Update the dimension dependent parameters.
If there are dimension-dependent parameters in the limit state function, we need to use the
iterative process. After a new value
0
d
of the mean
d
in an iterative step is available, we can
use
0
d
to update dimension-dependent parameters such as the static stress concentration factor
K
t
, the fatigue stress concentration factor K
f
, or the size modification factor k
b
if necessary.
en, we can go back to Step 2 to calculate a new dimension
1
d
.
3.2. DIMENSION DESIGN WITH REQUIRED RELIABILITY 125
Step 4: Convergence condition.
For a dimension design, since we will obtain an approximate result, we can use the following
convergence condition:
abs.
1
d
0
d
/ < 0:001
00
: (3.7)
If the convergence condition is not satisfied, we go back to Step 2 until the convergence condition
is satisfied.
If the convergence condition (3.7) is satisfied, the
1
d
will be the mean of the dimension
with the required reliability, that is,
d
D
1
d
: (3.8)
If the limit state function is a linear function of all normally distributed random variables, the
FOSM method will provide an accurate result. However, if the limit state function is a nonlinear
function of all normally distributed random variables, the FOSM method will only provide an
approximate result [1].
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.
1
8
R "
Ø 3.250"
d
Figure 3.1: Schematic of the segment of a stepped bar.
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-round fillet. Per Table 3.2, we have the preliminary static stress con-
centration factor,
K
t
D 1:9: (a)
126 3. THE DIMENSION OF A COMPONENT WITH REQUIRED RELIABILITY
(2) e limit state function of this stepped bar.
For this problem, the critical section will be on the stress concentration section, that is, the
stepped section. e normal stress on the critical section 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
1
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)
We will use “allowable force” to form another version of the limit state function as shown in
Equation (d), which will be much easier to solve
0
d
g
2
S
y
; F; K
t
; d
D S
y
d
2
4
K
t
F D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(d)
In the limit state function, there are four normally distributed variables. e mean and standard
deviation of the static stress concentration factor K
t
can be determined per Equation (1.3). For
the dimension, the dimension standard deviation
d
can be determined per Equation (1.1).
eir distribution parameters in the limit state function (d) are listed in Table 3.4.
Table 3.4: Distribution parameters for Example 3.1
S
y
(ksi) F (klb) K
t
d (in)
μ
S
y
σ
S
y
μ
F
σ
F
μ
K
t
σ
K
t
μ
d
σ
d
32.2 3.63 28.72 2.87 1.9 0.095
μ
d
0.00125
(3)
0
d
in the first iterative process.
Per Equations (3.4) and (3.5), the mean and standard deviation of the limit state function
are:
g2
D g
2
S
y
;
F
;
K
t
;
d
D
S
y
0
d
2
4
K
t
F
(e)
g2
D
v
u
u
t
0
d
2
4
S
y
!
2
C
.
K
t
F
/
2
C
.
F
K
t
/
2
C
s
y
0
d
2
d
!
2
: (f)
3.2. DIMENSION DESIGN WITH REQUIRED RELIABILITY 127
e reliability index ˇ with the required reliability R D 0:99 per Equation (3.2) is
ˇ D ˆ
1
.
0:99
/
D 2:32635: (g)
Per Equation (3.6), we have:
ˇ D
g2
g2
D
S
y
0
d
2
4
K
t
F
v
u
u
t
0
d
2
4
S
y
!
2
C
.
K
t
F
/
2
C
.
F
K
t
/
2
C
s
y
0
d
2
d
!
2
: (h)
In Equation (h), only
0
d
is an unknown and can be solved. Equation (h) is based on the limit
state function (d). is is much simpler than the corresponding equation based on the limit state
function (c). Per Equation (h) with the current value, we can solve
0
d
:
0
d
D 1:7771
00
: (i)
(4) Update K
t
based on
0
d
D 1:7771
00
.
After we have dimension
0
d
D 1:7771, we have the geometric dimensions for the stress
concentration. en we can calculate the static stress concentration factor K
t
based on current
dimensions, that is, D D 3:25
00
; d D
0
d
D 1:7771
00
, and r D 1=8
00
. e updated K
t
in this case
is:
K
t
D 2:2039: (j)
(5) Create a MATLAB program to conduct the iterative process.
We can follow the procedure discussed above and the formula listed in this problem to
make a MATLAB program. e results of the iterative process from the program is listed in
Table 3.5.
From the iterative results, the dimension of d in this example is
d D 1:919 ˙ 0:005
00
:
Example 3.2
On the critical cross-section of a beam with a rectangular cross-section is subjected to a bending
moment M D 50:25 ˙4:16 (klb.in). e yield strength S
y
of this beam’s material follows a
normal distribution a mean
S
y
D 32:2 (ksi) and a standard deviation
S
y
D 3:63 (ksi). If the
width of the beam is b D 2:000 ˙ 0:010
00
. Determine the height h of the beam with a reliability
0.95 when the dimension tolerance is ˙0:010
00
.
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