134 3. THE DIMENSION OF A COMPONENT WITH REQUIRED RELIABILITY
In the limit state function, there are four normally distributed variables. For the dimension, the
dimension standard deviation
d
can be determined per Equation (1.1). e mean and standard
deviation of the static stress concentration factor K
t
are determined per Equation (1.3). eir
distribution parameters in the limit state function (c) are listed in Table 3.7. e mean and
standard deviation of K
t
in the table will be updated in each iterative step. (Note: e user-
defined subroutine for updating the stress concentration factor is required for this example.)
Table 3.7: Distribution parameters for Example 3.3
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) e reliability index ˇ with the required reliability R.
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: (d)
(4) Use the modified H-L method to determine the dimension.
Following the procedure and the flowchart in Figure 3.2 discussed above, we can compile a
MATLAB program. e program is listed in Appendix B.4 as “M-H-L-program-Example 3.3.”
e iterative results are listed in Table 3.8.
Table 3.8: e iterative results of Example 3.3 by the modified H-L method
Iterative #
S
y
*
F
*
K
t
*
d
*
|∆d
*
|
1 32.2 28.72 1.9 1.468913
2 26.20276 32.92314 1.969613 1.775099 0.306186
3 25.437 32.08462 2.286689 1.916347 0.141248
4 25.3237 32.1278 2.286277 1.921748 0.005401
5 25.31118 32.11423 2.286569 1.92194 0.000192
6 25.31073 32.11421 2.286546 1.921946 6.79E-06
According to the result obtained from the program, the mean of the diameter with a
reliability 0.99 is
d
D 1:922
00
: (e)
3.2. DIMENSION DESIGN WITH REQUIRED RELIABILITY 135
erefore, the diameter of the bar with the required reliability 0.99 under the specified loading
will be
d D 1:922 ˙ 0:005
00
:
Example 3.4
Use the modified H-L method to do 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 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). If the
width of the beam is b D 2:000 ˙ 0:010
00
. Determine the height h of the bar with a reliability
0.95 when the dimension tolerance is ˙0:010
00
.
Solution:
For this problem, there is not static stress concentration. So we do not need to do the preliminary
design.
(1) e limit state function.
e normal stress of the beam caused by the bending moment M is
D
Mh=2
bh
3
=12
D
6M
bh
2
: (a)
e limit state function of the beam is
g
S
y
; M; b ; h
D S
y
6M
bh
2
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(b)
e bending moment M can be treated as a normal distribution. Its mean and standard deviation
can be determined per Equation (1.2). For the dimension, the dimension standard deviation
d
can be determined per Equation (1.1). In the limit state function, there are four normally
distributed variables. eir distribution parameters in the limit state function (b) are listed in
Table 3.9.
(2) e reliability index ˇ with the required reliability 0.95.
e reliability index ˇ with the required reliability R D 0:95 per Equation (3.2) is
ˇ D ˆ
1
.
0:0:95
/
D 1:64485: (c)
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