150 3. THE DIMENSION OF A COMPONENT WITH REQUIRED RELIABILITY
the component with this dimension if the yield strength S
y
of the bar’s material follows a normal
distribution with the mean
S
y
D 34:5 (ksi) and the standard deviation
S
y
D 3:12 (ksi).
Solution:
In this example, there is no dimension-dependent parameter.
(1) e limit state function for question #1.
e deformation of the bar due to the axial loading F is
ı D
FL
EA
D
FL
Ed
2
=4
D
4FL
Ed
2
: (a)
e limit state function of the bar is
g
.
E; F; L; d
/
D 0:014
4FL
Ed
2
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(b)
In the limit state function, there are four normally distributed random variables. e mean and
the standard deviation of length L can be determined per Equation (1.1). e standard deviation
of normally distributed d can be determined per Equation (1.1). eir distribution parameters
in the limit state function (b) are listed in Table 3.17.
Table 3.17: Distribution parameters for the question #1 in Example 3.8
E (ksi) F (klb) L (in) d (in)
μ
E
σ
E
μ
F
σ
F
μ
L
σ
L
μ
d
σ
d
2.76×10
4
6.89×10
2
8.92 0.675 17.000 0.0025
μ
d
0.00125
(2) Use the modified Monte Carlo method to determine the mean
d
of the diameter d .
We can follow the procedure of the modified Monte Carlo method discussed above and
the flowchart shown in Figure 3.5 to compile a MATLAB program. e iterative results are
listed in Table 3.18.
According to the result obtained from the program, the mean of the diameter of the bar
with the required reliability 0.99 is
d
D 0:772
00
: (c)
erefore, the diameter of the bar with the required reliability 0.99 under the specified loading
will be
d D 0:772 ˙ 0:005
00
:
3.2. DIMENSION DESIGN WITH REQUIRED RELIABILITY 151
Table 3.18: e iterative results of Example 3.8 by the modified Monte Carlo Method
Iterative #
μ
d
*
R
*
∆R
*
1 0.729342 0.789484 -0.20052
2 0.730342 0.79973 -0.19027
… … … …
42 0.770342 0.989449 -0.00055
43 0.771342 0.990444 0.000444
(3) e limit state function for question #2.
e normal stress of the bar caused by the axial loading F is
D
F
A
D
F
d
2
=4
D
4F
d
2
: (d)
e limit state function of the bar is
g
S
y
; F; d
D S
y
4F
d
2
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(e)
In the limit state function, there are three normally distributed random variables. eir distri-
bution parameters in the limit state function (e) are listed in Table 3.19.
Table 3.19: Distribution parameters for the question #2 in Example 3.8
S
y
(ksi) F (klb) d (in)
μ
E
σ
E
μ
F
σ
F
μ
d
σ
d
34.5 3.12 8.92 0.675 0.772 0.00125
(4) Use the Monte Carlo method to calculate the reliability.
We can follow the procedure of the Monte Carlo method discussed and the flowchart in
Appendix A.3 to compile a MATLAB program. e reliability R of the bar with the dimension
d D 0:772 ˙ 0:005
00
is:
R D
15998345
15998400
D 0:999997:
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