122 3. COMPUTATIONAL METHODS FOR THE RELIABILITY OF A COMPONENT
We can use any one of Equations (3.9), (3.10), and (3.11) to calculate the reliability of the
component. Use Equation (3.9).
R D P
g
N
f
; n
s
0
D 1 NORM:DIST
.
0; 1:419; 0:6651; t rue
/
D 1 0:01644 D 0:9836:
3.3.4 COMPUTATION OF RELIABILITY WHEN BOTH ARE
EXPONENTIAL DISTRIBUTIONS
When both component strength index Sand component stress index Q follow exponential dis-
tributions, we can also use the interference method to calculate the reliability of the component.
e exponentially distributed component strength index S with the distribution parameter
S
has the PDF and the CDF as
f
S
.
s
/
D
S
e
S
S
0 s 1 (3.16)
F
S
.
s
/
D 1 e
S
S
0 s 1: (3.17)
e exponentially distributed component strength index Q with the distribution parameter
Q
has the PDF as
f
Q
.
q
/
D
Q
e
Q
q
0 q 1 (3.18)
F
Q
.
q
/
D 1 e
Q
q
0 q 1: (3.19)
Equations (3.5) or (3.6) can be used to calculate the reliability of the component. Use Equa-
tion (3.6) to run the calculation:
R D P
Œ
g
.
S; Q
/
0
D
Z
C1
0
f
S
.
s
/
F
Q
.
s
/
ds D
Z
C1
0
S
e
S
S
h
1 e
Q
s
i
ds
D
Z
C1
0
h
S
e
S
S
S
e
.
S
C
Q
/s
i
ds D
e
S
S
C
S
S
C
Q
e
.
S
C
Q
/s
ˇ
ˇ
ˇ
ˇ
1
0
D 1
S
S
C
Q
D
Q
S
C
Q
: (3.20)
Example 3.7
e failure mode of a component will be a static failure when the maximum stress at the critical
section is more than the ultimate material strength. It is assumed that both the maximum stress
max
and the ultimate strength S
u
follow an exponential distribution with a mean
max
D
12:4 (ksi) and a mean
S
u
D 62:5 (ksi). Calculate the reliability of this component.
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