3.3. RELIABILITY OF A COMPONENT WITH TWO RANDOM VARIABLES 115
e limit state function g
.
S
e
; h; b; M
/
is the function of four parameters S
e
, h, b; and M .
3.3 RELIABILITY OF A COMPONENT WITH TWO
RANDOM VARIABLES
When the limit state function of a component consists of only two mutually independent ran-
dom variables, we can have an explicit formula for determining the reliability of a component.
In this section, we will derive an explicit formula for a general case. en, four special cases will
be discussed, which are: both uniform distributions, both normal distributions, both log-normal
distributions, and both exponential distributions.
3.3.1 INTERFERENCE METHOD
Let f
S
.
s
/
and F
S
.
s
/
represent the PDF and CDF of component strength index S. Let f
Q
.
q
/
and F
Q
.
q
/
represent the PDF and CDF of component stress index Q. In the following, we
assume that S and Q are statistically independent.
e interference method is to use the following Equations (3.5) or (3.6) to calculate the
reliability of a component when there are only two statistically independent random variables in
a limit state function.
Now, we will derive these two equations. Based on Equation (3.2), the reliability of a
component is
R D P
Œ
g
.
S; Q
/
0
D
ZZ
SQ0
f
S
.s/f
Q
.q/dsdq D
ZZ
SQ
f
S
.s/f
Q
.q/dsdq: (3.4)
If we run the integration concerning the random variable S first, as shown in Figure 3.2,
the reliability of the component will become:
R D P
Œ
g
.
S; Q
/
0
D
ZZ
SQ
f
S
.s/f
Q
.q/dsdq D
Z
C1
1
f
Q
.q/
Z
C1
q
f
S
.s/ds
dq:
D
Z
C1
1
f
Q
.q/
1
Z
q
1
f
S
.s/ds
dq D
Z
C1
1
f
Q
.q/
Œ
1 F
S
.q/
dq: (3.5)