222 A. COMPUTATIONAL METHODS FOR THE RELIABILITY OF A COMPONENT
j th trial of the virtual experiment. Here, the subscript i in x
j
i
refers to the ith random variable
X
i
. e superscript j in x
j
i
refers to the j th trial. e x
j
i
is the sample value of the random
variable X
i
in the j th trial of the virtual experiment.
Step 2: Use x
j
i
.i D 1; 2; : : : ; n/ in the limit state function to get a trial value of the
limit state function. Per the definition of the limit state function, when the trial value:
g
x
j
1
; x
j
2
; : : : ; x
j
n
of the limit state function of the component is larger than or equal to
zero, the component is safe. When the trail value: g
x
j
1
; x
j
2
; : : : ; x
j
n
of the limit state func-
tion of the component is less than zero, the component is a failure. We can use VT
j
to represent
the trial result:
VT
j
D
8
<
:
1 when g
x
j
1
; x
j
2
: : : ; x
j
n
0
0 when g
x
j
1
; x
j
2
; : : : ; x
j
n
< 0;
(A.35)
where VT
j
is the trial result of the j th trial of the virtual experiment. e value “1” of the VT
j
indicates a safe status of the component. e value “0” of the VT
j
indicates a failure status of
the component.
Step 3: Repeat Steps 1 and 2 until enough number of trials N have been conducted.
Since the limit state function of a mechanical component is typically not too complicated, we
can use N D 15;998;400, which is big enough for a critical component with a reliability 0.9999.
Step 4: e relative frequency of the component with a safe status in total trial N will be the
probability of the event g
.
X
1
; X
2
; : : : ; X
n
/
0. erefore, the reliability of the component will
be
R D P
Œ
g
.
X
1
; X
2
; : : : ; X
n
/
0
D
P
N
j D1
VT
j
N
: (A.36)
e probability of the component failure F will be:
F D 1 R D 1
P
N
j D1
VT
j
N
: (A.37)
Step 5: Calculate the relative errors.
In the Monte Carlo method, the relative error between the true value of the probability of the
component failure and the estimated value in Equation (A.37) will become smaller when the
trail number N increases. For a 95% confidence level, the relationship [2, 4] between the relative
error " and the trial number N is:
" D 2
r
1 F
N F
; (A.38)
where " is the relative error of the probability of component failure with a 95% confidence level.