220 A. COMPUTATIONAL METHODS FOR THE RELIABILITY OF A COMPONENT
After the X
1
n
is obtained from Equation (A.30), Z
1
n
can be calculated through the conversion
Equation (A.31):
Z
1
n
D
X
0
n
X
n
X
n
: (A.31)
Now we have the new design point P
1
X
1
1
; X
1
2
; : : : ; X
1
n
in original normal distribution
space and the same design point P
1
Z
1
1
; Z
1
2
; : : : ; Z
1
n
in the standard normal distribution
space.
Step 8: Check convergence condition.
e convergence equation for this iterative process will be the difference
ˇ
ˇ
ˇ
0
ˇ
ˇ
between the
current reliability index and the previous reliability index. Since ˇ is a reliability index, the fol-
lowing convergence condition will provide an accurate estimation of the reliability:
ˇ
ˇ
ˇ
0
ˇ
ˇ
0:0001: (A.32)
If the convergence condition is satisfied, the reliability of the component will be:
R D P
Œ
g
.
X
1
; X
2
; : : : ; X
n
/
> 0
D ˆ
ˇ
0
: (A.33)
If the convergence condition is not satisfied, we use this new design point
P
1
X
1
1
; X
1
2
; : : : ; X
1
n
to replace the previous design point P
0
X
0
1
; X
0
2
; : : : ; X
0
n
,
that is,
X
0
i
D X
1
i
i D 1; : : : ; n
ˇ D ˇ
0
:
(A.34)
en, we go to Step 4 for a new iterative process again until the convergence condition is satis-
fied.
e program flowchart for the R-F method is shown in Figure A.2.
A.3 THE MONTE CARLO METHOD
A general limit state function g
.
X
1
; X
2
; : : : ; X
n
/
of a component is the function of random
variables X
1
; X
2
; : : : , and X
n
. erefore, it is also a random variable. Per the definition of prob-
ability, we can use the relative frequency to estimate the reliability when the number of sam-
ple data of the limit state function is sufficiently big [2]. e Monte Carlo method [2, 3, 5]
relies on repeated random sampling to obtain the numerical value of the limit state function
g
.
X
1
; X
2
; : : : ; X
n
/
for estimating the relative frequency.
Basic concepts and procedure for the Monte Carlo method are as follows.
Step 1: Uniformly and randomly generate one sample value for each random variable per its
corresponding probabilistic distribution. Let x
j
i
.i D 1; 2; : : : ; n/ to be the sample data in the
A.3. THE MONTE CARLO METHOD 221
Figure A.2: e program flowchart for the R-F method.
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.
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