150 3. COMPUTATIONAL METHODS FOR THE RELIABILITY OF A COMPONENT
where wblpdf .T
; ; ˇ/ and wblcdf .T
; ; ˇ/ are the commands for the PDF and the CDF of
the Weibull distribution with the scale parameter and the shape function ˇ at the design point
T
. norminv.z
T
) is the command for the inverse of standard normal distribution at the given
probability z
T
.
Per Equation (3.78), the Taylor series coefficients in this example are:
G
T
j
P
D
T
@g .T; d; S
sy
/
@T
ˇ
ˇ
ˇ
ˇ
atP
0
D
T
.1/
j
atP
0
D
T
G
d
j
P
D
d
@g.T; d; S
sy
/
@d
ˇ
ˇ
ˇ
ˇ
atP
0
D
d
S
sy
3d
2
16
!
ˇ
ˇ
ˇ
ˇ
ˇ
atP
0
D
d
S
sy
3
.
d
/
2
16
(f)
G
S
sy
ˇ
ˇ
P
D
S
sy
@g .T; d; S
sy
/
@S
sy
ˇ
ˇ
ˇ
ˇ
atP
0
D
S
sy
d
3
16
ˇ
ˇ
ˇ
ˇ
atP
0
D
S
sy
.
d
/
3
16
:
We can follow the provided procedure or the flowchart in Figure 3.8 to compile the MATLAB
program. e MATLAB program for this example is listed in Appendix A as “A.2: e R-F
method for Example 3.13.”
e data for the iterative process is shown in Table 3.3. e first column is the number
of iterative processes. e second to the fourth columns are the values of the design points. e
fifth column is the reliability index ˇ, and the last column is the convergence condition.
From the iterative results, the reliability index ˇ and corresponding reliability R of this
component are:
ˇ
0
D 2:3058; R D ˆ
ˇ
0
D ˆ
.
2:3058
/
D 0:9894:
e iterative results by the R-F method for Example 3.13.
Table 3.3: Iterative results by the R
F
method
# T
*
d
*
S
y
*
β
*
|∆ β
*
| < 0.0001
1 30.3613 2.125 16.11438 2.28222
2 54.64176 2.124941 29.00377 2.305446 0.023226
3 54.43365 2.124887 28.89548 2.305759 0.000314
4 54.44002 2.124888 28.89884 2.305756 2.85E-06
3.8 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. e reliability of the com-
ponent is the probability of the event g
.
X
1
; X
2
; : : : ; X
n
/
0. When the distributions of all
3.8. THE MONTE CARLO METHOD 151
random variables are given, we can use the interference method, the FOSM method, the H-L
method, or the R-F method to calculate the reliability, which has been discussed in Sections 3.3–
3.7. We can also use the relative frequency to estimate the reliability, which has been discussed
in Section 2.4. For example, when a sample value x
i
of each random variable is known, we can
use these sample values
.
x
1
; x
2
; : : : ; x
n
/
to calculate a sample value of the limit state function
g
.
x
1
; x
2
; : : : ; x
n
/
, which may be larger than and equal to, or less than zero. is process can be
called a trial in the virtual experiment. Per the definition of probability, we can use the relative
frequency to estimate the reliability when the number of sample data of the limit state func-
tion is sufficiently big. e Monte Carlo method 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/ be the sample data in the 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 function of the component
is less than zero, the component is a failure. We can use V T
j
to represent the trial result:
V T
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;
(3.87)
where V T
j
is the trial result of the j th trial of the virtual experiment. e value “1” of the
V T
j
indicates a safe status of the component. e value “0” of the V T
j
indicates a failure
status of the component.
Step 3: Repeat Step 1 and Step 2 until enough trials N have been conducted.
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
V T
j
N
: (3.88)
152 3. COMPUTATIONAL METHODS FOR THE RELIABILITY OF A COMPONENT
e probability of the component failure F will be:
F D 1 R D 1
P
N
j D1
V T
j
N
: (3.89)
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 (3.89) will become smaller when the
trail number N increases. For a 95% percent confidence level, the relationship [4, 5] between
the relative error " and the trial number N is:
" D 2
r
1 F
N F
; (3.90)
where " is the relative error of the probability of component failure with a 95% confidence level.
F is the estimated probability of the failure of the component per Equation (3.89). N is the trial
number in the Monte Carlo method. For example, if the estimated reliability is 0.975 and the
relative error of the probability of the component failure is 0.034, that is, 3.4%. In this case, the
estimated probability of the failure is F D 1 R D 1 0:975 D 0:025. Since the relative error
for F is 0.034, the F will have the value in the range 0:025 ˙ 0:025 0:034 D 0:025 ˙ 0:0008.
erefore, the range of reliability R will be 0:975 ˙ 0:0008.
We can rearrange Equation (3.90) to express the trial number N as a function of relative
percent error " and the estimated reliability R:
N D
4.1 F /
"
2
F
D
4R
"
2
.1 R/
: (3.91)
We can use Equation (3.91) to calculate the trial number N for the Monte Carlo method. After
completing the calculation by the Monte Carlo method, we can use Equation (3.91) to calculate
the relative error. To use Equation (3.91) to calculate the trial number N , we need to know the
pre-specified relative error " and the possible value of the reliability R. e possible value of the
reliability R can be guested according to its importance level in mechanical design, as shown in
Table 3.4.
If the pre-specified relative error of the probability of component failure is 0.05 with a
95% confidence level, the trial number N for the Monte Carlo method for the different level of
importance of a component is listed in Table 3.5.
e Monte Carlo method needs to have a huge amount of trials for an acceptable accuracy
of the estimated reliability. It is typically implemented through a computer program. is book
will use MATLAB as the computer program to implement the Monte Carlo method. We can
use the following command in the MATLAB software to generate a matrix 1 N of random
numbers of a specified distributed random variable:
RX
i
D random
0
name
0
; A; B; 1; N
; (3.92)
3.8. THE MONTE CARLO METHOD 153
Table 3.4: e levels of importance
Levels of Importance Reliability Notes
Critical component 0.9999 e failure of this component will cause death to the
operators or huge fi nancial loss.
Key component 0.999 e failure of this component will cause major fi nancial
loss or a long period of system shutdown.
Important component 0.99 e failure of this component might cause some fi nan-
cial loss or a short period of system shutdown.
General component 0.9 e failure of this component will not cause a big issue
to the system.
Table 3.5: e estimation of the trial number N
Levels of Importance Reliability Relative Error Trial Number N
Critical component 0.9999 0.05 15998400
Key component 0.999 0.05 1598400
Important component 0.99 0.05 158400
General component 0.9 0.05 14400
where RX
i
is a matrix 1 N with N of random samplings of a distributed random variable X
i
.
random is the MATLAB command for generating a random number.
0
name
0
is to specify the
type of distribution. A and B are the distribution parameters of the distributed random variables.
“1, N” in Equation (3.92) means that the matrix for these random number will be stored as one
row with N column. e
0
name
0
, A and B for several distributions are listed in Table 3.6.
Table 3.6: e
0
name
0
,
A
and
B
for several typical distributions
Distribution Input parameter A Input parameter B
Exponential distribution Mean μ = 1/λ /
Log-normal distribution Mean of logarithmic values,
μ
ln
Standard deviation of loga-
rithmic values, σ
ln
′
Normal distribution Mean μ Standard deviation σ
Uniform distribution Lower endpoint a Upper endpoint b
Weibull distribution Scale parameter η Shape parameter β
154 3. COMPUTATIONAL METHODS FOR THE RELIABILITY OF A COMPONENT
e flowchart of a MATLAB program for the Monte Carlo method is displayed in Fig-
ure 3.9.
Example 3.14
A key component is subjected to two axial loadings, as shown in Figure 3.10. e two axial
loadings P
1
(kip) and P
2
(kip) are statistically independent random variables. P
1
follows a
normal distribution with a mean
P
1
D 10:2 and a standard deviation
P
1
D 1:2. P
2
follows
a Weibull distribution with the scale parameter D 4:5 and the shape parameter ˇ D 1:5. e
yield strength S
y
of the component will be the component strength index and follows a normal
distribution with a mean
S
y
D 61:5 (ksi) and a standard deviation
S
y
D 5:95 (ksi). e di-
ameter d of the round bar can be described by a normal distribution with a mean
d
D 0:75
00
and the standard deviation
d
D 0:003
00
. Compile the MATLAB program to implement the
Monte Carlo method to estimate the reliability of this component with a 95% confidence level
and calculate the relative error. Show the range of the reliability of the component with a 95%
confidence level.
Solution:
In this example, the limit state function is:
g
S
y
; P
1
; P
2
; d
D S
y
4P
1
d
2
4P
2
d
2
: (a)
Let’s use 5% as the pre-specified relative error with a 95% confidence level. Because the com-
ponent is a key component, the trial number
N
for this Monte Carlo method from Table 3.2
is:
N D 1;598;400: (b)
We can compile the MATLAB program by following the procedure and the flowchart in Fig-
ure 3.9. e MATLAB program for this problem is listed in Appendix A with the name “A.3:
e Monte Carlo Method for Example 3.14.”
e estimated reliability of this component R is:
R D 0:9973: (c)
e estimated probability of the component failure F is
F D 1 R D 1 0:9973 D 0:0027: (d)
e relative error of the probability of the failure is
" D 0:0306: (e)
So, the range of the probability of the component failure with a 95% confidence level will be:
F D 0:0027 ˙ 0:0027 0:0306 D 0:0027 ˙ 0:00008: (f)
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