216 A. COMPUTATIONAL METHODS FOR THE RELIABILITY OF A COMPONENT
that is,
X
0
i
D X
1
1
Z
0
i
D Z
0
i
i D 1; : : : ; n (A.17)
ˇ D ˇ
0
:
en go to Step 4 for a new iterative process again until the convergence condition is satisfied.
Since the H-L method is an iterative process, we should use the program for calculation.
e program flowchart for the H-L method is shown in Figure A.1.
A.2 THE RACKWITZ AND FIESSLER (R-F) METHOD
When a limit state function of a component contains at least one non-normal distributed ran-
dom variables such as log-normal distribution or Weibull distribution, we need to use the R-
F (Rackwitz and Fiessler) method [2–4] to calculate the reliability of a component. e R-F
method is a modified H-L method. In the R-F method, any non-normally distributed random
variable at the design point will be first converted into an equivalent normally distributed random
variable. And then the H-L method is applied for calculating the reliability index. Two condi-
tions for calculating the equivalent mean and the equivalent standard deviation of the equivalent
normal distribution at the design point are: (1) the PDF of a non-normal distribution variable
at the design point will be equal to the PDF of its equivalent normal distribution at the design
point; and (2) the CDF of a non-normal distribution variable at the design point will be equal
to the CDF of its equivalent normal distribution at the design point.
e following is the general procedure for the R-F method.
Step 1: Calculate the mean for non-normal distributed random variables.
For a clear description of the R-F method procedure, we can rearrange the limit state func-
tion per Equation (A.18). In Equation (A.19), the first r random variables are non-normally
distributed random variables, and the rest .n r/ random variables are normally distributed
random variables.
g
.
X
1
; : : : ; X
r
; X
rC1
; : : : ; X
n
/
D
8
ˆ
<
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(A.18)
e surface of this limit state function is
g
.
X
1
; : : : ; X
r
; X
rC1
; : : : ; X
n
/
D 0: (A.19)
For non-normally distributed random variable, we can use their PDFs to calculate their means:
X
i
.i D 1; 2; : : : ; r/.
A.2. THE RACKWITZ AND FIESSLER (R-F) METHOD 217
Figure A.1: e program flowchart for the H-L method.
218 A. COMPUTATIONAL METHODS FOR THE RELIABILITY OF A COMPONENT
Step 2: Pick an initial design point P
0
X
0
1
; X
0
2
; : : : ; X
0
n
.
e initial design point could be any point. But it must be on the surface of the limit state
function as specified by Equation (A.19). We can use the mean values for the first n 1 variables,
as shown in Equation (A.20):
X
0
i
D
X
i
i D 1; 2; : : : ; n 1: (A.20)
X
0
n
can be determined through the surface of the limit state function of Equation (A.19). When
the actual limit state function is provided, we can rearrange express X
0
n
by using X
0
1
; X
0
2
; : : : ,
and X
0
n1
as shown in Equation (A.21):
X
0
n
D g
1
X
0
1
; X
0
2
; : : : ; X
0
n1
: (A.21)
Now, we have the initial design point P
0
X
0
1
; X
0
2
; : : : X
0
n1
; X
0
n
.
Step 3: Set ˇ D 0.
is setting is only for the MATLAB program. is setting will make sure that the iterative
process will have at least two iterative loops.
Step 4: e mean and standard deviation at the design point P
0
X
0
1
; X
0
2
; : : : X
0
n1
; X
0
n
.
For non-normally distributed random variables, we convert them into equivalent normal dis-
tributed random variables and calculate its equivalent mean and equivalent standard deviation
per Equation (A.22):
z
0
X
i
D ˆ
1
F
X
i
X
0
i
D norminv
F
X
i
X
0
i
X
i
eq
D
1
f
X
i
X
0
i
z
0
X
i
i D 1; 2; : : : ; r (A.22)
X
i
eq
D x
0
i
z
0
X
i
X
i
eq
;
where x
0
i
is the value of the non-normally distributed random variable X
i
at the design point
P
0
X
0
1
; X
0
2
; : : : X
0
n1
; X
0
n
. f
X
i
x
0
i
and F
X
i
x
0
i
are the PDF and the CDF of the non-
normally distributed random variable X
i
at the design point X
0
i
.
X
i
eq
and
X
i
eq
are the equiv-
alent mean and the equivalent standard deviation of the equivalent normally distributed random
variable at the design point x
0
i
.
Now every random variable in the limit state function in Equation (A.18) at the design
point P
0
are normally distributed random variables. e mean and standard deviation of these
normally distributed random variables are
X
i
D
8
<
:
X
ieq
i D 1; 2; : : : ; r
X
i
i D r C 1; : : : n
(A.23)
A.2. THE RACKWITZ AND FIESSLER (R-F) METHOD 219
X
i
D
8
<
:
X
ieq
i D 1; 2; : : : ; r
X
i
i
D
r
C
1; : : : n:
(A.24)
Step 5: Calculate the initial design point P
0
in the standard normal distribution space.
In the standard normal distribution space, the initial design point P
0
X
0
1
; : : : ; X
0
n
can be
expressed as P
0
Z
0
1
; Z
0
2
; : : : ; Z
0
n
in the standard normal distribution space through Equa-
tion (A.25):
Z
0
i
D
X
0
i
X
i
X
i
i D 1; : : : ; n: (A.25)
Step 6: Calculate the reliability index ˇ
0
at the design point P
0
Z
0
1
; Z
0
2
; : : : ; Z
0
n
.
Per Equation (A.26), we can calculate the Taylor series coefficients, and per Equation (A.27)
we can calculate the reliability index ˇ
0
:
G
i
j
P
0
D
X
i
@g
.
X
1
; X
2
; : : : ; X
n
/
@X
i
ˇ
ˇ
ˇ
ˇ
at P
0
.
X
0
1
;X
0
2
;:::;X
0
n
/
i D 1; 2; : : : ; n (A.26)
ˇ
0
D
P
n
iD1
Z
0
i
G
i
j
P
0
q
P
n
iD1
.
G
i
j
P
0
/
2
: (A.27)
Step 7: Determine the new design point P
1
Z
1
1
; Z
1
2
; : : : ; Z
1
n
for the iterative process.
e recurrence equations for the iterative process are the following equations:
Z
1
i
D
G
i
j
P
0
q
P
n
iD1
.
G
i
j
P
0
/
2
ˇ
0
i D 1; 2; : : : ; n 1: (A.28)
Since the new design point P
1
Z
1
1
; Z
1
2
; : : : ; Z
1
n
is on the surface of the limit state function
g
Z
1
1
; Z
1
2
; : : : ; Z
1
n
D 0, the Z
1
n
will be obtained from the surface of the limit state func-
tion. Since we typically still use the limit state function g
.
X
1
; X
2
; : : : ; X
n
/
D 0 to conduct the
calculation, we will use the following equations to get the Z
1
n
.
We can use the conversion Equation (A.25) to get the first n 1 values of the new design
point P
1
X
1
1
; X
1
2
; : : : X
1
n1
; X
1
n
, that is:
X
1
1i
D
X
i
C
X
i
Z
1
i
: (A.29)
Per the surface of the limit state function Equation (A.19), we express X
1
n
as the function of
X
1
1
; X
1
2
; : : : , and X
1
n1
, as shown in Equation (A.30):
X
1
n
D g
1
X
1
1
; X
1
2
; : : : ; X
1
n1
: (A.30)
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