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/.