213
A P P E N D I X A
Computational Methods for
the Reliability of a Component
A.1 THE HASOFER–LIND (H-L) METHOD
When all variables are statistically independent normally distributed random variables, the
Hasofer–Lind (H-L) method [1, 2] can be used to calculate the reliability of a component
with a nonlinear limit state function. e H-L method will linearize the non-limit state func-
tion at the design point. e design point is a point on the surface of the limit state function:
g
.
X
1
; X
2
; : : : ; X
n
/
D 0. Since the design point is generally not known in advance, the H-L
method is an iterative process to calculate the reliability of a component with a convergence
condition.
Consider the following general nonlinear limit state function, which consists of mutually
independently normally distributed random variables:
g
.
X
1
; X
2
; : : : ; X
n
/
D
8
ˆ
<
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure;
(A.1)
where X
i
.i D 1; 2; : : : ; n/ is a normal distributed random variable with corresponding a mean
X
i
and a standard deviation
X
i
. e following equation defines the surface of a limit state
function:
g
.
X
1
; X
2
; : : : ; X
n
/
D 0: (A.2)
e general procedure for the H-L method is explained and displayed here.
Step 1: Pick an initial design point P
0
X
0
1
; X
0
2
; : : : ; X
0
n
.
e initial design point must be on the surface of the limit state function as specified by Equa-
tion (A.2). We can use the mean values for the first n 1 variables, as shown in Equation (A.3):
X
0
i
D
X
i
i D 1; 2; : : : ; n 1: (A.3)
When the actual limit state function is provided, we express X
0
n
by using X
0
1
; X
0
2
; : : :, and
X
0
n1
. erefore, the X
0
n
can use the following equation to be calculated:
X
0
n
D g
1
X
0
1
; X
0
2
; : : : ; X
0
n1
: (A.4)
214 A. COMPUTATIONAL METHODS FOR THE RELIABILITY OF A COMPONENT
Step 2: Set ˇ D 0.
is setting is only for the MATLAB program. is setting will make sure that there are at least
two iterative loops for the iterative process.
Step 3: Calculate the initial design point in the standard normal distribution space.
We convert a normal distribution X
i
into a standard normal distribution Z
i
through the fol-
lowing conversion equation:
Z
i
D
X
i
X
i
X
i
i D 1; : : : ; n: (A.5)
e initial design point P
0
X
0
1
; X
0
2
; : : : ; X
0
n
in the original normal distributional space
can be expressed by P
0
Z
0
1
; Z
0
2
; : : : ; Z
0
n
in the standard normal distribution space through
Equation (A.5). Z
0
i
.i D 1; : : : ; n/ can be calculated per Equation (A.6).
Z
0
i
D
X
0
i
X
i
X
i
i D 1; : : : ; n: (A.6)
Step 4: Calculate the reliability index ˇ
0
at the design point P
0
Z
0
1
; Z
0
2
; : : : ; Z
0
n
.
In the H-L method, the limit state function g
.
Z
1
; Z
2
; : : : ; Z
n
/
is linearized at the initial design
point P
0
Z
0
1
; Z
0
2
; : : : ; Z
0
n
through the Taylor Series. e Taylor Series coefficient, in this
case, will be:
G
i
j
P
0
D
@g
.
Z
1
; Z
2
; : : : ; Z
n
/
@Z
i
ˇ
ˇ
ˇ
ˇ
at P
0
.
Z
0
1
;Z
0
2
;:::;Z
0
n
/
i D 1; 2; : : : ; n; (A.7)
where G
i
j
P
0
means the Taylor Series coefficient for the variable Z
i
at the design point
P
0
Z
0
1
; Z
0
2
; : : : ; Z
0
n
. According to the conversion Equation (A.5), we have:
@X
i
@Z
i
D
X
i
: (A.8)
Equation (A.7) can be rewritten as:
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
/
:
(A.9)
e reliability index ˇ
0
per Equations (A.6) and (A.9) will be:
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.10)
A.1. THE HASOFER–LIND (H-L) METHOD 215
Step 5: Determine the new design point P
1
Z
1
1
; Z
1
2
; : : : ; Z
1
n
.
e recurrence equation for the iterative process in the H-L method is the following equation.
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.11)
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.6) 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
per Equation (A.12):
X
1
i
D
X
i
C
X
i
Z
1
i
: (A.12)
e value X
1
n
is obtained per Equation (A.4), that is,
X
1
n
D g
1
X
1
1
; X
1
2
; : : : ; X
1
n1
: (A.13)
When the X
1
n
is obtained per Equation (A.13), Z
1
n
can be calculated through the conversion
Equation (A.14):
Z
1
n
D
X
0
n
X
n
X
n
: (A.14)
Now we have the new design point P
1
X
1
1
; X
1
2
; : : : ; X
1
n
in the original normal distribu-
tional space and the same design point P
1
Z
1
1
; Z
1
2
; : : : ; Z
1
n
in the standard normal distri-
butional space.
Step 6: Check convergence condition.
e convergence equation for this iterative process will be the difference
j
ˇ
j
between the cur-
rent reliability index and the previous reliability index. Since ˇ is a reliability index, the following
convergence condition will provide an accurate estimation of the reliability.
j
ˇ
j
0:0001: (A.15)
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.16)
If the convergence condition is not satisfied, we use this new design point
P
1
Z
1
1
; Z
1
2
; : : : ; Z
1
n
to replace the previous design point P
0
Z
0
1
; Z
0
2
; : : : ; Z
0
n
,
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