3.6. THE HASOFER–LIND (H-L) METHOD 133
3.6 THE HASOFER–LIND (H-L) METHOD
When all variables are statistically independent, normally distributed random variables, the
Hasofer–Lind (H-L) method [1] provides a more accurate and unique result of the reliabil-
ity of a component with a nonlinear limit state function. e main difference between the H-L
method and the FOSM method is that the 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, instead of the mean-value point. 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 nonlinear limit state function, which consists of mutually inde-
pendent, normally distributed random variables:
g
.
X
1
; X
2
; : : : ; X
n
/
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure;
(3.35)
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: (3.36)
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 could be any point, but it must be on the surface of the limit state
function as specified by Equation (3.36). We can use the mean values for the first n 1 variables,
as shown in Equation (3.37) and then determine the last one through Equation (3.38a):
X
0
i
D
X
i
i D 1; 2; : : : ; n 1 (3.37)
g
X
0
1
; X
0
2
; : : : X
0
n1
; X
0
n
D 0 (3.38a)
ere is only one unknown X
0
n
in Equation (3.38a). We can solve this unknown X
0
n
from
Equation (3.38b). When the actual limit state function is provided, we can rearrange the second
equation in Equation (3.38a) and express X
0
n
by using X
0
1
; X
0
2
; : : :, and X
0
n1
. Let’s use the
following equation to represent this:
X
0
n
D g
1
X
0
1
; X
0
2
; : : : ; X
0
n1
: (3.38b)
Step 2: Set ˇ D 0.