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.
134 3. COMPUTATIONAL METHODS FOR THE RELIABILITY OF A COMPONENT
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.
In the standard normal distribution space, we convert a normal distribution X
i
into a
standard normal distribution Z
i
through the following conversion equation:
Z
i
D
X
i
X
i
X
i
i D 1; : : : ; n: (3.39)
Now, the surface of the limit state function in the standard normal distribution space can be
expressed as:
g
.
Z
1
; Z
2
; : : : ; Z
n
/
D 0: (3.40)
e initial design point P
0
X
0
1
; X
0
2
; : : : ; X
0
n
in the original normal distribution space can
be expressed by P
0
Z
0
1
; Z
0
2
; : : : ; Z
0
n
in the standard normal distribution space through
Equation (3.39). Z
0
i
.i D 1; : : : ; n/ can be calculated per Equation (3.41):
Z
0
i
D
X
0
i
X
i
X
i
i D 1; : : : ; n: (3.41)
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 design
point P
Z
1
; Z
2
; : : : ; Z
n
through the Taylor Series. e Taylor Series coefficient, in this case,
will be:
G
i
j
P
D
@g
.
Z
1
; Z
2
; : : : ; Z
n
/
@Z
i
ˇ
ˇ
ˇ
ˇ
atP
.
Z
1
;Z
2
;:::;Z
n
/
i D 1; 2; : : : ; n; (3.42)
where G
i
j
P
means the Taylor Series coefficient for the variable Z
i
at the design point
P
Z
1
; Z
2
; : : : ; Z
n
. According to the conversion Equation (3.39), we have:
@X
i
@Z
i
D
X
i
: (3.43)
Equation (3.42) can be rewritten as:
G
i
j
P
D
@g
.
Z
1
; Z
2
; : : : ; Z
n
/
@Z
i
ˇ
ˇ
ˇ
ˇ
atP
.
Z
1
;Z
2
;:::;Z
n
/
D
@g
.
Z
1
; Z
2
; : : : ; Z
n
/
@X
i
@X
i
@Z
i
ˇ
ˇ
ˇ
ˇ
atP
.
Z
1
;Z
2
;:::;Z
n
/
D
X
i
@g
.
X
1
; X
2
; : : : ; X
n
/
@X
i
ˇ
ˇ
ˇ
ˇ
atP
.
X
1
;X
2
;:::;X
n
/
: (3.44)
Typically, we like to use Equation (3.44) to calculate the Taylor Series coefficient.
3.6. THE HASOFER–LIND (H-L) METHOD 135
Now, use the FOSM method to the limit state function g
.
Z
1
; Z
2
; : : : ; Z
n
/
at the ini-
tial design point P
0
Z
0
1
; Z
0
2
; : : : ; Z
0
n
to calculate the Taylor series coefficient G
i
j
P
0
per
Equation (3.45) and then calculate the reliability index ˇ
0
per Equation (3.46):
G
i
j
P
0
D
X
i
@g
.
X
1
; X
2
; : : : ; X
n
/
@X
i
ˇ
ˇ
ˇ
ˇ
atP
0
.
X
0
1
;X
0
2
;:::;X
0
n
/
i D 1; 2; : : : ; n (3.45)
ˇ
0
D
P
n
iD1
Z
0
i
G
i
j
P
0
q
P
n
iD1
.
G
i
j
P
0
/
2
(3.46)
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: (3.47)
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
; Z
2
; : : : ; Z
n
/
D 0, the Z
1
n
will be obtained from the surface of the limit state function.
Since we typically still use the limit state function g
.
X
1
; X
2
; : : : ; X
n
/
D 0 to conduct the calcu-
lation, we will use the following equations to get the Z
1
n
.
We can use the conversion Equation (3.39) 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 (3.48):
X
1
i
D
X
i
C
X
i
Z
1
i
: (3.48)
e value X
1
n
is obtained per Equation (3.38b), that is,
X
1
n
D g
1
X
1
1
; X
1
2
; : : : ; X
1
n1
: (3.49)
When the X
1
n
is obtained per Equation (3.49), Z
1
n
can be calculated through the conversion
Equation (3.39):
Z
1
n
D
X
0
n
X
n
X
n
: (3.50)
Now we have the new design point P
1
X
1
1
; X
1
2
; : : : ; X
1
n
in the original normal distribution
space and the same design point P
1
Z
1
1
; Z
1
2
; : : : ; Z
1
n
in the standard normal distribution
space.
Step 6: Check convergence condition.
e convergence equation for this iterative process will be the difference
j
ˇ
j
between
the current 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: (3.51)
136 3. COMPUTATIONAL METHODS FOR THE RELIABILITY OF A COMPONENT
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
: (3.52)
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
,
that is,
X
0
i
D X
1
1
Z
0
i
D Z
0
i
i D 1; : : : ; n (3.53)
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 3.6.
We will use the H-L method to calculate the reliability of component in Example 3.10.
Example 3.11 (Redo Example 3.10 by the H-L method)
A simple support beam is under a concentrated loading at the middle of the beam, as
shown in Figure 3.7. e yield strength S
y
, the concentrated load P , the beam span L and the
section modulus Z of the beam are all normal distributed random variables. eir distribution
parameters are:
S
y
D 6 10
5
(kN/m
2
),
S
y
D 10
5
(kN/m
2
);
P
D 10 (kN/),
P
D 2 (kN);
L
D 8 (m),
L
D 2:083 10
2
(m); and
Z
D 10
4
(m
3
),
S
y
D 2 10
5
(m
3
). Use the H-L
method to calculate the reliability of the beam based on the following two different limit state
function.
(1) e yield strength is the component strength index. So, the limit state function is:
g
S
y
; Z; P; L;
D S
y
PL
4Z
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(1)
(2) e allowable bending moment is the component strength index. So, the limit state func-
tion is:
g
S
y
; Z; P; L
D S
y
Z
PL
4
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(2)
3.6. THE HASOFER–LIND (H-L) METHOD 137
Start
,
,
3
Figure 3.6: e program flowchart for the H-L method.
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