3.7. THE RACKWITZ AND FIESSLER (R-F) METHOD 145
When the actual limit state function is provided, we can rearrange the above equation 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
: (3.71b)
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 MATLAB. 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
distributed random variables per Equations (3.61), (3.62), and (3.63).
For the first r random variables in the limit state function described in Equation (3.64),
we have:
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 (3.72)
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 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 (3.64) 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
(3.73)
X
i
D
8
<
:
X
ieq
i D 1; 2; : : : ; r
X
i
i D r C 1; : : : n:
(3.74)
Step 5: Calculate the initial design point P
0
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: (3.75)
146 3. COMPUTATIONAL METHODS FOR THE RELIABILITY OF A COMPONENT
Now, the limit state function in the standard normal distribution space can be expressed as:
g
.
Z
1
; Z
2
; : : : ; Z
n
/
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(3.76)
e 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 Equation (3.77):
Z
0
i
D
X
0
i
X
i
X
i
i D 1; : : : ; n: (3.77)
Step 6: Calculate the reliability index ˇ
0
at the design point P
0
Z
0
1
; Z
0
2
; : : : ; Z
0
n
.
is step is the same as Step 4 for the H-L method. We can use the FOSM method
to linearize the limit state function as shown in Equation (3.76) at the initial design point
P
0
Z
0
1
; Z
0
2
; : : : ; Z
0
n
. en, per Equation (3.78) we can calculate the Taylor series co-
efficients, and per Equation (3.79) 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
atP
0
.
X
0
1
;X
0
2
;:::;X
0
n
/
i D 1; 2; : : : ; n (3.78)
ˇ
0
D
P
n
iD1
Z
0
i
G
i
j
P
0
q
P
n
iD1
.
G
i
j
P
0
/
2
: (3.79)
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: (3.80)
Because the new design point P
1
Z
1
1
; Z
1
2
; : : : ; Z
1
n
is on the surface of the limit state func-
tion g
.
Z
1
; Z
2
; : : : ; Z
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 (3.75) 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
: (3.81)
3.7. THE RACKWITZ AND FIESSLER (R-F) METHOD 147
Per the surface of the limit state function Equation (3.65), we can rearrange the limit state
function in such a way that X
n
is expressed as the function of X
1
; X
2
; : : :, and X
n1
, that is,
X
n
D g
1
.
X
1
; X
2
; : : : ; X
n1
/
. e value X
1
n
is obtained per Equation (3.82):
X
1
n
D g
1
X
1
1
; X
1
2
; : : : ; X
1
n1
: (3.82)
After the X
1
n
is obtained from Equation (3.82), Z
1
n
can be calculated through the conversion
Equation (3.83):
Z
1
n
D
X
0
n
X
n
X
n
: (3.83)
Now we have the new design point P
1
X
1
1
; X
1
2
; : : : ; X
1
n
in 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 8: Check convergence condition.
e convergence equation for this iterative process will be the difference
ˇ
ˇ
ˇ
ˇ
0
ˇ
ˇ
ˇ
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:
ˇ
ˇ
ˇ
ˇ
0
ˇ
ˇ
ˇ
0:0001: (3.84)
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
/: (3.85)
If the convergence condition is not satisfied, we use this new design point
P
1
X
1
1
; X
1
2
; : : : ; X
1
n
to replace the previous design point P
0
X
0
1
; X
0
2
; : : : ; X
0
n
,
that is,
X
0
i
D X
1
i
i D 1; : : : ; n
ˇ D ˇ
0
:
(3.86)
en, we go to Step 4 for a new iterative process again until the convergence condition is satis-
fied.
e program flowchart for the R-F method is shown in Figure 3.8.
Example 3.13
e shear yield strength S
sy
(ksi) of a shaft follows a normal distribution with a mean
S
sy
D
31 (ksi) and a standard deviation
S
sy
D 2:4 (ksi). e diameter d (inch) of the solid shaft follows
a normal distribution with a mean 2.125
00
and the standard deviation 0.002
00
. e torque applied
on the shaft T (klb.in) follows a two-parameter Weibull distribution with the scale parameter
D 34 and the shape parameter ˇ D 3. Use the allowable torque to build the limit state function,
148 3. COMPUTATIONAL METHODS FOR THE RELIABILITY OF A COMPONENT
–
Calculate the new design point X
1
(i) and Z
1
(i), Equations (3.78) – (3.81)
Figure 3.8: e program flowchart for the R-F method.
3.7. THE RACKWITZ AND FIESSLER (R-F) METHOD 149
which is equal to the multiplication of the shear yield strength with the section modulus of the
shaft. en determine the reliability of this shaft.
Solution:
(1) Use the allowable torque to build the limit state function.
If we use the yield strength as the component strength index, the limit state function of
this example will be:
g
T; d; S
sy
D S
sy
T
16
d
3
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(a)
In this example, the allowable torque is the component strength index. erefore, we can have
the following limit state function:
g
T; d; S
sy
D S
sy
16
d
3
T D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(b)
Both Equations (a) and (b) are limit state functions of this example. When the R-F method
is used, the results will be the same by using any one of these two limit state functions. In this
example, we will use the limit state function described in Equation (b) to calculate the reliability.
We can establish the following equations for the preparation of compiling the MATLAB
program.
e mean for the Weibull distribution per Equation (3.68), we have:
X
D
1
ˇ
C 1
: (c)
For this limit state function, the explicit form of Equation (3.71b) for determining the value of
the last variable at the design point is
S
sy
D g
1
.
T; d
/
D
16T
d
3
: (d)
In this example, we have three random variables. Two random variables d and S
sy
are normal
distributions. e torque T is a Weibull distribution. Per Equation (3.72), the equivalent mean
T
eq
and the equivalent standard deviation
T
eq
of the torque T at the design point T
are:
z
T
D ˆ
1
F
T
T
D norminv
wblcdf .T
; ; ˇ/
T
eq
D
1
f
T
.T
/
z
T
D
1
wblpdf .T
; ; ˇ/
normpdf
z
T
(e)
T
eq
D T
z
T
T
eq
;
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