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,