128 3. COMPUTATIONAL METHODS FOR THE RELIABILITY OF A COMPONENT
B
D 36:3 (ksi) and a standard deviation
B
D 4:21 (ksi).
A
is a normally distributed ran-
dom variable with a mean
A
D 6:81 (ksi) and a standard deviation
A
D 2:65 (ksi). Calculate
the reliability of this component.
Solution:
In this example, the component strength index will be the ultimate material strength S
u
. e
component stress index will be the sum of normal stress
B
by bending moment and the normal
stress
A
by axial loading. erefore, the limit state function of this component will be:
g
.
S
u
;
B
;
A
/
D S
u
.
B
C
A
/
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(a)
Since all three random variables S
u
,
B
, and
A
are normally distributed random variables, and
the limit state function is a linear function of them, we can use Equations (3.25) and (3.26) to
calculate the mean and standard deviation of the limit state function. ey are:
g
D
u
B
A
D 61:5 36:3 6:81 D 18:39 .ksi/ (b)
g
D
q
2
u
C
2
B
C
2
A
D
p
5:91
2
C 4:21
2
C 2:65
2
D 7:7248: (c)
Per Equation (3.27), we can calculate the reliability index ˇ:
ˇ D
g
g
D
18:39
7:7248
D 2:3806: (d)
Per Equation (3.28), we can calculate the reliability of the component. We can use Excel to
calculate the reliability of this component
R D ˆ
.
2:3806
/
D NORM:DIST
.
2:3806; 0; 1; true
/
D 0:9914:
3.5.2 THE FOSM METHOD FOR A NONLINEAR STATE FUNCTION
When a limit state function is a nonlinear function of all normally distributed random variable,
we can obtain an approximate value of the reliability by using the FOSM method. In the FOSM
method, the nonlinear limit state function will be simplified by the first-order Taylor series
expansion at the mean point where all random variables have their means. e first-order Taylor