3.5. THE FIRST-ORDER SECOND-MOMENT (FOSM) METHOD 127
• e FOSM method for a nonlinear limit state function. When a limit state function is a
nonlinear function of all normally distributed random variables, the FOSM method will
only provide an approximate result of the reliability of a component.
3.5.1 THE FOSM METHOD FOR A LINEAR LIMIT STATE FUNCTION
A limit state function of a component, which is a linear function of the statistically independent
normally distributed random variables, can be express as:
g
.
X
1
; X
2
; : : : ; X
n
/
D a
0
C
n
X
iD1
.
a
i
X
i
/
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure;
(3.24)
where a
i
.i D 0; 1; : : : ; n/ is a constant. X
i
.i D 1; 2; : : : ; n/ is a normally distributed random
variable with a mean
X
i
and a standard deviation
X
i
.
Since all random variables are statistically independent normally distributed random vari-
ables, the limit state function g
.
X
1
; X
2
; : : : ; X
n
/
will be a normally distributed random variable.
e mean
g
and the standard deviation
g
of the limit state function will be:
g
D a
0
C
n
X
iD1
a
i
X
i
(3.25)
g
D
v
u
u
t
n
X
iD1
a
i
X
i
2
: (3.26)
e reliability index ˇ and the reliability R of the component will be:
ˇ D
g
g
D
a
0
C
P
n
iD1
a
i
X
i
q
P
n
iD1
a
i
X
i
2
(3.27)
R D ˆ.ˇ/ D ˆ
g
g
D ˆ
0
B
@
a
0
C
P
n
iD1
a
i
X
i
q
P
n
iD1
a
i
X
i
2
1
C
A
: (3.28)
Example 3.9
e component fails when its maximum normal stress is more than the ultimate material
strength. e ultimate material strength S
u
of this component is a normally distributed random
variable with a mean
S
u
D 61:5 (ksi) and a standard deviation
S
u
D 5:91 (ksi). e maximum
normal stress at the critical section is the sum of normal stress
B
by bending moment and the
normal stress
A
by axial loading.
B
is a normally distributed random variable with a mean
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