126 3. COMPUTATIONAL METHODS FOR THE RELIABILITY OF A COMPONENT
From Figure 3.4, in the triangle OPB, we have:
sin
.
a
/
D
ˇ
S
Q
Q
D
ˇ
Q
S
Q
: (g)
From Figure 3.4, in the triangle OPA, we have:
cos
.
a
/
D
ˇ
S
Q
S
D
ˇ
S
S
Q
: (h)
Square both sides of the Equations (g) and (h) and then add them together, we have
sin
2
.
a
/
C cos
2
.
a
/
D
ˇ
Q
S
Q
2
C
ˇ
S
S
Q
2
: (i)
Rewrite Equation (i), we can obtain:
ˇ D
S
Q
q
2
S
C
2
Q
: (j)
is result verifies that the shortest distance between the origin of the standard normally dis-
tributed space and the surface of the limit state function is equal to .
S
Q
/=
q
2
S
C
2
Q
which
is the value of the reliability index ˇ.
3.5 THE FIRST-ORDER SECOND-MOMENT (FOSM)
METHOD
For a limit state function containing more than two random variables, if all random variables are
normally distributed random variables, we can use the First-Order Second-Moment (FOSM)
method to calculate its reliability.
e First-Order Second-Moment (FOSM) method is a probabilistic method to determine
the reliability of a component when its limit state function is simplified by a first-order Taylor
series. en, the first and second moments (mean and standard deviation) of all random variables
are used for the calculation of the reliability.
When the type of distribution of a random variable is unknown due to the lack of data,
we can use its mean and standard deviation (the first and second moments) for estimating its
reliability by using the FOSM method. is result will be only a rough approximate result of
the reliability. In this section, we will only discuss the following two cases.
• e FOSM method for a linear limit state function: when a limit state function is a linear
function of all normally distributed random variables, the FOSM method will provide an
accurate result of the reliability of a component.
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