Civil Engineering Reference
In-Depth Information
Z
¼
g
ð
X
1
;
X
2
;
...
;
X
n
Þ
ð
5
:
1
Þ
Failure occurs when Z \ 0. Therefore, the probability of failure, p
f
, is given by the
integral,
p
f
¼
Z
...
Z
f
X
x
1
;
x
2
;
...
;
x
n
ð
Þ
dx
1
dx
2
...dx
n
ð
5
:
2
Þ
g
ðÞ
\0
in which f
X
x
1
;
x
2
;
...
;
x
ð Þ
is the joint density function for the basic random vari-
ables X
1
, X
2
, …, X
n
. If the random variables are statically independent, then the
joint probability density function may be replaced by the product of the individual
probability density functions in the integral. Evaluation of the multiple integral is
extremely complicated. One approach is to use analytical approximations of this
integral that are simpler to compute, such as FORM.
A Taylor series expansion of the performance function about the mean values
gives,
Z
¼
g l
ðÞþ
X
X
X
n
n
n
2
g
oX
i
oX
j
þ
1
2
X
j
l
X
j
og
oX
i
o
X
i
l
X
i
X
i
l
X
i
i
¼
1
i
¼
1
j
¼
1
þ
ð
5
:
3
Þ
where the derivatives are evaluated at the mean values of the random variables (X
1
,
X
2
,…,X
n
) and l
Xi
is the mean value of X
i
. Truncating the series at the linear terms,
we obtain the first-order approximate mean and variance of Z as
r
Z
X
n
X
n
;
2
g
oX
i
oX
j
Cov X
i
;
X
j
o
l
Z
g l
X
1
;
l
X
2
;
...
;
l
X
n
ð
5
:
4
Þ
i
¼
1
j
¼
1
where Cov(X
i
, X
j
) is the covariance of X
i
, X
j
. If the variables are uncorrelated, then
the variance is
2
Var X
ðÞ
r
Z
X
n
og
oX
i
5
:
5
Þ
i
¼
1
The probability of failure depends on the ratio of the mean value of Z to its
standard deviation. This ratio is defined as the safety index b:
b
¼
l
Z
p
f
¼
1
U
ðÞ
r
Z
;
where U is the CDF of the standard normal variant.
Hasofer and Lind [
4
] proposed the advanced first-order second moment
(AFOSM) method. This method is applicable for normal variables. It first defines
the reduced variables as,
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