Civil Engineering Reference
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process is simulated a large number of times and the probability of failure
is computed as the number of failures divided by the number of simulations.
However, such methods are not suitable for computing low failure proba-
bilities as is the case for well-designed engineering systems. Therefore, with
reference to Kumar
et al.
(2012), the next section presents the SSA formula-
tion that is computationally effi cient and provides accurate results even for
small failure probabilities.
16.3 Modeling of a general deterioration process using
the stochastic semi-analytical approach (SSA)
The effects of deterioration on demand and capacity can be modeled by
expressing the respective changes caused by deterioration using an additive
form as adopted in Kumar
et al.
(2012). Alternatively, it is also possible to
develop a multiplicative form by multiplying a factor to the demand or the
capacity. However, it is noted that one form can be obtained from the other
by using an exponential or logarithmic transformation depending on the
starting model form. While the specifi c process of damage accumulation is
modeled as a stationary process, the formulation presented here is more
general and capable of modeling a non-stationary {
D
t
n
} process. In addition,
it accounts for the correlation between
W
i
and {
D
t
n
}.
16.3.1 Modeling the effect of deterioration on demands
In general, owing to deterioration, {
D
t
n
} may not be a stationary process;
i.e.,
D
t
1
,
D
t
2
, . . . are not statistically independent and identically distributed
(SIID) random variables. The non-stationary nature of {
D
t
n
} can be modeled
as follows:
DY
t
=+
−
α
[16.6]
t
n
n
t
n
where {
Y
t
n
}
∈
(
−∞
,
∞
) is a stationary process independent of the deteriora-
tion process and
(
x
t
) is a stochastic process that captures the effect of
deterioration on {
D
t
n
}. The value of
α
t
=
α
α
0
is arbitrary and in most cases may
be assumed to be zero since at
t
=
0 deterioration has not yet occurred.
Considering the case where
x
t
n
=
x
0
(i.e., no deterioration), we have
D
(
x
0
,
S
t
n
)
α
0
. Since {
Y
t
n
} is a stationary process that
is independent of the deterioration process, we always have
Y
t
n
=
Y
t
n
+
α
0
or
Y
t
n
=
D
(
x
0
,
S
t
n
)
−
=
D
(
x
0
,
S
t
n
)
−
α
0
. Therefore, the cumulative distribution function (CDF)
F
Y
(
y
) for
Y
t
n
is
given by
P
[
D
(
x
0
,
S
t
n
)
0
<
y
]. Adopting an additive form, the total effect
of deterioration is written as follows:
−
α
−
=+ +
t
n
∫
()
αα
Δ
α
s
Δ
α
g
tt
d
[16.7]
−
−
t
t
t
n
−
1
n
n
1
+
t
n
1
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