Civil Engineering Reference
In-Depth Information
()
∏
Nt
⎡
⎣
⎤
⎥
(
)
(
)
=
[
()
]
=
Pt
>
t
Pn
>
Nt
W
<
w
E
1
F
1
−
W Z
[16.23]
∩
⎢
{
}
F
F
t
a
Ww
≤
ˆ
t
t
t
a
YZ
i
i
i
=
1
where 1
{
W
t
≤
w
a
}
=
1 if
W
t
≤
w
a
and 0 otherwise. If
N
(
t
) is a Poisson process,
then
n
⎡
⎢
∞
()
⎤
⎥
ν
t
e
!
−
ν
t
n
∑
∏
(
)
=
(
)
Pt
>
t
E
1
F
1
−
WZ
[16.24]
{
}
ˆ
F
Ww
≤
t
t
t
a
YZ
i
i
n
n
=
0
i
=
1
If there is no deterioration then
t
F
=
t
n
F
and
∞
∑
(
)
n
(
)
=
[
(
()
=
)
]
=
[
()
=
]
Pt
>
t
EPt
>
tNt
n
1
−
P
PNt
n
[16.25]
F
n
f
F
n
=
0
These solutions for the cases with deterioration are referred to as semi-
analytical, because the values of
E
[
] in Eqs. (16.17) to (16.24) have to be
computed by simulating the processes {
Ŷ
t
i
}, {
Z
t
i
} and
R
(
t
). The computation
of expectations in Eqs. (16.19), (16.21) and (16.23) is computationally more
effi cient than computing the expectation of a Bernoulli random variable
(takes only values 0 and 1; Ang and Tang, 2007) which is normally practiced
in a basic 'hit and miss' method to compute probabilities. This is because in
the above-mentioned equations, the expectations are computed for con-
tinuous random variables that converge faster than the expectation of
a Bernoulli random variable. The estimates of Eqs. (16.19), (16.21) and
(16.23) are found to converge fast even for small failure probabilities of the
order 10
−6
.
As the solution requires simulations, the computation of the expectations
may have simulation errors which must be reported. For assessing simula-
tion errors, variance of the expectation estimated through simulations can
be computed. This variance can be used to compute upper and lower bounds
on the estimate. If
X
¯
is the sample mean of
X
in the simulation, then Var(
X
¯
)
=
⋅
V
¯
(
X
)/
m
, where
V
¯
(
X
) is the sample variance of
X
and
m
is the number of
simulations. Typically,
X
()
±
2Var
X
can be used as upper and lower
bounds for estimate of
X
¯
.
16.4 Stochastic modeling of deterioration in
reinforced concrete (RC) bridges
As an illustration of the method described so far, this section models the
effects of deterioration on RC bridge columns caused by the occurrence of
earthquakes and corrosion of the reinforcing steel. For the purpose of this
illustration, we analyze an RC highway bridge that is assumed to be located
in San Francisco, CA, having the typical properties of a highway bridge
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