Civil Engineering Reference
In-Depth Information
[
]
{}
{}
{
()
}
Pn
>
n Z
,
t
,
Rt
[16.16]
F
t
i
i
i
≥
1
i
≥
1
i
≥
1
i
n
=
∏
ˆ
{}
{
}
{
( )
}
⎣
⎦
=
PY
≤ −
1
W
Z
,
t
,
Rt
t
t
t
i
i
i
≥
1
i
≥
1
i
≥
1
i
i
i
i
1
Now, taking the expectation of the expression in Eq. (16.16) over the dis-
tributions of {
Z
t
i
}, {
t
i
} and {
R
(
t
i
)}, we obtain
n
⎡
⎣
⎤
⎥
ˆ
=
∏
(
)
=
{}
{}
{
()
}
Pn
>
n
E
PY
⎣
≤ −
1
W
Z
,
t
,
Rt
⎦
[16.17]
⎢
F
t
t
t
i
i
i
i
i
i
≥
1
i
≥
1
i
≥
1
i
1
where
E
[
] is the expectation. Since
Ŷ
t
i
does not depend on {
t
i
} and {
R
(
t
i
)},
and depends only on
Z
t
i
, we have
⋅
ˆ
{}
{}
{
()
}
(
)
PY
⎣
≤−
1
W
Z
,
t
,
Rt
⎦
=
F
1
−
W Z
[16.18]
ˆ
t
t
t
i
i
t
t
i
≥
1
i
≥
1
i
≥
1
YZ
i
i
i
i
i
From Eqs. (16.17) and (16.18) we have,
n
⎡
⎢
⎤
⎥
=
∏
(
)
=
(
)
Pn
>
n
E
F
1
−
W Z
[16.19]
ˆ
F
t
t
YZ
i
i
i
1
It must be noted that in Eq. (16.19),
W
t
i
is a function of {
t
i
} and {
R
(
t
i
)} and
the expectation is obtained over the distribution of {
Z
t
i
}
i
≥1
, {
t
t
}
i
≥1
and {
R
(
t
i
)}
i
1
. The expression in Eq. (16.19) is general and as a special case, if
N
(
t
) is
a Poisson process with rate
v
, then
t
i
has a Gamma distribution (Ang and
Tang, 2007) with parameters
v
and
i
where the mean is
i
/
v
. In the case of
no deterioration,
P
(
n
F
>
n
)
≥
=
(1
−
P
f
)
n
, where
P
f
is the probability of failure
given the occurrence of a load.
•
Probability distribution of t
n
F
.
Considering failures due to excessive
demand only, the distribution for time to failure
t
n
F
is given by:
)
∏
(
Nt
⎡
⎢
⎤
⎥
ˆ
(
)
=
⎣
{}
{}
{
()
}
⎦
Pt
>
t
E
PY
≤ −
1
W
Z
,
t
,
Rt
[16.20]
n
t
t
t
i
i
F
i
i
i
i
≥
1
i
≥
1
i
≥
1
i
=
1
Therefore,
()
∏
Nt
⎡
⎢
⎤
⎥
(
)
=
(
)
Pt
>
t
E
F
1
−
W Z
[16.21]
n
ˆ
t
t
F
YZ
i
i
i
=
1
In particular, if
N
(
t
) is a Poisson process with rate
v
, then
()
n
⎡
⎢
∞
−
ν
t
n
⎤
⎥
ν
t
e
!
(
)
=
∑∏
(
)
Pt
>
t
E
F
1
−
WZ
[16.22]
n
ˆ
t
t
F
YZ
i
i
n
i
=
1
n
=
0
•
Probability distribution of t
F
.
Now considering the failures due to both
excessive demand and excessive deterioration, the distribution for time
to failure
t
F
is estimated as follows:
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