Civil Engineering Reference
In-Depth Information
The simplest recovery model is the
uniform cumulative distribution
recovery function (also known as the
linear model
). This model is usually
adopted when there is no information regarding the preparedness, resources
available, societal response, etc.
[
]
()
=+
(
)
⋅
(
)
Qt
Q
F tt t
,
+
T
Q
−−
Q
L
[11.9]
0
0
0
RE
R
0
0
where
Q
0
is the initial functionality after the drop;
L
0
is the initial total loss
of functionality after the drop;
Q
R
is the residual functionality after the
recovery process ends (see Fig. 11.1); and
F
(
t
/
t
0
,
t
0
T
RE
) is the uniform
cumulative distribution function (CDF) which is given by
+
(
)
tt
T
−
(
)
=
0
(
)
Ftt t
,
+
T
It t
,
+
T
[11.10]
00
RE
00
RE
RE
where
I(t
0
, t
0
+
T
RE
)
is the interval step function. The model is characterized
by only one parameter (Fig. 11.6) which defi nes the slope of the curve and
represents
rapidity
(Cimellaro
et al.
, 2010a). The model can also be general-
ized by dividing the recovery process in several time intervals using a
mul-
tilinear model
that is given by:
(
)
tt
−
∑
()
=+
(
)
i
(
)
Qt
Q
H t t
−
QQ
−
[11.11]
i
i
i
+
1
i
(
)
t
−
t
i
i
+
1
i
where
Q
i
is the residual functionality at the step
i
and
Q
i
+1
is the residual
functionality at the step
i
1;
H( )
is the Heaviside step function.
Alternatively,
lognormal cumulative distribution
(CDF) recovery func-
tion, can be adopted, having three parameters (
L
0
,
+
θ
,
β
), and it is given by
()
=+
(
)
⋅
[
(
)
]
Qt
Q
F t
θβ
,
Q
−−
Q
L
[11.12]
0
R
0
0
120
100
80
60
40
80% losses
60% losses
40% losses
20% losses
20
0
0
10
20
t
30
40
50
[months]
11.6
Uniform cumulative distribution recovery function.
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