Civil Engineering Reference
In-Depth Information
where
2
[
()
−
]
log
x
θ
t
2
1
2
e
2
β
(
)
=
−
∫
Ft
θβ
,
d
x
[11.13]
x
βπ
This model combines both the exponential recovery model proposed by
Kafali and Grigoriu (2005) and the trigonometric recovery model proposed
by Chang and Shinozuka (2004). The parameter
L
0
in Equation (11.12) can
be used to defi ne the initial total loss of functionality after the drop (Fig.
11.7a). The parameter
can be used to defi ne the time frame (Fig. 11.7b)
when the societal response and recovery are driven by lack or limited orga-
nization and/or resources. The parameter
θ
β
defi nes the rapidity of the recov-
ery process (Fig. 11.7c).
Another recovery model presented in Cimellaro
et al.
(2010a) is the
harmonically over-damped recovery function
having three parameters
(
L
0
,
ω
,
ζ
). It is defi ned as
⎩
⎡
⎢
⎛
⎜
αβ
β
+
⎞
⎟
⎛
⎜
βα
β
−
⎞
⎟
⎤
⎥
⎭
()
=+−
[
(
)
]
Qt
Q
1
e
−
α
t
e
β
t
+
e
−
β
t
Q
−−−
QL
0
0
R
0
[11.14]
2
2
where
L
0
defi nes the initial total loss of functionality after the drop (Fig.
11.8a);
(
)
α
=
ωζ
;
βωζ
=
2
−
1
;
ω
and
ζ
are related to the rapidity dimension
(
ζ
≥
1;
ω
≥
1). Furthermore, rapidity of recovery increases when either
ω
increases or
ζ
reduces as shown in Figs 11.8b,c. For critically damped systems
L
0
and
Q
.
(0)
(
ζ
=
1), placing the same initial condition
Q
(0)
=
1
−
=
0, the
solution is given by
()
=−
(
)
Qt
1
L
0
e
ω
−
t
1
+
ω
t
[11.15]
The second group of recovery models is called
short-term recovery models
and instead of using CDF shape models such as in the long term recovery
models, they use the probability density functions (PDF) shape models. The
simplest recovery model after the linear model proposed in Equation (11.9)
is the Rayleigh PDF recovery model, defi ned as:
()
()
ftb
ftb
()
=−
Qt
1
L
[11.16]
0
(
)
max
where
2
2
⎛
⎜
−
t
b
⎞
⎟
t
b
()
=
ftb
e
2
[11.17]
2
The model is calibrated using two parameters:
L
0
is related to the robust-
ness dimension (Fig. 11.9a), while
b
is related to the rapidity and the delay
in the recovery process (Fig. 11.9b).
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