Geology Reference
In-Depth Information
n
= =
(
)
A similar reasoning allows the determination
of the PDF for the arrival time
t
n
of the n
th
event,
and its mean value:
T
n
ν
D
P X
(
n
)
exp(
−
ν
T
)
(27)
T
D
!
D
Finally,
∞
∫
ν
n
t
n
−
1
n
f
( )
t
=
exp(
−
ν
t
)
→
t
=
t f
( )
t dt
=
t
n
t
(
n
1
)!
−
ν
n
n
0
(23)
T
(
)
n
ν
T
n
∞
∑
n
−
1
ν
i
D
=
∫
i
(
)
D
C
=
C DIES
(
)
ν
t
exp
− +
(
r
ν
)
t dt
exp(
−
ν
T
)
d
DIES
f
D
i
!
!
n
=
1
i
0
0
(28)
Under the assumption that the structure is fully
repaired after each event, the total expected cost
(at present values and conditional on the damage
index
DIES
) becomes:
In general, the costs
C
n
increase with
n
, but this
is compensated by the diminishing probability of
occurrence of an increasing number of
n
events in
T
D
. Thus, the outcome from Eq.(28) converges to a
finite number as
n
is increased. In the application
results in this Chapter, the summation in Eq.(28)
was truncated when the relative contribution of
the last term was less than 0.001.
The function relating the cost
C
f
to the damage
index
DIES
(which ranges between 0 and 1) was
assumed to be of the form
C
=
C
+
C
+ +
C
d
DIES
1
2
n
DIES
DIES
DIES
(24)
in which
T
D
∫
C
=
C DIES f
(
)
( )
t dt
1
f
0
t
DIES
1
0
T
D
∫
C
=
C DIES f
(
)
( )
t dt
b
DIES
2
f
0
t
(25)
DIES
C DIES
(
)
=
α
C
for
DIES
≤
0 60
.
2
f
0
0 60
.
0
C DIES
(
)
=
α
C
when
DIES
>
0 60
.
0
f
(29)
T
D
=
∫
C
C DIES f
(
)
( )
t dt
n
DIES
f
0
t
n
in which
C
0
is the total replacement cost and the
coefficient
α
= 1.20 is assumed to contemplate
extra costs for demolition and cleanup. Eq.(29)
further assumes that complete replacement would
be required should the damage index exceed a
value DIES
LIM
= 0.60. The exponent
b
needs to
be set by calibration to practical experience, and
in the applications in this Chapter has been as-
sumed to be
b
= 1.
Finally, the total expected repair cost must
consider the PDF for the damage index
DIES
,
0
Introducing Eq.(17) and (23) into Eq.(24) the
final expected cost
C
d
|
DIES
, conditional on a dam-
age level
DIES,
becomes
T
n
−
1
ν
i
D
=
(
)
∫
i
C
=
C DIES
(
)
ν
t
exp
− +
(
r
ν
)
t dt
d
DIES
f
i
!
i
0
0
(26)
Eq.(26) is still conditional on the number
n
of
earthquakes during the service life
T
D
.This number
is itself a random quantity and, assuming that it
obeys a Poisson distribution, the probability of
n
events in
T
D
is
∞
∫
0
C
(
x
=
)
C
⋅
f
(
DIES d DIES
)
⋅
(
)
d
d
d
DIES
DIES
(30)
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