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C
0
(
x
d
) and the damage repair cost
C
d
(
x
d
), assum-
ing that the structure will be completely repaired
after each earthquake that could occur during the
service life.
this damage occurs at a time
t
, the present value
C
f
0
(
DIES
) is given by
C DIES
0
(
)
=
C DIES
(
) exp(
−
r t
)
(17)
f
f
5.1 Initial Cost
in which
r
is the interest rate. The expected value
of
C
f
0
(
DIES
) can be calculated once the probability
density function (PDF) for the time
t
is known.
For this, a Poisson arrival process is assumed,
with mean arrival rate ν. Thus, the PDF for the
time
t
to the first earthquake event,
t
1
, is
For a reinforced concrete structure, the initial
cost is determined as a function of the concrete
volumes
V
and steel weights
W
in beams and
columns, and the unit costs of these items. Thus,
if
CUC
is the unit cost for concrete, and
CUS
is
the corresponding cost for steel,
f
t
1
( )
t
=
ν
exp(
−
ν
t
)
(18)
C
0
(
x
)
=
V
(
x
)
+
V
(
x
)
CUC W
+
(
x
)
+
W
(
xx
d
CUS
)
d
c beam
,
d
c col
,
d
s beam
,
d
s col
,
which allows the calculation of the mean for the
time
t
1
as
(16)
The concrete volumen and steel weight are
functions of the design parameters
x
d
. The unit
costs should include not only those for materials
but also for forming and labor. For the first appli-
cation example in this Chapter, unit costs of 1500
$/
m
3
and 500 $/
KN
have been used respectively,
for concrete and steel.
∞
∫
1
t
t f
( )
t dt
=
=
(19)
1
t
ν
1
0
The probability
P
(
t
2
>
t
) that the time to the
second event,
t
2
, be greater than a time
t
must
consider two posibilities: either that there are no
events before
t
or that there is only one. Thus,
5.2 Repair Costs
P t
(
2
> =
t
)
exp(
−
ν
t
)
+
ν
t
exp(
−
ν
t
)
The repair costs, converted to present values,
depend on the level of damage inflicted during
the earthquakes, the uncertainty as to when these
will occur and the number of earthquakes occur-
ring during the life time
T
D
, and the interest rate
to be applied to repair funds accumulated from
the time of construction.
The nonlinear dynamic analysis should include
a parallel calculation of the damage accumulated
during the shaking. The analysis software used for
the first application in this Chapter (Möller, 2001;
Möller and Foschi, 2003) quantifies the global
structural damage by means of an index
DIES.
If
C
f
(
DIES
) is the function used to estimate the cost
of repairing a damage
DIES
, and assuming that
(20)
from which the cumulative distribution
F
t
2
(
t
) for
t
2
,
and its PDF
f
t
2
(
t
) become
2
P t
(
≤ = − +
t
)
1
(
1
ν
t
) exp(
−
ν
t
)
=
F t
( )
→
f
( )
t
=
ν
t
exp(
−
ν
t
)
2
t
t
2
2
(21)
The expected time to the second event is
∞
∫
2
t
t f
( )
t dt
=
=
(22)
2
t
ν
2
0
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