Geology Reference
In-Depth Information
The LCC Model
ν(IM)
in Eqn. (3) defines the hazard curve, where
IM
is PGA for this study. The conditional prob-
ability of demand being greater than the capacity
(or fragility) is
Once the hazard and performance levels are
defined, the next step is the calculation of the
damage state probabilities, i.e. the probability of
the structure attaining the pre-defined damage
states throughout its lifetime. Once the damage
state probabilities are calculated the cost of repair
for each damage state is evaluated and the LCC
of the structure is easily found. In the following
the full derivation of a LCC model is provided.
The expected LCC of a structure is calculated as
∞
∫
(
)
=
(
)
( )
P
∆
>
∆
|
IM im
=
P
∆
>
δ
|
IM im f
=
δ
d
δ
,
,
D
C i
D
C i
0
(4)
where δ is the variable of integration and
f
C,i
is the
probability density function for structural capacity
for the i
th
damage state. This formulation assumes
that the demand and the capacity are independent
of each other (as discussed in the previous section).
Structural capacity is assumed to be lognormally
distributed with Δ
C,i
and
β
C
that are, respectively,
the mean and the standard deviation of the cor-
responding normal distribution. As an example,
the lognormal probability density functions for
structural limit states are shown in Figure 6(b). In
Figure 6(b), three limit states IO, LS, and CP with
threshold values of 1%, 2.5% and 7% interstory
drift, respectively, are assumed and
β
C
is taken as
0.3. The uncertainty in capacity (here represented
with
β
C
) accounts for factors such as modelling
error and variation in material properties. A more
detailed investigation of capacity uncertainty is
available in Wen et al. (2004) and Kwon and
Elnashai (2006).
The structural demand is also assumed to fol-
low a lognormal distribution and the probability of
demand exceeding a certain value,
δ
, is given by
N
( )
=
∑
E C
t
C
+ ⋅
t
C P
(1)
LC
i
i
0
i
=
1
where
C
0
is the initial construction cost,
t
is the
service life of the structure,
N
is the total number
of limit-states considered,
P
i
is the total probability
that the structure will be in the i
th
damage state
throughout its lifetime, and
C
i
is the corresponding
cost as a fraction of the initial cost of the structure.
P
i
is given by
(
)
−
(
)
P
=
P
∆
>
∆
P
∆
>
∆
1
i
D
C i
,
D
C i
,
+
(2)
where Δ
D
is the earthquake demand and Δ
C,i
is the
structural capacity, usually in terms of drift ratio,
defining the i
th
damage state, as described in the
previous section. The probability of demand being
greater than capacity is evaluated as
( )
−
ln
δ
λ
(
)
= −
D IM im
|
=
P
∆
>
δ
|
IM im
=
1
Φ
(
)
∞
∫
IM im
dv IM
dIM
D
β
(
)
=
(
)
P
∆
>
∆
P
∆
>
∆
|
=
dIM
D
D
C i
,
D
C i
,
(5)
0
(3)
where Ф[·] is the standard normal cumulative
distribution, λ
D
is the natural logarithm of the
mean of the earthquake demand as a function of
the ground motion intensity, and
β
D
is the standard
deviation of the corresponding normal distribu-
tion of the earthquake demand. Although,
β
C
and
β
D
are dependent on ground motion intensity, in
where the first term inside the integral is the con-
ditional probability of demand being greater than
the capacity given the ground motion intensity,
IM
.
This term is also known as the fragility function.
The second term is the slope of the mean annual
rate of exceedance of the ground motion intensity.
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