Civil Engineering Reference
In-Depth Information
0.1614
independently from the value of
T
N
. This implies that the rate of occurrence
of earthquakes
v
for the region is 0.1614.
Using the probabilistic deformation capacity model by Choe
et al.
(2007),
the capacity for deformation mode of the RC column in terms of ln(
using
OpenSHA
(Field
et al.
, 2003). The fi gure shows that
H
(0)
=
Δ
/
H
)
at time
t
is given by
C
t
=
C
d
(
x
t
,
Θ
C
), where
C
d
(
x
,
Θ
C
) is the probabilistic
deformation capacity model and
Θ
C
is a set of random model parameters.
Θ
C
and random modeling error in the probabilistic model. In this illustration,
the failure time analysis is conducted by conditioning on the value of
C
d
(
x
0
,
Θ
Note that
C
d
(
x
0
,
Θ
C
) is a random variable due to the uncertainty in
x
0
,
C
), estimates of the analysis
can be multiplied by the probability density function (PDF) of
C
d
(
x
0
,
C
). To account for the uncertainty in
C
d
(
x
0
,
Θ
Θ
C
)
and the product can be integrated over the domain of
C
d
(
x
0
,
Θ
C
). In this
case,
C
d
(
x
0
,
Θ
C
) is taken to be equal to the median capacity estimated fol-
lowing Choe
et al.
(2007). The median value of
C
d
(
x
0
,
Θ
C
) is found to be
−
0.05).
The probability of failure can be computed in the space of several defor-
mation variables such as
2.97 (i.e., drift capacity
≈
Δ
y
(ductility).
In addition, any monotonic transformation of these variables can also be
used. Having a positive value for
C
0
simplifi es the modeling process because
it allows setting
Δ
(displacement),
Δ
/
H
(drift), or
Δ
/
0
equal to 0 in Eq. (16.11) and still satisfying the assump-
tion made to write Eq. (16.12), i.e.,
C
0
α
−
α
0
> 0. Therefore, we choose
ln(100
Δ
/
H
) as the deformation variable which gives
C
0
=
C
d
(
x
0
,
Θ
C
)
+
ln(100)
=
1.64. Following the described probabilistic demand and capacity models
and using ln(100
Δ
/
H
) as the deformation variable, we have
C
t
=
C
d
(
x
t
,
Θ
C
)
+
ln(100) and
(
)
(
)
DD
=
⎣
x
,
SS
,
x
,
Q
⎦
+
ln
100
[16.26]
t
δ
−
at
−
D
n
t
n
t
n
n
16.4.3 Stiffness deterioration due to earthquakes
The deterioration in the lateral stiffness due to earthquakes results in the
change in the distribution of the deformation demand. This change can be
modeled as the stochastic process {
t
n
} (see Eq. (16.6)). In this illustration,
the shock deterioration in the deformation capacity {
Δ
α
s
Δ
C
t
n
}
=
0 and therefore,
t
n
/
u
0
. As seen
in Eqs. (16.19), (16.21) and (16.23), to perform the failure time analysis, we
need to compute the CDFs
F
Ŷ
(
y
),
F
Z
(
z
), and
F
Ŷ
|
Z
(
y
|
Z
t
n
). Now, assuming that
α
for the failure mode d, the total shock deterioration
Z
t
n
=
Δ
α
s
0
=
0 (as discussed earlier) we have
u
0
=
C
0
,
Z
t
n
=
Δ
α
t
n
/
C
0
,
Ŷ
t
n
=
D
d
[
x
0
,
S
a
(
S
t
n
,
x
0
),
Θ
D
]/
C
0
.
Now to compute
F
Z
(
z
), and
F
Ŷ
|
Z
(
y
|
Z
t
n
), we generate data representing the
relation between
Ŷ
t
n
and
Z
t
n
). For this purpose, we conduct nonlinear time-
history analyses of the bent for ground motions selected to generate
Θ
D
] and
Ŷ
t
n
=
D
d
[
x
0
,
S
a
(
S
t
n
,
x
0
),
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