Civil Engineering Reference
In-Depth Information
Using the expressions for
R
C
(
t
) and
R
α
(
t
), we obtain
Rt
()
=
(
)
−
2
(
)
−
4
−
4
3 516
.
×
10
tT
−
3 443
.
×
10
tT
−
[16.35]
corr
corr
[
]
(
)
−
4
+
0 61
.
ln
1
+
12 5
.
×
10
tT
corr
−
16.4.5 Modeling low-cycle fatigue damage due
to earthquakes
As mentioned earlier, the damage index
DI
accumulates with every passing
damaging earthquake and, a longitudinal reinforcement bar is expected
to fail due to low-cycle fatigue when
DI
> 1.0. Therefore, writing incre-
mental damage due to
S
t
n
as
Δ
DI
t
n
, we need to compute the distribution of
Δ
DI
t
n
to compute
n
F
and
t
F
. In this case, the computation of distribution
of
n
F
simplifi es to
Pn
n
⎛
⎜
⎞
⎟
=
∑
Δ
(
)
=
>
n
P
DI
t
<
1.
and for that of
t
F
to
F
n
i
1
()
∑
Δ
1
Nt
⎛
⎜
⎞
⎟
(
)
=
Pt
>
t
P
DI
t
<
1.
. In order to compute the CDF
F
ΔDI
(
d
) for
F
i
i
=
Δ
DI
t
n
, we fi rst develop a function
Δ
DI
t
n
=
Δ
DI
(
Ŷ
t
n
) and then compute
F
Δ
DI
(
d
)
DI
(
Ŷ
t
n
), we perform a regres-
sion analysis using the results of time-history analyses conducted earlier.
First, we compute
=
P
(
Δ
DI
t
n
<
d
)
=
P
[
Δ
DI
(
Ŷ
t
n
) <
d
]. To develop
Δ
DI
t
n
for each ground motion by analyzing the strain
cycles induced in the reinforcement bars by the ground motion and follow-
ing the methodology in Brown and Kunnath (2004) to compute low-cycle
fatigue. Figure 16.9 shows the data generated for
Δ
Δ
DI
t
n
and
Ŷ
t
n
(
Ŷ
t
n
for each
0.016
0.012
0.008
0.004
0
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Y
tn
16.9
Plot showing data regarding
Δ
DI
t
n
and
Ŷ
t
n
.
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