Geology Reference
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collapse, when
DI
PA
≥
1 .
. These criteria are
based on calibration of DIPA against experimen-
tal results and field observations in earthquakes
(Park et al., 1987). Note that Eq (5) reveals that
both maximum ductility and hysteretic energy
dissipation contribute to the structural damage
during earthquakes. Eq. (5) expresses damage as
a linear combination of the damage caused by
excessive deformation and that contributed by
repeated cyclic loading effect. Note also that the
quantities
x
tion which are selected to span satisfactory the
frequency range of
x t
g
( )
. In constructing critical
seismic inputs, the envelope function is taken to
be known. The information on energy E, peak
ground acceleration (PGA) M1, peak ground
velocity (PGV) M2, peak ground displacement
(PGD) M3, upper bound Fourier amplitude spec-
tra (UBFAS)
M
4
(
ω
,
and lower bound Fourier
amplitude spectra (LBFAS)
M
5
(
ω
are also taken
to be available which enables defining the fol-
lowing nonlinear constraints (Abbas & Manohar,
2002, Abbas, 2006):
max
,
depend on the loading history
while the quantities
β
, ,
E
H
x
u y
are independent of
the loading history and are determined from ex-
perimental tests. It should also be emphasized
that Eqs (2-5) can be used to estimate damage for
a member in a structure which defines the local
damage. To estimate the global damage of the
structure, a weighted sum of the local damage
indices need to be estimated (Park et al, 1987).
In this chapter, Eq (5) is adopted in quantifying
the structural damage. The next section develops
the mathematical modeling of critical future
earthquake loads.
1
2
∞
∫
x t dt
2
( )
E
≤
g
0
max |
x t
( ) |
≤
M
g
1
0
< <∞
t
max |
x t
( ) |
≤
M
(7)
g
2
0
< <∞
t
max |
x t
( ) |
≤
M
g
3
0
< <∞
t
M
( )
ω
≤
|
X
( ) |
ω
≤
M
( )
ω
5
g
4
Here,
X
g
(
ω
is the Fourier transform of
x t
g
( )
. Note that the constraint on the earthquake en-
ergy is related to the Arias intensity (Arias 1970).
The spectra constraints aim to replicate the fre-
quency content and amplitude observed in past
recorded accelerograms on the future earthquake.
The ground velocity and displacement are obtained
from Eq. (6) and seen in Box 2:
Making use of the conditions
x
g
( 0
3. DERIVATION OF WORST
FUTURE EARTHQUAKE LOADS
The worst future ground acceleration is repre-
sented as a product of a Fourier series and an
envelope function in Box 1.
Here,
A
0
is a scaling constant and the param-
eters
α α
1
=
and
,
impart the transient trend to
x t
g
( )
.
R
i
and
ϕ
i
are
2
N
f
unknown amplitudes and phase
angles, respectively and
ω
i
0
(Shinozuka & Henry, 1965), the
constants in the above equation can be shown to
be given as (Abbas & Manohar, 2002, Abbas,
2006):
lim ( )
t
g
x t
→∞
=
1 2
are
the frequencies presented in the ground accelera-
,
i
,
, ...,
N
f
Box 1.
N
N
f
f
∑
∑
1
x t
(6)
( )
=
e t
( )
R
cos(
ω
t
−
ϕ
)
=
A
[exp(
−
α
t
)
−
exp(
−
α
t
)]
R
c
os(
ω
t
−
ϕ
)
g
i
i
i
0
1
2
i
i
i
i
=
1
i
=
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