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In-Depth Information
E f M C M M S f M
a
( ,
)
=
(
)
( ,
)
(17)
methodology with near-fault ground motion
models (Mavroeidis and Papageorgiou,2003).
0
=
+
is the
seismic moment, and the factor
S
a
is the displace-
ment source spectrum given by (Atkinson and
Silva, 2000)
1 5
.
M
10 7
.
where
C
is a constant,
M M
(
)
10
0
Envelope Function
The envelope function for the ground accelera-
tion is represented by (Saragoni and Hart, 1974;
Boore, 2003)
1
−
ε
ε
(18)
S f M
( ,
)
=
+
a
2
2
if
if
if
if
a
t
t
n
2
1
+
1
+
t
t
−
a
3
e t M r
( ,
, )
=
a
e
(14)
1
a
b
n
where the corner frequencies
if
a
and
if
b
,
and the
weighting parameter
ε
are defined, respectively,
as
where
=
a
2
−
+
0 2
.
ln
ln
( .
0 05
)
a
e
a
=
)
,
a
=
2
,
a
2
3
1
log( )
if
a
=
2 181
.
−
0 496
.
M
,
1
0 2
. (
( . )
0 2
−
1
0 2
.
0 2
.
(15)
log( )
if
b
=
2 41
.
−
0 408
.
M
,
and
log( )
ε
=
0 605
.
−
0 255
M
.
.
The envelope function has a peak equal to
unity when
t
=
0 2
.
t
n
,
and
e t M r
( ,
, )
=
0 05
.
The constant
C
is given by
C UR VF
when
t
=
2 0. ,
where
T
gm
is the
duration of ground motion, expressed as a sum
of a path dependent and source dependent com-
ponent
T
t
=
, with
t
T
3
πρ β
where
U
is a unit
dependent factor,
R
Φ
is the radiation pattern,
V
represents the partition of total shear-wave en-
ergy into horizontal components,
F
is the effect
of the free surface amplification,
ρ
s
and
β
s
are
the density and shear-wave velocity in the vicin-
ity of the source, and
R
0
is a reference distance.
Next, the path effect
P f r
=
/
4
R
,
n
gm
Φ
s
s
0
2 2
. . / ,
where
r
is the epicentral distance, and the parameters
h
and
if
a
(corner frequency) are moment dependent
g i v e n b y
log( )
=
0 05
r
+ +
h
0 5
if
gm
a
h
=
0 15
.
M
−
0 05
a n d
.
( , )
which is another
component of the process that affects the spectrum
of motion at a particular site is represented by
functions that account for geometrical spreading
and attenuation
log( )
if
a
=
2 181
.
−
0 496
(Atkinson and Silva,
.
M
2000).
Ground Motion Spectrum
The total spectrum of the motion at a site
S f M r
−
π
fR r Q f
( )/ ( )
β
P f r
( , )
=
Z R r
( ( ))
e
(19)
s
( , , )
is expressed as the product of the
contribution from the earthquake source
E f M
( ,
),
path
P f r
( , ),
site
G
( )
and type of motion
I
( ),
where
R
( )
is the radial distance from the hypo-
center to the site given by
R r
i.e.
= +
2 2
The
attenuation quantity
Q
( )
is taken as
Q f
( )
r
h
.
S f M r
( ,
, )
=
E f M P f r G f
( ,
)
( , )
( ) ( )
I if
(16)
=
180
0 45
and the geometrical spreading
function is selected as
Z R r
.
( )
if
( ( ))
=
1
/ ( )
R r
if
The source component is given by
R
( )
<
70 0
km and
Z R r
.
( ( ))
=
1 70 0
other-
/
.
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