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4
4
3
3
2
2
1
1
0
0
-1
-1
-3
-2
-1
0
1
2
3
3 2 10123
ln (IM2=
S
a
V)
ln (IM1=
S
a
H)
(a)
(b)
20.11
Distribution of Type 2-VF with respect to (a) horizontal spectral
accelerations and (b) vertical spectral accelerations.
⎧
S
S
⎫
a
()
0
→≤
if
V
ln
c
⎪
⎪
⎪
⎪
1
a
H
(
)
=
ln
Type
1
−
VF
0
→≤
⎛
⎜
if S
S
S
0 1
.
⎨
⎬
a
H
⎪
⎪
⎪
⎪
⎪
⎡
⎢
⎞
⎟
−
⎤
a
⎥
×+×
(
[
)
]
±→
()
ln
V
ln
c
c
c
ln
S
σ
otherwise
1
2
3
a
H
VF
⎪
⎩
⎭
a
H
[20.19]
S
S
⎧
⎫
a
0
→≤
if
V
c
⎪
⎪
4
a
(
)
=
H
ln
Type
2
_
VF
[20.20]
⎨
⎬
+×
⎛
S
S
⎞
⎟
±→
⎪
⎪
⎪
⎪
a
cc
ln
V
σ
otherwi
se
⎜
5
6
VF
⎩
⎭
a
H
where
c
1
to
c
6
are the model coeffi cients determined by nonlinear regression
(Gülerce
et al.
, 2012). Since PSDMs depend on a vector of IMs, vector-
valued PSHA was required at this stage. Gülerce and Abrahamson (2010)
modifi ed the vector-valued hazard integral given in Equation 20.10 for
structural demand as:
⎧
∫
∫
∫
∫
()(
) ( ) (
)
⎫
N
⋅
f
M
f
M R f
,
ε
f
ε
ε
min
M
R
ε
H
ε
V
H
H
V
⎪
⎪
M
R
ε
ε
H
V
⎨
⎬
ˆ
(
(
)
=
[
(
)
ν
EDP
_
Stage
2
>
z
P EDP
_
Stage
2
>
z EDP S
M R
,
,
ε
,
⎪
⎪
⎪
⎪
a
H
H
(
)
]
)
SMR
,,
ε
,
σ
ddd d
MR
ε
ε
V
⎩
⎭
a
V
VF
H
V
[20.21]
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