Geology Reference
In-Depth Information
Figure 1. Envelope function, ground acceleration spectrum and sample ground motion for epicentral
distance r=20 km and moment magnitude M=7
form of linear and reciprocal approximations. In
particular, the following approximation is con-
sidered
linearization is not guaranteed to be conservative
in an absolute sense. That is, it is not known that
the approximations are more conservative than
the original functions. One way to affect the con-
servatism of the approximation is by considering
second-order terms. For example, introducing
diagonal quadratic terms in the convex approxi-
mation (20) yields
p y
({ })
≈
p
({ }) ({ })
({ })
y
=
p y
0
cl
0
∂
p y
y
∑
( ,
0
)(
0
)
+
δ
y y
y
−
y
∂
pl
i
i
i
i
i
i
(20)
∑
δ
0
0 2
p
({ })
y
=
p
({ })
y
+
( ,
y y
)(
y
−
y
)
cq
cl
pq
i
i
i
i
where
i
(22)
∂
p y
y
({ })
0
δ
pl
(
x y
,
0
)
=
1
if
≥
0
,
and
( ,
0
are the coefficients of the second-
order diagonal terms. These coefficients are de-
fined in terms of the second-order derivatives of
the convex approximation
p
where
δ
pq
y y
i
i
∂
i
i
i
0
0
y
y
∂
p y
y
({ })
0
δ
pl
(
x y
,
)
=
i
if
<
0
(21)
i
i
∂
i
i
cl
({ })
at the point
y
An attractive property of this approximation,
called convex linearization, is that it yields the
most conservative approximation among all pos-
sible combinations of direct/reciprocal variables
(Fleury and Braibant, 1986). In principle, convex
y
0
as
{ }
2
0
∂
p
({ })
y
δ
( ,
y y
0
)
=
χ
cl
(23)
pq
i
i
i
2
∂
y
i
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