Civil Engineering Reference
In-Depth Information
convective mode of the liquid and the foundation to the ground accelera-
tion. The term τ
R
(ω) represents the transfer function relating the rocking
motion of the foundation to the ground acceleration. All these relationships
are valid in terms of wavelet coefficients. At this point, let us assume the
expressions for τ
i
(ω), τ
c
(ω), τ
f
(ω) and τ
R
(ω), as mentioned below:
∑
∑
Wx ab
(,) ˆ
ψω=τ ω
()
()
Wxab
(,) ˆ
ψ
()
ω
(4.72)
ψ
ψ
i
j
i
ab
,
i
gj
i
ab
,
j
i
j
i
i
i
∑
∑
Wx ab
(,) ˆ
ψω=τ ω
()
()
Wx ab
(,) ˆ
ψ
()
ω
(4.73)
ψ
ψ
cj
i
ab
,
c
gj
i
ab
,
j
i
j
i
i
i
∑
∑
Wx ab
(,) ˆ
ψω=τ ω
()
()
Wxab
(,) ˆ
ψ
()
ω
(4.74)
ψ
ψ
f
j
i
ab
,
f
gj
i
ab
,
j
i
j
i
i
i
∑
∑
Wa
ψ
(,) ˆ
b
ψ ω=τω
()
()
W
xab
(,) ˆ
ψω
()
(4.75)
ψ
ψ
Rj
i
ab
,
R
gj
i
ab
,
j
i
j
i
i
i
In fact, the above expressions may even be simplified if we use the nam-
ing convention as described at the beginning of Section 4.4 and rewritten as
X
i
=τ ω
()
X
g
(4.76)
i
X
c
=τ ω
()
X
g
(4.77)
c
X
f
=τ ω
()
X
g
(4.78)
f
X
ψ
=τ ω
()
X
g
(4.79)
R
B
()
from Equations (4.37), (4.39), (4.41), (4.43) and (4.67) in Equation (4.64), one
may obtain the following expression for base shear in the wavelet domain:
On substituting the expressions for
xt
f
(),
ψ
t
R
(),
xt
i
(),
xt
c
() and
Qt
∑∑
Kb
a
WQ ab
(,)
ψ
( )
t
ψ
Bj
i
ab
,
j
i
j
i
j
Kb
a
Kb
a
∑∑
∑∑
=
m
Wx ab
(,)
ψ
( )
t
+
Wx ab
(
,)
ψ
()
t
i
ψ
i
j
i
ab
,
ψ
f
j
i
ab
,
j
i
j
i
j
j
i
j
i
j
∑∑
Kb
a
∑∑
Kb
a
+
h
Wa
ψ
(,)
b
ψ
( )
t
+
m
Wx ab
(
,)
ψ
()
t
i
Rj
i
c
cj
i
ψ
ab
,
ψ
ab
,
j
i
j
i
j
j
i
j
i
j
Kb
a
Kb
a
∑∑
∑∑
+
Wx ab
(,)
ψ
( )
th
+
Wa
ψ
(
,)
b
ψ
()
t
ψ
f
j
i
ab
,
c
ψ
Rj
i
ab
,
j
i
j
i
j
j
i
j
i
j
(4.80)
