Civil Engineering Reference
In-Depth Information
On taking the Fourier transform of both sides, Equation (4.48) becomes
∑∑
Kb
a
∑∑
Kb
a
(,)
ˆ
2
(,)
ˆ
2
Wx ab
ψω−ω +
( (
)
Wx ab
ψ ω−ω
( (
)
ψ
cj
i
ψ
j
i
ab
,
f
ab
,
j
i
j
i
j
j
i
j
i
j
∑∑
Kb
a
(,)
ˆ
2
+
h
Wa
ψ ψω−ω +
b
( (
)
ψ
c
Rj
i
ab
,
j
i
j
i
j
∑∑
Kb
a
∑∑
Kb
a
2
(,)
ˆ
(,)
ˆ
ω
Wx ab
ψ ω+
()
h
Wa
ψ
b
ψ
()
ω
c
ψ
cj
i
ab
,
c
ψ
Rj
i
ab
,
j
i
j
i
j
j
i
j
i
j
∑∑
Kb
a
,)
ˆ
+ζω
2
Wx ab
(
ψ ωω+
()()
i
cc
ψ
cj
i
ab
,
j
i
j
i
j
∑∑
Kb
a
(,)
ˆ
(4.49)
h
Wa
ψ
b
ψ ωω=
()() 0
i
C
ψ
Rj
i
ab
,
j
i
j
i
j
On collecting similar terms, Equation (4.49) takes the following form:
Kb
a
∑∑
(,)
ˆ
2
2
Wx ab
ψωω−ω+ ζωω+
()
2
i
ψ
cj
i
ab
,
c
C
c
j
i
j
i
j
Kb
a
∑∑
2
Wx ab
(,)
ˆ
[
−ω
]
ψω+
()
j
i
ψ
f
ab
,
j
i
j
i
j
Kb
a
∑∑
2
2
ω−ω+ ζωω
hh
2
i
h
Wa
ψ ψω=
(
,)ˆ
b
() 0
cc
c
c
c
c
ψ
Rj
i
ab
,
j
i
j
i
j
(4.50)
Now, we shall use the same technique here. First, we multiply both sides
of Equation (4.50) by ψω
ˆ
()
ab
*
,
kl
and use Equation (2.40) to obtain the fol-
lowing equation:
∑
∑
ω−ω+ ζωω
2
2
2
i
W xab
(
,)ˆ
ψ ω+−ω
() [
2
]
W xab
(,) ˆ
ψω
()
c
c
c
ψ
cj
i
ψ
j
i
ab
,
f
ab
,
j
i
j
i
i
i
∑
2
2
+ω−ω +ζωω
h
2
i
Wa
ψ
(
,)ˆ
b
ψ ω=
() 0
(4.51)
c
c
cc
ψ
Rj
i
ab
,
j
i
i
