Civil Engineering Reference
In-Depth Information
On collecting similar terms, Equation (4.45) reduces to
Kb
a
∑
∑
2
2
,)
ˆ
ω−ω+ ζωω
2
i
Wx ab
(
ψ ω+
()
i
i
i
ψ
i
j
i
ab
,
j
i
j
i
j
Kb
a
∑
∑
2
(,)
ˆ
[
−ω
]
Wx ab
ψω+
()
ψ
f
j
i
ab
,
(4.46)
j
i
j
i
j
Kb
a
∑
∑
2
2
ω−ω+ ζωω
hh
2
i
h
Wa
ψ ψω=
(
,)ˆ
b
() 0
i
i
i
i
i
i
ψ
Rj
i
ab
,
j
i
j
i
j
,
and using
Equation (2.40) of Chapter 2, the following equation is obtained:
Now, multiplying both sides of Equation (4.46) by ψω
a
kl
ˆ
*
()
∑
∑
,)
ˆ
(,)
ˆ
ω−ω+ ζωω
2
2
2
i
W xa b
(
ψ ω+−ω
() [
2
]
W xab
ψω+
()
i
i
i
ψ
i
j
i
ab
,
ψ
f
j
i
ab
,
j
i
j
i
i
i
∑
2
2
,)
ˆ
(4.47)
h
ω−ω+ ζωω
2
i
Wa
ψ ψω=
(
b
() 0
i
i
i
i
ψ
Rj
i
ab
,
j
i
i
Now, let us take up the next equation of system equilibrium, which is
Equation (4.13). On substituting the expressions for
x
()
c
,
xt
f
(),
ψ
txt
(),
(),
R
c
(),
() and ψ
t
R
() in Equation (4.13) and cancelling the common term,
ψ
txt
R
c
m
c
, one may get
∑
∑
Kb
a
∑
∑
Kb
a
Wx ab
(,)
ψ+
( )
t
Wx ab
(
,)
ψ+
()
t
ψ
ψ
cj
i
ab
,
f
j
i
ab
,
j
i
j
i
j
j
i
j
i
j
∑
∑
Kb
a
h
Wa
ψ
(,)
b
ψ
( )
t
+
ψ
c
Rj
i
ab
,
j
i
j
i
j
∑
∑
Kb
a
∑
∑
Kb
a
2
ω
Wx ab
(,)
ψ
( )
th
+
Wa
ψ
(
,)
b
ψ
()
t
+
c
ψ
cj
i
ab
,
c
ψ
Rj
i
ab
,
j
i
j
i
j
j
i
j
i
j
∑
∑
Kb
a
2
ζω
Wx ab
(
,)
ψ+
()
t
cc
ψ
cj
i
ab
,
j
i
j
i
i
Kb
a
∑
∑
(4.48)
h
Wa
ψ
(,)
b
ψ
( )
t
=
0
c
ψ
Rj
i
ab
,
j
i
j
i
j
