Civil Engineering Reference
In-Depth Information
Equation (5.66) may also be written in the form
∞
*
Kb
a
(
)
∑
∫
2
EW
3
(,).
abWab
(,)
ψωω
ˆ
()
d
ψ
0
j
i
ψ
0
j
i
ab
,
j
i
j
j
−∞
(5.68)
∞
*
Kb
a
∑
∫
{
}
(
)
2
2
2
()
=
EWxab
H
( ).
ˆ
ω ψωΦω
d
3
,
.
.
ψ
gm
j
i
1
ab
,
j
i
j
j
−∞
For the readers, there are some important points to note here. First, the
value of σ is close to 1 in the case of nonstationary ground motions (viz.
Section 2.4), and the integrands in Equation (5.68) also get closer to 1. Hence,
the integral sign may be dropped to obtain a point-wise equality in the rela-
tionship. In addition, as per the basic assumption that the frequency bands
are nonoverlapping in nature, the expressions under the summation sign
on both sides of Equation (5.68) are equal. Thus, on multiplying all terms
of this equation by the term ω
H
(
2
and summing up over all time instants,
the relation between the nondimensional displacement term and the ground
motion for a specific frequency band may be expressed as follows:
∑
2
() ()
3
ˆ
EW
(,).
abWabH
(,)
ωψ ω
ψ
0
j
i
ψ
0
j
i
2
ab
,
j
i
i
(5.69)
∑
{
}
2
2
2
() ()
.
ˆ
=Φ
3
EWxab
(
,) .
H H
( ).
ω
ω ψω
ψ
gm
j
i
1
2
ab
,
j
i
i
Combining Equations (5.57) and (5.69), the cubic nondimensional dis-
placement term may be avoided as follows:
∑
2
()
ˆ
EW
(,).
abWab
(,)
ψω
ψ
1
j
i
ψ
0
j
i
ab
,
j
i
i
(5.70)
∑
{
}
2
2
2
() () ()
.
ˆ
=Φ
3
EWxab
(
,) .
H H
ω
.
ω ψω
ψ
gm
j
i
1
2
ab
,
j
i
i
H
(
2
, consists of a real component and an
imaginary component, which can be easily derived from Equation (5.53)
as follows:
The transfer function, ω
(
)
2
1
−ω
ζω
−ω + ω
2
(
ω=−
H
+
i
2
(
)
(
)
2
2
(
)
2
(
)
2
(5.71)
2
2
1
−ω + ω
2
1
2
