Civil Engineering Reference
In-Depth Information
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11
⎡
⎤
ˆ
ˆ
()
( )
( )
(
)
J
∫∫
G
x
ˆ
G
x
ˆ
S
x
ˆ
,
dx dx
ˆ
ˆ
(2.110)
ω
=
⎢
⋅
⋅
Δ
ω
⎥
M
M
1
M
2
q
y
1
2
⎢
⎥
⎣
⎦
00
in which case the following is obtained:
1/2
⎡
∞
⎤
ˆ
2
()
LJ
∫
d
σσ
=⋅
⋅
ωω
(2.111)
⎢
⎥
Mq
M
y
⎢
⎥
⎣
⎦
0
Under ideal conditions Eqs. 2.111 and 2.105 should render identical results. Obviously,
Eq. 2.105 is the simpler choice, as 2.111 contains frequency domain integration as well
as spatial averaging.
The necessity of a transition from the product of two line integrals into a volume
integral in Eq. 2.101 (and similarly in Eq. 2.107), is better understood if the integral is
replaced by a summation, as illustrated in Fig. 2.15.
I.e., the load is split into
N
concentrated loads
()
(
)
Qt qxt x
,
=
⋅ Δ
(2.112)
k
y
k
Fig. 2.15
Calculation of base moment in cantilevered tower type of beam subject to
fluctuating wind
