Civil Engineering Reference
In-Depth Information
J
σ=⋅
(2.105)
Mq
M
y
()
M t
may be obtained by taking the Fourier
Similarly, the auto spectral density of
transform on either side of Eq. 2.99
L
()
()
(
)
∫
a
ω
=
G
x
⋅
a
x
,
ω
⋅
dx
(2.106)
M
M
q
y
0
and applying Eq. 2.67
1
()
*
()
()
S
lim
a
a
ω
=
⋅
ω
⋅
ω
M
M
M
T
π
T
→∞
L
L
⎛
⎞ ⎛
⎞
1
()
(
)
()
(
)
*
lim
∫
Gxax x
,
∫
Gxax x
,
=
⋅
⎜
⋅
ω
⋅
⎟ ⎜
⋅
⋅
ω
⋅
⎟
M
q
M
q
⎜
⎟ ⎜
⎟
T
y
y
π
T
→∞
⎝
⎠ ⎝
⎠
0
0
LL
1
⎡
⎤
()
()
*
(
)
(
)
∫∫
GxGx
lim
ax
,
ax
,
x x
=
⋅
⋅
⋅
ω
⋅
ω
⎢
⎥
M
1
M
2
q
1
q
2
1
2
T
y
y
π
⎣
T
⎦
→∞
00
LL
()
( )
( )
(
)
S
∫∫
GxGxS
x x x
,
⇒
ω
=
⋅
⋅
Δ
ω
(2.107)
M
M
1
M
2
q
y
1
2
00
(
)
(
)
where
q
y
Sx
,
is the cross spectral density of the fluctuating part
y
qxt
of the
,
Δ
ω
distributed load, and
Δ= −
is spatial separation. Integrating over the entire
frequency domain will then render the variance of
x
xx
21
()
M t
:
∞
∞
LL
⎡
⎤
2
()
( )
( )
(
)
∫
∫ ∫∫
σ
=
S
ω
df
=
G
x
⋅
G
x
⋅
S
Δ
x
,
ω
dx dx
d
ω
⎢
⎥
M
M
M
1
M
2
q
y
1
2
⎢
⎥
⎣
⎦
0
0
0 0
(2.108)
∞
LL
⎡
⎤
ˆ
2
()
()
(
)
∫∫∫
=⋅
σ
G
x
⋅
G
x
⋅
S
Δ
x
,
ω
dx dx
d
ω
⎢
⎥
q
M
1
M
2
q
1
2
y
y
⎢
⎥
⎣
⎦
000
ˆ
(
)
where
is defined above and
q
y
Sx
,
is the normalised (but not non-
σ
Δ
ω
q
y
(
)
dimensional) version of
q
y
Sx
,
, i.e.
Δ
ω
ˆ
(
)
(
)
2
Sx
,
Sx
,
/
Δ=
ω
Δ
ω
σ
(2.109)
q
q
q
y
y
y
Introducing
ˆ
ˆ
ˆ
ˆ
, then a normalised frequency
domain version of the joint acceptance function may be defined by
x
=
xL
/
and correspondingly
Δ=
x
xx
−
1
2
