Civil Engineering Reference
In-Depth Information
whose elements on row
i
column
j
are given by
1
⎡
⎤
()
*
()
()
S
lim
a
a
ω
=
ω
⋅
ω
ˆˆ
ˆ
ˆ
⎢
⎥
QQ
T
Q
Q
π
⎣
⎦
ij
T
i
j
→∞
T
⎛
⎞
T
() (
*
)
T
() (
)
⎡
⎤
∫
x
x
,
x
∫
x
x
,
x
φ a
ω
φ a
ω
⎜
⎟
i
q
⎣
j
q
⎦
1
⎜
⎟
L
L
exp
exp
lim
=
⋅
⎜
⎟
T
2
2
π
M
M
T
ω
ω
→∞
⎜
⎟
i
i
j
j
⎜
⎟
⎝
⎠
T
⎧
⎫
T
() (
*
)
T
() (
)
⎡
⎤ ⎡
⎤
∫∫
x
x
,
x
x
,
dx dx
φ
a
ω
⋅
φ
a
ω
⎪
⎪
⎣
i
1
q
1
⎦ ⎣
j
2
q
2
⎦
1
2
1
⎪
⎪
L
exp
lim
=
⎨
⎬
(
) (
)
T
2
2
π
T
MM
→∞
ω
⋅
ω
⎪
⎪
i
i
j
j
⎪
⎪
⎩
⎭
1
T
i
()
*
(
)
T
(
)
(
)
⎡
⎤
∫∫
φ
x
lim
a
x
,
a
x
,
φ
x
dx dx
⋅
ω
⋅
ω
⋅
1
q
1
q
2
j
2
1
2
⎣
⎦
T
π
T
→∞
L
exp
=
(
) (
)
2
2
MM
ω
⋅
ω
i
i
j
j
(4.76)
()
Thus, the elements of
S
are given by
ω
Q
T
i
()
(
) ()
∫∫
φ
x
x
,
x
dx dx
⋅
S
Δ⋅
ω
φ
1
qq
j
2
1
2
L
exp
()
⇒
S
ω
=
(4.77)
ˆˆ
(
) (
)
Q
ij
2
2
ω
MM
⋅
ω
i
i
j
j
(
)
where
Δ=
x
xx
−
, and where
S
Δ
x
,
ω
is the spectral density matrix of cross sec-
12
qq
(
)
(
)
*
(
)
T
(
)
⎡
⎤
tional loads, i.e.
S
Δ=
x
,
ω
lim 1 /
π
T
a
x
,
ω
⋅
a
x
,
ω
qq
⎣
q
1
q
2
⎦
T
→∞
⎡
*
*
*
⎤
⎡
aa aa aa
⋅
⋅
⋅
⎤
S
S
S
q
q
q
q
q
q
⎢
y
y
y
z
y
⎥
⎢
θ
qq
qq
qq
⎥
yy
yz
y
θ
1
⎢
⎥
⎢
(
)
*
*
*
⎥
(4.78)
S
x
,
lim
a
a
a
a
a
a
S
S
S
⇒
Δ=
ω
⋅
⋅
⋅
=
qq
⎢
q
q
q
q
q
q
⎥
⎢
q q
q q
q q
T
z
y
z
z
z
θ
z
y
z z
z
θ
⎥
π
T
→∞
⎢
⎥
⎢
⎥
S
S
S
*
*
*
aa aa aa
⋅
⋅
⋅
⎢
⎥
⎣
qq
qq
qq
y
z
⎦
q
q
q
q
q
q
θ
θ
θ
θ
⎣
y
z
⎦
θ
θ
θ
θ
[
]
...
...
Extracting from the mode shape matrix
Φφ φ φ
(see Eq. 4.8) a
=
1
i
N
x
three by
N
matrix associated with a chosen span-wise position
mod
