Civil Engineering Reference
In-Depth Information
2
⎧
⎡
D
⎪
⎤
ˆ
2
()
()
(
)
∫∫
J
=
2
C I
⋅
φ
x
⋅
φ
x
⋅
S
Δ
x
,
ω
+
⎨
⎢
⎥
ii
D
u
y
1
y
2
uu
i
i
i
B
⎣
⎦
⎪
⎩
L
exp
2
2
ˆ
⎡
(
)
() ()(
)
() ()
⎤
(
)
CI
′
⋅
φ
x
⋅
φ
x
+
BC I
′
⋅
φ
x
⋅
φ
x
⋅
S
Δ
x
,
ω
+
⎢
Lw
z
1
z
2
M w
θ
1
θ
2
⎥
ww
i
⎣
⎦
i
i
i
i
)
}
ˆ
() ()
() ()
2
⎡
⎤
(
C BC
′
′
I
x
x
x
x
S
x
,
dx dx
φ
⋅
φ
+
φ
⋅
φ
⋅
Δ
ω
LMwz
⎣
1
2
1
z
2
⎦
w
i
1
2
θ
θ
i
i
i
i
and
J
ˆ
ii
J
=
(
)
ii
2
2
2
∫
φφφ
++
dx
y
z
θ
i
i
i
L
Introducing the sinusoidal mode shapes, and
()
S
ω
x
⎛
ω
⋅Δ
⎞
ˆ
ˆ
ˆ
(
)
ui
(
)
(
)
i
Sx
Δ
,
ω
=
⋅
o
Δ
x
,
ω
where
Co
Δ
x
,
ω
=
exp
−
C
⋅
⎠
⎜
⎟
uu
i
uu
i
uu
i
uy
2
V
σ
⎝
u
()
S
ω
ω
⋅Δ
x
⎛
⎞
ˆ
ˆ
ˆ
(
)
wi
(
)
(
)
i
Sx
Δ
,
ω
=
⋅
o
Δ
x
,
ω
where
Co
Δ
x
,
ω
=
exp
−
C
⋅
⎠
⎜
⎟
ww
i
ww
i
ww
i
wy
2
V
σ
⎝
w
then
2
()
()
S
S
D
CaI
ω
ω
⎛
⎞
2
ui
() (
)
wi
()
2
⋅
⋅
ψ
ω
+
⎡
Ca
′
+
BC a I
′
⎤
⋅
⋅
ψ
ω
⎜
⎟
⎣
⎦
D
yu
u
i
L z
M
θ
w
w
i
B
2
2
⎝
⎠
σ
σ
ˆ
2
u
w
J
=
ii
(
)
2
2
2
2
aaa
++
y
z
θ
where
ˆ
()
()
⎡
(
)
⎤
Co
x
,
⎡
ψω
⎤
Δ
ω
1
π
π
u
w
uu
∫∫
=
⋅
sin
x
⋅
sin
x
⋅
⎢
⎥
dx dx
⎢
⎥
1
2
ˆ
1
2
2
ψω
L
L
(
)
⎢
Co
x
,
⎥
⎢
⎥
π
Δ
ω
⎣
⎦
⎛
⎞
⎣
⎦
L
ww
∫
sin
xdx
exp
⎜
⎟
L
⎝
⎠
L
This integral has previously been solved in Example 6.1, and thus
⎡
⎤
(
)
1exp
ˆ
L
ˆ
+
−
ω
ω
ω
⎢
⎥
()
u
i
exp
u
2
4
2
ˆ
C
ψω
=⋅
+
π
⋅
where
ω
=
⋅
⎢
⎥
ui
u
uy
f
2
2
2
(
)
V
ωπ
ˆ
+
2
2
ˆ
ωπ
+
⎢
u
⎥
u
⎣
⎦
⎡
⎤
(
)
1exp
ˆ
ω
L
ˆ
+
−
ω
ω
⎢
⎥
()
w
i
exp
w
2
4
2
ˆ
C
ψω
=⋅
+
π
⋅
where
ω
=
⋅
⎢
⎥
wi
w
y
f
2
2
2
ˆ
(
)
V
ωπ
+
2
2
ˆ
ωπ
+
⎢
w
⎥
w
⎣
⎦
2
2
⎡
⎤
3
BV
⎛ ⎞
ρ
ˆ
⎢
⎥
S
J
x
0
Thus,
=
⋅
⋅
is defined. At
=
⎜ ⎟
ˆˆ
ii
r
Q
ii
2
mB
⎢
ω
⎥
⎝ ⎠
i
i
⎣
⎦
πω
⋅
S
ˆˆ
i
QQ
(
)
(
)
(
)
ii
T
T
x
0
x
0
x
0
Cov
==
⋅
T β
⋅
=⋅
β
=⋅
T
(
) (
)
FF
r
i
r
i
r
R
i
41
−
κ
⋅
ζ
−
ζ
ae
i
ae
i
i
