Civil Engineering Reference
In-Depth Information
11
−
Δ
x
ˆ
1
⎡
⎤
xxx
ˆ
ˆ
ˆ
1
+
Δ
3
(
)
⎡
(
)
(
)
⎤
(
)
2
2
exp
J
2
L
∫∫
dx
ˆ
x d x
ˆ
ˆ
∫
2
3
x
ˆ
x
ˆ
x d
ˆ
x
ˆ
=
⎢
⋅
⎥
⋅
ψ
ΔΔ
=
−
Δ Δ ψ
+
ΔΔ
rr
kk
⎢
⎥
kk
⎣
⎦
mn
LL
3
⎢
⎥
⎣
exp
exp
⎦
00
0
The solutions to a good number of cases have been shown by Dyrbye & Hansen [21] and by
Davenport [14], who has also developed simple approximate expressions.
The most common cases are graphically illustrated in appendix B.
Example 6.2
Let us consider a typical single mode single component situation, where the three modes
kmn
,,
00
T
0
T
T
[
]
[
]
φ
=
⎣
⎡
φ
⎤
φ
=
0
φ
φ
=
00
φ
⎦
k
y
m
z
n
with corresponding eigen-frequencies
ωωω have been singled out for a response calculation.
Since the main girder cross section of many bridges are close to a flat plate, the load coefficient
properties
, ,
yz
θ
C
C
C
′
C
C
C
⎫
⎪
⎫
⎪
D
D
L
M
D
CC
B
′
0
,
0
and
′
≠
≈
⎬
⎪
⎬
⎪
⎭
L
D
L
′
⎭
M
are frequently encountered in bridge engineering. In that case
2
(
)
Sx
,
D
Δω
⎛
⎞
2
()
() ()
uu
⎠
∫∫
J
ω
=
2
CI
φ
x
⋅
φ
x
⋅
dxdx
⎜
⎟
y
D
u
y
1
y
2
1
2
2
B
σ
⎝
L
u
exp
(
)
Sx
,
Δω
2
2
() (
)
() ()
ww
J
ω
=
CI
′
∫∫
φ
x
⋅
φ
x
⋅
dxdx
z
L
w
z
1
z
2
1
2
2
σ
L
w
exp
(
)
Sx
,
Δω
2
2
() (
)
() ()
ww
J
ω
=
C
′
I
∫∫
φ
x
⋅
φ
x
⋅
dx dx
θ
Mw
θ
1
θ
2
1
2
2
σ
L
w
exp
Introducing:
ˆ
ˆ
(
)
( )
(
)
(
)
(
)
(
)
Sx
,
S o
x
,
Sx
,
S
o
x
,
Δ=
ω
ω
⋅
Δ,
ω
Δ=
ω
ω
⋅
Δ,
ω
uu
u
uu
ww
w
ww
2
2
2
K
Mmdx
=⋅
∫
nyz
, or
=⋅
ω
ω
φ
,
=
θ
n
n
n
n
n
n
L
and the non-dimensional joint acceptance functions
12
⎛
⎞
ˆ
⎜
ˆ
⎟
()
( ) ( )
(
)
2
J
∫∫
x
x
o
x
,
dx dx
∫
dx
ω
=
φ
⋅
φ
⋅
Δ
ω
φ
y
y
1
y
2
uu
1
2
y
⎜
⎟
L
L
⎝
⎠
exp
12
⎛
⎞
ˆ
⎜
ˆ
⎟
()
( ) ( )
(
)
2
J
∫∫
x
x
o
x
,
dx dx
∫
dx
ω
=
φ
⋅
φ
⋅
Δ
ω
φ
z
z
1
z
2
ww
1
2
z
⎜
⎟
L
L
⎝
⎠
exp
