WIND INDUCED STATIC AND DYNAMIC RESPONSE CALCULATIONS - Theory of Bridge Aerodynamics - page 132

Thus, in case of multi mode calculations there will be NN

⋅ such reduced joint

mod mod

ˆ ij

2

J acceptance functions , each defined by

{ }

ˆ ˆ ˆ

T

()

2

( )

T

()

∫∫

⎡ ⎤

φ BI S Bφ

x

⋅ ⋅ ⋅ Δω

x ,

⋅ ⋅

x dx dx

i 1 q v v q j 2 1 2

⎣ ⎦

L

ˆ

2 exp

J = (6.60)

ij

⎛ ⎞ ⎛ ⎞

T T

∫ ∫

φφ ⋅

dx

⋅ φφ ⋅

dx ⎜ ⎟ ⎜ ⎟

i i j j

⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ L L

2
ij

J

A fully expanded version of is given by

⎧ ⎡

2 2

⎤

D D

⎪ ⎛ ⎞ ⎛ ⎞

ˆ ˆ

() ()

2 2 2

J

∫∫

x x 2 C I S C

′

C I S = φ φ ⎢ + − ⎥

⎨

⎜ ⎟ ⎜ ⎟ ij y 1 y 2 D u uu D L w ww

i j

B B

⎢ ⎝ ⎠ ⎝ ⎠ ⎥

⎪

⎣ ⎦

L

⎩ exp

⎡

2

⎤

D

⎛ ⎞

() () (

)

2

ˆ ˆ

2 2

x x 2 CI S CCI S

B

′

+ φ φ ⎢ + + ⎥

⎜ ⎟ z 1 z 2 L u uu L D w ww

i j

⎢ ⎝ ⎠ ⎥

⎣ ⎦

() () (

⎡

)

2

⎤

ˆ

2

ˆ

2

( )

2

x x 2 BC I S BC

′

I S

+ φ φ +

1 2

⎢

Mu u Mw w

⎥

θ θ

i j

⎣ ⎦

⎡

D D D

⎤

⎛ ⎞⎛ ⎞

ˆ ˆ ⎡

() () () ()

⎤

2 2

+ φ

x

φ

x

+ φ

x

φ

x 4 CCIS

+

C

′ −

C C

′ +

C IS

⎢ ⎜ ⎟⎜ ⎟ ⎥ y 1 z 2 z 1 y 2 D Lu uu D L L D w ww

⎣ ⎦

i j i j

B B B

⎣ ⎝ ⎠⎝ ⎠ ⎦

D D ⎡ ⎛ ⎞ ⎤

ˆ ˆ

⎡ ⎤

() () () ()

2 2

x x x x 4 CBCIS C

′

C BCIS

′

+ φ φ + φ φ + −

⎢ ⎜ ⎟ ⎥

y 1 2 1 y 2 D M u uu D L M w ww

⎣

θ θ

⎦

i j i j

B B

⎝ ⎠

⎣ ⎦

⎫

⎡

D

⎤

⎛ ⎞ ⎪

ˆ ˆ

⎡

() () () ()

⎤

2 2

x x x x 4 CBCIS C

′

C BCIS

′

dx dx + φ φ + φ φ + +

⎬

⎢ ⎥

⎜ ⎟

z 1 2 1 z 2 L M u uu L D M w ww 12

⎣

θ θ

⎦

i j i j

B

⎝ ⎠ ⎪ ⎭

⎣ ⎦

(6.61)

and the corresponding reduced version is given by

2

J

ˆ

2 ij

J

= ⎛

(6.62)

ij

⎞ ⎛ ⎞

( )

( )

2 2 2 2 2 2

∫ ∫

φφφ ++

dx

⋅ φ φφ + +

dx

⎜ ⎟ ⎜ ⎟

⎜

y z

θ

⎟ ⎜

y z

θ

⎟ i i i j j j

⎝ ⎠ ⎝ ⎠

L L

ˆ uu ˆ ww

S S The reduced cross spectra and are defined by

ˆ

( )

2

⎫

SSx , / = Δωσ

⎪ ⎬

uu uu u

(6.63)

ˆ

( )

2

SSx , /

= Δωσ

⎪ ⎭

ww ww w