WIND INDUCED STATIC AND DYNAMIC RESPONSE CALCULATIONS - Theory of Bridge Aerodynamics - page 117

⎧ ⎫

2

VB 1 D D

ρ ⎡ ⎤

⎛ ⎞ ⎪ ⎛ ⎞ ⎪

()

* *

S lim

∫

2 Ca C

′

C a dx

ω = φ + −

⎨ ⎬

⎜ ⎟ ⎢ ⎜ ⎟ ⎥

y D u D L w

Q y

2

π

T B B

⎝ ⎠

T →∞

⎝ ⎠

⎣ ⎦

⎪ ⎪

L

⎩ ⎭

exp

(6.15)

⎧ ⎫

⎡

D D

⎤

⎪ ⎛ ⎞ ⎪

∫

2 Ca C

′

C a dx ⋅ φ + −

⎨ ⎬

⎢ ⎜ ⎟ ⎥

y D u D L w

B B

⎝ ⎠

⎣ ⎦

⎪ ⎪ L

⎩ ⎭

exp

Acknowledging that

m

1

⎫

( )

*

( ) ( )

Sx , lim

⎡

a x , a x ,

⎤

uw , Δω = ω ⋅ ω

⎦ where

= (6.16)

⎬ ⎭

mn

⎣

m 1 n 2

T n π

T →∞

and assuming that the cross spectra between flow components are negligible, i.e. that

( ) ( )

Sx , Sx , 0 Δω = Δω ≈

(6.17)

uw wu

then

2

⎡

2

⎤

VB ρ

() ()

S J

ω = ⋅ ω (6.18)

⎢ ⎥ y

Q y

2

⎢ ⎥

⎣ ⎦

where

2

⎧ ⎪⎛

( )

Sx ,

D

Δω

⎞ 2

() ( ) ( )

uu

J

∫∫

x x 2 C I

ω = φ ⋅ φ ⋅

⎨ ⎜

⎟

y y 1 y 2 D u

2

B

⎝ ⎠ σ

⎪ ⎩

u

L

exp

(6.19)

2

( )

⎫

Sx

Δω
σ

,

D CCI ⎡ ⎛ ⎞ ⎤ ⎪ ww

′

x x

+ −

⎬

⎢ ⎜ ⎟ ⎥

DLw 12

B

2

⎝ ⎠

⎣ ⎦

⎪ ⎭

w

is the joint acceptance function containing the span-wise statistical averaging of variance
contributions from the fluctuating u and w flow components.

I I

and are the

u w

corresponding turbulence intensities and Δ =− is the spatial (span-wise)

separation. Combining Eqs. 6.11 and 6.18, using

x xx

12

, and introducing the

2

K M = ω

y yy

modally equivalent and evenly distributed mass

2 2 2

mM ∫ ∫ x m x

∫

x

= φ φ φ (6.20)

y y y y y y

L L L

then the following expression is obtained for the standard deviation of the dynamic
response in the along wind y direction