DETERMINATION OF CROSS SECTIONAL FORCES - Theory of Bridge Aerodynamics - page 164

The variance of M is then defined by

z B

2

⎡ ⎤

⎧ ⎫

{ }

2

⎢ ⎪ ⎪ ⎥

⎡ ⎤

2

() ( ) () ( )

xEMxt , E

∫

Gxqxt x ,

σ = = ⋅

⎨ ⎬

⎢ ⎥

⎢ ⎥

Mr z r M y

z B z

B

⎣ ⎦

⎪ ⎪

⎢ ⎥

L

⎩ ⎭ (7.17)

exp

⎣ ⎦

() () ( ) ( )

∫∫

GxGxEqxt , qxt , x x = ⋅ ⋅ ⎡ ⋅ ⎤

⎣ ⎦

M 1 M 2 y 1 y 2 1 2

z z

L

exp

rendering a spatial and time domain averaging of the fluctuating cross sectional load.
Introducing Eq. 7.15 then this space and time domain averaging is given by

( ) ( )

Eq x t q x t

⎡

, ,

⎤

⋅ =

⎣ ⎦

y 1 y 2

2

VB

⎡

D D D D

⎤

ρ ⎧ ⎫⎧ ⎫

⎛ ⎞ ⎛ ⎞ ⎛ ⎞

ECu 2 CCw

′

2 Cu CCw

′

+ − ⋅ + −

⎨ ⎬⎨ ⎬

⎢ ⎥

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

D 1 D L 1 D 2 D L 2

2 B B B B

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎣ ⎩ ⎭ ⎩ ⎭ ⎦

(7.18)

( ) ( ) ( ) ( )

where . It is a usual

assumption in wind engineering that cross-covariance between different velocity
components is negligible, i.e. that

uuxt

1 ,

, uuxt

2 ,

and wwxt

1 ,

, wwxt

2 ,

= = = =

1 2 1 2

( ) ( ) ( ) ( )

Euxt wxt

⎡

,

⋅

,

⎤ ⎡ =

Euxt wxt ,

⋅

,

⎤

⎦

≈

0

(7.19)

⎣ ⎦ ⎣

1 2 2 1

in which case

( ) ( )

Eq x t q x t

⎡

, ,

⎤

⋅ =

⎣ ⎦

y 1 y 2

2 2 2

⎧ ⎫

VB D D

⎛ ρ ⎞ ⎛ ⎪ ⎞ ⎛ ⎞ ⎪

( ) ( ) ( ) ( )

2 CEuxt , uxt , CCEwxt

′

, wxt ,

⎡ ⋅ ⎤ + − ⎡ ⋅ ⎤

⎨ ⎬

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎣ ⎦ ⎣ ⎦

D 1 2 D L 1 2

2 B B

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎪ ⎪

⎩ ⎭

(7.20)

Introducing (see chapters 2.2 and 3.3)

( ) ( )

2

( )

E ux t ux t , , x

⎡ ⋅ ⎤ = σρ Δ ⋅

⎣ ⎦

1 2 u u

(7.21)

( ) ( )

2

( )

E wx t wx t

⎡

,

⋅

,

⎤ = σ ⋅ ρ Δ

x

⎣ ⎦

1 2 w w

where ρ and ρ are the covariance coefficients of the u- and w- components, and

uu ww

where x =− is spanwise separation, then xx

Δ

12