DETERMINATION OF CROSS SECTIONAL FORCES - Theory of Bridge Aerodynamics - page 186

are zero, and thus, the total covariance of cross sectional forces may be obtained as the
sum of the covariance contributions from each mode, i.e.

N

mod

()

∑

Cov

x

= Cov (7.82)

FF r FF

R R i

i 1

=

For an arbitrary mode i Eq. 7.76 is then reduced to

⎡

2

⎤

ˆ

( ) ( ) ()

TT

S

x ,

ω T β

H

ω

S

ω β T

=⋅ ⋅ ⋅ ⋅ ⋅ (7.83)

F r i

⎢

i ˆˆ i

⎥

i

R QQ

i

⎣

ii

⎦

T

where β = ⎣ ⎡ φφφφφφ ′′ ′′′ ′′ ′′′ ′ ′′′ ⎤ , and thus

⎦

i y y z z

θ θ

∞ ∞

ˆ

() ( )

TT

()

x

∫

x , d S

∫

H d Cov = S ωω = T ββ T ⋅ ⋅ ⋅ ⋅ ⋅ ω ω

FF r F r i i ˆˆ i

R R QQ

i i ii

0 0

(7.84)

S πω ⋅

i ˆˆ

QQ

TT

ii

= ⋅ T ββ T ⋅ ⋅ ⋅

( ) ( )

i i

41 − κ ⋅ ζ − ζ

ae i ae

i i

where

and are given in Eqs. 6.45 and 6.46, and where

κ ζ

ae i ae i

2

2

⎡ ⎤

3

BV

⎛ ⎞

ρ

ˆ

⎢ ⎥

S J = ⋅ ⋅ (7.85)

⎜ ⎟ ˆˆ ii

Q ii

2 mB

ω

⎢ ⎥

⎝ ⎠

i i

⎣ ⎦

{ }

ˆ ˆ ˆ

T

()

⎡

2

( )

⎤

T

()

∫∫

x x , x dx dx φ BI S Bφ ⋅ ⋅ ⋅ Δ ω ⋅ ⋅

i 1 q

⎣

v v i

⎦

q i 2 1 2

L

ˆ

exp

2

J

= (7.86)

ii

2

⎛ ⎞

T
i

∫

dx

⎜ φφ ⋅ ⎟

i

⎜ ⎟

⎝ ⎠

L