DETERMINATION OF CROSS SECTIONAL FORCES - Theory of Bridge Aerodynamics - page 184

⎡ ⎤

% $

⎢ ⎥

S ⎢

S

⎥

= ⎢ (7.73)

ˆ ˆ ˆ

Q Q Q

R i j

⎥

⎢ ⎥

$ %

⎣ ⎦

whose elements on row i column j are given by

T
i () ( ) ()

∫∫ φ

x

S

x ,

φ

x dx dx

⋅ Δ ⋅ ω

1 qq i j 2 1 2

L

exp

S

= (7.74)

ˆˆ

( ) ( )

Q ij

2 2

MM

ω ⋅ ω

i i j j

where

( )

x xx x ,

Δ= − , and where S Δ ω is the spectral density matrix of cross

1 2 qq i

sectional loads at the eigen-frequency ω (see Eq. 4.78), i.e.

i

⎡ ⎤

S S S

qq qq qq

⎢

yy yz y

⎥

θ

( )

⎢ ⎥

x , S S S

S Δ ω = ⎢ (7.75)

qq i q q q q q q

zy zz z θ

⎥

⎢ ⎥

S S S

qq qq qq

⎣ ⎦

θ y θ z θ θ

It is seen that S is frequency independent. The resonant part of the auto and cross

Q R

spectral density matrix of all force components is then given by

ˆ ˆ ⎡ ⎤

( ) ()

TT *

()

T

()

TT

x , S ω =⋅ T β S ⋅ ω ⋅ β TTβ HSHβ T ⋅ =⋅ ⋅ ω ⋅ ⋅ ω ⋅ ⋅ (7.76)

Fr η

⎢

η ˆ η

⎥

R R

⎣

Q

⎦

R

The corresponding matrix containing the resonant part of the variance and covariance of
cross sectional stress resultants is obtained by frequency domain integration, i.e.

∞

() ( )

x

∫

x , d

Cov = S ωω

FF r F r

R R

0

2

⎡

Cov Cov Cov Cov

⎤

σ

VV VV VM VM VM

yy yz yx yy yz

⎢ ⎥

⎢ ⎥

2

Cov Cov Cov σ

(7.77)

VV VM VM VM

⎢ ⎥

zz zx zy zz

⎢ ⎥ 2

Cov Cov

= ⎢ σ

⎥

MM MM MM

xx xy xz

⎢ ⎥

2

Sym . Cov σ

⎢ ⎥

MM MM

yy yz

⎢ ⎥

2

⎢ σ ⎥

⎣

M zz

⎦

R