WIND INDUCED STATIC AND DYNAMIC RESPONSE CALCULATIONS - Theory of Bridge Aerodynamics - page 116

2

()

∞

x

∞

φ

2

ˆ

2

() ( )

i r

() ()

x

∫

Sx , d

∫

H S d

σ = ω ω = ⋅ ω ⋅ ω ω (6.11)

r r

r r i

i i

2

Q

K

i

i

0 0

where

( )

1

()

*

S lim

T

a a

ω =

(6.12)

⋅

Q Q Q

T

π

i i i

→∞

a is the Fourier amplitude of the appropriate flow induced modal loading

and

Q i

K and the modal frequency-response-

component

q , q or q θ . The modal stiffness

i

ˆ i

()

function H are defined in Eqs. 4.19 and 4.24. As shown in Eq. 4.24, any motion ω

ˆ i H ω .
Let us for simplicity consider the displacement response in the along wind horizontal
direction r at

()

induced load effects are included in

x , and develop its variance contribution from one of the predominantly

00 T

y -modes, (e.g. the

contribution from the y -mode with lowest eigen-frequency). The flow induced modal
load is then given by (see Eqs. 4.19 and 5.12)

⎡ ⎤

, with corresponding eigen-frequency

φ ≈ ⎣ φ ωω =

⎦

i y

i y

() ( ) ( )

Qt

=

∫

φ

xqxtdx

⋅

,

y y y

L

exp

(6.13)

VB D D ρ ⎡ ⎤

⎛ ⎞

() ( ) ( )

= ⋅

∫

φ

x

⋅

2 Cuxt

⋅

,

+

C C wxt x

′ − ⋅

,

⎢ ⎜ ⎟ ⎥

y D D L

2 B B

⎣ ⎝ ⎠ ⎦

L

exp

where ex L is the flow exposed part of the structure. Taking the Fourier transform on
either side renders

() () ( )

a

∫

x a x , dx ω = φ ⋅ ω

y q

Q

y

y

L

exp

(6.14)

VB D D ρ ⎡ ⎛ ⎞ ⎤

() ( ) ( )

∫

x 2 Cax , C C ax , x

= ⋅ φ ⋅ ⋅ ω + ′ − ⋅ ω

⎢ ⎜ ⎟ ⎥

y D u D L w

2 B B

⎝ ⎠

⎣ ⎦

L

exp

and thus, the modal load spectrum is given by