WIND INDUCED STATIC AND DYNAMIC RESPONSE CALCULATIONS - Theory of Bridge Aerodynamics - page 145

The corresponding covariance response matrix

()

Cov

x for the dynamic response at

rr r

x is then given by

span-wise position

∞

() ( )

x

∫

x , d Cov = S ωω

rr r rr r

0

⎡

2

⎤

Cov Cov

2

() () () () ()

σ ⎡ ⎤

x x x x x φ φ ⋅ φ φ ⋅ φ rr rr rr

yy yz y θ y r y r z r y r θ r

⎢ ⎥

⎢ ⎥

N

mod

⎢ ⎥

2 2

() () ()

()

2

∑

⎢ ⎥

Cov x x x

= σ = φ φ ⋅ φ σ

η

rr rr z r z r θ r

⎢ ⎥

i

zz z θ

⎢ ⎥

i = 1

⎢ ⎥

2

Sym . x

2

⎢ φ ⎥

Sym .

σ θ r

⎢

rr

⎥ ⎣ ⎦

⎣ ⎦

i

θθ

(6.81)

∞

2

∫

S d where σ = ω (6.82)

η η

i i

0

is the variance contribution from an arbitrary mode i . Usually, vortex shedding induced
dynamic response is largely resonant and narrow-banded. It will then usually suffice to
only consider the resonant part of the frequency domain integration in Eq. 6.82, and
discard the background part. Thus,

()

S πω ⋅ ω

∞ ∞ ∞

2 2 i ˆ i

ˆ ˆ

Q i

2

() () () ()

∫

Sd

∫

H S d

∫

H d S

σ = ω = ω ⋅ ω ω ≈ ω ω ⋅ ω =

η η η

ˆ

η

ˆ

i

( )

i i i Q i Q

4

i i

ζζ −

0 0 0 i ae i

(6.83)

where (see Eqs. 6.77 and 5.33)

⎡ ⎤

⎢

()

2

()

2

⎥

2 DS

∫

dx S

∫

dx

λ ω φ + ω φ

q i i q i i

⎢

z z θ θ

⎥

L L

⎣ ⎦

exp exp

()

S

ω =

ˆ

i

Q i

( )

2

2

M

ω

i i

( )

2

2

⎧

2

VB /2

2

⎡ ⎤

ρ

σ

⎛

1/

⎞

2

λ

D

− ωω

⎪

q z

2

⎢

i s

⎥

∫

dx exp = ⋅ ⋅ φ ⋅ −

⎨ ⎜ ⎟

i z

2

( )

b b

⎢ ⎥

πω ⋅

2

⎝ ⎠

M

⎪

ω

z z

s L

⎣ ⎦

⎩

exp

i i

2

( )

⎫ 2

B

⎡ ⎤

σ

⎛

1/

⎞

− ωω

⎪

q

2

θ

⎢

i s

⎥

∫

dx exp + φ ⋅ −

⎜ ⎟ ⎬

i

b

θ

b ⎢ ⎥

⎝ ⎠ ⎪ θ θ

L

⎣ ⎦ ⎭

exp

(6.84)