DETERMINATION OF CROSS SECTIONAL FORCES - Theory of Bridge Aerodynamics - page 178

where

p

⎧

[ ]

I =

diag I I I j

= (7.58)

⎨ ⎩ j u v w

j

k

( ) [ ]

ρ Δ=

s diag

ρρρ (7.59)

pk pk uu vv w

The auto covariance functions of the fluctuating flow components u , v and w are
defined by (see Eq. 3.35)

⎧

12

⎫

u
nv
w

⎧ ⎪

2 2

⎡ ⎤

⎛

x

⎞ ⎛

z

⎞

Δ Δ

⎪ ⎪

( )

⎢

pk pk

⎥

s exp ρ Δ= − ⎜ ⎟ ⎜ + ⎟ where = (7.60)

⎨ ⎬ ⎨

⎪ ⎩

nn pk

⎜ ⎟ ⎜ ⎟

x z

⎢ ⎥

L L

⎪ ⎝ ⎠ ⎝ ⎠ ⎪ n n

⎣ ⎦

⎩ ⎭

and where

x xx Δ=−

pk p k

(7.61)

z zz

Δ=−

pk p k

are defined in Fig. 7.10.

Fig. 7.10 Spatial separation between nodes p and k

Example 7.2
Let us consider the simply supported beam shown in Fig. 7.11, and assume that it has been
subdivided into N elements with equal length L N . Let us for simplicity assume that all cross
sectional quantities ar e constan ts along the span of the bridge, and that it has a typical type of cross
section where

C ′

,

C

and

C

are negligible and

CDB C ′

⋅ . The reduced wind load

D M D L

vector associated with an arbitrary node p is then given by

BLN /

⋅

⎛ ⎞

ˆ

ˆ

2 θ B ψ Q =⋅ ⋅ ⋅ ⋅

⎜ ⎟

p m q m

2

m

⎝ ⎠