DETERMINATION OF THE JOINT ACCEPTANCE FUNCTION - Theory of Bridge Aerodynamics - page 276

Appendix B

DETERMINATION OF THE JOINT
ACCEPTANCE FUNCTION

B.1 Closed form solutions

The calculation of wind load effects, static or dynamic, will inevitably involve the
establishment of the joint acceptance function, normalised or non-normalised. As shown
in chapter 2.10, it represents the statistical averaging in space, and it contains the integral

11

() () () ( )

I

³³

fx fx ˆ ˆ exp xdxdx ˆ ˆ ˆ (B.1) β = ⋅ ⋅ − ⋅Δ β

1 2 1 2

00

()

f x is some influence function or mode shape, x is a non-dimensional ˆ

where,

x xx ˆ ˆ ˆ

coordinate between 0 and 1, Δ= − and

1 2

CLV
LL

/ if dynamic respons ω

- °
= ® ° ¯

mu w
ny z

or
or

- ° ® = mn exp

where (B.2) β

/

x

if static respons

=

° ¯

f f

exp m

Some closed form solutions (presented by Davenport [14], see also examples 6.1 and
6.2) are given below (and plotted in Figs. B.1 - B.3):

Influence function or
Mode shape,

() ()

f x ˆ I Reduced integral, β

( )

1

( )

2

2/

ββ ª −+ −

1 exp

β º

¬ ¼

( )

x

( ) ( )

2/

4

ª

3

/3

2

/2 1 exp 1

º

ββ β ⋅ − + − − ⋅ + ββ

¬ ¼

( )

( ) (

)

ˆ21 x −

8/

4

ª

3

/12

2

/4 1 exp

2

/4 1

º

ββ β ⋅ − + − − ⋅ ββ β + +

¬ ¼

-

( )
( )

2

½

2 n

1

π

° °

( ) ( ) ( )

ˆ 1exp cos n

sin nx

⋅ + β ⋅ − −⋅ ª β π º

π

® ¾

¬ ¼

( )

2 2

2

n

2

β + π β + π

° °

¯ ¿