INTRODUCTION - Theory of Bridge Aerodynamics - page 11

( )

()
()
()
()
()

M

⎡

Mx

⎤ ⎡

Mxt ,

,
,
,
,

⎡ ⎤ ⎢ ⎤

y y y

⎥ ⎢ ⎥

⎢ ⎥ ⎢

( )

M M x Mxt

⎥ ⎢

⎢ ⎥ ⎢ ⎥ z z z

⎥ ⎢ ⎥

⎢ ⎥

( )

M M x Mxt

= + (1.14)

x

⎢

x

⎥ ⎢

x

⎢ ⎥ ⎢ ⎥

( )

Q Qx Q x t

⎥ ⎢ ⎥

⎢ ⎥ ⎢

y y y

⎥ ⎢

⎢ ⎥ ⎥

( )

Q Qx Q x t

⎢ ⎥

⎣ ⎦ ⎢ ⎥

⎣ ⎦ ⎣ ⎦

z z

z

( )

y z are

Then the normal stress and shear stress components at cross sectional position

given by

⎡

M

⎤ ⎡

M

M M

⎤

y y

z z

z y z y

+ +

⎢ ⎥ ⎢

⎥

()
()

I I I I

s s ⎡

st

⎤

⎡⎤⎡⎤ ⎢ ⎥ ⎢

⎥

y z y z

x x x

= + = + (1.15)

⎢⎥⎢⎥ ⎢ ⎥

⎢ ⎥ ⎢

⎥

s s

st Q

QM

Q ⎢ ⎥

QM

⎣⎦⎣⎦⎣ ⎦ ⎢

yz yz yz

⎥ ⎢

y

y z x

⎥

z x

++

++

⎢ ⎥ ⎢

A AAt 2

⎥

AA At 2

⎣ ⎦

⎣ ⎦

y z m 0

y z m 0

where

I and I are moments of inertia associated with bending about y and z axis,
A and A are the cross sectional shear areas (rendering averaged values of shear
stresses) and, for simplicity assuming that we are dealing with a closed box type of cross
section, A is the sector area inscribed by the cross section and t is the material
thickness at position

y

( )

y z . The variance of the normal stress is then given by

2

⎡ ⎤

M

⎛ ⎞

M

y

2 2

()

⎢

z

⎥

⎡ ⎤

σ =

Es t

=

E z

+

y ⎜ ⎟

s

⎣ ⎦

x

⎜ ⎟

x

⎢

I I

⎥

⎝ ⎠

y z

⎣ ⎦

2

⎡

2

⎤

⎛

M

⎞ ⎛

M

⎞ ⎛

MM

⎞ ⎛ ⎞

⎢

y y

⎥

z z

E z 2 z y y = ⎜ ⎟ ⎜ + ⎟ ⎜ ⋅ + (1.16)

⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟

⎢

I I I I

⎥

⎝ ⎠ ⎝ ⎠

⎝ ⎠ ⎝ ⎠

y y z z

⎣ ⎦

2

2

Cov

σ

⎛ ⎞ σ

⎛ ⎞ M M M

M

y y z

z

z 2 yz y

= ⎜ ⎟ + +

⎜ ⎟

⎜ ⎟

⎜

I

⎟

I I I

⎝ ⎠

⎝

y

⎠

y z z

2 2 2 2

⎡ ⎤ ⎡ ⎤

EM EM Cov E M M

⎡ ⎤

where σ =

⎣ ⎦ ,

σ =

⎣ ⎦ and

=

⎦ , which may be

⎣

M y M z MM y z

y z yz

further developed into

2

2

σ σ

⎛ ⎞ ⎛ ⎞ ⎛ σ σ

⎞ ⎛ ⎞ M M

M M

y y

2 z z

z 2 z y y

σ = ⎜ ⎟ ⎜ + ⎟ ρ + (1.17)

⎜ ⎟ ⎜ ⎟

s

⎜ ⎟

M M

⎜ ⎟

x

⎜

I

⎟ ⎜

I

⎟ ⎝

I

y z

I

⎠ ⎝ ⎠

⎝

y

⎠ ⎝

y

⎠

z z