Along-Wind Dynamic Response - Wind Engineering - Lecture Slides, Slides of Environmental Law and Policy

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Along-wind dynamic response

• Significant resonant dynamic response can occur under wind actions

for structures with n 1 < 1 Hertz (approximate)

• All structures will experience fluctuating loads below resonant

frequencies (background response)

• Significant resonant response may not occur if damping is high enough
  • e.g. electrical transmission lines - ‘pendulum’ modes - high aerodynamic

damping

• Time history of fluctuating wind force

D(t)

time

• Time history of fluctuating wind force

D(t)

time

time

x(t)

High

n 1

• Time history of response :
  • Structure with high natural frequency

• Features of resonant dynamic response :
• Time-history effect : when vibrations build up structure response at

any given time depends on history of loading

• Stable vibration amplitudes : damping forces = applied loads

inertial forces (mass  acceleration) balance elastic forces in structure

effective static loads : ( 1 times) inertial forces

• Additional forces resist loading : inertial forces, damping forces

• Comparison with dynamic response to earthquakes :
• Earthquakes are shorter duration than most wind storms
• Earthquake forces appear as fully-correlated equivalent lateral forces

wind forces (along-wind and cross wind) are partially -correlated fluctuating

forces

• Dominant frequencies of excitation in earthquakes are 10-50 times higher

than wind loading

• Random vibration approach :
  • Uses spectral densities (frequency domain) for calculation :

• Along-wind response of single-degree-of freedom structure :
  • mass-spring-damper

system, mass small w.r.t.

length scale of turbulence

D(t)

k

c

m

2 mk

c η  m

k

2 π

n 1 

representative of large mass

supported by a low-mass column

mx  cxkxD(t)

• equation of motion :

• Along-wind response of single-degree-of freedom structure :
  • deflection : X(t) = X + x '( t )

spectral density :

mean deflection :

where the mechanical admittance is given by :

this is relation between spectral density

of deflection and approach velocity

k

D

X ^ k = spring stiffness

H(n) S (n) k

S (n) D

2 x (^2)

2

1

2

2 2

1

2

n

n 4 η n

n 1

H(n)

S (n)

U

4 D

H(n)

k

S (n) u

2

2 2 x (^2)

• Aerodynamic admittance:
  • Larger structures - velocity fluctuations approaching

windward face cannot be assumed to be uniform

where  2

(n) is the ‘aerodynamic admittance’

then :

S (n)

U

4 D

S (n) (n). u

2

2 2

D  Χ

• Aerodynamic admittance:

hence :

substituting D = kX :

. (n).S (n)

U

4 D

H(n)

k

S (n) u

2 2

2 2 x (^2)

H(n). (n).S (n)

U

4 X

S (n) u

(^22) 2

2

x  Χ

• Mean square deflection :

where :

 

 

 

0

u

(^22) 2

2

0

x

2 x H(n). (n).S (n).dn U

4 X

σ S (n).dn Χ

B R

U

4 X σ

.dn

S (n)

H(n). (n).

U

4 X σ

2

2 u

2

0

2 u

(^22) u 2

2 u

2 2 x ^   









0

2 u

(^2) u .dn σ

S (n) B Χ (n). 





0

2 2 u

u 1 1

2 H(n) .dn σ

S (n ) R Χ (n ).

assumes X^2 (n) and Su(n) are constant at X^2 (n 1 ) and

Su(n 1 ), near the resonant peak

independent of

frequency

• Gust response factor (G) :

Expected maximum response in defined time period /

mean response in same time period

g = peak factor

X X gσx

B R

U

1 2g

X

1 g

X

X

G

x u

2 log (υT)

g 2 log (υT)

e

 e 

 = ‘cycling’ rate (average frequency)

• Dynamic response factor (C dyn ):

Maximum response including correlation and resonant effects /

maximum response excluding correlation and resonant effects

This is a factor defined as follows :

U

1 2g

B R

U

1 2g

C

u

u

dyn

B = 1 (reduction due to correlation ignored)

R = 0 (resonant effects ignored)

Used in codes and standards based on peak gust (e.g. ASCE-7)