Chimneys and Masts - Wind Engineering - Lecture Slides, Slides of Environmental Law and Policy

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• Slender structures (height/width is high)
• Mode shape in first mode - non linear
• Higher resonant modes may be significant
• Cross-wind response significant for circular cross-sections

critical velocity for vortex shedding  5n 1 b for circular sections 10 n 1 b for square sections

• more frequently occurring wind speeds than for square sections

• Drag coefficients for tower cross-sections

Cd = 1.

Cd = 1.

Cd  0.6 (smooth, high Re)

• Drag coefficients for lattice tower sections

 = solidity of one face = area of members  total enclosed area

Australian Standards

0.0 0.2 0.4 0.6 0.8 1. Solidity Ratio 

Drag coefficient CD (q= 0 O)

e.g. square cross section with flat-sided members (wind normal to face)

includes interference and shielding effects between members

( will be covered in Lecture 23 )

ASCE 7-02 (Fig. 6.22) : CD= 4^2 – 5.9 + 4.

• Along-wind response - effective static loads

0

20

40

60

80

100

120

140

160

0.0 0.2 0.4 0.6 0.8 1. Effective pressure (kPa)

Height (m)

Combined

Resonant

Background Mean

Separate effective static load distributions for mean, background and resonant components (Lecture 13, Chapter 5)

• Cross-wind response of slender towers

For lattice towers - only excitation mechanism is lateral turbulence

For ‘solid’ cross-sections, excitation by vortex shedding is usually dominant (depends on wind speed)

Two models : i) Sinusoidal excitation ii) Random excitation Sinusoidal excitation has generally been applied to steel chimneys where large amplitudes and ‘lock-in’ can occur - useful for diagnostic check of peak amplitudes in codes and standards

Random excitation has generally been applied to R.C. chimneys where amplitudes of vibration are lower. Accurate values are required for design purposes. Method needs experimental data at high Reynolds Numbers.

• Cross-wind response of slender towers

Sinusoidal excitation model :

Equation of motion (jth mode):

j(z) is mode shape

G (^) j a  Cja  Kja  Qj ( t )

Gj is the ‘generalized’ or effective mass = 

h 0

2 m(z) j (z) dz

Qj(t) is the ‘generalized’ or effective force = 

h

0 f(z,t)^ j^ (z) dz

• Sinusoidal excitation model

Representing the applied force Qj(t) as a sinusoidal function of time, an expression for the peak deflection at the top of the structure can be derived :

(see Section 11.5.1 in book)



 ^   h 0

2 j

2

h 0 j 2 j j

2

h 0 j

2 max a 4 π Sc St (z) dz

C (z) dz 16 π G ηSt

ρ C b (z) dz b

y (h) 

 ^  

where j is the critical damping ratio for the jth mode, equal to j j

j G K

C

U(z )

n b U(z )

n b St e

j e

 s^ 

2 a

j ρ b

4 mη Sc

^ (Scruton Number or mass-damping parameter) m = average mass/unit height

Strouhal Number for vortex shedding ze = effective height ( 2h/3)

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• Random excitation model (Vickery/Basu) (Section 11.5.2)

Assumes excitation due to vortex shedding is a random process

A = a non dimensional parameter constant for a particular structure (forcing terms)

In its simplest form, peak response can be written as :

Peak response is inversely proportional to the square root of the damping

1 2 2

2 [( / 4 ) ( 1 )]

/ L

ao (^) y Sc K y

A

b

y

 

‘lock-in’ behaviour is reproduced by negative aerodynamic damping

yL= limiting amplitude of vibration

Kao = a non dimensional parameter associated with aerodynamic damping

• Random excitation model (Vickery/Basu)

Three response regimes :

Lock in region - response driven by aerodynamic damping

‘Lock-in’ Regime

‘Transition’ Regime ‘Forced vibration’ Regime 2 5 10 20

Scruton Number

Maximum tip deflection / diameter

• Scruton Number and steel stacks

Sc (or Ka) is often used to indicate the propensity to vortex-induced vibration

e.g. for a circular cylinder, Sc > 10 (or Ka > 0.8), usually indicates low amplitudes of vibration induced by vortex shedding for circular cylinders

American National Standard on Steel Stacks (ASME STS-1-1992) provides criteria for checking for vortex-induced vibrations, based on Ka

A method based on the random excitation model is also provided in ASME STS-1-1992 (Appendix 5.C) for calculation of displacements for design purposes.

Mitigation methods are also discussed : helical strakes, shrouds, additional damping (mass dampers, fabric pads, hanging chains)

• Helical strakes

For mitigation of vortex-shedding induced vibration :

Eliminates cross-wind vibration, but increases drag coefficient and along-wind vibration

h/

0.1b^ h

b

aeroelastic model (1/150)

• Case study : Macau Tower

• Case study : Macau Tower
• Combination of wind tunnel and theoretical

modelling of tower response used

• Effective static load distributions
  • distributions of mean, background and resonant wind loads derived (Lecture 13)
• Wind-tunnel test results used to ‘calibrate’ computer model