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Towers, chimneys and masts. Wind loading and structural response Lecture 21 Dr. J.D. Holmes. Towers, chimneys and masts. Slender structures (height/width is high) . Mode shape in first mode - non linear. Higher resonant modes may be significant.

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### Towers, chimneys and masts

Lecture 21 Dr. J.D. Holmes

Towers, chimneys and masts
• 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  5n1b for circular sections

10 n1b for square sections

- more frequently occurring wind speeds than for square sections

Towers, chimneys and masts
• Drag coefficients for tower cross-sections

Cd = 2.2

Cd = 1.2

Cd = 2.0

Towers, chimneys and masts
• Drag coefficients for tower cross-sections

Cd = 1.5

Cd = 1.4

Cd 0.6 (smooth, high Re)

4.0

3.5

3.0

2.5

2.0

1.5

Drag coefficient CD (q=0O)

Australian Standards

0.0 0.2 0.4 0.6 0.8 1.0

Solidity Ratiod

Towers, chimneys and masts
• Drag coefficients for lattice tower sections

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

ASCE 7-02 (Fig. 6.22) :

CD= 42 – 5.9 + 4.0

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

includes interference and shielding effects between members

( will be covered in Lecture 23 )

Towers, chimneys and masts
• Along-wind response - gust response factor

Shear force : Qmax = Q. Gq

Bending moment : Mmax = M. Gm

Deflection : xmax = x. Gx

The gust response factors for base b.m. and tip deflection differ - because of non-linear mode shape

The gust response factors for b.m. and shear depend on the height of the load effect, z1 i.e. Gq(z1) and Gm(z1) increase with z1

160

140

Resonant

Combined

120

100

Background

Height (m)

80

Mean

60

40

20

0

0.0

0.2

0.4

0.6

0.8

1.0

Effective pressure (kPa)

Towers, chimneys and masts
• Along-wind response - effective static loads

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

Towers, chimneys and masts
• 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.

Towers, chimneys and masts
• Cross-wind response of slender towers

Sinusoidal excitation model :

• Assumptions :
• sinusoidal cross-wind force variation with time
• full correlation of forces over the height
• constant amplitude of fluctuating force coefficient

‘Deterministic’ model - not random

Sinusoidal excitation leads to sinusoidal response (deflection)

Gj is the ‘generalized’ or effective mass =

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

Towers, chimneys and masts
• Cross-wind response of slender towers

Sinusoidal excitation model :

Equation of motion (jth mode):

j(z) is mode shape

Towers, chimneys and masts
• 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)

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

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

where k is a parameter depending on mode shapeTowers, chimneys and masts
• Sinusoidal excitation model

This can be simplified to :

The mode shape j(z) can be taken as (z/h)

For uniform or near-uniform cantilevers,  can be taken as 1.5; then k = 1.6

Towers, chimneys and masts
• Random excitation model (Vickery/Basu) (Section 11.5.2)

Assumes excitation due to vortex shedding is a random process

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

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

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

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

Kao = a non dimensional parameter associated with aerodynamic damping

yL= limiting amplitude of vibration

0.10

0.01

0.001

Maximum tip deflection / diameter

‘Lock-in’

Regime

‘Transition’

Regime

‘Forced vibration’

Regime

2 5 10 20

Scruton Number

Towers, chimneys and masts
• Random excitation model (Vickery/Basu)

Three response regimes :

Lock in region - response driven by aerodynamic damping

Towers, chimneys and masts
• Scruton Number

The Scruton Number (or mass-damping parameter) appears in peak response calculated by both the sinusoidal and random excitation models

Sometimes a mass-damping parameter is used = Sc /4 = Ka =

Clearly the lower the Sc, the higher the value of ymax / b (either model)

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

Towers, chimneys and masts
• 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

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

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.

h/3

h

0.1b

b

Towers, chimneys and masts
• Helical strakes

For mitigation of vortex-shedding induced vibration :

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

Towers, chimneys and masts
• Case study : Macau Tower

‘Pod’ with restaurant and observation decks

between 200 m and 238m

Steel communications tower 248 to 338 metres (814 to 1109 feet)

Concrete tower 248 metres (814 feet) high

Tapered cylindrical section up to 200 m (656 feet) :

16 m diameter (0 m) to 12 m diameter (200 m)

Towers, chimneys and masts
• Case study : Macau Tower

aeroelastic model (1/150)

Towers, chimneys and masts
• Case study : Macau Tower
• Combination of wind tunnel and theoretical modelling of tower response used
• distributions of mean, background and resonant wind loads derived (Lecture 13)
• Wind-tunnel test results used to ‘calibrate’ computer model
Towers, chimneys and masts
• Case study : Macau Tower
• Length ratio Lr = 1/150
• Density ratio r = 1
• Velocity ratio Vr = 1/3

Wind tunnel model scaling :

Towers, chimneys and masts
• Case study : Macau Tower
• Bending stiffness ratio EIr = r Vr2 Lr4
• Axial stiffness ratio EAr = r Vr2 Lr2
• Use stepped aluminium alloy ‘spine’ to model stiffness of main shaft and legs

Derived ratios to design model :

Towers, chimneys and masts
• Case study : Macau Tower

Mean velocity profile :

Towers, chimneys and masts
• Case study : Macau Tower

Turbulence intensity profile :

Towers, chimneys and masts

Case study : Macau TowerWind tunnel test results - along-wind b.m. (MN.m) at 85.5 m (280 ft.)

Towers, chimneys and masts

Case study : Macau TowerWind tunnel test results - cross-wind b.m.(MN.m) at 85.5 m (280 ft.)

Towers, chimneys and masts

Case study : Macau Tower

• Along-wind response was dominant
• Cross-wind vortex shedding excitation not strong because of complex ‘pod’ geometry near the top
• Along- and cross-wind have similar fluctuating components about equal, but total along-wind response includes mean component
Towers, chimneys and masts

Case study : Macau Tower

Along wind response :

• At each level on the structure define equivalent wind loads for :
• mean wind pressure
• background (quasi-static) fluctuating wind pressure