1. Towers, chimneys and masts
Wind loading and structural response
Lecture 21 Dr. J.D. Holmes
Towers, chimneys and masts

2. 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

3. Towers, chimneys and masts
Drag coefficients for tower cross-sections
Cd = 2.2
Cd = 1.2
Cd = 2.0

4. Towers, chimneys and masts
Drag coefficients for tower cross-sections
Cd = 1.5
Cd = 1.4
Cd  0.6 (smooth, high Re)

5. Towers, chimneys and masts
Drag coefficients for lattice tower sections
e.g. square cross section with flat-sided members (wind normal to face)
Australian Standards
Solidity Ratio d
4.0
3.5
3.0
2.5
2.0
1.5
Drag coefficient CD (q=0O)
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 )

6. 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

7. Towers, chimneys and masts
Along-wind response - effective static loads 20 40 60 80
100
120
140
160
0.0
0.2
0.4
0.6
0.8
1.0
Effective pressure (kPa)
Height (m)
Combined
Resonant
Background
Mean
Separate effective static load distributions for mean, background and resonant components (Lecture 13, Chapter 5)

8. 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.

9. 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)

10. Towers, chimneys and masts
Cross-wind response of slender towers
Sinusoidal excitation model :
Equation of motion (jth mode):
Gj is the ‘generalized’ or effective mass =
j(z) is mode shape
Qj(t) is the ‘generalized’ or effective force =

11. 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 in book)
where j is the critical damping ratio for the jth mode, equal to
Strouhal Number for vortex shedding ze = effective height ( 2h/3)
(Scruton Number or mass-damping parameter) m = average mass/unit height

12. Towers, chimneys and masts
Sinusoidal excitation model
This can be simplified to :
where k is a parameter depending on mode shape
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

13. Towers, chimneys and masts
Random excitation model (Vickery/Basu) (Section )
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

14. Towers, chimneys and masts
Random excitation model (Vickery/Basu)
‘Lock-in’
Regime
‘Transition’
‘Forced vibration’
0.10
0.01
0.001
Scruton Number
Maximum tip deflection / diameter
Three response regimes :
Lock in region - response driven by aerodynamic damping

15. 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

16. 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 ) 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 (Appendix 5.C) for calculation of displacements for design purposes.

17. Towers, chimneys and masts
Helical strakes
h/3 h
0.1b b
For mitigation of vortex-shedding induced vibration :
Eliminates cross-wind vibration, but increases drag coefficient and along-wind vibration

18. Towers, chimneys and masts
Case study : Macau Tower
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)
‘Pod’ with restaurant and observation decks
between 200 m and 238m
Steel communications tower 248 to 338 metres (814 to 1109 feet)

19. Towers, chimneys and masts
Case study : Macau Tower
aeroelastic model (1/150)

20. Towers, chimneys and masts
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

21. Towers, chimneys and masts
Case study : Macau Tower
Wind tunnel model scaling :
Length ratio Lr = 1/150
Density ratio r = 1
Velocity ratio Vr = 1/3

22. Towers, chimneys and masts
Case study : Macau Tower
Derived ratios to design model :
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

23. Towers, chimneys and masts
Case study : Macau Tower
Mean velocity profile :

24. Towers, chimneys and masts
Case study : Macau Tower
Turbulence intensity profile :

25. Towers, chimneys and masts
Case study : Macau Tower Wind tunnel test results - along-wind b.m. (MN.m) at 85.5 m (280 ft.)

26. Towers, chimneys and masts
Case study : Macau Tower Wind tunnel test results - cross-wind b.m.(MN.m) at 85.5 m (280 ft.)

27. 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

28. 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
resonant (inertial) loads
These components all have different distributions
Combine three components of load distributions for bending moments at various levels on tower
Computer model calibrated against wind-tunnel results

29. Towers, chimneys and masts
Case study : Macau Tower Design graphs

30. Case study : Macau Tower Design graphs
Towers, chimneys and masts
Case study : Macau Tower Design graphs

31. End of Lecture 21 John Holmes 225-405-3789 JHolmes@lsu.edu