Towers, chimneys and masts

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

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. 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 )

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. 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)

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. 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

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

12. where k is a parameter depending on mode shape Towers, 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

13. 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

14. 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

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-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.

17. 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

18. 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)

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 • Length ratio Lr = 1/150 • Density ratio r = 1 • Velocity ratio Vr = 1/3 Wind tunnel model scaling :

22. 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 :

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

26. Towers, chimneys and masts Case study : Macau TowerWind 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 TowerDesign graphs

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

31. End of Lecture 21John Holmes225-405-3789 JHolmes@lsu.edu