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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
• 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)
Towers, chimneys and masts
• Drag coefficients for lattice tower sections
δ = solidity of one face = area of members ÷ total enclosed area
Australian Standards
0.0 0.2 0.4 0.6 0.8 1.0
Solidity Ratio δ
4.0
3.5
3.0
2.5
2.0
1.5
Drag coefficient
CD (θ=0O)
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= 4δ2 – 5.9δ + 4.0
Towers, chimneys and masts
• Along-wind response - gust response factor
The gust response factors for base b.m. and tip deflection differ -
because of non-linear mode shape
Shear force : Qmax = Q. Gq
Bending moment : Mmax = M. Gm
Deflection : xmax = x. Gx
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
Towers, chimneys and masts
• 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 1.0
Effective pressure (kPa)
Height (m)
Combined Resonant
Background
Mean
Separate effective static load distributions for mean, background
and resonant components (Lecture 13, Chapter 5)
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)
Towers, chimneys and masts
• Cross-wind response of slender towers
Sinusoidal excitation model :
Equation of motion (jth mode):
φj(z) is mode shape
)(tQaKaCaG jjjj =++ &&&
Gj is the ‘generalized’ or effective mass = ∫h
0
2
j dz(z)m(z)φ
Qj(t) is the ‘generalized’ or effective force = ∫h
0j dz(z)t)f(z, φ
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)
∫∫∫
==h
0
2
j
2
h
0 j
2
jj
2
h
0 j
2
amax
dz(z)StSc4π
dz(z)C
StηG16π
dz(z)bCρ
b
(h)y
φ
φφll
where ηj is the critical damping ratio for the jth mode, equal to jj
j
KG
C
2
)(zU
bn
)(zU
bnSt
e
j
e
s ==
2
a
j
bρ
mη4Sc
π= (Scruton Number or mass-damping parameter)
m = average mass/unit height
Strouhal Number for vortex shedding ze = effective height (≈ 2h/3)
Towers, chimneys and masts
• Sinusoidal excitation model
This can be simplified to :
For uniform or near-uniform cantilevers, β can be taken as 1.5; then k = 1.6
The mode shape φj(z) can be taken as (z/h)β
2
max
.Sc.St4
k.C
b
y
πl=
where k is a parameter depending on mode shape
=∫∫
h
0
2
j
h
0 j
dz(z)
dz(z)
φ
φ
Towers, chimneys and masts
• 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
212
2
)]1()4/[(
ˆ
/
L
aoy
yKSc
A
b
y
−−=
π
‘lock-in’ behaviour is reproduced by negative aerodynamic damping
yL= limiting amplitude of vibration
Kao = a non dimensional parameter associated with aerodynamic damping
Towers, chimneys and masts
• 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
0.10
0.01
0.001
Scruton Number
Maximum tip deflection / diameter
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 /4π = Ka =
2
abρ
mη4Sc
π=
2
a bρ
mη
Sc (or Ka) are often used to indicate the propensity to vortex-
induced vibration
Clearly the lower the Sc, the higher the value of ymax / b (either model)
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
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)
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
h/3
h 0.1b
b
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)
Towers, chimneys and masts
aeroelastic
model
(1/150)
• Case study : Macau Tower
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
Towers, chimneys and masts
• Length ratio Lr = 1/150
• Density ratio ρρρρr = 1
• Velocity ratio Vr = 1/3
Wind tunnel model scaling :
• Case study : Macau Tower
Towers, chimneys and masts
• Bending stiffness ratio EIr = ρρρρr Vr2 Lr
4
• Axial stiffness ratio EAr = ρρρρr Vr2 Lr
2
• Use stepped aluminium alloy ‘spine’ to model stiffness of main shaft and legs
Derived ratios to design model :
• Case study : Macau Tower
Towers, chimneys and masts
0
50
100
150
200
250
300
350
0.0 0.5 1.0 1.5Vm /V240
Fu
ll-s
cale
He
igh
t (m
)
Wind-tunnel
AS1170.2
Macau Building Code
Mean velocity
profile :
• Case study : Macau Tower
Towers, chimneys and masts
MACAU TOWER - Turbulence
Intensity Profile
0
50
100
150
200
250
300
350
0.0 0.1 0.2 0.3Iu
Full-s
cale
He
igh
t (m
)
Wind-tunnelAS1170.2Macau Building Code
Turbulence
intensity
profile :
• Case study : Macau Tower
Towers, chimneys and masts
MACAU TOWER
0.5% damping
-500
0
500
1000
1500
2000
0 20 40 60 80 100
Full scale mean wind speed at 250m (m/s)
R.m.s. Mean
Maximum Minimum
Case study : Macau Tower Wind tunnel test results - along-wind b.m. (MN.m) at 85.5 m (280 ft.)
MACAU TOWER
0.5% damping
-2000
-1500
-1000
-500
0
500
1000
1500
2000
0 20 40 60 80 100
Full scale mean wind speed at 250m (m/s)
R.m.s. Mean
Maximum Minimum
Towers, chimneys and masts
Case study : Macau Tower Wind tunnel test results - cross-wind b.m.(MN.m) at 85.5 m (280 ft.)
Towers, chimneys and masts
• 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
Case study : Macau Tower
Towers, chimneys and masts
• 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
• Computer model calibrated against wind-tunnel results
• Combine three components of load distributions for
bending moments at various levels on tower
Case study : Macau Tower
Along wind response :
Towers, chimneys and masts
cracked concrete 5% damping
0
100
200
300
400
500
0 20 40 60 80 100Full scale mean wind speed at 250m (m/s)
Along-wind
bending
moment
at 200
metres
(MN.m)
Mean Maximum
Case study : Macau Tower Design graphs
Case study : Macau Tower Design graphs
Macau Tower Effective static loads
(s=0 m)
U mean = 59 7m/s; 5% damping
0
50
100
150
200
250
300
350
0 100 200
Load (kN/m)
He
igh
t (m
)
Mean
Background
Resonant
Combined
Towers, chimneys and masts
End of Lecture 21
John Holmes
225-405-3789 JHolmes@lsu.edu