Transcript Towers, chimneys and masts - LSU Hurricane Engineering

Wind loading and structural response
Lecture 21 Dr. J.D. Holmes
Towers, chimneys and masts
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
e.g. square cross section with flat-sided members (wind normal to face)
4.0
Drag
coefficient
CD (q=0O)
3.5
Australian
Standards
3.0
ASCE 7-02 (Fig. 6.22) :
2.5
CD= 42 – 5.9 + 4.0
2.0
1.5
0.0
0.2
0.4
0.6
Solidity Ratio 
0.8
1.0
 = 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
Towers, chimneys and masts
• Along-wind response - effective static loads
160
Height (m)
140
Resonant
Combined
120
100
Background
80
Mean
60
40
20
0
0.0
0.2
0.4
0.6
0.8
1.0
Effective pressure (kPa)
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):
  C j a  K j a  Qj (t )
Gj a
Gj is the ‘generalized’ or effective mass =

h
0
m(z) j (z) dz
2
j(z) is mode shape
Qj(t) is the ‘generalized’ or effective force =

h
0
f(z,t) j (z) dz
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
h
C   j (z) dz
y max (h) ρ a C b 0  j (z) dz
0


h 2
b
16π 2G jη jSt 2
4π Sc St 2   j (z) dz
2
0
where j is the critical damping ratio for the jth mode, equal to
n jb
nsb
St 

U(ze ) U(ze )
Sc 
4mη j
ρa b
2
Cj
2 GjK j
Strouhal Number for vortex shedding
ze = effective height ( 2h/3)
(Scruton Number or mass-damping parameter)
m = average mass/unit height
Towers, chimneys and masts
• Sinusoidal excitation model
This can be simplified to :
y max
k.C 

b
4 .Sc.St 2
where k is a parameter depending on mode shape





  (z) dz 



(z)
dz


h
j
0
h
0
2
j
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 :
yˆ
A

b [(Sc / 4 )  K (1  y 2
ao
yL
2
)]1 / 2
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
Towers, chimneys and masts
• Random excitation model (Vickery/Basu)
Three response regimes :
Maximum tip 0.10
deflection /
diameter
‘Lock-in’
Regime
0.01
‘Transition’
Regime
0.001
2
5
10
Scruton Number
Lock in region - response driven by aerodynamic damping
‘Forced
vibration’
Regime
20
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
Sc 
4mη
ρa b2
Sometimes a mass-damping parameter is used = Sc /4 = Ka =
mη
ρa b2
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 vortexinduced 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.
Towers, chimneys and masts
• Helical strakes
For mitigation of vortex-shedding induced vibration :
h/3
0.1b
h
b
Eliminates cross-wind vibration, but increases drag coefficient and along-wind
vibration
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
• 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
• 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
• Case study : Macau Tower
Wind tunnel model scaling :
• Length ratio Lr = 1/150
• Density ratio r = 1
• Velocity ratio Vr = 1/3
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
Towers, chimneys and masts
• Case study : Macau Tower
Wind-tunnel
AS1170.2
Macau Building Code
Full-scale Height (m)
Mean velocity
profile :
350
300
250
200
150
100
50
0
0.0
0.5
1.0
Vm /V240
1.5
Towers, chimneys and masts
• Case study : Macau Tower
350
300
Full-scale
Height (m)
Turbulence
intensity
profile :
Wind-tunnel
MACAU TOWER
- Turbulence
AS1170.2
Macau Building
Intensity
ProfileCode
250
200
150
100
50
0
0.0
0.1
Iu
0.2
0.3
Towers, chimneys and masts
Case study : Macau Tower
Wind tunnel test results - along-wind b.m. (MN.m) at 85.5 m (280 ft.)
R.m.s.
MACAU
TOWER Mean
Maximum
0.5% damping Minimum
2000
1500
1000
500
0
-500 0
20
40
60
80
100
Full scale mean wind speed at 250m (m/s)
Towers, chimneys and masts
Case study : Macau Tower
Wind tunnel test results - cross-wind b.m.(MN.m) at 85.5 m (280 ft.)
R.m.s.
MACAU
TOWER Mean
Maximum
0.5% damping Minimum
2000
1500
1000
500
0
-500 0
-1000
-1500
-2000
20
40
60
80
100
Full scale mean wind speed at 250m (m/s)
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
– 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
Towers, chimneys and masts
Case study : Macau Tower
Design graphs
cracked concrete 5% damping
Mean
Along-wind
bending
moment
at 200
metres
(MN.m)
Maximum
500
400
300
200
100
0
0
20
40
60
80
100
Full scale mean wind speed at 250m (m/s)
Towers, chimneys and masts
Case study : Macau Tower
Design graphs
Macau Tower Effective static loads
(s=0 m)
Height (m)
U m ean = 59 7m/s; 5% damping
350
300
250
200
150
100
50
0
Mean
Background
Resonant
Combined
0
100
Load (kN/m)
200
End of Lecture 21
John Holmes
225-405-3789 [email protected]