# Towers_ chimneys and masts by dffhrtcv3

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```									   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       3.5
coefficient
CD (q=0O)     3.0               Australian
Standards                  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      0.8   1.0
Solidity Ratio 

 = 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

140
Resonant              Combined
120
Height (m)

100         Background
80

60
Mean
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):

G j a  C j a  K j a  Q j (t )
      

h
       m(z)  j (z) dz
2
Gj is the ‘generalized’ or effective mass =           0

j(z) is mode shape

h
Qj(t) is the ‘generalized’ or effective force =   
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
y m ax(h) ρ a C  b 0  j (z) dz      C    j (z) dz
2

                                 0
b         16π 2 G jη jSt 2                  h
4π Sc St 2   j (z) dz
2
0

Cj
where j is the critical damping ratio for the jth mode, equal to
2 GjK j
nsb    n jb
St                           Strouhal Number for vortex shedding
U(ze ) U(ze )             ze = effective height ( 2h/3)

4mη j
Sc                     (Scruton Number or mass-damping parameter)
ρa b   2
m = average mass/unit height
Towers, chimneys and masts

• Sinusoidal excitation model

This can be simplified to :   y max      k.C

b     4 .Sc.St2


  (z) dz 
h

where k is a parameter depending on mode shape                    j

0
h         
      (z) dz 2
    0
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 :

ˆ
y                 A

b [(Sc / 4 )  K (1  y 2            )]1 / 2
ao               2
yL

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

‘Forced
vibration’
0.001                                    Regime
2              5      10       20
Scruton Number

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

4mη
Sc 
ρa b2
mη
Sometimes a mass-damping parameter is used = Sc /4 = Ka =
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 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.
Towers, chimneys and masts

• Helical strakes

For mitigation of vortex-shedding induced vibration :
h/3

h
0.1b

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

• 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
Mean velocity                                            AS1170.2
profile :                                                Macau Building Code
350

Full-scale Height (m)
300
250
200
150
100
50
0
0.0   0.5   1.0   1.5
Vm /V240
Towers, chimneys and masts
• Case study : Macau Tower

Wind-tunnel
Turbulence             MACAU TOWER - Turbulence
AS1170.2
intensity                       Macau Profile
Intensity Building Code
profile :                       350
300
Height (m)   250
Full-scale

200
150
100
50
0
0.0   0.1        0.2   0.3
Iu
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
0.5% damping Minimum
Maximum

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.)

MACAU
R.m.s.  TOWER Mean
0.5% damping Minimum
Maximum

2000
1500
1000
500
0
-500 0      20     40    60     80     100
-1000
-1500
-2000
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
• 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           Maximum
500
Along-wind
bending      400
moment       300
at 200
200
metres
(MN.m)      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
(s=0 m)
U m ean = 59 7m/s; 5% damping

350
300                                       Mean
Height (m)

250
Background
200
Resonant
150
100                                       Combined
50
0
0            100         200