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

Towers, chimneys and masts

Wind loading and structural response

Lecture 21 Dr. J.D. Holmes


• 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


• Drag coefficients for tower cross-sections

Cd = 2.2

Cd = 1.2

Cd = 2.0


• Drag coefficients for tower cross-sections

Cd = 1.5

Cd = 1.4

Cd 0.6 (smooth, high Re)


• Drag coefficients for lattice tower sections

= 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

4.0

3.5

3.0

2.5

2.0

1.5

Dra

g co

effic

ient

C

D (

=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= 42 – 5.9 + 4.0


• 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


• 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)

Hei

ght

(m)

CombinedResonant

Background

Mean

Separate effective static load distributions for mean, background and resonant components (Lecture 13, Chapter 5)


• 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.



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)



Sinusoidal excitation model :

Equation of motion (jth mode):

j(z) is mode shape

)(tQaKaCaG jjjj

Gj is the ‘generalized’ or effective mass = h

0

2j dz(z)m(z)

Qj(t) is the ‘generalized’ or effective force = h

0 j dz(z)t)f(z,


• 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

2j

2

h

0 j

2jj

2

h

0 j2

amax

dz(z)StSc4π

dz(z)C

StηG16π

dz(z)bCρ

b

(h)y

where j is the critical damping ratio for the jth mode, equal to jj

j

KG

C

2

)(zU

bn

)(zU

bnSt

e

j

e

s

2a

j

bρ

mη4Sc

(Scruton Number or mass-damping parameter)

m = average mass/unit height

Strouhal Number for vortex shedding ze = effective height ( 2h/3)


• Sinusoidal excitation model

This can be simplified to :

For uniform or near-uniform cantilevers, can be taken as 1.5; then k = 1.6

The mode shape j(z) can be taken as (z/h)

2max

.Sc.St4

k.C

b

y

where k is a parameter depending on mode shape

h

0

2j

h

0 j

dz(z)

dz(z)


• 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

/

Lao y

yKSc

A

b

y

‘lock-in’ behaviour is reproduced by negative aerodynamic damping

yL= limiting amplitude of vibration

Kao = a non dimensional parameter associated with aerodynamic damping


• 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

Ma

xim

um

tip

d

efle

ctio

n /

d

iam

ete

r


• 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 =

2abρ

mη4Sc

2abρ

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)


• 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)


• Helical strakes

For mitigation of vortex-shedding induced vibration :

Eliminates cross-wind vibration, but increases drag coefficient and along-wind vibration

h/3

h0.1b

b


• Case study : Macau Tower

Concrete tower 248 metres (814 feet) highTapered 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)


aeroelastic model (1/150)




• 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


• Length ratio Lr = 1/150

• Density ratio r = 1

• Velocity ratio Vr = 1/3

Wind tunnel model scaling :



• Bending stiffness ratio EIr = r Vr2 Lr

4

• Axial stiffness ratio EAr = r Vr2 Lr

2

• Use stepped aluminium alloy ‘spine’ to model stiffness of main shaft and legs

Derived ratios to design model :



0

50

100

150

200

250

300

350

0.0 0.5 1.0 1.5Vm /V240

Fu

ll-sc

ale

He

igh

t (m

)

Wind-tunnelAS1170.2Macau Building Code

Mean velocity profile :



MACAU TOWER - Turbulence Intensity Profile

0

50

100

150

200

250

300

350

0.0 0.1 0.2 0.3Iu

Fu

ll-sc

ale

H

eig

ht

(m)

Wind-tunnelAS1170.2Macau Building Code

Turbulence intensity profile :



MACAU TOWER 0.5% damping

-500

0500

10001500

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 TOWER0.5% damping

-2000-1500-1000-500

0500

100015002000

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 - cross-wind b.m.(MN.m) at 85.5 m (280 ft.)


• 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


• 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 :


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 TowerDesign graphs

Case study : Macau Tower Design graphs

Macau Tower Effective static loads (s=0 m)

U mean = 59 7m/s; 5% damping

050

100150

200250300350

0 100 200

Load (kN/m)

He

igh

t (m

)

Mean

Background

Resonant

Combined


End of Lecture 21

John Holmes225-405-3789 [email protected]

Documents

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