Wind loading and structural response Lecture 21 Dr....

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

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


• 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)

Height (m)

Combined Resonant

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):

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


• 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)


• 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


• 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


• 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


• 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)


• 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

h 0.1b

b


• 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)


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 ρρρρr = 1

• Velocity ratio Vr = 1/3

Wind tunnel model scaling :



• 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 :



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 :



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 :



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


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 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


End of Lecture 21

John Holmes

225-405-3789 [email protected]

Wind loading and structural response Lecture 21 Dr....

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