2nd PartConversion of wind speeds to wind loads,

aerodynamical coefficients, wind pressure andsuction and practical design methods

November 2002Júlíus Sólnes

dU s tdt

U sdU sds

dpds

dds

U sdU sds

dpds

p U p U

sz( , )( ) ( )

( )( )

( )( )

,

= + = − + ≈

+ = + = + =

01 1

0

10 0

12 0

2 12

2

ρ ρτ

ρρ ρ constant

Drag forces on a moving objectA bluff body is dragged through a liquid with a speed U(t)=U

U(t) nip

Liquid or air

Surface area of object is projected on aplane perpendicular to flow direction, Ap

s

Ap: the projection area,U0: undisturbed air speeds

F n pdAi iA

= ∫F U C Ap p= ⋅ ⋅12 0

2ρC

n pdA

A Up

iA

p=

⋅

∫1

12

2ρ

Force exerted onthe body:

Drag forces on a moving objectA bluff body is dragged through a liquid with a speed U(t)=U

U(t) nip

Liquid or air

Surface area of object is projected on aplane perpendicular to flow direction, Ap

Force exerted onthe plane:

s

U0: undisturbed air speedCp: a form coefficient

The form or aerodynamicpressure coefficient

q U Cp= ⋅ ⋅12 0

2ρ

For direct pressure/suction Cp=CD “the DragCoefficient”The pressure coefficients can be measureddirectly with pressure cells. They will show marketdynamical behaviour (rapid oscillations)The 10 minute average value will give the staticpressure coefficient

Pressure coefficientsThe pressure/suction per unit area (m2)

Measurement of aerodynamic coefficients© Jónas Þór Snæbjörnsson

Aerodynamical/Pressurecoefficients

Aerodynamic pressure coefficients are used to interpret windpressure/suction on bodies or surfaces. Mostly they are based

on in-situ or wind tunnel measurements.

Positiveinternalpressure

Suction(negative)

Suction(negative)

Pressure(positive)

Suction(negative)

Suction(negative)

Suction(negative)

Negativeinternalpressure(suction)

Suction(negative)

Pressure(positive)

Wind tunnel measurements of theLandsvirkjun office building

©Jónas Þór Snæbjörnsson

Wind blowsinto thegable

Wind blows into a main wallThe wind loads depend on openings in the structure and

can create very different results

Aerodynamical/Pressure coefficientsfor a typical shed building

The wind has to blow over the building creating eddies due toturbulence at the edges and roof top. This causes non-

uniform distribution of the pressure/suction forces Pressure(positive)

Suction(negative)

Pressure(positive)

Suction(negative)

Suction(negative)

Suction(negative)

Pressure(positive)

Suction(negative)

300 450

0,58 0,31 0,320,65

Q t C U t U t CADU t AD M p( ) [ ( ) ( ) & ( )]= +1

20

0

ρ ρ

When structural response is affectedby dynamical behaviour theacceleration can not be disregarded. The force exerted on the object canthen be written as

The Morrison equationFor wave forces on harbour structures as proposed by

Morrison 1932

U(t)

Q(t)D0 and A0 are the diameter and area ofa circle, which circumscribes theprojection area of the object, Ap.CD: the “drag” coefficientCM: the “added mass” coefficient

mY t cY t kY t C U t Y t CAD

U t Y t AD M p&&( ) &( ) ( ) [ ( ( ) &( )) ( & ( ) &&( ))]+ + = − + −1

22 0

0

ρ ρ

&&( )[ ] &( )[ ] ( )Y t m m Y t c c kY t Q QA A stat dyn+ + + + = +

Dynamical response of a simple structureThe drag force depends on the relative velocity and acceleration

Q(t)

U(z;t)=U(z)+u(x,y,z;t)

Y(t)

z=h

Insert U(t)=U+u(x,y,z;t), discard all second orderterms (u2(t), u(t)Y(t), Y2(t)) and rearrange to get:

&&( )[ ] &( )[ ] ( )Y t m m Y t c c kY t Q QA A stat dyn+ + + + = +

m CADAA M p= ⋅ρ 0

0

c C U c c c kmA D cr cr= = ⋅ =ρ λ, , 2

Q C U A YQk

Q C Uu t CADu t A

stat D p statstat

dyn D M p

= =

= +

12

2

0

0

ρ

ρ ρ

,

[ ( ) &( )]

Dynamical response of a simple structureThe response is composed of a static part (deflection caused by themean wind speed) and a dynamical part caused by the wind gusts

Y(t)=Ystat+y(t)

z=h

Added mass:

Aerodynamical damping:

The drag and mass coefficients are sensitive to thecharacteristics of turbulence (A.G. Davenport)

The form coefficientsIt is convenient to introduce the reduced frequency of thewind gusts (turbulent part u(x,y,z;t)), i.e. >=fD/UR whereT=1/f is the period of the wind gusts (0,5-30 sec.), D is a

reference diameter and UR the reference wind speed

P The lift coefficient CL is heavily dependent on theReynold’s number Re=U(z;t)D/<< z direction (vertical) makes airoplanes fly< y direction (horizontal) produces a Strouhal effect (cross

wind vibration) in towers and chimneys

Other form coefficientsThe drag coefficient CD is mostly related to the turbulent x-component u(x,y,z;t). The other components give rise to

different kind of excitations or actions

U(z;t)

P Static Design (structures not susceptible todynamical excitations)< Buildings and chimneys with H#200 m and bridges with

span L#200 m< Mean wind speed in the Qstat term is increased by a peak

factor g(g,<) to account for increased pressure in wind gusts< A dynamical factor Cd is also applied

P Dynamical Design (structure susceptible todynamical excitations)< Buildings and chimneys: H>200 m; bridges with L>200 m< All other structures susceptible to dynamical effects such as

aero-elasticity, vortex shedding, galloping and flutter

Static versus dynamical designEurocode 1 prescribes two different categories for design

Simple equivalent static designDesign wind speed is the reference wind speed plus a

mean gust value (extreme peaks Up): Udes(z)=U(z)+E[Up]

0 50 100 150 200 250 300 350 400 450 5008

10

12

14

16

18

20

22

24

Time in seconds

U(zR)

E[Up]Up,i

[ ]E TT

g Tee

Ξ = + =2 0 57722

log ( ) ,log ( )

( )νν

ν

Probability distribution of extreme peaks. Consider theturbulence component u(x,y,z;t) to be a stochasticprocess u(t) at point (x,y,z).In a time interval T, there will be a large number N ofextreme peaks Up. Introduce the dimensionlessvariable ==Up/FU where FU is the standard deviation ofthe wind gust process u(t). Then, the mean value E[=]is given (approx. for a narrow band process) by:

Extreme value statisticsStochastic Processes and Random Vibrations

g(<T) is a peak function; < is a zero crossing frequency andcan be substitued by the natural frequency of the structure

The peak function g(<T)In Eurocode 1, a fixed value g(<T)=3,5 has been set. Therefore the mean extreme gust speed E[Up]=3,5FU

g(<T)=3,5

S R e d R S e dU Ui

U Ui( ) ( ) ( ) ( )ω

πτ τ τ ω ωωτ ωτ= ↔ =−

−∞

∞

−∞

∞

∫ ∫1

2

[ ]R E u t u tU ( ) ( ) ( )τ τ= +

[ ]σ ω ωU U UE u t R S d2 0= = =−∞

∞

∫( ) ( ) ( )

The autocorrelation function of the wind gusts is

The power spectral density of the wind gusts is

The standard deviation of the windgusts FU

FU2=E[u(t)u(t)], :U=E[u(t)]=0

The variance of the wind gusts is therefore given by

The turbulence intensity IUThe turbulence intensity is defined as IU=FU/UR=VU (thecoefficient of variation). At height z m, IU(z)=FU/vm(z).

IU=q66.kr at 10 metre reference height, IU(z)=kr/(cr(z)qct) at z m

Longitudinal spectrumof horizontal gusts

c z c z ck

c z ce r tr

r t

( ) ( )( )

= ⋅ +

2 2 1 7

The unit wind force at height z metresqd(z)=½qDqvd

2(z)=ce(z)qref(z)=ce(z)½Dvref2

=cr2(z)ct

2[1+3,5IU(z)]2 vref2/1600 KN/m2

.cr2(z)ct

2[1+7,0IU(z)+(IU2(z).0)]½Dvref

2=ce(z)vref2/1600

The so-called exposure factor ce(z) is given inEurocode 1 by

The modified static wind pressurevd(z)=vm(z)+E[Up]=vm(z)+3,5FU=cr(z)ct@vref(z)[1+3,5IU(z)]

PTakes into account< Lack of correlation of pressure on a large surfaces< The magnification effect due to frequency content

of gusts close to natural frequence of structureP Is applied for< Buildings less than 200 m height< Bridges less than 200 m span length

The dynamic modification factor cd The introduction of the design wind speed vd(z)

presupposes that the equivalent static wind pressureengulfs a building facade in its entirety. Actually, the windgusts are not well correlated and will only affect portions of

the facade simultaneously

Dynamic factors for buildingsConcrete/masonry and steel buildings

5 10 20 50 100Breadth b (m)

200

10

20

100

50

The dynamic factor cd

1,051,0

0,95

0,9

5 10 20 100Breadth b (m)

200

10

20

100

50

50

The dynamic factor cd

Concrete and masonry buildings Steel buildings

1,15

1,10

1,0

0,95

1,05

0,9

Aerodynamical/Pressurecoefficients

Pressure coefficients are dependent on the magnitudeof the area considered. Eurocode 1 uses the referenceexternal pressure coefficient cpe,1 and cpe,10 for small and

large areas respectively

cpe

A m2

cpe=cpe,1+(cpe,10-cpe,1)log10A

Zone d/h#1 d/h$4

cpe,10 cpe,1 cpe,10 cpe,1

A -1,0 -1,3 -1,0 -1,3

B,B* -0,8 -1,0 -0,8 -1,0

C -0,5 -0,5 -0,5 -0,5

D +0,8 +1,0 +0,6 +1,0

E -0,3 -0,3 -0,3 -0,3

Building facadesAerodynamic/pressure coefficientswind

D

Pland

E b

A B CB*

windA

wind

Elevation

B C

Case d>ee

e/5

h

h

Case d#ee/5

A B*

e=b or 2h whichever is smaller

Zone

F G H I

cpe,10 cpe,1 cpe,10 cpe,1 cpe,10 cpe,1 cpe,10 cpe,1

Sharp eaves -1.8 -2.5 -1.2 -2.0 -0.7 -1.2 ±0.2

withparapets

hp/h=0.025 -1.6 -2.2 -1.1 -1.8 -0.7 -1.2 ±0.2

hp/h=0.05 -1.4 -2.0 -0.9 -1.6 -0.7 -1.2 ±0.2

hp/h=0.01 -1.2 -1.8 -0.8 -1.4 -0.7 -1.2 ±0.2

withcurvededges

r/h=0.05 -1.0 -1.5 -1.2 -1.8 -0.4 ±0.2

r/h=0.10 -0.7 -1.2 -0.8 -1.4 -0.3 ±0.2

r/h=0.20 -0.5 -0.8 -0.5 -0.8 -0.3 ±0.2

withmansardeaves

"=300 -1.0 -1.5 -1.0 -1.5 -0.3 ±0.2

"=450 -1.2 -1.8 -1.3 -1.9 -0.4 ±0.2

"=600 -1.3 -1.9 -1.3 -1.9 -0.5 ±0.2

Notes: (1) For roofs with parapets or curved eaves, linear interpolation can be used for intermediatevalues(2) For roofs with mansard eaves, linear interpolation between 300 and 600 may be used. For">600 linearly interpolate between the values for 600 and those for sharp edges(3) In Zone I, where positive/negative values are given, both apply(4) For the mansard eave itself, use coefficients for monopitch roofs(5) For the curved edge itself, use linear interpolation between the wall values and the roof

Flat roofsPressure coefficientsh

hp

Parapets

Curved andmansard eaves

edge of eave

hr

"

F

G

F

H I

d

e/4

e/4

bwind

reference height: ze=he=b or 2hwhichever is smaller

Wind action on vertical walls ofrectangular plan buildings

U(z) [m/s]

z [m]

b

h<b

hze=h

b

b<h<2b

ze=h

ze=b

h>2b

b

ze=h

ze=h-b

ze=z

ze=b

Terrain category II (kr=0,19, z0=0,05)Vref=27,6 m/sqref=27,62/1600=0,476 KN/m2

Wind loads on an actual buildingA building in Cologne, Germany (G. König)

40 m

35 m

wind

500 m

wind

Calculationscd=0,96 (from graph of dynamic coefficient)

The topography coefficient:M=H/L=100/250=0,4, Le=H/0,3,6ct=1+0,6@s

z/Le=120/333=0,36 ,x/Le=06s=0,46ct=1+0,6@0,4=1,24The roughness coefficients:cr(40)=0,19@ln(40/0,05)=1,27cr(80)=0,19@ln(80/0,05)=1,40

cr(120)=0,19@ln(40/0,05)=1,48The exposure coefficients:

ce(40)=1,272@1,242(1+7(0,19/(1,27@1,24))=4,57ce(80)=1,402@1,242(1+7(0,19/(1,40@1,24))=5,32ce120)=1,482@1,242(1+7(0,19/(1,48@1,24))=5,81

Wind pressures/suctions:we=qref@cd@ce(z)@cpewi=qref@cd@ce(z)@cpi

Height ze m Zone cpe,10*) ce(z) cd@qref we KN/m2

ze=120 A -1.0 5,81 0,457 -2,66

B* -0.8 -2,12

D +0.8 2,12

E -0.3 -0,80

ze=80 A -1.0 5,32 -2,43

B* -0.8 -1,95

D +0.8 1,95

E -0.3 -0,73

ze=80 A -1.0 4,57 -2,09

B* -0.8 -1,67

D +0.8 1,67

E -0.3 -0,63*) d/h=35/120=0,29#1

Final results: walls

Internal pressure: cpi=-0,25

b=40 m

d=35 m

wind

D

ze=120 m

ze=80 m

ze=40 m

e/5=b/5=40/5=8 m

AB*

E

we=cpe,10@ce(120)@cd@qref=cpe,10@5,81@0,457=cpe,10@2,655

Final results: roofRoof with parapets: hp/h=0,005 (use sharp edge)

b=40 m

wind

ze=120 mF

I

F GH

416

15

2010 10

we,F=-1,80@2,655=-4,78 KN/m2

we,G=-1,20@2,655=-3,19 KN/m2

we,H=-0,70@2,655=-1,86 KN/m2

we,I=±0,20@2,655=-0,53 KN/m2

PMonopitch roofsPDuopitch roofsPHipped roofsPCylindrical roofsPCylindrical domesP Internal pressure coefficients and coefficients

for walls and roofs with more than one skinPCanopy roofs and multispan roofsPBoundary walls, fences and lattice structures

and more

Other pressure coefficientsIn Eurocode 1 more pressure coefficients are presented

PSpecialist treatment, which requiresknowledge of random vibrations andstochastic processes

PThe coherence of the wind gusts andaerodynamical admittance functions comeinto play

Dynamical designStructures susceptible to dynamical effects (Tfundamental$3 sec.)

This concludes the treatment of wind loads