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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