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### Transcript of 2nd Part Conversion of wind speeds to wind loads, November ...

F:\VERA\kennsla\1127\Glærusýningar\Shows\ec1ES4.shwNovember 2002 Júlíus Sólnes
dU s t dt
dp ds
d ds
dp ds
ρ ρ ρ constant
Drag forces on a moving object A bluff body is dragged through a liquid with a speed U(t)=U
U(t) ni p
Liquid or air
Surface area of object is projected on a plane perpendicular to flow direction, Ap
s
F n pdAi i A
= ∫F U C Ap p= ⋅ ⋅1 2 0
2ρ C
n pdA
A Up
i A
Force exerted on the body:
Drag forces on a moving object A bluff body is dragged through a liquid with a speed U(t)=U
U(t) ni p
Liquid or air
Surface area of object is projected on a plane perpendicular to flow direction, Ap
Force exerted on the plane:
s
The form or aerodynamic pressure coefficient
q U Cp= ⋅ ⋅1 2 0

For direct pressure/suction Cp=CD “the Drag Coefficient” The pressure coefficients can be measured directly with pressure cells. They will show market dynamical behaviour (rapid oscillations) The 10 minute average value will give the static pressure coefficient
Pressure coefficients The pressure/suction per unit area (m2)
Measurement of aerodynamic coefficients © Jónas Þór Snæbjörnsson
Aerodynamical/Pressure coefficients
Aerodynamic pressure coefficients are used to interpret wind pressure/suction on bodies or surfaces. Mostly they are based
on in-situ or wind tunnel measurements.
Positive internal pressure
Wind blows into the gable
Wind blows into a main wall The wind loads depend on openings in the structure and
can create very different results
Aerodynamical/Pressure coefficients for a typical shed building
The wind has to blow over the building creating eddies due to turbulence 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 C A D U t AD M p( ) [ ( ) ( ) & ( )]= +1
2 0
ρ ρ
When structural response is affected by dynamical behaviour the acceleration can not be disregarded. The force exerted on the object can then be written as
The Morrison equation For wave forces on harbour structures as proposed by
Morrison 1932
U(t)
Q(t) D0 and A0 are the diameter and area of a circle, which circumscribes the projection area of the object, Ap. CD: the “drag” coefficient CM: the “added mass” coefficient
mY t cY t kY t C U t Y t C A D
U t Y t AD M p &&( ) &( ) ( ) [ ( ( ) &( )) ( & ( ) &&( ))]+ + = − + −1
2 2 0
ρ ρ
&&( )[ ] &( )[ ] ( )Y t m m Y t c c kY t Q QA A stat dyn+ + + + = +
Dynamical response of a simple structure The drag force depends on the relative velocity and acceleration
Q(t)
z=h
Insert U(t)=U+u(x,y,z;t), discard all second order terms (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 C A D AA M p= ⋅ρ 0
0
c C U c c c kmA D cr cr= = ⋅ =ρ λ, , 2
Q C U A Y Q k
Q C Uu t C A D u t A
stat D p stat stat
dyn D M p
,
[ ( ) &( )]
Dynamical response of a simple structure The response is composed of a static part (deflection caused by the mean wind speed) and a dynamical part caused by the wind gusts
Y(t)=Ystat+y(t)
z=h
Aerodynamical damping:
The drag and mass coefficients are sensitive to the characteristics of turbulence (A.G. Davenport)
The form coefficients It is convenient to introduce the reduced frequency of the wind gusts (turbulent part u(x,y,z;t)), i.e. >=fD/UR where T=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 the Reynold’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 coefficients The 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 to dynamical 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 to dynamical 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 design Eurocode 1 prescribes two different categories for design
Simple equivalent static design Design wind speed is the reference wind speed plus a
mean gust value (extreme peaks Up): Udes(z)=U(z)+E[Up]
0 50 100 150 200 250 300 350 400 450 500 8
10
12
14
16
18
20
22
24
log ( ) , log ( )
( )ν ν
ν
Probability distribution of extreme peaks. Consider the turbulence component u(x,y,z;t) to be a stochastic process u(t) at point (x,y,z). In a time interval T, there will be a large number N of extreme peaks Up. Introduce the dimensionless variable ==Up/FU where FU is the standard deviation of the wind gust process u(t). Then, the mean value E[=] is given (approx. for a narrow band process) by:
Extreme value statistics Stochastic Processes and Random Vibrations
g(<T) is a peak function; < is a zero crossing frequency and can 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 U i
U U i( ) ( ) ( ) ( )ω
−∞

−∞

[ ]R E u t u tU ( ) ( ) ( )τ τ= +

∫( ) ( ) ( )
The power spectral density of the wind gusts is
The standard deviation of the wind gusts FU
FU 2=E[u(t)u(t)], :U=E[u(t)]=0
The variance of the wind gusts is therefore given by
The turbulence intensity IU The turbulence intensity is defined as IU=FU/UR=VU (the coefficient 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 spectrum of horizontal gusts
c z c z c k
c z ce r t r
r t
The unit wind force at height z metres qd(z)=½qDqvd
2(z)=ce(z)qref(z)=ce(z)½Dvref 2
=cr 2(z)ct
.cr 2(z)ct
2=ce(z)vref 2/1600
The so-called exposure factor ce(z) is given in Eurocode 1 by
The modified static wind pressure vd(z)=vm(z)+E[Up]=vm(z)+3,5FU=cr(z)[email protected](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 structure P 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 pressure engulfs a building facade in its entirety. Actually, the wind gusts are not well correlated and will only affect portions of
200
10
20
100
50
200
10
20
100
50
50
1,15
1,10
1,0
0,95
1,05
0,9
Aerodynamical/Pressure coefficients
Pressure coefficients are dependent on the magnitude of the area considered. Eurocode 1 uses the reference external pressure coefficient cpe,1 and cpe,10 for small and
large areas respectively
Building facades Aerodynamic/pressure coefficientswind
Zone
Sharp eaves -1.8 -2.5 -1.2 -2.0 -0.7 -1.2 ±0.2
with parapets
with curved edges
with mansard eaves
"=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 intermediate values (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 roofs Pressure coefficientsh
b wind
reference height: ze=h e=b or 2h whichever is smaller
Wind action on vertical walls of rectangular plan buildings
U(z) [m/s]
z [m]
ze=h
ze=h-b
ze=z
ze=b
Terrain category II (kr=0,19, z0=0,05) Vref=27,6 m/s qref=27,62/1600=0,476 KN/m2
Wind loads on an actual building A building in Cologne, Germany (G. König)
40 m
35 m
Calculations cd=0,96 (from graph of dynamic coefficient)
The topography coefficient: M=H/L=100/250=0,4, Le=H/0,3,6ct=1+0,[email protected]
z/Le=120/333=0,36 ,x/Le=06s=0,46ct=1+0,[email protected],4=1,24 The roughness coefficients: cr(40)=0,[email protected](40/0,05)=1,27 cr(80)=0,[email protected](80/0,05)=1,40
cr(120)=0,[email protected](40/0,05)=1,48 The exposure coefficients:
ce(40)=1,[email protected],242(1+7(0,19/(1,[email protected],24))=4,57 ce(80)=1,[email protected],242(1+7(0,19/(1,[email protected],24))=5,32 ce120)=1,[email protected],242(1+7(0,19/(1,[email protected],24))=5,81
Wind pressures/suctions: [email protected]@ce(z)@cpe [email protected]@ce(z)@cpi
Height ze m Zone cpe,10 *) ce(z) [email protected] 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
B* -0.8 -1,95
D +0.8 1,95
E -0.3 -0,73
B* -0.8 -1,67
D +0.8 1,67
Final results: walls
A B*
we=cpe,[email protected](120)@[email protected]=cpe,[email protected],[email protected],457=cpe,[email protected],655
Final results: roof Roof with parapets: hp/h=0,005 (use sharp edge)
b=40 m
PMonopitch roofs PDuopitch roofs PHipped roofs PCylindrical roofs PCylindrical domes P Internal pressure coefficients and coefficients
for walls and roofs with more than one skin PCanopy roofs and multispan roofs PBoundary walls, fences and lattice structures
and more
Other pressure coefficients In Eurocode 1 more pressure coefficients are presented
PSpecialist treatment, which requires knowledge of random vibrations and stochastic processes
PThe coherence of the wind gusts and aerodynamical admittance functions come into play
Dynamical design Structures susceptible to dynamical effects (Tfundamental\$3 sec.)
This concludes the treatment of wind loads