Wind Loads. 2 nd Part Conversion of wind speeds to wind loads, aerodynamical coefficients, wind pressure and suction, and practical design methods

Transcription

1 Wind Loads nd Part Conversion of wind speeds to wind loads, aerodynamical coefficients, wind pressure and suction, and practical design methods March 007 Júlíus Sólnes, Professor of civil and environmental engineering, University of Iceland in Reykjavik

2 Drag forces on a moving object A bluff body is dragged through a liquid with a speed U(t)=U du ( s, t) U s du ( s ) 1 dp 1 dτ ( ) ( ) sz = 0 + = + ( 0) dt ds ρ ds ρ ds U s du ( ( ) s ) ds 1 dp = 0, p0 + ρu 0 = p + ρu = constant ρ ds Surface area of object is projected on a plane perpendicular to flow direction, A p A p : the projection area, U 0 : undisturbed air speeds Liquid or air U(t) n i p s

3 Drag forces on a moving object A bluff body is dragged through a liquid with a speed U(t)=U Force exerted on the plane: F = ρ U C A p p F i = A Surface area of object is projected on a plane perpendicular to flow direction, A p Force exerted on the body: n pda i The form coefficient or aerodynamic pressure coefficient C p = 1 F1 ρ A U p = 1 ni pda ρ A U U 0 : undisturbed air speed C p : a form coefficient Liquid or air A p 1 U(t) n i p s

4 Pressure coefficients The pressure/suction per unit area (m ) q = 1 ρ U 0 C p For direct pressure/suction C p =C D the Drag Coefficient The pressure coefficients can be measured directly with pressure cells. They will show marked dynamical behaviour (rapid oscillations) The 10 minute average value will give the static pressure coefficient

5 Measurement of aerodynamic coefficients Jónas Þór Snæbjörnsson

6 Wind tunnel measurements of the Landsvirkjun office building Jónas Þór Snæbjörnsson

7 Aerodynamical/Pressure coefficients Aerodynamical/pressure coefficients are used to interpret wind pressure/suction on bodies or surfaces. Mostly they are based on in-situ or wind tunnel measurements. Suction (negative) Suction (negative) Suction (negative) Suction (negative) Pressure (positive) Negative internal pressure (suction) Suction (negative) Pressure (positive) Positive internal pressure Suction (negative)

8 Wind blows into the gable

9 Wind blows into a main wall The wind loads depend on openings in the structure and can create very different results

10 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 nonuniform distribution of the pressure/suction forces Suction (negative) Suction (negative) Pressure (positive) Suction (negative) Pressure (positive) Suction (negative) Pressure (positive) Suction (negative) 0,58 0,31 0,65 0,3

11 The Morrison equation For wave forces on harbour structures as proposed by Morrison 193 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 U(t) Q(t) Q t C U t U t C A 1 0 ( ) = [ ρ ( ) ( ) + ρ D U & ( t )] A D M p 0 D 0 and A 0 are the diameter and area of a circle, which circumscribes the projection area of the object, A p. C D : the drag coefficient C M : the added mass coefficient

12 Dynamical response of a simple structure The drag force depends on the relative velocity and acceleration my&& t cy t ky t C U t Y t C A ( ) + & ( ) + ( ) = [ 1 ( ( ) & ( )) + 0 ρ ρ ( D U & ( t ) Y && ( t ))] A D M p 0 Insert U(t)=U+u(x,y,z;t), discard all second order terms (u (t), u(t)y(t), Y (t)) and rearrange to get: Y && ( t)[ m + m ] + Y & ( t)[ c + c ] + ky( t) = Q + Q A A stat dyn Y(t) Q(t) z=h U(z;t)=U(z)+u(x,y,z;t)

13 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)[ m + m ] + Y&( t)[ c + c ] + ky( t) = Q + Q A A stat dyn Added mass: z=h Y(t)=Y stat +y(t) m C A 0 = ρ D A A M p 0 Aerodynamical damping (sometimes negative!): c = ρc U, c = λ c, c = km A D cr cr 1 Q = ρc U A, Y = stat D p stat Q stat k Q C Uu t C A 0 = [ ρ ( ) + ρ D u &( t )] A dyn D M p 0

14 The form coefficients It is convenient to introduce the reduced frequency of the wind gusts (turbulent part u(x,y,z;t)), i.e. ξ=fd/u R where T=1/f is the period of the wind gusts (0,5-30 sec.), D is a reference diameter and U R the reference wind speed The drag and mass coefficients are sensitive to the characteristics of turbulence (A.G. Davenport)

15 Other form coefficients The drag coefficient C D 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 P The lift coefficient C L is heavily dependent on the Reynold s number Re=U(z;t)D/ν < z direction (vertical); makes airoplanes fly < y direction (horizontal) produces a Strouhal effect (cross wind vibration) in towers and chimneys U(z;t)

16 Stochastic processes and random vibrations The random nature of the wind gusts, that is, the turbulent part of the wind speed, makes it difficult to interpret unless reverting to the theory of stochastic processes. The wind gusts can be treated as a stochastic Gaussian process X(t) with a power spectral density S X (ω) and an autocorrelation function R X (J)=E[X(t)X(t+J)]

17 The stochastic character of the wind gusts 1 iωτ iωτ SU ( ω) = RU ( τ) e dτ RU ( τ) = SU ( ω) e dω π σ U =E[u(t)u(t)], µ U =E[u(t)]=0 The autocorrelation function of the wind gusts is [ τ ] R ( τ) = U E u ( t ) u ( t + ) The power spectral density of the wind gusts is The variance of the wind gusts is therefore given by [ ] U U U σ = E u ( t) = R ( 0) = S ( ω ) dω

18 The longitudinal spectrum of horizontal wind gusts The turbulence intensity is defined as I U =σ U /U R =V U (the coefficient of variation). At height z m, I U (z)=σ U /v m (z). I U =q66.k r at 10 metre reference height, I U (z)=k r /(c r (z)qc o ) at z m Longitudinal spectrum of horizontal gusts

19 Turbulence Intensity I U Measurements at Keilisnes (z R =10 m), Iceland I U.k r =0.15 (terrain category between I and II) Average Direction of Wind (E)

20 1 iωτ S XY ( ω) = RXY ( τ ) e dω π Coh XY The cross spectrum 1 Consider two stochastic processes X(t) and Y(t) A spectrum for the cross-correlation function R XY (τ)=e[x(t)y(t+j)] can be defined: S XY (ω)=co XY (ω)-iqu XY (ω), where Co XY (ω) is the cospectrum and the imaginary part Qu XY (ω) is the quadrature spectrum. The coherence function is defined as SXY ( ω) ( ω) =, 1 Coh XY ( ω) 1 S ( ω) S ( ω) X Y

21 The cross spectrum The phase angle Φ The phase angle is proportional to the reduced frequency ξ QuXY ( ω) ωrmn Φ XY ( ω) = Arc tan c = c ξ CoXY ( ω) U R where c is of the order Thus the real and imaginary parts of the cross spectrum can be written as Co ( ω) = Coh ( ω) S ( ω) S ( ω) cos Φ ( ω) XY XY X Y XY Qu ( ω) = Coh ( ω) S ( ω) S ( ω) sin Φ ( ω) XY XY X Y XY

22 The Coherence: Coh mn (ω) The coherence describes the correlation between two pressure points on the building facade in the frequency domain (- 1#Coh mn (ω)#1) n r mn m θ mn x m (z,t) x n (z,t) U R U(z) x(z,t) x(z,t) {U(z),0,0} z(z,t) y(z,t) Coh mn Loss of coherence between immediate wind velocities at two points on the facade (m,n) is exponentially related to the reduced frequency ξ ω r ( ξ ) = exp[ cξ ], ξ = U c = a ( horizontal separation), c = b ( vertical) ab ω r mn Cohmn ( ω) = exp U R a sin θmn + b cos θmn a 3 b in the northern hemisphere ( a 3.8, b 1.3) R mn

23 A tapered chimney subjected to wind loads X(z,t) is the lateral deflection in the along wind direction z y v(z,t) -Y(z,t) U(z) u(z,t) H U(z) y u(z,t) -X(z,t) x z D(z)

24 Loading functions and response of the chimney z z t X ( z, t) X ( z, t) * aero aero EI ( z) + m( z) = P ( z, t) C ( z) X& ( z, t) M ( z) X&& ( z, t) * P z t = P z + P z t (, ) ( ) (, ) 1 P( z) = ρc D ( z, 0) D( z) U ( z) π P z t ρc z ξ D z U z U z t ρ C z ξ D z U& z t 4 (, ) = D (, ) ( ) ( ) (, ) + M (, ) ( ) (, ) aero C ( z) = ρc ( z, ξ ) D( z) U ( z) D π M z ρ C z ξ D z 4 aero ( ) = M (, ) ( ) X ( z, t) = φ ( z) Q ( t), φ ( z) is the i th normal mode shape i= 1 i i i H H i ( ) = i ( ) i ( ), i ( ) i ( ), i ( ) (, ) i ( ) M = φ φ i M = i M 0 i 0 Q t L u h t u du M m z z dz L t P z t z dz

25 Mean response and gust response The gust response has to be treated as random processes * L X z t = P z + P z t [ (, )] ( ) (, ) 1 X z L P z L X z t z t * * ( ) = [ ( )], [ (, )] = (, ) E[ X ( z, t) X ( z, t + τ )] = R ( τ ) = φ ( z) φ ( z) E[ Q ( t) Q ( t + τ )] S ( ω, z) = φ ( z) φ ( z) S ( ω ) X i j QiQ i= 1 j = 1 X i j i i i = 1 j = 1 * Q ( ) ( ) ( ) ( ), ( ) iq = j i Li L j j i = M i M j ω ωi 0 0 j S ω H ω S ω H ω H ω ( ) + iλ ω The cross spectrum for the generalized force L H H = S ( ω ) φ ( z ) φ ( z ) S ( ω ) dz dz L L i m j n P P m n i j m n i i i

26 Stochastic gust response Having obtained the cross spectrum for the generalized forces L i, the cross spectrum for the direct wind loads P i can be obtained The power law (z/z R ) α is used instead of the logarithm wind speed profile k r qln(z/z 0 ) α zm z n P R mpn m n = D m m D n n zr zr S ( ω, z, z ) ρ U [ C ( z, ξ ) C ( z, ξ ) + 4 π ξ m ξ ncm zm ξ m CM zn ξ n 4 (, ) (, )] + D( z ) D( z ) S ( ω), ξ = m n U U m m n ωd( z ) πu m R

27 The aerodynamical admittance F(ξ) Introducing the coherence function Coh mn (ω) and the phase angle Φ mn (ω), the real part of the cross spectrum for the generalized forces L i can be directly related to the simple wind velocity spectrum S U (ω). Thus an aerodynamical admittance converts wind speeds into wind loads. Spatial separation r mn =(z m -z n ). H S L (,, ) ( ) m L ω z n m z n U S U ω = 4 Fij ( ω ), A = D ( z ) d z 1 ( A U ) U ρ R 0 R H H U 1 z m z n ij ω = D m m D n n A ξ ξ z 0 0 R F ( ) [ C ( z, ) C ( z, ) + π 4 4 C ( z, ξ ) C ( z, ξ )] D ( z ) D ( z ) M m m M n n m n bω r m n cω r m n + e x p c o s φ ( z ) φ ( z ) d z d z U R U R i m j n m n

28 Aerodynamical Admittance Homogeneous isotropic turbulence, a=b=1.7, c=0.0

29 Aerodynamical Admittance Homogeneous isotropic turbulence, a=b=1.7, c=0.8

30 Aerodynamical Admittance Homogeneous isotropic turbulence, a=3.8, b=1.7, c=0.8

31 X max =X+(X peak -X) Overload design factors Consider the response process X(t) (deflection) Introducing the relative extreme peaks = X =(X peak - X)/σ X Having accounted for the static mean response X, the probability distribution of extreme peaks = X is fairly narrow and a sensible design value can be taken as the average extreme peak value E[= X ]. Thus X max =X+σ X E[= X ] or X max =X(1+(σ X /X)q E[= X ])=GFqX

32 Numerical analysis of two chimney stacks Open country environment Symbol Stack 1 Stack Height (m) H Base diameter (m) D Tip diameter (m) D Natural frequency (rps) T Critical damping ratio Drag coefficient CD Added mass coefficient CM 0 0 Mean wind velocity (m/s) U Decay constant b Phase constant c Wind profile exponent " Roughness parameter Maximum Response values Alongwind response F/= Gust factor GF Across wind response: gusts F/= wake F/= Gust factor GF Resulting dynamic response factor GF

33 End nd Part