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

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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

2ρ

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

©Jónas Þór Snæbjörnsson

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

Added mass:

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

the facade simultaneously

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

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

2ρ

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

©Jónas Þór Snæbjörnsson

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

Added mass:

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

the facade simultaneously

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