Wind Effects onStructures

1st PartWind effects and distribution of wind speeds.Analysis of turbulent flow in a boundary layer

March 2007Júlíus Sólnes, Professor of civil and environmental

engineering, University of Iceland in Reykjavik

PMain topic is wind effects on structures

PSome building codes background

PDistribution of wind speeds; wind speedprofiles

PConversion of wind speeds into wind loads

PPressure coefficients

PDynamical effects

PPractical design examples

Main Topics

Green colour means one or more Inter-national Codes® currently enforced statewide

The International Building Codehttp://www.iccsafe.org/

Wind Loading of Stuctures © Spon Press 2001John D. Holmes

P EN1990 Eurocode 0: Basis of structural design (2001, 2004)

P EN1991 Eurocode 1: Actions on structures (self weight, imposedloads, snow, wind, accidents, thermal etc.) (2001, 2002, 2003,2004, 2005, 2006)

P EN1992 Eurocode 2: Concrete structures (2004, 2005)

P EN1993 Eurocode 3: Steel structures (2004, 2006)

P EN1994 Eurocode 4: Composite structures (steel and concrete)(2004, 2005)

P EN1995 Eurocode 5: Timber structures (2004, 2006)

P EN1996 Eurocode 6: Masonry structures (2004, 2005)

P EN1997 Eurocode 7: Geothechnical and foundation design (2004,2006)

P EN1998 Eurocode 8: Earthquake resistant design of structures(2004, 2005, 2006)

P EN1999 Eurocode 9: Aluminium structures (2006)

The EUROCODE system Year of ratification as European Standard

RS

p

R

q Sγ

γ= ⋅

PEurocode 0 (EN1990)< Probability distributions of environmental loads< Environmental loads (earthquakes, snow, wind etc.)< Anthropogenic loads (floor loads etc.)< The reliability concept (ps=P[R>S])< The safety index β< 1st, 2nd and 3rd level safety methods< Yield functions and safety margins< Safety factors and the partial factor system < Combination of actions and limit states< Design situations (ULS, SLS, ALS, ELS)

Background of the Eurocodes

Wind effects on structuresStatic and dynamic response of structures to wind loading

P Hurricane Severity Scale (Herbert Saffir and Robert Simpson). < Four criteria for each category: barometric pressure, wind speed, storm

surge, damage potential (Wind speeds are determining factor)– Category 1: 74–95 mph (33-43 m/s); storm surge 3–5 ft, some damage to shrubbery,

trees, and unanchored mobile homes; some flooding of low-lying coastal roads.

– Category 2: 96–110 mph (43–50 m/s); storm surge 6–8 ft, considerable damage toshrubbery with some trees being blown down, extensive damage to mobile homes,and inundation by rising water of coast roads and low-lying escape routes.

– Category 3: 111–130 mph (50–58 m/s); storm surge 9–12 ft; large trees blown down,some structural damage to small buildings, destruction of mobile homes, and floodingof sea-level coastland 8 mi (13 km) or more inland; requires evacuation of low-lyingshoreline residences

– Category 4: 131–155 mph (59–69 m/s); storm surge of 13–18 ft, severe damage toroofing materials, windows, and doors, complete destruction of mobile homes,flooding of low-lying areas as much as 6 mi (10 km) inland, and major damage tostructures near shore due to battering by waves and floating debris

– Category 5: >155 mph (>69 m/s); storm surge higher than 18 ft (5.6 m), completefailure of roofs on residences and industrial buildings, overturning or sweeping away ofsmall buildings, and major damage to structures less than 15 ft (4.6 m) above sealevel within 1,500 ft (457 m) of shore. Category 5 storm requires evacution of allresidential areas on low-lying ground within 5–10 mi (8–16 km) of shore

The Saffir-Simpson ScaleHurricane Severity and Damage Intensity: Categories 1-5

0

( , ) ( ) ( , , ; )

1( ) ( , ) / , 10min

1( , ) ( , ) ,

t T

basic

t

t

U z t U z u x y z t

U z U z t dt v m s TT

U z t U z s ds moving averaget

+

= +

= = =

=

∫

∫

Standard wind speed in m/s (100mphq0.45=45 m/s) is measured at10 m height and averaged over 10minutes. A moving average givesa smoothed slowly varying meanvalue

Wind speed measurementsKeilisnes, SW Iceland

Measurements and PhotoJónas Snæbjörnsson

zR=10 m

Wind speeds at Keilisnes, SW IcelandThe wind speed U(zR,t), the 10 min. average U(zR) and

the moving average U(zR,t), zR=10 m

0 50 100 150 200 250 300 350 400 450 5008

10

12

14

16

18

20

22

24

Time in seconds

U(zR)=vbasic

U(zR,t)

P vb,0-basic value as presented by appropriate windmaps in a National Annex (NA)

P CDIR-direction factor taken as 1,0 unless otherwisespecified in the NA

P CSeasonal-temporal (seasonal) factor taken as 1,0unless otherwise specified in the NA

P CALT-altitude factor taken as 1,0 unless otherwisespecified in the NA (was dropped in the final versionof Eurocode 1)

P ASCE Standard: ASCE 7-98 uses 3 sec. gust wind

The Basic Wind SpeedThe 10 min. average wind speed at 10 metre height (U(zR=10)

m/s) is the fundamental design parameter called vb,0 in Eurocode 1

vb=CDirCSeasonal (CALT)vb,0

V T VK T

Kref ref

e e

n

( ) (50)log [ log ( / )]

,=

− − −

+

1 1 1 1

1 3 902

1

1

P National authorities have to prepare wind maps (zonation)with maximum expected wind speeds V m/s

P Use extreme value statistics< P[V#v]=exp(-exp(-a(v-U))), the Gumbel distribution with the

attraction coefficients 1/a and U (equal to the “mode of thedata”). E[V]=U+0.5772/a, Var[V]=π2/6a2

.1,645/a2

< V(T)=U-1/a(loge(-loge(1-1/T))) is the maximum wind speedwith a return period of T years

< Usually V(50)=U+3,902/a-the 50 year wind< Eurocode 1 suggests the formula below for other periods T

Distribution of maximumwind speeds

For convenience vb,0 is represented by the random variable V m/s

K1=0,2 and n=0,5 if not otherwise specified in the NA

USA wind hazardFrom http://www.hazardmaps.gov

Expected wind speeds (mph) every 100 years

Map showing the modeUG m/s of the gradientwind speed VG m/s

European windregime (50 year

wind)

10 min. averagewind speed m/s

VG

V(z)

Height in metresabove ground

45

45

45

45

50

45

4540

45

40

45

35

35

3035 23

25

30

27

40

35

23

25

30

25

40

23

30

27

40

35

26

20

30

3030

23

23

( , , ; ) ( ) ( , , ; )

( , , ; ) 0 ( , , ; )

( , , ; ) 0 ( , , ; )

( ; ) ( ) ( ; )ii i i i i

U x y z t U z u x y z t

V x y z t v x y z t

W x y z z x y z t

or

U x t U x u x t

= +

= +

[ ]

[ ]

E U x t U x t U x

and

E u x t

i i i i

i i

( ; ) ( ; ) ( )

( ; )

= ≈

=

1 1

0

Wind as a boundary layer air flowMean wind speed plus a turbulent component, which

can be treated as a stochastic or random process

The x-axis (1-axis) is themain direction of the wind

Buildingfacade

Z(z,t)

X(z,t)

Y(z,t)

{U(z),0,0}

U(z) X(z,t)

UR

z

σ

σ τ τ

τ σ τ

τ τ σ

τ

τ τ τ

τ τ τ

τ τ τ

=

= =

x xy xz

yx y yz

yx yz z

ij

11 12 13

21 22 23

31 32 33

x xkk kk

k

==

∑1

3

,

j

j k

k

uu

x

∂

∂=

Flow stresses in tensor notationIn hydrodynamics, the stresses Jij often signify the

stress velocities Jij=Mτij/Mt

z,x3,3

y,x2,2

Jyx

Jxy

σzz=σz

x,x1,1

i,j,k0{1,2,3}

The summationconvention of Einstein.If an index is twofold itmeans a summation

Einstein used thisshort hand term forpartial derivatives

τ µ γ µ∂

∂τ µ ρνzx zx

u

zu u= ⋅ = ⋅ = = ⋅ = ⋅31 1 3 1 3, ,

The shear stressesKinematic and dynamic viscosity

ρ=1.25 Kg/m3, air densityµ=1.81 g/(cmqs), dynamicviscosity of airν=µ/ρ, kinematic viscosity of air

z,x3,3

x,x1,1∆z

∆u

γzx

Jzx

{ }& , , ,, , ,U U U p i ki k i k i ik k+ ⋅ = − + ∈1 1

1 2 3ρ ρ

τ

τ τ

∂

∂

∂

∂

∂

∂

∂

∂

τ∂τ

∂

∂τ

∂

∂τ

∂

ij xy

i ji

j

k i ki i i

ik ki i i

UU

x

U U UU

xU

U

xU

U

x

x x x

=

=

⋅ = + +

= + +

,

,

, ( )

1

1

2

2

3

3

1

1

2

2

3

3

Ui={U1,U2,U3}

is the wind speed (m/s)

p(xi ) is the air pressure

Jij is the boundary flowshear stress

(Jii=σii=0)

Fundamental equationsgoverning viscous flow

The Navier-Stokes equations

Tensor Notation:

[ ]E U U U E pi k i k i ik k&

, , ,+ ⋅ = − −

1 1

ρ ρτ

[ ]{ }

∂

∂ ρ

∂

∂

U

tU U p

E u u

xi k

ik i k i

i k

k

( )( )

, , ,, ,= + ⋅ = − +

− ⋅∈0

11 2 3

{ }& , , ,, , ,U U U p i ki k i k i ik k+ ⋅ = − + ∈1 1

1 2 3ρ ρ

τ

Taking the expectation of both sides of the NavierStokes equations gives the Reynold’s equation:

The Reynold’s equationU i(x i;t)=U i(x i)+u i(x i;t); E[U i(x i;t)]=U i(x i) , E[u i(x i;t)]=0

and disregarding the viscous shear stress τik

Compare with the original Navier-Stokes equation

Measurements show that:

J12=-ρE[u1u2].0

J23=-ρE[u2u3].0

J13=-ρE[u1u3]…………0 , that is,

only the turbulencecomponent in the direction ofthe mean wind speed u1 andthe vertical component u3

seem to be correlated

The Reynold’s shear stressThe last term of the Reynold’s equation can beinterpreted as a shear stress induced by the mixingof the turbulence components: Jij=-ρE[uiuj]

U1(x3)

x3

x1

x2

u1(xi;t)

Moreover: E[u1u3]#0 for higher wind speeds withoutmajor temperature influences

[ ]κ

τ

ρ= = −13

2

1 3

2

U

E u u

U

u E u u U

or

U

* [ ]= = − = ⋅

=

τ

ρκ

τ ρκ

131 3

13

2

The surface roughness coefficient6 is interpreted as a dimensionlessReynold’s shear stress and can bemeasured through E[u1u3]

The roughness coefficient κThe Reynold’s stress -ρE[u1u3] gives an indication of the

surface roughness; a rougher surface increases thecorrelation between the two components u1 and u3

The shear velocity orfrictional velocity isdirectly related to thesquare root of theroughness coefficienttimes the meanreference velocity

( )0.4 Rxz

U U zK U z

z z

∂ ∂τ ρ ρ κ

∂ ∂= = ⋅ ⋅

* 0.4R Ra aK k zu k z U z Uκ κ= = ⋅ = ⋅ ⋅

2

Rxz Uτ ρκ=

The shear stress of the turbubulent flow is given by

A simple model for turbulent flowDue to the surface roughness and the vortices produced by u1 andu3, the mean velocity decreases approaching the surface where itbecomes zero. This condition can be described by the K-modelturbulence theory of von Karman. For convenience let x=x1 and z=x3

K is the eddy viscosity dependent on the size and velocity ofthe vortices

where ka is the von Karman coefficient. Measurementsshow it to be approximately constant equal to 0.4. Forcomparison, the Reynold’s shear stress is written as

[ ]

2

0

0 0

1 3

0

( )ln ln ( ) ,

1exp , 6

0.4

R Rxz a xz

R a

r rR a

R r

r a R

dUU k z U

dz

dU dz

k zU

U z z zk c z z z

k z zU

E u uz z k

k k U

τ ρκ ρ κ τ

κ

κ

κκ

= = ⋅ =

= ⋅

= = = ≥

− = = = ≈

Wind velocity profilesThe two different versions of the shear stress τxz are put equal

10 min. averagewind speed m/s

Height in metresabove ground

UR

U(z)

z0 is a constant of integration, the roughness length. In Eurocode 1,kr is called the terrain factor and cr(z) the roughness coefficient.

Eurocode 1: Terrain categories

Terrain category kr z0

[m]zmin

[m]g

O

I

II

III

IV

Sea or coastal areas exposed to the open sea

Lakes or flat horizontal area with negligiblevegetation and without obstacles

Area with low vegetation such as grass and isolatedobstacles (trees, buildings) with separations of atleast 20 obstacle heights

Area with regular cover of vegetation or buildings orwith isolated obstacles with separations of maximum20 obstacle heights (such as villages, suburbanterrain, premanent forest)

Area in which at least 15% of the surface is coveredwith buildings and their average height exceeds 15 m

0.155

0.17

0.19

0.22

0.24

0.003 1 0.06

0.01

0.05

0.3

1.0

1

2

5

10

0.13

0.26

0.37

0.46

The above parameters have been fitted to available wind measurement data. The coefficient g is used forspecial dynamical analysis of inline response of structures

Wind measurements: KeilisnesJónas Þór Snæbjörnsson

0 50 100 150 200 250 300 350 4000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Average Direction of Wind (E)

Wind Measurements: ReykjavíkBústaðavegur: Jónas Þór Snæbjörnsson

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

100

200

300

400

500

600

700

Terrain Factor kr

Mean value ofkr=0.196

-14 -12 -10 -8 -6 -4 -2 0 20

100

200

300

400

500

600

700

ln(zo)

Mean value ofz0=0.0645

Terrain Category II:kr=0.19

Terrain Category II:z0=0.05

Structures located in isolated hills orescarpents suffer more wind loads due totightening of the wind pressure (velocity) lines.

The wind loads are increased by applying aspecial topography coefficient ct

Consult:

Jackson and Hunt 1975

Taylor and Lee 1984

Structures on hills orescarpments

1 0.05

1 2 0.05 0.3

1 0.6 0.3

o

for

c s for

s for

Φ <

= + ⋅ ⋅ Φ < Φ ≤ + ⋅ Φ >

The reference wind speedis multiplied by theorography coefficient coqvb

Topography Coefficient ctWind speed increases when blowing over isolated hills and

escarpments (not undulating and mountainous regions)

Wind

Φ

Φ

Situation I

Situation IIWind

M is the slope of thehill/escarpments is the hill parameter to beobtained from graphs

0,05 0.3

0.30.3

e

L for

L Hfor

< Φ <

= Φ ≥

The hill factor s:1Situation I: cliffs and escarpments

H

L

Φ

Downwind slope <0,05

-x +x

-x

z

z/Le

2,0

1,5

1,0

0,5

0,2

0,1

0,60

0,80

up wind down wind x/Le

0,0 0,5 1,0 1,5 2,0-0,5-1,0-1,5 2,5

The effective slope length:

Building on an upwind slope:The factor s

0.05 0.3

0.30.3

e

L for

L Hfor

< Φ <

= Φ ≥

The hill factor s:2Situation II: hills and ridges

H

L

Φ Downwind slope >0,05

L

-x +x

z

x

Building on a hill

The effective slope length:

z/Le

2,0

1,5

1,0

0,5

0,2

0,1

0,60

0,8

up wind down wind x/Le

0,0 0,5 1,0 1,5 2,0-0,5-1,0-1,5 2,5

The factor s

c zk

z

zfor z z m

c z for z z

r

r

r

( )ln min

min min

=

≤ ≤

<

0

200

After applying the orography coefficient co andcorrecting for the appropriate height aboveground z m, the mean wind speed is given by

Modified basic wind speed10 min. average wind speed: Eurocode 1

vm(z)=cr(z)covb

10 min. averagewind speed m/s

VG

V(z)

where cr(z) is the roughnesscoefficient

EndThis concludes the first part