Wind Effects on Structures 1 st Part Wind effects and distribution of wind speeds. Analysis of turbulent flow in a boundary layer March 2007 Júlíus Sólnes, Professor of civil and environmental engineering, University of Iceland in Reykjavik
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