Kamal Handa, 2006-04-04
EUROPEAN STANDARD FOR WIND LOADS
(Eurocode EN 1991-1-4 WIND ACTIONS)
1
General
The Eurocode for wind loads on building structures, EN 1991-1-2, has been issued by CEN and is
expected to replace the national documents in the very near future.
Application of the standard is rather complicated as each member country of the European Union is
permitted to include national annexes as a supplementary document. The national annexes can include
most of the design rules for calculating wind actions implying unique standard for each country.
A comparison of salient features of the Eurocode with the national code BSV 97 is of considerable
interest, as it will affect the future design rules.
This note discusses the proposed models in Eurocode for wind induced loads caused by gusts (along
wind) and vortex induced forces (cross wind) with reference to BSV 97. It does not dwell on the
whole document that consists of 146 pages
2.
Calculation of gust loads in along wind direction
2.1
Background
Ymax
σp
h
Mean Velocity
Profile
Mean Pressure
Profile
Figure 1:
Structure
Response
Deflection
Loading pattern on a structure
Depending on the height and slenderness of the building, wind forces can cause a structure to oscillate
at its natural frequency. Level of turbulence in the flow contributes significantly to the dynamic
response of the structure.
As the flow is random in character, maximum deflection at the top of the structure due to wind forces
is obtained in terms of mean, standard deviation and a peak factor given by:
X max ( h ) = xm ( h ) + k pσ x ( h )
(1)
The standard deviation σx is obtained from a quasi-static part (B2) and a dynamic part (resonant part
R2) as shown in figure 2. kp is the peak factor relating the mean and the standard deviation of the
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2
response. B2 is the background excitation caused by the wind gusts acting as a static force on the
structure. The resonant part R2 represents the interaction between the dynamic properties of the
structure and the wind gusts.
Background excitation
B2
Response
Spectrum
Resonance part
R2
n1
Figure 2:
frequency(Hz)
Response spectrum
Expressing equation (1) in terms of wind action, it can be shown that the maximum load is given by:
Fw ( z ) = f m ( z ).⎡1 + 2.k p .I v ( z ) B 2 + R 2 ⎤
⎢⎣
⎥⎦
1
f m ( z ) = ρ .v m2 ( z ).c f .Aref
2
(2)
(3)
Equations 2 and 3 form basis of the Eurocode and the Swedish code (BSV 97).
2.2
Eurocode EN 1991-1-4 WIND ACTONS
According to the Eurocode, wind load on a cantilever structure vibrating in the first mode is given by:
Fw = c s c d .c f .q p ( z e ). Aref
(4)
cs is the size factor defined as:
cs =
1 + 7.I v ( z s ). B 2
1 + 7.I v ( z s )
(5)
cd is dynamic factor consisting of quasi-static and resonant parts defined as:
cd =
1 + 2.k p .I v ( z s ). B 2 + R 2
(6)
1 + 7.I v ( z s ). B 2
For quasi-static wind loads, the resonant part R2 = 0 and equation (6) can be written as:
cd =
1 + 2.k p .I v ( z s ). B 2
(7)
1 + 7.I v ( z s ). B 2
b
d
h
zs=0.6.h
zs
Figure 3:
Reference height zs for calculating cs cd
3
Maximum velocity pressure qp(ze) is expressed in terms of mean velocity, surface roughness and
intensity of turbulence and is given by:
q p ( z e ) = [1 + 7.I v ( z e )].
1
ρ .v m2 ( z e ) = q b .c r ( z e ) 2 [1 + 7.I v ( z e )]
2
The exposure factor cr is equivalent to
(8)
C exp in BSV 97.
The intensity of turbulence Iv is defined as:
I v (ze ) =
1
ln ( z e / z 0 )
(9)
The exposure factor cr is governed by the surface roughness of the site and is defined as:
⎡z ⎤
c r ( z e ) = k r .ln ⎢ e ⎥
⎣ z0 ⎦
(10)
qb is the basic velocity pressure averaged over 10 minutes at a reference height of 10 meters for open
site (z0 = 0.05) conditions.
qb =
1
.ρ .v b2
2
(11)
h > 2b
b
cr(z)=cr(h)
ze=h
h
cr(z)=cr(zstrip)
ze=zstrip
cr(z)=cr(b)
ze=b
Velocity-pressure
distribution parameter
Structure
Figure 4:
Reference height ze depending on h and b and velocity pressure parameter.
Substitution of equations 5 to 11 into 4 gives:
∑
Fw = ⎡1 + 2.k p .I v ( z s ). B 2 + R 2 ⎤ . c f .q b .c r ( z e ) 2 .b
⎥⎦
⎢⎣
strip
2.3
(12)
Swedish code BSV 97
BSV 97 also employs equation (2), but the steps needed in calculating the wind loads are few and
straightforward. Rewriting equation (2) and defining dynamic factor:
Define
Fw ( z ) = f m ( z ).⎡1 + 2.k p .I v ( z ) B 2 + R 2 ⎤
⎥⎦
⎢⎣
1
f m ( z ) = ρ .v m2 ( z ).c f .Aref = q b .c r2 ( z ).c f Aref
2
C dyn = ⎡1 + 2.k p .I v ( h ). B 2 + R 2 ⎤
⎥⎦
⎢⎣
(2)
(3)
(13)
4
Substitution of equation (13) in equation (2) gives:
Fw ( z ) = C dyn .c f .q b .c r ( z ) 2 .Aref
(14)
kp is the peak factor and is approximately equal to 3 for overall loading on static structures.
Equation (13) for non-vibrating structures can now be written as:
Cdyn = 1 + 6 Iv(h)
(15)
z0 represents the surface roughness of the terrain and is given in table 1.
0
1
2
3
4
Site roughness
kr
z0
Coastal areas
Lakes
Open
Built up areas
City centres
0.16
0.17
0.19
0.22
0.24
0.003
0.01
0.05
0.3
1.0
zmin
Eurocode
1
1
2
5
10
zmin
BSV 97
2
4
8
16
Table 1. Parameters for different surface roughness
2.4
Comments on Eurocode
Main difference between Eurocode and BSV 97 lies in the definition of dynamic factor cs cd and the
velocity pressure. The Eurocode multiplies the exposure factor with intensity of turbulence to obtain
the maximum velocity-pressure, while the BSV 97 follows the mean velocity profile and any
contribution from gusts is included in the dynamic factor Cdyn. The author believes that the model
included in the BSV 97 represents interaction of wind and structure in a systematic manner.
Some critical remarks on the practical application of Eurocode are in order.
1.
The size factor cs is approximately equal to 1 except for very large structures. Substitution of
B2 = 1.0 will not result in any significant increase in the loads.
2.
Factor 7 in equations (5), (6) and (8) is based on kp = 3.5 and is relevant for 1-hour averaging
period for wind speeds. For 10-minutes mean wind speeds with 2-3 seconds gusts, a factor of 6 is
more appropriate in the calculation of structural wind loads. It can be further shown that a factor 8
should be used in the design of fixings.
3.
Substituting value of kp = 3 and replacing factor 7 by 6 in equation (7) will result in cd = 1.
4.
For a tall building (h>b), product of cscd is calculated at a height of zs= 0.6 h, while the
velocity-pressure distribution is approximated by dividing into strips as shown in figure 4. It is
unnecessary to complicate the design by employing different heights for calculation of dynamic factor
and velocity-pressure profiles.
2.5
Calculation of B2 and R2
Procedure to calculate quasi static part (B2) and resonant part (R2) of the response are taken from
BSV 97 and are included in the NAD. This replaces the annex B and C in the Eurocode.
1.
2.
Background Excitation
⎡
⎛ h ⎞ ⎧
⎟ + 1−
B 2 = exp ⎢− 0.05⎜
⎜ href ⎟ ⎨⎩
⎢
⎝
⎠
⎣
Resonance Part of Response
2πFφ b φ h
R2 =
δs +δa
⎛ h
b ⎫⎧⎪
⎬⎨0.04 + 0.01⎜⎜
h ⎭⎪
⎝ href
⎩
⎞⎫⎪⎤
⎟ ⎬⎥
⎟ ⎪⎥
⎠ ⎭⎦
(16)
(17)
5
3.
Peak Factor
k p = 2ln( 600ν ) +
4.
0.58
2ln( 600ν )
(18)
Apparent Frequency
R2
ν = n 1,x
B2 + R2
Intensity of Turbulence
1
Iv =
ln( h / z 0 )
Wind Energy Spectrum
4C
F=
5
5.
6.
[1 + 70.8(X ) ]
2 6
C=
8.
9.
(20)
(21)
Lc n 1 ,x
Lc ≈ 150 m
vm ( h )
Size effect (Breadth of the building)
1
φb =
⎡ 3.2n1,x b ⎤
1+ ⎢
⎥
⎣ vm ( h ) ⎦
Size Effect (Height of the Building)
1
φh =
⎡ 2 n 1,x h ⎤
1+ ⎢
⎥
⎣ vm ( h )⎦
Aerodynamic Damping
ρ c f b vm ( h )
δa =
n 1,x M e
7.
(19)
(22)
(23)
(24)
h
Me =
∫ m( z )φ ( z )
2
dz
0
(25)
h
∫ φ( z )
2
dz
0
Me is the equivalent mass.
m - mass per unit length (kg/m)
φ(z) = mode shape
2.6
Accelerations for serviceability assessments of a vertical structure
The maximum acceleration for a cantilever structure vibrating in the first mode is given by:
(26)
X&& max ( z ) = k p σ &x& ( z )
σ &x& ( z ) is the standard deviation of the acceleration and is expressed as:
σ &x& ( z ) = ( 2πn1,x ) 2 σ x ( z )
(27)
1
σx (z) is the standard deviation of displacement and is defined as:
σ x ( z ) 2 = σ x ,B ( z ) 2 + σ x ,R ( z ) 2
(28)
σx,B (z) and σx,R (z) are the quasi-static and resonant part of the response respectively as shown in
figure 1. The acceleration is caused by the resonant part of the response and the quasi-static part is
2
2
6
omitted from the calculations. The standard deviation of displacement due to resonant part of the
response is obtained from equation (2) as:
σ B ( z ) = 0.0
(29)
σ R ( z ) = 2I v R xm ( z )
The resonant part R is given by equation (17).
xm (z) is the mean deflection given by:
⎡
⎤
P1
⎥
xm ( z ) = φ1 ( z ) γ 1 ⎢
⎢⎣ (2πn1,x )2 M e ⎥⎦
(30)
P1 is the generalised force defined as:
h
∫c (z)
r
P1 = q b b c f
2
φ 1 ( z )dz
0
(31)
h
∫ φ ( z )dz
1
0
γ1 is a constant defined as:
h
∫ φ ( z )dz
1
γ1 =
0
h
(32)
∫φ ( z )
1
2
dz
0
φ1(z) - first mode shape for a cantilever.
For serviceability, the wind speeds are calculated for a probability of occurance of 1 in Ta years and is
given by:
⎧⎪
⎛
⎛
1
vTa = 0.75 v 50 ⎨1 − 0.2ln⎜ − ln⎜⎜ 1 −
⎜
T
⎪⎩
a
⎝
⎝
⎞ ⎞⎫⎪
⎟ ⎟⎬
⎟⎟
⎠ ⎠⎪⎭
(33)
where Ta is the number of years. For 10 years period, the wind speeds are given by:
vTa = 0.9 v50
(34)
v50 is the reference wind velocity with a probability of occurrence of 1 in 50 years.
3.
Vortex shedding
3.1
Reynolds number
Relationship between the Reynolds number, velocity and the size of the structure is given by:
Re =
v cr b
14.5×!0 −6
(35)
Range of Reynolds number defines if separation of vortices is regular, periodic or random in character.
For circular cylinders, there are three ranges of Reynolds number defined as:
Re < 3 × 105
(Sub-critical range)
(Super-critical range)
3 × 105 < Re < .3.5 ×106
(Trans-critical range)
Re > 3.5 × 106
7
In the sub critical and trans critical ranges the separation of vortices from the sides of a cylinder is
regular and a simple harmonic model can be used to calculate the forces by examining the response at
the natural frequency of the structure. In the super critical range, the separation of vortices is random
in character and a model based on spectral method is employed. However, if the amplitude of
vibration is large, then the harmonic model should be used as the response will be on safe side.
3.2
Strouhal Number and lift coefficients
The velocity at which the vortices are shed from the sides of a cylinder is called the critical velocity vcr
and is obtained from Strouhal number defined as:
St =
n st b
v cr
(36)
Where nst represents the frequency of vortex shedding. Resonance occurs when the natural and the
vortex shedding frequencies are close to one another.
3.3
Strouhal number for sharp-edged bodies
0,16
0,12
St
0,08
0,04
0
1
2
Figure: 5
3.4
3
4
5
6
7
8
9
10
d/b
Strouhal number for rectangular section
Lift coefficient for circular cylinders
0,8
0,7
0,6
CLat,0
0,5
0,4
0,3
0,2
0,1
0
3.5
4
10
1
5
10
10
6
10
100
Figure: 6
Lift coefficient and Reynolds number
7
10
1000
Re
Response calculations
Model for calculation of vortex-induced forces in Annex E of Eurocode is replaced by the method
given in BSV 97. Reason for non-acceptance of Eurocode is that the approach 1 can result in
8
underestimation of deformations in steel chimneys. The second method in annex E of Eurocode gives
widely different deformations depending on the intensity of turbulence. The method given in BSV 97
has been verified against full-scale behaviour of steel chimneys and gives reliable estimates of
deformation.
The method for calculation of vortex-induced forces given in BSV 97 is limited to slenderness ratio
h / b = 30. For h/b> 30, a detailed analysis is recommended. Vortex induced loading is given by:
⎛π ⎞
wek = p m (h) ⎜⎜ ⎟⎟
(37)
⎝δs ⎠
pm(h) = qcr(h) b Clat,0
(38)
qcr (h) = 0.5 ρ vcr (h)2
(39)
Maximum equivalent design load
wd = γw wek
γw is a safety factor
(40)
Maximum design displacement at the top is:
γ
⎛5⎞
y d = 2 w ⎜ ⎟ wek
ω Me ⎝3⎠
3.6
(41)
Check for aerodynamic stability
If the maximum reduced deflection at the top is less than 0.06, the chimney is considered to be
aerodynamically stable.
yd ( h )
≤ 0.06
b
3.7
(42)
Suppression of large displacement by use of helical strakes
For a single chimney with reduced amplitude larger than (
yd ( h )
≥ 0.06 ),mounting of helical strakes
b
reduce the level of oscillations to acceptable levels. The wind induced load becomes:
⎛ π
wek = p m (h) ⎜⎜
⎝ δ mech
h
⎤
⎞⎡
⎟ ⎢1 − spiral ⎥
⎟
h ⎦
⎠⎣
3
(43)
For chimneys in a group or in the vicinity of each other, the helical strakes are not very effective in
reducing large amplitudes. Liquid or mass dampers can be used in such cases.
3
⎡ h spiral ⎤
⎢1 −
⎥ wek
h ⎥⎦
⎣⎢
hspiral
ymax
wek
h
Plain chimney
Figure: 7:
h
Chimney with helical strakes
Load distribution on Chimneys with and without helical strakes
9
3.8
Number of load cycles caused by vortex induced forces
The total number of cycles N occurring during a period of T years is obtained from
N = 365 × 24 × 60 × 60 × T× n1 × P(v) (c/s)
(44)
T - life time of the structure
(years)
P(v) - probability of wind velocity occurring during the interval vcr and ε vcr.
Expression for P(v) is assumed to be given by Rayleigh distribution and can be written as:
⎡ ⎡ ⎧ v ⎫2 ⎤
⎡ ⎧ε v ⎫ 2 ⎤⎤
P( v ) = ⎢exp ⎢− ⎨ cr ⎬ ⎥ − exp ⎢− ⎨ cr ⎬ ⎥ ⎥
⎢ ⎢ ⎩ v( z ) ⎭ ⎥
⎢ ⎩ v( z ) ⎭ ⎥ ⎥
⎦⎦
⎣
⎦
⎣ ⎣
(45)
Coefficient ε van be taken as 1.25, provided the minimum interval between vcr and ε vcr is 2 m / s.
v(z) presents the yearly mean wind velocity at a height z. for a given roughness and is obtained from
equation (46)
v( z ) = 5.5.c r ( z )
4.
(46)
General Remarks
The note deals with the salient features of the Eurocode without going into details of the various
clauses. Two main items from the Eurocode are discussed in section 2 and 3, namely,
1.
2.
Derivation of gust loads
Derivation of vortex induced forces.
Examination of the two proposed methods for gust loading in Eurocode has shown that it is
unnecessarily complicated and includes data from different sources without presenting any
background documentation. This is particularly true of the peak values recommended in the gust
loading equations. In view of this, it is suggested that the method used in the BSV 97 should be
adopted for calculating gust-induced loads.
Derivation of vortex-induced forces in the Eurocode is based on two widely different approaches.
Application of the two methods to actual design of steel chimneys has shown that they can
underestimate the deformations of the structure. In view of this, it is recommended that the method
given in BSV 97 should be used in the design of steel chimneys.
The Eurocode does not specify any design rules for steel chimneys subjected to ovalling oscillations.
However, BSV 97 deals with this topic and it is recommended that it should be included in the NAD
document.
It should be noted that Eurocode is applicable to bridges with a span of less than 200 meters subjected
to quasi-static wind loads. This implies that pedestrian timber bridges or light bridges under 200meters spans subjected to oscillations need further dynamic analysis.
It is recommended that a national annex dealing with along wind and vortex induced oscillations of
light and pedestrian timber bridges should be formulated and included in the Eurocode EN 1991-1-4
WIND ACTIONS.
10
5.
Aref
b
h
href
B2
cf
cs
cd
cdyn
cr(ze)
Clat,0
Fw(z),
fm(z)
h
Iv(z)
kp
kr
m
Me
n1,x, n1,y
qp(ze)
P1
Re
R2
St
vb
vm(z)
vTa
xm(h)
Xmax(h)
z0
ze
zs
σx(h)
ρ
φ(z)
ν
δs,a δ
γw
Symbols
Reference area
Breadth
height
Reference height (10 meters)
Quasi-static excitation
Aerodynamic force factor
Size factor
Dynamic factor
Dynamic gust factor (BSV 97)
Exposure factor
Aerodynamic lift coefficient
Maximum wind force.
Mean wind force
Height
Intensity of turbulence
Peak factor
Velocity constant
Mass (kg/m)
Equivalent mass (Kg/m)
First natural frequency (c/s)
Maximum Velocity-Pressure
Generalised load
Reynolds number
Resonant part of the response
Strouhal number
10 minutes mean reference velocity at 10 m height (Open site)
Mean velocity
Mean velocity averaged over Ta years
Mean deflection
Maximum deflection
Surface roughness height
Height parameter
0.6 h
Standard deviation of the response
Air density
Mode shape
Apparent frequency
Structural and air logarithmic damping
Safety factor