A comparison of cross-wind response evaluation for chimneys followingdifferent international codes

Celso J. Muñoz Black1, Hugo Hernández Barrios2, Alberto López López3

1Researcher, Gerencia de Ingeniería Civil, Instituto de Investigaciones Eléctricas, Cuernavaca,Morelos, México, cjmb@iie.org.mx

2Professor, Escuela de Ingeniería Civil, Universidad de Michoacán, Morelia, Michoacán,México, hugohbarrios@yahoo.com.mx

3Researcher, Gerencia de Ingeniería Civil, Instituto de Investigaciones Eléctricas, Cuernavaca,Morelos, México, alopezl@iie.org.mx

ABSTRACT

It is well known that at certain intensities of wind flow velocity acting on a structure, theresponse of the latter in the transverse direction of flow is induced by alternating vortices. Themost important parameters that contribute to the cross wind response are: the intensity ofturbulence, the duration of the gusts and the magnitude of the wind speed in the floe direction.Nevertheless, there is not exist an unified criteria to evaluate the cross-wind response andseveral methodologies has been proposed. In this paper, two of those methodologies for thecalculation of total displacements in the transverse direction of the wind flow, are analyzed anddescribed. Also, some procedures proposed in different international design codes to evaluatethe cross-wind response of cylindrical structures, are applied to the case of a chimney forcomparison. Finally, it is concluded that the method that best estimates the cross response due tovortices, with respect to the results reported in the literature and obtained experimentally in fullscale prototypes, is the one proposed in the Danish code.

INTRODUCTION

A high speed flow passing around a body with arbitrary shape (Figure 1) produces a wake vortexon the back with alternating movement from one side to another, a phenomenon known as wakevortices Bénard - von Karman who are credited with this observation.

In many structures not only the dynamic response in the along-wind direction isimportant, but also the response due to vortex shedding in the leeward side that produce cross-wind displacements and which must be considered in the total response. The longitudinalvibrations of the structure are caused by the natural turbulence of the wind, but the cross-windones are caused, besides the natural turbulence of the wind, by vortex shedding. Thisphenomenon can be presented in structures like lattice towers with a high solidity ratio

)5.0( , in rectangular prismatic structures like buildings (Figure 2) and slim bridges, butmainly it appears in structures with cylindrical cross-sectional section like cylindrical towers,poles, masts and chimneys.

A great number of failures in structures, mainly with circular cross-section [1], have beenreported in literature. In steel chimneys, the cross-wind effects produce important displacementsperpendicular to the wind direction and these at the time, increase the cross-wind baseoverturning moment in the foundations, and therefore in the joint stresses, as they are theanchors, nuts and the base plate. Because the vortex shedding produced fluctuating forces, this istranslated in a series of cycles of load that can generate a fatigue failure in the material. Someexamples are shown in Figure 3.

Figure 1: Von Karman Vortex, behind the Guadalupe Island (www.daac.gstc.nasa.gov)

(a) Tower with high solidity ratio (b) Slender building Figure 2: Structures susceptible to transversal response to wind flow

(a) Failure in Foundation (b) Failure by cycles of loadFigure 3: Failure in chimney foundation and their attachment due to cross-wind effects

(www.mecaconsulting.com)

METHODS TO EVALUATE CROSS-WIND RESPONSE

Two basic methodologies for the calculation of the total displacements in the transverse direction of thewind flow in civil structures due to the vortex shedding exist: the spectral method and the resonance

vortex shedding method. Before describing these two methodologies, it is necessary to remarksome concepts described in the following.

The vortex shedding frequency depends on the body shape, flow velocity, surfaceroughness and the flow turbulence. The frequency of vortex shedding is given by

bVSt

crit (1)

Where tS is the Strouhal number (dimensionless), V (m/s) is the mean wind velocityand b (m) is the characteristics width of the cross-section; for circular cylinders thecharacteristics width is the mean external diameter. The vortex shedding effect on a circularcylinder depends on the Reynolds number, which is given by:

bVRe (2)

Where V and b are defined as in Equation 1 and is the kinematic viscosity of the air,which is approximately 15 x 10-6 m2/s corresponding to a temperature of 20° centigrade. Thepath of vortex wake in the leeward is important, mainly those that occur regularly and movealternately from one side to another side (Figure 4).

Figure 4: Experimental evidence of the vortex shedding on a leeward side of circular section

When Reynolds number increases the flow shifts from laminar to transitory turbulence.Achenbach [2] identified four intervals depending on the flow behavior of the boundary layer,these are: subcritical, critical, supercritical and transcritical. The Reynolds number that defineseach scheme has not been determined accurately because it depends on various factors like theroughness of the cylinder, the intensity of streamlines, and aspect ratio of the cylinder (Figure 5).In Table 1, this intervals and their corresponding approximated Reynolds number are related.

For circular cross-section, Strouhal number varies with flow velocity and therefore withReynolds number. In Table 2, Strouhal–Reynolds number empirical relationships are shown [3],where 310 x6.1ln eRy .

In general, Strouhal number ranges from 0.18 to 0.20. In most practical cases that arise instructures, it may be considered a constant Strouhal number equal to 0.2.

In the following paragraphs the spectral and resonance vortex shedding methodologiesare described.

SPECTRALMETHODThe spectral method gives adequate results for relatively rigid structures such as concrete

silos, concrete chimneys with large diameter and low height [4]. In this case, the procedure of thevortex shedding response is based on the spectral modal analysis.

The generalized force on the structure, due to the vortex shedding, is:

Figure 5: Vortex variation on leeward side in function of flow velocity

Table 1: Flow regime type– Reynolds numberRegime Reynolds Number

Subcritical 200 < eR ≤ 1.5 x 105

Critical 1.5 x 105 < eR ≤ 4.0 x 105

Supercritical 4.0 x 105 < eR ≤ 1.0 x 107

Transcritical eR > 1.0 x 107

Table 2: Strouhal – Reynolds number empirical relationshipsStrouhal Number Reynolds Number

et RS 0.42139.0 325 < eR ≤ 1.6 x 103

3.29.0exp0261.01853.0 yS t 1.6 x 103 < eR ≤ 1.5 x 105

5-4 10 x5.110 x6.81848.0 et RS 1.5 x 105 < eR ≤ 3.4 x 105

h

v zztzFtQ0

d)(),()( (3)

Where the inertia force per unit length in cross-wind direction, ),( tzFv , is given by:

),()()(),( tzCzbzqtzF Lv (4)

)(z Modal shape, andh Height of the structure

In the Equation 4, )(zq is the velocity wind pressure, )(zb is the external diameter and),( tzCL is a dimensionless factor. If it is considered that (Hz)e is the natural modal frequency,

then the structure deflexion can be written as)()(),( taztzy (5)

Where )(ta is the modal displacement, which is considered as stochastic process with apower spectral density given by

222

2 )()()(

J

BCbqh

HSrefe

refLa (6)

Where2)(H is the structure transfer function, is the modal frequency and

2)(J isthe aerodynamic admittance given by:

212120 0 122 dd)z,(),(),(1)( zzzzgzg

hJ

h h (7)

Where ),( 21 zz is the correlation function and ),( zg is defined by:

2

)()(

21exp

)()()()()()(),(

zBz

zzBB

CbqzzCzbzqzg s

s

ref

refL

L

(8)

If the correlation length is small, the joint aerodynamic admittance can be approximatedby:

zzghb

Jhref d),(2)(

2

022

(9)

Where refb is the correlation length.The standard deviation of the displacement for a white noise excitation can be

approximated by:

0

d)()()( ay Szz (10)

Or

2)2(

d),(1)(

2)( 2

0

2

,4

refrefref

ve

h

e

ref

refL

refrefy

bqh

bzzgh

BC

mzz

(11)

Where e (Hz) is the fundamental frequency of the structure and v is a dimensionlessfactor given by:

zm

zzmh

h

refrefv d)()(1

0 2

2

(12)

Where )(zm is the mass per unit length.The total logarithmic decrement of damping, , is equal to the sum of the logarithmic

decrement of structural damping, s , and the logarithmic decrement of aerodynamic damping,

a , which is given by:

e

refaa m

bK

2

2

(13)

Where is the air density, refb is the reference width, aK is an aerodynamic parameter,which is positive if the aerodynamic damping is negative and em is the equivalent mass per unitlength given by:

h

h

ezz

zzzmm

0

2

0

2

d)(

d)()(

(14)

The inertia force per unit length, )(zFv , acting perpendicular to the wind direction, canbe obtained by:

)()2()()( 2 zkzmzF ypev (15)

Where pk is the peak factor and )(zy is the standard deviation of the displacementgiven in Equation 11.

RESONANT VORTEX SHEDDINGMETHODThe modal force for a dynamic system is the same that is obtained with the Spectral

Method (Equation 3). Nevertheless, the Resonant Vortex Shedding Method [5] establishes thatinertia force per unit length, in cross-wind direction, is:

)(2sin)()()(),( ztzczdzqtzF sFv (16)

Where )(zq and )(zd was already defined in Equation 4, )(zcF is a dimensionless formfactor that describes the amplitude, s is vortex shedding frequency and )(z is a factor equal to0 or 1, which determines if the load has the same sign that the modal shape at all pointsthroughout the structure length. For modes with constant sign, 0)( z . The form factor, )(zcF ,depends on vibration amplitude, air turbulence, Reynolds and Strouhal numbers, cross-sectionand on the aspect ratio. The maximum cross-wind deflection is given by:

see

e

mFY

2.max )2( (17)

In previous equation, s is the logarithmic decrement of structural damping, (Hz)e isnatural frequency of structure in cross-wind direction, em is equivalent mass per unit length and

eF is equivalent inertia force given by:

h

0

2

h

0.max

d)(

d)()()()(

zz

zzzczbzqF

Fe

(18)

Where .max is the maximum amplitude of the modal shape. The Equation 17 can bewritten as:

22

h

0ref.ref.

.maxref.

.max 11

d)(4

d)()()()(

tc

F

SSzz

zzzcb

zbq

zq

bY

(19)

Where cS and tS are the Scruton and Strouhal numbers, respectively. Scruton number isgiven by:

2ref.

2b

mS esc

(20)

And Strouhal number is given by:

ref.

ref.

VbS e

t

(21)

The vibration amplitude and the limited correlation of load, described by the correlationlength, shows that maximum load does no occur simultaneously along the structure.Ruscheweyh [5] takes in account this last effect by integrating the maximum amplitude of loadalong the entire length, L , so that 2L is the correlation length that is equal to the integral ofcorrelation function from zero to infinite. The maximum amplitude of the load is calculated inthe nodal points near the maximum deflection; this is for considering the aeroelastic effects andthe wind action that produces the maximum response. If variations, throughout structure height,of wind pressure and width of structure are negligible and, still more, assuming that the modalshape has the same sign, the integral of the numerator of Equation 18 can be approximated by:

L

plat.

h

F zzkczzzc d)(d)()(0

(22)

Where lat.c is the standard deviation of the load. It has seen that the maximum load canbe equal to the standard deviation multiplied by the peak factor. Ruscheweyh [5] considers thepeak factor by means integration of the modal shape over the effective correlation length, eL ,defined by:

L

pL

zzkzze

d)(d)( (23)

Thus, the effective correlation length incorporates the influences of the correlation ofload and peak factor. Substituting Equations 22 and 23 in Equation 19, is obtained

2.ref.

.max 11

tclatw SS

cKKb

Y (24)

Where K and wK are constants. For example, in the Euro Code [6], these constants aredefined, respectively, as:

h

h

zz

zzK

0

2

0.max

d)(4

d)(

(25)

h

Lw

zz

zzK e

0d)(

d)(

(26)

For modes that do not have constant sign, it is assumed that load acts in the samedirection as the modal deflection, so the definition of K and wK should be modifying asproposed in Reference [4].

SUMMARY OF PROPOSED EXPRESSIONS IN VARIOUS INTERNATIONAL CODES

Some of parameters involved in expressions proposed in the codes have already been defined, soonly the new ones will be defined in the following corresponding expressions.

CANADIAN CODE [7]

This code is based on spectral method. The critical wind velocity is given by:

t

ecrit S

bV . (27)

The Strouhal number has a value of approximately 1/6 for chimneys.If the motion is stable, i.e. 2

2 )/( Cmb es , the vortex excited amplitude maxY can beestimated using the formula:

hb

mbCmbC

bY

es

e

)/()/(

22

23max

(28)

Where s is the structural damping, 6.02 C and 0.13 C .

AUSTRALIAN/NEW ZEALAND CODE [8]

The critical wind velocity is given by:bV ecrit 5. For circular cross-sections (29)

The maximum cross-wind deflection is:

c

t

SbKY .max (30)

Where K is the factor for maximum tip deflection, taken as 0.5 for circularcross-sections, and tb is the average breadth of the top third of the structure.

The equivalent static wind force per unit length for chimneys is given as follows:

.max2 )()2()()( YzzmzF ev (31)

Where )(z is the first mode shape as a function of height z , normalized to unity at

hz , which shall be taken as 2hz .

EURO CODE [6]

Spectral MethodThe maximum cross-wind deflection is given by:

ypkY .max (32)

The standard deviation of the displacement related to the width b at the point with thelargest deflection can be calculated by:

2211

2

cccb

y

(33)

Where the constants 1c and 2c are calculated using Equations 33 and 34, respectively.

a

cL

KSac4

12

2

1 (34)

hb

SC

Ka

mbc

t

c

a

L

e4

222

2

(35)

Where cC is the aerodynamic constant dependent on cross-sectional shape, and for acircular cylinder also dependent on the Reynolds number. aK is the aerodynamic dampingparameter, which decreases with increasing turbulence intensity. For a turbulence intensity of0%, this constant may be taken as .max,aa KK , which gives conservative estimation of

displacements. La is the normalized limiting amplitude given the deflection of structures withvery low damping.

For a circular cylinder, the constants cC , .max,aK , and La are given in Table 3 [6].

Resonant Vortex Shedding MethodThe maximum cross-wind deflection is given by:

wlat

ct

KKKSS

bY mod2.max11

(36)

Where modK is the mode shape factor, latK is the lateral force coefficient and wK is theeffective correlation length factor.

The mode shape factor is calculated with:

m

j lj

m

j lj

j

j

z

zz

K

1

2

1

mod

d4

d)(

(37)

Where m is the number of antinodes of the vibrating structure in the considered modeshape, )(zj , and jl is the length of the structure between two nodes. If one considers only thefirst vibration mode for a cantilever structure, 1j , 1m and hl j . If it assumed

that 21 )( hzz , then the previous equation gives 13.0125mod K .The lateral force coefficient is shown in Table 4. 0,,1. 1

4.23 latLcritlat KVVK

Table 3: Constants for determination of the effect of vortex shedding

Constant 510eR 510 x5eR 610eR

cC 0.02 0.005 0.01

.max,aK 2 0.5 1

La 0.4 0.4 0.4

The constants cC and .max,aK are assumed to vary linearly with the logarithm of the

Reynolds number for 55 10 x510 eR and for 65 1010 x5 eR .

Table 4: Lateral force coefficient, latK , versus critical wind velocity ratio,1,1. Lcrit VV

latK Critical wind velocity ratio

0,latlat KK 83.01,1. Lcrit VV

0,,1. 14.23 latLcritlat KVVK 25.183.0

1,1. Lcrit VV

0latK1,1.25.1 Lcrit VV

In Table 4,1,1 LV is the mean wind velocity in the centre of the effective correlation

length, 1L , which is obtained from Table 5 as a function of vibration amplitude for first vibrationmode, )( 1sY . The basic value 0,latK of the lateral force coefficient is given Figure 6 for circularcylinders.

The effective correlation length factor, for first vibration mode of a cantilever structure, isgiven by:

2

1

1

1

1

1

1 /31/

1/

3

bLbLbLK w Where bh1 (38)

Table 5: Effective correlation length 1L as a function of vibration amplitude )( 1sYbsY )( 1 bL1

1.0 60.6 to0.1 bsY )(2.18.4 1

6.0 12

Figure 6: Basic value 0,latK of the lateral force coefficient versus Reynolds number )( .crite VR forcircular cylinders

DANISHCODE [9]

Danish Code establishes that the effect vortex shedding shall be investigated when ratioof the largest to smallest cross-wind dimension of the structure, both taken in the planeperpendicular to the wind flow, exceeds 6. The rules provided in this code are valid only instructures with slightly varying cross-wind dimensions. If the structure is not heavily damped,the oscillations will increase when the vortex shedding is in resonance with a mode vibratingperpendicular to the wind. This occurs at wind velocities close to the resonance wind velocity,which is calculated by Equation 27. For a circular cylinder 16.0tS for 6/ bh and

20.0tS for 15/ bh . For bh / in the range between 6 and 15, tS is assumed to vary as thelogarithm of bh / . For structures with varying cross-wind dimensions, values in Equation 27corresponding to the point with maximum movement are used.

The effect of resonant vortex shedding depends on the turbulence intensity of the wind.For 10 min. mean wind velocities larger than approximately 15 m/s the turbulence intensity ofthe wind is calculated by:

min

0

zzifln

1)(

1)(

zzzc

zIt

v (39a)

minmin zzif)()( zIzI vv (39b)

Where )(zct is the topography factor, which is taken as unity in this paper work. Theroughness length 0z and the minimum height minz are shown in Table 6.

The inertia force per unit length and the maximum deflection in cross-wind direction aregiven by Equations 31 and 32, respectively. The standard deviation can be calculates by meansof Equations 33-35, except that:

Table 6: Definition 0f terrain categories and terrain parameters

Terrain categoryRoughness length

0z (m)Minimum height

minz (m)I Sea with breaking waves, lakes and inlets with

at least 5 km fetch upwind and smooth, flatcountry without obstacles.

0.01 2

II Farmland with boundary hedges, occasionalsmall farm structures, houses or trees. 0.05 4

III Suburban or industrial areas, rows of boundaryhedges. 0.3 8

IV Densely built-up urban areas with buildings ofaverage height above 15 m. 1.0 16

)(max, vvaa IhKK (40)

Where:

25.0)(0if)(31)( zIzIIh vvvv (41a)

25.0)(if25.0)( zIIh vvv (41b)

The turbulence intensity )(zI v is determined at the height where the movement of thestructure is at a maximum.

When the standard deviation of the deflection is less than approximately 2% of the cross-wind dimension, the peak factor can be calculated with:

)600ln(2577.0)600ln(2

eepk

(42)

For standard deviation exceeding approximately 20% of the diameter, the peak factor canbe taken as 2 .

CASE OF STUDY: CHIMNEY STRUCTURE

To compare the results obtained by applying expressions proposed in different internationaldesign codes for cross-wind response of cylindrical structures, is proposed as a case study,calculate the maximum displacement in the cross-wind direction due to the effects of vortexshedding in a steel chimney with constant thickness. It is assumed that foundation is quit rigid,which mean that chimney can be considered as cantilever beam without soil-structureinteraction. The design data are:

External diameter, m3bTotal height, m60hShaft mass, kg/m740smCoating mass, kg/m260cmTotal equivalent mass, kg/m000,1 cse mmmFrequency of cross-wind first mode, Hz73.01 Frequency of cross-wind second mode, Hz57.42 Mean wind velocity, m/s36V

Logarithmic decrement of structural damping, 025.0sStructural damping, 004.02 ss

Strouhal number, 18.0tSAir density, 3kg/m25.1Kinematic viscosity of the air, /sm10 x15 26c

EURO CODE [6]

Resonant Vortex Shedding Method

m/s17.120.18

)Hz73.0()m3(1, critV ; m/s17.76

0.18)Hz57.4()m3(

2, critV Equation 27

m/s45m/s)36(25.125.1m/s17.121, VVcrit The effect vortex shedding need beinvestigated

m/s45m/s17.762, critV The effect vortex shedding need not be investigated

0,1, 83.034.0

m/s36m/s17.12

latlatcrit KKV

V Table 4

2.010 x434.2/sm10 x15

)m/s17.12)(m3()( 0,6

26-1, latlatcrite KKVR Table 3

13.0mod K Equation 37

44.4)m3()kg/m25.1(

)kg/m000,1()025.0(223 cS Equation 20

m542.0)2.0()13.0(44.41

)18.0(1)m3( 2max ww KKY

Equation 36

800,10601

2020m3m60

2111

1LLLKbh w Equation 38

As 1L depends onf bYmax , one must proceed in an iterative way with equations thatincludes this values. Thus, the maximum cross-wind deflection is m366.0max Y .

Spectral Method

Because 4.0and1,01.01010 x434.2 max,66 Laace aKKCR

0517.0)1(4

44.412

)4.0( 2

1

c Equation 34

6-4

2223

2 10 x57.8m)60(

m)3()18.0()01.0(

1)4.0(

kg/m000,1)m3()kg/m25.1(

c Equation 35

1035.010 x57.8)0517.0(0517.0 6-22

b

yEquation 33

m965.03217.0

y

y

b

854.1)1(4

44.475.0arctan2.1124

75.0arctan2.112

a

cp K

Sk

m789.1)m965.0()854.1(max Y Equation 32

DANISHCODE [9]

For calculated the turbulence intensity we assumed that the terrain category is II. Thus,

m4m60141.0

m05.0m60ln

111)(

zI v Equation 39a

25.0141.00577.0)141,0(31)( vv Ih Equation 41a

577.0)577.0()1( aK Equation 40

031.0)577.0(4

44.412

)4.0( 2

1

c Equation 34

6-4

2223

2 10 x859.14m)60(

m)3()18.0()01.0(

577.0)4.0(

kg/m000,1)m3()kg/m25.1(

c Equation 35

0622.010 x859.14)031.0(031.0 6-22

b

yEquation 33

m748.02494.0

y

y

b

414.12 pk ( m6.0)m3(2.0m748.0 ) Equation 42

m058.1m)748.0()414.1(max ypkY Equation 32

AUSTRALIAN/NEW ZEELAND CODE [8]

m/s65.6)m3()Hz73.0(5. critV Equation 29

m338.044.4

)m3(5.0max Y Equation 30

CANADIAN CODE [7]

0039.001125.060.0m/kg000,1

m3m/kg25.1 23

s The motion is unstable

The Canadian Code and ISO 4354:1997(E) [10] point out that in this case, largeamplitudes up to value of b may results. Therefore, maxY could outcome 3 m.

A summary of the total displacements due to cross-wind response for a chimney withtypical characteristics, applying the wind design codes previously described, is shown in Table 7.In this table, it can be seen that the results obtained applying the Canadian and theISO 4356:1997(E) Codes, can be very conservative. Likewise, the spectral method proposed byEuro Code ignored the turbulence intensity influence; therefore the results obtained areconservative as well.

On the other hand, the Danish code certainly considers the turbulence intensity, reasonwhy the displacements acquired with this code are more reliable. About this, spite of it doesn’tappear in this paperwork, the results that have been obtained in other research works [5, 11-12]demonstrate that the Danish code provides more accurate values.

Table 7: Results obtained for the case of study

Code (m)maxYAustralian/New Zealand 0.338Canadian and ISO 4356:1997(E) ≤ 3EuroResonant vortex shedding MethodSpectral Method

0.3661.789

DanishTerrain category 2 1.058

CONCLUSIONS

In this work the methodology proposed by different wind design codes has been analyzed tocalculate the maximum response that can take place in a cylindrical structure in cross winddirection, due to the vortex shedding. Considering the turbulence intensity and, for hence, theroughness place in where the structure will be built; the results are more congruent between thedata obtained with experimental tests and the data resultant from the environment vibration testsmade on steel chimneys. In conclusion, the methodology proposed by the Danish code for vortexshedding effects is the more accurate.

REFERENCES

[1] Tranvik P. and Goran A., Dynamic behaviour under wind loading of a 90 m steel chimney, AlstomPower Sweden AB, Vaxjo, Report S-0141, 2002.

[2] Achenbach E., Influence of surface roughness on the cross-flow around a circular cylinder, J. FluidMechanics, 1971, 46, p. 321-335.

[3] Norberg C., Fluctuating lift on a circular cylinder: Review and new measurements, J. Fluids Struct.,2003, 17, 57.

[4] Dyrbye G. and Hansen S., Wind load on structures, John Wiley and Sons, ISBN 0-471-9565-1, 1997.

[5] Ruscheweyh H. and Sedlacek G., Crosswind vibrations of steel stacks-critical comparison betweensome recently proposed codes, Journal of Wind Engineering and Industrial Aerodynamics, 1988, 30,p. 173-183

[6] BS EN 1991-1-4-4:2005, Euro Code 1: Actions on structures, Part 1-4: General actions-Windactions, British Standard, 2005.

[7] NRCC 48192, National Research Council Canada, User´s Gudie-NBC 2005 Structural Commentaries(Part 4 of Division B), ISBN 0-660-19506-2, 1993.

[8] AS/NZS 1170.2:2002, Australian/Neo Zealand Standard, Structural design actions, Part 2: WindActions, 2005.

[9] DS410 E:2004, Code of Practice for Loads for the Design of Structures, Danish StandardAssociation, 2004.

[10] ISO 4354:1997(E), International Standard, Wind Actions on Structures, 1997.

[11] Ciesielski R., Gaczek M. and Kawecki J., Observation results of cross-wind response of towers andsteel chimneys, Journal of Wind Engineering and Industrial Aerodynamic, 1992, 41-44, p. 2205-2211.

[12] Vickery B. J. and Basu R. I., Simplified approaches to the evaluation of the across-wind response ofchimneys, Journal of Wind Engineering and Industrial Aerodynamics, 1983, Vol. 14, p. 153-166.