Fatigue prognosis of slender chimneys considering long...

ABSTRACT: Industrial chimneys are sensitive to wind actions because of their high slenderness. For some notch connection

cases, the random load due to the stochastic wind action can produce fatigue problems. To consider the buffeting fatigue, the

Eurocode 1991-1-4 presents in Annex B a curve for the calculation of the load cycles that exceed a given value ∆S during a

period of 50 years. Long-term wind data recorded in northern Germany shows the appearance of six different wind profile

shapes. Using this information, an extensive Monte Carlo simulation has been carried out to calculate the influence of the wind

statistic on the fatigue life of industrial chimneys. The spectral information of the response has been introduced in Dirlik’s

method to obtain the probability distribution of the stress amplitudes. The damage has been compared with the formulation

presents in Eurocode 1 and the results show a clear difference between both approaches. The standard yields highly conservative

damage due to its large safety grade.

KEY WORDS: Wind speed statistics; Buffeting loading; Monte Carlo simulation; Dirlik method; Fatigue prognosis.

1 INTRODUCTION

Slender structures such as industrial chimneys are very

sensitive to wind actions. The characteristics of the structural

response depend on the dynamic structural amplification, thus

the frequency and damping, as well as the incident gusty wind

field. The statistical wind data given in the Eurocode 1 [4] and

in its corresponding national annexes are indispensable for all

wind design projects. Clobes and Willecke [1] analysed full

scale, long term data of the natural wind field measured on a

344 m high guyed mast located in northern Germany. They

showed that mean wind speed profiles in moderate wind speed

conditions differ from the commonly used logarithmic or

power law wind profiles presented in the standards. Thus, six

mean wind speed profile shapes were identified. The use of

those realistic wind profiles for an analysis of vortex

excitation of steel chimneys leads to a significant reduction in

overestimating fatigue damages compared to the currently

recommended in the Eurocode [1].

Starting from the conclusions made in [1] a refined fatigue life

prognosis of industrial chimneys in case wind buffeting has

been developed. Long-term full scale measurements of wind

speed have been used to obtain a wind speed and wind profile

shape-dependent standard deviation of σu(z). A correct

identification of this parameter is fundamental due to its high

importance on fatigue process. Using a Monte-Carlo

simulation, the fatigue life of steel chimneys to frequent gust

excitation is analysed. Synthetic wind profiles for wind speed

and turbulence intensity are generated using a robust

statistical model, based on the results presented in [1].

For each generated profile, the buffeting response of a 150 m

high steel chimney is individually calculated. In order to

evaluate the material fatigue, Dirlik’s formula [15] is used to

determine a stress cycle count of the chimney from the

spectral information of the bending moment. This information

is used to build load collectives during the design life of the

chimney. On the contrary to the Eurocode, load collectives

from site-dependent wind parameters and/or structural

characteristics of the structure are calculated. Their

applications to the design of this kind of structures show a

lower fatigue damage compared to the procedure of Eurocode.

2 CLASSIFICATION OF MEASUREMENTS

2.1 Classification of the mean wind speed

Since 1989 the Institute of Steel Structures of the Technische

Universität Braunschweig operates a wind monitoring system

located on the 344 m guyed mast Gartow II (northern

Germany). There, the mean wind speed ( )U z , the standard

deviation of the wind turbulence ( )u zσ , temperatures and the

wind directions have been measured over the entire height of

the mast [1],[2]. During this period low, moderate and high

winds have been measured. A neural network technique has

been used to classify the measured wind data in six different

wind shapes as in Figure 1 [2]. The results shows that the

power law class profile, which is used for the along-wind

buffeting, is the most frequent wind profile shape with a

frequency of occurrence of 55.9%. The constant shape has an

occurrence ratio of 29.9%, the linear 9% and the jets and the

sinus classes less than 3% [3].

For each profile class c a statistical description based on the

corresponding selected profiles has been derived. The Weibull

distributions of the mean wind speed have been calculated for

the different profile classes and different heights. Large values

of the shape parameter k ( 3.5k ≈ ) are obtained and therefore,

the probability distribution tends to a symmetrical shape over

Fatigue prognosis of slender chimneys considering long-term wind profile statistics

Hodei Aizpurua Aldasoro, Mathias Clobes, Klaus Thiele

Institute of Steel Structures, Technische Universität Braunschweig, Beethovenstrasse 51, 38106 Braunschweig , Germany

email: [email protected], [email protected], [email protected]

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014Porto, Portugal, 30 June - 2 July 2014

A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)ISSN: 2311-9020; ISBN: 978-972-752-165-4

1399

the mode being also well approximated by a Gauss

distribution.

Figure 1. Characteristic shapes of profile classes [1]

This allows for a description of the statistical characteristics in

terms of a mean vector Uµ and a covariance matrix

UCOV over the height only [2].

2.2 Classification of the turbulence intensity

In addition to the statistical classification of the mean wind

speed profiles, a definition of the wind turbulence is needed

for a refined analysis of structures under buffeting wind

loading. The vibrations induced by the gusty wind can be

amplified due the coincidence of the resonant frequency range

of the structure with the energy provided by the wind in the

frequency domain. The energy of the turbulent wind is a

broad-band process in comparison with the resonant response

of the most structures. It is defined in the spectral domain by

the power spectral density function uuS , which depends on

the standard deviation of the incoming turbulence ( )u zσ . The

turbulence grade is mathematically described by the

turbulence intensity ( )uI z , which is directly related to the

mean wind speed ( )U z and the standard deviation of the wind

speed fluctuations ( )u zσ :

( )

( )( )

uu

zI z

U z=

σ (1)

In the design codes a constant value of uσ over the height is

accepted. Its value depends on the terrain category and wind

zone. Using the classification of the profile classes c made in

[3], uσ is also classified inside the classes in terms of the

mean wind speed value.

Firstly, for each wind profile class c, the total measured

profiles ( )cU z are separated in wind speed ranges of U∆ = 5

m/s at 156 m height. Secondly, the standard deviation profiles

( )cu zσ , associated to the wind speed profiles classified on

each range iU∆ , are selected. Thirdly, from this set of

profiles ( )i

cu U

z ∆σ , the mean standard deviation value

( )i

cu U

z ∆σ over the height z is calculated. The results show

that the variable ( )i

cu U

z ∆σ does not vary over the height.

Finally, due to this argument it is possible to calculate cuσ

µ ,

defined as a mean standard deviation over the height for each

class c and wind speed range iU∆ .

Figure 2 shows the results of the evaluation of the wind

turbulence. Each profile class c is plotted in a different colour.

The rounded points inserted on the lines coincide with the

calculated mean value of cuσ

µ for the different wind speed

ranges. The lines between points have been plotted assuming

a linear relationship.

The duration in years of the long-term measurements used for

this work (about 20 years) has been not sufficient long to

measure extreme wind speeds. Therefore, the dotted lines

symbolize the supposed performance of uσ for high wind

speed ranges. The performance of the jet profiles is quite

strange reducing the standard deviation uσ even if

U increases. The power law profile, equivalent to the

logarithmic profile given in the design standards, tends to

confirm an equivalent value of uσ as in Eurocode for the

place of Gartow. The mast is located in close proximity to the

Elbe River and the surrounding area is covered with low

vegetation. Results provided in [2] suggest a direction-

dependent terrain category II or III.

Figure 2. Tendency of cuσ

µ with the mean wind speed at 156

m for the six different profile classes [9]

The blue circles in Figure 3 depict the calculated turbulence

intensity profiles ( )uI z following equation (1) for different

wind speed ranges. The green lines show the turbulence

intensity profiles for a terrain category II and III in EN 1991-

1-4 [4]. Except in cases of extreme low velocities, the values

of ( )uI z obtained in Gartow are well comparable with a

turbulence intensity profile for terrain category between II and

III. This fact was also observed by Willecke [3] without

differentiation of mean wind speed ranges.

Although the values of uσ given in the EC1 correspond to a

50 years wind situation, the tendency for lower velocities is

also well comparable due to the concordance of the turbulence

intensity profile ( )uI z . In case of wind profile classes

different than the power law class, these conclusions could be

not completely assumed.

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

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Figure 3. Discretization of the turbulence intensity profiles in

different speed ranges (power law class)

This classification helps to provide the standard deviation

value cuσ associated to a synthetic wind profile ( )cU z which

has been artificially generated using Uµ and the covariance

matrix U

COV . That has a huge advantage in the calculation

of the buffeting response of the chimney for low and moderate

winds. Usually, this loading case is calculated only in case of

extreme wind situations without focusing on a fatigue

calculation.

3 SIMULATION OF THE BUFFETING RESPONSE

3.1 Structural characteristics

In order to investigate the influence of realistic wind statistics

on the buffeting response of an industrial chimney, a 150 m-

tall steel chimney with constant circular cross section has been

studied. The dynamic analysis has been carried out in the

frequency domain using the finite element technique

developed in Matlab. The diameter is constant over the height

and the wall-thickness varies between 20 mm near the

foundation and 14 mm near the tip.

The total mass of the chimney is 148 tons including a 2% of

mass concerning the structural connections elements. From a

modal analysis, the two first frequencies are determined in

1 0.12f = Hz and 2 0.73f = Hz. For the structural damping, a

logarithmic decrement of 0.02=Λ is selected. This arbitrary

value of damping includes the material and assembly damping

components.

The mechanical damping matrix mechD is calculated using the

classical Rayleigh damping, where D is a linear combination

of the mass matrix M and stiffness matrix K:

mech = ⋅ + ⋅D M Kα β (2)

The aerodynamic damping aeroD is also considered in the

calculation. In many occasions, the aerodynamic damping is

often of the same order of magnitude as the structural

damping. This effect is higher as the wind speed increases and

also its importance to the structural damping, if the mass ratio

of the structure decreases. Therefore, it gives significant

response reductions for light structures such as steel chimneys

or lattice towers [5].

Figure 4. Structural properties

The aerodynamic damping is introduced in the damping

matrix in terms of a diagonal matrix. It is taken into account in

wind direction at the eleven nodes of the chimney, being

proportional to the mean wind speed.

1 11aero

1 11

diag(2 ,..., 2 )F F

U U= ⋅ ⋅D (3)

Fluctuations in across wind direction are not considered

during the calculation. Finally, the complex mechanical

transfer matrix ( )fH of the system depends on the mass

matrix M, the damping matrix D and the stiffness matrix K.

This function is defined in the frequency domain as:

2 1( ) ( (2π ) i (2π ) )f f f −= − ⋅ + ⋅ ⋅ +H M D K (4)

3.2 Generation of synthetic wind profiles

To study the fatigue prognosis of steel chimneys under

buffeting load using realistic wind conditions, the long-term

wind statistics obtained in Gartow has been used. As

explained in [1] and [2], the measured data show a high

variability over the height but also inside the same profile

class. Therefore, the mean wind speed- and profile class-

dependent value of uσ makes more logical the application of

the Monte Carlo approach.

For the Monte Carlo technique, a large number of simulations

are needed to obtain a statistically firm data. Due to the

complex nature of some statement of the simulation’s process,

the original profiles measured in Gartow could not be

sufficient large in number to ensure statistically the results.

For this purpose, synthetic wind profiles of each class c can be

artificially generated using the statistical information

contained in mean vector cUµ and covariance matrix

cU

COV over the height [1], [3]. For each generated wind

speed profile, a value of uσ is assigned depending on its mean

wind speed value at 156z = m. Finally, the response of the

chimney has been separately calculated.


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3.3 Dynamic calculation in frequency domain

The response of the chimney under gusty wind has been

calculated using the considerations of the stochastic vibrations

theory. Early studies made by Davenport [6] show the

difficulty to obtain in a deterministic way the response of the

structure, reducing the response’s information to statistical

parameters in terms of mean values, standard deviations and

spectral density functions. If a linear system is considered, the

maximal response of a structure X can be divided into a

static mean part X , provoked by the mean wind speed, and a

superimposed dynamic part x Xσ obtained from the wind

fluctuations and statistically firm with the well-known peak

factor xg [10]:

ˆ xxX X g

X= + ⋅

σ (5)

The method of resolution for a SDOF system under stochastic

stationary wind loading has been implemented in the finite

element method.

Figure 5. MDOF system considered in the calculation (a) and

node-element interaction (b) [9]

The resolution of a MDOF is carried out extending the

formulation to a matrix based representation after [7] and [8].

The multi degree of freedom system MDOF is shown in

Figure 5. The stationary wind force acting on the whole

structure is transformed into a finite number of stochastic

forces acting at the different nodes as depicts Figure 5a.

Nodes and elements are shown in Figure 5b. The mean wind

force applied on the node i basically depends on the

corresponding mean wind speed iU provided by the synthetic

profile ( )cU z :

( )2 1

2 2i i

j j

iwind D

L LF U D C

++= ⋅ ⋅ ⋅ ⋅

ρ (6)

The height-dependent drag coefficient iDC is obtained using

the formula given in EN 1991-1-4 for circular cylinders with a

diameter D and equivalent roughness of 0.5 mm. For the

calculation of the fluctuating dynamic part, is necessary the

definition of a correlated cross-spectral density matrix

( )fffS . It is obtained from the diagonal matrix ( )ff fS ,

which contains the 11 power spectral density functions of the

wind forces ( )iffS f :

2

2

2( ) 4 ( ) ( )

i

iff uu i

i

FS f S f f

U= ⋅ ⋅ ⋅ χ (7)

The power spectral density function of the wind turbulence

( )uuS f depends on the assigned standard deviation of the

wind turbulence uσ to the synthetic wind profile. The same

expression as in EN 1991-1-4 has been used:

2 5/3

( ) 6.8 ( )

(1 10.2 ( ))

uu L

u L

f S f f f

f f

⋅ ⋅=

+ ⋅σ (8)

The nondimensional frequency Lf can be approximated by

L uf f T= ⋅ if the Taylor-Hypothesis is assumed. From the

integral length scale given in the EN 1991-1-4 for a terrain

category II, an integral time scale of 6.8uT = has been

selected. The coherence function ( )ij fγ proposed by

Davenport [10] is used to expand the diagonal matrix ( )ff fS

to a fully correlated cross-spectral density matrix. The

aerodynamic admittance function 2 ( )fχ is used as in the

formulation presented in the Eurocode 1 considering fully

correlation in along-wind direction:

2( ) ( ) ( )

i ii y zf R f R f= ⋅χ (9)

where:

( )2

2

1 1( ) 1

2

y

y

y y

R f e− ⋅

= − −⋅

η

η η (10)

( )2

2

1 1( ) 1

2

zz

z z

R f e− ⋅= − −

⋅

η

η η (11)

The frequency-dependent coefficients yη and zη depends on

the dimensions of the corresponding body surface, the decay

coefficients 11.5z yC C= = [11], the mean wind speed acting

on the node iU and the factors Ky and Kz.

( )y y j

y

i

K C Bf f

U

⋅ ⋅= ⋅η (12)

1

2( )

j j

z z

z

i

L LK C

f fU

++ ⋅ ⋅

= ⋅η (13)

These two factors are studied in detail by Solari [12]. They are

influenced by the mean wind speed profile acting on the

structure and its mode shape. A constant value of

0.4z yK K= = is chosen. The spectral matrix of the

displacements ( )fxxS can be calculated using the following

expression [8]:

* T( ) ( ) ( ) ( )f f f f= ⋅ ⋅xx ffS H S H (14)

Finally, the standard deviation of the response xσ is obtained

after integration of the response spectrum:

x xxS dfσ = ⋅∫ (15)


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3.4 Results of the Monte Carlo simulation

For each synthetic wind speed profile, the mean, fluctuating

and maximal response is calculated in terms of bending

moment yM at the foundation of the chimney. The

convergences of two variables have been considered. The

first, the convergence of the maximal response ˆ .yM The

second, the convergence of the fluctuating loading part in

terms of .yMσ With ˆ

yM andyMσ , the convergence of the

static and dynamic parts is statistically firm. About 180.000

total simulations have been carried and the participation of

each wind profile class c has been weighted with the

corresponding wind occurrence frequency .cH

Figure 6. Results provided by the Monte-Carlo simulation

Figure 6 shows the relative occurrence frequency cP of the

standard deviation of bending moment yMσ for each wind

class c. The power law and linear classes are the wind profiles

with highest wind speeds and therefore the largest responses

are obtained. The responses caused by the jet classes are very

small and concentrated on the low response ranges and their

influence is almost negligible. The sinus and constant classes

are located in the low-medium response range. Even though

the linear profile class produces large bending moments at the

base of the chimney, its participation in the global response is

low (H6 = 9%) and its importance is dramatically reduced in

comparison to the power law (H1 = 55.9 %) and constant

classes (H5

= 29.9 %). To consider the expected fatigue

damage of any structure, the low and medium wind situations

are more important due to their high occurrence frequencies.

Therefore, it could be stated that the power law and constant

profiles are the most important profile classes for a fatigue life

analysis.

4 FATIGUE ANALYSIS

The fatigue life of any structure under wind buffeting depends

on the number of load cycles caused by the gusts and on the

sequence in which these external loads are applied. The

stochastic nature of the wind makes this analysis complex and

the spectral characteristics of the incident loads determine the

form and number of load cycles acting on the structure during

its design life. Therefore, for a correct design approach of a

structure under wind fatigue, the knowledge of a realistic load

collective during its predetermined lifetime is necessary.

4.1 Number of loads caused by gusts according to EN

1991-1-4

For the number of load cycles caused by gusts, Eurocode 1

provides a simple approach. EN 1991-1-4 allows the

calculation of the number of time GN that a given value ∆S

of an effect of the wind is reached or exceeded during a total

period of 50 years:

2

( ) 0.7 log ( ) 17.4 log( ) 100G G G

k

SN N N

S

∆= ⋅ − ⋅ + (16)

The given value ∆S is defined by percentage of the maximal

effect kS on the structure due to a 50 years return wind

period. As stated in [13], the unconfined usage of EN 1991-1-

4 seems to be not precise. The mathematical background of

the curve is directly related to the mathematical method

proposed by Davenport [14]. He combined probabilities of

wind climates and response processes from wind tunnel

experiments with the mathematical estimation of the

upcrossing levels made by Rice. The formulation of the

problem is not only vague in the definition itself but also in

the conditions in which the corresponding expression can be

used. Any specifications about the site-dependent wind

parameters and/or structural characteristics are necessary to

use the formula. And of course, it does not take into account

the occurrence of different profile shapes.

4.2 Dirlik’s methodology

In case of an arbitrary stochastic process Dirlik [15] derived a

formula to count the number of cycles and amplitude ranges

based on the power spectra density function of the process:

2 2

21 2 2 232

0

( )2

Z Z Z

Q RD D Z

e e D Z eQ R

pm

− − −

⋅⋅

⋅ + ⋅ + ⋅ ⋅

∆σ = (17)

where:

( )2 21 1

1 22

3 1 2

0

23 2 1

21 1 1

2 1 2

0 40 4

4

02

2 1

11

∆1

2

1.25 ( )

1

( ) [ ]

m

m

m

nn

x D DD D

R

D D D Zm

D D R x DQ R

D D D

m m mx

m mm m

mm f S f df E P

m

∞

⋅ − − − += =

−+

= − − =

⋅ − − ⋅ − −= =

− − +

= = ⋅⋅

= ⋅ ⋅ =∫ σσ

γ γ

γ

σ

γ γ

γ

γ

(18)


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where nm are the statistical moments, E[P] symbolize the

number of peaks maxima per second or peak rate and γ is

known as the irregularity factor. The probability density

function of the rainflow ranges (∆ )p σ is based on the

weighted sum of Rayleigh, modified Rayleigh and

exponential probability distributions. The frequency of

occurrence of stress ranges '( )N ∆σ in an expected time T can

be determined with the Dirlik’s method in following terms:

[ ]'( ) ( )N E P T p∆σ = ⋅ ⋅ ∆σ (19)

This approach, defined entirely in the frequency domain,

provides an equivalent result than the rainflow counting

methodology defined in the time domain. This empirical

method can be used successfully for stochastic processes with

any band-width ranges. In other works [16] the verification of

the reliability of this spectral method has been carried out with

parallel calculations in time domain. The spectral estimations

matches excellent for gust excited structural responses.

4.3 Influence of the wind profile shape on the fatigue

The aerodynamic damping in equation (3), as well as the

coherence function ( )ij fγ , the aerodynamic admittance

function 2 ( )i fχ , the spectral density function of the wind

forces ( )iffS f and the standard deviation of the wind

turbulence uσ are variables strongly dependent on the mean

wind speed and thus on the shape of the mean wind speed

profile. The results show a high variability of (∆ )p σ between

synthetic profiles [9] and therefore, the Monte Carlo

simulation is again necessary.

Figure 7 shows the results using Dirlik’s method. The values

of the mean irregularity factor γ for each wind class c are

also plotted. This factor is defined between 0 and 1 and it used

to classify the bandwidth of a process. For a later treatment of

the fatigue life prognosis of the chimney, the frequencies of

occurrence of each profile class Hc given in Figure 6 are

considered.

The numerical solution of the Dirlik methodology in the

Monte-Carlo simulation has been considered in terms of a

mixture distribution:

6

realistic

1

(∆ ) (∆ )c

c

c

p H p

=

= ⋅∑σ σ (20)

where (∆ )cp σ are the probability distributions associated to

each profile class c and displayed in Figure 7. The mixture

distribution is represented with the pink dotted line. The

increase of (∆ )p σ for very low values of stress occurs due to

the contribution of the exponential distribution, defined in the

first term of the numerator in equation (17).

4.4 Determination of load collectives using realistic wind

profiles

In other publications, load collectives from gusts are

developed combining separately wind speed distributions and

responses at different load levels [16], [17]. In the presented

work, the wind speed distribution over the height and the

dynamics of the structure are implicit included in the Monte

Carlo simulation.

Figure 7. Results provided by Dirlik’s method in case

different profile classes

The load collective can be calculated integrating the

probability density function as follows:

realistic( ) [ ] ( )

i

i LifeN T E P p d

∞

∆σ

∆σ > ∆σ = ⋅ ⋅ ∆σ ⋅ ∆σ∫ (21)

where TLife is the designing lifetime in seconds. The main

problematic observed with equation (21) is the correct

coverage of the extreme values if a design proposal for a time

period of 50 years will be carried out. The parent Gaussian

distribution of the mean wind speed used for the definition of

realistic (∆ )p σ shows an excellent behavior in the description of

the low-and moderate wind conditions. But the distribution of

the extreme values is not well covered, because extreme

responses are always studied with other distributions, as the

Fisher –Tipett . Peil et al. [19] noted that for probabilities of

occurrence lower than 0.01, the Weibull distribution is not an

effective tool to represent extreme winds and the application

of equation (21) at theses ranges is highly questionable.

If a load collective for a return period of 50 years is

calculated, it is assumed that the maximal response

maxY occurs only once. This reference value is associated to

the peak response of the structure for a wind with 50 years

return period and it can be calculated applying equation (5).

A reason which contributes to the possible overestimation of

the real maximal response max∆σ can be primarily related

with the definition of the peak factor xg itself [6]. Davenport

considered that response of the structure to a Gaussian

stationary wind action can be described as a narrow-band

process. On the contrary, the MDOF system used in the

current work incurs in stochastic responses which diverge

from the ideal narrow-band case. To avoid the overestimation

in the response produced by a narrow-band assumption,

Cartwright and Longuet-Higgins [20] proposed a new

definition of peak factor xg with consideration of the

bandwidth:


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0.5772

2 ln( )2 ln( )

LHg TT

= ⋅ ⋅ ⋅ +⋅ ⋅ ⋅

γ υγ υ

(22)

Figure 8 shows the load collective of the 150 m high steel

chimney with fundamental frequency 0 0.123f = Hz and

structural damping of 0.02=Λ determined with the Dirlik

method for a time period of 50 years. The continuous blue line

represents the results obtained from the Monte-Carlo

simulation. There were obtained after calculation of the

response of the chimney for more than 100.000 synthetic wind

profiles. Due to the impossibility to cover the extreme values,

the curve should be extended up to the left part of the figure.

Therefore, two hypothetical dotted lines have been plotted

additionally, unifying the medium load range with the extreme

amplitudes. The maximal stress amplitude max∆σ has been

computed for a 50 years return period wind associated to a

terrain category II and basic wind speed 25 m/s and applying

the two peak factor expressions.

The results obtained from the Monte-Carlo simulation shows

lower amplitude levels for a same number of occurrences GN

than the proposal of Kemper [16]. Kemper developed his

method in a conservative way and does not take into account

the effects of the aerodynamic damping aeroD and

aerodynamic admittance function 2 ( )fχ . The chimney is 150

meters high and the absence of aeroD and 2 ( )fχ yields to a

large overestimation of the dynamic response.

Figure 8. Load collective within 50 years

4.5 Calculation of damage accumulation

The total damage is calculated using the Palmgren-Miner

hypothesis. The principle of operation of this methodology is

discretizing different load levels in separated damage cells

and adding linearly their influence on the structure over the

entire lifetime of the structure:

i

i

nD

N=∑ (23)

One advantage of working with probability density functions

is the possibility of the direct analytically calculation of the

fatigue damage. If the reduction of the fatigue strength ∆ Dσ

to its threshold ∆ Lσ is considered, a tri-linear Wöhler’s curve

of fatigue damage can be applied to the determination

of (∆ )iN σ . In case of engineering solutions, the probability

density function is discretized into a finite number m of stress

ranges of width∆ wσ . For the comparative study of the

expected damage between the current work and EN 1991-1-4,

a total 10 different stress levels have been selected for the

application of equation (23). Another four Monte-Carlo

simulations have been carried to calibrate the sensitivity of the

expected damage on the structural damping Λ .

Table 1 shows the reduction factor DR of the damage in case

of different structural damping ratios Λ for the chimney

depicts in Figure 4. The damage has been entirely calculated

at the chimney foundation supposing a fatigue detail category

of ∆ 71C =σ N/mm2 for a period of 50 years. This factor is

defined as follows:

EN 1991-1-4D

DR

D= (24)

The expected damage calculated using the above presented

method is considerable lower than the expected material

damage obtained if the Eurocode 1 is applied. The results

show that the factor DR increases if the structural damping

Λ also increases. This performance has been also observed

for other fatigue detail categories.

Table 1. Reduction factor RD of the expected damage.

Structural damping Λ [-]

0.01 0.02 0.03 0.04 0.05

DR 49.5 52.3 54.5 56.4 59.4

There are several reasons to explain the large differences in

the fatigue prognosis between the Eurocode proposal and the

present work. For the application of EN 1991-1-4, no more

information than the maximal stress amplitude max∆σ is

necessary. However, the current work has combined in a

MDOF system the influence of the site dependent

characteristics, the inclusion of different mean wind speed

profile classes, the aerodynamic damping, aerodynamic

admittance function and the Dirlik method. The combination

of all these factors yields to a large reduction of the expected

damage due to the precision inserted along the entire process

of calculation.

5 CONCLUSIONS

The response of slender industrial chimneys under wind

buffeting have been simulated in order to compare the fatigue

life prognosis according Eurocode 1 and the wind statistics

defined by Clobes et al. from the site Gartow [2]. A Monte

Carlo simulation has been used to provide statistically firm

data. Large number of synthetic wind profiles has been

generated starting from the mean vector Uµ and the

covariance matrixU

COV separately defined for each profile

class c. From the data obtained in Gartow, a classification of

the standard deviation of the wind turbulence uσ is made.


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This last variable is fundamental in the buffeting loading

process and therefore, a unique realistic value of uσ can be

separately assigned for each synthetic wind speed profile

depending on its profile class c and mean wind speed value.

The dynamic response of a 150 m high steel chimney has been

simulated applying the stochastic vibration theory on a multi

degree of freedom system. The model has been generated as

real as possible including the influence of the aerodynamic

damping, as well as the inclusion of several wind profile

shape-dependent variables, as the aerodynamic damping etc.

The results of the Monte Carlo simulation show a clear

differentiation between the profile classes c on the response of

the chimney. Power law class and linear class produce the

largest response but the effect of the latter is diluted in the

overall response due to its low occurrence frequency.

In order to study the expected fatigue damage of the structure,

the Dirlik method has been applied to calculate the probability

distribution of the stress amplitudes (∆ )p σ using the spectral

information of the bending moment at the foundation of the

chimney. This easy method is characterized by its precision

being a combination of different probability distributions. The

resulting probability distribution of the stress amplitudes has

been defined in terms of a mixture distribution considering the

occurrence frequency of each wind profile class c. With the

integration of this formula, is a load collective of the structure

for a lifetime of 50 years is obtained. This collective has been

implemented with the Palgren-Miner method to calculate the

expected damage of the chimney and the results have been

compared with those if the load collective given in Eurocode

is applied. The approach given in the standard shows an

overestimation of the damage in comparison with the actual

study. Therefore, the unique application of the approach of the

Eurocode 1 for each structure without consideration of

damping, site-dependent wind statistics and natural

frequencies of the structure can yield to a large overestimation

in the expected damage.

ACKNOWLEDGMENTS

The funding of the Basque Government and the CICIND are

gratefully acknowledged.

REFERENCES

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