Specification of the design wind load—A critical review of code concepts

ARTICLE IN PRESS J. Wind Eng. Ind. Aerodyn. 97 (2009) 335–357

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Specification of the design wind load—A critical review of code concepts Michael Kasperski  Ruhr-University Bochum, Department of Civil and Environmental Engineering Sciences, Bochum, Germany

a r t i c l e in fo

abstract

Article history: Accepted 18 May 2009 Available online 5 July 2009

In a simplified approach, the design wind load can be specified based on an appropriate small target value of the exceedance probability. For the ultimate limit state, the reasonable reference period is the projected design working life of the structure; for the serviceability limit state a suitable reference period is one year. Basically, at least the extreme wind speeds and the extremes of the aerodynamic coefficients have to be understood as random variables. Further random variables are the duration of a single storm and the relative intensity over the length of the storm. Neglecting these two parameters may lead to underestimations of the design wind load. The design values of the wind speeds are specified in codes with mainly two different concepts: either in terms of a product of the characteristic wind speed and a partial factor or directly as design value. The variable wind speed is represented in codes by gust wind speeds, by 10-min mean wind speeds or by hourly mean wind speeds. For the design value of the aerodynamic coefficient, mainly two concepts are used in codes: the mean value of the extremes or the 78%-fractile value, the latter known as ‘Cook–Mayne’ coefficient. The paper tries to sort out the differences between these approaches and tries to comment on one or the other shortcoming. Additionally, the complexity of the codification task is discussed when different wind climates have to be covered. & 2009 Published by Elsevier Ltd.

Keywords: Design wind load Target values exceedance probability Design working life Duration of storm Extreme wind speed Extreme aerodynamic coefficient

1. Introduction The modern approach to the analysis of the reliability of structures considers the resistance R and the action effects E as random variables. Basic design aim is to achieve a maximum target value of the failure probability pf which can be obtained from the so-called failure integral: Z 1 Z 1 f R ðRÞ f E ðEÞ dE dR (1) pf ¼ 0

R

where fR and fE are the probability densities of the resistance and the action effect. The appropriate reference period for the failure probability is the design working life, i.e. the period for which the structure is intended to perform its required functions. Typical values for the design working life are 1–5 years for temporary structures, 10–40 years for industrial structures and buildings, 60–80 years for residential buildings and 100–150 years for bridges. Most codes do not allow considering the design working life as an explicit design variable; instead, they often consider implicitly a general value of 50 years. For buildings and civil engineering structures, the required functions especially are structural safety and serviceability. The corresponding two limit states are the ultimate limit state (ULS) and the serviceability limit state (SLS).  Tel.: +49 234 3224148; fax: +49 234 3214317.

E-mail address: [email protected] 0167-6105/$ - see front matter & 2009 Published by Elsevier Ltd. doi:10.1016/j.jweia.2009.05.002

The design scenarios for the ULS are:

 loss of equilibrium of the structure, considered as a rigid body, e.g. overturning;

 attainment of the maximum resistance capacity of sections, members or connections by rapture or excessive deformations;

 rapture of members or connections caused by fatigue or other time-dependent effects;

 instability of the structure or part of it; and  sudden change of the assumed structural system to a new system. The design scenarios for the SLS are:

 local damage (including cracking) which may reduce the   

durability of the structure or affect the efficiency or appearance of structural or non-structural members; observable damage caused by fatigue or other time-dependent effects; unacceptable deformations which affect the efficient use or appearance of structural or non-structural elements or the functioning of equipment; and excessive vibrations which cause discomfort to people or affect non-structural elements or the functioning of equipment.

Basic result of the design process is the specification of the required resistance R. For a steel fastener which fixes the cladding to the structure, the resistance R can be specified in terms of a

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maximum tensile force which is obtained as the design value of the yielding stress times the required cross-section area; for the structural member of a steel portal frame, the resistance can be given as the plastic bending moment which is obtained as the product of the design value of the yielding stress and the required plastic cross-section modulus. Action effects are internal forces like bending moments or normal forces, or stresses from a combination of internal forces, or deflections or any other structural response which has to be considered in the design process. In the general case, several actions with different origin will occur simultaneously and will test the designed structure in regard to a sufficient resistance. Beside wind actions, usually at least actions induced by dead load have to be considered. Depending on the climate, snow actions may occur as well. Since the solution of the failure integral is too complex for the every-day engineering practice, codes separate the design problem in a simplified approach by comparing the design value of the resistance R to the design value of the action effect E in the following: Edes  Rdes

(2)

The design values Rdes and Edes can be obtained from statistics if corresponding target values for the exceedance probabilities are specified. It is worth mentioning that in Eq. (2) not the design values of actions are used but the design values of action effects. The paper presents for wind actions the methods which are required for the appropriate specification of these design values. In Section 2 of the paper, target values for the exceedance probability of the design value of the wind loads are given. Section 3 of the paper deals with the question which parameters and variables in the wind load have to be treated as random and which different command variables have to be considered to meet the basic demand of structural safety. The analysis of the wind climate and the corresponding code models are presented in Section 4; Section 5 discusses the extreme value statistics of the extreme aerodynamic coefficients. Finally, in Section 6, the convolution of all contributing random variables is performed, leading to adjusting factors which translate the mean extreme of the aerodynamic coefficient to the required design value.

2. Target values for the exceedance probability of the design wind load In a simplified approach, the reliability level in regard to wind resistance can be set with the choice of the exceedance probability of the design wind load. Respective target values can be specified for the reference period of the projected or design working life. These target values can be translated to corresponding yearly exceedance probabilities as follows: p1 ¼ 1  ð1  pL Þ1=L

(3)

pL is the exceedance probability in the design working life, p1 the annual exceedance probability and L the design working life in years. It is reasonable to classify structures or structural elements in regard to the consequences of a failure. These consequences may be losses of life or economic losses. The following classes can be distinguished: A. structures with a special post disaster function (hospitals, schools, transmission lines, bridges); B. buildings which as a whole contain people in crowds (high-rise buildings, stadia, concert halls);

Table 1 Tentative target values of the exceedance probability of the design wind load for the ultimate limit state (reference period: design working life) and serviceability limit state (reference period: one year). Structural class

A

B

C

D

ULS SLS

0.025 0.01

0.05 0.02

0.10 0.05

0.20 0.10

Table 2 Influence of the design working life on the yearly exceedance probability of the design wind load. L (years)

1 2 5 10 20 50 80 100

Structural class A

B

C

D

1/40 1/80 1/200 1/400 1/790 1/1975 1/3160 1/3950

1/20 1/40 1/100 1/200 1/390 1/975 1/1560 1/1950

1/10 1/20 1/50 1/95 1/190 1/475 1/760 1/950

1/5 1/10 1/25 1/45 1/90 1/225 1/360 1/450

C. normal structures (office buildings, commercial buildings, factories, residential buildings); and D. structures presenting a low degree of hazard to life and other properties (farm buildings, house chimneys, roofing tiles). The target value of the exceedance probability of the design wind load for structural safety is specified in the following for the design work life (Table 1). For the serviceability limit state respective target values are presented with reference to the period of one year. The influence of the design working life on the yearly exceedance probability of the design wind speed is presented in Table 2 for values of the design working life from 1 to 100 years. If the design working life extends over several decades, the target value of the yearly exceedance probability becomes fairly small, i.e. for the design rare to very rare events are required. Usually, meteorological observations are available only for a few decades, i.e. for the specification of the design wind load obviously larger extrapolations are required.

3. Basic variables and command variables 3.1. Elementary random contributions to the design wind load In the simplest approach, the wind load w is given by three variables w¼

1 2 rv c 2

(4)

where r is the air density, v the wind speed and c the aerodynamic coefficient. The question arises, which of these variables have to be considered as random variables. In Fig. 1, the variability of v and c are shown. For v, the maximum hourly mean wind speeds in strong frontal depressions observed at the meteorological station Du¨sseldorf, Germany from 1952 to 2000 are used. For c, the extreme aerodynamic coefficients on the upwind wall and at the leading edge at the centre bay of a low-rise building are taken from a set of independent repetitions of a wind tunnel test. Both

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extreme aerodynamic coefficient

max. hourly mean wind speed [m/s]

M. Kasperski / J. Wind Eng. Ind. Aerodyn. 97 (2009) 335–357

21 20 19 18 17 16 15 14 0

10

20

30 40 50 60 70 no. of storm extreme hourly mean wind speed v

4 3 2 1 -3 -4 -5 -6 -7 0

80

337

20

40 60 80 no. of wind tunnel run extreme aerodynamic coefficient c

100

1.40

1.40

1.35

1.35

air density [kg/m3]

air density [kg/m3]

Fig. 1. Typical scatter of extreme wind speeds and extreme aerodynamic coefficients.

1.30 1.25 1.20 1.15 1.10

1.30 1.25 1.20 1.15 1.10

0

5

10

15

14

20

15 16 17 18 19 hourly mean wind speed [m/s] storm situations

hourly mean wind speed [m/s] general

20

Fig. 2. Variation of air density with hourly mean wind speed.

parameters, v and c, show significant scatter, and the absolute value of the product of v2 and c is considerably influenced by these scatters. Hence, v and c have to be treated as random variables. It is important to note, that a limiting value of the product v2 and c may be exceeded by an infinite number of combinations of v and c values, i.e. the design wind load may be exceeded in a weak storm due to a rare very large c-value, or it is exceeded in a rare and strong storm due to a fairly frequent and low extreme c-value, or it is exceeded from mid-range values of c and v. The air density also is not a natural constant. The air density depends on the air temperature, the barometric pressure and, strictly speaking, on the relative humidity

r¼

p Rh T

(5)

where p is the barometric pressure in Pa, T the air temperature in K, T (K) ¼ T (1C)+273.15 and Rh the gas constant for humid air. A good approximation on the safe side is obtained by using the gas constant for dry air, which is 287.06 J/kg/K. In Fig. 2, the natural variation of the air density is shown for Du¨sseldorf based on observations of several decades. For smaller wind speeds, a large range of air densities is obtained. The range of observed air densities becomes smaller for increasing hourly mean wind speeds; for strong storm situations, i.e. for wind speeds above a threshold value of 14 m/s, the scatter of the observed air densities is small. As appropriate design value for large hourly mean wind speeds, rdes can be assumed as 1.25 kg/m3.

A straight-forward solution for the specification of the design value for wind loads is wdes ¼

1 r v2 c 2 des des des

(6)

where rdes is the design value of the air density, vdes the design value of the wind speed and cdes the design value of the aerodynamic coefficient. Most codes, however, ‘hide’ the design value of the wind speed by introducing a characteristic wind speed vk and a partial factor g: wdes ¼

1 r gv2 c 2 des k des

(7)

The ‘hidden’ design wind speed is obtained from vdes ¼

pffiffiffi gvk

(8)

It is important to note that there is only one basic equation to specify the design values of the contributing variables v and c (and eventually r) which is !

pðw4wdes Þ ¼ ptarget

(9)

Obviously, further restrictions and demands have to be specified to conclude from Eq. (8) to the required values of vk and g, or vdes, respectively, and cdes. For a general approach, as it is required in a code, it is reasonable to demand that the exceedance probability of vdes equals the target exceedance probability, leading to one and the same design wind speed for all structures in the same

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structural class and having the same design working life: !

pðv4vdes Þ ¼ ptarget

(10)

The basic equation to obtain the exceedance probability of the design value of the wind load is obtained from the following convolution integral of the probability densities of v and c: Z 1 Z 1 pðw4wdes Þ ¼ f v ðvÞ f c ðcÞ dc dv (11) v¼0

c¼clim

fv and fc are the probability density of the extreme wind speeds and extreme aerodynamic coefficients. The lower limit clim in the second integral depends on the value of the design wind load and the actual value of the wind speed in the first integral: clim ¼

2wdes v2des cdes ¼ rv2 v2

(12)

The second integral can be replaced by the cumulative probability distribution of c leading to the fundamental equation for specifying the design value of the wind load as follows: Z 1 ! pðw4wdes Þ ¼ f v ðvÞ½1  F c ðclim Þ dv ¼ ptarget (13) v¼0

with Fc the cumulative probability distribution of c. 3.2. Appropriate probability distributions for v and c The basic variables v and c have to be understood as extreme values; therefore it seems to be reasonable to fit observed data to one of the three extreme value distributions:

 type I—Gumbel distribution;  type II—Fre´chet distribution; and  type III—Weibull distribution. The distributions of types II and III each form a family of curves with specific characters. Compared to the type I distribution, they show as special feature a certain curvature when plotted in Gumbel probability paper (Fig. 3). Generally, a probability paper is a graph paper with one axis specially ruled to transform the

distribution function of a specified function to a straight line when it is plotted against the variate as the abscissa. While the curves for the type II distribution bend in a concave shape in respect to the axis of the reduced variate, the curves corresponding to type III show a distinct convex character. These two types are separated by the type I distribution which appears in the plot as a straight line. In an alternative approach (Naess and Clausen, 2001; Cook and Mayne,1980), which also has been adopted in the Eurocode, it is assumed that xk follows a type I distribution. Especially for k ¼ 2 applied to wind speeds there seems to be a physical justification for this concept, since the velocity pressure is obtained from squaring the wind speed. In Fig. 3, the traces of the nonexceedance probabilities of v2 following a type I distribution are shown for three different variation coefficients of v, namely 0.075, 0.125 and 0.175. All three traces, which are marked by dots, show a clear convex curvature; in other words: if v2 follows the type I distribution, this may indicate that v follows the type III distribution. A respective fit is shown in Fig. 3 as solid lines. The differences between these two approaches are quiet small, and fitting a type III distribution for v basically may cover the trace as good as fitting a type I distribution for v2. One important difference, however, is obtained in the tails of the distributions. While for types I and II the distributions have no upper limit—i.e. they extend to infinity—the distributions of type III have a finite upper tail. This feature seems to make them especially appropriate for the analysis of extreme phenomena which may be limited from geophysical reasons like extreme wind speeds or fluid-mechanical reasons like wind actions in terms of local pressures or global forces. Using the type III distributions may lead to more economic designs, which makes the application of type III distributions especially attractive. The cumulative probability for the type III distribution is given as follows:    x  m1=t FðxÞ ¼ exp  f 1  f 2

m is the mean value, s the standard deviation and t the shape parameter 40.

type I

0.999

(14)

s

cov(v) = 0.175 - 0.125 - 0.075

0.999

0.99

type II

non-exceedance probabilty

non-exceedance probabilty

type III

0.9

0.5

0.99

0.9

0.5

0.1

0.1

0.01 0.001

0.01 0.001 -2

0

2 4 6 reduced variate xred

traces for v following type I, II or III

8

-2

-1

0 1 2 3 reduced variate xred

traces for v2 following type I

Fig. 3. Traces of the extreme value limit distributions in a Gumbel probability paper.

4

5

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The coefficients f1 and f2 depend on the actual shape parameter

t and are given as follows:

f2 ¼

(15)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gð1 þ 2tÞ  f 21

(16)

G-Gamma function The maximum value which cannot be exceeded with probability of 1 is given as xmax ¼ m þ s

f1 f2

(17)

For t ¼ 0, the extreme distribution type I is obtained, i.e. in the limit Eq. (14) becomes:     p xm (18) FðxÞ ¼ exp  exp  g þ pffiffiffi 6 s

g is the Euler constant ¼ 0.5772. Shape parameters to0 lead to the extreme value distribution type II. This distribution is defined for values x40, i.e. instead of an upper limit, a lower limit is obtained. 3.3. Choice of partial factor

2.0

2.0

1.8

1.8

1.6

τ = 0.0

1.6

1.4

τ = 0.1

1.4

1.2

τ = 0.2 τ = 0.3 τ = 0.4

1.0 0.8

partial factor γ

partial factor γ

With the basic demand from Eq. (10), only the amplitude of the product of the characteristic wind speed and the partial factor can be specified. For sake of transparency, it is reasonable not to use characteristic wind speeds but to specify the design wind speed explicitly. Most of today’s codes, however, do not follow this recommendation. For the specification of vk, exceedance prob1 abilities with reference to the period of one year are used, e.g. 50 1 per year in the Eurocode (EN1991-1-4, 2005), or 100 per year in the Recommendations of the Architectural Institute of Japan (AIJ) (AIJ Recommendations for loads in buildings, 1996). For each specific wind climate, there is exactly one partial factor which translates the characteristic wind speed to the desired design wind speed. Hence, any value of p(v4vk) per year can be introduced, and the history of codification of wind loads gives a large variety of 1 1 1 (Germany), over 12:5 (Netherlands) to 33 examples from 10 (Canada). In Fig. 4, the basic relation between p(v4vk) per year and the partial factor g is shown for demanding 5% exceedance probability for the design value of the wind speed in a projected lifetime of 50 years. Obviously, there is one combination of p(v4vk) and g which forms a general solution for any given wind climate. This ‘optimum’ combination of characteristic value and partial factor is obtained for vk ¼ vdes and g ¼ 1.0. For a code intending to cover

a lot of different wind climates, like e.g. the Eurocode, this combination should be the obligate choice (EN1990, 2002). For sake of completeness, it is worth mentioning that there is a further reason to use g ¼ 1 and vk ¼ vdes. For structures which show considerable resonant responses in regard to wind induced vibrations, most codes offer methods to estimate the gust response factor GRF. Basically, the gust response factor shows an over-linear increase with wind speed. Then, the result from g times GRF(vk) will be smaller than GRF(g1/2vk). This defect will increase with increasing vibration sensitivity and may lead to a considerable underdesign. It is worth mentioning that the Australian code (AS 1170.2, 1989) has adopted this general concept already in its version from 1989, while e.g. the ‘modern’ Eurocode still follows the concept of a 50-year return wind speed and a partial factor of 1.5. 3.4. Representative values A closer look to different codes reveals that for the variables v-wind speed and c-aerodynamic coefficient considerably different representative values are used. The variable v may stand for the gust wind speed, the 10-min mean wind speed or the hourly mean wind speed. The question arises, which of these three values is able to represent best the intensity or strength of a storm. Basically, three different storm types may have to be covered in codes: strong frontal depressions, tropical cyclones and thunderstorms. For strong frontal depressions, all three values are able to indicate the intensity of a single storm event, and conversion factors have been introduced to relate the maximum gust wind speed and the maximum 10-min mean wind speed to the hourly mean wind speed. With the information given in ESDU— Engineering Science Data Unit (1985,1986), a sophisticated model for the turbulent wind fields in strong frontal depressions exists, which allows analysing the randomness in these conversion factors. In Fig. 5, the traces of the cumulative probabilities for the conversion factor of the maximum 10-min means and the maximum 3 s-gusts are shown. The traces are obtained from 1000 independent simulations, using the following input parameters: mean wind speed 20 m/s, turbulence intensity 0.19, integral length scale 100 m, Karman spectral density distribution. The mean value of the gust factor in these simulations is 1.64, the 90%-confidence interval extends from 1.56 to 1.76. For the ratio of the maximum 10-min mean to the maximum hourly mean, the 90%-confidence interval extends from 1.01 to 1.06, the mean value is 1.03. For thunderstorms, hourly mean wind speeds are not able to represent the intensity of the storm. A meaningful measure of the

τ = 0.3 τ = 0.4

1.2 1.0 0.8

1.0 0.8 0.6 0.4

0.2

0.2

0.2

0.0 0.0001

0.0 0.0001

p (v>vk) cov (v) = 0.075

cov (v) = 0.125

0.1

τ = 0.4

1.2

0.4

0.001 0.01 p (v>vk)

τ = 0.3

1.4

0.4

0.1

τ = 0.2

1.6

τ = 0.2

0.6

0.01

τ = 0.1

1.8 τ = 0.1

0.6

0.001

τ = 0.0

2.0

τ = 0.0

partial factor γ

f 1 ¼ Gð1 þ tÞ

339

0.0 0.0001

0.001 0.01 p (v>vk)

0.1

cov (v) = 0.175

Fig. 4. Possible combinations of partial factor g and p(v4vk) to achieve a basic exceedance probability of 5% in 50 years for different wind climates.

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0.999

non-exceedance probabiltiy

non-exceedance probability

0.999

0.99

0.9

0.5

0.1

0.99

0.9

0.5

0.1

0.01 0.001 0.90

0.95

1.00

1.05

0.01 0.001 1.4

1.10

1.5

1.6

max. 10-min mean / hourly mean

1.7 1.8 gust factor

1.9

2.0

Fig. 5. Example of cumulative probabilities for the conversion factors for maximum 10-min means and maximum 3 s-gust in strong frontal depressions.

1.2 -

0.6

0.3

-

-

+

0.8

influence line (qualitatively)

0.3

typical wind load distribution

Fig. 6. Influence line for the bending moment in the corner of the frame and typical wind load distribution.

intensity of thunderstorms probably only is obtained with the gust wind speed which can be translated to an artificial or equivalent 10-min mean wind speed based on the conversion factors above. Finally, for tropical cyclones, it is believed that the hourly mean wind speed is a less stationary value; hence, the intensity of these storms can be represented by 10-min mean wind speeds or gust wind speeds (Fig. 6). In a similar way, the values behind the aerodynamic coefficients c may have different origin. The obsolete approach is based on mean values of the aerodynamic coefficients and leads to the concept of an enveloping gust which is thought to influence all surface pressures in the same way, i.e. it is assumed that the basic shape of the pressure distribution remains unchanged during a storm. This assumption is far from the real behaviour of surface pressures. Furthermore, this concept neglects all body-induced turbulence effects, which may lead to severe underestimations of local action effects especially in areas with separated flow. A consistent approach has been developed by Cook and Mayne, who recommended—based on the wind climate in the UK—to use the 78%-fractile value of the extreme aerodynamic coefficients. Other approaches partially adopted in codes use the mean extreme value, or 80% of the largest extreme from N independent runs, or the largest extreme from N independent runs or simply the largest extreme from a single run, which gives a quiet confusing picture. Basically, the absolute value of extremes increases with increasing observation period, i.e. when analysing extreme values, a reference period has to be specified. It is quiet seducing simply

to adopt either the 10-min or 1 h period, whichever value has been used for the wind speeds. The consistent approach requires considering the duration of the each storm, which may be considerable longer than 10 min or 1 h. A similar trap hides in the analysis of the extreme wind speeds. Basically, the target exceedance probability for the characteristic or the design value is specified with reference to the period of one year. This may seduce to use yearly extremes of the wind speeds to describe the strong wind climate. As will be shown in Section 4, yearly extremes form an inadequate basis for both interpolation and extrapolation purposes. Hence, the list of basic variables has to be extended by introducing the following parameters:

 the duration of a storm in terms of multiples of the basic reference sub-period 10 min or 1 h;

 the relative intensity of the 2nd-, 3rd-, 4th- to kth-strongest sub-period; and

 the number of storms per year. The accumulation of the exceedance probability of the design wind load over the duration of the storm is obtained as follows: Y pðw4wdes Þ ¼ 1  pj ðw  wdes Þ (19) j

pj the probability in storm hour j.

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3.5. Required design scenarios for wind induced actions From the basic design scenarios specified for the ultimate limit state, the following information on the wind load has to be provided:

 global wind loads, i.e. drag, lift and overturning moment in regard to a possible loss of equilibrium;

 local loads for the design of the fasteners of the cladding and small-size cladding elements; and

 structural loads for the design of members and the design of large-size cladding elements. Generally, local peaks will not occur simultaneously. The intuitive solution to use the envelope of extreme local actions as structural loads then may lead to two basic problems: an overestimation of the structural responses or an underestimation of the structural responses. This is explained in the following on the example of a portal frame for a low-rise building. The main dimensions are h/d ¼ 0.4 (h—eaves height and d—span), the roof pitch is 51. In Fig. 7, the qualitative influence line for the bending moment in the left corner is shown. Additionally, a typical load distribution as it might be specified in a wind load code is shown for wind blowing normal to the ridge. The structural response, i.e. the bending moment, is obtained by integration of the weighted load amplitudes over the frame, the weighting coefficients are obtained from the influence line. For the bending moment in the upwind corner of the frame, the positive pressures on the upwind wall are weighted with positive values of the influence line, thus leading to positive contributions to the load effect. The negative pressures along the roof—except a very small section at the leading edge—and along the downwind wall are weighted with negative values, leading to positive contributions to the action effect. If the load distribution is given as the envelope of the local peaks and peaks along the roof will not occur simultaneously, the action effect is overestimated. Then, the design will be uneconomic. For the bending moment in the downwind corner of the frame, the integration of the weighted load amplitudes over the frame leads to a different situation. Now, the contributions from the walls have a negative sign, while the contributions from the roof have a positive sign. Using the envelope of the local peaks will lead to an underestimation of the negative action effect, and the design will turn out to be unsafe. The wind induced bending moment in the upwind corner of the frame usually has a larger absolute value than the bending moment in the downwind corner of the frame. However, it would be wrong to conclude that the wind-induced downwind bending moment may not be important for the design. In Fig. 7, qualitative

-

distributions of the bending moment are shown for dead and/or snow load and for wind loads using for the latter the load distribution from Fig. 6. In the upwind corner of the frame, the action effect which has to be compared to the resistance is obtained as the absolute value of the sum from the negative bending moment induced by dead load and/or snow and the positive wind induced bending moment. For the downwind corner, the action effect is the absolute value of the sum of the negative values from wind and dead load and/or snow. Hence, for a light roof construction in a climate without snow, the upwind bending moment usually becomes decisive for the design; for heavier roof constructions and climates with a considerable snow load, usually the downwind bending moment is governing the design. A general approach to design wind loads therefore has to consider all design-decisive so-called command variables, which are the local action effects, the global action in terms of drag, lift and overturning moment and all structural load effects which have to be compared to a resistance value. It is worth mentioning that most codes nevertheless prefer the solution to specify one load distribution and then give some additional rules to balance the shortcomings of this approach. 4. Analysis of the wind climate and corresponding code models 4.1. Wind climate model for the ultimate limit state and the serviceability limit state Basically, there are two different aspects which a code has to cover in regard to the wind climate. For applications considering the serviceability limit state, like e.g. fatigue effects from vortex induced vibrations or acceleration amplitudes which might lead to discomfort of the users, a model of the relative frequency of wind speeds is required for the lower to middle range of wind speed amplitudes. A second model required for the Ultimate Limit Design deals with extreme wind speeds and aims in specifying the design value of the wind speeds in terms of a sufficiently rare event. The lower range of wind speeds often is called parent distribution, suggesting that strong storms are born from small winds (‘they sow the wind and reap the whirlwind’). Basically, the observed hourly wind speeds have to be understood simply as a measure of the velocity of moving air, and the driving forces behind these movements in terms of weather systems or pressure differences may have considerable different physical origins. Hence, there is definitely no reason to assume that a sea-breeze induced by increasing temperature differences between the land and the water is the same feature or phenomenon as a strong frontal depression which has been induced by instabilities of the polar front.

-

+

+

341

-

-

+ -

dead load / snow

wind

Fig. 7. Distribution of bending moments induced by dead load, snow and wind.

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A widely accepted model for the relative frequency of hourly mean wind speeds is given with the Weibull distribution. Its probability density and corresponding cumulative probability distribution are given as follows:    k ! k x k1 x (20) exp  f ðxÞ ¼ x0 x0 x0  k ! x FðxÞ ¼ 1  exp  x0

(21)

0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00

pk is the probability for individual storm type k. 4.2. Sampling ensemble for extreme wind speeds The classical approach, intensely promoted since Gumbel presented his method in the mid-1950s of the last century, uses annual maxima. This seems to be attractive, since the statistics are aiming to predict rare events with reference to periods in years, e.g. the once-in-a-100-years event. It is nevertheless a bit surprising, how detached the implementations of Gumbel’s method in wind engineering and e.g. hydrology have developed. In wind engineering, it is common practice to sample extremes for the western calendar year, although it is obvious that the 1st January cuts the storm season in e.g. Europe and Canada just at its peak. Hydrologists, coping with the same problem, have therefore introduced the so-called hydrologic or water year, which generally extends in the Northern Hemisphere from the 1st of October of one year to the 30th of September of the consecutive year and has been specified for Germany to last from the 1st of November of one year to the 31st of October of the consecutive year. For wind engineering purposes, similarly a continuous twelve-month period has to be defined in which a complete annual climatic cycle occurs, and which is selected to provide a more meaningful comparison of observed wind data. Nevertheless, the ‘aeolian’ year is a rather new idea in wind engineering. For Germany, an appropriate aeolian year extends from the 1st of May to the 30th of April. In Fig. 9, the identified 12-month cycle maxima are shown for the observed data at Du¨sseldorf airport from 1952 to

5 10 15 hourly mean wind speed [m/s]

0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00

20

1

1

0.1

0.1

0.01

0.01

1 - Fv (v)

1 - Fv (v)

0

1E-3 1E-4

(22)

k

fv (v)

fv (v)

with k the shape parameter and x0 the scale parameter. It is important to note, that the Weibull distribution is an efficient expression to describe the observed relative frequencies of wind speeds; however, it is not the ‘true’ distribution of ‘parent’ winds. This is illustrated based on the observed wind speeds at the meteorological station Du¨sseldorf airport for the years from 1952 to 1999 in Fig. 8. A first solution is obtained from a leastsquare fit of the counted relative frequencies in a linear range for f(x). This solution is an appropriate model for wind speeds from 0 to 10 m/s. The model is less appropriate for higher wind speeds up to 15 m/s. A second fitting method uses the number of hours above a threshold value and performs the least-square fit in the log-scale for (1F(x)). The resulting solution is an appropriate model for wind speeds in the range from 7.5 to 15 m/s; however it fails in the lower wind speed range. For the analysis of extreme wind speeds it is widely accepted that for so-called ‘mixed’ wind climates extremes value statistics have to be performed for each independent storm phenomenon, e.g. strong frontal depressions and thunderstorms. The resulting exceedance probability of a reference wind speed level then is obtained from the following combination of k individual non-

exceedance probabilities: Y pk ðv  vref Þ pðv4vref Þ ¼ 1 

0

5 10 15 hourly mean wind speed [m/s]

20

0

5 10 15 hourly mean wind speed [m/s]

20

1E-3 1E-4

1E-5

1E-5 0

5 10 15 hourly mean wind speed [m/s] v0 = 4.36 m/s ⋅ k=1.78

20

v0 = 3.97 m/s ⋅ k = 1.60

Fig. 8. Fitting the Weibull-distribution to observed hourly mean wind speed (Data from Du¨sseldorf airport 1952–1999).

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22

343

22 Western calendar year German aeolian year

12 month-cycle maxima aeolian year [m/s]

12-month cycle max.hourly mean wind speed [m/s]

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20

18

16

14

12

10 1950

20

18

16

14

12

10 1960

1970 1980 observation year / season

1990

2000

12 14 16 18 20 12-month cycle maxima calendar year [m/s]

10

22

14

22 20

total number of years

hourly mean wind speed [m/s]

Fig. 9. Twelve-month cycle maxima for the Aeolian year and the western calendar year (Data from Du¨sseldorf airport 1952–1999).

18 16 14 yearly extremes

additional extremes

12

12 10 8 6 4 2 0

0

10

20

30 40 50 60 current number of storm

70

0 1 2 3 4 5 6 7 8 9 number of storms per calendar year

80

Fig. 10. Comparison of the ensemble ‘yearly extremes’ and extremes above a threshold value for Du¨sseldorf airport 1952–1999.

1999 for the calendar and the German Aeolian year. The ensembles are clearly different, i.e. they contain different storm events. In 48 years observation period, there are only 27 events occurring in both ensembles of 12-month cycles. This illustrates already that sampling 12-month cycle maxima is less appropriate to get the full information about storm intensity and frequency. A closer look to the wind climate in Europe and Germany, respectively, reveals that sampling maxima from 12-month cycles is leading to two basic shortcomings. First, extreme storms tend to occur in clusters or families, i.e. during one season more than one extreme storm may be obtained. In Europe, famous examples of such destructive families are Daria, Vivian and Wiebke in 1990 and Anatol, Lothar and Martin in 1999. If only the strongest storm per 12-month cycle is sampled important information might get lost, since the second-strongest storm in one year may be considerably stronger than the strongest storm in another year. A further characteristic of the European/German wind climate is that there are years without any strong storm at all. Then, sampling stolidly annual maxima is going to mix unimportant information into the ensemble of ‘extremes’. In Fig. 10, the series of observed independent storms in the period from 1952 to 1999 at Du¨sseldorf airport is shown. A storm is defined using a threshold value of 14 m/s. The yearly extremes are marked separately. The ensemble of independent extremes contains 43 samples more than the ensemble of the yearly extremes. Additionally, the observed number of storms per calendar year is shown. There are 13 years where no storm has occurred, and statistics based on yearly extremes then are going to use 13 events with hourly mean wind speeds not caused by a strong storm.

4.3. Order statistics For the observed extremes, order statistics can be performed to plot the observed trace of non-exceedance probabilities in an appropriate probability paper. The ensemble firstly is sorted in ascending order and then the non-exceedance is estimated from: f rel ðx  xi Þ ¼



 ia N  2a þ 1

(23)

i is the rank in list of ascending order, highest value rank N, lowest value rank 1 and N the ensemble size. The parameter a defines the ‘plotting position’. In Gumbel’s original concept, a is zero. A least square fit in Gumbel probability paper, however, then is leading to biased estimates of the parameters of the Gumbel distribution. Therefore, for acceptance tests with Gumbel probability paper, in Gringorten (1963) the recommended a-value is 0.44. The analogous number for the type III distribution is a ¼ 0.30 (Cunnane, 1978). For the translation of the non-exceedance probability per event to the non-exceedance probability per year, Cook and Mayne (1980) proposed the following approach: f rel ðx  xi Þ ¼



N=K ia N  2a þ 1

(24)

K is the length of the observation period in years. The exponent N/K in Eq. (24) is the average number of extreme events per year. To avoid any biasing influence from seasonal

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effects, it is usually recommended to use an ensemble which is based on whole years.

and the non-exceedance probability per event as follows: pðv  vref jyearÞ ¼

1 X

pðNÞ pðv  vref ÞN

(25)

N¼0

4.4. A refined model Strictly speaking, the approach in Eq. (24) is appropriate only for large values of the non-exceedance probability. This leads to a considerable influence of the choice of threshold value on the left tail of the trace (Kasperski, 2002). A more consistent approach is obtained with combining the probability of the number of storms

p(N) is the probability of N storms per year and p(vrvref) the nonexceedance probability per event. The consequences of the basic shortcomings of the statistics based on yearly extremes can be illustrated by comparing the obtained trace of the yearly non-exceedance probability to the observed statistics obtained by simply counting the number of

10000

1000

N (v>Vref|year) [h]

100

input for plot in

10

probability paper

1

0.1

0.01

1E-3 0

5

10 15 hourly mean wind speed vref [m/s]

20

25

Fig. 11. Average number of storms per year with an hourly mean wind speed above a threshold value vref (Du¨sseldorf airport 1952–1999).

0.999 yearly extremes

non-exceedance probability

counted rel. frequency of non-exceedance

0.99

0.9

0.5

0.1 0.01 0.001 10.0

12.5

15.0

17.5 vref [m/s]

20.0

22.5

25.0

Fig. 12. Trace of the non-exceedance probability of the yearly extremes of the hourly mean wind speed compared to the counted relative frequency of non-exceedance per year (Du¨sseldorf airport 1952–1999).

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hours above a certain threshold as shown in Fig. 11. All counted events with a relative yearly frequency smaller than 1 can be used as input for the plot in the probability paper for the yearly nonexceedance probability. It is worth mentioning that this approach requires no sophisticated statistics at all, but only a sufficient large observation period of several decades. As can be seen in Fig. 12, the only agreement between the two different traces is obtained at the basic anchor point which is obtained from the largest event in the observation period. For all other wind speed levels, the statistics based on yearly extremes are not able to predict the observed non-exceedance probability per year. The question may arise why not simply using the trace of the observed non-exceedance probabilities to get a model for the extreme wind climate. The answer is: the trace contains correlated data since a single storm at Du¨sseldorf lasts on average for 3 h. Extreme value statistics, however, require independent ensemble entries. Furthermore, the observed trace at the right tail is biased to lower values since the future strongest storm is not only going to add a new highest value but is also going to mix in the existing trace on average two further entries. The consistent model therefore analysis the intensity of the independent strongest hour of individual storms in an appropriate extreme value statistic and considers the adjacent storm hours in the convolution integral used to calculate p(w4wdes). In Fig. 13, the trace of the non-exceedance probability of vref in the strongest storm hour of strong frontal depressions at Du¨sseldorf airport is shown with reference to the period of 12 month. The trace shows a considerable curvature and a leastsquare fit leads to the following identified parameters: m ¼ 15.7 m/s, s ¼ 2.07 m/s, and t ¼ 0.11. Similar statistics have to be performed for the other storm phenomena which may influence the strong wind climate. In Germany, at least thunderstorms as they usually occur in the summer have to be considered. Merging the individual statistics, however, may become difficult when the intensities of the different storm types are described by different representative values, e.g. for thunderstorms the observed gust wind speeds and

345

in the consistent model for strong frontal depressions the strongest hourly mean wind speeds. Strictly speaking, merging of the individual statistics is first allowed after the convolution with the extreme aerodynamic coefficients, i.e. merging should be performed on the basis of the individual non-exceedance probabilities of the design wind load obtained from each contributing storm phenomenon, considering the different durations and different flow fields of each storm type. If, however, a combined exceedance probability of wind speed is wanted, the gust wind speeds have to be translated to equivalent hourly mean wind speeds. As translation factor, the mean gust factor can be used, which gives the ratio of the 3 s-gust wind speeds to the hourly mean wind speed. A deeper analysis of the observed extreme wind speeds during strong frontal depressions in Germany reveals that there is a further phenomenon which is not yet covered with the above model. As already mentioned, ESDU provides a sophisticated model for the structure of the turbulent wind field in strong frontal depressions. This or equivalent models form the basis of wind tunnel tests which are the major source of aerodynamic coefficients. The completeness of the model can be examined by comparing observed gust factors to gust factors obtained by simulations. The expected scatter for open terrain roughness at 10 m height above ground is shown in Fig. 14 using the simulation data from Fig. 5. The typical range for the gust factor is from 1.5 to 1.8. Additionally, the observed gust factors in strong frontal depressions are shown. They are considerably larger than to be expected with the ESDU model. The corresponding weather systems inducing these high gust factors are gust fronts induced by convective processes which are superimposed to the frontal depression system; separate statistics are required for these additional gust speeds. It is important to note that similar to the wind fields in thunderstorms the corresponding wind fields in gust fronts are not yet fully understood or known. For the further analysis, again, the gust wind speeds can be replaced by equivalent mean wind speeds by applying the mean value of the gust factor.

0.999

non-exceedance probability

0.99

0.9

0.5

0.1 0.01 0.001 10

15

20

25

vmax [m/s] Fig. 13. Trace of the non-exceedance probability of the strongest hourly mean wind speed in a storm induced by strong frontal depressions with reference to a single year (Du¨sseldorf airport 1952–1999).

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2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.5 1.4

observed gust factor

gust factor

346

0

10 20 30 40 50 60 70 80 90 100 current number of storm typical expected scatter

2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.5 1.4 14

15

16

17

18

19

20

21

22

hourly mean wind speed [m/s] observed values

Fig. 14. Example of the expected range of gust factors in strong frontal depressions and observed values for Du¨sseldorf airport (1952–1999).

Fig. 15. Traces of the non-exceedance probabilities of the extreme pressure coefficients for a low-rise building (all values normalized with the mean wind velocity pressure at eaves height).

5. Analysis of extreme aerodynamic coefficients 5.1. General remarks Most of the information on aerodynamic coefficients is obtained from scaled model tests in boundary layer wind tunnels. Basically, this technique allows sampling any required number of independent peaks. The unofficial world record on repeated tests reaches 5100 runs (Holmes and Cochran, 2003). Then, the problem of statistical stability is obviously solved. For most tests with commercial background, however, the number of repeated tests will be considerably smaller. It is important to note that a stable identification of a type III distribution requires several 100 runs. A conservative approach is obtained assuming that the extreme aerodynamic coefficients follow the type I distribution; then, the required fractile value can be estimated with an ensemble size smaller than 100. If the ensemble size becomes too small, adjusting factors may be required. It is important to note that generally the variation coefficient of the extreme aerodynamic coefficient is unknown; for the derivation of an adjusting factor, however, this parameter is required. Today’s boundary layer wind tunnel techniques usually are only able to model the wind fields as they are obtained in strong frontal depressions. It remains to some extent unknown, if the respective aerodynamics are also valid for the flow fields of the other storm types like tropical cyclones, thunderstorms and gust fronts. As already mentioned above, extremes generally will increase with increasing sampling period, i.e. extremes of the aerodynamic coefficient sampled from 10 min periods will be smaller than

extremes sampled from 1 h periods. A mismatch between the sampling period and the intended reference period can be corrected (Kasperski, 2003), supposing that the chosen sampling period is not too short to become unrepresentative for the analysed process. The demand to statistical stability finally has to be met for the ‘true’ duration of the storm phenomenon under consideration, and failing in sampling the extremes from the appropriate period may lead to considerable underestimations of the design wind load. The appropriate sampling period in the wind tunnel is obtained applying to the desired reference period the time scale lT which itself depends on the geometric scale lL and the velocity scale lv. Basically, any scale l thereby is obtained as the ratio of the wind tunnel variable to the full-scale variable. The time scale is

lT ¼ lL =lv

(26)

Strictly speaking, the geometric scale of the model has to meet the geometric scale of the flow; a mismatch may lead to considerably biased results. Since the expected wind speeds in full-scale have a range, the velocity scale itself is specified by a range. As an engineering approach, the velocity scale for specifying the sampling period can be based on the ratio of the mean wind speed in the wind tunnel to the design value of the mean wind speed in full-scale conditions (Kasperski, 2003). In its original concept, Cook and Mayne supposed that the extremes of the aerodynamic coefficient follow a type I distribution. This assumption is usually on the safe side. In the following some examples are presented indicating which type of distribution is to be expected for which type of aerodynamic coefficient.

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347

0.999

non-exceedance probability

stagnation

0.99

0.9

0.5 0.1 0.01 0.001 0

1

2

3 cp, max

4

5

6

Fig. 16. Example of extreme pressures at stagnation taken from full-scale experiments.

Fig. 17. Traces of the non-exceedance probability of drag and lift for a 451-gable roof.

5.2. Local pressures A first example of extreme local pressures is taken from an extensive wind tunnel study performed at the Ruhr-University Bochum. Object of the study is a low-rise building with span to width to height ¼ 1.6/1.0/0.4. The roof pitch is 51. Simultaneous pressures have been measured at the centre bay (Kasperski, 2000) for 1200 independent runs each corresponding to 10 min in full scale. Fig. 15 shows the traces of the positive extreme pressures for the stagnation and the negative extreme pressures for the leading edge. The trace for the positive pressures shows a clear curvature, the identified curvature parameter is 0.1. For the extremes of local pressures at the leading edge, no curvature is observed, i.e. the observed trace is best represented by a type I distribution. In Fig. 16, some full-scale results for the Silsoe-cube are shown. Based on 81 independent runs each with a length of about 7 min, a considerable curvature is observed for the pressures at stagnation (Kasperski and Hoxey, 2008). The extremes of positive pressures on the upwind face can be represented by a type III distribution with m ¼ 2.63, s ¼ 0.51 and t ¼ 0.26.

5.3. Global wind forces Examples for the traces of global wind forces are taken from a study on gable roof pressures. In Fig. 17, the drag and the lift for the centre bay of a 451 roof are shown based on 60 individual runs corresponding to 10 min in full scale. While the extremes of the drag coefficient show a considerable curvature and therefore are

Table 3 Typical behaviour of the characteristic values of the extreme drag and lift of a highrise building with b/d/h ¼ 40/40/160 (m)-flow normal to the front face (values normalized to mean velocity pressure at the height h of the building). z/h

0.05 0.15 0.30 0.45 0.60 0.75 0.85 0.95

Drag

Lift

Mean value

rms value

Curvature

Mean value

rms value

Curvature

1.740 1.820 1.896 1.929 1.976 2.090 2.161 2.121

0.211 0.205 0.181 0.181 0.167 0.179 0.179 0.164

0.23 0.18 0.22 0.18 0.12 0.17 0.17 0.14

71.122 71.252 71.287 71.279 71.248 71.135 71.083 70.977

0.238 0.247 0.247 0.237 0.218 0.201 0.206 0.160

0.02 0.01 0.06 0.07 0.08 0.10 0.04 0.06

best represented by a type III distribution, the extreme lift coefficients follow more or less a straight line and are best represented by the type I distribution. For a high rise building with square cross-section shape 40  40 m and height 160 m, drag and lift have been analysed at eight different levels z/H. The study is based on 240 independent runs each corresponding to 10 min in full scale. In Table 3, the identified parameters of the extreme value distributions are summarized. For the extreme drag coefficient, the curvature parameter are quiet large, while for the lift coefficients the curvature parameters usually remain small.

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Fig. 18. Traces of the non-exceedance probabilities of the extreme bending moments for a low-rise building (all values in kN m/m frame spacing per kPa mean velocity pressure at eaves height).

non-exceedance probability

0.999

0.99

0.9

0.5 0.1 0.01 0.001 0.0

2.5

5.0

7.5

10.0

12.5

Mmax Fig. 19. Irregular trace of the non-exceedance probability of a bending moment in an arched roof structure.

5.4. Wind induced action effects On the example of a low-rise building, Fig. 18 shows the traces of extreme wind induced action effects in terms of the bending moments at the centre bay for flow normal to the ridge. While the extremes of the bending moment in the upwind corner show a clear curvature, the extremes in the downwind corner are—if at all—best represented by a straight line. It is important to note that a homogenous trace only can be expected if there is only one physical origin for extremes. For extremes of local pressures and global forces, this might generally be the case. For the downwind bending moment, clearly different pressure distributions may induce extreme values of the structural response, and strictly speaking the ensemble then has to be sub-divided to separate the different situations of wind actions. A further example of a probably mixed distribution is shown in Fig. 19, which presents the trace of extreme bending moments in an arched roof construction.

6. Convolution 6.1. Accumulation of the exceedance probability of the design wind load The basic interaction between the two random variables v and c in regard to the accumulation of the exceedance probability of the design wind load is shown in Fig. 20. If both the extremes for v

and for c show considerable scatter, an exceedance of the design wind load amplitude wdes may be obtained for any wind speed level. The respective contributions to the exceedance probability are obtained as the product fv(v)  (1Fc(v)). The area under this curve has to meet the target value of the exceedance probability of the design wind load. The required design value of the aerodynamic coefficient cannot be obtained by an explicit solution of Eq. (13), but has to be obtained iteratively. In the example, a target value of 0.001 for the exceedance probability of the design wind load is used, assuming in a first step that a storm exactly lasts for only 1 h. The first important point to notice is that the range of wind speeds contributing to the final value of the exceedance probability is fairly large. First significant contributions are obtained for wind speeds which are considerably lower than the design wind speed. Altogether, in this example about 50% of the target exceedance probability is obtained for wind speeds lower than the design wind speed. Consequently, 50% of the damages are to be expected in this range of wind speeds. The second important finding is that the 2nd or 3rd strongest storm hours have a significant contribution to the exceedance probability. In the example, the relative intensity of the second hour is 0.97, for the third hour 0.93. The accumulated exceedance probability considering the additional storm hours will be about 40% larger. The required increase of the design wind load to meet 1 will be discussed later. the initial target value of 1000 The range of wind speeds which contribute to the accumulated exceedance probability strongly depends on the variation

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0.4

1.00

0.3

0.75

349

assumptions:

fv

1-Fc

0.50

0.2

0.25

0.1

0.0

Fv ⋅(1−fc)

10

15

20 25 wind speed [m/s]

30

0.00 35

1 - Fc (clim)

fv (v)

ρ = 1.25 kg/m3 v follows type I with m = 16 m/s and σ = 2 m/s p(v ≤ vdes) = 0.999 ⇒ vdes = 25.9 m/s

c follows type I with m = 1 and σ = 0.15

3.0x10-6

p(c ≤ cdes) = 0.78

2.5x10-6

⇒ cdes = 1.095

2.0x10-6

wdes = 0.458 kN/m2

1.5x10-6 duration of storm: one hour

1.0x10-6 p (w> wdes) = 0.00096

5.0x10-7 0.0 10

15

20

25

30

35

3.0x10-6

Fv ⋅(1−fc)

2.5x10-6

duration of storm: three hours

2.0x10-6

rel. I1 = 1.00 rel. I2 = 0.96 rel. I3 = 0.93

1.5x10-6 1.0x10-6 5.0x10

p (w> wdes) = 0.00137

-7

0.0 10

15

20

25

30

35

Fig. 20. Accumulation of the exceedance probability of the design wind load, influence of the duration of a storm.

coefficient of the extreme aerodynamic coefficient. This is shown in Fig. 21. For very small variation coefficients (cov(c) ¼ 0.01), the accumulation more or less sharply starts beyond a threshold value, which is the design wind speed. As the variation coefficient increases the accumulation curve becomes broader and the maximum is shifted to lower wind speeds. The relative accumulation from 0 to 1 for the full exceedance probability is shown in Fig. 22. The level of the design wind speed with p(v4vd) ¼ 0.001 and the level of the characteristic wind speed with p(v4vk) ¼ 0.02 are marked in the diagram by vertical lines. If a storm with the characteristic wind speed occurs, the exceedance probability of the design wind load for a structure having a wind load with cov(c) smaller or equal 0.1 is practically zero. For larger variation coefficients, the fraction of the total exceedance probability increases to 3% and 10% for variation coefficients of 20% and 30%, respectively. These different accumulation curves can be understood as indicators for damages to be expected at different wind speed levels. It is important to note, that the expected amount of wind induced damages in a storm of a specific intensity will not be distributed homogeneously for all structures, but that the variation coefficient of c ‘triggers’ the amount of damages from lower to higher values.

In Fig. 22, additionally the normalized rate of expected damage is shown, using as reference the fraction of exceedance probability for cov(c) ¼ 0.1. For the level of the characteristic wind speed, expected damages for cov(c) ¼ 0.20 and 0.30 will be about 20–55 times higher than for cov(c) ¼ 0.10. For the level of the design wind speed, the respective ratios still are considerably different with roughly a factor of 2 for the expected damages in case of higher variation coefficient of c.

6.2. Influence of the distribution of v and c on the design value of the aerodynamic coefficient The original concept by Cook and Mayne is based on the wind climate in the UK considering for each storm only the largest hourly mean wind speed. The Cook–Mayne approach recommends using the 78%-fractile value assuming that the extreme aerodynamic coefficients follow the type I distribution. The 78%fractile can be obtained from the mean value and the standard deviation of the extremes as follows: x78% ¼ mx þ 0:636sx ¼ mð1 þ 0:636V x Þ

(27)

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Fig. 21. Influence of the variation coefficient of the extreme aerodynamic coefficient on the range of velocities contributing to the exceedance probability of the design wind load.

vk

vd

vk

100

vd

0.9 0.8

relative amount of damages

fraction of total exceedance probability

1.0

0.7 0.6 0.5 0.4 cov(c) =

0.3

30% 20% 10%

1%

0.2

cov(c) = 20%

30%

10

0.1 1

0.0 16

18

20

22 24 26 28 30 hourly mean wind speed [m/s] fraction of exceedance probability

32

34

16

18

20

22

24

26

28

30

32

34

hourly mean wind speed [m/s] expected damage rates normalized to cov(c) = 0.10

Fig. 22. Influence of the variation coefficient of the extreme aerodynamic coefficient on the fraction of accumulated exceedance probability, expected normalized damage rates.

with mx the mean value of the extreme aerodynamic coefficients, sx the standard deviation of the extreme aerodynamic coefficients, and Vx the variation coefficient ¼ sx/mx. The term in brackets in Eq. (27) can be understood as an adjusting factor to translate the mean value of the extremes to the required design value. Strictly speaking, the required or ‘optimum’ adjusting factor depends on the respective variation coefficients of the extreme wind speeds and the extreme aerodynamic coefficients. This dependency is shown in Fig. 23. Additionally, the simplified conversion factor to obtain the ‘Cook–Mayne coefficient’ is shown. The ‘Cook–Mayne’ coefficient is leading to an underestimation of the ‘optimum’ design value if the variation coefficient of the extreme aerodynamic coefficient becomes large and the variation coefficient of the extreme wind speeds becomes small (Fig. 24). The adoption of ‘Cook–Mayne’ coefficients therefore requires an analysis of the extreme wind climate in regard to the variation coefficient of extreme wind speeds. A change in the type of distribution for either v or c from types I to III has the following effects: For increasing curvature parameter tv, the isoline structure is rotating counter-clockwise, i.e. the required adjusting factors are increasing (Fig. 25). For increasing curvature parameter tc, the isoline structure is rotating clock-wise, i.e. the required adjusting factor is decreasing (Fig. 26).

The combinations of cov(v) and cov(c) leading to an under-design are marked in the respective figures as shaded areas. It is important to note that trying to exploit the favourable effects of assuming a type III distribution for v leads to larger design values of the aerodynamic coefficient. The effective gains in terms of a reduction of the design wind load will be considerably smaller than is suggested when comparing the design values of the wind speeds assuming either type I or III distribution. The question on how large the possible gains finally may become is answered in Fig. 27. The isolines show the ratio of the ‘optimum’ design wind load to the conservative approach assuming that both variables follow the type I distribution. For small variation coefficients of c and large variation coefficients of v, the possible gains are quiet large. However, as explained in Section 4, the statistical instability of the identified curvature parameter of v unfortunately inhibits the exploitation of these favourable effects. The basic gains for cases where the extreme coefficients follow a type III distribution generally are less profitable. For a curvature parameter of tc ¼ 0.1, the range of possible gains is between 5% and 10% for the wind climate in Germany (cov(v) ¼ 0.125). If the curvature parameters increases to tc ¼ 0.2 and larger, a refinement of the design value of the aerodynamic coefficient seems to be beneficial in terms of

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Fig. 23. Adjusting factor for the mean extreme aerodynamic coefficient to obtain a ‘Cook–Mayne’ coefficient and the ‘optimum’ design value.

Fig. 24. Ratio of the ‘Cook–Mayne’ coefficient to the ‘optimum’ design value.

Fig. 25. Adjusting factor to obtain the optimum extreme coefficient (extreme value distribution v: type III-extreme value distribution c: type I).

Fig. 26. Adjusting factor to obtain the optimum extreme coefficient (extreme value distribution v: type I-extreme value distribution c: type III).

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Fig. 27. Reduction of the design wind load for different combinations tvtc compared to the worst combination tvtc ¼ 0.0–0.0.

1.40

6.0

1.35

5.5

1.30

5.0 mismatch time scale

adjusting factor

1.25 1.20 1.15

1

1.10 1.05 1.00 0.95

2

1.25

4.5 4.0

1.20

3.5

1.15

3.0 1.10

2.5 2.0

0.90

1.05

1.5

0.85 0.80 0.00

1.30

0.05

0.10

0.15

0.20

0.25

cov (c) 1 - adjusting factor for 78%-fractile one hour 2 - mismatch of 10-minute design load

0.30

1.0 0.00

0.05

0.10

0.15 0.20 0.25 0.30 cov (c) required adjusting factor due to mismatch in time scale

Fig. 28. Required adjusting factors for mismatches in storm duration or time scale.

comparing possible gains of up to 20% to the increased experimental effort to identify the type III distribution of c with sufficient statistical stability.

6.3. Influence of the duration and the time scale As mentioned above, it is important to consider the duration of the storm events. The duration is thereby counted in multiples of the basic reference period, which is 1 h for strong frontal depressions and 10 min for tropical cyclones. Additionally, the relative intensity of the 2nd, 3rd to kth strongest period has to be considered. The following example is based on assuming for the extremes of v and c type I distributions. The first question deals with the error that occurs if the duration of a frontal depression erroneously is assumed to be 10 min instead of 1 h. The respective 78%-fractile in 10 min is considerable smaller than the 78%-fractile in 1 h. To compare the

two load models, additionally, the ‘translation’ factor between the 10 min mean value and the hourly mean value has to be applied. For the wind climate of Germany, 1.06 has been applied to the original statistics for hourly mean wind speeds (Kasperski, 2002). In Fig. 28, the adjusting factor for translating the 78%-fractile value for 10 min to the 78%-fractile value is shown. Additionally, the ratio between the 1-h design wind load and the 10-min design wind load is presented. The larger the variation coefficient of c becomes, the larger will be the error in the design wind load when applying instead of the 1 h storm duration only 10 min. In Fig. 28, additionally the influence of a mismatch in the time scale is shown. The shorter the time window used for sampling the peaks has been chosen, the larger will be the error in the estimated fractile-value. It is important to note, that the ‘optimum’ design wind load may differ from the 78%-fractile approach, hence, the total error produced by a mismatch of the time scale or the duration may be considerable larger than obtained from Fig. 28.

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at Brunsbu¨ttel, 4 h at Bremen, 3 h at Du¨sseldorf and only 2 h at Frankfurt. The required adjusting factors for the four stations are shown with reference to the 78%-fractile value. They are considerably different for larger values of the variation coefficient of c. While for the favourable wind climate at Frankfurt the adjusting factor remains smaller than 1.1 up to a variation coefficient of 0.23, for the severe wind climate at Brunsbu¨ttel the adjusting factor exceeds 1.1 already for a variation coefficient of 0.18. The differences between Bremen and Du¨sseldorf, on the other hand remain small, which leads to the conclusion to use for the specification of the design value of c the average wind climate, i.e. for Germany 3 h of duration and relative intensities of 0.97 and 0.93 for the 2nd and 3rd strongest hour. In the example so far, the influence of the range of winds speeds has not yet been considered. This influence will be discussed in the final section of the paper, when the question is answered how to implement the design value of the aerodynamic coefficient in a code.

The next example illustrates the influence of a second storm hour or more generally of a second time period of the storm. The variation coefficient of the wind climate has been set to 0.125, which more or less corresponds to strong frontal depressions in Western Europe and Germany. In Fig. 29, the required adjusting factor for the strongest storm hour for a code concept based on mean extreme or the 78%-fractile of extremes is shown. A second storm hour has been considered varying its relative intensity from 0.85 to 1.00. The influence of a third hour is shown in Fig. 30. The example assumes that the 2nd storm hour has a relative intensity of 0.97; therefore, the relative intensity of the 3rd (strongest) storm hour is varied from 0.85 to 0.97 (Fig. 30 left). The complete ‘chain’ of adjusting factors is also shown with reference to the mean extreme value of the aerodynamic coefficient (Fig. 30, right). Usually, the wind climate over a larger area will differ in regard to the duration of storms. For Germany, storms tend to last longer at the coast and tend to be shorter further inland. In the following example this is illustrated based on four typical stations in Germany. The first station Brunsbu¨ttel lies at the mouth of the Elbe-river and belongs to the strongest wind zone IV in Germany, the second station Bremen is situated in the second strongest wind zone III, while Du¨sseldorf and Frankfurt are situated in the weakest wind zone I. A map in Fig. 31 illustrates where these cities are situated. Additionally, in Fig. 31 the relative intensities of the respective storm hours are shown. Storms last on average 5 h

6.4. Specification of the design value of the aerodynamic coefficient in a code Basically, there are two further problems which have to be addressed. The first question deals with the appropriate reference pressure. Strictly speaking the choice of the reference pressure is

1.35 mean value

78%-fractile

1.30

rel. int. 2nd hour

cad (1st hour)

1.25 1.20 1.15 1.10 1.05 1.00 0.95 0.00

0.05

0.10

0.15 cov (c)

0.20

0.25

353

0.30

1.00 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 0.90 0.89 0.88 0.87 0.86 0.85 0.00

1.08 1.07 1.06 1.05 1.04 1.03 1.02 1.01

0.05

0.10

0.15

0.20

0.25

0.30

cov (c)

adjusting factor for 1st hour

additional adjusting factor for 2nd storm hour

1.00 0.99 0.98 0.97

1.04 hour

1.03 1.02 1.01

1.45

0.96

rd

1.00 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 0.90 0.89 0.88 0.87 0.86 0.85 0.00

rel. int. 3

rel.int. 3rd hour

Fig. 29. Adjusting factors for the 1st and 2nd storm hour.

1.40 1.30 1.35 1.25

0.95 0.94 0.93

1.20

0.92

1.15

0.91 0.90

1.10

0.89

1.05

0.88 0.87 0.86 0.05

0.10

0.15 cov (c)

0.20

0.25

0.30

0.85 0.00

1.01 0.05

0.10

0.15

0.20

0.25

cov (c)

additional adjusting factor for 3rd hour

st

nd

adjusting factor for 1 , 2

Fig. 30. Influence of the 3rd storm hour.

and 3

rd

storm hour

0.30

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Brunsbüttel

Bremen

Düsseldorf

Frankfurt

1.0

relative intensity

0.8

0.6 Brunsbüttel

0.4

Bremen 0.2

0.0 1st

2nd 3rd 4th ranked average storm hour

5th Düsseldorf

1.45 1.40 1.35

1 - Brunsbüttel 2 - Bremen 3 - Düsseldorf 4 - Frankfurt

Frankfurt am Main

1.30 1.25 cad

1

1.20

2 3

1.15

4

1.10 1.05 1.00 0.95 0.00

0.05

0.10

0.15

0.20

0.25

0.30

cov (c)

Fig. 31. Influence of the duration of storms on the example of four stations in Germany.

completely free, however, with the only restriction that the same reference pressure has to be used in the code when transforming the experimentally obtained aerodynamic coefficient into a local pressure, global force or structural response. At least for wind tunnel experiments, the easiest way of getting aerodynamic coefficients is to refer the measured pressures or forces or structural responses to a mean velocity pressure. The choice of the height of the reference pressure remains free, and consequently local pressures for e.g. gable roofs have been referenced to the pressure at eaves height, or mid-roof height or the height of the ridge. Depending on the slope of the roof, the reference pressures may be considerably different, leading to differences in the aerodynamic coefficients. Using the mean velocity pressure as reference, however, will lead to complete different numbers as are actually used in our codes. In the historic approach to the codification of wind loads, gust wind speeds have been used, and practising engineers are rather familiar with a maximum local pressure coefficient of e.g. 1.0 for a vertical upwind wall and might get confused with the numbers presented in Fig. 1, which specifies for the local pressure at stagnation extreme values in the range from 2.5 to 3.0. For sake of high recognition value, it is therefore reasonable to use gust wind pressures to obtain aerodynamic coefficients for the code. These transformed aerodynamic coefficients have been named pseudo-steady pressure coefficients (Cook, 1990). Using gust velocity pressures as reference is also thought to minimize the variability with terrain roughness, i.e. aerodynamic coefficients which have been obtained for the standard terrain ‘open country’ can be used as good approximations for other terrain roughness. Strictly speaking, the choice of the transformation factor is free, as long as the code reproduces the applied transformation factors in terms of the squared gust factor. To obtain for the 78%-fractile of the extreme local pressures at stagnation the desired pseudosteady coefficient of 1.0 requires a gust factor of 1.69. If observed

gust factors for an averaging time of 3 s do not agree to this value, a shorter averaging time allows to ‘trigger’ the gust factor until the desired value is obtained. The simulations used in Fig. 6 lead to an average gust factor of 1.64 for the 3 s-gust and 1.69 for the 1 s-gust. In the next step, the transformation factors of some leading codes are compared in Fig. 32 for the reference terrain open country. The terrain roughness ‘open country’ still leaves to some extent space for interpretations. Therefore, in Fig. 32 additionally the gust pressure profiles as published in the ESDU-document for roughness heights from z0 ¼ 0.02 to 0.07 m are shown as shaded area. It is important to note that these gust pressures profiles have been obtained by ESDU applying different values for the integral length scale than have been used for the above simulations and are based on a modified Karman-spectrum. The differences in the proposed profiles are quiet large, and there are profiles lying well outside the range predicted by the ESDU-model. The question which of these profiles represents the ‘truth’ is obviously difficult to answer; therefore the Eurocode decided to leave this decision to the national committees and gives only a recommendation for a gust pressure profile. This decision leaves the perfect impression of complete arbitrariness in regard to this decisive parameter and is thought to cause further confusion amongst practising engineers. Assuming that at least some of the aerodynamic coefficients published in the Eurocode have been obtained applying the gust pressure profile of the British Standard leads to the conclusion that additional factors are required to adjust for the differences between the profile used to obtain the pseudo-steady aerodynamic coefficient and the profile going to be recommended by the individual national committees. To adjust e.g. the DIN-profile to the BS-profile requires a factor of about 1.2. It is worth mentioning that at least the differences in the gust pressure profiles as recommended in the Eurocode and the gust

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3

1

4

2

6 5 1 - DIN 1055 Teil 4 : 03.2005

45

2 - EN 1991-1-4: 04.2005

40

3 - ISO 4354 - DIS: 05.2007 4 - ASCE 7-02: 2002

35

5 - AS /NZS1170.2: 2002

30

6 - BS 6399-2: 1997

25 20 15 U

height above ground [m]

50

355

ES D

10 5 1.0

1.5

2.0 2.5 3.0 3.5 factor for gust velocity pressure

4.0

1.0

1.0

0.9

0.9

0.8

0.8

0.7

0.7 relative intensity

relative intensity

Fig. 32. Profiles of the gust velocity pressure as proposed in some leading codes compared to the expected range based on the ESDU-model (normalized to the hourly mean wind speed at 10 m above ground-open country).

0.6 0.5 0.4

0.6 0.5 0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.0

0.0 50

52

54 56 58 60 62 64 66 68 current no. of measured 15-minute interval

70

Rita

10

12

14

16

18

20

22

24

26

28

30

current no. of measured 15-minute interval Wilma

Fig. 33. Example of the time history of 15-min mean wind speeds for moderately severe tropical cyclones over Florida (max. v15 min ¼ 35 m/s).

profile specified in the German DIN 1055 can be explained. The DIN-profile has not been obtained to represent the ESDU-model, but aims in reproducing directly global actions and therefore has implemented a general reduction factor of 0.9 for favourable correlation effects. To reproduce with the reduced reference pressure the appropriate amplitudes for the local actions, a factor 1 is required, supposing the EC-profile times the aerodynamic of 0:9 coefficients leads to the correct wind induced loads. Consequently, in its original concept, the DIN-document introduced an additional load factor of 1.1 for local pressures. Unfortunately, during the objection and revision phase, this factor has got lost, which means that in the worst case local pressures now are underestimated with the DIN-concept. The second question deals with the problem that some codes have to cover a large variety of different wind climates, e.g. the Eurocode has to deal with climates which are governed by strong frontal depressions having their origin over the North-Atlantic, with climates governed by thunderstorms, with climates additionally influenced by katabatic winds like the Mistral in France or the Bora in the Adriatic, climates governed by strong synoptic storms like in the Eastern Aegaeis and so on. The probably worst example of different wind climates is obtained for Australia, for

which the Australian Standard specifies four regions A to D. With the additional information provided in Holmes (2001), the following descriptions for the different wind climates can be given: A. thunderstorms with 3 s-gust wind speeds between 30 and 45 m/s and synoptic storms with 10-min means between 20 and 30 m/s; B. weakening tropical cyclones with 10-min means in the range from 30 to 35 m/s; C. moderately severe to severe tropical cyclones with 10-min means between 35 and 45 m/s; and D. severe tropical cyclones with 10-min means between 45 and 55 m/s. When specifying the design values of the aerodynamic coefficients, at least the different levels of wind speed and the different durations and corresponding relative intensities of the respective storm types have to be considered. In Fig. 33, the time-histories of 15-min mean wind speeds for two category III hurricanes (Rita and Wilma, 2005) in Florida are shown (Florida Coastal

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0.30

0.30 0.94

0.65

0.25

0.25

0.95

0.70

0.15

0.80 0.85

0.10

0.96

0.20

0.75

cov (c)

cov (c)

0.20

0.97 0.98

0.15

1.0 0.99

0.10

1.0

0.90

0.05

0.05 0.95 thunderstorms

strong frontal depressions

0.00

0.00 20

25 30 equivalent mean wind speed [m/s]

35

20

0.30

0.30

25 30 hourly mean wind speed [m/s]

35

0.86

0.96

0.25

0.90

0.97

0.98

0.15 0.10

0.20

1.0

cov (c)

0.20 cov (c)

0.88

0.25

0.99

0.92

0.15 0.94

0.10

1.0

0.05

0.96

0.05

0.98 severe tropical cyclones

moderate tropical cyclones

0.00

0.00 35

40 45 10-minute mean wind speed [m/s]

50

40

45 50 10-minute mean wind speed [m/s]

55

Fig. 34. Tentative adjusting factors to cover the different wind climates in Australia.

Monitoring Program, 2007). For both storms, similar maximum values for the 15 min means have been observed, classifying both storms as moderately severe tropical cyclones. The contributing storm sub-periods of the two storms are considerably different. While for Rita there is only one sub-period contributing to the exceedance probability of the design wind load, for hurricane Wilma at least four sub-periods have to be considered. Based on this example it can be generally concluded that describing tropical cyclones with only one single sub-period is not sufficient. In Fig. 34, an example of tentative adjusting factors covering the different wind climates in Australia is shown. As reference, the mean extreme value obtained for an hourly mean wind speed of 30 m/s is used. For each region A to D it is assumed that only one storm phenomenon is dominant and that the appropriate design value can be specified by the 78%-fractile value referring to the respective durations of the different storm types. As a first approach, for each storm type the same variation coefficient of 0.125 for the extreme wind speeds is introduced. The following scenarios are used to obtain the adjusting factor:

 thunderstorms with an average duration of 10 min and equivalent mean wind speeds from 20 to 35 m/s;

 strong frontal depressions with an average duration of 1 h and hourly mean wind speeds from 20 to 35 m/s;

 weakening or moderate severe tropical cyclones lasting with constant intensity for four sub-periods with 10 min mean speeds between 35 and 50 m/s; and

 severe tropical cyclones with two sub-periods of 10 min and maximum 10-min mean wind speeds between 40 and 55 m/s. Due to the short duration of thunderstorms, the design values of the aerodynamic coefficient could be reduced assuming that the structure of the turbulence in thunderstorms is sufficiently similar to that observed in strong frontal depressions. For a typical variation coefficient of c between 0.1 and 0.15 the possible reduction is about 15%–20%. The different intensities of the storm have only an influence of about 5% in this range. For strong frontal depressions, the influence of the intensity of the storm on the 78%-fractile value can also be neglected. Estimations on the safe side are obtained when the upper value of the wind speed to be covered in the code is introduced as reference. If the range is not too large, the mean value of the range can be used as well. Tropical cyclones on the other hand have a considerably higher intensity than strong frontal depressions. This basically will lead to larger design values of the aerodynamic coefficient. However, this effect is balanced by the probably shorter duration of the phases with highest intensity. With the above introduced tentative values, the adjusting factor for moderate tropical cyclones is almost 1.0 for the range of the variation coefficient from 0.1 to 0.15. For severe tropical cyclones, assuming that they have only two 10-min periods of strongest intensity, the design value of the aerodynamic coefficients could be reduced. In conclusion, the design value of the aerodynamic coefficient referring to strong frontal depressions in this example leads to a conservative approach for the other wind climates.

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7. Conclusions The findings so far can be summarized in some concluding recommendations for the codification of wind induced actions:

 In the design process, generally more than one design task has





to be met, e.g. design values have to be specified for local loads, global loads and structural loads. The usual codification practice of specifying only one set of coefficients and giving some remedy rules to balance the shortcomings of this approach is obsolete. The recent published model code (ISO 4354 Wind Action on Structures—DIS 05, 2007) solves this problem by specifying the extreme aerodynamic coefficients for local pressures in tables and presenting the required simultaneous distributions of aerodynamic coefficients in figures. The projected design working life should be considered as explicit design variable. For the ultimate limit state, appropriate target values for the exceedance probability of the design wind load have to be specified with reference to this period. Target values for the serviceability limit state may be referred to a single year. It is reasonable to specify the design targets in dependence on the consequences of a failure, i.e. to introduce structural classes. Basically, design values should be specified explicitly and should not be hidden behind characteristic wind speeds and partial factors. The analysis of the extreme wind climate has to be based on separate storm mechanisms, e.g. strong frontal depressions or other synoptic storms, gust fronts, thunderstorms and tropical cyclones. Yearly extremes are usually not leading to an exhaustive ensemble of extreme wind speeds. Instead, for the specification of the design value of the wind speed, maximum wind speeds from individual storms have to be sampled. Individual storms are identified as events above a sufficiently high threshold value and are separated by a minimum time lag appropriate to the considered storm type. The number of storms per 12-month cycle (aeolian year) then becomes a basic variable which has to be considered in the further method of specifying the design wind load. The non-exceedance probability per event can be translated to the required yearly nonexceedance probability by convolving the probability per event with the probability of the numbers of storms per year. Generally, an individual storm may last longer than the representative period used to sample the wind speeds (e.g. 10 min or 1 h). Consequently, a consistent model of the extreme wind climate has to consider the average duration of the storm and the average relative intensities of the 2nd to kth strongest storm sub-period as well. Since the observation periods are far too short to get a stable estimate of the parameters of the type III distribution, the maximum wind speeds should be fitted to a type I distribution. To avoid additional adjusting factors for

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meeting the target 75%-confidence interval, the original Gumbel-concept with a ¼ 0 should be used for the estimation of the non-exceedance probability per event. The design values of the aerodynamic coefficient should be specified as appropriate fractile values of the respective extremes. When assuming a type I distribution, usually the 78%- or 80%fractile value of the extremes can be used as design value, however, with reference to the entire duration of a storm. If the variation coefficient of the aerodynamic coefficient becomes large, say it exceeds 20%, the use of higher fractile values is strongly recommended. The estimation of the fractile values should be based on a sufficient large number of independent runs which have the appropriate duration when translated to full-scale conditions. The stable identification of the parameters of the type III distribution requires a very large number of independent runs. For codification purposes therefore the type I distribution is also recommended as generally appropriate model for the aerodynamic coefficients. References AIJ Recommendations for Loads in Buildings, 1996. Architectural Institute of Japan. AS 1170.2, 1989. Australian Standard—Minimum Design Loads on Structures, Part 2: Wind Loads Standards Australia, Sydney. Cook, N.J., 1990. The Designer’s Guide to Wind Loading of Building Structures. Part 2: Static Structures. Butterworths, London. Cook, N.J., Mayne, J.R., 1980. A refined working approach to the assessment of wind loads for equivalent static design. Journal of Wind Engineering and Industrial Aerodynamics 6, 125–137. Cunnane, C., 1978. Unbiased plotting positions—a review. Journal of Hydrology 37, 205–222. EN1990, 2002. Eurocode: Basis of Structural Design. EN1991-1-4, 2005. Eurocode 1: Actions on Structures—Part 1–4: General Actions–Wind Actions. ESDU—Engineering Science Data Unit, 1985, 1986. Characteristic of atmospheric turbulence near ground. Single point data for strong winds, ITEM 85020. Variations in space and time for strong winds, ITEM 86010. Florida Coastal Monitoring Program, 2007. Collected Data: /http://users.ce.ufl. edu/fcmp/collected_data/data.htmS. Gringorten, I.I., 1963. A plotting rule for extreme probability paper. Journal of Geophysical Research 68, 813–814. Holmes, J.D., 2001. Wind Loading of Structures. Spon Press. Holmes, J.D., Cochran, L.S., 2003. Probability distributions of extreme pressure coefficients. Journal of Wind Engineering and Industrial Aerodynamics 91, 893–901. ISO 4354 Wind Action on Structures—DIS 05, 2007. Kasperski, M., 2000. Specification and codification of design wind loads. Habilitation Thesis, Department of Civil Engineering, Ruhr-University Bochum. Kasperski, M., 2002. A new wind zone map for Germany. Journal of Wind Engineering and Industrial Aerodynamics 90, 1271–1287. Kasperski, M., 2003. Specification of the design wind load based on wind tunnel experiments. Journal of Wind Engineering and Industrial Aerodynamics 91, 527–541. Kasperski, M., Hoxey, R., 2008. Extreme value analysis for observed peak pressures on the Silsoe-cube. Journal of Wind Engineering and Industrial Aerodynamic, 994–1002. Naess, A., Clausen, P.A., 2001. In: Corotis, R.B., et al. (Eds.), Effects of Data Accuracy on the POT Estimates of Extreme Values Structural Safety and Reliability. Swets & Zeitlinger, ISBN 90 5809197 X.