ACTIONS ON STRUCTURES

WIND LOADS

CIB Report Publication 193

Actions on Structures

April 1996

O CIB

PREFACE

The Task of CIB Working Commission W81

The Task of the commission is to develop a set of stochastic models for actions which are

mutually consistent and which can be used both in probabilistic design and analysis and as

a basis for deterministic models for actions.

This work will be described in a series of reports having a common general title "Actions

on Structures". There will be one report dealing with general principles which are common

for many kinds of actions and a number of reports each describing a specific action.

The documents published in these series are:

Actions on Structures

Actions on Structures

Actions on Structures

Actions on Structures

Actions on Structures

Actions on Structures

Actions on Structures

Actions on Structures

Self-weight Loads

Live Loads in buildings

Snow Loads

Fire

lmpact

Loads in car parks

Wind loads

Traffic loads on bridges

CIB Publication 1 15

CIB Publication 1 16

CIB Publication 141

CIB Publication 166

CIB Publication 167

CIB Publication 194

CIB Publication 193

CIB Publication 195

Actions on structures - Wind loads

This report describes those aspects of windload that are considered to be important in

probabilistic load modelling. It has not been the intention that the report includes detailed

information on wind loads at specific sites since this can be obtained from literature and

windtunnelexperiments. An extensive list of references has been included for this purpose.

This Report seeks to review and summarise current methods of representing Wind Loads

in the design of ordinary buildings and structures. These methods, in many respects, are

also relevant to the design of more complex structures, but full and complete details for

the design of such special structures are outside the scope of this Report.

This report was prepared by E.C. Klaver of the Technical University Delft, Department of

Civil Engineering, taking into account an earlier draft of N.J. Cook, and contributions by

A. Vrouwenvelder, R. Rackwitz and A. Larsen.

Keywords: wind, statistics, building structures

LIST OF SYMBOLS

pressure coefficient cpE = pE /q

force coefficient cm = FE /(q qef )

moment coefficient cME = ME /(q qef Gef )

coherence function of u l and u2

zero plane displacement 2

reduced spectrum FD = o, S,, If

cumulative distribution of the mean wind speed V

frequency

natural frequency of structure

Coriolis parameter (=lo4 rad/s for temperate latitudes)

probability density function of At-mean pressure q

probability density function of the annual maximum of the At-mean of

windpressure qE = 0 . 5 ~ v ~

Gumbel distribution of the maxima (or minima) of the pressure coefficient

CE = F/q in a period At

structural frequency response function

aerodynamic transfer function

(longitudinal) turbulence intensity

influence function used in the engineering model

aerodynamic coefficient

length scales of turbulence in x-direction relating to u, v and w components

mean wind pressure q = 0.5pV 2

representative value of the windpressure

denotes point of application (x, y, z)

variance spectrum of the inwind fluctuations u

variance spectrum of the cross wind fluctuations v

cross spectrum of ul and u2

inwind fluctuations of windspeed

friction velocity

fluctuating components of windspeed along x, y and z axes respectively.

parameters describing the Gumbel distribution PCE of the maxima (or

minima) of the pressure coefficient cE = p(t)/q

parameters describing the Gumbel disEibution of the annual maximum of

the 10 min-mean (or At-mean) windpressure q = 0.5pV 2

parameters describing the Gumbel distribution of the annual maximum of

the load effect due to wind

10 minute-mean of (undisturbed) windspeed

annual maximum of the mean windspeed

often VI0 , mean windspeed at 10 meter height

system of rectangular cartesian co-ordinates with x-axis defined in direction

of mean wind

height above groundlevel

aerodynamic roughness length of the surface

reference height

gradient height (=u1/6fCor)

exponent in power law description of V(z)

mass density of air

ow standard deviation u, v and w components

damping ratio

von Karman constant (=0.4)

circular frequency (=2xf)

reduction coefficient, gamma function

aerodynamic admittance (or joint admittance)

CONTENTS

1 INTRODUCTION

2 CLIMATIC BACKGROUND

2.1 Atmosferic circulation

2.2 Depressions

2.3 Tropical cyclones

2.4 Tornadoes

2.5 Dustdevils

2.6 Thunderstorms

2.7 Katabatic winds and lee waves

3 WINDSTRUCTURE

3.1 Introduction

3.2 Longterm stationary model (Macrometeorologic model)

3.3 Short term model (Micrometeorologic model)

4 PRESSURES, FORCES AND RESPONSE OF STRUCTURE

4.1 Wind Force models

4.2 Kinetic pressure and loading coefficient

4.3 Models for building areodynamics

5 THE BLUFF BODY MODEL

5.1 Probabilistic model

5.2 Codification

6 ENGINEERING MODEL

6.1 Static structures

6.2 Dynamic structures

6.3 Aeroelasticity

6.4 Model uncertainties

REFERENCES

APPENDIX A: MEASUREMENT OF LOADING DATA (GENERAL)

APPENDIX B: THE VON KARMAN SPECTRUM

APPENDIX C: GENERAL DESCRIPTION OF THE ENGINEERING MODEL

1. INTRODUCTION

Wind Engineering covers a wide field of engineering including the subject of the forces on

buildings and structures, wind flows around buildings (environmental winds) and the

dispersal of pollutants. In this CIB. report we shall be concerned only with the first of

these.

That part of the subject of Wind Engineering concerned with the design of structures to

resist wind forces can be divided into three distinct sub-parts corresponding to the physical

processes involved.

1. The first concerns the mechanisms which produce strong winds and the description

of the characteristics of these winds when blowing over the surface of the Earth.

This part of the subject is treated in Chapters 2 and 3.

The second concerns the forces and pressures which the wind exerts on structures

over which it flows. These forces are fluctuating, not only because the incident

flow is turbulent, but also because of the detailed flows induced by the structure

itself. Since the forces are intimately related to flows over the surface of the

structure, which are in turn determined both by the characteristics of the incident

flow and the shape of the structure, another large body of experimental data is

required to provide comprehensively for the many situations met with in design.

This part of the subject is treated in Chapters 4 and 5.

3. The third part concerns the static and dynamic response of the structure to these

wind forces. In many cases the dynamic part of the response is negligible and

design is then concerned only with ensuring the stability of the structure and its

component parts under the action of the most severe quasi static wind loading

expected in the lifetime of the structure. In a smaller number of cases the dynamic

response of the structure is not negligible. Then the design must achieve control of

the response so as to produce a serviceable structure. Here design may be concer-

ned with the problems of human comfort and tolerance, with compliance with

performance criteria and sometimes with fatigue. In a very small class of structures

the dynamic response is such that it actually alters significantly the forces which

are producing the response and then the designer is dealing with so-called 'Aeroe-

lastic' structures. The subject of structural response is covered in Chapter 6. This is

done by the description of a commonly used engineering model.

2. CLIMATIC BACKGROUND

2.1 Atmospheric circulation

The energy which produces winds comes ultimately from the Sun which heats the

atmosphere unevenly. This uneven heating occurs at several widely differing scales from a

global scale down to scales of a few metres, and it results in corresponding different

scales of atmospheric circulation. Beginning at the largest global scale, the atmosphere is

warmed more near the equator than near the poles. At the same time, due to the rotation

of the Earth, the angular momentum of the rotating atmosphere is greatest at the equator.

Conservation of mass and momentum is responsible for atmospheric circulation on a

global scale.

The basic circulation pattern in each hemisphere consists of three main cells as shown in

Figure 1 . (The depth of the atmosphere is not shown to scale.) Mean winds at the surface

Figure 1 : Global circulation

of the Earth are divided into three bands on either side of the Equator corresponding to

these three cells. Between each band, and at the poles, is a region either of 'convergence'

(where winds blow towards a confined area which thus must be a low pressure region) or

of 'divergence' (where winds blow away from a certain area which thus must be a high

pressure region). Conservation of momentum induces these winds to turn to the right in

the northern hemisphere and to the left in the southern hemisphere producing the pattern

shown in Figure 1. The temperate latitudes of Europe lie within the band of prevailing

south-westerly winds just south of a band of low pressure between Scotland and Iceland.

2.2 Depressions

The instability of the cold and warm air masses along the line of the sub-polar low

pressure bands (Figure 1) gives rise to the frontal depressions which are 500 km to 1000

km in diameter. They are responsible for the strongest winds in these latitudes. The

depressions move with the mean circulation, tracking from west to east, but each individu-

al track is influenced by the position and strength of neighbouring weather systems.

Sometimes their progress is blocked by large stationary anti-cyclones.

2.3 Tropical cyclones

Tropical cyclones (hurricanes or typhoons) are typically 200 km in diameter, with the

region of strongest winds around a nearly windless "eye" of between 10 km and 20 km in

diameter. They form over the sea when the surface temperature is in excess of 27" and

where there is a sufficient component of the Earth's rotation to drive the circulation.

Accordingly, these resmctions limit their range to the band of latitudes from 5" to 30"

either side of the Equator. Smaller and more intense than depressions, tropical cyclones

drift from east to west with the Trade Winds, gaining in strength over the warm sea, but

dissipating rapidly over land so that they are a problem only within a 70 km-wind strip

around the coast.

2.4 Tornadoes

Although tornadoes are much smaller than tropical cyclones, they can produce the

strongest and most destructive winds known. They can form when strong high-level winds

blow over the top of strongly convective cells (usually frontal thunderstorms). The high

altitude wind draws air upwards from the core of the cell thus reducing the internal static

pressure. A funnel-shaped cloud grows downward from the cloud-base as the reducing

pressure lowers the condensation level. The air drawn into the tornado at its base causes

strong convergence near the ground, so that the tornado rotates rapidly in the cyclonic

direction. The best known region for the occurrence of destructive tornadoes is probably

the central-southern United States.

Differential solar heating also occurs on much smaller scales due to the different absorpti-

ve characteristics of the Earth's surface. A local hot area may produce enough convection

to cause strong local convergence and the formation of cyclonic circulation. On the

smallest scale this may result in 'dust-devils'. These are not usually considered in

structural design although there is some evidence that they cause damage to small

buildings.

2.6 Thunderstorms

In temperate latitudes, thunderstorms may be associated with fronts or they may be

generated by direct solar heating on sunny summer days, but their wind speeds are

generally exceeded in depressions. In the equatorial regions the latter mechanism is

responsible and here they may be the principal source of strong winds. Thunderstorms

form discrete convective cells, typically 5 km in diameter and 10 km high, each marked

by a characteristic anvil-shaped cloud, comprising a small cyclonic cell, in which warm

moist ground level air is carried up through the lower levels of the atmosphere to very

great heights, and a complementary larger anti-cyclonic cell in which cold dry high air

decends from high levels to the surface, sometimes called the "downburst"

2.7 Katabatic winds and lee waves

These two forms of wind climate are associated with mountains. A continuous range of

mountains act as a barrier to a dense mass of cold air as it attempts to displace a warmer

air mass. The cold air accumulates behind the mountain range until it is able to pour over

the top, accelerating under gravity to produce strong down-slope or katabatic winds.

These are significant in parts of Switzerland and Italy, along the Rocky Mountains and

other mountainous regions.

3. WIND STRUCTURE

3.1 Introduction

In this report the main subject is windloads on structures. Therefore only strong winds,

with average windspeeds > I0 rn/s are considered.

The majority of wind speed data throughout the world is acquired by measuring surface

winds within the boundary layer by means of anemometers mounted on masts. It is

standard practice to reduce these data to the common base of a standard height above a

standard terrain, usually a height of 10 m above flat or gently undulating open country

with very few surface obstructions (zo = 0.03 m, see 3.2.2). Where possible, the anemo-

meters are sited at this standard height in open country; for exampIe on airfields. In this

case, little or no correction is needed. Otherwise the data are corrected to the standard

exposure using the known characteristics of the boundary layer appropriate to the

particular exposure.

The wind speed data thus acquired contain contributions resulting from the passage of

large-scale weather systems and from the boundary layer itself. A record of the wind

speed will in principle look Iike the diagram shown in Figure 2. It has long term fluctuati-

ons, 1, associated with the large-scale weather systems and short term fluctuations, g, due

to the turbulence in the boundary layer. The total windspeed can thus be broken down in

two parts. a mean windspeed 1 (averaged over At and in the mean winddirection) and

windfluctuations g = (u.v,w) around that mean. In vector notation:

These two parts can be readily identified from the data record because of the widely

differing physical scales which result in the production of wind speed fluctuations of

, Time V tor

(a) Instantaneous wind speed

Time v, (b) 1 0-minute mean wind speed (moving window)

(c) 10-minute mean wind speed (fixed window)

Figure 2: Wind Speed Records (schematic)

widely differing frequencies. This is best shown by displaying the data in the 'frequency

domain'. The first comprehensive "spectrum" of this type was compiled by Van der

Hoven 111 from wind records taken at Brookhaven, Long Island, NY, USA, and is

reproduced in Figure 3.

0.001 0.01 0.1 1 10 100 1000 Frequency [ 1 Iuur ]

Figure 3: Van der Hoven spectrum

The Van der Hoven shows three distinct features:

1. A major peak at a centre frequency of 0.01 cycleshour, which corresponds to the

typical Cday transit period of fully developed weather systems, usually called the

macrometeorological peak.

2. A second major peak comprising a range of higher frequencies which are associa-

ted with the turbulence of the boundary layer and which range in period from

about 10 minutes to less than 3 seconds. This is usually called the micrometeorolo-

gical peak. (For the calculation of the dynamic response of structures, this part of

the spectrum is important because the lowest frequencies of structures fall within

its range.)

3. The "spectral" gap between these two peaks in which there is little wind fluctuation

over a range of frequency of about one order of magnitude.

From the form of this spectrum it follows that in most cases the assessment of the

magnitudes of wind loads on structures in any particular storm, which depend on the

exposure of the structure (i.e. its situation) and the shape of the structure, can be separated

from the prediction of the severity and frequency of occurrence of the storms themselves.

Macro- and micrometeorological fluctuations (or long and short term fluctuations), for this

reason, may be discussed separately.

3.2. Long term stationary model (Macrometeorological peaks)

3.2.1. The average wind speed proces

When wind speed data are collected over a long period of time (usually at a height of 10

m), these data accumulate to form a statistical distribution in the 'amplitude domain'. The

large body of available data shows that for temperate latitudes generally the distribution of

average wind speeds is very well modelled by the Weibull distribution:

where FV(x) gives the probability of a At-mean wind speed less than x and Vo and k are

parameters defining the scale and shape of the distribution. The shape parameter k ranges

between 1.7 and 2.5 in most temperate areas, but values close to 1 have been measured in

areas of Italy dominated by katabatic winds. The solid line in Figure 4 shows the shape

of the Weibull distribution for the special case of the Rayleigh distribution, for which k =

2, corresponding to the middle of the observed range.

The wind velocity is often modelled as consisting of N stationary subprocesses in a year,

each of a duration of usually 10 minutes, characterised by the 10 min-mean windspeed V,

in a fixed window and taken over all winddirections. Figure 5(a) gives an example (a

realisation). However in search of the relationship between the parent distribution of

windspeed and the annual maxima (see also 3.2.2.) we must be aware of the correlation

between these 10 min-mean windspeeds in the successive subprocesses.

1 2

Non-dimensional wind speed VIPv

Figure 4: Weibull distribution with k = 2 (Rayleigh) and distribution of extreme.

R = mean value of the 10 rnin average windspeed

A possible approximation to take this persistency into account is shown in Figure 5(b),

where the windclimate is considered to consist of n independent stationary subprocesses

in a year, each of duration having a At-mean windspeed. The point is that when At is

chosen somewhere between 3 and 12 hours, the mean wind velocity various periods At

may in some calculations be considered as independent.

An alternative model of the windclimate is a process where the At-moving average

windspeed is taken as a continuous function of t, characterised by statistical means (among

others a correlation function). See Figure 5(c).

The windload on a structure depends on the windspeed and the wind direction. For

describing the statistics of the windload it is therefore necessary to take into account the

distribution of wind directions j, commonly divided in 12 sectors. A first approximation

for describing the windvelocities is a varying vector with an intensity V, described by (I),

and a direction statistically described by fraction of time that this vector is within a sector.

10 min

P I

Figure 5: Representation of windclimate

3.2.2. Distribution of the annual maxima of At-mean windspeed.

The probability distribution for annual maxima FVm,() can be connected to the parent

distribution Fv() of the At-mean windspeed V as follows:

where n is the (suitable chosen) number of independent stationary subprocesses in a year

(as in Figure 2(c) or 5(b)). The typical shape of the distribution of extremes from a

Weibull parent is also shown in Figure 4 as a dashed line, indicating how the distribution

moves from the parent to higher wind speeds.

Unfortunately the value of n in (3) is not always known at a site, so that the distribution

function of Vmax, FVm,(u) is usually directly estimated from observations.

The essence of this method to estimate the annual maximum, is the extraction of indepen-

dent samples from the parent to recover maxima, in this case the maximum wind speed in

each year. However, most sites have fairly short data records, 10 to 50 years, so that the

distribution of annual maxima will only contain 10 to 50 values, insufficient to determine

the shape accurately. In order to make a reasonable estimation, the shape of the dismbuti-

on needs to be determined independently, so that the observations are used to fit only its

parameters. A Weibull parent distribution leads asymptotically to a Type 1 distribution

(Gumbel):

There is some indication that the Gumbel dis~ibution given by (4) overestimates the

probabilities in the very high windspeeds [49]. In case the distribution functions FVm,(x)

and FV(x) are known, the value of n can be estimated from (3).

Notes:

Seasonal and Directional Extreme Value Analysis.

Since the seasonal variations of climate can be averaged over a period of years, the

wind extremes for any one given month should belong to the same parent populati-

on and, being from separate storms, also be independent of the populations for

other months. Thus separate estimates of extreme winds for each calender month

can be obtained by taking all the maxima for a particular month, year by year, and

ranking these in the way already described for the overall annual maxima. This

yields separate estimates for the risk of extreme winds for each month of the year.

Such data are of use for the assessment of design loads on temporary structures,

structures during construction and for occasional structures such as marquees and

pavilions which are erected only for specific sub-annual periods. Since the winter

months are more susceptible to high winds in the temperate latitudes of Europe,

structures erected only for the summer would be significantly over-designed by

using annual extreme winds.

If continuous wind data are available, the basic data can be grouped in sectors with

respect to direction. For practical reasons these are usually 30" sectors. However,

the standard method of extracting annual extremes when applied to individual

sectors is no longer valid because the partition of the number of hours of wind is

different in each sector and also varies more from year to year. Hence there is no

longer a constant annual parent sectorial distribution from which the extremes can

be drawn. Attempts are frequently made to extract directional information from

discontinuous data; for example, from monthly (daily, or storm) maxima for which

the direction is also known - but this is invalid because it neglects the fact that

important data for other directions in the same month (day or storm) are excluded

by the process of recording only the maximum and discarding everything else.

However, the method of using the independent storm data can be applied if the

basic data are continuous; i.e. wind speeds and wind directions are available for

every hour. Each sector is treated separately, and individual sectorial storm maxima

are extracted. Most storms will contribute extremes to several sectors but this

correlation between sectors does not invalidate the method because extremes within

one particular sector will be selected from separate storms and thus remain

independent. The number of extremes in each sector will not be the same. The

average annual rate of occurrence of independent maxima is calculated for each

sector separately; this average may also differ from sector to sector. The CDF of

independent maxima for each sector is raised to the power of the annual rate of

occurrence for that sector thus yielding the CDFs of the independent sectorial

maxima which occur at an average rate of one per annum in each sector. These are

the required distributions for assessing risk within a sector and comparing this with

other sectors.

Rijkoort [49] studied the windclimate in the Netherlands and used subdivisions of

wind data in order to get more or less statistical homogeneous groups of data. He

distinguished day- and nighttime, 6 seasons of two month each and 12 30"-sectors

denoted by 0, 30, etc., where 0 refers to the sector 345"-15". In doing so, he

arrived at 26.12 = 144 groups of data that could be characterised by a frequency

distribution of the form (2) (for daytime) or a variant of (2) (for nighttime). From

there on Rijkoort could describe the windclimate taking into account the correlation

between the hourly mean of windspeed in succesive hours, the period (for instance

one year) and the winddirection. In [49] Rijkoort gives several parent distributions

and extreme value distributions of windspeed (see also 3.2.3.2.).

Extremes in mixed climates

The wind climate in Temperate latitudes is generally dominated by fully-developed

weather systems, other storm mechanisms either producing less strong winds or

else having a risk of occurrence sufficiently small to be neglected in design.

Elsewhere in the world, the balance between these storm mechanisms will differ. In

the Tropics and sub-Tropics, which are subject to a high risk of tropical cyclones

or tornadoes, the problem of the estimation of extreme winds is less tractable.

Moreover, the extent, form and quality of wind speed data in these regions is

variable.

In the case of a meteorological station in a region where there are two major storm

producing mechanisms, the collection of annual maximum wind speed values and

the plotting of these extremes by the standard Gumbel method results in a graph

represented by the solid symbols in Figure 6. The points will not lie on a straight

line but curve upwards towards the largest extremes. Until recently, the pragmatic

approach was to fit the data to a different extreme value distribution, having a

more slowly decaying upper tail. The Fisher-Tippett Type I1 distribution was a

strong candidate and there were extensive debates about the validity of this

procedure. Simiu and Filliben [12] concluded that attempts to fit other model distri-

butions were not satisfactory. The problem was resolved in 1978 when Comes and

Vickery [13] demonstrated that when a mixed climate is separated into its compo-

nent storm mechanisms, the data from each separate mechanism converge to a

Fisher-Tippett Type I distribution.

The schematic example of Figure 6 shows two populations of maxima. The data

represented by the circular symbols come from one storm mechanism (A), while

those represented by the square symbols come from another (B). In any particular

year the annual maximum for the mixed climate (M) will be the larger of the

maxima from A or B occurring in the same year. The largest values of the mixed

distribution generally come from B and the smallest from A, but values from A

and B are interleaved in the region where the two Type I lines cross, so that the

mixed distribution follows the upper branch of each line.

v- 11 v

Reduced variate y= 7 v

Figure 6: Gumbel plot for annual maximum wind speeds in a two population mixed

climate

3.3. Short term model (The Micrometeorological Peak)

3.3.1. General description of the atmospheric boundary layer.

When winds are produced by any of the climatic mechanisms described in the previous

Chapter, the air movement is resisted by the frictional effects of the rough surface of the

Earth: sea, sand, grass, trees, small and large buildings, and hills and mountains. This

frictional resistance ultimately dissipates the energy in the wind. The surface drag slows

down the air flow near to the ground thus forming a 'Boundary layer' extending from the

surface itself, where the flow velocity is zero, up to a height called the 'Gradient height'

where the ground friction ceases to have any effect and the flow is controlled solely by

the high-level pressure gradients. The depth of this boundary layer is dependent upon the

scale of the surface roughness being deeper for larger scales of roughness. The depth is

typically between 900m and 2000m. However, the wind must blow over a considerable

distance, called 'fetch', before the full depth of boundary layer is established in equilibri-

um with the ground roughness. Each change of roughness disturbs any established

boundary layer and begins the growth of a new boundary layer displacing the old one

from the ground upwards. Thus equilibrium boundary layers are rarely established to their

full depth unless there is a long fetch of uniform terrain. In particular, city centre

boundary layers seldom reach full depth since these need the longest fetch of all to

achieve equilibrium; up to lO0krn in extreme cases. However, in the majority of urban

areas, smaller fetches will usually produce part- depth boundary layers sufficiently deep

to immerse all but the tallest of buildings.

The flow in the atmospheric boundary layer is turbulent, with components in the three

principle orthogonal directions. The range of eddy size extends from the largest which can

be accomodated within the boundary layer of the order of 1000 m down to the small

vortices which whirl leaves about of the order of 1 m and beyond. A cascade is formed

which transfers energy from larger to progressively smaller eddies right down to the point

where the final energy dissipation is at a molecular level in the form of heat. This

specuum of turbulence produces the familiar gustiness of the wind which changes both

speed and direction from moment to moment. The level of gustiness corresponds to the

turbulence intensity, and is greatest in regions of the roughest terrain and least in regions

of the smoothest terrain.

In an equilibrium boundary layer there exists a velocity profile, (i.e, an increase of

velocity with height) which can be standardised to a universal non-dimensional form, and

a turbulence profile, (i.e. a functional relationship between turbulence intensity and height)

which can similarly be standardised with respect to appropriate parameters. The structure

of the boundary layer about to be described therefore strictly only applies to strong winds

in temperate latitudes, in which turbulent mixing ensures that the atmosphere is neutrally

bouyant.

A full-depth boundary layer is not established in case of tornadoes and thunderstorms due

to their small scale relative to the length of fetch needed to establish equilibrium. These

wind producing mechanisms destroy the existing boundary layer in the localised region

that they influence.

A good mathematical model of an equilibrium boundary layer is that due to Harris and

Deaves [14]. The model is based upon the principle of 'asymptotic similarity'. The model

has been verified against experimental data. Harris and Deaves [14] have also extended

their model to cover the transitional conditions associated with abrupt changes of surface

roughness.

3.3.2 Profile of mean wind speed.

Most investigations of wind speeds have been conducted in the lowest 200 m of the

atmosphere since this region is most accessible to measurement. In this layer it has been

found that the systematic change of mean wind direction with height, (the Ekman spiral),

is negligible so that the the mean velocity vector can be considered in terms of its scalar

magnitude (i.e. the mean wind speed).

Log-law Model.

Boundary layer theory indicates that in the lowest layer of the atmosphere a logarithmic

profile law is to be expected. In measurements of mean wind profiles in the atmosphere, it

has been found that over level terrain of uniform roughness, in suong winds (i.e. neutral

stability), the results can be represented by:

where

V(z) is the mean wind speed at height z,

do is the zero-plane displacement

u* is the friction velocity ( = \ l ( ~ p ) )

p airdensity

Z~ shear stress at ground level

K von Karmans' constant (=0.4)

z is the height above ground

z0 is the aerodynamic roughness length of the surface.

The parameters u* and z0 can be related theorectically to other flow variables:

In practice u* and zg are measured from a log/linear plot of the observed profiles.

Note: The logarithmic-law model does not apply above the surface layer. Ln order to

extend it higher into the boundary layer an additional term is needed:

where

z is the gradient height z =u*/6fcor S S

fcor Coriolis parameter (=lo4 rad/s for temperate latitudes)

K I is a constant in the range between 4 and 7.

With this correction term the profile is accurately represented to a height of about 300 m.

In the full model of the boundary layer developed by Harris and Deaves [14] there are

several more additional terms involving higher powers of z/z S'

Power-law Model.

For many years the velocity profile has been empirical represented by a power-law model.

where

V(z) is the mean wind speed at height z,

z is the height above ground,

zref is the reference height (e.g. 10 metres),

do is the zero-plane displacement (traditionally do = 0)

a is the exponent of the power-law.

For some values see Table 1.

Although a popular model for engineering applications owing to its simplicity, the power

law has two major disadvantages: (1) it has no theoretical justification, and (2) it models

the upper part of the boundary layer quite well, but it is a poor fit near the surface which

is where it is most needed.

3.3.3. Turbulence intensities and spectra

The fluctuating parts of the wind u represents the wind turbulence. The turbulence is a

three dimensional rotational air movement resulting from the roughness of the earth

surface. For relatively long fetches and shon periods the turbulence can be described as a

zero mean stationary and homogeneous Gaussian random velocity field. A full description

of such fields requires the definition of a 3*3 mamx of covariance functions (or alternati-

vely spectra), each covariance function depending on the time gap At and the spatial

distance h-. In descriptions of this type usually the assumption of a "frozen isotropic

field" is made (see [53] and annex B). In reality the wind velocity field, however, is far

from istropic.

In this document we maintain the assumption of the "moving frozen field", but isotropy is

not assumed. We shall, however, in this main text not present a full description, but

resmct ourselves to the "autospectra" for the individual components of the vector u. Only

for the inwind velocity, information on the spatial correlation pattern shall be given. For a

more general model. reference is made to Annex B.

Standard deviations

The amount of wind turbulence in respectively the x-, y- and z direction is expressed by

the standard deviations 0, , ov and ow . In first approximation holds (Lurnleyl Panofksy

[541)

which demonstrates the anisotropy of the turbulent wind. The smaller values are valid for

rough surfaces (See Table 1). There is also a tendency to decrease with height above the

ground. Figure 7 gives a more detailled picture of pi as a function of the height.

height z (m)

Figure 7a: Standard deviations (oi = piu*) (ref. [58])

3

CE L

0 + 0

a j

3 -

1.5

1 . '

- P u

- /---\ = 5: \ \

- \

- 5

t - A

V,,=20m/s; z ,=0. 1 m

10 20 JO 80 10 320

height z-do in meter Figure 7b: Standard deviations (o, = piu*) (according to Deaves and Harris)

Turbulence intensity

The longitudinal turbulence intensity I, is defined by:

Approximately holds:

based on the log-law model.

For some information about the various parameters, see Table 1.

Autospectra (Spectral densities)

The spectrum for the turbulent part of the windvelocity is often expressed as a reduced

spectrum FD(f):

Herein is Sii ( f ) the variance spectrum of the inwind fluctuations in i- direction, defined in

such a way that:

Table 1: Characteristic roughness and profil parameters

*) Eurocode [56]

Note: various authors may give different values for some of the variables.

Von Karman 1101 proposed spectra equations which are generally accepted as the best

analytical representation of isotropic turbulence. The effect of departure from isotropic

turbulence near the ground is allowed for by the variation with height and surface

roughness of the appropriate variance oi2 and length parameter Li since they typify the

intensity and size of eddies constituting turbulence. The von Karrnan spectral equation for

the in-wind each velocity is given by (see Annex B for the mathematical background):

fS,,(f) FDu = F D = - = 4xu

2 (Von Karman)

2 516 (13)

ou (1 +70.8xu)

The corresponding spectral equations for the lateral and vertical velocities are:

2 fSvv(f) 2xJI +188.6xv)

FDv =- = 2 2 1116 (Von Karman)

ov (1 +70.8xv)

2 fsww(f) - 2xW(1 +188.6xw)

FDw = - - 2

(Von Karman)

Ow (1 +70.8x;)l

In these spectra x, . x, and x, are given by:

Values of oi follow from oi = Oi u* ; see Table 1. Values of I,, , Lv and L, are, for

example, presented by ESDU [58]:

Note that in [58] some definitions are slightly different, which may give rise to different

numerical values. Other suggestions might be:

Note that the longitudinal length scales vary appreciably up to a factor of 2. The ESDU

formula appears to reflect observations on the lower side and therefore is considered as

conservative with respect to the velocities spectra in the high frequency range.

For the inwind fluctuations u of the windspeed, a number of empirical spectra are com-

monly used. The most popular spectra are:

(Davenport)

(Harris)

(Kaimal)

6 . 8 ~ FD(x) = (Eurocode)

(1 + 1 0 . 2 ~ ) ~ / ~

Herein is:

x a dimensionless frequency (fL/Vref)

f the frequency in Hz

L Characteristic length:

Davenport: L = 1200 m, Vref = V10

Harris: L = 1800 m, Vref = VI0

Kaimal: L = 50z, Vref = V,

Eurocode: L = 300 ( ~ 1 3 0 0 ) ~ , for E see table 1. z and L are in [ml

Vref = average windspeed at equivalent height zequ (see [56])

VIO 10 min-mean of wind speed at 10 m height

All these spectra are autospectra. They all have the same characteristic at higher frequen-

cies:

sUu(f) = f -5'3 for f+- (24)

This follows from theoretical considerations [52] and is fairly well confirmed by experi-

ments [53], [59].

Figure 8 shows the typical form of the spectra for the inwind, crosswind and vertical

turbulence components u, v and w, taken from [58]. The empirical Davenport spectrum

(20) and the more scientific Von Karman spectrum (13) compare very well. Differences

must be seen in the light of the limitations of the engineering model (Chapter 6) where

these spectra are used.

- von Karman -+- Davenport

8 %

'\

Harris - \ , - Simiu

*, a * Eurocode

V,t =25m/s

frequency in Hz

Figure 8a.l: Windspectra in terms of f S(f) / 2, z = 10 m.

0.001 0.2 0.4 0.6 0.8 1 1.2 1.4 1.b 1.8 2 33 2.4 2.6 1 8 3 2.3 3.4 3.6 2.3

frequency in Hz

- von Karman . Davenport - Harris

S ----~t---- Simiu - Eurocode

zO=0.05m eps=0.26

Figure 8a.2: Windspectra in terms o f f S(f), logscale, z = 10 m.

von Karman - Davenport Harris

A Simiu Eurocode

V,,, =25m/s m '* r

z=90m ~ - m . 2, =0.05m u \

j.2 0.4 0.b 0.8 1 1.2 1.4 1.6 1.8 1- 2 2 L4 2 6 28 3 3.2 2.4 3.6 3.8

frequency in Hz

Figure 8b.l: Windspectra in terms of f S(f) / cf, z = 90 m.

0.001 0.9 0.4 0.6 0.8 1 1.1 1.4 1.6 1.8 2 L2 2 4 2.6 3 8 3 2.1- 3.4 3.6 3.8

frequency in Hz

t F - von Karman ', - Davenport

P --c- Harris - Simiu - - Eurocode

-

z, =0.05m eps=0.26

Figure 8b.2: Windspectra in t e r n of f S(f), z = 90 m, logscale

Cmss spectra

The cross spectrum for windspeeds in the points i and j is mostly written in the form:

Herein is:

S4uJ(f) the cross specmum for fluctuations u at points i and j

Surui(f) autospectrum for fluctuations u at point i

whulU,(f) coherence function for the fluctuations ui and u. J

The coherence function takes into account the windspeed fluctuations at different points.

Coherence data can be sufficiently accurate represented by a relatively simple exponential

expression of the form:

Herein is

y.z the coordinates

coherence constant z-direction (in the order of 6-10)

C y coherente constant y-direction (in the order of 6-16)

windspeed fluctuation u at yi , zi

U . 1

windspeed fluctuation u at y. . zj 1

In (26) sometimes J[vi(z) vj(z)] is used in stead of V l o . Annex B gives a number of

alternative expressions for coherence functions based on the von Karman model (equations

(15) - (19). In figure 9 examples of these coherence functions are given and compared

with the simplified form (26).

frequency in Hz

0. S

0. - 0.0

0.5

Figure 9a: Coherence functions, C, = 6, C,, = 6.3, Vlo = 10 m/s

t - a,

I I + ESDU --t ESDU (appr.)

- :, -A- E-power A 8

- *.

m -*I - ESDU

A + ESDU (appr.) ,

- I ) + E-power \ \

I\.

- '4,

A \ z, =0,1 om - Z , =50m ' & Z, =40m

- ;\ y , =o \ \Q Y, =o

-

-

* _ _ - - , - - - -

0.05 0.15 0.3 0.35 0.45 0.55 0.65 0.75 0.85 0.95

frequency in Hz

Figure 9b: Coherence functions, C, = 8.5, Cy = 9, VI0 = 10 m/S

A *' - ¤ - ESDU

A - ESDU(appr.) - E-power

8 - A

\

A - * A

z,=O,lOm -

A Z i =50m z, =40m

- . L, y i =o - Y, =o

-

0.05 0.15 0 . 3 0.35 0.45 0.55 0 .6 0.75 0.85 0.95

frequency in Hz

Figure 9c: Coherence functions, C, = 8.5, Cy = 9, VI0 = 25 4 s

4. PRESSURES, FORCES AND RESPONSE OF STRUCTURES

4.1 Wind force models

Some wind force models commonly used are:

Lattice plate model

Consider a lattice plate formed by a fine mesh of thin rods or bars and held normal to the

wind. Provided that lattice is not very dense. the wind flows relatively freely through the

lattice with very little divergence and the flow around any one element of the lattice

depends on the incident wind only at that point. This makes the loading of a lattice plate

relatively simple to analyse, since each elemental wind force is in one-to-one correspon-

dence with the local kinetic pressure and the total forces may be obtained by summing all

the elemental forces, including the effects of gusts of varying sizes with respect to the size

of the lattice plate.

The range of real structures for which the lattice plate model is a good model includes

lattice masts and towers, scaffolding and building frames during construction. The lattice

plate model is also extended to less favourable situations, subject to appropriate correcti-

ons.

Strip model

Two forms of strip model are relevant. The first is obtained when a lattice plate is

elongated so that the plate tends towards a line, as with a tall guyed mast. The lattice

plate model applies, but now only the dimension along the line is relevant when conside-

ring the action of gusts and summation of the elemental forces.

The second form is obtained when an elongated structure is not a lattice, but is solid, like

a tall chimney stack or long bridge deck. The structure sheds a turbulent wake and vortex

shedding is very likely to occur unless steps are taken to prevent it. Independence is

maintained between any plane along the length of the structure, so that the structure may

be divided into short sections or "strips" and the loading of each strip considered separate-

ly. This is a standard technique in the design of road decks for long-span suspension

bridges and is called "strip theory".

Structures of both these types that have their structural properties as well as their

aerodynamic properties concentrated along a line are often called "line-like structures". A

characteristic of line-like structures is that their structural stiffness is low, unless externally

braced or guyed, making them likely candidates for large dynamic response or aeroelastic

instability.

Bluff body model

Whenever a structure is not elongated sufficiently for the strip model and is relatively

solid, so that the flow must diverge around it in a threedimensional manner, the structure

is "bluff' and the bluff body model applies. This includes the vast majority of all buildings

and habitable structures. The aerodynamics of bluff bodies is the most complex and

resistant to aerodynamic analysis, and the knowledge of their behaviour has been built up

from experience of full-scale and model-scale testing. The bluff body model is the only

model where the effect of the building on the mean windspeed and the wind spectrum is

taken into account.

Numerical computation of flow around bluff objects have received a fair amount of

attention over the past decade. The main features of the flow (i.e. mean wind loading on

walls and wake properties) can be calculated with fair accuracy for the 2D as well as the

3D case. Finer details such as the peak suction at the eaves and comers of low pitch roofs

are not yet satisfactorily accounted for. References are [62] to [64] and [67].

4.2 Kinetic pressure and loading coefficients

The pressure acting on a structure can be described by the reference windpressure q

connected to the mean windspeed V, and a dimensionless pressure coefficient c, .

where p=p(r,t) is the windpressure on the suucture at point L and q@ is defined with:

where V@ is the mean undisturbed windspeed at point and p is the mass density of air.

Note that in the definition (27) c is considered as a function of time. The characteristics P

of this function are very complicated. In general it will depend on the point of application

and for higher frequencies the behaviour of p and that of c are not necessarly the same. P However, the (statistical) characteristics of c are considered as being independent of the P mean wind speed.

The same procedure is used to define the force coefficient cF and the moment coefficient

chq (both dimensionless):

where %ef is a reference area, most conveniently the area under load, and 4ef is a

reference length.

Conventionally, the force coefficient in the direction of the wind is called the drag

coefficient, c,, , and the force coefficients normal to the wind are called lift coefficients,

cL . both terms having been borrowed from aircraft aerodynamics.

4.3 Models for building aerodvnamics

It is convenient to model the range of structures by these three categories of their response

to the fluctuating turbulent wind forces:

Structures with a static response

When a structure is very stiff, deflection under the wind load will be very small

and the structure is said to be static. As the lowest natural frequencies of the

structure are high, usually above 5 Hz where is little energy in the spectrum of

atmospheric turbulence to excite resonance, the only design loading parameter of

importance is the maximum load experienced in its lifetime.

Structures with a dynamic response

When a structure is sufficiently flexible, with the lowest natural frequencies below

about 5 Hz, the energy of the atmospheric turbulence excites resonant response of

the structure in the normal modes corresponding to each natural frequency. The

amplitude of the response depends on the spectrum of the turbulence and the model

parameters of the structure: frequency, mode shape, mass, stiffness and damping.

Aeroelastic structures

When the motion of a dynamic structure becomes large it tends to change the

aerodynamics, and when this change increases the loads the motion may increase to

an unacceptable, or even catastrophic, level. The aim with such structures is to

suppress any aeroelastic instabilities completely.

The vast majority of all structures are static and are designed for strength with a check on

maximum deflection and, possibly, on low-cycle fatigue. A proportion of the taller

buildings, tall chimneys and light flexible structures such as masts, towers and long-span

bridges, will be dynamic and are designed for deflection with checks on acceleration (for

comfort of occupants) and on high-cycle fatigue. Line-like structures are the most

susceptible to aero elastic effects, typically tall chimneys, suspension bridges and other

similar specialised structures.

5 THE BLUFF BODY MODEL

5.1 Probabilistic model

In this chapter, the windclimate is thought to consist of n independent stationary subpro-

cesses in a year, each of duration At (see figure 5(b)). In this way some persistency of

mean windspeeds is taken into account (see also section 2.2.1). The value of At should be

in the order of 3 to 12 hours.

The windloading (p, F. M) on the structure will be described by the At-reference wind-

pressure q=0.5 p ~ 2 and a loading coefficient c (cp , cF , cM ) defined by (see also 4.2.):

Due to windspeed fluctuations within a period At the load coefficient will also fluctuate. A

maximum (minimum) loading coefficient can now be defined by

where pE , FE and ME are relative maxima (minima) taken from a continuous wind loa-

dingrecord (load effect p, F, or M).

Figure 10 shows typical results for measurements of pressure coefficients for minima

(peak suctions) of durations Is, 4s, and 16s. This shows that a fit to a Fisher Tippett Type

I distribution is excellent with observation periods as short as ten minutes. We assume

here that this also holds for the maxima (peak pressures).

0.0

-0.5

d

C .$ -1.0 .- - - a 0

-1.5 !? 7 ln ln

2? -2.0 P

f E -2.5 .- C

5 -3.0

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Reduced variate y : a(V-U)

Figure 10: Gumbel plot of minimum pressure coefficient (maximum suction).

The distribution of the relative maxima (minima) of the loading coefficients CE,] ( c ~ ~ , ~ . c ~ ~ , ~ or c ~ ~ , ~ ) in a stationary subprocess of duration At and winddirection j can thus be

expressed by:

Herein is:

acE, a measure of the dispersion of CE

UCE,] the modus of the maximum CE in a period At

j winddirection (with respect to structure)

UCE,] and acEj can be determined for instance in a windtunnel as a function of the flow

direction with respect to the orientation of the structure.

Consider now the At reference wind pressure q; and describe the upper tail of the

probability density function, fq,,(x), of the At-mean reference pressure ( 0 . 5 ~ ~ ~ ) in wind di-

rection j, by the following Gumbel distribution:

Herein is

a q . , a measure of the dispersion of the At-mean wind pressures q .

Uq , the modus of the At-mean wind pressure q ., in a subprocess

The probability density function f q j ( ) has a relation with the probability density function

f,,,, (or distribution Fq.,,) of the annual maximum of the At-mean wind pressure. In this

relation the distribution of wind directions and the correlation between mean windspeeds

of succesive stationary subprocesses play a role.

where

p, factor depending on the distribution of wind directions; pj=1/12 (12 wind directions

are considered)

fq the probability density function of q, over all wind directions

n the number of independent stationary subprocesses in a year:

n=T/At with T=l year (so if At = 6 hours then n = 1460).

Note:

For fg,,, (T=l year) measurements in the Netherlands (At= 1 hour and wind directions

not distinguished) give values of the order of l/aq = 70 ~ / m * U, = 325 ~ / m ~ [50].

The probability that the load effect F=cE q exceeds the value x in a period At follows now

from:

or:

Let us now assume that (the upper tail of) FF(x) resembles a Gumbel distribution,

characterised by UF (the modus) and aF (a measure of the dispersion), derived by for

instance a least square method:

The modus of the load effect with a reference period of R year follows then from (remem-

ber that At=l/n year):

For describing the forces (moments) due to wind just the same approach as given above

can be followed. For forces (or moments) (36) must be read as the distribution of the

maxima (or minima) of forces (or moments). The expressions (29) and (30) give the

appropriate forms of the force and moment coeficient respectively. The bluff body theory

can play a role to a better understanding of the pressure coefficient c and the factor B to

be used in the engineering model which is described in chapter 6.

The bluff body model is a complete model, however it requires wind tunnel experiments

in order to get information on FCEj (see (33)).

5.2 Codification

In codes a representative pressure coefficient crep can be defined as foIlows:

with

where Uq and aq are determined from the distribution of the annual maximum of At-mean

wind speed independent of the direction of the wind (In this case At=10 rnin). With the

above given definitions the design windload can be expressed as follows:

herein is

q r e ~ a representative wind pressure

'rep a representative pressure coefficient

YF a load factor

It is also possible to calculate the design values by choosing the right return period R,

corresponding to the required reliability.

6 ENGINEERING MODEL OF WINDLOADING

In this chapter a commonly used "engineering model" is summarized. Davenport proposed

[18] and subsequently refined [19. 201 the statistical concepts that form the basis of this

method. The method works in the frequency domain as shown in figure 11 and extends

naturally to dynamic structures by adding the final two steps the transfer function of the

structure, leading to the spectrum of structural response.

Starting-points are (1) the undisturbed velocity profile as if the structure considered and

the structures nearby are not present; (2) the relation-ship between windspeed and wind-

pressure is linearised around the mean windspeed, (3) the pressure coefficient is taken as

constant in time and (4) interaction, because of structures nearby, is not considered. These

assumptions are not always justified and therefore the engineering model, described below,

has its limitations . On the other hand the engineering model needs less wind tunnel

experiments.

6.1 Static structures

6.1.1 Local windpressure (windpressure at a point)

To calculate the load effect (pressure) due to wind on a relatively small area the following

linearisation is commonly used

The linearisation is accurate if u/V<<l, so for relatively low turbulence.

The mean b, the variance op2 and the spectrum Spp of the loadeffect p (in this case a

pressure p=cq) are then given by:

1 p, = - p c v 2

The pressure coefficient c depends on many factors for instance: point of application, the

inluence of the building itself and of the buildings nearby, the frequency-content of the

wind speed fluctuations etc). Yet a common assumption is still a pressure coefficient

constant in time.

p is approximately a Gaussian process. The maximum p in a period At (pAt) can be

expressed as

where g is a random variabele, characterised by p, and o,

An estimate of the pressure p, with a frequency of exceedence once in At, indicated by pAt

, is given with

Wind spectrum

Aerodynamic 1 adrniTnce 1 ' Load

spectrum

Structural transfer function

Dynamic response spectrum

Figure 11: Davenports admittance approach.

The peakfactor pg is given by

with nl the number of peakpressures within the period considered.

Note: For a period At of 10 minutes to one hour: pg=3.5, 0,=0.35

6.1.2. Windload on an aera A

For larger areas A loaded by wind (figure 12) the wind effect F is expressed as

with

as the mean load and J a mean aerodynamic coefficient:

ch is the pressure coefficient at some height h

Vh is the At-mean windspeed at reference height h

i is an influence coefficient depending on the load effect considered

The fluctuating part of the windload effect for dAj at some point j (j = 1,2) follows from

d F . = pc .V ,u .i .dA. = (pc .V .i .dA ,)u, J J J J J J J J J J J (51)

Herein is:

Vj = V(yj , zj ) = At-mean windspeed

C . J = c ( y , z. ) = pressure coefficient

J J dAj = dyj , dzj = elementary area

'J = i(yj , zj ) = influence coefficient at point j

With this, the spectrum of the windloads can be expressed as

with

and

Substituting (53) and (54) in (52) and knowing that Vh is a function of V10 gives:

Note: In the integration (55) the total area of interest is involved. This could be only the

front area (in this case for the pressure coefficient is taken cfrO,pcbxk) or the total area

(front and back =2A). In the last case four coherence functions have to be distinguised

depending on where dA1 and dA2 are taken. (dAl, dA2 both on the front; dA1 on the

front and dA2 on the back; dA2 on the front and dA1 on the back; dA1 and dA2 both on

the back).

Figure 12: Static windload on an area A.

Written in another form (55) becomes

with as the joint admittance:

and Ha the aerodynamic transferfunction:

In (57) X2 is a very complex function depending on pressure coefficient c and the

coherence function(s) needed. An empirical expression for x where all these influences

(and other physical phenomena not included in (57)) are taken into account is given by:

where A is the front area. In figure 13 the empirical expression of x() (aero dynamic

admittance) is compared with the joint admittance (57). Hereby is (57) simplified to

with i=l; cl=c2=ch ; VI=V2=Vh=VI0 and oul=o,2=~uh .

Figure 13: Aerodynamic admittance; comparison of (59) and (60).

The variance of the fluctuating windload follows now with

or more compact, taking the square root

frequency in Hz

Note: For a small area both r and X 2 go to 1; if A approaches infinity l- and x go to 0.

6.2 Dvnarnic structures

In case of assessing dynamic structures (figure 14) equation (56) changes into:

with Ha is the aerodynamic transfer function given by (58) and H, is the structural

frequency response function. For a single degree of freedom system holds:

In (64) F, is the equivalent quasi static load on the structure. (For the static case: F=Fv),

f, the natural frequency of the structure and 5 the damping ratio (clc,,). The dampingratio

c includes aerodynamic damping as well: c=c,+c, with 6, as the structural damping ratio

and Ca the aerodynamic damping depending on V and the building mass.

The variance of F, follows from:

where X2 follows from (57).

With the approximation

it follows from (66) that

where

Figure 14: Mass and spring system loaded by wind.

and

The standard deviation of the fluctuating windload Fv follows now from:

The probability that Fv exceeds F,* follows from

where

with

I - , . , .

Herein is:

fo the dominant frequency of the quasistatic wind load fluctuations (estimate)

f, the natural frequency of the structure

feq the central frequency (estimate)

f ~ h the parent distribution of the At-means of wind speed (incident flow)

p a reduction factor because of varying wind directions

Only for A=O ( ~ ~ = 1 ) and the commonly used windspectra Suu some convergence

problems can occur in the calculation of fo according to (75).

According to (75) the frequency fo depends on several factors such as the dimensions of

area A (b,h). See figure 15 taken from the Eurocode 1, part 2.3 [56] as an example.

It must be kept in mind that due to a frequency depending c the dominant frequency of

the quasi static windload will be influenced.

Figure 15: Expected frequency of gust loading of rigid structures

p = (bh~ ' .~ / L , V = mean wind speed at the equivalent height zeg, L =

length of turbulence at the equivalent height zeq, b = width and h = height.

Given (b+h)/L and p results in an intermediate variable S (see left part of

the figure); S and V/L results in the dominant frequency f, .

The engineering model described above can be seen as a reference model. It is thought,

however, that the approach given in chapter 5 (bluff body model) gives the possibility to

arrive at a better understanding of the pressure coefficient c and the factor P to be used in

the engineering model.

For a more general description of the engineering model see appendix C .

The general equation for the linear elastic behaviour of a structure under mechanical loads

is given by:

In here M is the mass matrix, C the damping matrix, K the stiffness matrix and F denotes

the external load vector; y, y and y are respectively the structural displacement, velocity

and acceleration vector. If, in the case of wind loading, the external load depends on the

movements of the structure, aero- elastic interaction is said to be present. If the structure is

very rigid, the interaction terms in F(t) can be neglected. If not so, a fruitful strategy in

many cases is to separate the various interaction terms as follows:

%

F(t,y,y,ji) = F, - May - Cay - Kay (77)

The interaction terms can then be transferred to the other side of the equation:

- Ma is called the aerodynamic mass

- Ca is called the aerodynamic damping

- K, is called the aerodynamic stiffness

In general, structural mass and stiffness are large, while structural damping is small. As a

consequence the aerodynamic damping is of most practical significance. If the aerodyna-

mic damping is positive, it will help to stabilize the structure. However, aerodynamic

damping may also be negative, and have an increasing effect on the structural movements.

Problems may occur if the absolute value of the (negative) aerodynamic damping is

greater than the structural damping. In that case the system is highly unstable. Damping is

then only to be expected from nonlinear effects at larger displacements. As a result often

catastrophic damage is to be expected.

Some aerodynamic phenomena will now be discussed in more detail.

6.3.1. Inwind aerodynamic damping

The most simple aero-elastic phenomenon occurs during the inwind loading of structures.

If we accept Morrison's approach we may write, as an extension of the load calculation

(48):

Here CD, CM1 and CM2 are the dimensionless drag and inertia coefficients which should

follow from wind tunnel experiments and which can be found in many textbooks and

publications. Vo is the volume of the structural element. The point is that the force in

reality depends on the velocity difference between wind and structure (the relative wind

velocity), rather then on the wind velocity alone. As stated before, the most important

term is the linear velocity term [ -p A CD V y], which can be interpreted as a damping

term.

6.3.2. Divergence

Divergence is a purely static form of instability. Consider a beam under constant windloa-

ding as indicated in figure 16. If the beam has a small torsional rotation 0, the wind may

produce a torsional couple per unit length fT, which may be written as:

with

V = mean wind speed

B = width of the beam

Cy = coefficient to be found from wind tunnel experiments

0 = torsional rotation of the beam

Figure 16: Beam under constant wind loading, divergence galloping and flutter.

The load fT is proportional to 8 and has as a consequence the nature of a

stiffness term. If the wind velocity is large enough, the factor [ % ~ v ~ B ~ L c ~ ]

exceeds the torsional stiffness of the beam and "divergence" is said to occur.

6.3.3. Galloping

Galloping is a dynamic form of beam instability in a constant wind stream. Consider the

same beam of figure 16. Lf the beam moves up and down, the relative wind velocity has a

vertical component depending on the vertical velocity of the beam. The resulting vertical

force per unit length of the beam may be written as:

The quotient (y/V) is in fact the relative angle of the wind velocity. We see that the load

now has a pure damping term nature. In most cases the coefficient CG is negative and

stability is assured. However, for some cross sections CG is positive. In those cases, given

that the wind velocity is high enough, a small initial movement will cause large vertical

vibrations.

A clear trend as to the effect of turbulence on galloping and flutter is not eminent.

Experience indicates that turbulence have a slightly destabilizing effect on the flutter of

semi-streamlined bridge deck sections, but a stabilizing effect on H- and prismatic

sections. Galloping of structures composed of square, octagonal and D-type sections is in

general stabilized by the presence of turbulence in the flow. Rain fall on the other hand

has got a pronounced effect on the aeroelastic haviour on the galloping of bridge cables.

Evidence is found in [66].

6.3.4. Flutter

Flutter is a similar phenomenon as galloping, however, the beam performs both vertical

and rotational movements. This requires, among others, that the natural frequencies of the

rotational and vertical vibration modes have the same order of magnitude. Flutter may

occur for slender cable stayed bridges. The at that time in civil engineering unknown

flutter mechanism destroyed the Tacoma Narrow bridge in 1940.

6.3.5. Vortex shedding

Vortex shedding is a phenomenon that is a particular important for tower and chimney

type structures. Due to the development of a Van Karman vortex trail, these structures

may suffer from loads in directions perpendicular to the direction of the wind velocity (see

figure 17). This loading (per unit height) can be denoted as:

1 fy(z,t) = -~v(z) ' D CL sin(2xfWt + cp) with f, = SV(z)/D

2 (82)

Figure 17: Vortex shedding.

The coefficient CL depends on Reynolds number and the surface roughness; cp is a phase

angle and fw the vortex frequency, S is the Strouhall number which equals 0.2 for circular

cross sections and may vary between 0.1 and 0.2 for other cross sections. As long as the

structure reacts as a star object, the loads are relatively harmless: the vortices leave the

structure in small packages with a thickness of 2 to 6 times the diameter D; the various

packages need not to be vertical, need not to be in phase and may even have different

frequencies due to the presence of a vertical wind profile and turbulence, so cp and fw are

time dependent functions of the vertical coordinate z.

However, if the structure starts vibrating, the movements of the structure tends to have a

coordinating effect on the packages: all packages may then show a preference for the

frequency of the vibrating system (lock in). Given the fact that the structure has a

preference for vibrating in its natural frequency, this effect will lead to a frequency fw(z) =

f, over the total height of the structure. For this to happen, the mean wind speed has to be

approximately equal to the critical wind speed V,, = DfJS. In addition, while increasing

the amplitude of the vibration, all packages will show a preference for the same phase in

time and space. This will lead to an increase in the "effective" load. It is often assumed

that the effective load is proportional to the vibration amplitude:

As the preferred phase angle is such that the load is in phase with the velocity, we may

also write:

Of course this load reaches a maximum as soon as all packages leave the structure

periodically as one coordinated column over the complete height.

Note that in (84) damping (aerodynamic, structural) is not yet taken into account. Note

further that the lift coefficient CL depends to some extend on ymax . For further informati-

on see [67].

6.4 Model uncertainties

In structural analysis the wind velocity itself is widely recognised as a random variable.

However, in the total model of the wind loading on a structure many more uncertainties

are present. Some of these are of the "parameter uncertainty type", some of the "model

inaccuracy type". Examples are "the value of the pressure coefficient C" and the "assump-

tion of sufficient fetch and a homogeneous wind field" respectively.

Systematic research into all these uncertainties has never been made. In the following

table a number of the main model parameters have been listed and some subjective

quantification mainly based on engineering judgement have been given. In most cases a

lognormal distribution seems to be the most appropriate.

In addition to the model uncertainties, a hugh amount of statistical uncertainty is present

in the description of the velocity process, especially as far as the estimation of spectral

densities and coherence functions are concerned. The uncertainty in the long term velocity

can be estimated in a classical Bayesian manner on the basis of the number of observati-

ons. Other uncertainties may arise if long term data has to be interpolated between various

observation stations.

CoV = coefficient of variation in the mentioned parameter

Parameter

roughness z,

mean velocity V(z) given VI0 and z,

turbulence intensity I, given z,

turbulence length L,

coherence parameters c, , c Y

wind pressure coefficients Cp , CM , CF

CoV

0.10

0.10

0.10

0.30

0.30

0.15

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New York, 1985.

APPENDIX A MEASUREMENT OF LOADING DATA

A.l Full-scale tests

The earliest measurements of wind loading of stuctures were made exclusively at full

scale. Initially, the aim was to provide reliable values directly for use in design. In 1884,

reporting his experiments made for the design of the Forth Rail Bridge, and with the Tay

Bridge disaster still of major concern, Baker [23] wrote that the design:

"necessarily involved many matters of pure conjecture, which rendered it impossi-

ble to state with precision what factor of safety would belong to the Forth Bridge.

The same remark of course applies ever now with equal force to every other

bridge, because there exists a lamentable lack of data respecting the actual pressure

of the wind on large structures. Mr Fowler and I have spared no pains during the

past two year to contribute something to the general fund of information; and other

engineers, doubtless are experimenting - for experiments, and not speculations,

are wanted."

Early measurement equipment was quite crude, but ingenuity of approach was certainly

not lacking as when Irrninger [24] measured the wind loads on a large gas holder of the

floating type by tethering the gas holder gainst the wind and measuring the overall drag by

the tension in the tether and the overall lift by the change in gas pressure. Later experi-

ments were progressively better equipped and measurements became more extensive and

accurate. By the 1930s experiments were beginning to be made at model scale in wind

tunnels, and the role of full-scale tests had changed to providing "benchmark" data to

verify developing theories and models. The peak of full-scale experimentation occurred in

the decade 1970-80. A survey [25] in 1974 showed 103 current of recently completed

experiments on a wide range of structures.

Full-scale experiments are very expensive to conduct, especially in comparison with wind

tunnel or numerical modelling, and the number of current experiments has declined

considerably from the peak. Nevertheless, the need for full-scale data always remains

because new theories and analysis techniques must be verified by full-scale measurements

using the same contemporary analysis techniques.

A.2 Model-scale tests

In 1893, just before his full-scale experiment with the gas holder, lnninger constructed

what was probably the world's first wind tunnel by attaching a duct to the base of a 30m

high chimney. The updraft of the chimney drew air through the duct at speeds between

7.5 and 15 4 s . In 1927, in his eightieth year, he began a comprehensive series of tests

on building shapes in collaboration with Npkkentved which took a decade to complete [26,

271. Unfortunately, all these data were collected in smooth uniform flow and it is now

known from the later work of Jensen [28] are that results from wind tunnels without a

good simulation of the atmospheric boundary layer are very different from what actually

occurs in nature. The only good data that survive from this time is the work by Flachsbart

[29, 30, 311 on lattice frames in uniform flow, because of the properties of the lattice plate

model described earlier in nr. 4.1, these data can be transcribed to apply the the turbulent

atmospheric boundary layer.

The early work in wind tunnels was done in tunnels designed for aircraft tests which

produced smooth uniform flow everywhere in the test section, except for a very thin

turbulent boundary layer next to the tunnel walls and floor. The effects of these thin

boundary layers confused early researchers, since each wind tunnel produced a floor

boundary layer of a different thickness, it was found that the same building model tested

in each wind tunnel produced different results. Unique among the early data is the work

by Bailey and Vincent [32], who found that the floor boundary layer of their wind tunnel

was a good scale model of the atmospheric boundary layer at 11240 and their data

compares well with modern experiments, except that only mean values were measured.

Inappropriate model tests using smooth uniform flow continued well into the 1960s and

most of these data found their way into codes of practice which were still in force in

1990. It was the work of Jensen [28], who demonstrated the effect of surface roughness

on the wind speed profile and its consequent effects on building pressures, that led to the

the accurate scale models of the atmospheric boundary layer in use today.

Most aeronautical wind tunnels are too short to develop a deep enough floor boundary

layer to give a good simulation of the atmospheric boundary layer, so special "bounda-

ry-layer wind tunnels" were developed for this purpose. The first of these tunnels were

very long in order to grow a deep enough boundary layer naturally over scaled surface

roughness. In other laboratories, where access to such long tunnels was not possible,

attempts were made to accelerate the growth of the floor boundary layer, leading to the

contemporary techniques called "roughness, barrier and mixing-device methods" [33].

Development of these methods has continued to increase their accuracy and the trend

towards their use in longer wind tunnels has tended to blur the distinction between

completely natural and accelerated growth methods.

A typical arrangment of hardware is shown in Figure 18. The role of the roughness is the

same as in the atmospheric boundary layer; it represents the roughness of the ground

surface to the correct linear scale. The roughness is the most important in that it establis-

hed the values of the three log-law parameters in Equation (3, z,, d, and u*. The barrier

and mixing-device are the "artificial" part of the simulation: the barrier giving an initial

deficit of momentum and depth to the boundary layer, which is mixed into the developing

simulation by the turbulence generated by the mixing device which, in this case, is a grid

of slats. The flow is tricked by the barrier into believing the fetch of roughness to be

longer, and by the mixing device into believing the banier is not there at all!

Figure 18: Typical hardware used to simulate the atmospheric boundary layer.

I

The random or regular arrays of roughness elements used in the simulation methods

produce the correct general flow conditions, both mean velocity profile and turbulence

spectra, approaching the site. However, the conditions at a particular site are more

strongly influenced by the local terrain than by the terrain far upwind. At some distance

from the site, it is necessary to change from a general simulation to a detailed representa-

tion of the site. usually called the "proximity model". The construction of detailed

proximity models is time consuming and expensive, particularly for urban areas. If too

small an area is represented, the site conditions will not be accurately reproduced. A

rational approach to this problem is given in reference [34] along with a more detailed

description of the methodology and criteria for quality assurance.

W i n dr-N 3 Jurbulence grid

A.3 Measurement techniques

Measurements of wind loading and response at full and model scale are made in four main

areas:

1. Measurements of wind speed and turbulence: the agent,

2. Measurements of surface pressures: the actions,

3. Measurements of forces and moments: the action effects, and

4. Measurement of the response in deflection, velocity or acceleration: the effects.

A.3.1 Wind speed measurements.

In full scale, cup anemometers give the magnitude of the horizontal wind vector and

weathercock vanes give the corresponding direction and are the standard for meteoro-

logical stations, resolving gusts down to about Is duration in strong winds. If the compo-

nents of turbulence are required at full scale, propellor anemometers mounted in groups of

three (one for each orthogonal axis) will resolve frequencies up to about 2Hz. Even

higher frequencies, up to 30Hz, may be resolved using ultrasonic anemometers which have

become available in recent years at prices comparable with the older techniques.

Most model-scale methods can also be used at full scale, although the instruments may not

be rugged enough for long-term exposure. The hot-wire anemometer works on the

principle that the heat lost by a fine wire kept at a constant high temperature is a function

of the wind speed normal to the wire. Used in pairs (X-probes), they can resolve the

turbulence components i n a plane which includes the mean wind speed (ie. u with v, or u

with w, but not v with w). Used in threes, they can resolve the instantaneous velocity

vector in all three dimensions. The frequency response of hot-wire anemometers is very

high, up to lOOkHz typically. A completely different approach is used by the pulsed-wire

anemometer: here a sudden pulse of electricity down a central wire heats the air around

the wire which advects with the flow. Another pair of wires, either side of the heated

wire and aligned normal to it, are fast-acting thermometers which detect the pulse of

heated air. The value of the wind speed vector normal to the heated wire is proportional

to the time delay between pulse and detection, and the direction of the flow is given by

whichever of the two detectors sense the pulse. Unlike the conventional hot-wire

anemometer, which gives a continuous analogue signal, the pulsed-wire anemometer is a

digital instrument passing inividual values at a rate up to 50Hz. A recently-developed

optical technique uses two intersecting beams from a single laser to illuminate small dust

particles in the flow. Interference fringes in the region of intersection result in periodic

illumination of the moving particles. The frequency of illumination is proportional to the

velocity vector.

A.3.2 Pressure measurements.

In the earliest experiments at full and model scale pressures were measured using liquid

manometers which were read by eye. One side of the manometer was connected to the

pressure to be measured and the other to a static datum. The simplest manometer is a

U-tube containing water or alcohol, resolving to about 1Pa. The easiest to read is the

"Betz" or projection manometer in which a scale under a float projects the value onto a

screen, resolving to about O.1Pa. The most accurate are the zero-balancing types such as

the "Chattock", resolving to 0.02Pa, but these require the pressure to remain steady while

the manometer is adjusted to balance. Recent developments in transducers have enabled

direct-reading electronic manometers with a resolution comparable to the "Betz", about

0.1 Pa.

Pressure transducers are devices which generate a voltage signal in response to applied

pressure, ideally linearly with amplitude and constant with frequency. When the UK

Building Research Establishment embarked on its programme of full-scale pressure

measurements in the early 1960s, there were no suitable transducers in existence so it was

necessary to develop one specifically for the purpose [35]. In essence, the active pressure

was applied to one side of a diaphragm and a steady reference pressure to the other.

Deflection of the diaphragm, proportional to applied pressure, was detected by strain

gauges. Modem transducers working by similar diaphragm principles but exploiting

modern electronics are now widely available, ranging in size and sensitivity, suitable for

measurements at both full and model scale.

At full scale, many pressure transducers are distributed over the surface of the building,

with either the diaphragm flush with the building surface or connected to a hole in the

building surface by a short length of tube. Multiple pressure transducers are required

because the changing weather conditions in storms require simultaneous measurements at

all locations. At model scale, the transducer is generally too large to mount flush with the

model surface, unless the average pressure over a large area is required, so the active side

is usually connected by tubing to a hole in the surface called a "tapping". As incident

wind conditions can be kept constant in the wind tunnel, simultaneous measurements are

not usually necessary and are often made sequentially using a pressure-scanning switch to

connect tubes in turn to a single transducer. In recent years, trandsucer technology has

advanced so that as many as 100 tranducers, each with an individual connection tube on

the active side, can be made on a single silicon chip 25mm square. This has allowed

many measurements to be made simultaneously at model scale.

The use of tubes to duct the pressure from the tapping to the transducer "colours" the

signal in that the frequency response of the tube is not constant, but acts like an organ

pipe imposing harmonic resonances on the signal. A theory was developed in 1965 which

enables the transfer function of any arbitrary tubing system to be calculated [36]. At

about the same time, physical methods of modifying the transfer function of the tubing to

give a constant magnitude and linear phase with frequency (constant time delay) were

being developed and the theory helped in the development and verification of the methods.

The most common method is to insert a restriction about half-way down the tube.

Electronic or numerical correction methods are also possible, but for cost effectiveness a

small brass restrictor in a length of plastic tube is hard to beat!

A.3.3 Force and moment measurements.

On most bluff bodies, the normal pressure forces far exceed the shear stresses from

friction, so that overall forces and moments can be reliably obtained by pneumatic or

numerical summation of surface pressures. In 1971 Surry and Stathopoulos [37] proposed

that if a number of pressure tappings were connected by tubing to a common point, the

pressure at that point would be the instantaneous average of the individual pressures,

supporting the case with experimental measurements. More recently, Gurnley [38]

developed the pressure tube theory [36] to account for multiple tubes and verified the

method analytically. With pneumatic averaging a different set of specially spaced tap-

pings is required for each measurement of force or moment. The advent of the mul-

ti-channel pressure transducers in conjuction with microcomputers has enabled numerical

averaging using arbitrary weighted averages to synthesise forces and moments from a

single set of tappings.

The alternative to integration of surface pressures is direct measurement using some form

of balance or dyanamometer. This is the only practical solution when friction forces are

significant, when the structure is a lattice, or there is some other reason that pressure

tappings are unsuitable. A high frequency range is necessary in order to study peak loads

on static structures and load spectra for dynamic structures, and this demands that the

balance be stiff. The majority of balances use strain gauges as the transducers and this

stiffness mitigates against sensitivity. Typically, strain-gauge balances have a range of

three orders of magnitude, so that a 10N range balance will resolve about 0.1N. Transdu-

cers based on piezo-electric crystals are many orders of magnitude stiffer and offer a

major advantage in range and frequency response. For the same 0.1N resolution a

ION-range strain-gauge balance is easy to damage, whereas the equivalent piezo-electric

balance 1391 would have a range of 25kN and be virtually unbreakable.

A.3.4 Dynamic response measurements.

Three methods are used to measure or infer the dynamic response of models:

1. Rigid models. Dynamic response calculated from force spectra using equations of

motion of building.

2. Semi-rigid models. Response in fundamental modes represented by rigid model of

required mass mounted on gibals and springs.

3. Fully-dynamic models. Models consEucted to behave as near full-scale as

possible, often by using the same materials scaled down precisely.

In the second and third cases the motion of the model is measured in terms of deflection

or acceleration (occasionally, velocity). With semi-rigid models, the motion can be

inferred from strains in the spring system or directly from the underside of the model.

With fully-dynamic models, detection of the motion is more of a problem: accelerometers

may be mounted in the model (and their mass accounted for in the modelling) or the

motion may be detected optically using laser interfereometry or video tracking techniques.

Of these techniques, semi-rigid models were developed first, fully-dynamic models are

extremely expensive and time-consuming, whereas rigid model methods are quick and

cheap. With rigid models the motion is not represented, so that any aerodynamic feedback

is suppressed and this would not be acceptable for aeroelastic structures. However, the

response can be calculated for a range of dynamic characteristics from one set of load

spectra. With semi-rigid models, the support stiffness and model mass must be changed

and the measurements repeated for each set of dynamic characteristics. A separate

fully-dynamic model is required for each set of dynamic characteristics. At first sight, it

might be thought that calculating the response from the load spectra would be less accu-

rate than direct measurements, but the reality is that the semi-rigid and fully-dynamic

models are only physical representations of the equations of motion deduced from

knowledge of the structure, so are inherently less accurate since the physical representation

will be less than perfect.

APPENDIX B THE VON KARMAN TURBULENCE SPECTRUM

Isotropic Turbulence

The three-dimensional velocity wind field g(rJ on a certain point in time, given the

assumptions of homogenenity and normality, is fully described by the covariance-functi-

ons.

q j (hJ = E {ui@ ujQ + hJ) (B1)

ui@ = wind velocity component i at point = {x,, x2, x3]; it is assumed that x, is the

direction of the average velocity field.

Using Fourier transforms the nine covariance functions can be transferred into a set of

nine spectra:

dx, dx2 dx3

&- = (Axl ,h2 ,h3) and k = (kl,k2,k3) = wave number vector

For incompressible isotropic turbulence (that is: o(u,) = o(u2) = cT(u3)) this spectrum can

be written as [53]:

2 k = ~ ( k : + 6 + k3 ) = the wave number vector

'i j = Kronecker delta

Von Karman [lo] proposed in 1948 an expression for a function E(k) which fulfils the

theoretically correct behaviour for extreme frequencies, i.e. that for k + 0 the energy

spectrum grows from E(0) = 0 with k4 and falls as k-5/3 for large k:

where:

a = a dimensionless constant (- 1.7)

L = turbulence length scale.

E T dV - u.2 = the rate of viscious energy dissipation (=- - - -1 p dz kz

Given the expressions (3) and (4) one may derive the xl-one-dimensional spectral matrix

from:

Elaboration for the main diagonal terms leads to:

The isotropic variance can be calculated as:

In (B7) I-( ) is the Gamma-function.

In the above description the time variabiIity has not been considered. We now assume that

we may convert the space variabiIity into a time variability by means of Taylor's

hypothesis [61] of the "moving frozen field". In that case we have:

It can be shown that for the time spectra the following relation hold:

2 1 ~ 2xf Sl l(f) = - Fll(kl) with kl = v

+a0

This spectrum is defined such that = I SII (f)df. -

As we want to have a2 = Sll(f)df, we have to Multiply by 2: I 0

4n 2nf Sll(f) = - F I 1 (kl) with k l = - v v

Using (B6a) and (B7) we arrive at:

Now define P as the turbulence scale in xl -direction:

1 i kl Ax I = - I Rll(Ax) dAx with Rll(Ax) = ~ ~ ~ ( k ~ ) e dkl

2 0

It can be shown that:

The turbulence length scaIe is difficult to determine experimentally. In practice it can be

determined from the location of the spectral peak of f S(f). For the longitudinal spectrum

it is P=0.92&,.

Inserting (B13) into (B11) leads to:

In a similar way:

These are the formules (13) - (16) in the main text.

The cross spectra of u , , u2 and u3 for different locations can be defined in terms of the

coherence function. For example:

where P1 and P2 two points in the y-z-plane. In isotropic turbulence it is sufficient to

determine the coherence Cob(.,.,.) = (x(.,.,.)12. The functions x(.) are analytic (Mann et

al. 1991) for the three directions:

2 2 21R 2 2 with Y=(?kl +r /L ) , r =A r + and kl=27tf/V and where %(.) are the modified

Besselfunctions of order p. For larger r/L the coherences do not approach unity for k l + 0 which can be important in applications. All other cross spectra can be assumed to

vanish.

The normalized spatial autocorrelation function can be evaluated by the Wiener-Chintchin

relationships

with p = r/(aP). The autocorrelation function in time is again obtained by making use of

Taylor's hypothesis setting T = r/T. Then the time scale of correlation is

Corrections for non isotropicv

In the atmospheric boundary layer isotropy can at most be assumed in heights above 500

m or in the high frequency range. It is however, possible to adjust the spectra obtained

under isotropic conditions. Anisotropy means that o(ul) + o(u2) + o(u3) and that there is

a non zero cross spectrum FI3(kl) + 0. It is just the reason for the height dependent mean

velocity profile. As a consequence the turbuIent eddies are stretched in the wind direction.

One observes o(ul) > o(u2) > o(u3) , i.e. o(ul)/o(u3) = 2 and o(ul)/o(u2) = 1.25. One

also has to expect that the turbulence length scales are different for the longitudinal and

the lateral directions. Furthermore these length scales as well as the turbulence intensities

should be height dependent.

In fnst approximation it is (see [54]):

which demonstrates the pronounced anistropy of the turbulent wind. The small values are

valid for rougher surfaces. Recommended values are collected in table 1 in the main text.

Also, there is (see (5) in the main text):

with K = 0.4 (the Von Karman constant) and zg and do from table 1 in the main text. In

principle any other reference height can be chosen. The longitudinal turbulence intensity is

defined by

According to measurements cr,(z) grows f i s t rapidly with height and decays gradually

above the height of the roughness elements. This dependency usually is not taken into

account and is experimentally not always verified. Even if a, is assumed independent of

height, I&) decreases with height as i ( z ) increases. Approximately it is

The longitudinal length scale is (adapted from [57-501)

The vertical length scale L: = L: is about half of this value and the lateral length scale;^:

is somewhere in between.

In first approximation <iiw> = 0.3 a, a, is found for the cross correlation and there is a

cross spectrum Suw (0. In aeronautics the following formulae is proposed [lo]:

For wind loads on buildings it is not very important.

APPENDIX C GENERAL DESCRIPTION OF THE ENGINEERING MODEL

In this paragraph a general description is given of the engineering model applied to a

multi modal srmctural system. Considering a vibrationmode Qi and a natural frequency

Wei .

d ) = d i d i i i c M ~ = 1 eSQi ldvOl (C1) vol

Herein is:

d the displacement function

c the pressure coefficient as function of r, constant in time

p, the mass per unit volume, as a function of r

Mi the ie generalised mass

qj time function for vibrationmode Qi

r (x,y,z)

Along the lines in section 6.1.2 it can be derived that

m

and

(C4)

where

From this it can be derived

with

Using the mode superposition method gives:

where F$i is the loadeffect (force, moment, stress) associated with mode mi.

The probability of exceeding rs* follows from

where

Herein is:

f, the i-th natural frequency of the structure

T the period considered

fVIO the parent distribution of the incident flow

p reductionfactor because of variing winddirections

rs response (d, d" or F,)

rs* limit value for rs

with

The engineering model described above can be seen as a reference model. It is thought

that the approach given in chapter 5 gives the possibility to arive at a better understanding

of the pressure coefficient c and the factor B to be used in this model.

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