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- Herausgeber: Wiley-VCH
- Kategorie: Wissenschaft und neue Technologien
- Serie: Beton-Kalender Series
- Sprache: Englisch
The wind energy industry in Germany has an excellent global standing when it comes to the development and construction of wind turbines. Germany currently represents the world's largest market for wind energy. The ongoing development of ever more powerful wind turbines plus additional requirements for the design and construction of their offshore foundation structures exceeds the actual experiences gained so far in the various disciplines concerned. This book gives a comprehensive overview for planning and structural design analysis of reinforced concrete and pre-stressed concrete wind turbine towers for both, onshore and offshore wind turbines. Wind turbines represent structures subjected to highly dynamic loading patterns. Therefore, for the design of loadbearing structures, fatigue effects - and not just maximum loads - are extremely important, in particular in the connections and joints of concrete and hybrid structures. There multi-axial stress conditions occure which so far are not covered by the design codes. The specific actions, the nonlinear behaviour and modeling for the structural analysis are explained. Design and verification with a focus on fatigue are adressed. The chapter Manufacturing includes hybrid structures, segmental construction of pre-stressed concrete towers and offshore wind turbine foundations. Selected chapters from the German concrete yearbook are now being published in the new English "Beton-Kalender Series" for the benefit of an international audience. Since it was founded in 1906, the Ernst & Sohn "Beton-Kalender" has been supporting developments in reinforced and prestressed concrete. The aim was to publish a yearbook to reflect progress in "ferro-concrete" structures until - as the book's first editor, Fritz von Emperger (1862-1942), expressed it - the "tempestuous development" in this form of construction came to an end. However, the "Beton-Kalender" quickly became the chosen work of reference for civil and structural engineers, and apart from the years 1945-1950 has been published annually ever since.
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Seitenzahl: 269
Veröffentlichungsjahr: 2013
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Contents
Cover
Series Page
Title Page
Copyright
Editorial
Chapter 1: Introduction
Chapter 2: Actions on Wind Turbines
2.1 Permanent Actions
2.2 Turbine Operation (Rotor and Nacelle)
2.3 Wind Loads
2.4 Height of Sea Level
2.5 Hydrodynamic Environmental Conditions
2.6 Hydrodynamic Analysis
2.7 Thermal Actions
2.8 Sea Ice
2.9 Icing-Up of Structural Members
Chapter 3: Non-Linear Material Behaviour
3.1 General
3.2 Material Laws for Reinforced and Prestressed Concrete
3.3 Bending Moment-Curvature Relationships
3.4 Deformations and Bending Moments According to Second-Order Theory
3.5 Design of Cross-Section for Ultimate Limit State
3.6 Three-Dimensional Mechanical Models for Concrete
Chapter 4: Loadbearing Structures and Detailed Design
4.1 Basis for Design
4.2 Structural Model for Tower Shaft
4.3 Investigating Vibrations
4.4 Prestressing
4.5 Design of Onshore Wind Turbine Support Structures
4.6 Design of Offshore Wind Turbine Structures
4.7 Ultimate Limit State
4.8 Analysis of Serviceability Limit State
4.9 Fatigue Limit State
4.10 Design of Construction Nodes
4.11 Foundation Design
Chapter 5: Construction of Prestressed Concrete Towers
5.1 Introduction
5.2 Hybrid Structures of Steel and Prestressed Concrete
5.3 Prestressed Concrete Towers with Precast Concrete Segments
5.4 Offshore Substructures in Concrete
References
Index
All books published by Ernst & Sohn are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.
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The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at <http://dnb.d-nb.de>.
© 2013 Wilhelm Ernst & Sohn, Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Rotherstraße 21, 10245 Berlin, Germany
All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.
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Editorial
The “Concrete Yearbook” is a very important source of information for engineers involved in design, analysis, planning and production of concrete structures. It is published on a yearly basis and offers chapters devoted to various subjects with high actuality. Any chapter gives extended information based on the latest state of the art, written by renowned experts in the areas considered. The subjects change every year and may return in later years for an updated treatment. This publication strategy guarantees, that not only the most recent knowledge is involved in the presentation of topics, but that the choice of the topics itself meets the demand of actuality as well.
For decades already the themes chosen are treated in such a way, that on the one hand the reader is informed about the backgrounds and on the other hand gets acquainted with practical experience, methods and rules to bring this knowledge into practice. For practicing engineers, this is an optimum combination. Engineering practice requires knowledge of rules and recommendations, as well as understanding of the theories or assumptions behind them, in order to find adequate solutions for the wide scope of problems of daily or special nature.
During the history of the “Concrete Yearbook” an interesting development was noted. In the early editions themes of interest were chosen on an incidental basis. Meanwhile, however, the building industry has gone through a remarkable development. Where in the past predominantly matters concerning structural safety and serviceability were in the centre of attention, nowadays an increasing awareness develops due to our responsibility with regard to society in a broader sense. This is reflected e.g. by the wish to avoid problems related to limited durability of structures. Expensive repair of structures has been, and unfortunately still is, necessary because of insufficient awareness of deterioration processes of concrete and reinforcing steel in the past. Therefore structural design should focus now on realizing structures with sufficient reliability and serviceability for a specified period of time, without substantial maintenance costs. Moreover we are confronted with a heritage of older structures that should be assessed with regard to their suitability to safely carry the often increased loads applied to them today. Here several aspects of structural engineering have to be considered in an interrelated way, like risk, functionality, serviceability, deterioration processes, strengthening techniques, monitoring, dismantlement, adaptability and recycling of structures and structural materials, and the introduction of modern high performance materials. Also the significance of sustainability is recognized. This added to the awareness that design should not focus only on individual structures and their service life, but as well on their function in a wider context, with regard to harmony with their environment, acceptance by society, the responsible use of resources, low energy consumption and economy. Moreover the construction processes should become cleaner, with less environmental nuisance and pollution.
The editors of the “Concrete Yearbook” have clearly recognized those and other trends and offer now a selection of coherent subjects which resort under a common “umbrella” of a broader societal development of high relevance. In order to be able to cope with the corresponding challenges the reader is informed about progress in technology, theoretical methods, new findings of research, new ideas on design and execution, development in production, assessment and conservation strategies. By the actual selection of topics and the way those are treated, the “Concrete Yearbook” offers a splendid opportunity to get and stay aware of the development of technical knowledge, practical experience and concepts in the field of design of concrete structures on an international level.
Prof. Dr. Ir. Dr.-Ing. h.c. Joost Walraven, TU Delft
Honorary president of the international concrete federation fib
1
Introduction
The wind energy industry in Germany has an excellent global standing when it comes to the development and construction of wind turbines. Germany currently represents the world's largest market for wind energy. So far, more than 21 000 wind turbines with a total output of approx. 25 000 MW have been installed across the country. And at the moment that figure is growing by approx. 2000 MW every year [1]. Developments in land-based installations are moving in the direction of more powerful turbines with more than 3 MW per installation and towers exceeding 140 m in height.1)
However, the number of lucrative sites on land (onshore) is dwindling. Therefore, it is planned to construct wind turbines at sea (offshore) in the coming years. The plans provide for offshore wind farms in the North Sea and Baltic Sea and are intended to increase substantially the proportion of renewable energies in electricity generation. The target for the medium-term is installations in the North Sea and Baltic Sea with a total output amounting to some 3000 MW. By 2030 it is hoped that offshore wind turbines with a total output of about 20 000 to 25 000 MW will have been built [2].
Figure 1.1 shows the results of a study carried out by DEWI, the German Wind Energy Institute. It shows the annual installed wind energy output for each year since 1990 plus the forecast up to the year 2030. According to the study, the decline in onshore installations should be compensated for by the anticipated development in offshore wind farms and by the repowering of land-based installations, leading to a doubling in the annual installed output by the year 2020.
Figure 1.1 Installed wind energy output per year in Germany[3]
The towers supporting onshore wind turbines are mainly of steel or prestressed concrete with internal or external prestressing. Steel lattice masts are also used in isolated instances. The prestressed concrete towers make use of both in situ and precast concrete. In recent years, the use of hybrid towers, consisting of a prestressed concrete shaft and a steel top section, has proved to be a very economical solution, especially for wind turbines in the multi-megawatt category. The choice of a suitable tower design is governed by the conditions at the site (fabrication, transport, erection, etc.). Figure 1.2 illustrates typical towers for onshore wind turbines.
Figure 1.2 Typical onshore tower designs for wind turbines
Both shallow and deep foundations can be used for onshore wind turbines. Soil improvement measures can be employed to upgrade subsoil properties to those required for shallow foundations [4,5]. Driven piles of steel or concrete with appropriate toe forms are frequently used as deep foundations.
So far, about 25 wind farms have been approved for construction off the German coast in the North Sea and Baltic Sea within the 12-mile zone and the exclusive economic zone (EEZ) for water depths of up to 45 m. But the better wind conditions at sea call for a greater technical input for the loadbearing structure and the fabrication and erection of the wind turbines [6]. Besides the depth of the water, the choice of a suitable offshore structure is especially dependent on the wave and current conditions plus the subsoil beneath the seabed. Concrete structures in the form of gravity bases are economic propositions for nearshore sites and for greater depths of water, see [7]. Such foundations are built in a dock, for example, then floated out to their final position and sunk. Resolved designs with individual members made from prestressed high-strength concrete are also feasible. An overview of the offshore foundation concepts currently under discussion can be found in Section 5.
The ongoing development of ever more powerful wind turbines plus additional requirements for the design and construction of their offshore foundation structures exceeds the actual experience gained so far in the various disciplines concerned. Wind turbines represent structures subjected to highly dynamic loading patterns. The load cycles of onshore installations can reach N = 109, but those of offshore installations can be exposed to further load cycles of up to N = 108 due to the sea conditions. Therefore, for the design of loadbearing structures, fatigue effects – and not just maximum loads – are extremely important. This can lead, in particular, to multi-axial stress states arising in the connections and joints of concrete and hybrid structures (see Sections 3.6 and 4.9), which have considerable effects on the fatigue strength and so far have not been addressed in the applicable design codes.
On the whole, there is still a great need for further research in the various disciplines involved in the planning, design and construction of wind turbines. It was for this reason that the Centre for Wind Energy Research ForWind (www.forwind.de) was set up at the Carl von Ossietzky University of Oldenburg and the Leibniz University of Hannover in 2003, thus enabling engineers from different disciplines to work together on research into wind energy. Supported by the Lower Saxony Ministry of Science and Culture, the objective of the centre is to pool research activities. Construction technology research into offshore wind turbines began at the University of Hannover as long ago as 2000 in the shape of the GIGAWIND (www.gigawind.de) joint project sponsored by the Federal Ministry for the Environment, Nature Conservation and Nuclear Safety. These research activities are divided into three areas: actions due to wind and waves, design of loadbearing structures (including foundations) and environmental technology aspects. GIGAWIND alpha ventus is a project associated with the RAVE (Research atAlphaVentus) research initiative and therefore has access to the extensive programme of measurements carried out at the Alpha Ventus test site, Germany's first offshore wind farm. At European level, the University of Hannover participates in the European Academy of Wind Energy (www.eawe.eu). The objective here is to promote research and development and to train PhD students in the field of wind energy in various European countries.
The basic concepts for the planning, design, analysis and construction of tower structures, focusing on wind turbines especially, will be explored in the next chapters.
Many aspects of these basic concepts also apply to the structural and constructional requirements of other tower-type structures, for example
For more information on these structures please refer to Beton-Kalender 2006 Teil 1, pp. 103–223 [8].
Note
1. Source: Bundesverband der Windenergie e.V. (www.wind-energie.de).
2
Actions on Wind Turbines
2.1 Permanent Actions
In addition to the typical dead loads of the plant (rotor and nacelle) and the structure (tower and foundation), there are also other loads that are classed as permanent actions: for example the loads of items fitted inside the tower (cables, intermediate platforms, etc.), and those due to further electrical equipment, for example transformers, ventilation systems.
And when it comes to offshore wind turbines there are yet further dead loads to be considered such as external platforms, boat moorings or cathodic corrosion protection.
For the dynamic analysis in particular, the masses of the individual items and components must be known and taken into account accurately in the design.
2.2 Turbine Operation (Rotor and Nacelle)
The actions due to the operation of the turbine are determined by means of numerical simulations (see also Section 4.9.1). In addition to various wind load models, with the superposition of wave action effects where applicable, such simulations must also take into account particular operating situations, for example starting and stopping procedures.
The load case combinations to be investigated are laid down in the relevant codes and guidelines, for example the DIBt guideline for onshore wind turbines [9], see Section 4.5.3, and DIN EN 61400-3 for offshore wind turbines [10]. Load combinations are also defined in the guidelines published by a number of certification bodies, for example the GL Guideline [11], see Section 4.6.4.
Note: The GL Guideline for offshore wind turbines [11] is based on Rules and Guidelines, IV Industrial Services – 1 Guideline for the Certification of Wind Turbines dating from 2003/04, which in July 2010 was republished in a revised edition.
2.3 Wind Loads
2.3.1 Wind Loads for Onshore Wind Turbines
According to DIN 1055-4 [12], the environmental conditions in Germany (including the German Bight) can be divided into four wind zones (Figure 2.1).
Fig. 2.1 Wind zones to DIN 1055-4 [12]
The reference values (vref; qref) in the table are valid for
The relationship between reference values for wind speed vref and dynamic pressure qref is given by the following equation:
When designing towers, only the reference dynamic pressures for terrain categories II (inland) or I (wind zone 4 directly on the coast) should be assumed. Less onerous terrain categories (III and higher) can be ruled out because the effects of the various ground roughnesses decrease as the height of the structure increases.
Therefore, the combined profiles given in DIN 1055-4 [12] for structures up to 50 m in height should not be used either (see also Beton-Kalender 2006 [8]).
Prior to the introduction of DIN 1055-4 [12], the wind loads for tower-type structures were calculated according to DIN 1056 [13] or annex A of DIN 4131 [14] or annex A of DIN 4228 [15]. Beton-Kalender 2006 [8] compares the wind loads according to the old standards and DIN 1055-4 [12].
2.3.1.1 Wind Loads According to the DIBt Guideline
According to DIN 1055-4, the following basic parameters apply (see Figure 2.2 and Table 2.1):
Fig. 2.2 Angle of attack for the rotor of a wind turbine
Table 2.1 Wind conditions for onshore wind turbines in terrain category II according to [12]
Additional parameters (DIBt guideline [9]):
2.3.1.2 Checking the Susceptibility to Vibration
According to DIN 1055-4 [12] Section 10, the wind forces acting on structures not susceptible to vibration are based on the peak dynamic pressure, which is averaged over a gust duration of 2–4 s (Table 2.2).
Table 2.2 Peak dynamic pressure (to DIN 1055-4 [12] Table B.2)
According to DIN 1055-4 [12] 6.2 (2), the wind loads for structures acting as cantilevers may be determined according to the simplified method for structures not susceptible to vibration (see below) provided the following condition is satisfied:
where
2.3.1.3 Example of Application
Prestressed concrete wind turbine structure, hub height 130 m (see Section 5.2):
The tower is therefore susceptible to vibration.
According to [9] 8.3.1, the vibration effect of the tower in the direction of the wind caused by the gustiness of the wind for a wind turbine in the “non-operational” condition (see Section 4.5.2) must be taken into account by way of an equivalent static load, which according to [9] Section B.3 or DIN 1055-4 [12] annex C may be calculated as follows:
Resultant equivalent static wind load in structure segment j ([12] C.2)
where
Average dynamic pressure (10-min average) ([12] C.2 (3))
Table 2.3 Average wind speed (to DIN 1055-4 [12] Table B.2)
Figure 2.3 shows the associated wind speed profiles.
Fig. 2.3 Average wind speeds for various wind zones
The gust response factor (G) is related to the average dynamic pressure qm. DIN 1055-4 [12] C.3 (1) contains the following formula:
where
Table 2.4 Turbulence intensity (to DIN 1055-4 [12] Table B.2)
These parameters are explained below.
Figure 2.4 shows the associated turbulence intensity profiles.
Fig. 2.4 Turbulence intensities for various terrain categories
Peak factor (Figure 2.5) according to DIN 1055-4 [12] C.3 (2):
where
Fig. 2.5 Peak factor
Expected value for frequency of gust response to DIN 1055-4 [12] C.3 (3):
where
where
Table 2.5 Integral length Li (z) of turbulence (to DIN 1055-4 [12] C.3 (4))
Basic gust component Q0, squared ([12] C.3 (5))
Resonance response component Rx, squared ([12] C.3 (6))
where
Dimensionless spectral density function RN ([12] C.3 (7))
where
Aerodynamic transfer functions Rh and Rb ([12] C.3 (8))
These are specified for the fundamental vibration mode with identical sign (deformation in the same direction) and are calculated, starting from RL, as follows:
where
Logarithmic damping decrement δ ([12] F.5)
Estimate of the logarithmic damping decrement for the fundamental flexural vibration mode to DIN 1055-4 [12] F.5 (1):
Structural damping δs see Table 2.6.
Table 2.6 Structural damping (to DIN 1055-4 [12] F.5 (2)) δs = a1 · n1 + b1 ≥ δmin where n1 = fundamental flexural vibration frequency [Hz]. Parameters a1, b1, δmin to 1055-4 [12] Table F.2 (extract)
Aerodynamic damping ([12] F.5 (3))
where
(see DIN 1055-4 [12] F.4)
where ζ = 2 for towers and masts
Additional damping decrement δd ([12] F.5 (4))
Where special measures are provided for increasing the damping (e.g. vibration dampers),δdis to be calculated with the help of suitable theoretical or experimental methods.
Aerodynamic force coefficient for towers with a cylindrical cross-section ([12] 12.7.1 (1)):
where
Fig. 2.6 Basic force coefficients for cylinders, see DIN 1055-4 [12] Figure 19
Fig. 2.7 Slenderness reduction factor ψλ to DIN 1055-4 [12] Figure 26, for a solidity ratio ϕ = 1.00
Table 2.7 Equivalent roughnesses (to DIN 1055-4 [12] Table 11)
Reynolds number ([12] 12.7.1 (2)):
where
Effective slenderness λ for cylinders with segment length L and outside diameter b to DIN 1055-4 [12] Table 16:
Intermediate values may be obtained by linear interpolation.
Reference area of segment “j” considered (to DIN 1055-4 [12] 12.7.1. (5)):
Calculation of aerodynamic force coefficient ([12] 12.7.1)
Example: prestressed concrete wind turbine structure, hub height 130 m (see Section 5.2)
Determining the equivalent mass [12] annex F
Example: prestressed concrete wind turbine structure, hub height 130 m (see Section 5.2)
The following sample calculation compares the results according to DIN 1055-4 [12] with those of the DIBt guideline [9] which arise as a result of the different approaches.
Calculation of gust response factor
Example: prestressed concrete wind turbine structure, hub height 130 m (see Section 5.2)
Dynamic pressures taking into account the gust response
The simplified calculation is on the safe side for the example shown in Figure 2.8.
Fig. 2.8 Comparison of dynamic pressures taking into account the gust response according to DIN 1055-4 [12] and DIN 1056 [13]
2.3.2 Wind Loads for Offshore Wind Turbines
2.3.2.1 Classification of Wind Turbines
The definition of a wind turbine class is practical for designing the machinery (rotor – topsides structure) of an offshore wind turbine [11].
The values for the wind speed and turbulence intensity parameters should represent the characteristics of numerous different locations, the aim being to determine clearly defined levels of robustness (Table 2.8).
Table 2.8 Wind conditions for offshore wind turbines to [11] Table 4.2.1.
The design of the tower and foundation (support structure) for an offshore wind turbine must be based on the representative environmental conditions – including the sea conditions – at the respective location.
The design working life of an offshore wind turbine should be at least 20 years.
A rotor (turbine) designed according to one of the wind turbine classes given in Table 2.8 can withstand environmental conditions in which the 10-min average of the extreme wind speed for a 50-year return period is not greater than the given reference wind speed (Vref) at hub height.
The average wind speed (Vave) is the statistical mean of the momentary wind speed values averaged over a certain period – ranging from a few seconds to several years. In [11] Vave is the annual average wind speed over many years. This value is used in the Weibull or Rayleigh functions for the wind speed distributions.
2.3.2.2 Determining the Wind Conditions (Wind Climate)
The following basic parameters for the wind actions must be determined for the draft design and the location [11] 4.2.2.2:
The averaging time used in [11] for the reference wind speed (Vref) is 10 minutes (see Table 2.8). The wind conditions (wind climate) may be determined from measurements taken at the location provided the measurement period is at least six months. The effects of seasonal fluctuations must be taken into account where these have a substantial effect on the wind climate.
Here, I15 is the characteristic value of the turbulence intensity (I15(k)). It is calculated by adding the measured standard deviation to the measured mean value (I15(m)) of the turbulence intensity. If the standard deviation has not been calculated from measurements, then the characteristic turbulence intensity for Vhub = 15 m/s may be calculated as follows:
The value I15(k) should be determined from measured data when wind speeds exceed 10 m/s. In agreement with GL Wind (Germanischer Lloyd WindEnergie GmbH), the relevant characteristic values of the wind climate may be determined by numerical methods as an alternative.
2.3.2.3 Normal Wind Conditions
Wind Speed Distribution
The local distribution of the 10-min average of the wind speed at hub height (Vhub) is significant for the design of an offshore wind turbine because this determines the frequency of occurrence of individual load components.
A Weibull distribution (PW) must be derived from in situ measurements verified by long-term measurements in the immediate vicinity:
where
When k = 2, the Weibull distribution produces a Rayleigh distribution which can be used for calculating the wind loads to [11] (Figure 2.9):
From this we get the probability density for the wind speeds:
Fig. 2.9 Rayleigh distribution (function P and density f) for wind speeds
Normal Wind Profile Model (NWP)
The following power law equation should be assumed for the wind profile V(z):
where
This wind profile is used to define the average wind shear force on the area swept by the rotor. This model is based on neutral atmospheric stability. Taking a constant surface roughness length of 0.002 m, then α = 0.14.
Normal Turbulence Model (NTM)
The turbulence of the wind is represented by the energy that is transported by turbulence eddies and for which a spectral distribution is assumed. The following parameters are among those that characterise the natural turbulence of the wind over a relatively short period in which the spectrum remains unchanged:
The values of the turbulence intensity are defined for the height of the hub. The spectral energy densities of the random wind speed vector field must satisfy the following requirements for the wind turbine classes of Table 2.8:
where
Fig. 2.10 Standard deviation of wind speed and turbulence intensity
The turbulence scale parameter is to be taken as follows:
Every load simulation with the normal turbulence model (NTM) must be carried out for a period of 10 minutes at least. Furthermore, a series of further general requirements must be taken into account for load calculations, see [11] 4.2.2.3.3 (7).
2.3.2.4 Extreme Wind Conditions
Extreme wind conditions are assumed in order to determine extreme wind loads on offshore wind turbines. These conditions include peak wind speeds during storms and sudden changes to wind speed or direction. The extreme wind conditions also include the possible effects of turbulence, with the exception of the extreme wind speed model (EWM).
Extreme Wind Speed Model (EWM)
The EWM must be based on in situ studies. Alternatively, the following data may be used.
The EWM can be either a steady or a turbulent wind model. The basic parameters are the reference wind speed Vref (10-min average of extreme wind speed with a 50-year return period) and a certain standard deviation σL. The wind loads are described by applying power law equations over the height:
Table 2.9 Conversion factors for wind speeds based on the 10-min average.
Extreme Operating Gust (EOG)
The gust speed VgustN at hub height, with a return period of N years, is calculated as follows for standard wind turbine classes:
where (as for the NTM)
The change in wind speed over time for a return period of N years is determined using the following equation:
where (NWP, see above)
Figure 2.11 shows an example of how an extreme operating gust changes over time.
Fig. 2.11 Change in an extreme operating gust over time (N = 1, turbulence category A, D = 42 m, zhub = 30 m, Vhub = 25 m/s)
Extreme Wind Direction Change (EDC)
The magnitude of the extreme wind direction change θeN for a return period of N years should be calculated as follows (see Figure 2.12):
where
Fig. 2.12 Example of the magnitude of the extreme wind direction change
The way the extreme wind direction change θN (t) varies over time for a return period of N years should be calculated as follows:
Here, T = 6 s is the duration of the chronological progression of the extreme change in wind direction. The sign should be chosen in such a way that the loading gives the most unfavourable progression. Afterwards, a constant wind direction is assumed. Figure 2.13 shows an example of an extreme wind direction change.
Fig. 2.13 How an extreme wind direction change varies over time (N = 50, turbulence category A, D = 42 m, zhub = 30 m, Vhub
