Analyse av vind-indusert respons av en hengebro på flytende fundamenter

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I

DET TEKNISK-NATURVITENSKAPELIGE FAKULTET

MASTEROPPGAVE

Studieprogram/spesialisering:

Master i teknologi – Konstruksjoner og materialer

Fordypning Byggkonstruksjoner

Vårsemesteret, 2017

Åpen Forfatter: Per Simon Sjölander

………

(signatur forfatter)

Fagansvarlig: Jasna B. Jakobsen (UiS) Veileder(e): Jasna B. Jakobsen (UiS)

Bruno Villoria (Statens Vegvesen) Tittel på masteroppgaven:

Analyse av vind-indusert respons av en hengebro på flytende fundamenter Engelsk tittel:

Analysis of wind induced response of a suspension bridge on floating foundations

Studiepoeng: 30 Emneord:

Hengebro Sulafjorden Vindanalyse Parameterstudie ABAQUS

FEM

Sidetall: 73 + vedlegg/annet: 23

Stavanger, 15.06.2017

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Abstract

As a part of a nation-wide project to create a ferry-free coastal highway route, several fjord- crossings need to be evaluated for construction of long spanned bridges. Such a crossing lies between the municipalities of Sula and Hareid in the county of Møre og Romsdal. A proposition for such bridge is the Sulafjorden bridge, which would consist of three suspended spans, two land towers and two towers on floating support.

The purpose of this thesis is to investigate how difference in girder discretization of such a long-spanned suspension bridge would affect the response due to wind turbulence. With the purpose of setting a theoretic foundation for the thesis, a literature review is initially presented about wind related themes. Further, a parametric study of how modal loading affects different discretization levels of the span of a simplified bridge model is done.

Moreover, an existing FEM model of Sulafjorden bridge is remodeled and analyzed with respect to static and dynamic wind conditions. Lastly, a sensitivity study in frequency domain is implemented to evaluate how changes in discretization of the span of Sulafjorden bridge may affect the resulting power spectral density.

As a result of these investigations, it would seem that an increase in discretization of the bridge girder from 20m to 10m girder element length would result in a reduction of response of the standard deviation of the lateral displacement at center of span by 0.2% and for vertical displacement by 1.5%. Further, mean lateral and vertical displacement is proved to have similar characteristics in response as the standard deviation.

Moreover, difference in standard deviation of vertical displacement between 10m and 40m girder element length is determined to be 3% at center of span, which could suggest that such a discretization level might not sufficiently capture enough details in the turbulent vertical wind field compared to in-situ natural conditions.

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IV

Preface

This thesis is submitted as a final project in the 2-year Master’s education Constructions and Materials at the University of Stavanger.

The project has been supported by the Norwegian Public Road Administration and has primarily been executed in collaboration with parties from the University of Stavanger.

I wish to express thanks to main supervisor Prof. Jasna Bogunovic Jakobsen for valuable guidance and critic during the progress of the project, Ph.D. Jungao Wang and Ph.D. Etienne Cheynet for appreciated assistance with software and general guidance. I would also like to express gratitude to previous master student Sondre Aspøy for valuable software related information.

At last, I wish to thank supervisor Bruno Villoria at the Norwegian Public Road Administration for the opportunity to write about this topic.

Per Simon Sjölander June 2017

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V

List of Figures

Figure 1. Model of Sulafjorden suspension bridge. Here depicted with floating ellipse

foundations.[2] ... 1

Figure 2. Wind directions in relation to Sulafjorden bridge... 3

Figure 3. Logarithmic- and power law mean wind speed profile divided by U(z, zref=10m).[4] 4 Figure 4. Static and dynamic wind components. Standard deviation of fluctuating wind component.[4] ... 4

Figure 5. Stationary short term random stochastic process with corresponding Gaussian- probability distribution.[5] ... 5

Figure 6. Turbulence intensity for u-, v- and w-components. ... 6

Figure 7. Resonant and background part of the wind load. [5] ... 7

Figure 8.The cross-correlation of longitudinal turbulence at z=13.5m.[4] ... 7

Figure 9. Spatial interpretation of integral length scales.[5] ... 8

Figure 10. Integral length scales for ideally homogeneous wind flow conditions... 9

Figure 11. Horizontal longitudinal spectral density functions of von Karman- and Kaimal forms. ... 12

Figure 12. Vertical spectral density functions of von Karman- and Kaimal forms. ... 13

Figure 13. Root-coherence function as function of fixed frequency (left) and separation (right). ... 15

Figure 14. Normalized co-spectrum for arbitrary decay coefficient. ... 16

Figure 15. Mean pressure around the cross-section of a bridge girder, with angle of incidence α=0 [8]. ... 17

Figure 16. Change in value for shape-factors for varying angle of incidence [8]. ... 18

Figure 17. Wind components, structural motion, angle of incidence and instantaneous wind velocity[8]. ... 19

Figure 18. Linearization of a lift shape-factor[8]. ... 20

Figure 19. Aerodynamic admittance function developed by Vickery. ... 22

Figure 20. Illustration of the spectral approach to dynamic response.[4] ... 23

Figure 21. Kaimal spectra vs. simulated u- and w-components. ... 29

Figure 22. Simulated time-series of along-wind speed u, here depicted as superimposed upon U. ... 30

Figure 23. Simulated time-series of vertical wind speed component w... 30

Figure 24. Discretization of fundamental wind-field simulation for a 2000m long span. ... 31

Figure 25. Modal load for 1^st mode-shape at corresponding nodal position, I, and time-step, t. ... 32

Figure 26. Normalized total modal drag force, where decay coefficient c indicates both Cuy and Cuz. ... 33

Figure 27. Normalized total modal drag force, where decay coefficient c indicates both Cuy and Cuz. ... 33

Figure 28. Normalized total modal lift force, where decay coefficient c indicates both Cwy and Cwz. ... 34

Figure 29. Total normalized modal drag force for 10m and 250m element length. ... 36

Figure 30. Total normalized modal lift force for 10m and 250m element length. ... 36

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Figure 31. FE-model of Sulafjorden bridge relative to its global coordinate system. ... 38

Figure 32. Sulafjorden suspension bridge on floating foundation with tension leg platforms.[1] ... 38

Figure 33. Details of the FE-model with 20m girder element length discretization. ... 39

Figure 34. Logarithmic mean wind speed profile. ... 42

Figure 35. Empirical wind shape-factors for box girder.[1] ... 43

Figure 36. Horizontal displacement in z-direction due to static force. ... 47

Figure 37. Rotational displacement about y-axis due to static force. ... 48

Figure 38. Rotation about x-axis, a, angle of incidence. ... 48

Figure 39. Time-series of lateral response of node 1110 at middle of bridge span. ... 49

Figure 40. Mean lateral response for the 30min time-series and normalized comparison of mean lateral displacement for girder element length of 10m, 20m and 40m. ... 50

Figure 41. Maximum lateral response for the 30min time-series and normalized comparison of maximum lateral displacement for girder element length of 10m, 20m and 40m. ... 50

Figure 42. Minimum lateral response for the 30min time-series and normalized comparison of maximum lateral displacement for girder element length of 10m, 20m and 40m. ... 51

Figure 43. Standard deviation of lateral response at each node for the 30min time-series. Normalized comparison of STD of lateral displacement for girder element length of 10m, 20m and 40m. ... 51

Figure 44. Time-series of vertical response of node 1110 at middle of bridge span. ... 52

Figure 45. Maximum vertical response for the 30min time-series and normalized comparison of maximum lateral displacement for girder element length of 10m, 20m and 40m. ... 52

Figure 46. Minimum lateral response for the 30min time-series and normalized comparison of maximum lateral displacement for girder element length of 10m, 20m and 40m. ... 53

Figure 47. Standard deviation of vertical response at each node for the 30min time-series. Normalized comparison of STD of lateral displacement for girder element length of 10m, 20m and 40m. ... 53

Figure 48. Frequency-domain PSD of lateral response for Δx=10m and Δx=100m discretization level. ... 54

Figure 49. Normalized PSD of lateral response with respect to 10m girder element length discretization level. ... 54

Figure 50. Normalized PSD of lateral response; mode-shape 2,5,12 and 21 relative to discretization level. ... 55

Figure 51. Frequency-domain PSD of vertical response for Δx=10m and Δx=100m discretization level. ... 55

Figure 52. Normalized PSD of vertical response with respect to 10m girder element length discretization level. ... 56

Figure 53. Normalized PSD of vertical response; mode-shape 3,9,19 and 22 relative to discretization level. ... 56

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List of Tables

Table 1. Applied exponential decay coefficients specified by NPRA Handbook N400. ... 15

Table 2. Change in normalized standard deviation for increasing element length in relation to 10m element length. For 1^st mode-shape and c=5. ... 35

Table 5. Cross-sectional properties of the beam elements in the bridge model.[1] ... 40

Table 6. Applied aerodynamic damping for different discretization level and element length of the girder, including damping for cable and floating tower. ... 40

Table 7. Aerodynamic shape-factors for.[1] ... 43

Table 8. Mode-shapes and eigen frequencies. ... 45

Table 9. Horizontal displacement in z-direction due to static force. ... 47

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XI

Nomenclature

Roman symbols

𝑐_𝑖 Coherence decay coefficient

c Damping

𝐶_𝐷 Aerodynamic drag coefficient

𝐶′_𝐷 Derivative of aerodynamic drag coefficient 𝐶_𝐿 Aerodynamic lift coefficient

𝐶′_𝐿 Derivative of aerodynamic lift coefficient 𝐶_𝑀 Aerodynamic pitching moment coefficient

𝐶′_𝑀 Derivative of aerodynamic pitching moment coefficient 𝐹_𝑖 Force

f Frequency

𝐼_𝑖 Turbulence intensity k Stiffness

𝐿_𝑖 Integral length scale

𝑀_𝑖 Moment

n Frequency

q Wind velocity pressure 𝑆_𝑖 Spectral density function s Distance

𝑇_𝑖 Time scale

t Time

U(t) Total wind speed U Mean wind speed u(t) Dynamic wind speed

u Along-wind turbulence component v Across-wind turbulence component w Vertical-wind turbulence component z Height above sea level

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XII Greek symbols

α Angle of incidence, also parameter in Rayleigh damping β Parameter in Rayleigh damping

𝛾_𝑖 Root-coherence 𝜉 Damping ratio

ρ Air density, also correlation coefficient function 𝜎_𝑖 Standard deviation

τ Time lag

χ Aerodynamic cross-sectional admittance 𝜔 Angular frequency

Abbreviations

FEM Finite element modelling

NPRA Norwegian public road administration PSD Power spectral density

STD Standard deviation TLP Tension leg platform

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1

1. Introduction

1.1 Background

As a part of a national project led by NPRA to introduce a ferry-free coastal highway route, fjord-crossings need to be evaluated for construction of long spanned bridges. Such a crossing lies between the municipalities of Sula and Hareid in the county of Møre og Romsdal.

Multiconsult AS has done a feasibility study[1] of how such a bridge may be constructed. One of several suggestions is a three-spanned suspension bridge on floating supports as depicted in Figure 1.

Figure 1. Model of Sulafjorden suspension bridge. Here depicted with floating ellipse foundations.[2]

A turbulent wind field that varies in time and space is characterized by several parameters.

The size of an idealized wind gust may be described by the integral length scale and is a rather fixed parameter defined by standards. The coherence property of separate wind components separated by a distance is another parameter that describes how correlated the wind components are with respect to each other. The coherence parameter may also be defined by standardization by the coherence decay coefficient. The applied wind spectra that describes how much energy each frequency is imbued with is another fixed parameter. A parameter that may vary is how detailed the turbulent wind field is represented, a property that may have implication on the resulting response of a structure such as a long-spanned suspension bridge.

The fundamental hypothesis that will be continuously returned to throughout this thesis is the idea how a turbulent wind field that is discretized into discrete representative quantities may have a changing effect on response.

1.2 Objective

A FE-model of the Sulafjorden bridge is available and previous analysis of the model has been done. Further research into the response of large-spanned bridges might be to extend the analysis with emphasis on various levels of discretization of the bridge girder span.

The general objective of this thesis is to evaluate how response might alter as a result of analysis when applying different discretization levels of the bridge girder span.

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1.3 Thesis structure

Introductive in Chapter 2, a literature review is done to set the theoretical fundament to further analysis. The theory consists of wind related literature, wind forces and the application of damping on a suspension bridge. Chapter 3 presents a parametric study with corresponding methodology, results and discussion. In chapter 4, the main FE-analysis is presented followed by a frequency domain analysis, where methodology, results and discussion is presented separately.

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3

2. Literature review

2.1 Wind

Wind can be defined as movement of air in relation to the surface of the planet, where the sun heats the atmosphere and differentials in pressure occurs. The atmospheric boundary layer is more evident closer to the surface, where friction will reduce the speed of the wind and make it more turbulent. Thus, the turbulence reduces with increasing height. [3]

The wind velocity vector is split into three fluctuating components, u in the main flow along–

wind direction, and v and w in the across-wind horizontal and vertical directions. The u- and w-components have directions that are normal to the span and v is parallel to the span. For Sulafjorden bridge the directions are expressed as in Figure 2.

Figure 2. Wind directions in relation to Sulafjorden bridge.

The total along-wind speed 𝑈(𝑡), Eq. 2.1, can be split into a mean value U that increases with height above ground level and the time-dependent fluctuating part u(t). The mean wind speeds of v and w is normally assumed to be zero. Bernoulli’s equation, Eq. 2.2, is commonly applied to describe instantaneous wind velocity pressure.

𝑈(𝑡) = 𝑈 + 𝑢(𝑡)

Eq. 2.1

𝑞_𝑈 = 1

2𝜌𝑈(𝑡)²

Eq. 2.2

2.1.1 Mean wind speed profile

The mean wind speed profile may commonly be given by logarithmic- or power law, Eq. 2.3 and Eq. 2.4, and increases with increasing height. Figure 3 illustrates the difference between the two laws where z0=0.02m and α=0.128. [4]

𝑈(𝑧) = 𝑢1 𝜅𝑙𝑛 𝑧

𝑧_𝑜

Eq. 2.3

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4 Where: u – friction speed given as √^𝜏_𝜌⁰𝜏₀ (𝜏₀ - shear stress at ground level)

Κ – von Karman’s constant (~0.4) z – height above ground

𝑧_𝑜 – roughness length

𝑈(𝑧) = 𝑈(𝑧_𝑟𝑒𝑓)( 𝑧 𝑧_𝑟𝑒𝑓)^𝛼

Eq. 2.4

Where: 𝑧_𝑟𝑒𝑓 – reference height

𝛼 – height and roughness related parameter given as 1/ log (^𝑧_𝑧^𝑟𝑒𝑓

𝑜 )

Figure 3. Logarithmic- and power law mean wind speed profile divided by U(z, zref=10m).[4]

The dynamic wind component can be expressed as superimposing the mean wind speed, Figure 4. The decrease of turbulence with height is here expressed with decreased standard deviation of the dynamic wind component.

Figure 4. Static and dynamic wind components. Standard deviation of fluctuating wind component.[4]

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5 2.1.2 Stochastic process

A stochastic process may be described as a process of which its numerical outcome at any position in space at any time is random and may only be predicted by a probability. In principle, an infinite set of realization of the process may be probable but none are identical.[5]

One may distinguish between short and long-term statistics, where short-term outcome may be interpreted as time-domain characteristics of a shorter period, e.g. the process that is represented in Figure 5. Thus, long term statistics can be interpreted as data of a large set of short term processes. In terms of wind components and the applicability for engineering purposes, it is critical that the properties of the short-term statistics are regarded as homogeneous and stationary. Depending on structural properties and what time period that may be of importance for analysis, such a short-term period, T, is commonly set to 10, 30 or 60minutes.[5]

The characteristics of a certain stochastic process may be described by its predetermined inherent properties. Figure 5 describes a unique realization of a such a process where fluctuations are superimposed upon a mean value, similar to how the turbulent u-component fluctuates around a mean wind velocity U. The figure also depicts how a probability distribution is applied to generate fluctuations throughout the time-series. [5]

Figure 5. Stationary short term random stochastic process with corresponding Gaussian-probability distribution.[5]

2.1.3 Turbulence

The general level of turbulence in a wind field can be measured by its standard deviation of its wind component, Eq. 2.5.

𝜎_𝑢 = {1

𝑇∫ |𝑈(𝑡) − 𝑈|²

𝑇 0

}

1/2

Eq. 2.5

Where the expression is integrated over time, T.

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6 The ratio of the standard deviation of each fluctuating component to the mean value is known as the turbulence intensity of that component. Since the mean wind speed increases with increasing height and the fluctuating wind speed reduces with height, the intensity of the turbulence decreases with increasing height. The lateral and vertical turbulence components are generally lower in magnitude than the corresponding longitudinal value.[5]

𝐼_{𝑢,𝑣,𝑤} =𝜎_{𝑢,𝑣,𝑤} 𝑈

Eq. 2.6

𝐼_𝑢 = 𝑘_𝐼 𝑐₀ln (𝑧/𝑧₀)

Eq. 2.7

Where 𝑘_𝐼 is the turbulence factor, 𝑐₀ the orthogonality factor and 𝑧₀ the roughness length.

The turbulence intensity of u-, v- and w-components may be described as Eq. 2.6 - Eq. 2.7. For ideally homogeneous flow conditions, Eq. 2.8 describes the relation between the turbulence intensity components[6]. Figure 6 displays how the turbulence intensity vary with height, where one can clearly observe that the intensity of the turbulence is high closer to the surface.

[_𝐼^𝐼^𝑣

𝑤]= [3/4 1/4]^𝐼^𝑢

Eq. 2.8

Figure 6. Turbulence intensity for u-, v- and w-components.

The dynamic wind fluctuates with time and the fluctuating period can be relatively instantaneous to several minutes. The resulting wind loading creates a response from the structure. Figure 7 illustrates how the background part of the wind load generates a quasi-

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7 static response of the structure. These frequencies are different from the structural eigenfrequency and affects the structure in a static manner. The same figure also depicts the resonant part of the wind loading where the fluctuations in the wind occurs in frequencies close to the eigenfrequency. These fluctuations creates a resonant response of the structure that can lead to critical structural integrity. [5]

Figure 7. Resonant and background part of the wind load. [5]

2.1.4 Correlation

Wind components may be more or less correlated in time and space. The correlation characteristics for longitudinal velocity components is relative to the separation distance, where the synchronization of the turbulence reduces with an increasing distance. The correlation coefficient function, Eq. 2.9, may be applied to describe correlation. The function is +1 when the separation is zero and approaches 0 for larger separations. Figure 8 displays an example of the correlation coefficient function compared to observations of longitudinal velocity turbulence[4].

𝜌 ≈ exp [−𝑐|𝑧₁− 𝑧₂|]

Eq. 2.9

Figure 8.The cross-correlation of longitudinal turbulence at z=13.5m.[4]

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8 2.1.5 Time scale

The time scale, Eq. 2.10, expresses the average duration of an u, v or w wind gust. [5]

𝑇_𝑛 = ∫ 𝜌^∞ _𝑛(𝜏)𝑑𝜏

0

Eq. 2.10

Where: n – u, v or w components 𝜏 – average time lag

2.1.6 Integral length scale

If assuming that the turbulence convection in the main flow direction takes place with the mean wind flow velocity u, it can be assumed that the average length scale of the u- component in the x-direction is as described in Eq. 2.11. Figure 9 depicts a spatial interpretation of the meaning of integral length scales, where an idealized gust size may be expressed. [5]

𝐿_𝑛,𝑥 = 𝑈𝑇_𝑛 = 𝑈 ∫ 𝜌_𝑛(𝜏)𝑑𝜏

∞

0

Eq. 2.11

Figure 9. Spatial interpretation of integral length scales.[5]

The magnitude of the fluctuating load is a function of the turbulence intensity and its length scale. The turbulence intensity governs the magnitude of fluctuation while the turbulence length scale determines how well the fluctuations are correlated over the structure [7].

For a line- or plate-like structure, the integral length scale governs the total load on the structure. The integral length scale might be considered as the size of perfect turbulence.

NPRA Handbook N400[6] defines the integral length scale in the along-wind direction, x, Eq.

2.12.

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9 𝐿_𝑢

𝑥 = {

𝐿₁(𝑧

𝑧₁)^0.3, 𝑧 > 𝑧_𝑚𝑖𝑛 𝐿₁(𝑧_𝑚𝑖𝑛

𝑧₁ )^0.3, 𝑧 ≤ 𝑧_𝑚𝑖𝑛

Eq. 2.12

Where: L1 – 100m Z1 – 10m

Zmin - minimum height, specified by the terrain category

For ideally homogeneous flow conditions, other integral length scales are given by Eq. 2.13 and plotted in Figure 10. In general, the dimension of an idealized gust increases with increasing height.

[ 𝐿^𝑦 ^𝑢

𝐿_𝑢

𝑧

𝐿_𝑣

𝑥

𝐿_𝑣

𝑦

𝐿_𝑣

𝑧

𝐿_𝑤

𝑥

𝐿_𝑤

𝑦

𝐿_𝑤

𝑧 ]

=

[ 1/3 1/5 1/4 1/4 1/12 1/12 1/18 1/18]

𝐿_𝑢

𝑥

Eq. 2.13

Figure 10. Integral length scales for ideally homogeneous wind flow conditions.

2.1.7 Probability density function of turbulence

The wind speed variation is naturally a random occurring process which do not repeat in time.

As previously described, the turbulence is caused by vortices within the air flow and is moving

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10 along at the mean wind speed. These three-dimensional vortices are never identical, and statistical methods are applied to describe the characteristics of the gustiness. As characteristics of the stochastic process, measurements have shown that the wind velocity components in the atmospheric boundary layer closely follow the Gaussian probability density function, Eq. 2.14.[4]

𝑓_𝑢(𝑈) = 1

𝜎_𝑢√2𝜋exp [−1

2(𝑢 − 𝑈 𝜎_𝑢 )

2

]

Eq. 2.14

2.1.8 Wind spectra

While the probability density function describes the overall variation in wind velocity, the wind spectra describes how the wind velocity varies with time.

2.1.8.1 Power spectral density

The power spectral density (PSD) function shows the strength of the variations, or energy, as a function of frequency. The function describes at which frequencies variations are stronger and weaker. The unit of PSD is (m/s)²/Hz, or energy per frequency. The energy within a specific frequency range can be determined by integrating the PSD within that frequency range. [4]

Frequency is essentially transformation of time, i.e. observation of variations in frequency domain is just another method to look at variations of time-series data. Calculation of PSD is done directly by the Fast Fourier Transform (FFT) method or by determination of the autocorrelation function followed by transformation. Knowledge that is determined about the most common wind frequencies in addition to information about eigen frequencies of a bridge is required for design purposes.[4]

The spectral density function Su(n) is applied to describe the distribution of turbulence with frequency. The relation between variance and spectral density is described in Eq. 2.15 where Su(n) is integrated over the frequency range n to n+dn [4].

𝜎_𝑢² = ∫ 𝑆_𝑢(𝑛)𝑑𝑛

∞

0

Eq. 2.15

2.1.8.2 Horizontal longitudinal wind spectra

A commonly applied form of wind spectra for the longitudinal velocity component is the von Karman/Harris -form. The non-dimensional form is often written:

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11 𝑛𝑆_𝑢(𝑛)

𝜎_𝑢² = 4𝑛𝐿_𝑢,𝑥 𝑈 [1 + 70.8 (𝑛𝐿_𝑢,𝑥

𝑈 )

2

]

56

Eq. 2.16

Where Lu,x is the integral length scale of turbulence and n is the frequency. The value of Lu,x

determines the value of n/U at which the peak of the function occurs. The higher Lu,x is, the higher the value of n/U at the peak becomes, λ – ‘peak wavelength’.

The Kaimal spectra, Eq. 2.17, is another common form of longitudinal wind spectra that is applied by NPRA in Handbook N400[6].

𝑛𝑆_𝑢(𝑛)

𝜎_𝑢² = 6.8𝑛𝐿_𝑢,𝑥 𝑈 [1 + 10.2 (𝑛𝐿_𝑢,𝑥

𝑈 )

2

]

53

Eq. 2.17

2.1.8.3 Vertical wind spectra

The vertical velocity component of atmospheric turbulence, w, has a spectral density of a different characteristic than the longitudinal. The spectrum of vertical turbulence is especially interesting for horizontal structures such as bridges with wide and flat horizontal bridge deck girders that are affected by vertical wind effects.

The vertical wind-spectrum can be described in the von Karman form:

𝑛𝑆_𝑤(𝑛)

𝜎_𝑤² = 4𝑛𝐿_𝑤,𝑥

𝑈 (1 + 755.2 (𝑛𝐿_𝑤,𝑥 𝑈 )

2

) [1 + 283.2 (𝑛𝐿_𝑤,𝑥

𝑈 )

2

]

116

Eq. 2.18

Where integral length scale 𝐿_𝑤,𝑥 is previously defined as ₁₂¹ 𝐿_𝑢,𝑥

The vertical wind-spectrum can also be defined in Kaimal form, which is applied by NPRA in Handbook N400:

𝑛𝑆_𝑤(𝑛)

𝜎_𝑤² = 9.4𝑛𝐿_𝑤,𝑥 𝑈 [1 + 14.1𝑛𝐿_𝑤,𝑥

𝑈 ]

53

Eq. 2.19

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12 Figure 11 - Figure 12 displays horizontal and vertical wind spectra in von Karman and Kaimal forms as functions of frequency.

It can be shown that the von Karman horizontal wind spectra has more energy than the Kaimal form in the approximate frequency interval 0.01-0.1Hz for altitudes between 20m and 200m.

For the vertical spectra, the von Karman spectra is more energetic than the Kaimal in the approximate frequency range 0.07-10Hz.

Figure 11. Horizontal longitudinal spectral density functions of von Karman- and Kaimal forms.

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13 2.1.9 Cross-spectral density, co-spectral density and coherence

Determination of correlation between two separate points as a function of frequency is of interest when analyzing resonant response due to turbulence. Frequency-dependent correlation can be described by functions called cross-spectral density, co-spectral density and coherence.[4]

Figure 12. Vertical spectral density functions of von Karman- and Kaimal forms.

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14 The cross-spectral density, Eq. 2.20, is a complex function with real and imaginary components. The real part is called co-spectral density and can be regarded as a frequency- dependent covariance. The imaginary part is called quad-spectral density and describes the wind component 90 out of phase. The real part describes the simultaneous in-phase increase and decrease of wind components in two points.[4]

|𝑆_𝑢1,𝑢2| = √𝑅𝑒(𝑆_𝑢1,𝑢2)²+ 𝐼𝑚(𝑆_𝑢1,𝑢2)²

Eq. 2.20

Hence, the co-coherence function of longitudinal turbulence component at two points separated by distance s is defined as:

𝛾_𝑢 = 𝑅𝑒 [ 𝑆_𝑢1,𝑢2(𝑓, 𝑠)

√𝑆_𝑢1(𝑓)𝑆_𝑢2(𝑓)]

Eq. 2.21

Coherence may be regarded as a normalized magnitude of the cross-spectrum and is approximately equivalent to a frequency-dependent correlation coefficient.

Normalized co-spectrum is comparable to coherence but does only include the real component. When regarding wind forces on structures due to turbulence, only the quantity of the real part is considered. Normalized co-spectra is also called root-coherence.[4]

The root-coherence can simplified be characterized by the exponential function:

√𝑐𝑜ℎ(𝑓, 𝑠) = exp (−𝑐𝑓𝑠 𝑈 )

Eq. 2.22

Where: c – decay coefficient; empirical constant used to fit measured data f – frequency

s – separation between two points

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15 The normalized co-spectra, Figure 13, for the longitudinal wind direction can in a simplified manner be described by the exponential function, Eq. 2.22, for vertical and horizontal separation between the points considered.

The coherence decay coefficient, c, is applied to fit the exponential function to measured data.

As depicted in Figure 14, the normalized co-spectra will subside faster for higher frequencies and larger separations. Further, lower values of the decay coefficient will produce higher displacement.

Given by NPRA in Handbook N400, the decay coefficients for estimation of wind coherence are specified according to Table 1.

Table 1. Applied exponential decay coefficients specified by NPRA Handbook N400.

𝑐_𝑢𝑦 𝑐_𝑣𝑦 𝑐_𝑤𝑦 𝑐_𝑢𝑧 𝑐_𝑣𝑧 𝑐_𝑤𝑧

10 6.5 6.5 10 6.5 3.0

Figure 13. Root-coherence function as function of fixed frequency (left) and separation (right).

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16

Figure 14. Normalized co-spectrum for arbitrary decay coefficient.

2.1.10 Time- and frequency domain analysis

In time domain analysis, a general signal is described in terms of how amplitude changes as a function of time. Based on the stochastic process described in subchapter 2.1.2, a time- domain analysis can be implemented, where wind speed due to turbulence is described as a function of time.

A signal may be converted between time and frequency domain by transformation operators.

A commonly applied operator is the Fourier transform, that converts the time domain signal into a sum of sinusoidal waves of different frequencies. The subsequent spectrum, or in wind terminology wind spectrum, may be interpreted as the frequency domain representation of the signal. Hence, a reversed transform process converts the frequency domain signal back into the time domain.[5]

In process of frequency domain analysis, it is not required to generate individual turbulence components for each time increment, the process therefore becomes less time consuming than time domain analysis. The frequency domain analysis requires a frequency domain description of the wind field such as Von Karman or Kaimal wind spectra. The process also involves frequency domain transfer functions from the wind field velocity pressure distribution to the corresponding load as well as from load to structural response. Further, eigen-modes and corresponding eigen-frequencies of the system in question as well as structural properties are required as input. The output of the analysis can be interpreted as a power spectral density function where the magnitude of energy at corresponding frequency is represented as spectral peaks.[5]

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17

2.2 Wind loading

Wind force, Eq. 2.24, is determined from the wind velocity and air density according to Bernoulli’s equation for velocity pressure, Eq. 2.23. A, represents the wind affected area and C is the aerodynamic shape-factor. Shape factors are applied to consider the aerodynamic characteristics of the structure and are determined in wind-tunnel tests with a model of the actual structure.

𝑞 =1 2𝜌𝑈²

Eq. 2.23

𝐹 =1

2𝜌𝑈²𝐶𝐴

Eq. 2.24

Figure 15. Mean pressure around the cross-section of a bridge girder, with angle of incidence α=0 [8].

Figure 15 illustrates how the mean wind pressure creates a pressure distribution around a bridge cross-section. The pressure coefficient, cp, represents how the wind pressure normal to the girder changes with pressure differentials at discrete positions around the cross- section. A non-dimensional shape factor for drag pressure can be determined experimentally to represent the change in pressure around the bridge-girder.[8]

2.2.1 Drag force

Drag force, Eq. 2.25, is the load wind exerts on a body in the flow direction. Drag force originates from the combined effects of the components of pressure and shear forces acting on each elemental area of the body in the flow direction. Drag can be reduced by streamlining the cross-section, which may be a critical structural parameter when designing bridge girders.[7]

𝐹_𝐷=1

2𝜌𝑈²𝐶_𝐷𝐻𝐿

Eq. 2.25

Geometry for an arbitrary cross-section of a bridge girder, where H is the height, L is the length of the element and B is the width.

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18 2.2.2 Lift force

Lift force, Eq. 2.26, occurs when wind passes under or over the structure, thus creating increased or decreased pressure close to the structure. For a typical bridge girder design, the cross-section will be pulled by the lift force towards the reduced pressure.

𝐹_𝐿 = 1

2𝜌𝑈²𝐶_𝐿𝐵𝐿

Eq. 2.26

2.2.3 Torsional moment

Due to the structural property that the drag- and lift force will not act in the center of the cross-section, rather in the shear center, a torsional moment will occur, Eq. 2.27.

𝑀_𝑥 =1

2𝜌𝑈²𝐶_𝑀𝐵²𝐿

Eq. 2.27

2.2.4 Buffeting theory

The buffeting theory may adopt the quasi-steady assumption, which suggests that wind fluctuations instantaneously adapts to the moving bridge span. This implies that the aerodynamic coefficients and their first derivatives are independent of the frequency.[9]

As depicted in Figure 16 for an arbitrary cross-section, due to rotation of the cross-section, the angle of incidence, α, is changing each analyzed time-step. As a result, the shape-factors are altered about their mean value.

Figure 16. Change in value for shape-factors for varying angle of incidence [8].

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19 As described in Figure 17, by defining mean and fluctuating wind components, the structural motion of the cross-section, instantaneous velocity and angle of incidence as well as instantaneous and time-invariant coordinate system, the relative instantaneous velocity, Eq.

2.28, can be defined. The equation is simplified and less significant terms are removed. [8]

𝑈_𝑟𝑒𝑙² = (𝑈 + 𝑢 − 𝑟̇_𝑥)²+ (𝑤 − 𝑟̇_𝑧)² ≈ 𝑈²+ 2𝑈𝑢 − 2𝑈𝑟̇_𝑥

Eq. 2.28

Where 𝑟̇_𝑥 and 𝑟̇_𝑧 are the instantaneous structural velocity in corresponding direction.

Figure 17. Wind components, structural motion, angle of incidence and instantaneous wind velocity[8].

Where 𝜃_𝑡 = 𝜃̅ + 𝜃 is the torsional movement of the cross-section, px and pz instantaneous forces, 𝑝_𝜃 the torsional moment per unit length.

The relative instantaneous wind velocity acts at the relative angle of attack α, where its mean value 𝛼̅ represent the mean torsion 𝜃̅. The angle of incidence due to turbulence is:

𝛼_𝑓 = 𝜃 + tan⁻¹( 𝑤 − 𝑟̇_𝑧

𝑈 + 𝑢 − 𝑟̇_𝑥) ≈ 𝜃 +𝑤 − 𝑟̇_𝑧 𝑈

The simplification is valid for small angle of attack when mean wind speed is dominating over turbulence and structural velocity. Due to the property that shape-coefficients are dependent on its mean angle of incidence, as illustrated in Figure 18, shape-factors can be simplified by linearization about the mean angle of attack, Eq. 2.29. [8]

𝛼 = 𝛼̅ + 𝛼_𝑓

Eq. 2.29

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20

Figure 18. Linearization of a lift shape-factor[8].

Shape-factors can then be linearized, Eq. 2.30 - Eq. 2.32, and further applied when determining forces in buffeting conditions.

𝐶_𝐷(𝛼) = 𝐶̅̅̅̅ + 𝐶_𝐷 _𝐷^′(𝛼_𝑓)

Eq. 2.30

𝐶_𝐿(𝛼) = 𝐶̅̅̅ + 𝐶_𝐿 _𝐿^′(𝛼_𝑓)

Eq. 2.31

𝐶_𝑀(𝛼) = 𝐶̅̅̅̅ + 𝐶_𝑀 _𝑀^′ (𝛼_𝑓)

Eq. 2.32

Where 𝐶̅_𝑖 (i=D, L, M) is the mean shape-factor.

Fluctuating forces per unit length, Eq. 2.33, can then be stated as an expression consisting of force due to turbulence and force due to motion of the structure.

𝑝 =1 2𝜌𝑈²[

2𝐶̅̅̅̅𝐻 𝐶_𝐷 _𝐷^′𝐻 − 𝐶̅̅̅𝐵_𝐿 2𝐶̅̅̅𝐵 𝐶_𝐿 _𝐿^′𝐵 + 𝐶̅̅̅̅𝐻_𝐷 2𝐶̅̅̅̅𝐵_𝑀 ² 𝐶_𝑀𝐵²

] [ 𝑢 𝑤𝑈 𝑈

] +1 2𝜌𝑈²[

−2𝐶̅̅̅̅𝐻 −(𝐶_𝐷 _𝐷^′𝐻 − 𝐶̅̅̅𝐵) 𝐶_𝐿 ̅̅̅̅𝐻_𝐷

−2𝐶̅̅̅𝐵 −(𝐶_𝐿 _𝐿^′𝐵 + 𝐶̅̅̅̅𝐻) 𝐶_𝐷 _𝐿^′𝐵

−2𝐶̅̅̅̅𝐵_𝑀 ² −𝐶_𝑀𝐵² 𝐶_𝑀^′𝐵² ]

[ 𝑟̇_𝑥 𝑈𝑟̇_𝑧 𝑈𝜃 ]

Eq. 2.33

The second term in the expression which defines the forces due to motion of the structure will be further evaluated as aerodynamic damping in chapter 2.4.2 and is neglected in the expression. Hence, only the first term, Eq. 2.34, is applied as fluctuating forces. [8]

[ 𝑝_𝑥 𝑝_𝑧 𝑝_𝜃] =1

2𝜌𝑈²[

2𝐶̅̅̅̅𝐻 𝐶_𝐷 _𝐷^′𝐻 − 𝐶̅̅̅𝐵_𝐿 2𝐶̅̅̅𝐵 𝐶_𝐿 _𝐿^′𝐵 + 𝐶̅̅̅̅𝐻_𝐷 2𝐶̅̅̅̅𝐵_𝑀 ² 𝐶_𝑀^′ 𝐵²

]

Eq. 2.34

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21 2.2.5 Aerodynamic admittance

The aerodynamic admittance function is applied to account to the phenomenon that high frequency wind gusts do not affect the entire bridge width[4]. In other words, the effectiveness of a structure in capturing the wind is determined by its aerodynamic admittance [7]. Vickery presented an empirical function that is commonly applied today, Eq.

2.35, that could be determined from curve-fitting of experimental data [10].

The function is associated to the geometry of the girder cross-section as well as the mean wind speed and is frequency dependent. The function is determined from wind tunnel experiments with the relevant section model. It can be determined directly from pressure tap measurements around the border of the cross section or from time series of drag, lift and moment forces on the model [5]. For cross-section specific aerodynamic admittance functions for drag- and lift force as well as moment components, several other functions not presented here may be applied.

𝜒²(𝑓) = [

1 1 + [2𝑓√𝐴

𝑈 ]

43

]

2

Eq. 2.35

Where: f – frequency

A – cross-sectional area, including hollow area U – mean wind speed

Figure 19 depicts the function of Eq. 2.35 for Sulafjorden bridge. The bridge has eigen frequencies that are approximately ranging between 0.01 and 0.9 Hz for the first 150 mode- shapes[1]. For the lowest eigenfrequencies one can argue that the function has reduced influence on the lift- and drag forces as well as moment, where a frequency below 0.1Hz has a function-value higher than 0.97. Although, other functions (not specified in this thesis) can be applied that are more representative for wind-tunnel test data of the cross-section of the Sulafjorden box-girder.

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22

Figure 19. Aerodynamic admittance function developed by Vickery.

If cross-sectional aerodynamic admittance is taken into account when defining the force components due to turbulence, Eq. 2.24 may be modified into Eq. 2.36.

[ 𝑝_𝑥 𝑝_𝑧 𝑝_𝜃] =1

2𝜌𝑈²[

2𝐶̅̅̅̅𝐻 ∙ 𝜒_𝐷 _𝑢_𝑥_𝑥 (𝐶_𝐷^′𝐻 − 𝐶̅̅̅𝐵) ∙ 𝜒_𝐿 _𝑤𝑥 2𝐶̅̅̅𝐵 ∙ 𝜒_𝐿 _𝑢_𝑥_𝑧 (𝐶_𝐿^′𝐵 + 𝐶̅̅̅̅𝐻) ∙ 𝜒_𝐷 _𝑤𝑧 2𝐶̅̅̅̅𝐵_𝑀 ²∙ 𝜒_𝑢_𝑥_𝜃 𝐶_𝑀^′𝐵²∙ 𝜒_𝑤𝜃

]

Eq. 2.36

2.2.6 Mechanical admittance

The mechanical admittance function may be considered as a dynamic amplification factor, DAF, which is of relevance when the harmonic excitation force to the response of a single degree of freedom system is considered. Furthermore, mechanical admittance may be regarded as a transfer function between the spectral density of the aerodynamic forces and the spectral density of the structural response as depicted in Figure 20. [4]

2.2.7 Spectral approach to dynamic response

Figure 20 displays a schematic illustration of the main principles of how the response of a structural system may be determined by the wind velocity and wind force components by spectral computation.

The resulting fluctuating response is calculated from the spectral density of the response, which is in turn determined from the spectra of the aerodynamic forces through the mechanical admittance function. The spectral density of aerodynamic forces is determined from the spectral density of the turbulence itself through the aerodynamic admittance function. As previously described, the aerodynamic and mechanical admittance functions are

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23 frequency-dependent, where the mechanical admittance function may amplify the response due to a resonant frequencies between the structure and turbulence.[4]

Figure 20. Illustration of the spectral approach to dynamic response.[4]

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24

2.3 Mechanical vibration

When applying Newton’s second law of motion, Eq. 2.37, an expression for the equation of motion for free vibration consisting of additional stiffness and damping terms can be established for a structural system, Eq. 2.37 - Eq. 2.41.

𝐹 = 𝑚𝑥̈

Eq. 2.37

𝐹_{𝑑𝑎𝑚𝑝𝑖𝑛𝑔} = −𝑐𝑥̇

Eq. 2.38

𝐹_{𝑠𝑡𝑖𝑓𝑓𝑛𝑒𝑠𝑠} = −𝑘𝑥

Eq. 2.39

𝑚𝑥̈ = −𝑐𝑥̇ − 𝑘𝑥

Eq. 2.40

𝑚𝑥̈ + 𝑐𝑥̇ + 𝑘𝑥 = 0

Eq. 2.41

Where m is mass, c is damping [Ns/m], k is stiffness [N/m], x is position, 𝑥̇ speed and 𝑥̈

acceleration.

Free vibration occurs if a system, after an initial disturbance, is left to vibrate on its own. Eq.

2.41 describes the equation of motion for such a system. This system may be expressed as the characteristic equation:

𝑚𝑠²+ 𝑐𝑠 + 𝑘 = 0

Eq. 2.42

Which has the roots:

𝑠_1,2 = − 𝑐

2𝑚± √( 𝑐

2𝑚)²− 𝑘 𝑚

Eq. 2.43

Hence, the general solution of such a single degree of freedom-system can be expressed as:

𝑥(𝑡) = 𝐶₁exp[𝑠₁𝑡] + 𝐶₂exp[𝑠₂𝑡]

Eq. 2.44

Where C1 and C2 are determined by initial conditions of the system.

Critical damping, cc, is defined as the value of the damping constant c for which radical (under root sign of Eq. 2.43) becomes zero or as:

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25 𝑐_𝑐 = 2𝑚𝜔_𝑛

Eq. 2.45

Where 𝜔_𝑛 is the eigenfrequency of the system.

A critical damped system will have the smallest damping required for a periodic motion.

Further, the damping ratio is defined as:

𝜉 = 𝑐

𝑐_𝑐 = 𝑐 2𝑚𝜔_𝑛

Eq. 2.46

Depending on system properties with respects to response, systems may be regarded as underdamped (𝜉 < 1), critically damped (𝜉 = 1) or overdamped (𝜉 > 1).[11]

2.3.1 Eigen frequency

Eq. 2.47 describes the equation of motion for forced vibration.

𝑚𝑥̈ + 𝑐𝑥̇ + 𝑘𝑥 = 𝐹(𝑡)

Eq. 2.47

Forced vibration occurs when a system is subjected to an external force. If the frequency of the external force is equal to the natural frequency of the system, resonance occurs and the resulting mode-shape increases and the system will undergo critical structural oscillations.

Failures of structures due to wind force such as bridges and buildings have been associated with the occurrence of resonance. The vibration of a structure is described by the eigenfrequency of the structure and its associated mode-shape. [11] Analysis in frequency domain is done to determine the most significant frequencies.

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26

2.4 Damping

The total damping on a structure such as analyzed in this thesis is a combination of structural- , aerodynamic- and hydrodynamic damping.

2.4.1 Structural damping

Structural damping is the damping contribution from the structure itself. For the current model that is analyzed in this thesis, Rayleigh-damping is applied where the alpha-coefficient controls the damping from mass and beta-coefficient controls damping from stiffness.

𝑐_{𝑟𝑎𝑦𝑙𝑒𝑖𝑔ℎ} = 𝛼𝑀𝛽𝐾

Eq. 2.48

𝜉_𝑖 =1 2(𝛼

𝜔_𝑖+ 𝛽𝜔_𝑖)

Eq. 2.49

𝛼 = 𝜉2𝜔₁𝜔₂ 𝜔₁+ 𝜔₂

Eq. 2.50

𝛽 = 𝜉2 𝜔₁+ 𝜔₂

Eq. 2.51

Where ω1 and ω2 represents the lowest and highest eigenfrequency of the structure, M is the mass matrix and K is the stiffness matrix. [12]

2.4.2 Aerodynamic damping

The motion-induced forces presented in chapter 2.2.4, can be interpreted as aerodynamic damping. Based on the fundamental damping relationship, Eq. 2.52, Eq. 2.33 can be rewritten such that we have Eq. 2.53.

𝐹_{𝑑𝑎𝑚𝑝𝑖𝑛𝑔} = 𝑐𝑥̇

Eq. 2.52

The diagonal terms in the aerodynamic damping matrix represents modifications of the distributed structural damping and stiffness terms, while the negative terms increase the structural resistance forces. Further, a positive slope of the lift curve increases the vertical damping, while a positive slope of the moment curve decreases the effective torsional stiffness.[8]

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27 𝐹_{𝑑𝑎𝑚𝑝𝑖𝑛𝑔} =1

2𝜌𝑈²[

2𝐶̅̅̅̅𝐻 𝐶_𝐷 _𝐷^′𝐻 − 𝐶̅̅̅𝐵 −𝐶_𝐿 ̅̅̅̅𝐻_𝐷 2𝐶̅̅̅𝐵 𝐶_𝐿 _𝐿^′𝐵 + 𝐶̅̅̅̅𝐻 −𝐶_𝐷 _𝐿^′𝐵 2𝐶̅̅̅̅𝐵_𝑀 ² 𝐶_𝑀𝐵² −𝐶_𝑀^′𝐵²

] [

𝑟̇_𝑥 𝑟̇𝑈_𝑧 𝑈𝜃 ]

Eq. 2.53

Where the aerodynamic damping term can be expressed as Eq. 2.54.

𝑐_𝑎𝑒 =1 2𝜌𝑈 [

2𝐶̅̅̅̅𝐻_𝐷 𝐶_𝐷^′𝐻 − 𝐶̅̅̅𝐵 0_𝐿 2𝐶̅̅̅𝐵_𝐿 𝐶_𝐿^′𝐵 + 𝐶̅̅̅̅𝐻 0_𝐷 2𝐶̅̅̅̅𝐵_𝑀 ²𝑈 𝐶_𝑀𝐵²𝑈 0 ]

Eq. 2.54

The aerodynamic stiffness term can be expressed as Eq. 2.55. For calculations that analyses wind speeds around the characteristic wind speed, the stiffness term can be regarded as non- significant compared to the aerodynamic damping term and may be excluded. [5]

𝑘_𝑎𝑒 =1 2𝜌𝑈 [

0 0 −𝐶̅̅̅̅𝐻_𝐷 0 0 −𝐶_𝐿^′𝐵 0 0 −𝐶_𝑀^′𝐵²𝑈

]

Eq. 2.55

To express a damping term for rotation, Eq. 2.56 may be applied. The coefficient 𝑘_𝜃 may be regarded as the horizontal distance between the aerodynamic and shear center and may be set to ¼ of the width of the bridge girder [13].

1

2𝜌𝑈𝐶_𝑀^′ 𝐵²𝑘_𝜃𝑟̇_𝜃

Eq. 2.56

The aerodynamic damping matrix can be further simplified and regarded as non-coupled, Eq.

2.57:

𝑐_𝑎𝑒 =1 2𝜌𝑈 [

2𝐶̅̅̅̅𝐻_𝐷 0 0

0 𝐶_𝐿^′𝐵 + 𝐶̅̅̅̅𝐻_𝐷 0 0 0 𝐶_𝑀^′𝐵²𝑘_𝜃

]

Eq. 2.57

The equation of motion can then be expressed as, Eq. 2.58, where the aerodynamic damping alters the total damping for the system.

𝑴𝒓̈ + (𝑪 + 𝑪_𝒂𝒆)𝒓̇ + (𝑲 + 𝑲_𝒂𝒆)𝒓 = 𝑭

Eq. 2.58

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28 Where M is the mass matrix, 𝒓 = [

𝑟_𝑥 𝑟_𝑧

𝑟_𝜃] and F represents both static and dynamic forces applied to the system. The aerodynamic damping components can for further analysis be defined as horizontal, vertical and torsional components per unit length, Eq. 2.59 - Eq. 2.61.

𝑐_𝑎𝑒

𝑥 = 𝜌𝑈𝐶̅̅̅̅𝐻 _𝐷

Eq. 2.59

𝑐_𝑎𝑒

𝑧 =1

2𝜌𝑈(𝐶_𝐿^′𝐵 + 𝐶̅̅̅̅𝐻) _𝐷

Eq. 2.60

𝑐_𝑎𝑒

𝜃 = 1

2𝜌𝑈𝐶_𝑀^′𝐵²𝑘_𝜃

Eq. 2.61

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29

3. Parametric study of discretized bridge span

3.1 Method

A parametric sensitivity study has initially been done to examine how normalized total modal force, normalized modal reaction force at support and normalized modal bending moment varies with difference in discretization of a long-spanned simply supported structure. Such a structure may be interpreted as a long-spanned suspension bridge, such as Sulafjorden bridge.

3.1.1 Simulated wind field characteristics

The software MATLAB (matrix laboratory) has been applied in this work since it is significant applicable when computing large numerical data.

The MATLAB-script, WindSim[14], is applied to simulate wind-fields for the u- and w- components. The time-series of turbulence that are simulated are based on the following parameters and are defined in the script before the start of the simulation:

Sampling frequency 10 Hz

Time step 0.1s

No. time steps 2¹⁵ (32768 steps)

Length of wind series 3277s Spectral density function Kaimal Reference mean wind speed 29m/s Terrain roughness category 0

Geometry length, height 2000m, 85m

No. of nodes 401

Mean wind speed 46.39m/s

Integral length scale, 𝐿_𝑢𝑥 190m Turbulence intensity, Iu 9,4%

Iw 4,3%

Coherence decay coefficients Cuy=5, 10, 15 Cuz=5, 10, 15 Cwy=5, 10, 15 Cwz=5, 10, 15 Figure 21 depicts the Kaimal spectra for the u- and w components in comparison with the simulated spectra.

Figure 21. Kaimal spectra vs. simulated u- and w-components.

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30 Figure 22 and Figure 23 illustrates simulated time-series of along-wind u-component and across-wind vertical w-component at altitude z=85m.

Figure 22. Simulated time-series of along-wind speed u, here depicted as superimposed upon U.

Figure 23. Simulated time-series of vertical wind speed component w.

Six different wind field realizations of the same wind field properties are simulated. Except for the dynamic wind speed component, the only variable in the wind field is the coherence decay factor, c, which is set equal to 5, 10 and 15. All other wind properties are kept constant. The

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31 output are matrices for along-wind component u and across-wind vertical component w at each node and time-step.

The original wind component matrices serve as a fundamental wind fields for the parametric study. Depending on the relevant element size, as depicted in Figure 24, nodes are extracted from the original matrices for further manipulation in MATLAB. This procedure has the implication of applying the same wind field for each discretization level of the span. The average value of the six wind simulations is determined and applied in the calculations below.

Figure 24. Discretization of fundamental wind-field simulation for a 2000m long span.

As depicted in Figure 24, the fundamental discretized span has the element length of 5m.

Analyzed element lengths are: 10m, 20m, 40m, 80m, 100m, 200m and 250m.

A simplified modal load analysis is applied, where modal drag- and lift force components are assumed to attack nodal positions along the span corresponding to the relevant element length, Eq. 3.1 - Eq. 3.2.

𝑄_{𝑢,𝑖𝑗}(𝑡) = 𝜑_𝑖[U + u(t)_𝑖𝑗]Δx

Eq. 3.1

𝑄_{𝑤,𝑖𝑗}(𝑡) = 𝜑_𝑖[w(t)_𝑖𝑗]Δx

Eq. 3.2

𝑄𝑡𝑜𝑡_{𝑢,𝑖𝑗}(𝑡) = ∑ 𝜑_𝑖[U + u(t)_𝑖𝑗]Δx

𝑛

𝑖=1

Eq. 3.3

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32 𝑄𝑡𝑜𝑡_{𝑤,𝑖𝑗}(𝑡) = ∑ 𝜑_𝑖[w(t)_𝑖𝑗]Δx

𝑛

𝑖=1

Eq. 3.4

Where: φ – mode-shape Δx – element length

U – mean wind speed at 85m altitude u – horizontal along wind component w – vertical wind component

i – 1-5, mode-shapes j – 1-32768, time-steps

Figure 25. Modal load for 1^st mode-shape at corresponding nodal position, I, and time-step, t.

As depicted in Figure 25, the modal force components of each nodes are added and presented as a total force for each time-step, Eq. 3.3 - Eq. 3.4. A standard deviation is calculated of the total modal force terms at each time-steps. This standard deviation is normalized with respect to the standard deviation of 10m element length, and plotted as a function of both element length and number of total nodes along the bridge span.

The calculation procedure for modal reaction force at support is simply by taking the moment equilibrium at one support and solving for the reaction force at the other support.

Analyse av vind-indusert respons av en hengebro på flytende fundamenter

MASTEROPPGAVE

Abstract

Preface

Table of Contents

List of Figures

List of Tables

Nomenclature

1. Introduction

1.1 Background

1.2 Objective

1.3 Thesis structure

2. Literature review

2.1 Wind

2.2 Wind loading

2.3 Mechanical vibration

2.4 Damping

3. Parametric study of discretized bridge span

3.1 Method