DET TEKNISK-NATURVITENSKAPELIGE FAKULTET MASTEROPPGAVE

Transcription

1 DET TEKNISK-NATURVITENSKAPELIGE FAKULTET MASTEROPPGAVE Studieprogram/spesialisering: Master i teknologi Konstruksjoner og materialer Fordypning Byggkonstruksjoner Forfatter: Per Simon Sjölander Fagansvarlig: Jasna B. Jakobsen (UiS) Vårsemesteret, 2017 Åpen (signatur forfatter) 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, I

2 II

3 Abstract As a part of a nation-wide project to create a ferry-free coastal highway route, several fjordcrossings 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. III

4 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 IV

5 Table of Contents Abstract... III Preface... IV List of Figures... VIII List of Tables... X Nomenclature... XI 1. Introduction Background Objective Thesis structure Literature review Wind Mean wind speed profile Stochastic process Turbulence Correlation Time scale Integral length scale Probability density function of turbulence Wind spectra Power spectral density Horizontal longitudinal wind spectra Vertical wind spectra Cross-spectral density, co-spectral density and coherence Time- and frequency domain analysis Wind loading Drag force Lift force Torsional moment Buffeting theory Aerodynamic admittance Mechanical admittance Spectral approach to dynamic response V

6 2.3 Mechanical vibration Eigen frequency Damping Structural damping Aerodynamic damping Parametric study of discretized bridge span Method Simulated wind field characteristics Results Normalized total force Discussion parametric study Finite element analysis of Sulafjorden bridge Method FE-model in Abaqus Structure of Abaqus input-file Cross-sectional properties Boundary conditions Damping Eigen frequency and modal analysis Static analysis Dynamic analysis Frequency domain analysis Results Static analysis Dynamic analysis Horizontal displacement Vertical displacement Frequency-domain analysis Discussion: FE-analysis of Sulafjorden bridge Conclusion References Appendix Normalized total modal force VI

7 6.2 Normalized total modal reaction force at support Normalized total modal bending moment at center of span Time-series of normalized reaction force at support Time-series of normalized bending moment at center of span Manual calculations Mode-shapes VII

8 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 Gaussianprobability distribution.[5]... 5 Figure 6. Turbulence intensity for u-, v- and w-components 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 Figure 12. Vertical spectral density functions of von Karman- and Kaimal forms Figure 13. Root-coherence function as function of fixed frequency (left) and separation (right) Figure 14. Normalized co-spectrum for arbitrary decay coefficient Figure 15. Mean pressure around the cross-section of a bridge girder, with angle of incidence α=0 [8] Figure 16. Change in value for shape-factors for varying angle of incidence [8] Figure 17. Wind components, structural motion, angle of incidence and instantaneous wind velocity[8] Figure 18. Linearization of a lift shape-factor[8] Figure 19. Aerodynamic admittance function developed by Vickery Figure 20. Illustration of the spectral approach to dynamic response.[4] Figure 21. Kaimal spectra vs. simulated u- and w-components 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 Figure 24. Discretization of fundamental wind-field simulation for a 2000m long span Figure 25. Modal load for 1 st mode-shape at corresponding nodal position, I, and time-step, t Figure 26. Normalized total modal drag force, where decay coefficient c indicates both Cuy and Cuz Figure 27. Normalized total modal drag force, where decay coefficient c indicates both Cuy and Cuz Figure 28. Normalized total modal lift force, where decay coefficient c indicates both Cwy and Cwz Figure 29. Total normalized modal drag force for 10m and 250m element length Figure 30. Total normalized modal lift force for 10m and 250m element length VIII

9 Figure 31. FE-model of Sulafjorden bridge relative to its global coordinate system Figure 32. Sulafjorden suspension bridge on floating foundation with tension leg platforms.[1] Figure 33. Details of the FE-model with 20m girder element length discretization Figure 34. Logarithmic mean wind speed profile Figure 35. Empirical wind shape-factors for box girder.[1] Figure 36. Horizontal displacement in z-direction due to static force Figure 37. Rotational displacement about y-axis due to static force Figure 38. Rotation about x-axis, a, angle of incidence Figure 39. Time-series of lateral response of node 1110 at middle of bridge span 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 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 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 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 Figure 44. Time-series of vertical response of node 1110 at middle of bridge span 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 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 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 Figure 48. Frequency-domain PSD of lateral response for Δx=10m and Δx=100m discretization level Figure 49. Normalized PSD of lateral response with respect to 10m girder element length discretization level Figure 50. Normalized PSD of lateral response; mode-shape 2,5,12 and 21 relative to discretization level Figure 51. Frequency-domain PSD of vertical response for Δx=10m and Δx=100m discretization level Figure 52. Normalized PSD of vertical response with respect to 10m girder element length discretization level Figure 53. Normalized PSD of vertical response; mode-shape 3,9,19 and 22 relative to discretization level IX

10 List of Tables Table 1. Applied exponential decay coefficients specified by NPRA Handbook N Table 2. Change in normalized standard deviation for increasing element length in relation to 10m element length. For 1 st mode-shape and c= Table 3. Change in normalized standard deviation for increasing element length in relation to 10m element length. For 1 st mode-shape and c= Table 4. Change in normalized standard deviation for increasing element length in relation to 10m element length. For 1 st mode-shape and c= Table 5. Cross-sectional properties of the beam elements in the bridge model.[1] Table 6. Applied aerodynamic damping for different discretization level and element length of the girder, including damping for cable and floating tower Table 7. Aerodynamic shape-factors for.[1] Table 8. Mode-shapes and eigen frequencies Table 9. Horizontal displacement in z-direction due to static force X

11 Nomenclature Roman symbols c i c C D C D C L C L C M C M F i f I i k L i M i n q S i s T i t U(t) U u(t) u v w z Coherence decay coefficient Damping Aerodynamic drag coefficient Derivative of aerodynamic drag coefficient Aerodynamic lift coefficient Derivative of aerodynamic lift coefficient Aerodynamic pitching moment coefficient Derivative of aerodynamic pitching moment coefficient Force Frequency Turbulence intensity Stiffness Integral length scale Moment Frequency Wind velocity pressure Spectral density function Distance Time scale Time Total wind speed Mean wind speed Dynamic wind speed Along-wind turbulence component Across-wind turbulence component Vertical-wind turbulence component Height above sea level XI

12 Greek symbols α β γ i ξ ρ σ i τ χ ω Angle of incidence, also parameter in Rayleigh damping Parameter in Rayleigh damping Root-coherence Damping ratio Air density, also correlation coefficient function Standard deviation Time lag Aerodynamic cross-sectional admittance Angular frequency Abbreviations FEM Finite element modelling NPRA Norwegian public road administration PSD STD TLP Power spectral density Standard deviation Tension leg platform XII

13 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. 1

14 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. 2

15 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 2 ρu(t)2 Eq 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= [4] U(z) = u 1 κ ln z z o Eq

16 Where: u friction speed given as τ 0 ρ τ 0 (τ 0 - 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 ref z o ) Figure 3. Logarithmic- and power law mean wind speed profile divided by U(z, z ref=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] 4

17 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] Turbulence The general level of turbulence in a wind field can be measured by its standard deviation of its wind component, Eq σ u = { 1 T Where the expression is integrated over time, T. T 1/2 U(t) U 2} 0 Eq

18 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 I u = k I c 0 ln (z/z 0 ) Where k I is the turbulence factor, c 0 the orthogonality factor and z 0 the roughness length. Eq. 2.6 Eq. 2.7 The turbulence intensity of u-, v- and w-components may be described as Eq Eq 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 v I 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- 6

19 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] 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 z 1 z 2 ] Eq. 2.9 Figure 8.The cross-correlation of longitudinal turbulence at z=13.5m.[4] 7

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

21 x L u = { L 1 ( z 0.3 ), z > z z min 1 L 1 ( z min z 1 ) 0.3, z z min Eq 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 and plotted in Figure 10. In general, the dimension of an idealized gust increases with increasing height. [ y L u z L u x L v y L v z L v x L w y L w z L w 1/3 1/5 1/4 1/4 = 1/12 1/12 1/18 [ 1/18] ] x L u Eq Figure 10. Integral length scales for ideally homogeneous wind flow conditions 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 9

22 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 [4] f u (U) = 1 σ u 2π exp [ 1 2 U (u ) ] 2 σ u Eq 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 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 where Su(n) is integrated over the frequency range n to n+dn [4]. σ 2 u = S u(n) dn 0 Eq 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: 10

23 ns u (n) σ u 2 = 4 nl u,x U [ ( nl u,x U ) 2 ] 5 6 Eq 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 u 2 = 6.8 nl u,x U [ ( nl u,x U ) 2 ] 5 3 Eq 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) 4 nl w,x U = σ2 w 2 ( (nl w,x U ) ) [ ( nl 2 w,x U ) ] 11 6 Eq Where integral length scale L w,x is previously defined as 1 12 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 w 2 = 9.4 nl w,x U [ nl w,x U ] 5 3 Eq

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

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

26 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 frequencydependent 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 Hence, the co-coherence function of longitudinal turbulence component at two points separated by distance s is defined as: S u1,u2 (f, s) γ u = Re [ S u1 (f)s u2 (f) ] Eq 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 Where: c decay coefficient; empirical constant used to fit measured data f frequency s separation between two points 14

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

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

29 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 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 2 ρu2 Eq F = 1 2 ρu2 CA Eq 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 crosssection. A non-dimensional shape factor for drag pressure can be determined experimentally to represent the change in pressure around the bridge-girder.[8] 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 2 ρu2 C D HL Eq 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. 17

30 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 2 ρu2 C L BL Eq 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 M x = 1 2 ρu2 C M B 2 L Eq 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]. 18

31 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] 2 U rel = (U + u r x) 2 + (w r z) 2 U 2 + 2Uu 2Ur x Where r x and r z are the instantaneous structural velocity in corresponding direction. Eq Figure 17. Wind components, structural motion, angle of incidence and instantaneous wind velocity[8]. Where θ t = θ + 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 α represent the mean torsion θ. The angle of incidence due to turbulence is: α f = θ + tan 1 w r z ( ) θ + U + u r x w r 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 [8] α = α + α f Eq

32 Figure 18. Linearization of a lift shape-factor[8]. Shape-factors can then be linearized, Eq Eq. 2.32, and further applied when determining forces in buffeting conditions. Where C i (i=d, L, M) is the mean shape-factor. C D (α) = C D + C D (α f ) C L (α) = C L + C L (α f ) C M (α) = C M + C M (α f ) Eq Eq Eq 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 2C H D C u D H C B L 2 ρu2 [ 2C B L C L B + C H D ] [ U w] + 1 2C H D (C r x D H C B) L C H D U 2C B 2 M C M B 2 2 ρu2 [ 2C B L (C L B + C H) D C L B ] r z U 2C B 2 M C M B 2 C M B 2 U [ θ ] Eq 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 and is neglected in the expression. Hence, only the first term, Eq. 2.34, is applied as fluctuating forces. [8] p x [ p z ] = 1 2C H D C u D H C B L p 2 ρu2 [ 2C B L C L B + C H D ] [ U w] θ 2C B 2 M C M B 2 U Eq

33 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. χ 2 (f) = [ 2f A [ U ] ] 2 Eq Where: f frequency A cross-sectional area, including hollow area U mean wind speed Figure 19 depicts the function of Eq for Sulafjorden bridge. The bridge has eigen frequencies that are approximately ranging between 0.01 and 0.9 Hz for the first 150 modeshapes[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 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. 21

34 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 may be modified into Eq p x [ p z ] = 1 2C H D χ ux x (C D H C B) L χ u wx p 2 ρu2 [ 2C B L χ ux z (C L B + C H) D χ wz ] [ U w] θ 2C B 2 M χ ux θ C M B 2 χ wθ U Eq 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] 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 22

35 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] 23

36 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 Eq F = mx F damping = cx F stiffness = kx mx = cx kx mx + cx + kx = 0 Eq Eq Eq Eq Eq Where m is mass, c is damping [Ns/m], k is stiffness [N/m], x is position, x speed and x acceleration. Free vibration occurs if a system, after an initial disturbance, is left to vibrate on its own. Eq describes the equation of motion for such a system. This system may be expressed as the characteristic equation: Which has the roots: ms 2 + cs + k = 0 s 1,2 = c 2m ± ( c 2m ) 2 k m Eq Eq Hence, the general solution of such a single degree of freedom-system can be expressed as: x(t) = C 1 exp[s 1 t] + C 2 exp[s 2 t] Where C1 and C2 are determined by initial conditions of the system. Eq 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: 24

37 c c = 2mω n Eq 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 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] Eigen frequency Eq describes the equation of motion for forced vibration. mx + cx + kx = F(t) Eq 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. 25

38 2.4 Damping The total damping on a structure such as analyzed in this thesis is a combination of structural-, aerodynamic- and hydrodynamic damping 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 ξ i = 1 2 ( α ω i + βω i ) Eq α = ξ2ω 1ω 2 ω 1 + ω 2 ξ2 β = ω 1 + ω 2 Eq Eq 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] 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 can be rewritten such that we have Eq F damping = cx Eq 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] 26

39 F damping = 1 2C H D C r x D H C B L C H D U 2 ρu2 [ 2C B L C L B + C H D C L B ] r z 2C B 2 M C M B 2 C M B 2 U [ θ ] Eq Where the aerodynamic damping term can be expressed as Eq c ae = 1 2C H D C D H C B L 0 2 ρu [ 2C B L C L B + C H D 0] 2C B 2 M U C M B 2 U 0 Eq The aerodynamic stiffness term can be expressed as Eq For calculations that analyses wind speeds around the characteristic wind speed, the stiffness term can be regarded as nonsignificant compared to the aerodynamic damping term and may be excluded. [5] k ae = C H D 2 ρu [ 0 0 C L B ] 0 0 C M B 2 U Eq To express a damping term for rotation, Eq 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 2 ρuc M B 2 k θ r θ Eq The aerodynamic damping matrix can be further simplified and regarded as non-coupled, Eq. 2.57: c ae = 1 2C H D ρu [ 0 C L B + C H D 0 ] 0 0 C M B 2 k θ Eq The equation of motion can then be expressed as, Eq. 2.58, where the aerodynamic damping alters the total damping for the system. Mr + (C + C ae )r + (K + K ae )r = F Eq

40 r x Where M is the mass matrix, r = [ rz ] and F represents both static and dynamic forces applied r θ to the system. The aerodynamic damping components can for further analysis be defined as horizontal, vertical and torsional components per unit length, Eq Eq x c ae = ρuc H D Eq z c ae = 1 2 ρu(c L B + C H) D Eq θ c ae = 1 2 ρuc M B 2 k θ Eq

41 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 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 2 15 (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. 29

42 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 30

43 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 Eq Q u,ij (t) = φ i [U + u(t) ij ]Δx Q w,ij (t) = φ i [w(t) ij ]Δx n Qtot u,ij (t) = φ i [U + u(t) ij ]Δx i=1 Eq. 3.1 Eq. 3.2 Eq

44 n Qtot w,ij (t) = φ i [w(t) ij ]Δx 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 , 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 Eq 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. 32

45 The maximum bending moment is computed at center of span with known support force and nodal force contributions with corresponding distances to center of span. 3.2 Results The graphs in Figure 27Figure 26 and Figure 28 displays normalized standard deviation of total modal drag- and lift force for the first mode-shape and presented as a function of element size as well as total number of nodes separately. For comparative reasons, the standard deviation is normalized with respect to the first element length of 10m. The normalized standard deviation is expressed with maximum and minimum standard deviation of the six wind simulations. Total no. of nodes Total no. of nodes Total no. of nodes Figure Normalized total modal drag force, where decay coefficient c indicates both Cuy and Cuz. 33

46 Total no. of Total no. of Total no. of Figure 28. Normalized total modal lift force, where decay coefficient c indicates both Cwy and Cwz. The graphs containing normalized modal reaction force and normalized modal bending moment for mode-shape 1 are similar in characteristics as normalized total modal force and are presented in appendix Table 2 - Table 4 displays the variation of normalized standard deviation for increasing element length in relation to 10m element length. For mode-shape 1 and c=5, 10 and 15. The values in the tables are extracted from the graphs in Figure 27 and Figure 28 as well as appendix

47 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. Total modal force Modal reaction force Modal bending moment Δx [m] Drag 1,0000 1,0021 1,0049 1,0209 1,0245 1,0731 1,0948 Lift 1,0000 1,0094 1,0345 1,0906 1,1344 1,3063 1,4052 Drag 1,0000 1,0018 1,0048 1,0224 1,0232 1,0763 1,1014 Lift 1,0000 1,0100 1,0356 1,0954 1,1395 1,3137 1,4184 Drag 1,0000 1,0025 1,0048 1,0259 1,0279 1,0751 1,0962 Lift 1,0000 1,0098 1,0376 1,1024 1,1461 1,3230 1,4192 Table 3. Change in normalized standard deviation for increasing element length in relation to 10m element length. For 1 st mode-shape and c=10. Total modal force Modal reaction force Modal bending moment Δx [m] Drag 1,0000 1,0022 1,0116 1,0358 1,0386 1,1028 1,1412 Lift 1,0000 1,0235 1,0816 1,2217 1,2914 1,6073 1,7589 Drag 1,0000 1,0034 1,0123 1,0387 1,0426 1,1152 1,1554 Lift 1,0000 1,0231 1,0848 1,2248 1,2897 1,6097 1,7546 Drag 1,0000 1,0029 1,0116 1,0431 1,0383 1,1142 1,1454 Lift 1,0000 1,0259 1,0906 1,2405 1,3143 1,6377 1,7987 Table 4. Change in normalized standard deviation for increasing element length in relation to 10m element length. For 1 st mode-shape and c=15. Total modal force Modal reaction force Modal bending moment Δx [m] Drag 1,0000 1,0033 1,0159 1,0544 1,0586 1,1594 1,2017 Lift 1,0000 1,0401 1,1272 1,3197 1,4179 1,8248 1,9921 Drag 1,0000 1,0043 1,0169 1,0600 1,0669 1,1758 1,2114 Lift 1,0000 1,0377 1,1294 1,3182 1,4128 1,8152 1,9822 Drag 1,0000 1,0044 1,0170 1,0634 1,0610 1,1752 1,2115 Lift 1,0000 1,0427 1,1356 1,3419 1,4448 1,8573 2,

48 3.2.1 Normalized total force The normalized modal drag- and lift force for the 1 st mode-shape is here expressed with a discretization of 10m and 250m element length. It can be observed that the total modal load when the bridge-span is discretized with 250m elements is considerable more fluctuating compared to a discretization of 10m. The coherence decay coefficient is set to c=10. The increase in standard deviation of both lift- and drag force for increasing element length can be visualized in Figure 29 - Figure 30. For visualization purposes, the modal force is plotted for a sixty second time-series. The graphs for modal reaction force and modal bending moment are presented in appendix , as they are very similar in characteristics as for total modal load. Figure 29. Total normalized modal drag force for 10m and 250m element length. Figure 30. Total normalized modal lift force for 10m and 250m element length. 36

49 3.3 Discussion parametric study Generally, the normalized STD of total force, reaction force and bending moment calculated by the implemented time domain procedures are increasing for courser element discretization. As previously described in the literature review and illustrated in Figure 13, root-coherence is a measure how to how two wind components are correlated. As the separation between the components increases, the correlation decreases and the components will differ more with respect to each other. In addition, it can be seen that as the decay coefficient increases from 5, 10 to 15, the wind-components become less correlated and thus varies more with respect to each other. Hence, the standard deviation of the total force increases with reduced coherence. Further, when also implicating the 1 st mode-shape into the analysis the resulting total force may crudely be interpreted as proportional to a physical response of the structure itself. When comparing different element lengths, the standard deviation for the u-component implies that it converges for increasing discretization of the span. It can also be noted that the difference in STD in total force due to u-component between 10m- and 20m girder element length discretization for c=10 is merely 0.22%. As for the standard deviation of the vertical w- component, the convergence is more noticeable for lower values of the coherence decaycoefficient and a clearer convergence may even be more noticeable for an even higher discretization than what is analyzed in this thesis. Further, the difference in STD in total force due to w-component between 10m- and 20m girder element length discretization for c=5 is 0.94%, implying that a higher discretization is more significant when evaluating vertical response. The evaluation of total force when applying c=10 for u-component and c=5 for w-component is of comparative relevance since NPRA is applying Cuy=10, Cuz=10, Cwy=6.5 and Cwz=3 in handbook N400, decay coefficients that will be applied further in chapter 4. The response due to differences in discretization level of a bridge span is to be further analyzed for the Sulafjorden bridge in chapter 4. 37

50 4. Finite element analysis of Sulafjorden bridge 4.1 Method FE-model in Abaqus Abaqus is a software for FEM, visualization and process automation. The software has been applied in this thesis for remodeling of an existing model as well as job analysis execution. An existing Abaqus FE-model of Sulafjorden suspension bridge on floating foundation has been provided. The original model has a discretized girder element length of 20m, a total of 188 nodes along the bridge girder span. Figure 31 illustrates the model of Sulafjorden bridge relative to its global coordinate system, where z=0 is defined at the location of the bridge girders, y=0 at the base of the towers and x=0 at the left end of the model. Figure 31. FE-model of Sulafjorden bridge relative to its global coordinate system. The three-dimensional B31-Timoshenko element is applied for all elements, where the designation of a Timoshenko-element implies that shear deformation is applicable. Moreover, linear interpolation is applied to determine displacement of the nodes.[15] Figure 32 displays the global geometry of the bridge and Figure 33 depicts the structure of the model with respect to elements and nodes. The girder element is meant to represent the structural properties of the box girder itself, where the purpose of the dummy element is to provide unity to the FE-model. Thus, the dummy element has neglectable mass and extremely high stiffness. Figure 32. Sulafjorden suspension bridge on floating foundation with tension leg platforms.[1] 38

51 Figure 33. Details of the FE-model with 20m girder element length discretization Structure of Abaqus input-file The Abaqus model can be represented as an input-file, a text-based file where one can alter Abaqus-commands and properties of the model. The provided input file of the bridge model has the following defining content: Nodes and elements in relation to the global coordinate system. Cross-sectional properties. Boundary conditions and coupling properties. Dead load for the entire structure. Aerodynamic damping as dashpot-elements at corresponding nodes. Eigenfrequency- and modal analysis. Static analysis, where static wind components are defined. Dynamic analysis, where both static- and dynamic wind components are defined. 39

52 4.1.3 Cross-sectional properties Cross-sectional properties for the beam elements that are applied in the model are described in Table 5. Table 5. Cross-sectional properties of the beam elements in the bridge model.[1] Boundary conditions The pylons in axis 2/5 are fixed in all six degrees of freedom. The pylons in axis 3/4 are fixed with respect to the floating foundation. The stiffener in axis 2/5 is fixed in y-, z-direction and moment about x. Axis 3/4 is fixed in x-, z-direction and moment about y. The three supports of the viaduct at each ends of the bridge are fixed in y- and z-directions. The two main cables are fixed in all degrees of freedom at anchor Damping The aerodynamic damping that is applied in the analysis is calculated according to subchapter in appendix 6.6 and presented in Table 6. The aerodynamic damping is applied as dashpot elements at corresponding nodes for three different discretization levels of the bridge girder. Due to lack of data from wind tunnel experiments for the relevant box girder crosssection, applied shape-factors and its derivatives are empirical and taken from tests of similar cross-sections. Table 6. Applied aerodynamic damping for different discretization level and element length of the girder, including damping for cable and floating tower. Aerodynamic damping Δx [m] No. of nodes on bridge girder Bridge girder Horizontal [Ns/m] Vertical [Ns/m] Torsional [m Ns/m] Cables Horizontal [Ns/m] Floating pylon Horizontal [Ns/m] Multiconsult AS specifies a preferred effective structural damping ratio, ξ, of 0.5%[1]. The structural damping is defined as Rayleigh-damping in the model and can in theory be determined according to subchapter However, since effective structural damping varies for each mode-shape, focus has been to adopt a more correct damping ratio for the first 40

53 mode-shapes as they are assumed to give the greatest response. This is achieved through iterative steps of complex frequency analysis and adjustments of the Rayleigh-damping parameters in the input-file. Hydrodynamic damping is applied for the horizontal direction and is added as a dashpot element in the floating foundation. Multiconsult AS specifies a preferred damping ration, ξ, of 2%[1]. Since the damping coefficient, c, is a function of eigen frequency it will therefore alter for different mode-shapes. The first lateral mode-shapes are assumed to be most significant when it comes to lateral displacement and the effective damping ratio is therefore simplified determined by the eigenfrequencies of theses mode-shapes Eigen frequency and modal analysis The modal analysis in Abaqus determines the mode-shapes and corresponding eigenfrequencies of the structure. Further, this output may be applied in a frequency-domain analysis Static analysis A static analysis has been done to compare response when applying static wind load in the following order: 1. Load on girder 2. Load on girder and cables 3. Load on girder, cables and hangers 4. Load on girder, cables and hangers, floating towers A 188-node discretization of the bridge girder is applied for the static analysis, i.e. 20m girder element length. A logarithmic mean wind profile, Eq. 4.1[16], has been applied on the bridge model to calculate drag force, lift force and torsional moment for each node along the bridge girder and drag force for each node on cables, hangers and floating towers. The calculations of mean wind speed for z=85m can be seen in appendix 6.6. v m = c r c o v b c r = k r ln ( z z 0 ) k r = 0.19 ( z 0 z 0II ) 0.07 Eq. 4.1 Eq. 4.2 Eq. 4.3 v b = c dir c season v b0 Eq. 4.4 Where v m is the position wind velocity, c r roughness factor, k r terrain roughness factor and v b the basic wind velocity. 41

54 In addition: z 0 = 0.003, roughness length for terrain category 0 z 0II = 0.05 c o = 1.0, terrain form factor z - height above sea level c dir =1.0, directional factor c season =1.0, seasonal factor v b0 =29m/s, reference wind speed for Sula municipally Figure 34 displays the logarithmic mean wind speed profile consisting of the parameters given above. Figure 34. Logarithmic mean wind speed profile. According to Figure 35 and Eq Eq. 4.6, the shape-factors for lift and moment are derived by an updating process of the angle of incidence, α, of the box-girder. Full static load is applied on the entire structure in a preliminary static analysis in Abaqus, the angle is determined and the shape-factors updated. The iteration continues until the angle of incidence converges, for this case three iterations was needed. 42

55 Figure 35. Empirical wind shape-factors for box girder.[1] C L (α) = C L + C L (α f ) = C L + ([α 4th updated in radians] 180 π ) C L Eq. 4.5 C M (α) = C M + C M (α f ) = C M + ([α 4th updated in radians] 180 π ) C M Eq. 4.6 Table 7 displays the applied linearized aerodynamic shape-factors[1] for pylons, hangers and cables. Table 7. Aerodynamic shape-factors for.[1] 43

56 The static force components, Eq Eq. 4.9, are then generated in a MATLAB-scipt for each relevant node into an input-file for further static analysis in Abaqus. Where drag force is applied on the entire structure, but lift force and moment only applied on the girder. An example of calculations of static forces for the box-girder can be seen in appendix 6.6. F D = 1 2 ρu2 C D HL F L = 1 2 ρu2 C L (α)bl M x = 1 2 ρu2 C M (α)b 2 L Eq. 4.7 Eq. 4.8 Eq Dynamic analysis A dynamic analysis will be run to include both static and turbulent wind forces. The analysis is run for the three different cases of discretization levels of the bridge girder representing 10m, 20m and 40m girder element length. Since the objective in this thesis is to discern differences in response with respect to differences in discretization of the bridge girder, dynamic wind forces are only applied on the girder. The MATLAB-script WindSim[14] is once again applied to generate a wind-field consisting of the following properties: Sampling frequency 4 Hz Time step 0.25s No. of time steps 2 13 (8192 steps) Length of wind series 2048s (34.13 minutes) Spectral density function Kaimal Reference mean wind speed 29m/s (50-year return period) Terrain roughness category 0 (exposed to open sea) Geometry length, height 3750m, 85m Number of nodes 375 (element length of 10m) Mean wind speed 46.39m/s Integral length scale, L ux 190m Turbulence intensity, Iu 9.22% Iw 4.33% Coherence decay factors Cuy=10, Cuz=10, Cwy=6.5, Cwz=3 As the length and number of nodes implies, the discretization of the bridge girder span will result in an element length of 10m. Turbulence components from all nodes will be extracted for the 10m-analysis. As for the 20m- and 40m-discretized models, turbulence components from every second respectively fourth node will be extracted. The implication of this is that all the analyses will have turbulence components that are generated from the same wind-field simulation. Further, the first 7200 time-steps (out of 8192) are extracted to represent a 30- minute time-series. 44

57 In addition to the static forces and moment described in Eq Eq. 4.9, the dynamic forces and moment that are generated in MATLAB into an input-file and applied at selected nodal positions at each time increment are Eq Eq Due to the circumstance that the dominant eigenfrequencies of the bridge are around and below 0.1Hz, aerodynamic admittance functions are assumed to be of limited relevance and is hereby neglected in the calculations. Moreover, mechanical admittance is neither considered in the analysis. F D = 1 2 ρ[2uu(t)c H D + Uw(t)(C D H C B)]L L F L = 1 2 ρ[2uu(t)c B L + Uw(t)(C L B + C H)]L D M x = 1 2 ρ[2uu(t)c B M 2 + Uw(t)C M B 2 ]L Eq Eq Eq Where L indicates the length of the bridge girder element. The dynamic wind components u(t) and w(t) varies with each 0.25s step, hence the dynamic drag- and lift forces as well as moment varies with each time increment. In addition, the dynamic analysis is configured with a time step-size of 0.15s which may also be automatically altered by algorithms according to the efficiency of convergence within the analysis. An example of calculations of dynamic forces for the box-girder can be seen in appendix Frequency domain analysis An analysis in frequency domain will be done to analyze how difference in discretization of the bridge span will affect the resulting power spectral density in vertical and horizontal directions. Etienne Cheynet s MATLAB script[17] for generation of PSD in frequency domain is applied. The input is mode-shape data for each mode and relevant node, corresponding eigenfrequency as well as structural data of the bridge girder. Table 8 presents the applied mode-shapes and corresponding eigenfrequency. Illustrations of the mode-shapes are found in appendix 6.7. The mode-shapes with corresponding values and eigen-frequencies are determined in Abaqus when modal-analysis is run for the model. Table 8. Mode-shapes and eigen frequencies. Horizontal mode-shapes Vertical mode-shapes No. Eigen frequency No. Eigen frequency

58 10 different discretization models have been run that represents 10m, 20m, 30m, 40m, 50m, 60m, 70m, 80m, 90m and 100m girder element length. Similar to the parametric study, relevant nodal values of the shape-functions are extracted to represent a courser discretization. The PSD-analysis is set to compute data representing the middle of the bridge span. PSD output of the different discretization-levels will be compared by normalization about the discretization-level of 10m girder element length. The idea is to determine convergence levels for increased discretization. 46

59 4.2 Results Static analysis Figure 36 displays how displacement due to static force varies along the bridge span with respect to applied force. Table 9 displays that force on the floating towers adds the most to total displacement and hangers the least. Table 9. Horizontal displacement in z-direction due to static force. Static force applied on: Maximum displacement [m] Compared to total displacement Bridge girder % Incl. Cables % Incl. Hangers % Incl. Floating towers (total static force) % Figure 36. Horizontal displacement in z-direction due to static force. Figure 37 shows rotation about the y-axis along the bridge span. It can be noted that the rotation is zero at the center of the span and the force against the floating towers contributes the most to the rotation. Similar to the lateral displacement, the static force on the floating towers contributes the most to the rotation. 47

60 Figure 37. Rotational displacement about y-axis due to static force. Figure 38 displays the rotation about the x-axis along the bridge span. Also known as the angle of incidence, the angle has been applied to calculate the updated lift- and moment shapefactors that has been used for the static- and dynamic analysis. It can be noted that the angle peaks at three locations at the spans between the towers, where the angle is the largest at the center between the floating towers. Figure 38. Rotation about x-axis, a, angle of incidence. 48

61 4.2.2 Dynamic analysis Horizontal displacement Figure 39 illustrates the time-series of the lateral dynamic load sequence at the middle of the bridge span. Due to a probable programming error of the Abaqus input-file of the bridge model, the static load sequence starts at the same time-increment as the dynamic, which manifests as large oscillations in the beginning and throughout the time-series. Procedures for a more correct analysis would be to first apply the full static load and let the dynamic analysis hold until the static lateral displacement converges. Since dynamic analyses are timeconsuming and the data might still hold some analytical interest, it was decided to keep the current data. However, the end of the time-series should depict more correct how the lateral displacement oscillates due to dynamic loading. Figure 39. Time-series of lateral response of node 1110 at middle of bridge span. 49

62 Figure 40 - Figure 42 depicts the mean-, maximum and minimum lateral response for each node throughout the time-series. It is shown that the difference in lateral response for various levels of discretization of the bridge girder is small. Although, a more courser discretization will result in a slight increase in displacement. 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. 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

63 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. Figure 43 illustrates the standard deviation of the lateral response for each node. As a measure of how much the lateral displacement alters with the turbulent forces, it can be shown that the difference in standard deviation between discretization levels of 10m, 20m and 40m is very small. 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

64 Vertical displacement Figure 44 shows the time-series of vertical displacement at the middle of span for different discretization levels. Figure 44. Time-series of vertical response of node 1110 at middle of bridge span. Figure 45 - Figure 47 illustrates how the maximum, minimum and standard deviation of vertical displacement varies along the span of the bridge. It can be shown, in contrast to the lateral displacement, that the variation of vertical response due to difference in discretization level is considerably larger. It can be shown a 1.5% increase in standard deviation between 10m and 20m discretization levels for vertical displacement compared to 0.2% increase for lateral displacement. 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

65 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. 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

66 4.2.3 Frequency-domain analysis As previously described, the frequency-domain analysis focuses on eight lateral and eight vertical mode-shapes, where power spectral densities of the response are evaluated. The dominant lateral mode-shapes are visible as spectral peaks in Figure 48. H1/H2 H4 H6 H16 H5 H21 Figure 48. Frequency-domain PSD of lateral response for Δx=10m and Δx=100m discretization level. Figure 49 - Figure 50 displays the normalized PSD of lateral response, where one can observe that the overestimation effect of the spectral power increases with reduced discretizationlevel. Figure 49. Normalized PSD of lateral response with respect to 10m girder element length discretization level. 54

67 Figure 50. Normalized PSD of lateral response; mode-shape 2,5,12 and 21 relative to discretization level. The dominant vertical mode-shapes are visible as spectral peaks in Figure 51. V3 V9 V13/V17 V19 V18/V19/V20 Figure 51. Frequency-domain PSD of vertical response for Δx=10m and Δx=100m discretization level. 55

68 Figure 52 - Figure 53 displays the normalized PSD of vertical response. Similar to the PSD of lateral response, the overestimation effect of the spectral power increases with reduced discretization-level. Figure 52. Normalized PSD of vertical response with respect to 10m girder element length discretization level. Figure 53. Normalized PSD of vertical response; mode-shape 3,9,19 and 22 relative to discretization level. 56

69 4.3 Discussion: FE-analysis of Sulafjorden bridge The result of the static analysis implies that applied static force on the floating towers alone contribute to more than 50% to the lateral displacement of the bridge. It can therefore be of interest to reduce the drag-force that affects the pylon legs. A more streamlined design of the pylon cross-section might be an approach to optimize the drag shape-factors to reduce the magnitude of the aerodynamic drag force. When increasing the number of evaluation nodes of a turbulent wind field, one might expect that the variation of wind speed between adjacent nodes would reduce. This was indeed the case with the results of the dynamic analysis, which showed an increase in STD of lateral displacement by 0.18% between 10m and 20m discretization levels. In comparison to the parametric study, where an 0.22% increase in STD of total drag force could be noticed, a resemblance may be noted. Moreover, when comparing STD of vertical displacement and STD of total lift force for 10m and 20m discretization levels, the corresponding values of increase are 1.5% and 0.9%. The difference may partly be explained by the use of difference coherence decay coefficients, where the dynamic analysis applied Cwy=6.5 and the parametric study applied Cwy=5. In comparison, the total lift force from the parametric study for c=10 showed a 2.35% increase in STD between 10m and 20m girder element length. Although differences in STD of vertical response and total lift force are not perfectly alike, the larger magnitude in difference between 10m and 20m girder element length compared to lateral STD may indicate that a higher discretization level can have a greater impact on change in vertical response than change in lateral response. When comparing 10m to 40m discretization level, the increase in STD of vertical response and total lift force are 3% and 3.5% respectively. STD for lateral response and total drag force for the same comparison are 0.3% and 1.2% respectively. Thus, one can argue that increasing girder element length to 40m might not sufficiently capture the details in the turbulent wind field, thus not give a sufficiently descriptive response in relation to natural in-situ conditions. When running a dynamic analysis for a large spanned suspension bridge such as Sulafjorden bridge, when increasing the number of nodes in a FE-model, the expected time of analysis may also increase. It may therefore be important in each case to weight the benefits of an increased discretization to the time it takes to run the computations. The purpose with the frequency domain analysis was to evaluate a third comparison of the output due to differences in discretization level. Although a coarse convergence could be observed toward reduced girder element length, a more detailed result proved less productive. When observing spectral peaks that consists of several mode-shapes, one could argue that an amplifying effect effect of the PSD has taken place. Such and amplification may affect a further normalization, e.g. V19 in Figure 53, that a comparison would be of less significance. 57

70 In retrospection, one could initially have applied mode-shapes as input that had eigen frequencies that were not that close in magnitude. Thus, the amplifying effect of the PSD would not have been that large and a more detailed sensitivity study for difference in discretization level in frequency domain would perhaps have been more productive. In general, sources of error may always be present. Assuming that analytical expressions are correct and since you can always manually check calculations of analytic formulas before applying them in analysis, one may argue that these are of less significance when sources of error are reviewed. Due to a lack of fundamental knowledge regarding the FEM software Abaqus, the application of this software may contribute to a major source of error. 58

71 4.4 Conclusion The analysis that has been conducted in this thesis suggests that the standard deviation of the turbulence increases for reduced discretization of the bridge girder. Moreover, the standard deviation of the vertical turbulence component increases significantly more than the horizontal along-wind component for the same discretization. In other words, applying a finer discretization, from 20m to 10m girder element length, would not significantly alter the horizontal response but it would have a greater effect on the vertical response. 59

72 5. References [1] Multiconsult, Feasibility study Crossing of Sulafjorden, [2] M. AS, No Title. [Online]. Available: [Accessed: 03-Jun-2017]. [3] Y. Xu, Wind Effects on Cable-Supported Bridges. Wiley, [4] J. D. Holmes, Wind loading of structures, 1st ed. New York: Taylor & Francis, [5] E. N. Strømmen, Theory of Bridge Aerodynamics. Springer, [6] Statens Vegvesen, N400 Bruprosjektering. Vegdirektoratet, [7] Y. Tamura and A. Kareem, Advanced structural wind engineering, 1st ed. Japan: Springer, [8] J. Jakobsen B., Fluctuating wind load and response of a line-like engineering structure with emphasis on motion-induced wind froces, NTH, [9] E. Cheynet, Wind-induced vibrations of a suspension bridge, University of Stavanger, [10] B. J. Vickery, On the flow behind a coarse grid and its use as a model of atmospheric turbulence in studies related to wind loads on buildings. Aero Reports. National Physical Laboratory (UK), [11] S. R. Singiresu, Mechanical Vibrations, 5th ed. Pearson Education, [12] A. K. Chopra, Dynamics of structures: Theory an applications to earthquake engineering. Pears Education, [13] J. Wang, C. Etienne, J. B. Jakobsen, and J. Snærbjørnsson, Time-domian analysis of windinduced response of a suspension bridge in comparison with full-scale measurements, in Prodeedings of the ASME th international conference on ocean, offshore and arctic engineering OMAE [14] E. Cheynet, WindSim [15] D. Systemes, Abaqus 6.12 Keywords Reference Manual. Dassault Systemes, [16] NS-EN :2005+NA:2009. Standard Norge. [17] E. Cheynet, Buffeting response of a suspension bridge - a frequency domain approach

73 6. Appendix 6.1 Normalized total modal force 6.2 Normalized total modal reaction force at support 6.3 Normalized total modal bending moment at center of span 6.4 Time-series of normalized reaction force at support 6.5 Time-series of normalized bending moment at center of span 6.6 Manual calculations 6.7 Mode-shapes 61

74

75 Appendix 6.1: Normalized total modal force Normalized total modal drag force, 1st mode-shape Total no. of nodes Total no. of nodes Total no. of nodes

76 Normalized total modal drag force, 1st-5th mode-shape Total no. of nodes Total no. of nodes Total no. of nodes

77 Normalized total modal lift force, 1st mode-shape Total no. of nodes Total no. of nodes Total no. of nodes

78 Normalized total modal lift force, 1st-5th mode-shape Total no. of nodes Total no. of nodes Total no. of nodes

79 Appendix 6.2: Normalized reaction force at support Normalized modal reaction force at support due to drag force, 1st mode-shape

80 Normalized modal reaction force at support due to drag force, 1st-5th mode-shape

81 Normalized modal reaction force at support due to lift force, 1st mode-shape

82 Normalized modal reaction force at support due to lift force, 1st-5th mode-shape

83 Appendix 6.3: Normalized bending moment at center of span Normalized modal bending moment at center of span due to drag force, 1 st mode-shape

84 Normalized modal bending moment at center of span due to drag force, 1 st -5 th mode-shape

85 Normalized modal bending moment at center of span due to lift force, 1 st mode-shape

86 Normalized modal bending moment at center of span due to lift force, 1 st -5 th mode-shape

87 Appendix 6.4: Time-series of modal reaction force at support Normalized modal reaction force at support can be calculated from total modal force and the graphs in Figure 1 and Figure 2 has similar characteristics as the graphs for modal force. Figure 1. Normalized modal reaction force at support due to drag force for 10m and 250m element length and 60 second time-series. Figure 2. Normalized modal reaction force at support due to drag force for 10m and 250m element length and 60 second time-series.