Aeroelastic Analysis of Super Long Cable-Supported Bridges

Transcription

1 Aeroelastic Analysis of Super Long Cable-Supported Bridges Zhang, Xin SCHOOL OF CIVIL AND ENVIRONMENTAL ENGINEERING NANYANG TECHNOLOGICAL UNIVERSITY 3

2 Aeroelastic Analysis of Super Long Cable-Supported Bridges Zhang, Xin SCHOOL OF CIVIL AND ENVIRONMENTAL ENGINEERING A Thesis Submitted to Nanyang Technological University in Fulfilment for the Degree of Doctor of Philosophy 3

3 CONTENTS Acknowledgement Abstract List of Tables List of Figures Nomenclature i ii iii iv vi Chapter Introduction. Long-Span Bridges. Motivation for the Study.3 Organization 3 Chapter Aeroelasticity and Aerodynamics of Bridge Decks 4. Introduction 5. Thin Airfoil Aeroelasticity.3 Aeroelastic and Aerodynamic Forces on Long-Span Cable-Supported Bridge Decks 4.3. Formulation of the Self-Excited Forces 5.3. Buffeting Forces.4 Analytical Method in Frequency Domain.4. Analytical Method for Flutter Analysis.4. Governing Equations of Flutter Buffeting Analysis 9 Summary 3 Chapter 3 Wind Tunnel Experiment to Extract Flutter Derivatives 3 3. Introduction Similitude in the Experiment Other Model Types Extraction of Flutter Derivatives The Experiment The Wind Tunnel Sectional Models 4

4 3.3.3 The Experimental Setup The Experimental Procedure Calibration Basic Measurements 48 Summary 5 Chapter 4 Method Used to Identify Flutter Derivatives 5 4. Introduction Basics of ERA History of ERA The Method of ERA 54 Summary 6 Chapter 5 Experimental Detection of Nonlinearity in Self-Excited Forces 6 5. Introduction 6 5. Relative Amplitude Effect Physical Significance of Flutter Derivatives with Different Relative Amplitudes Use of Output Covariance as Markov Parameters Numerical Considerations for the Computation of Output Covariance The Experiment Results and Discussion 7 Summary 79 Chapter 6 Numerical Flutter Analysis 8 6. Introduction 8 6. The Suspension Bridge and Modeling Method to Solve the Aeroelastically Influenced Eigenvalue Problem Approximating the Impedance Matrix Description of the Analysis Analytical Cases Effect of Relative Amplitude 97

5 6.5.3 Effect of Lateral Flutter Derivatives 98 Summary 99 Chapter 7 Time Domain Formulation of Self-Excited Forces on Bridge Decks for Wind Tunnel Experiments 4 7. Introduction 5 7. Relative Amplitude Effect on the Transformation of Flutter Derivative Model to Time Domain State Space Model for SEF Generation System The Model Relation to Flutter Derivative Model The Transformation in Modal Coordinates Suggestions for Future Experiments 7 Summary 9 Chapter 8 Errors in the Identification of Flutter Derivatives 8. Introduction 8.. Errors Due to Non-White Noise 8.. Errors Due to Nonlinearity in the Self-Excited Forces 4 8. Evaluation Based on Block Hankel Matrix Data From Experiment 9 Summary 3 Chapter 9 Conclusions and Future Work Conclusions Suggestions for Future Work 37 Appendix I 38 Appendix II 39 Reference 4

6 Acknowledgements After going through almost three years of hard work it is time to thank all those who have pulled me through this period and made my stay at NTU a pleasant one. I would like to express my sincere gratitude and thanks to Prof. James Brownjohn for his invaluable guidance and moral support. My special thanks go to Dr. Piotr Omenzetter for the inspiring discussions and valuable suggestions. I take this opportunity to thank Mr. Tay Lye Chuan for the help in operating the wind tunnel and setting up the experimental devices. My Thanks also go to Mr. Phua Kok Soon for the help in manufacturing the sectional model and suspension system. My special gratitude is due to all my friends for making my time spent at NTU an unforgettable memory. I would like to thank the School of Civil and Environmental Engineering for the full financial support and the research facilities they provided during my study. i

7 Abstract A study on properties of interactive wind forces on bridge sectional models is presented in this thesis. Two and three-dimensional sectional model tests in the wind tunnel were carried out to detect nonlinearity in the self-excited wind forces. The transformation of a frequency-time domain hybrid flutter derivative model to either time or frequency domain usually requires the linearity assumption of the self-excited wind forces, which has not been investigated thoroughly. The self-excited wind forces on a bridge deck can be nonlinear even when the vibration amplitude of the body is small. Through the concept of relative amplitude, i.e. the amplitude of the externally triggered free vibration relative to the magnitude of the ambient response of an elastically supported rigid sectional model, nonlinearity in the self-excited wind forces is studied. The effect of relative amplitude on flutter derivatives and on the flutter boundary reveals, from the structural point of view, a complex relationship between the self-excited forces and the structural vibration noise due to buffeting forces relating to signature turbulence. Although the aeroelastic forces are linear when the body motion due to an external trigger is not affected significantly by the turbulence, they are nonlinear when the noise component in the vibration due to the turbulence is not negligible. The effect of lateral motion related derivative on flutter boundary is also studied by using flutter derivatives identified from and 3 degree of freedom (DOF) experiments. A time domain model for the self-excited forces generation mechanism is suggested with the objective in view to offer more flexibility for experimental studies of the self-excited forces. This expression can be linked to the frequency-time-domain hybrid flutter derivative model. A transform relationship between the two models is suggested. ii

8 List of Tables Table 3. Table 3. Table 3.3 Table 6. Table 6. Table 6.3 Table 6.4 Intensity of Lateral Turbulence Experimental Information Derivatives of Respective Static Force Coefficients Material Properties of the Humber Bridge Dynamic Properties of the Bridge Flutter Speeds & Frequencies in Different Combinations Participation Factors of Major Modes at Flutter iii

9 List of Figures Figure. Damping Driven Flutter Figure. Coalescence Flutter Figure 3. Conventions Figure 3. Power Spectral Density of Lateral Turbulence U=7.4m/s Figure 3.3 Twin Deck Bluff Model Figure 3.4 Streamlined Box Girder Model Figure 3.5 Set Up for Free Vibration Test Figure 3.6 Set Up for Static Force Coefficient Measurement Figure 3.7 C L of Model A Figure 3.8 C M of Model A Figure 3.9 C D of Model A Figure 3. C L of Model B Figure 3. C M of Model B Figure 3. C D of Model B Figure 5. The Definition of Relative Amplitude Figure 5. Non-Stationary Flutter Boundary Figure 5.3a. Transient Signal of Model B at U=7.5m/s Figure 5.3b. FFT of Transient Signal at U=7.5m/s Figure 5.4a. Ambient Vibration of Model B at U=7.5 m/s Figure 5.4b. FFT of Ambient Vibration at U=7.5 m/s Figure 5.5. Output Covariance of Model B at U=7.5 m/s Figure 5.6a. DOF H (Model A) Figure 5.6b. DOF A (Model A) Figure 5.7a. DOF H (Model B) Figure 5.7b. DOF A (Model B) Figure 5.8a. 3DOF H (Model B) Figure 5.8b. 3DOF A (Model B) Figure 5.8c. 3DOF P (Model B) Figure 6. Plot of the Bridge Figure 6..a Structural Modes of the Bridge Deck iv

10 Figure 6..b Figure 6..c Figure 6..d Figure 6..e Figure 6.3 Figure 6.4 Figure 6.5 Figure 6.6 Figure 6.7 Figure 6.8 Figure 6.9 Figure 6. Figure 6. Figure 6. Figure 7. Figure 7. Figure 7.3 Figure 7.4 Figure 8. Figure 8. Figure 8.3 Figure 8.4 Structural Modes of the Bridge Deck Structural Modes of the Bridge Deck Structural Modes of the Bridge Deck Structural Modes of the Bridge Deck Sensitivity of E-matrix to Damping Ratio Singular Values at Flutter (D FD Case), st Mode E-Matrix of D FD E-Matrix From D FD By Deleting P Related FD E-Matrix From 3D FD First Flutter Mode from D FD Second Flutter Mode from D FD The st Flutter Mode (D FD from 3D FD) The nd Flutter Mode (D FD From 3D FD) First Flutter Mode from 3D FD Indicial Functions of Different Kinds Simulation Diagram of the SEF Model FRF Matrix of the HB Bridge Section via Flutter Derivatives (Transient) FRF Matrix of the HB Bridge Section via Flutter Derivatives (Ambient) Singular Values at U=4m/s Error Index for 3D Transient and Ambient Vibration Testing (HB) Error Index for D Transient and Ambient Vibration Testing (HB) Error Index for D Transient and Ambient Vibration Testing (TK) v

11 Nomenclature A A c A, H, P m m m State Matrix of Discrete State Space Model State Matrix of Continuous State Space Model Flutter Derivatives A ij (K) Variables in E ij f A s, A s State Matrix of Rigid Body System B B c Input Matrix of Discrete State Space Model or Width of the Bridge Deck Input Matrix of Continuous State Space Model B f, B s, B cov Input Matrix of SEF, Rigid Body and Covariance Dynamics System B ij (K) Variables in E ij C [ C ] str, [ C ] aero Output Matrix, [ C ] eff Structural, Aeroelastic and Effective Damping Matrix C (k) Theodorsen Circulation Function C i (k) Output Covariance C f Output Matrix of SEF System C, C, C Static Wind Force Coefficient L C s Cov D ae D ae M ae Output Matrix of Rigid Body System Covariance Estimation Feed Through Matrix D, L, M Aeroelastic Forces D, L, M Buffeting Forces b dm b ae Infinitesimal Mass E Impedance Matrix E [] Expectation Operator E ij Element in Impedance Matrix vi

12 E r (n) Error Signal Matrix f (s) State Vector of SEF f buff Buffeting Force f sef Self-Excited Forces F State Matrix of SEF System F ( k), G( k) Functions in Aerodynamic Coefficient G Input Matrix for Covariance Dynamics G r i s j Modal Integral h, p,α Displacements of the Rigid Body in Vertical, Lateral and Rotational Direction, Respectively h, α, i i p i th i Vertical, Rotational and Lateral Mode, Respectively H (k) Block Hankel Matrix [ H (k)] Flutter Derivative Matrix [ ] K, [ ] aero K, [ ] effect K Structural, Aeroelastic and Effective Stiffness I i K l Generalized Inertia Reduced Frequency Bridge Deck Length L, M Lift and Moment Forces of Wind [ M ] Structural Mass Matrix M ae Aeroelastic Moment P α Observability Matrix p (t) Buffeting Force Q β Controllability Matrix r ( ) Rank Operation [ R ] FRF Matrix s Dimensionless Time Tr ( ) Trace Operation U Wind Speed vii

13 v (t) Measurement Noise X State Vector X (s) State Vector of Rigid Body Motion Y Y i Displacement Markov Parameters Y (s) Output Vector of Rigid Body State Space Model Z (τ ) Structural Function Ξ,Ξ Power Matrix of the Error Signal Σ, Θ Σ n Singular Value Matrix Signal Power Matrix Θ (k) Sears Function Ω,Ω Ratio of Error to Signal Power τ ω i Time Circular Frequency ξ η i Participation Factor Vector of Structural Modes at Flutter The I th Full Bridge Mode Shape φ (s) Wagner Function ψ Kussner Function χ χ, χ L D M, Admittance Function viii

14 CHAPTER ONE Introduction to the Research. Long-Span Bridges Long-span suspension bridges or cable-stayed bridges are highly susceptible to wind excitations because of their inherent structural flexibility and low damping ratios. The collapse of the center span of Tacoma Narrows Bridge in 94 at a relatively low wind speed of 4 mph is the most dramatic incident of wind-induced failure of bridges. This incident caused investigators to examine many of existing suspension bridges built in the same area for the possibility of excessive wind-induced vibrations. Up to now, the driving force to build bridges of this kind is still obvious due to its elegant appearance and economy. Fast developments in the state-of-the-art design over the last two decades have brought about a new stage of the construction of such structures. The ambitious Akashi-Kaikyo Bridge has a center span up to m. The Strait of Messina Bridge with a center span of 3.3km will stand as the landmark bridge of st century.

15 . Motivation for the study To have a better design, the study of the wind load on bridge decks is of vital importance. The wind load is classified in two categories: motion dependent (selfexcited) forces and motion independent forces. One of the main tasks of bridge aeroelasticity is to formulate the wind load on the structure when the body is in motion. With increasing length of bridges, the structure becomes more flexible when the span is longer. There is a transition of the analytical method from frequency domain to time domain to overcome the difficulties in dealing with structural nonlinearity. Other researchers also used the time domain approximation of self-excited wind load to study control of bridge vibrations. The transformation of the frequency-time domain hybrid flutter derivative model to either time or frequency domain usually requires the linearity assumption of self-excited wind forces, which, unfortunately, is yet to be proven either by theoretical or experimental means. Furthermore, current analytical methods for the buffeting analysis use flutter derivatives identified experimentally for flutter instability analysis, assuming that in these two cases, the interactive forces are the same in their properties. It is also based on the linearity assumption of self-excited forces. The possible existence of nonlinearity in the self-excited forces could have a fundamental impact on the state of the art understanding of the interactive wind load. Experiments in this research are efforts to test whether or not the self-excited wind forces can be treated linearly. Because usually aeroelastic analysis is meant to predict the structural behavior when the structural vibration amplitude and the angle of attack of the oncoming wind are both small, it is important to detect the existence of nonlinearity in the self-excited forces under the small amplitude condition. Previous tests (Scanlan, 997; Falco, et al. 99) did not take this factor into consideration.

16 The existence of nonlinearity in self-excited wind forces will demand more efforts to be exercised in the future to formulate the interactive wind load. It must be recognized that a frequency-time domain hybrid flutter derivative model works as a linear model under specific conditions. Any extension of the model to perform analysis under other conditions will need experimental verification..3 Organization After a brief review in Chapter on background literatures on bridge and airfoil aeroelasticity, the design for experiments to extract flutter derivatives in and 3 dimensions is presented in Chapter 3. The identification method selected in this thesis is eigensystem realization algorithm (ERA); it is introduced in Chapter 4. Through the concept of relative amplitude effect, the detection of the nonlinearity in self-excited wind force by experimental means is described in Chapter 5, where most of the experimental results are presented. Flutter boundary prediction is subsequently described in Chapter 6 to illustrate the effects of the nonlinearity in the self-excited wind force and the lateral flutter derivatives on aeroelastic instability analysis. Because of nonlinearity in the self-excited force, new considerations on the interactive force modeling is needed and an alternative model is proposed in Chapter 7 with the objective in view to offer more flexibility to manipulate the experiment and the empirical model. In Chapter 8, a new error index is presented to evaluate the identification of experimental results. Conclusions and suggestions are given in Chapter 9. 3

17 CHAPTER TWO Aeroelasticity and Aerodynamics of Bridge Decks Abstract This section is devoted to a review of the past work on aerodynamics and aeroelasticity of bridge decks relevant to the present work. The discussion begins with a few definitions followed by the classification of aeroelastic phenomena. After a brief introduction of thin airfoil aeroelasticity, current methods for analyzing the aeroelastic and aerodynamic behavior of bridge decks are reviewed. 4

18 . Introduction Researches are booming in the area of aerodynamics of civil structures, which are not usually designed to influence or accommodate the airflow over them, but rather with other objectives in view. The aerodynamics of such structures is characterized by separated flow and turbulent wakes exhibiting widely varying degrees of flow organizations. A body immersed in a fluid flow is subjected to surface pressures induced by the flow. If the oncoming flow is turbulent, this will be one of the sources of time dependent surface pressure. If the body moves or deforms appreciably under the induced surface pressure, these deflections, changing as they do the boundary conditions of the flow, will affect the fluid forces, which in turn will influence the deflections. Aeroelasticity is the discipline concerned with the study of the phenomenon wherein aerodynamic forces and structural motions interact significantly (Simiu and Scanlan, 996). If the body in the fluid flow deflects under some forces and the initial deflection gives rise to successive deflections of oscillatory and/or divergent character, aeroelastic instability is said to be produced. All aeroelastic instabilities involve aerodynamic forces that act on the body as a consequence of its motion. Such forces are termed self-excited. A body is said to be aerodynamically bluff when it causes the wind flow around it to separate from its surface leaving a significant trailing wake. In contrast, wind flow around a streamlined body remains tangential and attached to its entire surface, leaving a narrow trailing wake. Most civil engineering structures, including the bridge sections of the long span bridges qualify as bluff bodies, while the shapes of an airfoil belong to the category of a streamlined body. The fundamental aspects of aeroelastic phenomena that need to be taken into 5

19 account in the design of certain structural members, towers, stacks, tall buildings, suspension bridges, cable roofs piping system and power lines are not completely understood. In most investigations empirical models are set up because pure theoretical computations based on CFD can hardly produce reliable results. The corresponding analytical models usually include just enough parameters to match the strongest observed feature of the phenomena. Such models are minimally descriptive, but not explanatory in the sense of revealing basic physical causes; subtle but important details of the actual fluid-structure interaction may in certain cases be left unattended. According to these models, aeroelastic phenomena fall into the following categories: Vortex Shedding and the Lock-in Phenomenon. Under certain conditions a fixed bluff body sheds alternating vortices (Ehsan 988; Hartlen and Currie 97; Iwan and Blevins 974; Nakamura and Nakashima 986 and Ongoren and Rockwell 988). The primary frequency of the vortex shedding is according to the Strouhal Relation: N D s = S (..) U where the Strouhal number S depends on body geometry and the Reynolds number, D is the across-wind dimension, U is the mean velocity and frequency of the vortex shedding. N s is the primary If the body is elastically supported and being driven periodically by the vortices shed in its wake, it will experience small response unless the Strouhal frequency of the alternating pressure approaches the across-flow mechanical frequency of the structure. At this stage, the body interacts strongly with the flow. The mechanical frequency controls the vortex shedding even when variations in flow velocity displace the nominal Strouhal frequency away from the natural mechanical frequency by a few percent. This phenomenon is known as lock-in. 6

20 Galloping Galloping (Novak 97 and Van Oudheusden 995) is an instability of typically slender structure with a special cross section shape such as a rectangular or a D shapes. The structure exhibits large amplitude vibration in the direction normal to the flow at frequencies much lower than those of vortex shedding from the same section. In the across wind galloping, the relative angle of attack of the wind to the bridge section depends directly on the across wind velocity of the structure. Mean lift and drag force coefficients of the cross section obtained under static condition, as functions of angle of attack, suffice as a basis upon which to build the analytical description. Wake galloping is due to the turbulent wake of the upstream cylinder, and may occur only under conditions where the frequencies of response of the downstream cylinder are lower compared to its vortex-shedding frequencies and to those of the upstream cylinder. It is also governed by parameters describing mean rather than instantaneous aerodynamic phenomena. 3 Torsional divergence Under the effect of wind, the structure will be subjected to drag, lift and pitching moment. As the wind speed increases, the twisting moment may also increase, twisting the structure further. This condition may also, by increasing the effective angle of attack, further increase the twisting moment. Then additional deflections occur. Finally, a velocity is reached at which the magnitude of wind-induced moment together with the tendency for twisting to demand additional structural reaction creates an unstable condition and the structure twists to destruction. 4 Flutter In the context of bridge engineering, flutter is usually of single aeroelastic mode, a typical self-excited oscillation (Jain et al. 996; Matsumoto et al. 994; and Chen et al. ). The flutter mode changes from a pure structural mode to an aeroelastic mode incorporating the effect of aeroelastic coupling. The structural system by 7

21 means of its deflection and time derivatives taps off energy from the wind flow. If the system is given an initial disturbance, its motion will either decay or diverge according to whether the energy of motion extracted from the flow is less than or exceeds the energy dissipated by the system through mechanical damping. The dividing line between the decay and divergent case, namely, sustained sinusoidal oscillation, is recognized as the critical flutter condition, the threshold of negative damping. Therefore, the criterion for the flutter to occur is based on eigenvalues, i.e. whether or not one or more eigenvalues move, as functions of aerodynamic parameters, from the left-hand to the right-hand side of the s-plane (Figure.). In the figure, s = σ + iω is the complex eigen-frequency of the aeroelastic system. ω s = σ + iω σ Figure. Damping Driven Flutter Together with the change of eigenvalue(s), there could also be a change of flutter mode(s) from the pure structural mode(s) to aeroelastic mode(s) due to the fact that the aeroelastic coupling could be strong to change not only the modal damping but also mode shapes. This is classified in some literatures as coupled mode flutter. However, the term coupled essentially means the coupling of pure mechanical modes not the aeroelastic modes. However for coalescence flutter, two modes of vibration are required and damping is not necessarily present in the system. When coalescence flutter occurs, the energy required to drive the instability is extracted from one of the stable modes, and this energy is fed in a non-conservative manner to the other mode, which then becomes unstable. It can be classified as a non-conservative problem. In contrast to the more familiar conservative problems, the non-conservative problems have 8

22 non-self-adjoint characteristics and are inherently unstable. If we define the set of aeroelastically influenced stiffness matrix as a family of matrices depending on parameters e.g. reduced frequency, coalescence flutter is defined by the bifurcation position of the matrix family in a matrix bundle. The frequency degeneracy is not a sufficient condition for coalescence flutter. If the degenerated eigenvalues are encountered, one has to inspect the corresponding eigenvectors or, equivalently, the eigenvalue matrix at the point of eigenvalue degeneracy (Figure.). If the eigenvectors are not linearly independent, i.e. the angle between two eigenvectors becomes zero as shown in the figure or, equivalently, the eigen-matrix is a Jordan matrix, then coalescence flutter occurs (Afolabi, 994). The stiffness matrix, in this case, can no longer be diagonalized with the eigenvector matrix. There is a shortage of eigenvectors. If generalized eigenvectors are used, the resultant diagonal matrix appears in Jordan canonical form. It is well known that this operation is not stable. Coalescence flutter instability has not yet been studied extensively in bridge A set of matrices is called a bundle if all the matrices belonging to it have Jordan normal forms differ only by their eigenvalues, but for which the set of distinct eigenvalues and the order of the Jordan blocks are the same. For example, all the diagonal matrices with simple eigenvalues define one bundle. Families of matrices are in general position if they are transversal to all the bundles and in exceptional position if they are not. The matrices in general positions are called generic, while those are not in general positions are called degenerate. Corresponding to the decomposition of the space of matrices into bundles, the parameter space of the family decomposes into sub-manifolds. In a family in general position, almost all the matrices have simple eigenvalues. The exceptional parameter values to which there correspond matrices with multiple eigenvalues define a subset of the parameter space. This is called bifurcation diagram (Arnold, 97). A generic matrix has structural stability, and does not change its qualitative properties or behavior under small perturbations. A degenerate matrix, on the other hand, is structurally unstable. An arbitrary small perturbation will cause it to bifurcate into two or more generic matrices. Coalescence flutter happens on such bifurcation point. As a result of this instability, degenerated objects are unobservable, and are almost always not encountered in engineering practice. If they are encountered in mathematical model, it is only because one has made a theoretical assumption, which is not qualitatively valid in the actual physical problem. 9

23 engineering. Therefore, most of the past and current works concentrated on the prediction of the negative damping boundary with or without considering the change of pure structural modes to aeroelastic modes. Angle Between Eigenvectors Non-Dimensional Eigenvalues 9 Aerodynamic Parameter 5 Buffeting Figure. Coalescence Flutter Buffeting is defined as the unsteady loading of a structure by velocity fluctuations in the oncoming flow. If these velocity fluctuations are clearly associated with the turbulence shed from the wake of an upstream body, the unsteady loading is referred to as wake buffeting. However, the buffeting force is usually due to the atmospheric turbulence. In this study, the work is focused on flutter instability, i.e. the identification of the flutter derivative model for self-exited forces on the bridge deck and the prediction of the damping driven (single mode) flutter boundary. In the following part, some related work based on flutter derivative models done in the past decades on long span bridges are introduced. First, there is a review of some historical studies on thin airfoils.

24 . Thin Airfoil Aeroelasticity The aerodynamic forces acting on a thin airfoil undergoing complex sinusoidal motion h and α : t i e h h ω = (..) t i e ω α α = (..) in two-dimensional incompressible flow are given by Theodorsen (935) from basic principles of potential flow theory. The expressions for h L and α M are linear in h, α and their first and second derivatives: ) ) ( )( ( ) ( α α πρ α α πρ & & && && & a b h U k UC ba h U b L = (..3) = α α π ρ α α πρ & & && && & ) ( ) ( ) ( ) 8 ( ) ( a b h U k C a Ub abh a b Ub a b M (..4) where U b k / ω = is the reduced frequency, b is the half-chord of the airfoil, ab is the distance between the mid chord and the rotation point, ρ is the air density, U is the flow velocity and ω is the circular frequency of oscillation. The complex function ) ( ) ( ) ( k ig k F k C + = is Theodorsen s circulation function. The coefficients in the expression, referred to as aerodynamic coefficients, are defined in terms of two theoretical functions ) (k F and ) (k G, [ ] [ ] [ ] [ ] ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( k J k Y k Y k J k J k Y k Y k Y k J k J k F = (..5) [ ] [ ] ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( k J k Y k Y k J k J k J k Y k Y k G = (..6)

25 in which J, J are Bessel functions of the first kind, Y,Y the second kind. are Bessel functions of This equation is in frequency and time domain hybrid format. There were also efforts to transform the expression from unsteady aeroelastic force to time domain. Wagner (95) showed that the lift evolution with dimensionless time s = Ut / b acting on a theoretical flat airfoil given a step function change α in angle of attack is given by L = ρ U (b)(π ) α φ ( s) (..7) where φ (s) is the Wagner function: C( k) iks φ ( s) = e dk. (..8) π i k C(k) is the Theodorsen circulation function and k is the reduced frequency. For arbitrary motion, the lift force is given as s L( s) = ρ U (b)(π ) α 3/ 4( σ ) φ ( s σ ) dσ (..9) d where α 3 / 4 = α 3 / 4 ( s) and α 3 / 4 is the effective angle of attack, ds h& & α α 3 / 4 = α + + b( a). (..) U U h & is the vertical velocity and ab is the distance from the mid-chord to the reference point at which deflection and rotation angle are measured.

26 Jones [94] introduced rational approximation of the unsteady loads on a typical airfoil section in incompressible flow in order to ease the difficulties in flutter stability analysis. In dimensionless time domain,.455s.3s φ ( s).65e.335e. (..) Kussner (936) considered the problem of an airfoil with forward flight velocity U penetrating a uniform vertical gust of infinite downstream extent and vertical velocity w. He determined the lift due to this circumstance to evolve according to the description: w L( s) = ρu B(π ) ψ ( s) (..) U with L and w considered positive upward, and ψ (s) is the Kussner function defined approximately by Jones(94).3s s ψ ( s).5e. 5e. (..3) For the gust of arbitrary velocity distribution w (s), the lift generated by an airfoil advancing through it will be given as s L( s) = πρ UB w( σ ) ψ ( s σ ) dσ. (..4) For a gust velocity distribution that is sinusoidal of the form iks w( s) = we, Sears (94) derived the corresponding oscillatory lift on the airfoil in the form w L ) U iks ( s) = ρ U B(π ) Θ( k e (..5) 3

27 where Θ(k) is a complex frequency-domain function known as Sears function. It is clear that the Kussner function and Sears function are a Fourier transform pair: Θ ik ( k) = ik ψ ( σ ) e σ dσ. (..6) It was further shown by Fung (955) that Sears function Θ (k) is related to Theodorsen circulation function C(k) as follows [ J ( k) J ( k) ] ij ( ) Θ ( k ) = C( k) + k (..7) where J and Jare Bessel functions of argument k. Spectral forms of L (s) are also available, but will not be reviewed. In the case of bridge engineering, however, because of the complexity of the bluff body aerodynamics, special considerations are needed for the formulation of selfexcited forces on bridge decks..3 Aeroelastic and Aerodynamic Forces on Long-Span Cable-Supported Bridge Decks A basic task in the study of the bridge aeroelasticity is to formulate the forces of wind on the structure. The total lift force L, drag force D and moment M are decomposed to motion dependent force and motion independent force: aeroelastic forces (ae) and buffeting force (b): L = L ae + L b (.3.) D = D ae + D b (.3.) M = M ae + M b (.3.3) 4

28 .3. Formulation of the Self-Excited Forces Special considerations are needed for the formulation of self-excited force for bridge decks. The signature turbulence, in the case of efficient airfoils in smooth flow, is intentionally reduced by careful streamlining with notable attention to the introduction of a sharp trailing edge. For bluff bodies, however, the situation is different. The use of Theodorsen aerodynamics for such bluff bodies is not guaranteed correct. In view of this, the formulation of self-exited forces on civil engineering structures, such as a bridge deck, is more experimental than theoretical. Scanlan and Tomko (97) suggested reduced frequency dependent flutter derivatives be used in the modeling of self-excited wind load on bridge deck. This is the counterpart of Theodorsen theory in the experimental bridge aerodynamics. The flutter derivative format representation of self-excited wind forces, after being expanded from two degree of freedom to three degree of freedom to take into consideration the lateral vibration, now takes the form: L ae h& B & α h p& = ρu B KH + KH + K H 3α + K H 4 + KH 5 + K U U B U H 6 p (.3.4) B D ae p& B & α p h& h = ρ U B KP + KP + K P3 α + K P4 + KP5 + K P6 (.3.5) U U B U B M ae h& B & α h p& p = ρ U B KA + KA + K A3α + K A4 + KA5 + K A6 (.3.6) U U B U B In which h,α, p are deck deflection components in vertical, torsion and horizontal direction, respectively. H, P, A, m =,,3,4,5, 6 are reduced frequency dependent m m m aerodynamic flutter derivatives, K = Bω / U is the dimensionless frequency. ω is circular frequency, B is deck width, U is wind velocity, ρ is the density of air. 5

29 The flutter derivative model works only for sinusoidal or exponentially modified sinusoidal motion with decay rate less then %. Among the flutter derivatives, H, H, H 5, A, A, A 5, P, P and P 5 describe aerodynamic forces in phase with bridge deck velocity. Therefore they are damping terms. H 3, H 4, H 6, A 3, A 4, A 6, P 3, P 4 and P 6 describe aeroelastic forces in phase with the bridge deck displacement. They are stiffness terms. A better understanding of the flutter derivatives is due to quasi-static theory in the following way (Simiu and Scanlan, 996). The static wind force coefficients are defined as non-dimensional numbers: L =, ρu Bl C L D =, ρu Bl C D M = (.3.7) ρu B l C M where C, C, C are mean lift, drag and moment force coefficients respectively L D M L is lift force, D is drag force, M is moment, B model width, U is wind speed, l is model length, ρ is the density of air. For small angle of attack α, h & Bα& α = or α = (.3.8) U U the typical term in equation (.3.4~.3.6) can be viewed in the classical patterns of expressions for aerodynamic lift force per unit span. dc L L = ρu (B) CL ρu (B) α (.3.9) dα Formally, term KH is analogous of the lift coefficient derivative dc L / dα. These flutter derivatives should be referred to as motional derivatives and they go over into steady-state derivatives only for zero frequency, i.e. K. The general expressions of flutter derivatives in the form of quasi-static theory are as follows: 6

30 H U ' = C L (.3.) π nb H H 3 ' C L U = (.3.) 4 π nb U = π nb 5 C L (.3.) A U ' = C M (.3.3) π nb A A 3 ' C M U = (.3.4) 4 π nb U = π nb 5 C M (.3.5) P U = π nb C D (.3.6) P U ' = C D (.3.7) π nb P P 3 5 ' C D U = (.3.8) 4 π nb U ' = C D (.3.9) π nb where C ' ' ' L, CD, CM are corresponding first derivatives of force coefficients with respect to angle of attack α at α = ; n is structural frequency. Like the researchers in the airfoil aeroelasticity, civil engineering researchers are trying to expand the time-frequency domain hybrid format model to time domain. A more general understanding of the unsteady aeroelastic force is found by recognizing that the indicial function expression can be seen as a modification of quasi-static nominal form of wind force under turbulent condition. The wind lift 7

31 8 force is given by [ ] + + = U u C B C U L L L ) ( ) ( α α α ρ (.3.) or u UBC u UBC BC U BC U L L L L L α α ρ α ρ α α ρ α ρ ) ( ) ( ) ( ) ( = (.3.) This is recognized to be only a nominal form that may hold for very slow changes in the angle of attack and wind speed, but is strictly incorrect due to the known lag of interaction force behind their angle of attack or wind velocity changes (Scanlan, 993). Hence modification is needed: Φ Φ + Φ + Φ + Φ = s Lu s L L s Lu L s L L L L d s u d s UBC u d s u UBC d s BC U s BC U s L ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( σ σ σ σ σ σ α α ρ σ σ σ α ρ σ σ σ α α ρ α ρ α α α (.3.) The first term represents an initial transient, arriving for s at the steady state lift, and can be considered as constant. The second term represents self-excited lift, the third horizontal impulse lift, and the last, interaction between the two lift forces. It should be very small since the correlation between the fluctuation part of the oncoming flow and the bridge motion is very weak. An expression is also available for vertical movement related wind load. In the time domain formulation of unsteady self-exited forces on a bridge deck, indicial functions still remain the most important tool when the structure is subjected to arbitrary motion. Scanlan et al. (974) studied the aeroelastic moment on a bluff bridge deck due to indicial angular movement. The characteristic of indicial function corresponding to A of a bridge, according to their experiment, is

32 strongly different from those of the corresponding functions of airfoils. They showed the relationship between the flutter derivatives and the indicial function by recognizing that for a sinusoidal motion, the Duhamel integral is of the nature of a Fourier transform and the inverse transform of frequency domain expression should then produce the indicial function. The direct measurement of indicial function is neither easy, nor conventional in the sense of modern dynamic experiment techniques. Yoshimura and Nakamura (979) suggested, in their study on the measurement of the indicial aerodynamic moment response of moving bluff prismatic sections in still air, that since the aeroelastic moment arises from the relative motion between the fluid and body, it might be expressed more conveniently by the time derivatives of the state variables. By assuming the superposition of small disturbances to a linear aerodynamic system, the moment due to the angular motion is decomposed into two parts, namely the moment due to the angle of attack α(s) and the angular velocity of the body axis relative to the fixed coordinated q (s) : M α ( s), q( s)) = M ( α ( s)) q= const + M ( q( s)) α = const ( (.3.3) and the indicial dynamic moment response is also decomposed into two terms: Φθ ( s) = Φα ( s) + dφ q / ds. (.3.4) The first term is the indicial aerodynamic moment response for the angle of attack motion and the second term is the indicial aerodynamic moment response for the angular velocity motion. Three types of indicial motion were used. In the reported study, it was found that the contribution of the angle of attack motion dominates, while the second term contribution to the overall indicial function is small and negligible. 9

33 .3. Buffeting Forces Buffeting force on bridge decks is also an important topic. The quasi-static buffeting forces due to turbulence are (Scanlan 988): L b = u w ρ U B C + ( L CL + CD ) (.3.5) U U M b = u w ρ U B C + M CM (.3.6) U U where L D D M b = u w ρ U B C + D CD (.3.7) U U C, C, C are mean lift, drag and moment coefficients, and the primes denotes their first derivative with respect to angle of attack ( ) α at α =. For an accurate description, these expressions must be modified by aerodynamic admittance factors (Davenport, 96; Kumarasena, 989): L b L = L χ (K) (.3.8) b M b M = M χ (K) (.3.9) b D b D = D χ (K) (.3.3) b χ, L M D, χ χ are aerodynamic admittance functions of lift, moment and drag buffeting forces. These functions are characteristic of the bridge deck shape and in fact are approximately related to the flutter derivatives (Kumarasena, 989) in the following

34 way when the body has long after-body geometry such that the flow reattachment might occur: and α = ( H + ) L K χ ( K) = 3 ih (.3.3) dcl + CD dα α = ( A + ) M K χ ( K) = 3 ia dcm. (.3.3) dα Scanlan () showed how important central characteristics of admittances can be seen to be inherent in the measured flutter derivatives, and the buffeting forces can be formulated by flutter derivatives: w u L = b ρ U B KH ( K) + KH 5 ( K) U U ; (.3.33) w u M = b ρ U B KA ( K) + KA 5 ( K) U U and (.3.34) w u D = b ρ U B KP ( K) + KP 5 ( K) U U. (.3.35) In these expressions, the following replacements have been affected for the quasistatic buffeting force terms: C L = KH 5 ( K) (.3.36) L ( C + C ) χ = KH ( K) (.3.37) L C D D D χ = KP ( K) (.3.38) D C χ = KP 5 ( K) (.3.39) D

35 C M M χ = KA 5 ( K) (.3.4) M C χ = KA ( K) (.3.4) M All coefficients on the left are associated with zero angle of attack for a horizontal wind, or to any other desired reference position. Because these terms are seen to be functions of K rather than being, in general, simple constants, they reflect frequency dependency and thus incorporate aerodynamic admittance effects. In other words, aerodynamic admittance is inherently expressible in this context as a function of the flutter derivatives..4 Analytical Method in Frequency Domain In the last several decades, the most significant advances have been made in understanding aeroelastic phenomena. Most of current efforts are concentrated on developing methods to alleviate the flutter instability, vortex-induced vibration and buffeting. Modern approaches to address these issues are based on a combination of state-of-the-art analytical, numerical and experimental techniques..4. Analytical Method For Flutter Analysis Techniques predicting flutter boundary of cable-supported bridges have been developing in two parallel ways: one in frequency domain (Davenport 96; Scanlan 978; Jain et al, 996), the other in time domain (Matsumoto et al. 994; Chen et al. ). All these methods are developed to solve negative damping driven flutter, due to the fact that for civil engineering structures, the wind speed is rare to reach such a high value to bring about coalescence flutter. Frequency domain analysis has dominated in the past due to the efficiency of computation, especially when handling the unsteady aeroelastic forces that are functions of reduced frequency. The nature of flutter analysis is generally a

36 complex eigenvalue problem, while buffeting analysis is conducted mainly by mode-by-mode approach ignoring the aerodynamic coupling among modes. As mentioned in the proceeding paragraph, the flutter instability of cable-supported bridges is defined with respect to a negative damping threshold. It is reasonable to postulate that a single mode will approximate the total response. This assumption is justifiable from observation of the fact that typically just one predominant mode will become unstable and dominate the flutter response of a three-dimensional bridge model in the wind tunnel. The so-called two-degree-of-freedom flutter analysis method supposes that there is coherence between bending and torsional mode shapes along the span, and does not consider the possible influence of transverse displacements. The small lack of coherence between the bending and torsional mode shape in conventional suspension bridges may have a non-negligible influence on critical wind velocity, as has already been notice by other authors (Irwin 979; Scanlan 987; Lin and Yang 983; Miyata et al. 99). This effect is usually more important for shorter bridges especially for cable-stayed bridges. Other authors proposed three-dimensional flutter analysis on the basis of flutter derivatives (Scanlan and Tomko 97; Scanlan 989; 993). The main point in studying the fully 3D stability consists in taking into consideration the degrees of freedom in the lateral direction. Then the equations are more difficult to solve, since the relations between the vertical, rotational and lateral displacement and the aeroelastic forces become quite complex, as they depend on the deformation patterns of the full bridge. In any case it will be supposed that there is no aerodynamic coupling between these forces along the deck, so that the sectional description will be integrated along the full bridge length to get total forces. It has been common to use the combination of a set of mechanical modes, namely the modes of the bridge structure under non-wind condition, as the flutter mode to perform the flutter analysis. It is clear, however, due to the aeroelastic effects, the 3

37 combination of a limited number of the mechanical modes is only an approximation of what happens in wind. Direct FEM flutter analysis by Miyata and Yamada (988), Miyata et al. (995), and later development of the mode tracing method by Dung et al. (996, 998) could serve as a better representation of the dynamic behavior of the long-pan bridges in terms of complex flutter mode. Complex eigenanalysis is made for an integrated system consisting of the 3-D FEM model of a bridge and the aeroelastic force caused by the wind flow. To solve the complex mode is an iterative procedure, tracing down the evolution of each aerodynamic complex mode with step-by-step increment of wind speed. Finite element method is the most common choice in this circumstance (Miyata, and Yamada 988; Agar 988; 989; Namini 99; Namini et al. 99; Starossek 993). The deck is usually modeled by beam elements located along the bridge axis. Plate elements could also be used provided that aeroelastic force is applied along the elastic axis of the deck. By assuming harmonic oscillation, the self-exited force on a unit length of bridge deck is incorporated into the element matrix. With the usual FE procedure, the governing equations of the aeroelastically-influenced structure can be established. Eignvalues and eigenvectors need to be found by iterative method since the governing equation is reduced frequency dependent. In one step, a set of natural frequencies of the aeroelastically-influenced structure is obtained with fixed wind speed. The procedure repeats with a different wind speed covering the speed range of interest. More recently, an efficient scheme for coupled multimode flutter analysis has been proposed introducing the unsteady self-exited aerodynamic forces in terms of rational function approximations (Matsumoto et al. 994; Chen et al. ). This has led to a convenient transformation of the equation into a state space format independent of reduced frequency. A significant feature of this approach is that an iterative solution for determining flutter boundary is unnecessary because the equations are independent of reduced frequency K = ωb / U where ω is the circular frequency, B is the deck width and U is the wind velocity. In general, frequency domain methods are restricted to linear structures excited by the stationary wind load without aerodynamic nonlinearities. To include nonlinearities 4

38 of structural and aerodynamic origins, the time domain approach is more appropriate. Time domain methods, however, involve the transformation of flutter derivatives into indicial functions, which have inherent deficit, as will be shown in the thesis. The effectiveness of time domain analysis in calculating buffeting response depends on the establishment of an effective time domain model for the self-excited wind force. In this thesis, the traditional frequency domain analysis will be used for the flutter instability analysis in chapter 6. Therefore, it will be reviewed in detail in the following part..4. Governing Equations of Flutter (Jain et al, 996) Deck deflection components can be expressed by generalized mode coordinates ξ i (t). If B is the bridge deck width, h ( x, t) is vertical displacement, α ( x, t) is torsion displacement and p ( x, t) is lateral displacement. Deck deflections are expressed in the following forms: h ( x, t) = h ( x) Bξ ( t) (.4.) i i i α ( x, t) = α i ( x) ξi ( t) (.4.) i p ( x, t) = pi ( x) Bξ i ( t) (.4.3) i in which, h i (x), (x) α and (x) each direction respectively. i p i are dimensionless representations of i th mode in The governing equation for the bridge deck motion can then be deduced as: I (& ξ + ζ ω & ξ + ω ξ ) q ( t) (.4.4) i i i i i i i = i 5

39 The generalized force is defined as l qi ( h, p, α, t) [ Lhi B + DpiB + Mα i ]dx, (.4.5) = where l is the deck span length; the generalized inertia is I ( x, y, z) dm( x, y ) = η z (.4.6) i i, where ηi is the full bridge mode, ω i is the circular frequency and ζ i is damping ratio-to-critical and dm is infinite small mass. The lift force L, drag force D and moment M in the governing equations are decomposed to motion dependent force and motion independent force: aeroelastic forces (ae) and buffeting force (b) as in Equation (.3.~.3.3), Substituting (.4.5), (.4.6) into (.4.4), we have the dimensionless time domain governing equation for the bridge deck motion: Iξ + Aξ + Bξ = Q ( x, s) (.4.7) b where s = Ut / B is the dimensionless time; ξ is the generalized coordinate vector; a prime denotes the derivative with respect to dimensionless time s ; I is the identity matrix and the general terms of matrix A, B and Q b are 4 ρ B lk Aij ( k) = ζ ikiδ ij [ HG I + P G i p α i j 5 h h + P G i i j p h + H j + A G G h α i i j α h j + H 5 + A G G h p i α α i j j + P G + A G 5 p p i i j α p j ] (.4.8) 6

40 B ( k) = K ij 4 ρ B lk δ ij I i i + P G 4 [ H G p p i j 3 h α + P G i 6 j + H p h i j 4 G 3 h h + A G i j α α + H i j 6 G 4 h p + A G i j α h i + P G j 3 6 p α + A G i j α p i j ] (.4.9) 4 ρ B l l dx ( x, s) = { Lb ( x, s) hi + Db ( x, s) pi + M b ( x, s) αi} I l Qb i i (.4.) where K = Bω / U is the reduced frequency and K = Bω U is the reduced i i / frequency of mode i, H m, Am, Pm, ( m =, L,6) are flutter derivatives and δ ij is the Kronecker delta function defined as: i = j δ ij =. (.4.) i j The modal integrals Gr i s j are obtained by integration over the deck, which is the primarily aerodynamic load source G = l r i s j ri x) s j ( x) dx ( (.4.) l where r, i = hi pi or i α ; s, j = h j p j or j α. Note that the off-diagonal terms in equation (.4.7) represent the aeroelastic coupling through the flutter derivatives and mechanical coupling through the cross mode integrals among different modes. The new equation is Fourier transformed in to reduced frequency ( K ) domain (Scanlan and Jones 99) by f iks ( K) = f ( s) e ds (.4.3) and is represented as 7