Bridge Aerodynamic Stability

Transcription

1 Bridge Aerodynamic Stability Instability Phenomena and Simplified Models André Miguel da Silva Rebelo de Campos Dissertation for the degree of Master of Science Civil Engineering President: Professor Luís Manuel Coelho Guerreiro Internal examiner: Professor Pedro António Martins Mendes Supervisor: Professor António José Luís dos Reis NOVEMBER 2014

2 This page left blank intentionally.

3 This page left blank intentionally.

4 This page left blank intentionally.

5 Abstract With the collapse of the Tacoma Narrows Bridge a new area of study was born within the branch of Civil Engineering. This area, called Aeroelasticity, considers the effects of a bridge deck immerse in a wind flow. This has become more and more important due to the ever increasing length of spans in cable supported bridges. Even though this is a new subject within Civil Engineering, Aeroelasticity had already been studied for some time by the Aeronautical Industry. Because of this, most theories applied nowadays to deck cross sections have been adapted from aerofoils and flat plates behaviour. Aeroelasticity provides answers for what kind of displacements and forces occur when a bridge deck is subjected to a flow of air. It also defines what instability phenomena can occur in this situation. Furthermore, experimental procedures and simplified models have been developed to help with the evaluation of the stability of bridge decks. These are presented in this dissertation and are applied to practical cases to evaluate their validity. Keywords: Aeroelasticity, Instability Phenomena, Wind Tunnel Testing, Approximated Formulas, Bidimensional Models i

6 This page left blank intentionally. ii

7 Resumo Com o colapso da ponte de Tacoma Narrows, surgiu, na Engenharia Civil, uma nova area de estudo chamada Aeroelasticidade. Esta área considera os efeitos provocados num tabuleiro de uma ponte imerso num escoamento de vento. Estes fenómenos são cada vez mais importantes devido ao comprimento cada vez maior dos vãos de pontes suspensas e atirantadas. Apesar de ser uma nova disciplina na área da Engenharia Civil, a Aeroelasticidade já era objecto de estudo da Industria Aeronautica há algum tempo. Isto traduziu-se na adaptação das teorias usadas para prever o comportamento de aerofólios e placas finas para os tabuleiros das pontes. A aeroelasticidade explica quais os movimentos e forças que surgem nestes tabuleiros quando são sujeitos a um escoamento de vento. Também identifica quais os fenómenos de instabilidade passiveis de ocorrer. Além disso, existem procedimentos experimentais e modelos simplificados que têm sido desenvolvidos de modo a verificar a estabilidade deste tipo de estruturas. Neste relatório, estes são apresentados e aplicados a casos práticos para avaliar a sua validade. Palavras-chave: Aeroelasticidade, Fenómenos de Instabilidade, Ensaios em Túneis de Vento, Formulas Aproximadas, Modelos Bidimensionais iii

8 This page left blank intentionally. iv

9 Acknowledgements To professor António José Luís dos Reis, for his availability and readiness in providing references as well as help with any obstacle that crossed the way. To my friends, whose support I couldn t have done without. To my parents, for their unconditional support and for all the help and care they provided. To my brother and sister, for all the help and stress relief moments. And especially to my friend Ana Raquel Crespo, for her love and all the hours spent supporting me no matter what. v

10 This page left blank intentionally. vi

11 Table of Content Abstract... i Resumo... iii Acknowledgements... v Table of Content... vii List of Figures... xi List of Tables... xiii List of Symbols... xv Chapter 01 Introduction Introduction The Tacoma Effect Objectives Structure and Methodology Chapter 02 Bridge Aeroelasticity - Instability Phenomena and Wind Tunnel Testing Analytical Aeroelasticity Aeroelastic Forces Movement equations Instability Phenomena Aerostatic Instability Static Divergence Aerodynamic Instability Vortex Shedding Galloping Flutter Theodorsen s Theory applied to a flat plate Flutter Derivatives The Flutter Phenomenon Flutter in a single vibration mode vii

12 01.3. Wind Tunnel Testing Types of wind tunnel Wind tunnel composition Scale Models Bridge Scale Models Complete Aeroelastic Models Sectional Models Tant Tube Models Chapter 03 Analytical Methods Approximated Formulas The Eurocode (EN) Vortex Shedding Galloping Flutter in a single vibration mode Flutter in a single torsional mode - the CECM formula Flutter in a torsional and bending mode - The Selberg formula Approximated Models (Multimodal Flutter Analysis) Modal Analysis Bidimensional Approach Non-dimensional simplification Computer Models Chapter 04 Implementation of the Flutter Stability Methods for Practical Cases Cable-stayed Bridge of the highway in Funchal Cross Section Parameters The EN formula The CECM formula for torsional flutter The CECM formula for classical flutter Bidimensional Models viii

13 03.2. The Third crossing over the Tagus River Cross Section Parameters The EN formula The CECM formula for torsional flutter The CECM formula for classical flutter Bidimensional Models Result analysis Cable-stayed Bridge of the highway in Funchal The Third crossing over the Tagus River Chapter 05 Conclusions and References Conclusions References Chapter 06 Annexes Annex A. Table with Strouhal numbers for different cross-sections from the Eurocode Annex B. Auxiliary elements, from the Eurocode, for the use of Approach 1 for calculating the maximum amplitude of Vortex Shedding instability Annex C. Factor of Galloping instability provided by the Eurocode Annex D. Derivative of the moment force coefficient in order of the angle of attack,, provided by the Eurocode Annex E. Factor for the CECM formula for torsional flutter Annex F. Factor for the Selberg formula provided by the CECM Annex G. Values of the Theodorsen Function ( ) Annex H. Flutter derivatives for an aerofoil shape ix

14 This page left blank intentionally. x

15 List of Figures Figure Asymmetrical torsional mode of the Tacoma Narrows Bridge...03 Source: Jurado, J. A. [2011] Figure Deck cross-section of the Golden Gate Bridge before (left) and after (right) reinforcement...03 Source: Jurado, J. A. [2011] Figure 02.1 Aeroelastic forces...09 Source: Mendes P. [1997] Figure 02.2 A schematic of the shedding of vortices when the wind flow goes around a cylindrical body...14 Source: Selvam, R. P. [2001] Figure Qualitative trend of vortex shedding frequency with wind velocity during Lock-In...14 Source: Selvam, R. P. [2001] Figure 02.4 Two bodies with different behaviour for vertical oscillations: one with stable geometry (above) and one with instable geometry (below)...17 Source: Mendes, P. [1995b] Figure Displacement of an aerodynamic shape...17 Source: Jurado, J. A. [2011] Figure Movements considered for a flat plate...18 Source: Jurado, J. A. [2011] Figure Real and imaginary parts of the Theodorsen function...19 Source: Jurado, J. A. [2011] Figure Schematic picture of a Close-Return tunnel...26 Source: Figure Illustration of the dimensions used for the first criteria for the verification of flutter instability...35 Source: Portuguese Normative Figure Evolution of the imaginary part of the eigenvalues in an example by Jurado [2011] (4 vibration modes)...42 Source: Jurado, J. A. [2011] xi

16 Figure Evolution of the real part of the eigenvalues in an example by Jurado [2011] (4 vibration modes)...43 Source: Jurado, J. A. [2011] Figure Cable-stayed viaduct of the highway in Funchal...52 Source: Reis, A. [2014] Figure Bridge spans and both proposed deck cross sections: reinforced concrete (on the left) and composite box girder (on the right)...52 Source: Reis, A. [2014] Figure Force coefficients of the bridge in Funchal in function of the angle of attack...54 Source: Reis, A. [2014] Figure Evolution of the value of with the wind speed for the reinforced concrete solution...57 Source: Author Figure Evolution of the value of with the wind speed for the composite box girder solution...57 Source: Author Figure Evolution of the damping with the wind speed for the reinforced concrete solution...58 Source: Author Figure Evolution of the damping with the wind speed for the composite box girder solution...58 Source: Author Figure The Third crossing over the Tagus River...59 Source: Rave [2009] Figure Deck cross section for the TTT...59 Source: Rave [2009] Figure Force coefficients of the TTT Bridge in order of the angle of attack...61 Source: Rave [2009] Figure Evolution of the value of with the wind speed for the TTT...64 Source: Author Figure Evolution of the damping with the wind speed for the TTT...64 Source: Author xii

17 List of Tables Table Comparison of various aeroelastic studies on the Great Belt Bridge...38 Source: Adapted from Jurado, J. A. [2011] Table Cross section parameters for both designs of the bridge in Funchal...53 Source: Author Table Fundamental linear and circular frequencies for both designs of the bridge in Funchal...53 Source: Author Table 04.3 Stiffness parameters for both designs of the bridge in Funchal...54 Source: Author Table Critical wind speed for torsional flutter (EN formula) for the bridge in Funchal...55 Source: Author Table Critical wind speed for torsional flutter (CECM formula) for the bridge in Funchal...56 Source: Author Table Ratios for both solutions...56 Source: Author Table Critical wind speed for classical flutter (Selberg and CECM formula) for the bridge in Funchal...56 Source: Author Table Critical wind speed for classical flutter (Bidimensional Model) for the bridge in Funchal..57 Source: Author Table Cross section parameters for the TTT...60 Source: Author Table Fundamental linear and circular frequencies for the TTT...60 Source: Author Table Stiffness parameters for the TTT...61 Source: Author Table Critical wind speed for torsional flutter (EN formula) for the TTT...62 Source: Author Table Critical wind speed for torsional flutter (CECM formula) for the TTT...63 Source: Author xiii

18 Table Ratios for the TTT...63 Source: Author Table Critical wind speed for classical flutter (Selberg formula) for the TTT...63 Source: Author Table Critical wind speed for classical flutter (Bidimensional Model) for the TTT...64 Source: Author Table Critical wind speed for flutter for both cross sections of the bridge in Funchal...65 Source: Author Table Critical wind speed for flutter for the Third crossing over the Tagus River...67 Source: Author xiv

19 List of Symbols,, drag force, lift force and pitching moment air density average wind speed angle of attack of the wind flow,, drag force coefficient, lift force coefficient and moment force coefficient deck width, deck s mass and mass moment of inertia per unit length radius of gyration of the deck cross section,, horizontal displacement, vertical displacement and rotation of the deck cross section,, mass, damping and stiffness matrices of the deck cross section per unit length,, damping coefficients for horizontal, vertical and rotation vibrations,, logarithmic decrease for horizontal, vertical and rotation vibrations, aerodynamic damping and stiffness matrices,, aerodynamic damping coefficients for horizontal, vertical and rotation vibrations,, circular frequency of vibration for horizontal, vertical and rotation modes reduced frequency,, with 1,2,,6 flutter derivatives Reynolds Number Strouhal Number Scrouton Number xv

20 This page left blank intentionally. xvi

21 Chapter 01 Introduction The collapse of the Tacoma Narrows Bridge opened a new chapter in the design of suspension and cable-stayed bridges with great spans. Due to the increased slenderness of the deck s cross-section, seismic and wind loads have become much more relevant than the loads produced by general traffic or even the structural weight. 01

22 01.1. Introduction With the increasing span length of cable-stayed bridges and suspension bridges, the effects of the service loads produced by roadway and railway traffic on the structure started to become less important than seismic and wind loads. The study of the effects of wind flow on these types of structures was born with the collapse of the Tacoma Narrows Bridge [1940]. In the field of Civil Engineering, very little of these effects was known. However, in Aeronautical Engineering the theories of Fluid Dynamics and Deformable Solid Mechanics were already being combined to study the dynamic behaviour of the shapes adopted for aircraft wings submitted to wind flow loads The Tacoma Effect On the 1 st of July of 1940, the construction of the Tacoma Narrows Bridge had finished and it was open to the public. This bridge, situated in the state of Washington, just east of the city of Tahoma, was designed using top of the art design technology to withstand winds of 161 / and a static horizontal pressure of winds 146 /. (Scott, R. [2001]) However, later that same year, the bridge collapsed under winds of half that speed and about one sixth of that wind pressure. The Tacoma Narrows started showing irregular behaviour during its construction phase. Several workers complained about nausea and fits of vomiting due to the decks movements and, after its conclusion, these oscillations continued to occur. The bridge displayed a symmetrical bending vibration mode with amplitudes reaching 1,5 until one morning, after some cable bands supposedly had broken near the centre of the main span, the oscillatory movement changed to an asymmetrical torsional mode. This originated a series of reactions which eventually lead to the collapse of Galloping Gertie, a nickname given to the bridge due to its behaviour. With the collapse of the Tacoma Narrows Bridge, Civil Engineering entered a new period of bridge design. Even though the dynamic behaviour observed in this bridge was recognized as similar to what happens to aerofoils, aeroelasticity had never been applied to the field of Civil Engineering. Because of this, several of the bridges designed after this event were excessively over dimensioned and even some built prior to it were greatly reinforced. Such is the case of the Golden Gate Bridge in San Francisco, finished in 1937, and later reinforced, in 1955, with a new trussed surface (Figure 01.2). Nevertheless, this event also contributed to the introduction of aeroelasticity applied to bridge design. To this day, several of the theories applied in Aeronautical Engineering have already been adapted to bridges and, even though the computation is still somewhat arduous, hybrid theories have been 02

23 Figure Asymmetrical torsional mode of the Tacoma Narrows Bridge proven to yield fairly approximate results when compared to fully experimental testing. These theories are mostly analytical but are based on experimental results. Figure Deck cross-section of the Golden Gate Bridge before (left) and after (right) reinforcement Objectives The main objective of this dissertation is the study of aeroelastic instability phenomena of bridge decks. Within this main thematic, it is necessary to understand which instability phenomena are prone to occur and what conditions trigger them. It is also important to determine which tools are available nowadays to ensure the design of a bridge is considered safe to these phenomena. The following scheme details all the objectives this dissertation proposes to achieve: 03

24 Study Object - The aerodynamic stability of suspension and cable-stayed bridge decks. General Objectives - Study the behaviour of suspension and cable-stayed bridge decks when immerse in a wind flow; - Analyse the tools available to verify the stability of suspension and cable-stayed bridge decks. Specific Objectives - Understand the instability phenomena prone to occur on these types of structures and their importance in the process of designing them; - Determine what kinds of efforts have been made to better model the behaviour of these structures when subjected to a wind flow; - Analyse how experimental work can help with the computation of methods adopted to predict the behaviour of the structures; - Verify how the normative, essentially the Eurocode, recommends the safety verification is made for these phenomena; - Apply the methods to find the critical wind speed of flutter, determined in the second point, to practical cases Structure and Methodology In order to achieve the proposed objectives, this dissertation is divided into five main chapters. The first chapter, the present one, is the Introduction. Within this chapter it is possible to encounter a small introduction to the presented thematic as well as the proposed objectives, the structure of the dissertation and the methodologies adopted for its completion. The second chapter is Bridge Aeroelasticity - Instability Phenomena and Wind Tunnel Testing. This is a theoretical chapter. It starts by introducing aeroelasticity applied to bridge decks and evolving to the types of aerostatic and aerodynamic phenomena that are prone to occur on suspension and cablestayed bridges with great main spans. Finally, there is also a section describing wind tunnel testing, an important part of the hybrid theories adopted to predict these phenomena. Its execution was done by mainly collecting information from several sources and compiling them. 04

25 The third chapter is Analytical Methods. This is a mostly theoretical chapter with practical appliance. It replicates some of the formulas and models adopted to predict the phenomena explained in the second chapter. This is done by studying the Eurocode and other references and further developing some of the existing models. The fourth chapter is a practical chapter. Its title is Implementation of the Flutter Stability Methods for Practical Cases and it consists of the application of the methods presented in the third chapter for finding the critical wind speed of flutter to some bridge design cases. This is done through the compilation of an algorithm to help solve iterative models. The fifth, and final chapter, represents the Conclusions and References. This chapter summarizes the acquired knowledge from the previous chapters to achieve the objectives proposed in section

26 This page left blank intentionally. 06

27 Chapter 02 Bridge Aeroelasticity - Instability Phenomena and Wind Tunnel Testing In order to adequately design a bridge which is acted upon by a wind flow, one shall first learn how the structure would react to it, what phenomena would it be subjected to and what kind of testing could be done to assist in the computation. 07

28 02.1. Analytical Aeroelasticity Before understanding the instability phenomena, it is first necessary to know how a bridge deck reacts to wind flow loads. This is called aeroelastic behaviour. The basis of this was first studied, as stated before, by the Aeronautical branch of Engineering. In this area, the effects of wind flow on aerofoils are analysed to better develop airplanes wings. The Theodorsen s theory for flat plates works with this type of aerodynamic shapes and it can be considered the inspiration behind later formulations made in the field of cable-supported bridges [1]. (Jurado J.A. [2011]) Aeroelastic Forces If a fixed section of a bridge deck is inserted in a uniform wind flow with average speed and angle of attack, three kinds of aerodynamic forces may be considered. These forces, expressed per unit length, are Drag, acting along the direction of flow, Lift, acting perpendicular to it, and, if the centre of pressure does not coincide with the centre of rotation (which is the most common case), a Torsional Moment (Figure 02.1). These forces can be expressed by (Jurado [2011]): 1 2 (01a) 1 2 (01b) 1 2 (01c) The variables and are the mass density of the fluid and the deck width, respectively, and the coefficients, and can be called form or force coefficients. These coefficients are obtained for several angles of attack through wind tunnel testing. The transformation to the horizontal and vertical axis of these forces can be made by (Mendes [1995b]): cos sin sin cos (02a) (02b) [1] Cable-supported bridges suspension and cable-stayed bridges 08

29 Figure 02.1 Aeroelastic forces However, in this reference frame, and can also be rewritten in terms of their own form coefficient (Mendes [1995b]). 1 2 (03a) 1 2 (03b) with: 1 cos cos sin (04a) 1 cos sin cos (04b) Because the origin does not undergo any type of displacement, the pitching moment does not change Movement equations A deck cross-section immersed in a turbulent wind flow, with horizontal () and vertical () speed fluctuations, is bound to start vibrating. These vibrations cause displacements along the x and y axis as well as the rotation of the deck,,. These displacements, along with their respective speeds,,, cause variations of the effective wind speed and its angle of attack. 09

30 (05a) (05b) However, this is only true if the following hypotheses are verified (Mendes P. [1995b]): 1) The aerodynamic behaviour of the structure can be considered quasi-stationary; 2) The wind speed fluctuations are small when compared to the average wind speed; 3) The effects of the wind speed fluctuations along the deck s axial direction can be ignored; 4) The induced forces in each section are only dependent on the wind flow acted upon them. Using equations (03) and (05), it is possible to represent the variation of the aerodynamic forces as (Jurado [2011]): 1 2 d d 1 2 d d 1 2 d d (06a) 1 2 d d 1 2 d d 1 2 d d (06b) 1 2 d d 1 2 d d 1 2 d d (06c) If the forces and displacement vectors, and respectively, are defined by: (07a) (07b) it is possible to rewrite equations (06) in their matrix form: (08) The matrix has the parcels related to the wind turbulence and is not defined here. and are the aerodynamic damping and stiffness, respectively, and their components are (Jurado [2011]): 10

31 1 2 d d 1 2 d d 1 2 d d d d d d d d (09) All the non-diagonal components of these two matrices correspond to coupling aerodynamic forces. Including equation (08) in the equation that defines the oscillatory movement of a body subjected to an external force the following result is obtained (Mendes [1995b]): (10) where, and are the mass, damping and stiffness matrices per unit length. Moving these parcels on the right to the left side: (11) which means the forces exerted by the wind flow on the structure modify the global damping and stiffness matrices. Considering each degree of freedom separately [2], it is evident the global damping [3] is positive for every value of average wind speed and angle of attack for oscillations along the horizontal axis (Mendes [1995b]). 2 (12) In this case, is the natural frequency of a generic oscillation of the deck along the horizontal plane. The second term, related to aerodynamic damping, can be rewritten with the same form as the first one. 2 2, 2, (13) If the deck has the same geometry and mass along its whole span then:, 2 (14) [2] All aerodynamic coupling forces are null which implies every non-diagonal component of the aerodynamic damping and stiffness matrices are also null. [3] Global damping = Structural damping + Aerodynamic damping 11

32 For equation (14), it is noted the only way aerodynamic damping may become negative is if 0. Returning to equations (04a) it is possible to acknowledge that with a small angle of attack, the contribution of the Drag force coefficient, which is always positive, is bigger than the Lift force coefficient. However, even if the latter one has a negative value, it is multiplied by a factor 1 which means is always greater than 0. Hence, the vibrations along the direction of the acting wind can never reach an instability point. It is also important to note this damping is proportional to the average wind speed and inversely proportional to the mass per length unit and the vibration mode frequency which mean that for higher vibration modes the aerodynamic damping decreases in value. However, this cannot be confirmed for vibrations along the vertical axis, leading to a phenomenon known as Galloping, which is explained further ahead. Finally, the forces induced by the wind speed fluctuations of a turbulent flow are directly proportional to the force coefficients of the deck, for horizontal fluctuations, and to its respective derivative in order of the angle of attack, for vertical ones. These forces, on the right side on equation (11), have no contribution for the instability phenomena Instability Phenomena Aerostatic Instability Static Divergence Static divergence is one of the most relevant aerostatic instability phenomena. However, its importance within the global aeroelastic instability phenomena is minor and, therefore, many of the European codes don t require its verification if the bridge safety is guaranteed for aerodynamic phenomena. The occurrence of static divergence is triggered when the torsional parameter of the stiffness matrix is nulled by the effect of wind on the structure. This happens for high wind speeds relative to other aerodynamic instability phenomena. Taking the torsional equation for the oscillatory movement of the deck cross-section subjected to the wind flow: 0with 0 (15) and by using the stiffness parameters and the relation, with being the radius of gyration: 12

33 1 2 d d 0 (16) This equation can be rewritten so that both terms have the same general form. In this way it is easier to evaluate the occurrence of this phenomenon. (17) refers to an aerodynamic radius of gyration which modifies the global one. This radius can be written as: d d 2 (18) By looking at equation (17) it is possible to infer that, for static divergence to occur, the derivative in order to the angle of attack of the pitching moment force coefficient has to have a big enough positive value. This leads to the following necessary (but not sufficient) condition for the occurrence of static divergence: 0 d d 0 (19) Aerodynamic Instability Vortex Shedding Considering a body immersed in a uniform wind flow, it is easy to see this body creates a local disturbance in it - there is a separation of the flow that contours the body. This produces a pressure force, due to the encounter with the obstruction, on the upwind side and a suction force, due to the detachment of the flow, on the downwind side. This force results in the creation of vortices, with alternating rotations, in the wake region, causing the body to vibrate. The shed vortices are convected downwind by local wind speed and viscous diffusion but will also interact to form large-scale structures ( Selvam R. P. [2001]). The response of the structure depends on the frequency of vortex shedding. This frequency rises with the average wind speed. If vortex shedding frequency (or wake frequency) is below the structural natural frequency, the deck behaves as if rigidly fixed. However, if the shedding frequency reaches the natural frequency, the cross-section starts vibrating with increased amplitude. Although this amplitude can be high enough to cause discomfort to pedestrians and even enhance material fatigue, but it is not 13

34 Figure 02.2 A schematic of the shedding of vortices when the wind flow goes around a cylindrical body high enough ( rarely exceeding half of the across wind dimensions of the body (Selvam R. P. [2001])) to provoke the collapse of the bridge. Also, it is possible to control this amplitude of vibration by raising the damping of the structure. If the wind speed continues to rise, the wake frequency remains locked to the structural natural frequency for a restricted range of wind speeds, eventually breaking away, making the structure to behave, once again, as if rigidly fixed. This phenomenon is denominated Lock- In. Figure Qualitative trend of vortex shedding frequency with wind velocity during Lock-In. There is a tight relation between the dimension of the trail of vortices and the Reynolds number. (20) The variables, and are properties of the wind flow (density of the fluid, average speed relative to the immerse body and dynamic viscosity of the fluid, respectively) and is the diameter of the body. 14

35 According to Simiu and Scanlan [1986], if the Reynolds number is low enough, the wind flow just circumvents the body immersed in it. As the Reynolds number rises, there is a detachment around the edges and vortices start to appear in the immediate wake. A further increase in the Reynolds number leads to the creation of cyclically alternating eddies which are carried over with the flow downstream. After a certain point, the inertial effects become dominant over the viscous effects and the turbulence sets in, resulting in shear of the flow Galloping Galloping is a type of oscillating divergent phenomenon with vertical displacements with low frequencies. This type of phenomenon usually happens for wind speeds greater than the ones that cause flutter. This instability happens when aerodynamic damping reaches the same absolute value as structural damping for a vertical vibration mode, making the global damping null for this mode. This creates growing oscillation amplitudes leading to the eventual collapse of the structure. 0with 0 (21) Developing the second part of equation (21) (Mendes [1995b]) it is obtained: d d 0 (22) This equation can also be rewritten with the aerodynamic damping coefficient, the same way it was done with the horizontal vibrations. 2 2, 2, (23) is once again the natural frequency of a generic vibration mode, although this time it is a vertical one. The aerodynamic coefficient, for a bridge deck with the same mass and geometry along its length, can be written as:, d d 2 (24) This shows that global damping for vertical oscillations can only be negative, resulting in this divergent phenomenon, if the derivative of is positive in order of the angle of attack and leads to a large enough aerodynamic coefficient. 15

36 For small amplitudes of oscillations, resulting in a near zero angle of attack: d d d d (25) which leads to the Glauert-den Hartog criteria that states the following necessary (but not sufficient) condition for the occurrence of galloping., 0 d d 0 d d 0 (26) In conclusion, it is advised to avoid deck sections with a great negative derivative of the Lift force coefficient in order of the attack angle. In these types of sections, if the angle of attack rises, leading to downwards vertical velocities, a Lift force with the same direction is induced on the deck. These forces tend to help the movement instead of damping it, leading to this divergent behaviour called galloping. (Mendes P. [1995b]) Figure 02.4 shows two bodies with different behaviour related to this kind of input. The first one has a stable geometry, which leads to a damped oscillatory behaviour. The second one has an instable geometry, which leads to growing oscillatory amplitudes with every cycle leading to an eventual collapse Flutter In the design of bridges with great span, flutter is the most relevant type of aerodynamic instability. This phenomenon is characterized by a divergent oscillatory movement of the bridge deck for increasing values of wind speed. Flutter can be multimodal, also called classic flutter, or it can be in a single vibration mode. The first one, often being associated with aerofoils, is being studied since the 1930s by the aeronautical industry. The study of flutter on bridge decks started with the collapse of the Tacoma Narrows Bridge. Since then, some of the theories applied in the aeronautical industry have been adapted and further developed to better (and easier) understand the deck s behaviour. 16

37 Figure 02.4 Two bodies with different behavior for vertical oscillations: one with stable geometry (above) and one with instable geometry (below) Theodorsen s Theory applied to a flat plate The original Theodorsen s Theory is applied to an aerodynamic shape with cord length 2 which undergoes a horizontal displacement at speed and arrives at a generic position after a certain time has passed. Figure Displacement of an aerodynamic shape. 17

38 However, this theory can be adapted for a flat plate with length of 2 acted by a wind flow of average speed. This body undergoes a generic vertical displacement and a pitching rotation around a localized axis at a generic distance from the centre of the plate. Figure Movements considered for a flat plate. It is assumed that the plate movements are oscillatory in nature and can be expressed as and. The variables and represent the amplitude and phase angle of the both movements, respectively, and are complex in nature and represents the circular frequency. According to Theodorsen s Theory, the aeroelastic forces along this type of plate, using the non-dimensionalized parameters as a time variable and as a reduced frequency response, are lift and moment and can be defined as (Jurado [2011]): (27a) (27b) These forces are expressed per unit of length. The parameter is the air density and refers to the Theodorsen function ( ). It is important to denote that is the imaginary unit of reference ( 1) and is not to be confused with, the radius of gyration ( ), which is used further ahead. 18

39 Figure Real and imaginary parts of the Theodorsen function. This method can be adopted to obtain approximate results when applied to suspension or cablestayed bridge decks. However, because the geometry of those decks is not considered an aerodynamic shape, a complete analytical formulation is impossible. (Jurado J.A. [2011]) Flutter Derivatives The Scanlan Model - two degrees of freedom The geometry of bridge deck sections makes it impossible to use Theodorsen s theory for flat plates to analytically determine their oscillatory movement. Because of this, aeroelastic forces on these types of sections have a tendency to be expressed as a linear function of the same two degrees of freedom considered, affected by coefficients known as flutter derivatives. This formulation was first proposed by Scanlan[1969]. Flutter derivatives cannot be obtained analytically and have to be obtained through wind tunnel experimenting. This type of testing is further developed in another chapter. The equations that define these aeroelastic forces are (Jurado [2011]): 1 2 (28a) 1 2 (28b) 19

40 The parameters and are called flutter derivatives. It is also noted that the term related to length is now defined as the total length of the section ( 2) and, consequently, the reduced frequency is now twice the value adopted for the flat plate ( 2 2). Additionally, the coefficients and relate to vertical and torsional oscillation damping, respectively, whereas the coefficients and relate to the stiffness of those same oscillatory movements. Terms,, and are coupling coefficients between the two degrees of freedom. It is possible to obtain the exact flutter derivatives of the flat plate by transforming equations (27), expressing them according to the new variables, and comparing with equations (28) (29a) (29b) For plates with symmetrical section ( 0) the results are (Jurado [2011]): (30) The functions, and continue to be in order of 2. The altered sign of the variable can be explained by the change in direction of both displacements (vertical displacement and pitching rotation) from the Theodorsen s theory to the Scanlan model as shown in Figure

41 Aeroelastic Forces in Three Degrees of Freedom When adding a third degree of freedom to the model, the horizontal displacement following the direction of the wind appears. The associated aeroelastic force is called drag and it is here represented by the letter. This horizontal displacement is important for the model s accuracy because these long span structures are usually very flexible in this direction and the first natural vibration mode of suspension bridges is often in the transverse direction. In addition, equations (28) now have two extra parameters which relate lift and moment, respectively, to the horizontal displacement and speed (Jurado [2011]). 1 2 (31a) 1 2 (31b) 1 2 (31c) These flutter derivatives are all functions of the reduced frequency. There are total of 18 flutter derivatives, 6 for each force., and multiply the displacement, which means they are stiffness parameters, whereas, and multiply the speeds, and are, therefore, related to damping. All other 12 parameters are related to coupling forces and illustrate the non-conservative nature of the aerodynamic forces The Flutter Phenomenon It is also possible to group the three aeroelastic equations into a single matrix equation the same way it was done before for Galloping. (32) and are the aerodynamic damping and stiffness, respectively: (33a) 21

42 (33b) and and are defined as: (34) Following the same steps of equations (10) and (11): (35a) (35b) It is possible to see, in equations (33), that none of the terms has a direct correlation with the time variable. This means the aerodynamic instability does not have the same nature as the classic behaviour of resonance induced by an external force. The occurrence of flutter, whether classic or in a single vibration mode, happens when the parameter reaches zero. This can happen for multiple vibration modes at the same time, resulting in the first kind of flutter, or just for the torsional mode, resulting in the second kind Flutter in a single vibration mode Considering a deck cross-section acted upon the above aeroelastic forces, if each degree of freedom is considered separately, the movement equations for each of them become: (36a) (36b) (36c) If all members on the right side are moved to the left side, it is possible to see the stiffness and damping of the system is affected. Several of the phenomena that can occur due to these modifications in the properties of the system have already been attended. Such is the case of Static Divergence, which is the annulment of the global stiffness, and Galloping, which is the annulment of the global damping in the heave vibration mode. The horizontal one can also be left apart because it has already been 22

43 showed that there is no instability in this degree of freedom as well. All that is left is the torsional vibration mode: 2 0 (37) The occurrence of flutter in this vibration mode corresponds to a null or negative global damping value: 2 0 (38) which means an aerodynamic damping equal or greater that the structural damping: 2 (39) Wind Tunnel Testing As stated before, the usage of completely analytical methods to study the stability of cable supported bridge decks is impossible due to the geometry of their cross-section. Because of this, experimental methods like wind tunnel testing should be done to complement this analysis. These methods help study the global bridge stability preventing instability phenomena from happening. Also, by simulating the effect of the wind flow around the structure, it is possible to evaluate the effects of these types of actions on it. One example of these effects is the aeroelastic forces that arise within the structure. The usage of wind tunnel testing helps determine the force coefficients, and, defined in section , and the flutter derivatives, both in order to the angle of attack and mean wind speed. However, there are several types of wind tunnels employed for several types of reduced models. These experiments have different goals associated to each of them. In this section, the types of wind tunnels are listed as well as some of the principles of the Similarity Theory adopted in the construction of reduced models. 23

44 Types of wind tunnel Wind tunnels can be classified within four major categories. These categories are: 1) Type of circuit; 2) Type of flow; 3) Position of the fan; 4) Tunnel purpose. Within the first category, if the air is expelled to the atmosphere after passing through the test section, it is called an Open-Return Tunnel, whereas, if the air is forced to recirculate afterwards, it is called a Closed-Return Tunnel. The construction Open-Return Tunnels is simpler and less expensive than its counterpart. However, the amount of power needed for the same outflow is bigger, leading to greater energy consumption and noisier atmosphere. This means that, even though the initial cost may be lower, the maintenance cost is significantly higher. Also, because of its opening, the temperature and humidity are harder to regulate. The construction area required for these structures is longer but substantially narrower. Based on the type of wind flow, tunnel can be considered Laminar Boundary-Layer Tunnels or Turbulent Boundary-Layer Tunnels. As the name implies, a Laminar Boundary-Layer Tunnel expels a laminar wind flow in to its test section whereas a Turbulent one expels a turbulent flow. If the fan is positioned downwind from the test section, the tunnel is called a Suckdown Tunnel. These types of tunnel are not ideal for testing because the incoming air enters with significant turbulence. However, its counterpart, the Blower Tunnel, also has a small amount of turbulence. In this case, the fan is positioned upwind of the test section. The turbulent flow can be controlled, as is explained later, by proper devices placed before the test section. Finally, if the wind tunnel is vast and can simulate the atmospheric surroundings of the whole structure, it is called a Boundary-Layer Tunnel. In these types of tunnel, the three-dimensional effects of the wind flow on the structure can be evaluated. A complete model of the bridge, as well as a map of its immediate surroundings, is implemented for this type of test. If, however, the tunnel is small and only used to study the behaviour of the deck s cross-section under wind action, it is called an Aerodynamic Wind Tunnel. These are the types of tunnels employed to obtain the force coefficients and flutter derivatives stated above Wind tunnel composition Wind tunnels, whether they are Open-Return or Closed-Return, have many components in common. These components are the fan, the settling chamber, the contraction cone, the test section and the diffuser. 24

45 The fan is the element that produces the wind flow deployed for the tests. In open return tunnels, it can be positioned on the leeward side of the test section, sucking the air into the tunnel (Suckdown Tunnels), or it can be positioned on the windward side, blowing the air into the tunnel (Blower Tunnels). The fan needs to be connected to an electric engine with the appropriate power to function correctly. Before entering the test section, the air flow goes through the settling chamber. In here turbulence is partially reduced by screens placed within. This leads to a more uniform flow of wind which results in more reliable results. After leaving the settling chamber, the flow still has to go through the contraction cone. This element has a smaller cross-section at its aimed to increase the speed of the flow. This reduces the turbulence even further. However, special care has to be taken when building this element because the variation of the cross-section cannot lead to the detachment of the flow from the walls. Therefore, the variation has to decrease in value downstream. The test section is where the actual testing occurs. This chamber has to be isolated from the contraction cone and the diffusor, through the existence of joints, to avoid the transmission of mechanical vibrations that could influence the experiments. An extra screen can be introduced within the first joint to produce some turbulence, in case it is required for the trials. There are also glass windows on the sides to allow direct observation. If this chamber is too long, the ceiling should be able to be adapted. This is because the attrition between the air flow and the section walls increases the width of boundary-layers downwind. Adjusting the ceiling to have a wider cross-section at the end cancels the reduction of the flow section and subsequent increase in speed, which would lead to a higher drag force. The diffuser facilitates the flow transition after it has exited the test section. In Suckdown Tunnels, it is positioned right before the fan, whereas in Blower Tunnels, it is the last element the air goes through before being expelled to the atmosphere. In closed return tunnels, the diffuser works as a transitioning element between the test chamber and the fan, where the flow cycles restarts. In Closed-Return tunnel, there are more elements called vanes. These are placed on the corners and their function is to change the direction of the flow. These tunnels also need to have joints to keep the whole chamber at the atmospheric pressure. A schematic picture of a Close-Return tunnel can be seen on Figure

46 Figure Schematic picture of a Close-Return tunnel Scale Models Scale models adopted in wind tunnel testing are built according to the purpose of the experiment. These models can be rigid, aeroelastic or rigid elastically supported. Rigid models are to be adopted in static trials. The main aim of these trials is to find the aeroelastic pressures along the contour of the tested structure and the global forces acted upon said structure. These trials can also be employed to test the effects of the oscillatory movement on pedestrian comfort and for pollution studies. These models can be complete or partial. Aeroelastic models are adopted for dynamic testing. These are employed to study the behaviour of the prototype when subjected to a wind flow. They can be exact replicas of the actual prototype or be built from simple structural systems to match its dynamic characteristics with the real structure. The first ones can also be developed to determine the natural frequencies and vibration modes. Lastly, the rigid elastically supported models can also be adopted in dynamic tests. Well-known examples of these are the sectional models of bridges employed to study their stability. 26

47 Bridge Scale Models Bridge modelling can have different purposes. Therefore, several of the models stated before can be adopted for this kind of testing. As an example, the study of vortex shedding and the influence of atmospheric turbulence on the deck have tridimensional effects. However, the oscillatory movement caused by the interaction between the flow and the deck movement can be considered in a bidimensional model. This means that different models can and should be employed to study these phenomena, adapted to the results that want to be obtained Complete Aeroelastic Models As the name suggests, these models replicate the whole prototype, including not only the deck but the cables and towers as well, to match a set of geometric and dynamic properties. Their geometric scale is determined based on similarity conditions involving the atmospheric turbulence that can be produced by the tunnel. This turbulence is influenced by the height of the testing section, which determines the height of the boundary-layer, and by the overall terrain roughness. Although the Reynolds similarity can be overlooked, special care should be taken with the cables diameter to rectify the resulting drag force. This is because the violation of the Reynolds similarity has more effect on round bodies than edged bodies. The construction of these models is made out of wood. Afterwards, metallic bars can be added to the interior to correct weight or elastic properties differences. The cables stiffness is assured through strained springs which can also be exploited to further correct some weight relative deviations. In addition, the surrounding terrain can be built and if it has any relevant obstacles which could influence the flow of wind, these can also be built in case the testing section is big enough. Other obstacles can also be built, with variable sizes and shapes, with the sole purposed of creating various types of turbulence. The goal of testing these models is to ascertain the effect of turbulent boundary-layer wind flows on the deck. To simulate the effect of various angles of attack of the wind, the model can be placed on top of a rotating platform which turns according to the pretended angle. According to the Similarity Theory, there are equations that relate the elastic behaviour of model with the prototype. These equations are essential to translate the results obtained, which are related to the scale model, to the actual prototype. For the deck, there are two equations, one for the bending stiffness (55a) and one for the torsional stiffness (55b), and for the cables there is one, for the axial stiffness (55c). 27

48 (40a) (40b) (40c) The and indexes represent the model and the prototype variables, respectively. and are the bending and torsional stiffness of the bridge deck, respectively, and is the axial stiffness of the cables. The variables and have already been defined as the volumetric mass density of the fluid (atmospheric air) and the mean velocity of the wind flow. The measurements and are the deck width in the direction of the flow and the diameter of the supporting cables. In addition, there are also two equations that represent the similarity related to the inertia effect. These equations are only relevant for the bridge deck. (41a) (41b) and are the mass and the mass moment of inertia of the deck, respectively Sectional Models Unlike the previous ones, these models only reproduce a limited extension of the whole deck. This is done with a rigid body which is then supported by suspension mechanisms which simulate the dynamic properties such as the natural vibration frequencies of the first bending and torsional modes. Even though testing can be cheaper, the usage of these models has to be done with special care. By using just a small portion of the whole structure, a series of assumptions are taken into account which leads to a series of limitations. These assumptions are: 1) The influence of the towers and cables on the overall aeroelastic force is smaller enough than the deck s influence and is, therefore, irrelevant. Even though this can be the case for situations, in the construction phase of suspension bridges the influence of the cable parcel for this force can become relevant. 28

49 2) The aerodynamic force in each section of the deck is considered a linear function of the displacement and velocity of that same section. This means the force is rule by an equation with the following form. 1 2 (42) and are generic variables that multiply the displacement and the velocity, respectively. This was shown to be the case for pre-flutter behaviour. However, after reaching flutter, the oscillatory behaviour cannot be considered to follow this type of behaviour. In addition, if the flow has a sufficiently elevated level of turbulence, the assumption becomes incorrect as well. For these reasons, testing with sectional models is often employed as an auxiliary tool for analytical studies of the behaviour of bridges under the influence of wind flow. Moreover, these experiments need to be done with the lowest turbulence possible. This results in more conservative results because the presence of atmospheric turbulence can help the stability of the deck, mobilizing all vibration modes making it more difficult for instability to occur within a single one. Also, the width of bridge decks is usually sufficiently inferior to the typical values of the turbulence scale and the length is sufficiently superior to the typical values of the lateral turbulence scale. The consequence of these two details is the possibility of using a quasi-stationary model for the stability analysis and the rapid loss of correlation along the span between the speed fluctuations. The attainment of the aerodynamic forces can done through the direct measurement of the forces transmitted to the suspension springs or by integration of the aerodynamic pressures existent on the contour of the cross-section Taut Tube Models Taut Tube Models are built from several rigid geometric replicas of the prototype united by one strained string on each side. These rigid segments, which do not contribute to the stiffness of the model, are used only for their geometry and mass. The transformation of the natural frequencies of some of the most relevant vibration modes from the prototype to the scale model is done by adjusting the strain of the string, its length and the spacing [4]. The transformation formulas, one for bending modes and another for torsional modes, for vibration mode are: [4] This nomenclature was used by Mendes [1995a] 29

50 , 2 (43a), 2 4 (43b) The terms and present in these equations refer to the bending and torsional stiffness of the prototype, respectively. These parcels can be ignored due to their irrelevance when compared to the parcel containing the force. In these equations, is the number of the vibration mode and is not to be confused with the radius of gyration (represented by the same symbol) or the imaginary unit. When subjected to the wind flow, the model oscillates in a sinusoidal half wave. This vibration mode is closer to the real natural vibration modes of great-span bridges, unlike the one seen in sectional models. In addition, it is possible to change the horizontal angle of attack and to simulate tridimensional responses from interaction with the wind flow. Much like the other types of scale models, it is possible to place a body upwind from the model, with variable sizes and shapes, to create the desired level of turbulence. 30

51 Chapter 03 Analytical Methods After studying the phenomena a bridge can be subjected to when immerse in a wind flow, its stability need to be analysed properly. This can be done through a series of procedures which produce slightly different results based on the approximations and assumptions made a priori. 31

52 03.1. Approximated Formulas The Eurocode (EN) Vortex Shedding According to Annex E: Vortex shedding and aeroelastic instabilities of the EN , the effect of vortex shedding on a structure should be investigated when the ratio of the largest to the smallest crosswind dimension of the structure, both taken in the plane perpendicular to the wind, exceeds 6. However, even if this condition is verified, the verification for vortex shedding can be avoided if the wind speed on the site of the structure is low enough. The EN defines this low enough as:, 1,25 (44) The parameter on the left (, ) is the critical wind speed for Lock-In for vibration mode, whereas the parameter of the right ( ) is defined in the normative as the characteristic 10 minutes mean wind velocity and is specified in section (1) of EN The critical wind speed for vibration mode,,, is the wind speed at which the frequency of vortex shedding equals the natural frequency for the vibration mode. This mode is characterized for being a bending vibration mode. To find this wind speed it is first necessary to define the Strouhal number. This number represents a non-dimensional frequency of the appearance of eddies [1]. (45) Here, is the reference width of the cross-section immerse in the flow (in the case of a bridge deck, it is the height ), is the average wind speed and is the frequency of shedding. As explained before, for the Lock-In phenomenon to occur, the frequency must be the same as one of the natural frequencies for a bending vibration mode. Re-arranging equation (45):, (46) This is the expression adopted in the EN to find the wind speed at which resonant vortex shedding occurs. A table with the Strouhal number for different kinds of cross-sections is also provided (Annex A). [1] Vortices. 32

53 If the structure is prone to the phenomenon of Lock-In, its effect needs to be properly considered. The vibrations induced by the shedding of eddies generate a force of inertia per unit length on the crosssection. This force acts perpendicular to the wind direction at a generic position and it is given by: 2 (47) here is the oscillating mass per unit length, and are the natural frequency and the mode shape [2] for the bending vibration mode and is the maximum displacement for that vibration mode, the amplitude of the oscillation. The EN proposes two approaches for finding this amplitude. However, Approach 2 is reserved for cantilevered structures, which means, in the case of bridges, it can only be applied for the effects during certain stages of the construction phase. For this reason, this approach is not explained in full detail. Approach 1 For the calculation of the maximum amplitude of oscillation, this method first requires the knowledge of the Strouhal number (explained before) and the Scruton number. The latter one represents the susceptibility of the object to vibrations and is represented by: 2, (48) This expression depends on the structural damping through the use of the logarithmic decrease and the air density. It also depends on a parameter called equivalent mass per unit length for vibration mode (, ) which is expressed in the EN as: is the length of the structure and all other variables have already been defined. (49) Another parameter that needs to be defined is the lateral force coefficient. This coefficient is a function of the critical wind velocity ratio,,. The denominator in this ratio is the mean wind velocity in the centre of the effective correlation length. This length is represented in section E of Annex E of the EN [2] In this case, the vibration mode needs to be normalized to 1 at the point with the maximum displacement. 33

54 The final two parameters for the calculation of are the effective correlation length factor and the mode shape factor (not to be confused with the reduced frequency ). These can be obtained through the table given in the EN. It is important to underline that is dependent on the correlation length. All tables and graphics provided in the EN for this procedure are presented in Annex B. All the parameters for calculating the maximum amplitude of oscillation have been defined and the final step of this approach is just to use the following expression:, (50) Approach 2 This approach has a much more compact final expression: (51) However, the definition of these two parameters is more arduous. The parameter is the peak factor and is the standard deviation of the displacement and it is to be calculated at the point with the largest deflection. Within this standard deviation, there are some coefficients which are hard to be determined, like the aerodynamic constant, the aerodynamic damping parameter and the normalised limiting amplitude, for non-cylindrical structures such as deck cross-sections Galloping According to Annex E: Vortex shedding and aeroelastic instabilities of the EN , the wind speed at which the structure starts to show this divergent phenomenon called galloping is defined by the following simple expression:, 2, (52) Again, the width of the body corresponds to the height in bridge decks. is the Scruton number which has already been defined in section Approach 1 and, is the natural frequency for the first cross-wind vibration mode. The only missing coefficient is. This is the factor of galloping instability and it is dependent on the configuration of the cross-section. The EN provides a table with several cross-sections and the corresponding value of the factor of galloping instability, which is presented in Annex C. However, if this 34

55 coefficient is unknown, the EN also predicts the use of 10 which produces a much lower value of the critical wind speed for galloping. If the following condition is true:, 1,25 (53) where is the mean wind velocity at the point where galloping process is expected, then the structure is considered safe for this kind of phenomenon. However, if the critical wind speed for galloping, is close enough to the critical wind speed for vortex shedding : 0,7, 1,5 (54) then it is extremely likely the interaction between these two phenomena occurs. In this case, the EN advises the use of external specialized references to verify the safety of the structure Flutter in a single vibration mode In the case of flutter, the EN provides conditions based on simple structure criteria. If these conditions are verified, the structure is considered safe for this type of phenomenon. However, if this is not the case, the EN advises the consultation of specialists. These criteria established by the EN determine if the structure is prone to flutter instability. They are to be verified in the order given and if at least one of them fails, the structure is considered as not prone to this phenomenon. These criteria are: 1) The cross-section of the structure has an elongated shape with 0,25; Figure Illustration of the dimensions adopted for the first criteria for the verification of flutter instability. 2) The torsional axis is parallel to the plane of the plate and normal to the wind direction and the distance between the windward edge of the plate and the torsional centre is at least /4; 35

56 3) The lowest vibration mode is a torsional mode. If this is not the case, then the lowest translational vibration mode shall have a natural frequency which is at least 2 times bigger than the lowest torsional natural frequency. If all of these criteria are met, the EN provides a simple expression for the calculation of the critical wind speed on the onset of flutter divergence: 2 d d (55) is the torsional stiffness and is the air density. The length is the one defined in Figure 03.1 and d d is the derivative of the moment force coefficient, defined in section of this report, in order of the angle of attack. The EN also provides a graphic to define d d for rectangular shaped decks. This, however, produces greater values than experimental technique which lead to lower critical wind speeds. This graphic is presented in Annex D. Finally, this verification is considered successful if the critical wind speed from expression (55) is bigger than two times the mean wind speed. 2 (56) Flutter in a single torsional mode - the CECM formula Much like the EN, the CECM also provides a simple formula to solve the critical wind speed for torsional flutter. This formula very simple, based on the deck geometry and the fundamental natural torsional frequency: Γ (57) and have already been defined as the deck s width and the fundamental natural torsional frequency. The parameter Γ is based on the section geometry and ranges from 2 to (which is the case of the aerofoil a stable structure for this type of instability). Annex E contains the table from the CECM with the values of Γ. 36

57 Flutter in a torsional and bending mode - The Selberg formula In 1961, A. Selberg proposed a simple formula for calculating the critical wind speed for flutter instability: 3,71 1 (58) Even though this formula can only be applied to thin plates, it is possible to adapt it to deck crosssections by applying a factor, provided in the CECM, which depends on the geometry of the deck. This factor ranges from 0,10 to 1,00 (a table with this factor for several geometrical shapes is provides in Annex F). However, this formula only produces reliable result under certain conditions. The conditions are: 1) The ratio between the fundamental torsional natural frequency and the fundamental bending natural frequency should be at least 1,5; 1,5 (59) 2) The product 2, which represents the logarithmic decrease, should be around 0,05. If these two conditions are met, the usage of the Selberg formula produces an acceptable value of the critical wind speed Approximated Models (Multimodal Flutter Analysis) Modal Analysis With the objective of finding the critical wind speed for flutter a modal analysis of the bridge deck can be carried out. For this analysis, a previous calculation of the structural natural frequencies ( ) and vibration modes ( ) is needed. This is done by solving equation (60). (60) For cable stayed bridges, this equation results in a good enough approximation. However, when doing this step for suspension bridges, it is advised to use the matrix instead of just, where is the geometric stiffness matrix. This matrix is obtained through a second degree analysis of the stiffness of the bridge. Without doing this step, the natural frequencies obtained can be very different from the real ones. 37

58 To solve equation (60), it is necessary to model the entire bridge (deck, cables and towers) which results in degrees of freedom. However, only the components related to the deck s degrees of freedom should be applied in the rest of the process of finding the critical flutter wind speed (i.e. to build the modal matrix). Also, this matrix can be built with vibration modes instead of the total ones ( ). Using the most significant ones it is possible to achieve acceptable approximate results. However, a certain caution is needed when choosing which modes to use because it can produce fluctuations in the results obtained. As an example, for the Great Belt Bridge, a suspension bridge opened in 1998 over the Great Belt in Denmark with an amazing 1624 meter span, Jurado [2011] adopted two and eighteen modes on his trials. Cobo [1998] also adopted two modes on one of his trials, using only six on his other one. Larsen [1993] on the other hand, did some experimental testing using only a cross-section and using the whole bridge model. In addition, a trial with nine modes was also conducted. The results for critical wind speed of flutter for these authors are presented in Table As a result, instead of components, the modal matrix has components.,,,,,,,,, (61) Using the modal matrix the movement of the deck can be rewritten as a combination of vibration modes: (62) Table Comparison of various aeroelastic studies on the Great Belt Bridge Analysis Critical Flutter Speed (/) Larsen, cross-section test Larsen, whole model test Larsen, 9 modes 75 Cobo, 2 modes 82,75 Cobo, 6 modes 96,75 Jurado, 2 modes 89,91 Jurado, 18 modes 62,41 Replacing equation (62) in equation (35b) and multiplying by on the left: 38

59 (63) It is possible to simplify this equation by defining the following matrices [3] : (64) The matrices and are the reduced global damping and stiffness matrices, respectively. Equation (63) becomes: (65) There are many solutions for this equation. However, since the goal of this procedure is to find the wind speed at the onset of flutter instability, all solutions that are oscillatory in nature can be discarded and the solution can be assumed to be exponential in nature (characteristic of divergent behaviour). (66) and so, it s first and second derivative are: (67) Using the above solution and another trivial equation (employed to help the process of solving equation (65)) a system of two equations appears. (68) Changing it into its matrix format: (69) This equation can be written in a more appealing and clean way by defining the variables and as: (70a) (70b) [3] The matrix can only be the identity matrix if, and only if, the vibration modes where previously normalized in terms of mass. 39

60 The solution for this equation has to be found through an iterative procedure. This is because some of the prerequisite variables, the flutter derivatives employed to build the matrix, are dependent on the reduced frequency. In addition, this variable is also dependent on the damping frequency ( ) which is one of the outputs of this procedure, as demonstrated further ahead. This equation is a non-linear problem of eigenvalues and eigenvectors in which the solutions are only dependent on the average wind speed. There are 2 eigenvalue solutions for this equation, where is the number of vibration modes used to build the modal matrix. However, these solutions are complex in nature and can, therefore, be grouped in conjugate pairs where to each pair of conjugate eigenvalues corresponds a pair of conjugate eigenvectors: (71) The vector has 2 components which can be used to get the vector with only components by using the left expression (70a). With this vector and its corresponding eigenvalue, and using expressions (62) and (66), the damped oscillatory movement of the deck becomes defined. (72) The coefficient represents which eigenvalue solution is being applied and can be within the range 1,2. The real and imaginary parts of this equation can be separated to better correlate it with other oscillatory movement equations. cos sin sin sin (73) If the expression of the free reign oscillations of a linear oscillator with one degree of freedom is taken into account: cos sin (74) it is possible to see the real part of equation (73) is of the same form. Comparing both equations, it is evident that the imaginary part of the eigenvalues takes to role of the damping frequency of a linear oscillator, and the real part of the eigenvalues is associated with the product. (Jurado J.A. [2011]) 40

61 (75) From the above equations, it is observed that from the initial 2 solutions of equation (70b), only produce different results in terms of response frequency and damping coefficient, hence, different damped oscillatory motions (each pair of conjugate eigenvalues results in the same final parameters). Now that this iterative process is defined, a number of steps to find its output, according to Jurado [2011], is presented. 1) An initial value of the circular frequency is assumed, for the vibration mode. It is of good practice to choose the natural vibration frequency for each mode. This value is compared with the final result to check if the process as terminated or not; 2) The eigenvalue problem from equation (70b) is resolved, bearing in mind that to build the matrix from the right expression (70a) the frequency parameter should be the one defined in the step before; 3) From the 2 eigenvalues obtained, the one with the minimum difference between its imaginary part and the frequency value adopted to build is the output of this process (the eigenvalue and its complex conjugate that correspond to the mode ); 4) If the difference described in step 3) is below the established tolerance, the process is over. However, if this is not the case, steps 2) and 3) are repeated with the new circular frequency found in step 3). Much like Galloping, this divergent instability phenomenon called flutter can only happen when the global damping of the structure reaches a negative value, leading to an oscillatory movement with ever increasing amplitude until the eventual collapse. From equation (75), it is possible to see that the damping coefficient only has this nature when the real part of the eigenvalue solutions obtained for equation (70b) has a positive value. As stated before, this value is only dependent of the average wind speed. If the wind speed is low enough, the value of has a positive nature, leading to an oscillatory movement damped over time. However, as the wind speed rises, becomes smaller and smaller until it eventually becomes null. This circumstance identifies the critical condition for flutter and the wind speed for which it occurs is called critical wind speed. If the speed of the flow raises even more, the damping coefficient becomes negative, leading to the instability phenomenon. To find this critical wind speed, one can use the above procedure for solving the eigenvalue problem of equation (70b). Starting with a low enough wind speed that would not cause any instability, the process is run to find the value of,. Then, the wind speed is raised by an increment Δ. The process is run again, with the result now being,.if this value is still negative, the wind speed is 41

62 incremented again. This is done until, [4] becomes either null or positive. If it becomes null, the critical wind speed for flutter is Δ and the frequency is, whereas, if it becomes positive, it is only possible to deduce that the critical wind speed is in the range 1Δ; Δ. According to Jurado [2011], for the first calculation for the wind speed, it is advised to use the natural frequency for each mode as the first guess for the iterative process. As the wind speed increases, the values for the frequency should be based on the evolution of frequencies from previous steps: ω, 2ω, ω, (76) Bidimensional Approach Even though this is a generic model to be adopted with any amount of degrees of freedom, vibration modes and deck nodes, the modal analysis is often employed as an approximated method using only two degrees of freedom. These are the vertical displacement and the pitching rotation. This implies the usage of the two fundamental vibration modes associated with these displacements, bending and torsional modes. However, the amount of nodes adopted to model the deck is chosen based on the computation power available to the user. Figure Evolution of the imaginary part of the eigenvalues in an example by Jurado [2011] (4 vibration modes) [4] here is the number of steps taken to find,. 42

63 Figure Evolution of the real part of the eigenvalues in an example by Jurado [2011] (4 vibration modes) The first step, as stated before, is to find the fundamental natural frequencies and vibration modes for the torsional and bending modes. If the deck is divided into nodes, these can be expressed in their vector format by:,,, (77a),,, (77b) Building the modal matrix and using it to transform the movement variables, the following is obtained:,1,1,2,2,, (78) The mass, damping and stiffness matrices have 22 components. These are represented in equations (79). 43

64 (79) To solve equation (70b) there is only one more step that needs to be taken. This step is another transformation of the movement variables so they represent a divergent type of movement. This is done through expression (80). (80) The solution for equation (70b) is a pair of pairs of conjugate eigenvalues and eigenvectors, as in equations (71), represented by: (81a) (81b) Non-dimensional simplification The above method is sufficient to calculate the critical flutter speed. However, a small error in the input can lead to imperceptible variations in the output which can be carried on. This model is prone to those input error because every variable has to be in the appropriate dimensions. To overcome this liability, it is possible to simplify the model so that every variable in the input has a non-dimensional nature. Stating with the equation of the oscillatory movement of the deck subjected to an aerodynamic force, equation (35a): and expanding all its components: (82) (83) The parameters and are the damping coefficient for the vertical and rotation displacements, respectively, and are the stiffness parameters for the vertical and rotation displacements, respectively, and and are the mass coefficients for the vertical and rotation displacements, the mass and the mass moment of inertia, respectively. All of these coefficients are expressed per unit length. Using 44

65 equations (28a) and (28b), it is possible to further develop the variables on the right side of equation (83) and rewrite them in their matrix format: (84) Replacing equation (84) in equation (83): (85) The first step to non-dimensionalize is to remove the mass through the use of the following equivalences: (86a) 2 2 (86b) (86c) Equation (85) becomes: The next step is transforming the displacement into non-dimensional ones, and reusing concept of a non-dimensionalized time variable : (87) 45

66 (88a) (88b) d d d d d d (88c) Substituting in equation (87) and removing, the deck s width, from the first equation, in between steps: (89a) (89b) (89c) 46

67 (89d) Looking at the mass matrix, there is still need to remove the dimensions from the radius of gyration. Hence, another non-dimensionalized variable arises, the non-dimensional radius of gyration, and it is defined by: (90) Using it in equation (89d) the following is obtained: (91) After obtaining the equation above, there is only one step missing which is to simplify it. Looking at the global damping matrix, there is a coefficient which is common to every element. By rearranging this coefficient, it can be rewritten as a product of two other non-dimensional parameters. These are the non-dimensional wind speed and a ratio that relates the wind mass with the decks mass. The same process can also be done for the global stiffness matrix. By doing this, the writing of equation (91) becomes much clearer and easier to comprehend leading to an easier computation of the critical wind speed problem. This process is illustrated in equations (92). The variable is the assigned variable to represent the ratio between the fundamental torsional natural frequency and the fundamental bending natural frequency from now on. 47

68 2 (92a) (92b) (92c) The final form of equation (91) is: (93) This equation can be contracted to its matrix format: (94) Re-doing the same kind of steps done from equation (65) to (70b) it is possible to transform the equation above in the same kind of system to be solved iteratively. Multiplying by on the left: (95a) (95b) The next step is doing the movement variable transformation as in equation (66). (96) Since the variable is a generic variable, with the transformation to the non-dimensionalized time variable, from equation (88b), emerges. The derivatives of equation (96), in order of, are: (97) Using equations (94), (96) and (97) and a trivial equation of the same form as the upper part of the system (68), the following system is obtained: 48

69 (98) which in matrix format becomes: (99) This equation can be written in a more appealing and clean way by defining the variables and as: (100a) (100b) Equation (100b) is once more a non-linear problem of eigenvalues and eigenvectors, which can only be solved through an iterative procedure. The solution for this problem continues to be a pair of pair of conjugate eigenvalues and eigenvectors. (101a) (101b) Since the variable is non-dimensional and using the same deduction made from equation (72) until equation (75), the vibration frequencies on the onset of flutter instability and the damping coefficient associated to it can be obtain from equations (102). (102) Computer Models Another alternative for evaluating bridge stability is to do it through computer models. These are based on the accurate modelling of the wind flow through the Navier-Stokes equations, which can become extremely arduous. First, the bridge cross-section is drawn as a rigid body supported by fictional elements that will assume its elastic properties, like stiffness and damping. Then, a complex algorithm is deployed to generate a grid in the domain of the fluid flow around the structure. The generation of the grid is important and critical in order to get the correct results. Because of this, several studies have 49

70 been made and algorithms have been created and refined to facilitate the computation without compromising the results. The next step is to solve the structure and fluid equations simultaneously, incorporating the fluid-structure interaction. For this, the fluid is assumed viscous and incompressible. Because of the complexity involved in building these models, they fall outside of the area of study of this report. 50

71 Chapter 04 Implementation of the Flutter Stability Methods for Practical Cases With flutter being the most important instability phenomena in the aeroelasticity field, its verification will be shown for two practical examples. The first example is a small cable-stayed bridge built in the city of Funchal, Madeira, Portugal. The second example is a bridge designed for the Tagus river in Lisbon, the third crossing over the Tagus River. Even though this last example has not been built, its Reference Design was completed. 51

72 04.1. Cable-stayed Viaduct of the highway in Funchal This bridge was opened in September It belongs to a highway that surrounds the city of Funchal in Madeira, Portugal. Its cable-stayed design was picked due to the high density urban occupation and the railway that goes under it. Figure Cable-stayed viaduct of the highway in Funchal The bridge has a total of 200 in length and a main span of 92, which is quite small compared with some well-known bridges of the same design typology. Two deck cross sections were studied both with an unusual shape, which resembles a triangle, with 21,5 width and a height of just 2. The first one was fully made of prestressed concrete whereas the second one was a composite box girder cross section. These can be seen in Figure Figure Bridge spans and both proposed deck cross sections: reinforced concrete (on the left) and composite box girder (on the right) 52

73 Cross Section Parameters To calculate the critical wind speed for flutter it is necessary to define several parameters of the deck. Even though some of them vary from method to method, most of them are employed by all of them. All of the parameters are presented below. [1] Table Cross section parameters for both designs of the bridge in Funchal Reinforced Concrete Composite Box Girder width () height () 21,5 2,0 mass () / / mass moment of inertia ( ) 6, / 1, / There is also the need to define the damping coefficient. However, this is done by applying the value of 2% to the logarithmic decrease. To calculate the damping coefficient expression (103a) is adopted. These are the same for both solutions. 2 2 (103a) 0,32% 0,32% (103b) The circular fundamental frequencies for bending and torsion are also needed. These can be found through their respective linear frequencies by the use of equation (104). 2 (104) Table Fundamental linear and circular frequencies for both designs of the bridge in Funchal Reinforced Concrete Composite Box Girder 0,83 1,31 1,63 1,92 5,22 / 8,23 / 10,24 / 12,06 / [1] Source: Reis, A. [2014] 53

74 These fundamental frequencies can be employed to define the stiffness parameters need for the use of the EN formula. This is done through equation (105). (105) Table 04.3 Stiffness parameters for both designs of the bridge in Funchal Reinforced Concrete Composite Box Girder 678,81 // 1355,74 // 6,65 10 // 2,24 10 // Lastly, the force coefficients for the cross section were determined through wind tunnel testing. Only the reinforced concrete cross section was tested. However, since both cross sections have very similar shapes, these results can be applied for the composite cross section as well. These are presented in Figure The air density considered was 1,225 /. Figure Force coefficients of the bridge in Funchal in function of the angle of attack ( ) 54

75 The EN formula As already stated in section , the cross section has to fulfil three simple criteria to be eligible to be evaluated by the formula given. These criteria are: 1) The cross-section of the structure as an elongated shape with 0,25; Using the data from the section above, the width and height of both cross sections are: 21,5 2,0 (106a) 0,0930 0,25 (106b) 2) The torsional axis is parallel to the plane of the plate and normal to the wind direction and the distance between the windward edge of the plate and the torsional centre is at least /4; The first part of this criterion is true for every deck cross section. As for the second part, since the cross section is symmetric, the torsional centre is at a distance of /2 of the windward edge of the section. 3) The lowest vibration mode is a torsional mode. If this is not the case, then the lowest translational vibration mode should have a natural frequency which is at least 2 times bigger than the lowest torsional natural frequency. In this case, the lowest vibration mode is a bending mode for both cross sections (Table 04.2). However, since there is no information about the lowest translational mode, it will be assumed, for academic purposes, that it its frequency is at least 2 times bigger than the lowest torsional natural frequency to fulfil this criterion. With all criteria fulfilled, it is possible to use equation (55) to find the critical wind speed for flutter according to the Eurocode. Using the graphic provided by the EN, the derivative of the moment force coefficient for this cross section has the value of 1,51. However, looking at Figure 04.3, it is possible to see that even in the most conservative value is far inferior (d d 0,04) which results in greater value of the critical wind speed. The results for both value of d d are presented in Table Table Critical wind speed for torsional flutter (EN formula) for the bridge in Funchal Reinforced Concrete Composite Box Girder d d 1,51 394,4 / 229,0 / d d 0, ,4 / 1406,9 / 55

76 The CECM formula for torsional flutter To use the CECM formula for flutter in a single mode (torsional), there is only the need to define the Γ factor. Even though the shape of the deck cross-section is unusual, it is possible to define a lower limit for this factor (Γ 9). Table Critical wind speed for torsional flutter (CECM formula) for the bridge in Funchal Reinforced Concrete Composite Box Girder 315,4 / 371,5 / The CECM formula for classical flutter The use of the Selberg formula also provides more accurate results if two criteria are met. These criteria, also explained in section , are: 1) The ratio between the fundamental torsional natural frequency and the fundamental bending natural frequency should be at least 1,5; Table Ratios for both solutions Reinforced Concrete Composite Box Girder 1,9617 1,4654 As seen from the table above, the reinforced concrete cross section meets this criterion whereas the composite cross section doesn t. However, since the ratio is close to meeting the criterion, the formula will be adopted to compare with the other results. 2) The product 2, which represents the logarithmic decrease, should be around 0,05. Since the logarithmic decrease adopted is 2% for both modes, this criterion is also considered fulfilled. However, when evaluating the results of the Selberg formula, this consideration as to be taken into account. The factor adopted, due to the cross section geometry, was approximately 0,6. Table Critical wind speed for classical flutter (Selberg and CECM formula) for the bridge in Funchal Reinforced Concrete Composite Box Girder 359,5 / 239,1 / 215,7 / 143,5 / 56

77 Bidimensional Models To solve the eigenvalue and eigenvector problem to find the critical wind speed for flutter, it is necessary to build the mass, damping and stiffness matrices which have been defined in There is also the need to build the aerodynamic damping and stiffness matrices. These involve the definition of the flutter derivatives. The flutter derivatives are usually found through wind tunnel testing. However, these were not available for the sections above and, therefore, equations (30) were employed to define them. The use of equations (30) implies the use of the Theodorsen function (represented in Figure 02.7). Annex H provides a graphic with the flutter derivatives for different value of the reduced frequency. Table Critical wind speed for classical flutter (Bidimensional Model) for the bridge in Funchal Reinforced Concrete Composite Box Girder 372 / 205 / Figure Evolution of the value of with the wind speed for the reinforced concrete solution Figure Evolution of the value of with the wind speed for the composite box girder solution 57

78 Figure Evolution of the damping with the wind speed for the reinforced concrete solution Figure Evolution of the damping with the wind speed for the composite box girder solution Even though no aerodynamic instability were detected, the lowest potential critical wind speeds obtained by this method are 372 / and 205 / for the concrete and composite cross-sections respectively. These values are higher than the values obtained by the CECM (Selberg) approach for classical flutter. 58

79 04.2. The Third crossing over the Tagus River To help with the ever increasing traffic over the Tagus river (both roadway and railway), a third bridge was designed to add to the 25 de Abril Bridge and the Vasco da Gama Bridge. However, due to the lack of funding, this project never made it to the construction phase but the aerodynamic stability of this solution was tested. Figure The Third crossing over the Tagus River The TTT (Terceira Travessia do Tejo) has a total of 1140 in length and a main span of 540. The deck is held 50 above the water level by pairs of cables, with 15 of spacing between each other, connected to two main pillars (cable-stayed solution). The cross section consists of two levels, with the top level, allocated for roadway traffic, having just over 30 width and the bottom level, allocated for railway traffic, having just below 21. This is illustrated in Figure Figure Deck cross section for the TTT 59