POLITECNICO DI MILANO
School of Industrial and Information Engineering Master of Science in Mechanical Engineering
A Time-Domain Approach Based on
Rheological Models for the Buffeting
Response of Long-Span Bridges
Supervisor: Prof. Daniele Rocchi
Co-supervisors: Ing. Simone Omarini
Ing. Tommaso Argentini
Master Thesis of:
Andrea BRUSAMOLINO
ID Number: 882509
A mia nonna, ora mio angelo custode. Spero tu sia ugualmente fiera di questo mio traguardo.
Abstract
The need to overcome obstacles on very long distances typically implies the design of “streamlined” closed-box bridge decks. The size and lightness characterizing these structures make them very sensitive to dynamic problems and special care must be put in the design against the wind action. Dealing with this aspect, many difficulties are involved both in the wind description and in the computation of the bridge response, making the definition of good numerical codes a fundamental step. Nowadays, numerical models for the design of long-span bridges are mainly lin-earized models carried out both in frequency and time domain. However, some non-linear models can be found in the time domain. This models are, in fact, more appropriate to describe all the complex aspects related to the aerodynamic forces generated by the high fluctuations of the angle of attack. Unfortunately, such models are strongly limited by their applicability range since they would be more appropriate only to describe the non-linearities associated with the low frequency fluctuations of the incoming wind turbolence. An attempt to take into account also the non-linear contributions of the high frequencies is presented in this thesis where a new numerical approach based on the definition of rheological models has been developed and finally validated.
Various simulations have been run to test the numerical procedure and compar-isons were made with wind-tunnel tests. Code validation was done in a systematic way, starting from a simple linearized model and increasing the degree of complex-ity until the procedure was assessed robust enough to sustain the analysis of a full non-linear problem.
The wind-tunnel tests were performed at Politecnico di Milano and were part of an international benchmark project conducted by the Politecnico di Milano and promoted by the International Association for Bridge and Structural Engineering (IABSE).
La necessit`a di sorpassare ostacoli su distanze molto lunghe favorisce la progettazio-ne di ponti a cassoprogettazio-ne molto sprogettazio-nelli. Le dimensioni e la leggerezza che caratterizzano queste strutture le rendono molto sensibili ai problemi dinamici e, in particolare, occorre prestare molta attenzione nella progettazione contro l’azione del vento tur-bolento. In tal senso, si riscontrano molte difficolt`a sia nella descrizione del vento che nel calcolo della risposta del ponte, rendendo pertanto fondamentale la definizione di un buon codice numerico.
Al giorno d’oggi, i modelli numerici connessi allo studio di ponti di grande luce sono per la maggior parte modelli linearizzati definiti sia nel dominio delle frequenze sia nel dominio del tempo. Si possono comunque trovare modelli non lineari definiti nel dominio del tempo. Questi modelli sono infatti necessari se si vogliono analiz-zare tutti gli aspetti complessi legati alle forze aerodinamiche generate dalle grandi fluttuazioni dell’angolo d’attacco. Tuttavia, questi metodi sono fortemente limita-ti dal dominio di applicabilit`a poich´e adatti a descrivere le non linearit`a associate solamente alla variazione in bassa frequenza del vento turbolento. Un tentativo di estendere questo approccio anche alle non linearit`a legate alle alte frequenze `e pre-sentato in questo lavoro di tesi in cui un nuovo modello basato sulla definizione di modelli reologici `e stato sviluppato ed infine validato.
Varie simulazioni sono state condotte per testare il metodo numerico e confronti sono stati fatti con le prove sperimentali eseguite in galleria del vento. La validazione del modello `e stata svolta in maniera sistematica, partendo da formulazioni semplici e aumentando il grado di complessit`a fino a quando la procedura `e stata ritenuta sufficientemente robusta per l’analisi di un modello completamente non lineare.
Le prove in galleria del vento sono state condotte presso il Politecnico di Milano e sono inserite all’interno di un benchmark internazionale condotto dal Politecnico di Milano e promosso dall’Associazione Internazionale di Ponti e Grandi Strutture (IABSE).
Abstract . . . I Sommario . . . II List of Figures . . . X List of Tables . . . XI
Introduction 1
1 Wind Action on Bridges 4
1.1 Introduction . . . 4
1.1.1 Main Aspects of Wind-Bridge Interaction . . . 4
1.2 Description of the Aerodynamic Forces . . . 7
1.2.1 Quasi-Steady Theory . . . 7
1.2.2 Linearized Quasi-Steady Theory . . . 9
1.2.3 Flutter Derivatives and Admittance Functions . . . 12
1.3 Band Superposition Approach . . . 16
1.3.1 A New Model Idea . . . 19
2 Analytical Modeling of the New Non-Linear Approach 21 2.1 Introduction . . . 21 2.2 Linearized Formulation . . . 21 2.2.1 Modal Equations . . . 22 2.2.2 Aerodynamic Forces . . . 22 2.3 Non-Linear Formulation . . . 27 2.3.1 Modal Equations . . . 28 2.3.2 Aerodynamic Forces . . . 29 2.4 Numerical Approach . . . 33 2.4.1 Moving Average . . . 35 2.4.2 Integration Method . . . 39
3 Parameter Identification of the Rheological Models 42 3.1 Introduction . . . 42
3.2 Rheological Models . . . 43
3.3 Constrained Optimization . . . 44
3.3.1 Frequency Response Function of the Rheological Models . . . 44
3.3.2 Objective Function and Constraints . . . 46
3.3.3 Lower and Upper Boundaries . . . 48
3.3.4 BB3 Considerations . . . 49
3.4 Identification Approach for the Non-Linear Model . . . 53
3.5 Results . . . 56
3.5.2 BB3 Sectional Model . . . 58
4 Numerical Model Validation 71 4.1 Introduction . . . 71
4.2 Wind-Tunnel Experimental Set-up . . . 71
4.2.1 PoliMi Wind Tunnel . . . 71
4.2.2 Test Configurations . . . 72
4.2.3 Dynamic Tests . . . 73
4.2.4 Suspended Model Tests . . . 76
4.3 Free Motion - Eigenvalue Problem . . . 84
4.4 Forced Motion . . . 93 4.4.1 Linearized Model . . . 93 4.4.2 Non-Linear Model . . . 94 4.5 Final Considerations . . . 106 Conclusions 116 Appendices 119 A Assembly of the Global Matrices 120 A.1 Self-excited Forces . . . 120
A.2 Buffeting Forces . . . 124
B Numerical Integration Methods 128 B.1 Crank-Nicolson . . . 128
B.2 4th order Runge-Kutta . . . 129
1.1 The Yavuz Sultan Selim Bridge, or Third Bosphorus Bridge (BB3). . 4 1.2 Incoming wind profile on a suspension bridge . . . 5 1.3 Wind reference system. . . 7 1.4 Aerodynamic forces on a generic deck section: sign conventions. . . . 8 1.5 Measured aerodyamic coefficients (left) and derivatives (right) of the
BB3 sectional model. . . 9 1.6 Comparison between the last values of the experimental flutter
de-rivatives a∗3 of the BB3 sectional model with the corresponding QST values of KM for dfferent mean angles of attack. . . 14
1.7 Trends of B1z and B1θ of the BB3 sectional models against reduced
velocity for different mean angles of attack. . . 15 1.8 Davenport’s expression of the real function A versus reduced velocity
V∗ and reduced frequency f∗. . . 16 1.9 Flutter derivatives of the Messina bridge [5]. a∗2 (a) and a∗3 (b) vs. V∗
for nine different mean angles of attack α. . . 17 1.10 Schematization of the main algorithm of the band superposition
ap-proach [5]. . . 19 2.1 Schematization of the elastically suspended deck section. . . 22 2.2 Comparison of the self-excited force contribution between the
defin-ition of the actual whole state vector z and the defindefin-ition of a new high frequency state vector zHF. The moment Mse was divided by
B/2 so as to be consistent with the lift Lse. The comparison suggests
that the proposed formulation (Eq.2.37), which considers the whole state vector z, is a good approximation. . . 31 2.3 Example of time histories of the input forcing vectors, ψ and ˙ψ, of
Eq.2.38. . . 32 2.4 Block diagram of the main algorithm of the new non-linear model
discussed in this chapter. A comparison should be done with the one reported in Fig.1.10 (Section 1.3). . . 34 2.5 Example of vertical and horizontal time histories highlighting the
three major contributions: vertical turbolent component w, hori-zontal turbolent component u and wind mean speed V . . . 35 2.6 Time history and frequency spectrum of the incoming wind relative
angle ψ. The signal is finally filtered to obtain only the low frequency component. . . 36 2.7 Moving average examples: (a) effects of the window length on the
filtered signal; (b) complete filtering effect due to perfect window match. . . 36
2.8 Modulus and phase of the transfer function of the moving average (MA) considering different time windows. . . 37 2.9 Transfer function of the moving average considering a 1s time window. 39 2.10 Transfer function comparison (modulus and phase) among a simple
moving average (SMA - pass 1), a multi-passing simple moving aver-age (SMA - pass 2) and an exponential moving averaver-age (EMA). The EMA is the one with less filtering power but introduces smaller delays in the low frequency region. . . 39 2.11 Integration results for different time steps considering Crank-Nicolson
method. Solution converges for small dt proving the major effect due to the choice of the time step. . . 40 2.12 Effect of the time step on the integration results. Comparison between
Crank-Nicolson (CN) and Runge-Kutta (RK) methods of the high frequency component of the deck rotation. Decreasing dt, CN tends to RK proving that RK is less sensitive to the choice of the time step (provided that it is stable). . . 41 3.1 Mechanical oscillators used to describe the aerodynamic transfer
func-tions. . . 44 3.2 Comparison of the high frequency response of the deck rotation θ
between the two strategies adopted in the definition of the upper and lower boundaries: ±10 (NoMod) and deck modal parameter (Mod). The contribution given by the rhological models is the same. . . 49 3.3 PoliMi lift flutter derivative h∗3 for the BB3 sectional model. . . 50 3.4 Example of parameter trends against the angle α. . . 51 3.5 Fititious extensions used for the transfer function identfications. . . . 52 3.6 Comparison of the lift transfer function L/θ at 0◦ between the
non-linear approach (new) and the non-linearized approach (old). . . 55 3.7 Flutter derivatives h∗1−4 and a∗1−4. Theoretical expression for a flat
plate (PoliMi notation). . . 57 3.8 Identification results for the flat plate: L/(− ˙z/V ). Solution
compar-ison and final choice. . . 58 3.9 Identification results for the flat plate: M/(− ˙z/V ). Final choice and
curve behaviour for V∗ → ∞. . . 59 3.10 Identification results for the flat plate: L/θ. Final choice. . . 60 3.11 Identification results for the flat plate: M/θ. Final choice. . . 61 3.12 Identification results for the BB3 sectional model: L/(− ˙z/V ) at 0◦. . 61 3.13 Identification results for the BB3 sectional model: L/(− ˙z/V ) at 2◦. . 62 3.14 Identification results for the BB3 sectional model: L/(− ˙z/V ) at −2◦. 62 3.15 Identification results for the BB3 sectional model: L/(− ˙z/V ) at 4◦. . 63 3.16 Identification results for the BB3 sectional model: L/(− ˙z/V ) at −4◦. 63 3.17 Identification results for the BB3 sectional model: M/θ at 0◦.
Solu-tion comparison and final choice. . . 64 3.18 Identification results for the BB3 sectional model: M/θ at 2◦. Final
choice. . . 65 3.19 Identification results for the BB3 sectional model: M/θ at −2◦. Final
3.21 Identification results for the BB3 sectional model: M/(− ˙z/V ) at −4◦.
Final choice. . . 66
3.22 Identification results for the BB3 sectional model: lift transfer func-tions. All angles. . . 67
3.23 Identification results for the BB3 sectional model: moment transfer functions. All angles. . . 68
3.24 Identification results for the BB3 sectional model: lift and moment transfer functions of the flutter derivatives minus the QST. All angles. 69 3.25 Identification results for the BB3 sectional model: lift and moment transfer functions of the admitance functions minus the QST. All angles. 70 4.1 PoliMi Wind Tunnel: closed circuit facility. . . 72
4.2 PoliMi Wind Tunnel: axial fans for wind generation. . . 72
4.3 Deck section dimensions and shape. . . 73
4.4 Test configuration with oil-dynamic actuators. . . 73
4.5 Test configuration with steel cables. . . 74
4.6 Oil-dynamic actuators. . . 74
4.7 PoliMi Wind Tunnel: axial fans for wind generation. . . 74
4.8 Suspended sectional model forced by the active turbulence generator. 76 4.9 PoliMi lift flutter derivative h∗1. . . 78
4.10 PoliMi lift flutter derivative h∗2. . . 78
4.11 PoliMi lift flutter derivative h∗3. . . 79
4.12 PoliMi lift flutter derivative h∗4. . . 79
4.13 PoliMi lift flutter derivative a∗1. . . 80
4.14 PoliMi lift flutter derivative a∗2. . . 80
4.15 PoliMi lift flutter derivative a∗3. . . 81
4.16 PoliMi lift flutter derivative a∗4. . . 81
4.17 PoliMi lift admittance function χ∗Lw: amplitude and phase. . . 82
4.18 PoliMi moment admittance function χ∗M w: amplitude and phase. . . . 83
4.19 Vertical decay at 8 m/s mean wind speed. Vertical (top) and torsional (bottom) displacement comparisons. Linearied model. Configuration 1. . . 85
4.20 Vertical decay at 8 m/s mean wind speed. Vertical (top) and tor-sional (bottom) displacement comparisons. Non-linear model. Con-figuration 1. . . 86
4.21 Torsional decay at 10 m/s mean wind speed. Torsional (top) and vertical (bottom) displacement comparisons. Linearized model. Con-figuration 1. . . 87
4.22 Torsional decay at 10 m/s mean wind speed. Torsional (top) and ver-tical (bottom) displacement comparisons. Non-linear model. Config-uration 1. . . 88
4.23 Vertical decay at 10 m/s mean wind speed. Vertical (top) and tor-sional (bottom) displacement comparisons. Linearized model. Con-figuration 2. . . 89
4.24 Vertical decay at 10 m/s mean wind speed. Vertical (top) and tor-sional (bottom) displacement comparisons. Non-linear model. Con-figuration 2. . . 90 4.25 Torsional decay at 8 m/s mean wind speed. Torsional (top) and
ver-tical (bottom) displacement comparisons. Linearized model. Config-uration 2. . . 91 4.26 Torsional decay at 8 m/s mean wind speed. Torsional (top) and
ver-tical (bottom) displacement comparisons. Non-linear model. Config-uration 2. . . 92 4.27 Example of mono-harmonic (top) and bi-harmonic (bottom) vertical
turbolent component of the incoming wind at 8 m/s. Time histories and spectra. . . 94 4.28 Example of bi-harmonic (top) and tri-harmonic (bottom) vertical
tur-bolent component of the incoming wind at 10 m/s. Time histories and spectra. . . 96 4.29 Test 936. Mono-harmonic excitation at 2.2 Hz, 8 m/s. Vertical (top)
and torsional (bottom) motion comparisons. . . 97 4.30 Test 944. Mono-harmonic excitation at 3.5 Hz, 10 m/s. Torsional
(top) and vertical (bottom) motion comparisons. . . 98 4.31 Test 953. Bi-harmonic excitation at 2.2-3.5 Hz, 15 m/s. Vertical
(top) and torsional (bottom) motion comparisons. . . 99 4.32 Example of incorrect compensation caused by lacks of information.
Numerical vs. experimental comparison of θ, test 960, (left) and low frequency variations of the angle of attack, αrheol and deck rotation,
θLF (right). . . 100
4.33 Test 965. Bi-harmonic excitation at 0.1-2.2 Hz, 10 m/s. Vertical (top) and torsional (bottom) motion comparisons. . . 101 4.34 Test 964. Bi-harmonic excitation at 0.1-3.4 Hz, 10 m/s. Vertical
(top) and torsional (bottom) motion comparisons. . . 102 4.35 Test 961. Tri-harmonic excitation at 0.1-2.2-3.4 Hz, 10 m/s. Vertical
(top) and torsional (bottom) motion comparisons. . . 103 4.36 Test 960. Tri-harmonic excitation at 0.1-2.2-3.5 Hz, 10 m/s. Vertical
(top) and torsional (bottom) motion comparisons. . . 104 4.37 Comparison between the corresponding value of the linearized TQS
at 0◦ for the moment admittance function χ∗M w of the BB3 sectional model with the actual value passed to the integration (V∗ = 85). . . . 108 4.38 Comparison between the corresponding value of the linearized TQS
at 0◦ for the lift admittance function χ∗Lw of the BB3 sectional model with the actual value passed to the integration (V∗ = 85). . . 108 4.39 Test 959. Frequency spectrum of the z motion [mm]. Comparison
between experimental data (Exp) and numerical solutions (Num) for both linear and non-linear models. . . 110 4.40 Test 959. Frequency spectrum of the θ motion [deg]. Comparison
between experimental data (Exp) and numerical solutions (Num) for both linear and non-linear models. . . 110 4.41 Test 964. Frequency spectrum of the z motion [mm]. Comparison
between experimental data (Exp) and numerical solutions (Num) for both linear and non-linear models. . . 110
both linear and non-linear models. . . 111 4.43 Test 970. Frequency spectrum of the z motion [mm]. Comparison
between experimental data (Exp) and numerical solutions (Num) for both linear and non-linear models. . . 111 4.44 Test 970. Frequency spectrum of the θ motion [deg]. Comparison
between experimental data (Exp) and numerical solutions (Num) for both linear and non-linear models. . . 111 4.45 Test 964. Comparison between the low (left) and high (right)
fre-quency bands of the moment between experimental data (exp) and numerical solutions for both linear (lin) and non-linear (non-lin) models.112 4.46 Test 964. Comparison between the low (left) and high (right)
fre-quency bands of the moment between experimental data (exp) and numerical solutions for both linear (lin) and non-linear (non-lin) models.112 4.47 Test 964. Frequency spectrum of the moment for the non-linear
model. Contributions of the different forces acting on the deck: self-excited forces (Fse), buffeting forces (Fbuf f) and non-linear QST forces
(FQST). . . 112
4.48 Test 964. Frequency spectrum of the moment for the linearized model. Contributions of the different forces acting on the deck: self-excited forces (Fse) and buffeting forces (Fbuf f). . . 113
4.49 Test 964. Frequency spectrum of the moment. Comparison between experimental data (Exp) and numerical solutions (Num) for both linear and non-linear models. . . 113 4.50 Test 964. Frequency spectrum of the lift. Comparison between
ex-perimental data (Exp) and numerical solutions (Num) for both linear and non-linear models. . . 113 4.51 Test 964. Comparison among the frequency spectrum of the moment
(left) and lift (right) for the four deck sections (from 1 to 4). . . 114 4.52 Test 964. Frequency spectrum of the z motion [mm]. Comparison
between experimental data (Exp) and numerical solutions (Num) for both linear and non-linear models. The linear model has been “cor-rected” with the theoretical admittance functions for V∗ ≥ 15. . . 114 4.53 Test 964. Frequency spectrum of the θ motion [deg]. Comparison
between experimental data (Exp) and numerical solutions (Num) for both linear and non-linear models. The linear model has been “cor-rected” with the theoretical admittance functions for V∗ ≥ 15. . . 114 4.54 Test 964. Frequency spectrum of the lift. Comparison between
ex-perimental data (Exp) and numerical solutions (Num) for both linear and non-linear models. The linear model has been “corrected” with the theoretical admittance functions for V∗ ≥ 15. . . 115 4.55 Test 964. Frequency spectrum of the moment. Comparison between
experimental data (Exp) and numerical solutions (Num) for both linear and non-linear models. The linear model has been “corrected” with the theoretical admittance functions for V∗ ≥ 15. . . 115
List of Tables
2.1 Comparison among static deflection results coming from integration (Rheol), non-linear static problem (NL St) and experimental data (Exp) for different wind mean velocities. . . 27 2.2 Comparison among static deflection results coming from integration
(Rheol), non-linear static problem (NL St) and experimental data (Exp) for different wind mean velocities, considering the correction in the angle of attack, α0. . . 28
2.3 Numerical and experimental comparisons of the frequency contents for test 965 with (“corrected”) and without the correction on the delay. 38 4.1 Deck modal properties. Configuration 1. . . 77 4.2 Deck modal properties. Configuration 2. . . 77 4.3 Numerical and experimental comparisons of the frequency contents
for the tests conducted at 8 m/s and 15 m/s. The error is given with respect to the numerical results. The numbering refers to the PoliMi nomenclature adopted for the benchmark. . . 95 4.4 Numerical and experimental comparisons of the frequency contents
for the tests conducted at 10 m/s and 11.8 m/s. The error is given with respect to the numerical results. The numbering refers to the PoliMi nomenclature adopted for the benchmark. (∗) tests were made with reference to configuration 2. . . 95 4.5 Numerical and experimental comparisons of the frequency contents
for the tests taken in exam. The error is given with respect to the numerical results. The numbering refers to the PoliMi nomenclature adopted for the benchmark. . . 105 4.6 Numerical and experimental comparisons of the frequency contents
for the tests conducted at 11.8 m/s. The error is given with respect to the numerical results. The numbering refers to the PoliMi nomen-clature adopted for the benchmark. Tests were made with reference to configuration 2. . . 106
Introduction
The study of wind action on long-span bridges represents a difficult task due to various aspects both related to the complexity of wind structure interaction and to the randomness of wind phenomena. The high slenderness characterizing these structures makes them very sensitive to wind buffeting dynamic action that might induce large oscillations in the response of the bridge. These significant bridge mo-tions can even be the cause of fluid-structure interaction phenomena, referred to as aeroelastic phenomena. These effects occur when the motion of the considered structure is such as to modify, in a not negligible way, the flow field and the pressure induced by the wind. Aeroelasticity might also introduce energy in the system, de-creasing the overall stiffness and/or damping of the structure and producing critical conditions of incipient instability.
Various methods for buffeting analysis of long-span bridges have been developed in the last decades both in frequency ([1], [2]) and time domain ([3, 4, 5]). However, the validation of numerical methods is not a simple task. Many difficulties are in fact encountered due to the lack of exhaustive full scale measurements on existing bridges and to the complexity in the scaling of full aeroelastic models in wind tunnel tests. Moreover, it must be considered that these validation techniques present high economical costs.
In absence of a direct numerical comparison, code validation represents a crucial problem and, for these reasons, a common interest in the definition of reliable numer-ical standards for bridge buffeting analysis has spread among the wind engineering community at the global level. Recently (November 2016), the International Associ-ation for Bridge and Structural Engineering (IABSE) has launched an internAssoci-ational benchmark project in this field. A large number of participants - from universities to engineering companies - joined in, demonstrating the great amount of interest in this subject. Models based on different methodologies have been gathered by the work group members. Codes presented either work in time domain or in frequency do-main. Politecnico di Milano (PoliMi; Milan, Italy) is participating as leader member nominating professor G.Diana as chairman of the work group. Historically, PoliMi has focused on the definition of time domain methods [6, 7] able to capture the non-linearities connected to wind-bridge interactions. Considering this scenario, I have seized the opportunity to develop and finally validate a new numerical model intended to be an alternative to the one currently present in the PoliMi database. The validation is performed comparing the results against proper wind-tunnel tests.
At the moment, there is still not a full non-linear numerical methodology to study the aeroelastic effects on bridges since the Quasi-Steady Theory (QST) [8], presented more in detail in Chapter 1, is not able to take into account the reduced
around an angle of attack that varies slowly according to the deck response to the low-frequency part of the wind velocity spectrum that can be computed by a QST approach. The small oscillations are based on the definition of the aerodynamic transfer functions (flutter derivatives and admittance functions) and, since these functions are measured and defined in the frequency domain, a rheological approach is thus used in order to exploit the same informations through the definition of several mechanical oscillators which can be integrated in time. We will see that the advantage of a time domain method not only is to assess the stability of the problem and the steady-state response but also to study its transient and to consider the actual combination of the self-excited loads with the buffeting forces. But, most importantly, we will see how a non-linear model is necessary to describe the buffeting response in presence of a high-amplitude slow-varying fluctuation in the angle of attack. For this reason, a frequency domain approach results inappropriate for this purpose.
What is presented in this thesis wants to be an extension of the current band superposition method [5] developed at PoliMi. In particular, the following steps has been followed:
• the analytical formulation of the new model is first given in a linearized context. The rheological models (RM) are thus introduced to describe the linearized action of the aerodynamic forces (only flutter derivatives will be considered). The static deflection is still modeled to achieve a “one-shot” integration; • the validation of this first model is performed comparing the results against
wind tunnel tests. The integration is simple and represents a first starting point to assess the fairness of the RM. Its formulation is very similar to a frequency method with the difference, though, of having the possibility of studyng transients and decays;
• the aerodynamic forces are generally non-linear and this is more true if large fluctuations of the angle of attack are introduced [12]. Considering this aspect, a non-linear formulation is thus introduced. The idea is always to have a “one-shot” integration and, to do so, the following aspects are considered:
– definition of a non-linear QST to describe the low frequency (LF) deck response;
– definition of the RM linearized contributions (for both flutter derivatives and admittance functions) to describe the high frequency (HF) response. In this case, the RM are provided with a new formulation since the QST is already considered;
– the HF response is modulated by the LF one, avoiding the direct super-position of the two. A moving average filter is thus introduced.
• the validation is extended also to the non-linear model comparing the results against wind tunnel tests.
The analytical formulations of the proposed models are given in Chapter 2 where all the several aspects of the modeling are discussed in detail. The validation
of the numerical codes is reported in Chapter 4 for both linearized and non-linear models. Decays for different mean wind speeds are also considered.
Chapter 1, instead, gives a closer view on the aerodynamic forces and their effects on the bridge behavior. Aerodynamic forces are, in fact, complex to model as they show a dependency on the deck motion (x, ˙x), the wind angle of attack (α) and on the reduced velocity (V∗):
Faero = Faero(x, ˙x, α, V∗)
The identification of the functions defining the aerodynamic forces requires a series of specific wind tunnel tests. Collected measurements must then be elabor-ated. They define a discrete data set and so results must be interpolelabor-ated. Deck aerodynamics identification was carried out with a series of wind tunnel tests at the PoliMi facility [26]. Since the aerodynamic forces are defined at the sectional level, the experiments were performed on a BB3 (Third Bosphorus Bridge) deck rigid sectional model. These experimental tests were part of the IABSE project. The experimental setup and procedure will be briefly described in Chapter 3 along with the results obtained.
As it will be seen, the definition of the aerodynamic forces is done at a frequency base. In order to be integrated in time, they has to be identified through the use of rheological models. In Chapter 3 this identification is presented along with the results obtained. At first, reference is made to the flat plate which represents the basic starting element due to its simple theoretical expression for the aerodynamics. In such a way it was possible to assess the fairness of the optimizator. Then, a BB3 sectional model is analyzed whose aerodynamic transfer functions were measured in the wind tunnel.
Finally, some general considerations are also provided as possible guidelines for future analyses.
Wind Action on Bridges
1.1
Introduction
In this first chapter, an overview of the definition of the non-linear aerodynamic forces on a generic bridge section will be given along with the sign conventions followed in this numerical procedure. Moreover, the flutter derivatives and admit-tance functions notations will be here presented as well as the theoretical expressions adopted inside the code.
As the wind action has a strong impact only on very slender structures, reference is made to long-span bridges. They are typically built as cable-stayed bridges or sus-pension bridges. As an example, Figure1.1 shows the Yavuz Sultan Selim Bridge, also known as Third Bosphorus Bridge (BB3), which is the bridge considered in the present work.
1.1.1
Main Aspects of Wind-Bridge Interaction
In order to understand the different problems related to the wind action on bridges, it is necessary to define the aerodynamic forces. When a 3D structure is exposed to an air flow, three force and three moment components can be generally con-sidered. However, since bridges are extended only in one direction, the primary concern regards its behaviour when wind comes perpendicular to its longitudinal
1.1. INTRODUCTION 1. WIND ACTION ON BRIDGES
Figure 1.2: Incoming wind profile on a suspension bridge
axis (Fig.1.2) and so only a total number of three actions needs to be considered: lift force FL, drag force FD and pitching moment M .
The most sensitive part of the bridge to the wind action is the deck. As a matter of fact, both the static and dynamic behaviour of the structure due to the wind action is mainly governed by the deck aerodynamics.
The aerodynamic forces applied to the generic bridge section are complex to model as they show a dependency on the deck motion (x, ˙x), the wind angle of attack (α) and on the reduced velocity (V∗):
Faero = Faero(x, ˙x, α, V∗)
The concepts of reduced velocity and angle of attack are the main keyword to describe the wind interaction and they are now introduced:
• the reduced velocity is defined as:
V∗ = V
f · B (1.1)
and it represents the ratio between the period T = 1/f associated to the body vibration and the time B/V needed by the fluid particle to move through the deck width B at a mean wind speed V . The frequency f in the reduced velocity expression can also represent – in case that the turbulence effect is considered – the frequency of fluctuation of the turbulence spectrum. The inverse of the reduced velocity is referred to as reduced frequency:
f∗ = 1 V∗ =
f · B
V (1.2)
• the angle of attack is defined with respect to a reference axis (see Fig.1.4) as:
α = θ + ψ = θ + tan−1 w − ˙z − B1 ˙ θ V + u − ˙y ! (1.3)
and it represents the angle between the relative velocity of the wind and the deck horizontal axis.
If small displacements and velocities are considered, then its linearized formu-lation can be written as:
α = θ + ψ ∼ θ + w − ˙z − B1 ˙ θ
V + u − ˙y (1.4)
Problems related to the wind action on a bridge can be distinguished into static and dynamic:
• static problems are related to the static loads exerted by the average wind speed. Such loads are function of the angle of attack;
• dynamic problems are related to the turbulence of the incoming wind and to the aerodynamic forces function of deck motion, angle of attack and reduced velocity (aeroelastic problem).
Static Problem
As previously stated, the average wind speed produces a static load that acts on all the parts of the bridge. For very long bridges, the load on the deck is the most important. In the case of suspended bridges, it is transferred, through the hangers, to the main cables and to the top of the towers, producing a very high bending moment that has a strong impact on the tower and overall bridge design. For this reason, the deck drag must be kept as small as possible, in order to reduce the load at the top of the towers.
For the sake of the analysis, the crucial aspect of the static problem stands in the definition of the configuration of static equilibrium, around which it is possible to linearize the aerodynamic forces. In practice, what really matters is the static rotation of the deck θ that determines the aerodynamic parameters that must be used to describe the aeroelastic phenomena and the forces due to the turbulence of the incoming wind.
Aerodynamic Problems
The incoming wind is actually characterized by turbulent components in the three directions, respectively u(t), v(t) and w(t), to be added to the average wind velocity (see Fig.1.3-1.4). Equation 1.3 shows that turbulence components have an impact on the definition of the wind angle of attack and consequently on the aerodynamic forces, that, therefore, change randomly in time. This variation of the aerodynamic forces produces a bridge motion induced by turbulence, called buffeting.
If the deck or any part of the bridge is moving with a given velocity in the wind flow, the forces applied to the body are functions of the relative velocity of the wind with respect to the body and the same expressions can be applied introducing Vrel
instead of V . The motion of the bridge has also an effect on the angle of attack α. More precisly, the rotation of the deck θ and the deck velocities ˙z and ˙θ concur in the definition of α, as one can observe in Eq.1.3. Body motion has therefore an impact on the aerodynamic forces. Depending on the shape of the deck, if the motion-dependent (self-excited) aerodynamic forces act in favour of the motion, they introduce energy in the system and the motion is amplified. In other words, the bridge might become unstable. Different kinds of instability can be classified:
1.2. AERODYNAMIC FORCES 1. WIND ACTION ON BRIDGES
Figure 1.3: Wind reference system.
one degree of freedom instability in the vertical or torsional mode, or two degrees of freedom instability, which results from the coupling of vertical and torsional motions. The second type of instability is also known as flutter instability.
As it will be better discussed in Section 1.2, self-excited forces can be seen as equivalent damping and stiffness terms that modify the structural properties of the bridge. Typically long-span bridges behave in the following way:
• at zero wind speed the overall damping is due only to the structural part (usually structural damping assumes values about 3-5%);
• as the wind speed increases, the contribution of the aerodynamic forces be-comes very important and the overall damping bebe-comes slightly higher; • a further increase in wind speed causes a reduction in the overall damping.
The wind speed at which the overall damping becomes negative is defined as the flutter velocity.
The prediction of the buffeting response to turbulent wind is generally secondary to the question of aerodynamic stability. However, when the bridge is proved to be stable, the bridge response to wind gusts is important for the design of the superstructure and the assessment of the user comfort by predicting the acceleration level. Moreover, the large vibrations reached by the structure may give rise to fatigue problems. Vibration amplitudes associated with buffeting can be controlled by increasing the aerodynamic damping or equivalently, by increasing the stability of the bridge. From this point of view it is important to have a high critical flutter velocity not only to be conservative on stability conditions, but also to increase the aerodynamic damping so as to reduce the turbulence-induced motions.
1.2
Description of the Aerodynamic Forces
Different analytical approaches are available to model the aerodynamic behaviour of bridges. In the following, reference is made to a single isolated deck section of length L with three degrees of freedom: horizontal displacement (y), vertical displacement (z) and rotation (θ).
1.2.1
Quasi-Steady Theory
The Quasi-Steady Theory (QST) is the most suitable approach to understand the physics of the problem [8, 9]. The QST is based on the assumption that the aer-oelastic forces, acting on the bridge deck, are not influenced by the frequency of the
Figure 1.4: Aerodynamic forces on a generic deck section: sign conventions.
deck motion. Considering this, the QST provides refined results when the reduced velocity V∗ assumes values greater than 15 − 20. High reduced velocity indicates that the time needed by a particle to cross the body is very small compared to the period of oscillation of the body. In that sense, the aeroelastic forces are the same static forces that are measured in the wind tunnel on still sectional models. With reference to Fig.1.4 their expression is:
FD = 1 2ρSV 2 relCD(α) (1.5) FL= 1 2ρSV 2 relCL(α) (1.6) M = 1 2ρSBV 2 relCM(α) (1.7)
where ρ is the air density, Vrel is the relative wind velocity, S is a reference body
area, B is the deck width, α is the angle of attack and CD(α), CL(α), CM(α) are
respectively the drag, lift and moment aerodynamic coefficients. As an example, Fig.1.5 reports the aerodynamic coefficients measured in the wind tunnel and their derivatives as function of the angle of attack for the BB3 sectional model.
The square of the relative velocity can be expressed as (see Fig.1.4):
Vrel2 = (V + u − ˙y)2+ (w − ˙z − B1θ)˙ 2 (1.8)
while the angle of attack α is still provided by Eq.1.3.
Making reference to the physical coordinates y (horizontal displacement), z (ver-tical displacement) and θ (rotation), the equations of motion of the system are:
myy + r¨ yy + k˙ yy = Fy = FDcos(ψ) − FLsin(ψ) (1.9) mzz + r¨ z˙z + kzz = Fz = FDsin(ψ) + FLcos(ψ) (1.10) mθθ + r¨ θθ + k˙ θθ = M (1.11)
where my,z,θ represent the effective inertias of the deck for the horizontal, vertical
and torsional motions, ry,z,θ the effective viscous damping and ky,z,θ the effective
1.2. AERODYNAMIC FORCES 1. WIND ACTION ON BRIDGES -10 -5 0 5 10 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 C D C L C M -10 -5 0 5 10 -2 -1 0 1 2 3 4 5 6 K D K L K M
Figure 1.5: Measured aerodyamic coefficients (left) and derivatives (right) of the BB3 sectional model.
By substituting Eq.1.8 into Eq.1.5-1.6-1.7, the equations of motion in Eq.1.9-1.10-1.11 become: myy + r¨ yy + k˙ yy = 1 2ρS (V + u − ˙y)2+ (w − ˙z − B1yθ)˙ 2 · · (CD(α)cos(ψ) − CL(α)sin(ψ)) (1.12) mzz + r¨ z˙z + kzz = 1 2ρS (V + u − ˙y)2+ (w − ˙z − B1zθ)˙ 2 · · (CD(α)(sin(ψ) + CL(α)cos(ψ)) (1.13) mθθ + r¨ θθ + k˙ θθ = 1 2ρSB (V + u − ˙y)2+ (w − ˙z − B1θθ)˙ 2 CM(α) (1.14)
where α and ψ are defined in Eq.1.3.
This formulation is actually referred to as Corrected Quasi-Steasy Theory (QSTC), since the values taken for the reference body dimensions B1y,z,θ are chosen and
“cor-rect” after taking into consideration the flutter derivatives (see Section 1.2.3), and it represents a non-linear approach to simulate the bridge response to the incoming turbulent wind, assuming that the motion frequency of the bridge does not influence the aerodynamic forces acting on it (V∗ > 15).
1.2.2
Linearized Quasi-Steady Theory
Once the aerodynamic forces are defined, a linear formulation may be considered if the hypothesis of small variations of the parameters is assumed.
By considering the displacement vector
X = Y Z Θ (1.15)
Eq.1.12-1.13-1.14 can be formulated in matrix form as:
where FA= {Fy Fz M }T and b = {u/V w/V }T.
Starting from the static deformation, the equilibrium position around which the linearization is performed is computed solving the corresponding non-linear static problem. In this way, the aerodynamic forces are function only of X0 and the mean
wind speed V ( ˙X0, b0 = 0): [Ks]X0 = FA(X0, V ) (1.17) kyy0 = 1 2ρV 2BC D(θ0) (1.18) kzz0 = 1 2ρV 2BC L(θ0) (1.19) kθθ0 = 1 2ρV 2B2C M(θ0) (1.20)
The static vector X0 = {y0z0θ0}T is calculated by means of an iterative
proced-ure starting from Eq.1.20 which is non-linear in θ0. Once θ0 is found, it is passed
to the aerodynamic coefficients of the remaining equations so as to finally compure y0 and z0.
Once X0 is calculated, the linear expression of the aerodynamic forces can be
for-mulated considering the hypothesis of small variation of deck motion and turbulent wind fluctuations close to it:
FA(X, ˙X, V, b) = FA(X0, ˙X0, V, b0) + ∂FA ∂X 0 (X − X0) + ∂FA ∂ ˙X 0 ˙ X + ∂FA ∂b 0 b (1.21) being x = y z θ = X − X0
the dynamic part of the coordinate vector.
Considering the QST formulation, the non-linear aerodynamic forces are:
Fy =
1 2ρSV
2
rel(CD(α)cos(ψ) − CL(α)sin(ψ)) (1.22)
Fz =
1 2ρSV
2
rel(CD(α)sin(ψ) + CL(α)cos(ψ)) (1.23)
M = 1 2ρSV
2
relB2CM(α) (1.24)
Now, linearizing all the terms:
CD,L,M(α) = CD,L,M(α0) + KD,L,M(α0) · (α − α0)
= CD,L,M(α0) + KD,L,M(α0) · (θ + ψ)
KD,L,M(α) =
dCD,L,M
1.2. AERODYNAMIC FORCES 1. WIND ACTION ON BRIDGES ψk ≈ w − ˙z − Bkθ˙ U + u − ˙y ≈ w − ˙z − Bkθ˙ U Vrel2 ≈ V2+ 2V u − 2V ˙y cos(ψ) ≈ 1 sin(ψ) ≈ ψ
and substituting in Eq.1.22-1.23-1.24, what is obtained is
Fy = 1 2ρV 2B ( CD0− " 2CD0 ˙ y V + (KD0− CL0) ˙z V + Byθ˙ V ! − KD0θ # + + " 2CD0 u V + (KD0− CL0) w V #) (1.25) Fz = 1 2ρV 2B ( CL0− " 2CL0 ˙ y V + (KL0+ CD0) ˙z V + Bzθ˙ V ! − KL0θ # + + " 2CL0 u V + (KL0+ CD0) w V #) (1.26) M = 1 2ρV 2B2 ( CM 0− " 2CM 0 ˙ y V + (KM 0) ˙z V + Bθθ˙ V ! − KM 0θ # + + " 2CM 0 u V + (KM 0) w V #) (1.27)
which can be expressed in matrix formulation as follows
FA,Stat = Fy0 Fz0 M0 = 1 2ρV 2B CD0 CL0 B CM 0 (1.28) FA,Dyn = Fy Fz M = −1 2ρV 2B 0 0 −KD0 0 0 −KL0 0 0 −B KM 0 y z θ −1 2ρV B 2CD0 (KD0− CL0) (KD0− CL0)By 2CL0 (KL0+ CD0) (KL0+ CD0)Bz 2CM 0B KM 0B KM 0BBθ ˙ y ˙z ˙ θ +1 2ρV 2B 2CD0 (KD0− CL0) 2CL0 (KL0+ CD0) 2CM 0B KM 0B u V w V (1.29) Finally, a compact notation can be adopeted
FA= FA,Stat+ FA,Dyn
= FA,Stat+ Fse+ Fbuf f
= FA,Stat− ([Kaero]x + [Raero] ˙x) + [Am]b
where FA,Stat is the static component due to the mean velocity, Fseis the self excited
component due to the deck motion and Fbuf f is the force component related to the
turbolence effects.
The linearized equations of motion can be thus derived my 0 0 0 mz 0 0 0 Jθ ¨ y ¨ z ¨ θ + ry 0 0 0 rz 0 0 0 rθ ˙ y ˙z ˙ θ + ky 0 0 0 kz 0 0 0 kθ y + y0 z + z0 θ + θ0 = FA (1.31) [Ms] ¨x + [Rs] ˙x + [Ks](x + x0) = FA (1.32)
Substituting Eq.1.30, we obtain:
[Ms] ¨x + ([Rs] + [Raero]) ˙x + ([Ks] + [Kaero])x = [Am]b (1.33)
Notice that the contribution of the static load cancels out with the static deflec-tion [Ks]x0 since it is calculated considering Eq.1.17.
The structural damping and stiffness matrices can be evaluated knowing the structural natural frequencies and damping ratios.
[RS] = ry 0 0 0 rz 0 0 0 rθ = 4πζ my· fy 0 0 0 mz· fz 0 0 0 Jθ· fθ [KS] = ky 0 0 0 kz 0 0 0 kθ = (2π) 2 my · fy2 0 0 0 mz· fz2 0 0 0 Jθ· fθ2
Should be noted that [Kaero], [Raero] and [Am] matrices provided by the linearized
QST does not depend on the reduced velocity. Respectively, [Raero] ˙x and [Kaero]x
define the aerodynamic forces function of the deck motion (self-excited forces), while [Am]b identifies the aerodynamic forces function of the incoming turbulence
(buf-feting forces). The presence of the equivalent stiffness matrix [Kaero] modifies the
natural frequencies of the bridge with respect to those computed from [Ks] and [Ms]
in absence of wind. Similarly, the equivalent damping matrix [Raero] modifies the
system overall damping. Extra-diagonal elements in [Raero] and [Kaero] represent
coupling terms. The coupling introduced between the different types of motion de-pend on the values of the aerodynamic coefficient of the deck section and on the average wind speed. Depending on their values, the terms contained in [Raero] and
[Kaero] might give rise to 1 d.o.f. or 2 d.o.f. instability.
1.2.3
Flutter Derivatives and Admittance Functions
One of the major limitations of the Quasi-Steady Theory is the fact that is valid only for high reduced velocities (V∗ ≥ 15−20) which, commonly, means for low frequency fluctuations. This, consequently, brings us to a formulation of the aerodynamic forces which is no more dependent on V∗. In fact, the aerodynamic coefficients CD,
1.2. AERODYNAMIC FORCES 1. WIND ACTION ON BRIDGES
In order to overcome this problem and to extend the formulation also to the low reduced velocity range, the self-excited and buffeting forces can be expressed through the use of the flutter derivatives and admittance functions which are trasfer functions experimentally measured in the wind tunnels for different mean angles of attack. In fact, in common practice, the definition of the aerodynamic coefficients of the equivalent damping/stiffness and admittance matrices is done through specific wind tunnel tests [26]. We will see in Section 4.2 how they are experimentally measured making reference to the BB3 sectional model.
Of course, the conclusions that have been worked out making reference to the QST are still valid.
Flutter Derivatives
According to PoliMi notation [13], the aerodynamic self-excited forces can be written as: Dse= 1 2ρU 2B −p∗ 1 ˙z U − p ∗ 2 B ˙θ U + p ∗ 3θ + 2π3 V∗2p ∗ 4 z B − p ∗ 5 ˙ y U + 2π3 V∗2p ∗ 6 y B ! (1.34) Lse= 1 2ρU 2B −h∗ 1 ˙z U − h ∗ 2 B ˙θ U + h ∗ 3θ + 2π3 V∗2h ∗ 4 z B − h ∗ 5 ˙ y U + 2π3 V∗2h ∗ 6 y B ! (1.35) Mse= 1 2ρU 2B2 −a∗ 1 ˙z U − a ∗ 2 B ˙θ U + a ∗ 3θ + 2π3 V∗2a ∗ 4 z B − a ∗ 5 ˙ y U + 2π3 V∗2a ∗ 6 y B ! (1.36)
where p∗j are the flutter derivatives for the drag force, h∗j are the flutter derivatives for the lift force and a∗j the flutter derivatives for the pitching moment.
Since they are measured in a linearized context, the flutter derivatives have the same meaning of the aerodynamic stiffness and damping matrices presented in the linearized QST (see Section 1.2.2). In fact, in matrix formulation, they can be expressed as: Fse= 1 2ρV B p∗5 p∗1 p∗2B h∗5 h∗1 h∗2B a∗5B a∗1B a∗2B2 ˙ y ˙z ˙ θ + 1 2ρV 2B p∗6 π 2V∗ ω 2B p ∗ 4 π 2V∗ ω 2B p ∗ 3B h∗6 π 2V∗ ω 2 B h ∗ 4 π 2V∗ ω 2 B h ∗ 3B a∗6B a∗1B a∗3B2 y z θ (1.37) [RA] = 1 2ρV B p∗5 p∗1 p∗2B h∗5 h∗1 h∗2B a∗5B a∗1B a∗2B2 [KA] = 1 2ρV 2B p∗6 π 2V∗ ω 2B p ∗ 4 π 2V∗ ω 2B p ∗ 3B h∗6 π 2V∗ ω 2B h ∗ 4 π 2V∗ ω 2B h ∗ 3B a∗6B a∗1B a∗3B2 where Vω∗ = V∗/(2π).
-6 -4 -2 0 2 4 6 1 1.1 1.2 1.3 1.4 0 5 10 15 20 0.9 1 1.1 1.2 1.3 1.4 -4° -2° 0° 2° 4°
Figure 1.6: Comparison between the last values of the experimental flutter derivatives a∗3 of the BB3 sectional model with the corresponding QST values of KM for dfferent mean angles of
attack.
It is important to compare the [KA] and [RA] matrix coefficients to those
eval-uated through the QST. At high values of V∗, in fact, they tend to converge. An example is reported in Fig.1.6, where the trend for V∗ → ∞ of the experimental flutter derivatives a∗3 of the BB3 sectional model is compared with the corresponding QST value to which should tend, that is the derivative of the moment coefficient KM. This is highlighted for the different mean angles of attack available. We see
that the results are in agreement even if the match is not perfect. However, it should be noticed that a right comparison has to be done for V∗ → ∞, condition which cannot be experimentally satisfied as we will see. Thanks to that, it is possible to define the values of B1y, B1z and B1θ of the QSTC by the ratio of the [Raero]
matrix coefficients experimentally identified at high reduced velocity. An example is given in Fig.1.7 where the B1z and B1θ coefficients of the BB3 sectional model
are reported against reduced velocity for five different mean angles of attack. More precisely they are defined as:
B1y = p∗2 p∗ 1 B1z = h∗2 h∗ 1 B1θ = a∗2 a∗ 1 (1.38) This formulation, that uses the flutter derivatives to correct the QST, is what it is called Correct Quasi-Steady Theory (QSTC). If the QSTC is used, the following relationships, at high reduced velocity, hold:
• flutter derivatives from p∗
1 to p∗6
p∗1 = KD − CL p∗2 = (KD− CL)B1y p∗3 = KD
p∗4 = 0 p∗5 = 2CD p∗6 = 0
(1.39)
• flutter derivatives from h∗
1 to h ∗ 6 h∗1 = KL+ CD h∗2 = (KL+ CD)B1z h∗3 = KL h∗4 = 0 h∗5 = 2CL h∗6 = 0 (1.40)
• flutter derivatives from a∗
1 to a ∗ 6 a∗1 = KM a∗2 = KMB1θ a∗3 = KM a∗4 = 0 a∗5 = 2CM a∗6 = 0 (1.41)
1.2. AERODYNAMIC FORCES 1. WIND ACTION ON BRIDGES 0 5 10 15 20 25 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 -4° -2° 0° 2° 4° 0 5 10 15 20 25 0.2 0.25 0.3 0.35 0.4 -4° -2° 0° 2° 4°
Figure 1.7: Trends of B1zand B1θof the BB3 sectional models against reduced velocity for
different mean angles of attack.
Admittance Functions
The same considerations apply for the buffeting forces. The turbulence charac-teristics of the wind are combined with aerodynamic functions, called admittance functions, which convert the air flow properties into wind loads on the structure. They can be expressed as:
Dbuf f Lbuf f Mbuf f = 1 2ρBV 2 χ∗Du χ∗Dw χ∗Lu χ∗Lw Bχ∗M u Bχ∗M w u Vw V (1.42)
where χ∗(V∗) are complex functions called admittance functions and are represented as amplitude and phase: χ∗ = |χ|eiφ.
Since they are measured in a linearized context, the admittance functions have the same meaning of the aerodynamic admittance matrix [Am] presented in the
linearized QST (see Section 1.2.2).
Admittance functions χ∗ can either be measured (wind tunnel tests) or approx-imated using analytical expressions derived for a thin airfoil. Various formulations are available in literature but one of the most commonly used is the Davenport’s expression. In the early sixties, Davenport (1962) proposed a model for calculat-ing fluctuatcalculat-ing wind load on a structure located in the atmospheric boundary layer ([10], [11]). Its expression was used to compensate lack of informations in the ex-perimental data. According to Davenport, the admittance functions can be written as:
χ∗yu = 2CD · A(f∗) χ∗zu= 2CL· A(f∗) χ∗θu = 2CM · A(f∗)
χ∗yw = (KD− CL) · A(f∗) χ∗zw= (KL+ CD) · A(f∗) χ∗θw = KM · A(f∗)
(1.43) where f∗ is the reduced frequency (f∗ = 1/V∗) and A(f∗) is a real function that weights the quasi-steady values of buffeting forces:
A(f∗) = 2 (7f∗)2(7f
∗− 1 + e−7f∗
The value of A(f∗) as a function of the reduced frequency/velocity is reported in Fig.1.8. 0 10 20 30 40 50 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1
Figure 1.8: Davenport’s expression of the real function A versus reduced velocity V∗ and reduced frequency f∗.
1.3
Band Superposition Approach
In order to better understand the concepts of the new non-linear method presented in this thesis, a short overview is now given on the main aspects introduced by this model making reference to the Band Superposition (BS) approach developed at PoliMi [5]. The main objective is to understand that the non-linear aerodynamic forces may be seen as a low frequency contribution modeled by a non-linear QST and as a high frequency contribution modelled by several rheological models.
The band superposition approach was developed following the consideration that even if the bridge deck aerodynamics is dependent on the amplitude of the fluctu-ations of the angle of attack and on the reduced velocity, it is possible to separate the LF (low frequency) response of the bridge to turbulent wind from the HF (high frequency) response.
The separation is feasible since the two regions show different features. The features of the LF response are as follows:
• the aerodynamic forces show a small dependence on the reduced velocity, while they depend on the angle of attack. As an example to support this statement, Fig.1.9 reports the trend of the unsteady aerodynamic moment coefficient (flutter derivatives for torsional motion) as a function of the reduced velocity, for different mean angles of attack;
• large variations of the instantaneous angle of attack are mainly related to the fluctuation of the wind velocity components, since small oscillations of the bridge are expected for stable aerodynamic solutions. These variations are caused by the large turbulent scales present in the atmospheric turbulent wind. The large turbulent structures that are able to generate fluctuations of the angle of attack that could be effective on the bridge deck are, in fact, those having large length scale (usually several times larger than the deck chord) and long time scale (low frequency).
1.3. BAND SUPERPOSITION APPROACH 1. WIND ACTION ON BRIDGES
The features of the HF response are as follows:
• the aerodynamic forces show large dependence both on the reduced velocity and on the mean angle of attack;
• the fluctuation of the high reduced frequency wind velocity components are characterized by small turbulence length scales with respect to the deck chord length. These produce a flow field variation around the deck section that does not contribute to the global angle of attack.
(a)
(b)
Figure 1.9: Flutter derivatives of the Messina bridge [5]. a∗2(a) and a∗3 (b) vs. V∗ for nine
different mean angles of attack α.
The previous considerations imply that, when large fluctuations of the angle of attack occur, in the LF range, a linear modeling of the aerodynamic forces is not appropriate in presence of strong non linear dependence of the forces on this parameter [12]. In this case, a time domain approach, such as a quasi-steady theory (QST), that is able to consider the force dependence on large fluctuations of the angle of attack but not on the reduced velocity is appropriate, and this is why a frequency-domain method cannot be implemented.
The basic hypothesis of the BS approach is that the mean angle of attack, per-ceived by the deck in the definition of the aerodynamic force components acting in the HF range, is not only the static mean value but it is also related to the fluctu-ation that happens at low frequency with high amplitude (perceived as static region for the HF terms). Therefore the method suggests that the HF aerodynamic forces are related to the instantaneous angle of attack computed considering the LF deck response and wind spectrum and not to the static mean angle of attack. If we now consider that the fluctuation of the instantaneous angle of attack in the HF range are small, this means that the problem may be modeled using a linearization of the aerodynamic forces evaluated around the LF instantaneous angle of attack. Under this assumption, it is possible to model the aerodynamic forces in the HF range by means of the aerodynamic transfer functions that are commonly expressed in bridge aerodynamics through the flutter derivatives and admittance function coefficients, usually measured by wind tunnel tests at different reduced velocities and mean angles of attack. Therefore, the unstedy HF aerodynamic forces can be written as:
Funsteady(t) = Fse(αLF(t), xHF, ˙xHF) + Fbuf f(αLF(t), wHF, uHF) (1.45)
Eq.1.45 highlights that unsteady forces depend explicitly on the LF angle of attack and on HF deck motion and HF incoming turbulence.
If the BS hypothesis held, the bridge deck response to turbulent wind might be performed by dividing the problem in two parts: a non-linear problem at low fre-quency modeled using a corrected quasi-steady approach, and a non-linear problem at high frequency modeled using a modulation of the aerodynamic transfer functions performed on the basis of the LF fluctuation of the angle of attack.
Moreover, the BS approach exploits rheological models (RM) to contemporary reproduce the reduced velocity and the angle of attack dependence in time domain on the whole HF band without asking for a HF band decomposition. The developed RM, adopted for the HF band response, is a mechanical model that is made by a group of mechanical elements whose response to a deformation, proportional to the fluctuation of the angle of attack, produces a force that represents the aerodynamic load acting on the deck. The identification of the parameters of the single mechanical element in this case is based on the information contained in the flutter derivatives and admittance function coefficients measured at different mean angles of attack and different reduced velocity.
With a single morphology of the mechanical system it is possible to reproduce the aerodynamic transfer function dependence on the reduced velocity and angle of attack by using parameters of the RM that are function of the angle of attack. The RM are therefore able to provide a continuous interpolation of the aerodynamic transfer function in the field where V∗ and α were varied during flutter derivatives and admittance functions measurements.
The basic algorithm of the BS approach proposed by PoliMi [5] is reported in Fig.1.10 in which three main steps are highlighted: identification of a threshold for separeting the wind bands, solution of the LF deck reponse by means of a non-linear corrected QST and solution of the HF deck response by introducing the rheological models.
The threshold separating the LF range from the HF range has to be defined in terms of reduced velocity V∗ or reduced frequency f∗. The threshold delimits
1.3. BAND SUPERPOSITION APPROACH 1. WIND ACTION ON BRIDGES
Figure 1.10: Schematization of the main algorithm of the band superposition approach [5].
the region, at high reduced velocity, where the flutter derivatives coefficients and the aerodynamic admittance function coefficients show a small dependence on the reduced velocity. Once the V∗ threshold is defined, it can be used to separate the wind spectrum into two bands.
1.3.1
A New Model Idea
The model proposed in this thesis is intended to be an exstension of the approach reported in Fig.1.10. It is similar in many ways but it tries to avoid a sharp separation between LF and HF bands. In fact, the new approach introduces this modifications:
• no separation of the wind bands;
• no separated integrations.
The idea is to have a “one-shot” integration which gathers all the contributions at once made of:
• static deflection; • LF response; • HF response.
Static deflection and LF response are modeled by a non-linear QST, while the HF response is modeled by the RM and it is modulated by the LF variation of the angle of attack. Moreover, since in the new model the QST is forced to work also in the HF band, the RM will be considered in a new formulation. They are therefore introduced as linearized terms intended to provide the right compensation and de-pendence on V∗. The linearization of the HF contributions is possible since the HF fluctuations are small compared to the LF ones as seen in the previous section. Of course, the LF variation of the angle of attack is still needed in order to modulate the HF response and, for this purpose, a moving average is introduced. In this way, it is possible to provide a countinously updating low-passing filter avoiding the ne-cessity of separating the integration.
Now, in the present chapter all the main aspects have been provided, which are: • description of the non-linear aerodynamic forces;
• non-linear QST;
• flutter derivatives and admittance functions; • BS approach.
In the following chapter, these aspects are further investigated in order to provide the conceptual and analytical model of the new non-linear approach. This, of course, will be done sistematically, starting with a linearized fomulation of the problem so as to decrease the complexity and to have an initial validation of this procedure. Finally, the non-linear formulation will be provided in detail.
The only missing ingredient so far is the definition of the RM. Their introduction is an important aspect and it will discuss in a dedicated chapter (see Chapter 3). We will see that the RM aim to exploit the same informations of the aerodynamic transfer functions (flutter derivatives and admittance functions) in the time domain. This, however, was one of the possible solutions which could be adopted to deal with the problem. Other approaches can be found in literature, such as impulse methods [14] or Volterra models [15]. Historically, PoliMi focused his attention on the definition of rheological models which could be described through the definition of several mechanical oscillators. In this way, through a constrained optimization, it is possible to tune the oscillators to have the same frequency content of the aerodynamic trasfer functions so as to implicitly maintain the dependency on the reduced velocity. On the other hand, this comes with a price. In fact, we will see that the introduction of the RM imposes to write more equations of motion since we are adding more degrees of freedom to the problem.
Chapter 2
Analytical Modeling of the New
Non-Linear Approach
2.1
Introduction
Once all the main aspects related to the definition of the non-linear aerodynamic forces have been introduced, in the following chapter the analystical formulation of the new proposed approach will be presented. We will focus on the equations which govern the motion of a generic deck sectional model analyzing all the several aspects connected. More in detail, we will discuss how they were derived and managed in order to get a compact notation that could be numerically integrated in time. This will be done gradually.
In the first part, a linearized formulation will be analyzed considering small oscillations of the angle of attack so as to better understand the behaviuor and the contribution of the rheological models. For this porpuse, a similar formulation of the linearized QST will be presented.
In the second part, the integration will be done adding the non-linear corrected QST. The idea is to describe the low frequency oscillation of the deck by using the Quasi-Steady Theory which proves itself to be suitable for this purpose and to compensate the high frequency oscillations by introducing the rheological models.
2.2
Linearized Formulation
In the following, the new numerical model will be presented in a linearized context where we want to invetigate the non-linear contributions for small reduced velo-cities. For this purpose, a linearized formulation of the problem will be presented through the definition of a linearized QST. However, despite this, it should be poin-ted out that the static deflection is still considered in order to achieved a “one-shot” integration. So, such a model can detect slow variations in the static loads.
The rheological model will be introduced when dealing with the self-excited forces. Once the parameters are identified, it is possible to built the equivalent aerodynamic damping and stiffness matrices which are finally assembled with the structural ones.
Figure 2.1: Schematization of the elastically suspended deck section.
2.2.1
Modal Equations
The bridge has its natural modes to vibrate defined as vertical, horizontal and tor-sional motions. To understand the bridge behaviour under turbulent wind conditions and the flutter instability it is sufficient to consider the first vertical, torsional and horizontal modes.
The problem can be therefore modeled as in Fig.2.1 where y and z stand for the horizontal and vertical displacement components and θ is the rotation.
The equations of motion can be thus derived: my 0 0 0 mz 0 0 0 Jθ ¨ y ¨ z ¨ θ + ry 0 0 0 rz 0 0 0 rθ ˙ y ˙z ˙ θ + ky 0 0 0 kz 0 0 0 kθ y z θ = Fy Fz M = FA [Ms] ¨x + [Rs] ˙x + [Ks]x = FA
where FA is the vector of the aerodynamic forces which, due to the linearized
ap-proach, can be expressed as:
FA= Fstat+ Fse+ Fbuf f (2.1)
in which Fstat represents the static forces, Fse the self-excited forces and Fbuf f the
buffeting forces (see Chapter 1).
2.2.2
Aerodynamic Forces
As seen in the previous section, the aerodynamic forces can be expressed as
FA= Fstat+ Fse+ Fbuf f (2.2)
in which Fstat represents the static forces, Fse the self-excited forces and Fbuf f the
buffeting forces.
Although we are dealing with a linearized formulation, please notice that the contribution of the static forces is still present since we want to achieve a “one-shot” integration.
2.2. LINEARIZED FORMULATION 2. ANALYTICAL MODELING
Self-excited Forces
The self-excited forces, which are the ones related to the induced motion of the deck, can be written in matrix form if a linearized approach is used. Since we want to investigate the contributions for small reduced velocities, the use of the flutter derivatives is more appropriate. However, the flutter derivatives are trasfer functions defined and measured in the frequency domain (see Section 4.2), consequently, in order to exploit the same information in the time domain, the rheological models are introduced. The use of the rheological models seen as mechanical oscillators, in fact, allows to define and build the same equivalent aerodydamic matrices but taking into account the values of the springs, dampers and masses which define the type of oscillator. In this way, the models can be easily integrated providing, implicitly, the dependency on the reduced velocity.
Once these parameters are defined and properly tuned, we end up with matrices which can be finally assembled with the structural ones. Unfortunately, the intro-duction of the rheological models forces us to add more equations of motion to the problem since they are providing additional degrees of freedom. This means that a new state vector must be introduced along with a new formulation of the matrices involved. Once this is done (see Appendix A), as a final result, the whole problem can be summarized as follows:
[Atot] ˙z + [Btot]z = Ftot (2.3)
where z = ˙ x x ˙ η η µ (2.4)
is the state vector, and Ftot is the vector of the remaining acting forces given as
Ftot = F0+ F0∗+ FB (2.5) F0 = Fstat 0 F0∗ = −[Dtot]z0 FB = Fbuf f 0 (2.6) where F0∗ represents a compensation of the static loads as will be better explained later when dealing with the static forces.
Notice that η and µ represent the additional degrees of freedom introduced by the rheological models.
Buffeting Forces
The buffeting forces Fbuf f are due to the turbolent motion of the incoming wind and
they can be represented by introuducing the admittance functions. So, they can be written as
![Figure 1.10: Schematization of the main algorithm of the band superposition approach [5].](https://thumb-eu.123doks.com/thumbv2/123dokorg/7525021.106403/35.892.166.703.115.779/figure-schematization-main-algorithm-band-superposition-approach.webp)
