POLITECNICO DI MILANO
School of Civil, Environmental and Land Management Engineering
Master of Science Degree in Civil Engineering
A Frequency Domain Method for Buffeting Analysis of
Long-Span Bridges
Supervisor:
Prof. Daniele Rocchi Co-Supervisors:
Ing. Tommaso Argentini Ing. Simone Omarini
Dissertation of: Alex Pellegri 862673 Andrea Tornielli 862663
i
Abstract
The need to overpass obstacles on very long distances typically implies the desi-gn of "streamlined" closed-box bridge decks. The size and lightness characterizing these structures makes them very sensitive to dynamic problems and special care must be put in the design against the buffeting action of wind turbulence. Dealing with this aspect, many difficulties are involved both in the wind description and in the computation of the bridge response. These challenges are addressed in the pre-sent thesis with the implementation of a numerical model working in the frequency domain.
Numerical wind generation was carried out with a frequency domain re - elabora-tion of an existing procedure based on empirical expressions. A series of wind-tunnel tests, performed at Politecnico di Milano, provided the parameters needed to descri-be the aerodynamic forces acting on the bridge. These experimental tests 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).
Various simulations have been run to test the numerical procedure and compa-risons were made with other methods developed by the participants of the IABSE project. Code validation was done in a systematic way, starting from very simple models and increasing the degree of complexity until the procedure was assessed robust enough to sustain the analysis of a full bridge. Furthermore, specific wind-tunnel experiments were used to have a direct comparison between experimental and numerical response.
Finally, some parametric analysis were conducted to deepen the understan-ding of finer aspects involved in long-span bridge buffeting analysis. Some general considerations were later assessed aiming to become guidelines for future analysis.
Sommario
Il progresso tecnologico apre costantemente nuove possibilità nella costruzione di ponti di grande luce e al giorno d’oggi nuove sfide ingegneristiche vengono affron-tate in ogni parte del mondo. La necessità di sorpassare ostacoli su distanze molto lunghe favorisce la progettazione di ponti a cassone molto snelli. Le dimensioni e la leggerezza che caratterizzano queste strutture le rendono molto sensibili ai proble-mi dinaproble-mici e, in particolare, occorre prestare molta attenzione nella progettazione contro l’azione del vento turbolento. In tal senso, si riscontrano molte difficoltà sia nella descrizione del vento che nel calcolo della risposta del ponte. Queste problema-tiche vengono approfondite nella presente tesi con l’implementazione di un metodo numerico nel dominio della frequenza.
La generazione numerica del vento è stata eseguita attraverso una rielaborazione in frequenza di una procedura esistente basata su espressioni empiriche. I parame-tri necessari a descrivere le forze aerodinamiche agenti sul ponte sono stati ricavati grazie a una serie di prove sperimentali in galleria del vento, svoltesi presso il Poli-tecnico di Milano. Tali prove fanno parte di un progetto internazionale per la defini-zione di benchmark condotto dal Politecnico di Milano e promosso dall’International Association for Bridge and Structural Engineering (IABSE).
Diverse simulazioni sono state condotte per testare il metodo numerico e sono stati eseguiti confronti con altri metodi sviluppati dai partecipanti al progetto IAB-SE. La validazione del codice è stata eseguita in maniera sistematica, a partire da modelli molto semplici e incrementando man mano il grado di complessità fintanto-ché la procedura è stata ritenuta sufficientemente robusta per l’analisi di un ponte intero. Inoltre, alcune prove in galleria del vento sono state utilizzate per fare dei confronti numerico-sperimentali.
Infine, sono state svolte alcune analisi parametriche per indagare in maniera più approfondita alcuni aspetti legati all’analisi di buffeting dei ponti di grande luce. Al termine di ciò, sono state fatte alcune considerazioni al fine di delineare delle linee guida per analisi future.
Contents
List of Figures vii
List of Tables xiii
Introduction 1
1 Wind Action on Bridges 5
1.1 Introduction . . . 5
1.1.1 Main aspects of wind-bridge interaction . . . 6
Static problem . . . 7
Aerodynamic problems . . . 8
1.2 Description of the aerodynamic forces . . . 10
1.2.1 Quasi-Steady Theory . . . 10
Linearised equation of motion . . . 12
1.2.2 Forces identification by means of wind tunnel tests . . . 15
Flutter derivatives . . . 15
Admittance functions . . . 20
1.3 Response in the frequency domain . . . 22
1.3.1 Frequency Response Function . . . 22
2 Wind Numerical Generation 25 2.1 Introduction . . . 25
2.2 Wind description . . . 26
2.2.1 Mean wind velocity profile . . . 28
2.2.2 Wind turbulence . . . 29
Turbulence intensity . . . 29
Integral length scales . . . 29
Power spectral density . . . 30
2.2.3 Point coherence of wind turbulence . . . 32
2.2.4 Spatial coherence of wind turbulence . . . 33
Spatial coherence of the same turbulence components . . . 34
Spatial coherence of different turbulence components . . . 35
Spatial coherence of the aerodynamic forces . . . 36
2.3 Generation of buffeting input . . . 37 iii
2.3.1 Input for time domain methods . . . 37
AR and ARMA methods . . . 37
Shinozuka-Deodatis method . . . 38
2.3.2 Input for frequency domain methods . . . 42
Modified Shinozuka-Deodatis method . . . 43
3 Computation of the Full Bridge Response 49 3.1 Introduction . . . 49
3.2 Modal structural information . . . 49
3.3 Equations of motion with the modal approach . . . 52
3.3.1 Static forces . . . 54
3.3.2 Self-excited forces . . . 55
3.3.3 Buffeting forces . . . 57
3.4 Methods of analysis . . . 59
3.4.1 Frequency domain methods . . . 59
3.4.2 Time domain methods . . . 62
Rheological model approach . . . 62
3.5 Post-process . . . 65
4 Wind Tunnel Testing 67 4.1 Introduction . . . 67
4.2 Polimi Wind Tunnel . . . 68
4.2.1 Boundary layer test section . . . 69
4.3 Experimental Setup . . . 69
4.3.1 Deck sectional model . . . 71
4.3.2 Forcing system . . . 73
Hydraulic actuators . . . 73
Active turbulence generator . . . 73
4.3.3 Wind flow measurements . . . 75
Pitot tube . . . 75
Cobra Probe . . . 76
4.3.4 Force measurements . . . 77
Surface pressure measurements . . . 77
Force balance . . . 79
4.3.5 Motion measurements . . . 80
4.4 Tests on the Third Bosphorus Bridge Sectional Model . . . 81
4.4.1 Static tests . . . 81
4.4.2 Dynamic tests . . . 87
Vertical harmonic motion . . . 87
Torsional harmonic motion . . . 91
QSTC flutter derivatives . . . 94
4.4.3 Admittance functions . . . 94
CONTENTS v
5 Numerical Model Validation 101
5.1 Introduction . . . 101
5.2 Sectional Model . . . 102
5.2.1 Numerical models comparison . . . 102
5.2.2 Numerical-Experimental validation . . . 106
Response to mono-harmonic turbulence excitation . . . 108
Response to bi-harmonic turbulence excitation . . . 110
5.3 Multi-sectional model . . . 113
5.3.1 Numerical models comparison . . . 115
5.3.2 Rigid and deformable models comparison . . . 117
6 Parametric Analysis 121 6.1 Introduction . . . 121
6.2 Wind Grid Resolution . . . 128
6.3 Different Wind Turbulence Components Coherence . . . 138
6.4 Span-wise Variation of Mean Wind Velocity . . . 145
Conclusions 155 A Aeroelastic problems 159 A.1 One degree of freedom instability . . . 159
A.1.1 Vertical dynamic instability . . . 160
A.1.2 Torsional dynamic instability . . . 160
A.2 Flutter instability . . . 161
B Definition of Modal Matrices 163
C Structural and Wind Grids 165
D IABSE benchmark project 167
List of Figures
1 Incoming wind profile on a suspension bridge. . . 2
2 Third Bosphorus Bridge, FEM model. . . 3
1.1 Storebaelt Bridge, Denmark 1998 [1]. . . 5
1.2 Aerodynamic forces on a generic deck section. . . 6
1.3 Wind reference system. . . 8
1.4 Aerodynamic forces: sign convention. . . 10
1.5 Elastically suspended deck section. . . 11
1.6 Theodorsen circulatory function C(f∗), real and imaginary parts. . . 17
1.7 Components of Theodorsen circulatory function versus reduced fre-quency f∗ and reduced velocity V∗. . . 18
1.8 Flutter derivatives h∗ 1−4, theoretical expression for a flat plate (PoliMi notation). . . 19
1.9 Flutter derivatives a∗ 1−4, theoretical expression for a flat plate (PoliMi notation). . . 19
1.10 A(f∗) versus reduced frequency f∗. . . 21
1.11 A(V∗) versus reduced velocity V∗. . . 21
2.1 Yavuz Sultan Selim Bridge: even world’s tallest bridges are fully im-mersed in the ABL. . . 25
2.2 Spectrum of horizontal wind speed after Van der Hoven. . . 27
2.3 Wind profiles on different terrain roughness [2]. . . 28
2.4 Integral length scales of the longitudinal turbulence component u. . . 30
2.5 Non-dimensional power spectral density according to Von Karman and Eurocode 1. . . 31
2.6 Spectra of the three components of wind turbulence compared with the structural frequencies of some slender structures. . . 32
2.7 Space coherence function: measured data in wind tunnel simulation of the atmospheric boundary layer. . . 34
2.8 Space coherence function of different turbulence components: mea-sured data in wind tunnel simulation of the Akashi-Kaikyo bridge full aeroelastic model test. . . 35
2.9 Generation of a separation bubble in the flow over a sharp-edged body. 36 2.10 Shinozuka-Deodatis method: outer frequency discretization. . . 40
2.11 Shinozuka-Deodatis method: inner frequency discretization. . . 41
2.12 Shinozuka-Deodatis method: partitioning of the wind energy. . . 41
2.13 Shinozuka-Deodatis method (three nodes): FFT of wind turbulence component at first node over a single frequency interval (∆f). . . 44
2.14 Shinozuka-Deodatis method (three nodes): FFT of wind turbulence component at second node over a single frequency interval (∆f). . . . 45
2.15 Shinozuka-Deodatis method (three nodes): FFT of wind turbulence component at third node over a single frequency interval (∆f). . . 45
2.16 Shinozuka-Deodatis method (three nodes): complete set of input FFT for the frequency domain method. . . 46
2.17 Graphic representation of the analysis input for time domain methods (time histories of the aerodynamic forces). . . 48
2.18 Graphic representation of the analysis input for our frequency domain method (FFT of the aerodynamic forces). . . 48
3.1 Flow chart of the numerical procedure implemented in our frequency domain method. . . 50
3.2 FEM of a bridge with beam elements. . . 51
3.3 From the continuous bridge to the discrete one: (a) portion of a bridge subdivided in rigid deck sections (wind sections); (b) i-th wind section with the applied aerodynamic forces. . . 53
3.4 Frequency domain: PSD of the response of a rigid deck section with increasing velocity; flutter velocity V = 63m/s. . . 61
3.5 Scheme of the rheological model. . . 63
3.6 Comparison of low and high frequency contributions on the angle of attack α [3]. . . 63
4.1 Closed Circuit Facility. . . 67
4.2 Axial Fans in parallel [4]. . . 68
4.3 Boundary Layer Test Section. . . 69
4.4 Model linked to oil dynamic actuators. . . 70
4.5 Model suspended by means of steel cables. . . 70
4.6 Suspended configuration: (a) control room side; (b) spring fixed at the base of the steel frame. . . 71
4.7 Deck section dimensions and shape. . . 71
4.8 Deck section dimensions [mm]; top view. . . 72
4.9 End-plates placed at the extremities of the model. . . 72
4.10 Oil dynamic actuators: (a) actuators L2 and L3 ; (b) actuator L1 (control room side). . . 73
4.11 Active turbulence generator made by a horizontal array of 10 NACA0012 profile airfoils. . . 74
4.12 Details of the active turbulence generator: (a) array of NACA0012 profile airfoils ; (b) connection with the brushless motor. . . 74
LIST OF FIGURES ix 4.13 Standard pitot tube: (a) installed pitot tube; (b) picture from Pope,
low-speed wind tunnel testing. . . 75
4.14 Series 100 Cobra Probe (Turbulent flow instrumentation). . . 76
4.15 Turbulent flow measurements: (a) four installed Cobra Probes ; (b) zoom on Cobra Probe A. . . 76
4.16 Pressure and shear stress on a point of the body surface. . . 77
4.17 Position of the pressure taps on ring n◦1. The position of the taps on the other rings is equivalent. . . 78
4.18 One of the multi-channel pressure transducers. Their small dimen-sions allow to place them into the bridge deck, close to the measuring points. . . 78
4.19 Force balance during assembly stage: (a) middle section of the model; (b) force balance installed on the removable central module. . . 79
4.20 Laser transducers: (a) three set of laser transducers; (b) set of laser transducers below pressure ring n◦3. . . 80
4.21 Aerodynamic forces: local reference system for measurements. . . 82
4.22 Drag offset in local coordinates. . . 83
4.23 Drag correction in local coordinates. . . 83
4.24 Drag coefficient measurements in the four pressure taps rings. . . 84
4.25 Lift coefficient measurements in the four pressure taps rings. . . 84
4.26 Moment coefficient measurements in the four pressure taps rings. . . 85
4.27 Static aerodynamic coefficients of the Third Bosphorus Bridge deck (pressure taps). . . 86
4.28 Static aerodynamic coefficients derivatives of the Third Bosphorus Bridge deck (pressure taps). . . 86
4.29 Procedure to define the aerodynamic transfer functions for the z d.o.f. 88 4.30 "PoliMi" lift flutter derivatives; h∗ 1. . . 89
4.31 "PoliMi" lift flutter derivatives; h∗ 4. . . 89
4.32 "PoliMi" moment flutter derivatives; a∗ 1. . . 90
4.33 "PoliMi" moment flutter derivatives; a∗ 4. . . 90
4.34 Procedure to define the aerodynamic transfer functions for the θ d.o.f. 91 4.35 "PoliMi" lift flutter derivatives; h∗ 2. . . 92
4.36 "PoliMi" lift flutter derivatives; h∗ 3. . . 92
4.37 "PoliMi" moment flutter derivatives; a∗ 2. . . 93
4.38 "PoliMi" moment flutter derivatives; a∗ 3. . . 93
4.39 Procedure to define the admittance functions. . . 95
4.40 Definition of the wind angle of attack α due to the incoming turbu-lence w. . . 95
4.41 Lift admittance function χLw; amplitude and phase. . . 96
4.42 Moment admittance function χM w; amplitude and phase. . . 97
4.43 Suspended sectional model forced by the active turbulence generator. 98 5.1 Example of a rigid bridge deck section; sign convention. . . 102
5.3 Numerical models comparison (sectional model): PSD of the response, V = 15 m/s. . . 105 5.4 Numerical models comparison (sectional model): PSD of the response,
V = 30 m/s. . . 105 5.5 Numerical models comparison (sectional model): PSD of the response,
V = 45 m/s. . . 106 5.6 Numerical models comparison (sectional model): PSD of the response,
V = 60 m/s. . . 106 5.7 Example of recorded wind spectra for vertical mode excitation; V =
8.22 m/s and f1 = 2.2 Hz. . . 107
5.8 Example of recorded wind spectra for vertical and torsional mode excitation; V = 15.07 m/s and f1 = 2.2 Hz, f2 = 3.5 Hz. . . 108
5.9 Numerical (num) and experimental (exp) comparison: PSD of the response; V = 8.22 m/s and f1 = 2.2 Hz. . . 109
5.10 Numerical (num.) and Experimental (exp.) comparison: PSD of the response; V = 15.06 m/s and f1 = 2.2 Hz. . . 109
5.11 Numerical (num) and Experimental (exp) comparison: PSD of the response; V = 8.29 m/s and f1 = 3.5 Hz. . . 110
5.12 Numerical (num) and Experimental (exp) comparison: PSD of the response; V = 15.05 m/s and f1 = 3.5 Hz. . . 111
5.13 Numerical (num) and Experimental (exp) comparison: PSD of the response; V = 8.29 m/s and f1 = 2.2 Hz, f2 = 3.5 Hz. . . 112
5.14 Numerical (num) and Experimental (exp) comparison: PSD of the response; V = 15.07 m/s and f1 = 2.2 Hz, f2 = 3.5 Hz. . . 112
5.15 Multi-sectional model used for the validation of the modal approach. 114 5.16 Numerical models comparison (sinusoidal model): PSD of the
re-sponse, V = 15 m/s. . . 115 5.17 Numerical models comparison (sinusoidal model): PSD of the
re-sponse, V = 30 m/s. . . 116 5.18 Numerical models comparison (sinusoidal model): PSD of the
re-sponse, V = 45 m/s. . . 116 5.19 Numerical models comparison (sinusoidal model): PSD of the
re-sponse, V = 60 m/s. . . 117 5.20 Rigid and deformable model comparisons: mass, damping, stiffness
and buffeting force Lagrangian component ratios for the vertical mode.119 5.21 Rigid and deformable model comparisons: ratios for the response of
the vertical mode. . . 119 5.22 Rigid and deformable model comparisons: mass, damping, stiffness
and buffeting force Lagrangian component ratios for the torsional mode.120 5.23 Rigid and deformable model comparisons: ratios for the response of
the torsional mode. . . 120 6.1 Structural reference system (Storebaelt bridge: B=31 m; L=2696 m). 121 6.2 Storebaelt mode shapes (modes: 1, 2, 3 and 4). . . 122
LIST OF FIGURES xi 6.3 Storebaelt mode shapes (modes: 5, 6, 7 and 8). . . 123 6.4 Storebaelt mode shapes (modes: 9, 10, 11 and 12). . . 123 6.5 Adopted "PoliMi" lift flutter derivatives; a∗
1, a∗2, a∗3 and a∗4. . . 124
6.6 Adopted "PoliMi" moment flutter derivatives; h∗
1, h∗2, h∗3 and h∗4. . . . 124
6.7 Trend of the coherence functions for ∆x = B and ∆z = 0. . . 126 6.8 Trend of coherence functions for f = 0.099 [Hz] (second vertical mode).126 6.9 Trend of coherence functions for f = 0.278 [Hz] (first torsional mode). 126 6.10 Example of output of the numerical code: spectral density chart of
the vertical displacement; ∆f = 0.0017 Hz and 45 m/s mean wind velocity. . . 127 6.11 Example of output of the numerical code: spectral density chart of
the torsional displacement; ∆f = 0.0017 Hz and 45 m/s mean wind velocity. . . 127 6.12 Buffeting response for different resolutions of the wind grid. PSD at
mid-span: 15 m/s average wind velocity. . . 129 6.13 Buffeting response for different resolutions of the wind grid. PSD at
mid-span: 30 m/s average wind velocity. . . 130 6.14 Buffeting response for different resolutions of the wind grid. PSD at
mid-span: 45 m/s average wind velocity. . . 131 6.15 Buffeting response for different resolutions of the wind grid. PSD at
mid-span: 60 m/s average wind velocity. . . 132 6.16 Isolation of multiple resonance peaks of the response for the
calcula-tion of the RMS. . . 133 6.17 Graphical representation of the error percentage in terms of RMS at
mid span computed with respect to the 86 sections model. . . 134 6.18 PSD of the vertical and torsional displacement at 45 m/s average
wind velocity. Comparison between 86 and 173 sections model at mid span. . . 135 6.19 Graphical representation of the error percentage in terms of RMS at
mid span computed with respect to the 173 sections model. V = 45 m/s. . . 135 6.20 Buffeting response for different resolutions of the wind grid. PSD at
x = -455 m: 60 m/s average wind velocity. . . 137 6.21 Effect of different wind turbulence components coherence. PSD at
mid-span: 15 m/s average wind velocity. . . 139 6.22 Effect of different wind turbulence components coherence. PSD at
mid-span: 30 m/s average wind velocity. . . 140 6.23 Effect of different wind turbulence components coherence. PSD at
mid-span: 45 m/s average wind velocity. . . 141 6.24 Effect of different wind turbulence components coherence. PSD at
mid-span: 60 m/s average wind velocity. . . 142 6.25 Three different views from the bridge site: a) from the South-West,
b) from the East, c) from the lidar location on the West, and d) their position on the map. [5] . . . 145
6.26 Mean of the radial wind velocity derived from the PPI scan (wind lidar) recorded on 2014-05-22 between 09:30:03 and 10:13:16, with an elevation of 1.8°. The wind was blowing from N-NE with Vx = 6.0m/s
at the bridge centre. [5] . . . 146 6.27 Mean of the along-beam wind velocity derived from PPI scans (wind
lidar) recorded on 2014-05-22 between 16:50:22 and 17:25:44 with an elevation of 3.2°. The wind was blowing from S-SW with Vx = 8.0m/s
at the bridge site. The towers are shown as thick crosses. [5] . . . 146 6.28 Mean wind profile: cosine variation in the central span ranging from
+15% V to −15% V . . . 147 6.29 Effect of span-wise variation of the mean wind velocity. PSD at
mid-span: 15 m/s average wind velocity. . . 148 6.30 Effect of span-wise variation of the mean wind velocity. PSD at
mid-span: 30 m/s average wind velocity. . . 149 6.31 Effect of span-wise variation of the mean wind velocity. PSD at
mid-span: 45 m/s average wind velocity. . . 150 6.32 Effect of span-wise variation of the mean wind velocity. PSD at
mid-span: 60 m/s average wind velocity. . . 151 6.33 Static rotation at 60 m/s average wind velocity. Effect of span-wise
variation of the mean wind velocity. . . 152 A.1 Frequency as a function of the wind speed an airfoil type deck. . . 162 C.1 Example of structural grid (blue) and wind grid (red). . . 165 C.2 Graphic representation of the rectangular/trapezoidal integration rule.166 D.1 PSD of vertical displacement, 15 m/s wind mean speed. . . 169 D.2 PSD of equivalent torsional displacement, 15 m/s wind mean speed. . 169 D.3 PSD of vertical displacement, 30 m/s wind mean speed. . . 170 D.4 PSD of equivalent torsional displacement, 30 m/s wind mean speed. . 170 D.5 PSD of vertical displacement, 45 m/s wind mean speed. . . 171 D.6 PSD of equivalent torsional displacement, 45 m/s wind mean speed. . 171 D.7 PSD of vertical displacement, 60 m/s wind mean speed. . . 172 D.8 PSD of equivalent torsional displacement, 60 m/s wind mean speed. . 172 D.9 RMS of vertical displacement versus mean wind speed: frequency/time
domain results. . . 173 D.10 RMS of equivalent torsional displacement versus mean wind speed:
List of Tables
4.1 Correct Quasi-Steady theory (QST); drag flutter derivatives. . . 94 4.2 Correct Quasi-Steady theory (QST); lift flutter derivatives. . . 94 4.3 Correct Quasi-Steady theory (QST); moment flutter derivatives. . . . 94 4.4 Selected suspended model tests. . . 99 5.1 Main structural properties adopted for numerical models comparison. 103 5.2 Main structural properties of the Third Bosphorus Bridge model. . . 107 5.3 Mono-harmonic excitation: numerical and experimental comparison;
maximum amplitude error. . . 111 5.4 Bi-harmonic excitation: numerical and experimental comparison;
max-imum amplitude error. . . 113 5.5 Main structural properties adopted for the modal approach validation.114 6.1 Storebaelt structural modal properties. . . 122 6.2 Wind grid resolution. . . 128 6.3 Wind grid resolution: root mean square comparison at mid span.
Errors are computed with respect to the 86 sections model (1B). See Fig.6.17. . . 134 6.4 Wind grid resolution: root mean square comparison at mid span.
Errors are computed with respect to the 173 sections model (0.5B). . 135 6.5 Wind grid resolution: root mean square comparison at x = -455 m.
Errors are computed with respect to the 86 sections model (1B). . . . 136 6.6 Different wind turbulence components coherence: root mean square
comparison at mid span. . . 144
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 displacements and/or velocities of the considered structure are 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, decreasing 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. However, universities and companies interested in the study of these problems have written their own in-house codes without any information from reliable databases. In absence of a direct numerical comparison, code validation represents a crucial problem. Many difficulties are in fact encountered due to the lack of exhaustive full scale measurements and the complexity in the scaling of full aeroelastic models. Moreover, it must be considered that these validation techniques present high economical costs. For these reasons, a common interest in the defini-tion of reliable numerical standards for bridge buffeting analysis has spread among the wind engineering community at the global level. Recently (November 2016), the International Association for Bridge and Structural Engineering (IABSE) has launched an international benchmark project in this field. A large number of par-ticipants - 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 domain. Politecnico di Milano (PoliMi; Milan, Italy) was participating in the project with a time domain method based on a rheological approach [6, 7]. Historically, PoliMi has in fact focused on the definition of time domain methods able to capture the non-linearities connected to wind-bridge in-teraction. Considering this scenario, we have seized the opportunity to develop a frequency domain numerical model for buffeting analysis of long-span bridges, that was lacking in the PoliMi code database.
The definition of a numerical procedure for bridge buffeting analysis represents a difficult task. In fact, aerodynamic forces are complex to model as they show a dependency on the wind angle of attack α and on the reduced velocity V∗:
Faero = Faero(α, V∗) (1) The identification of the functions defining the aerodynamic forces requires a series of specific wind tunnel tests. Collected measurements must then be elaborated. They define a discrete data set and so results must be interpolated. Frequency do-main methods work with a specific frequency resolution and the aerodynamic forces must be defined at each frequency step inside an iterative process.
The method developed in this thesis is based on the subdivision of the bridge deck into a set of rigid sections and on the definition of the aerodynamic forces at the sectional level. Computations are performed considering only the deck since this is the part of the bridge most sensitive to wind action and its aerodynamics mainly governs the structure overall static and dynamic behaviour. In general, when a 3D structure is exposed to an air flow, three force components and three moment components can be considered. However since bridges are extended only in one direction, the primary concern regards its behavior when wind comes perpendicular to its longitudinal axis (see Fig.1) and so only a total number of three actions needs to be considered: lift force FL, drag force FD and pitching moment M. A closer view
on the aerodynamic forces and their effects on the bridge behavior will be given in Chapter 1.
Figure 1: Incoming wind profile on a suspension bridge.
A crucial aspect of the analysis consists in the numerical simulation of the wind acting on the bridge. The great size of the structures under consideration requires that special care is put in representing correctly the spatial correlation of the phe-nomenon. This is not a straight-forward task. In this thesis we present an innovative
3 method for the representation of point and spatial coherence of wind turbulence us-ing Fourier transforms (FFT). Differently from existus-ing frequency methods that work in terms of power spectral densities (PSD) [8, 9, 10], our model makes use of the FFT complex form to carry the information regarding wind turbulence coherence. In particular, wind numerical generations are carried out with a frequency domain re-elaboration of an existing procedure developed by Shinozuka [11] and Deodatis [12]. Chapter 2 gives all the details on the issues behind the numerical generation of wind turbulence components for the frequency domain method developed in this thesis.
The computation of the global response is performed following a classical modal approach, since it allows some significant simplifications. Our method is in fact able to consistently reduce the degree of freedom (d.o.f.) of the problem by considering just a compact set of first modes of the bridge. In this way, computations are performed considering few modal coordinates, instead of a large number of physical coordinates. This produces a drastic reduction of the computational cost. Bridge modal information are obtained by making use of a provided Finite Element Method (FEM) model. It is common practice to develop finite element modelling of the bridge by means of beam elements for deck and tower, while taut string or tensioned beam elements are used for the cables (see Fig.2). However, a more sophisticated modelling of the deck and tower can be pursued by employing plate elements. More details regarding the modal approach will be provided in Chapter 3.
Figure 2: Third Bosphorus Bridge, FEM model.
Deck aerodynamics identification was carried out with a series of wind tunnel tests at the PoliMi facility. Since the aerodynamic forces are defined at the sectional level, the experiments were performed on a bridge deck rigid sectional model. These experimental tests were part of the IABSE project. The experimental setup and procedure will be described in Chapter 4 along with the results obtained.
Various simulations to validate the code are presented in Chapter 5. Validation process is carried out in a systematic way, starting from very simple models and increasing the degree of complexity until the procedure is assessed robust enough for the analysis of full bridge models. This is done in order to examine every step of the procedure. At first, reference is made to sectional models which represent the basic elements composing any generic full bridge model. In such a way it is possible to assess the fairness of the aerodynamic forces identification. Sectional models are characterized at the beginning with simple theoretical expressions for the aerodynamics and later on with more complex experimental aerodynamic func-tions measured in the wind tunnel. Afterwards, the degree of complexity is further increased by considering a multi-sectional bridge model. A simple but yet complete multi-mode model is used to assess the fairness of the modal approach. Compar-isons are made both between alternative numerical methods supplied by IABSE project participants and between several wind tunnel experimental measurements. An overview of the principles behind the alternative methods used for comparisons is given in Chapters 2-3.
To conclude, the validated numerical code is used to execute parametric analysis in Chapter 6. These simulations are run to investigate which degree of detail is needed for the analyst to obtain representative results optimizing the computing time. Three aspects of the problem are discussed: the influence of the wind grid resolution, the effect of different wind turbulence-components coherence and the ef-fect of span-wise variation of mean wind velocity. The bridge model used for these analysis is built considering the highest level of complexity, since multiple coupled modes are taken into account and wind tunnel measured aerodynamics characterize the deck. As for the investigation of the effect of span-wise variation of mean wind velocity, the analysis were inspired from the research activity by Jakobsen [5], who provided extensive lidar monitoring data of an existing bridge in Norway. Some general considerations are also provided as possible guidelines for future analysis.
Chapter 1
Wind Action on Bridges
1.1
Introduction
In this first Chapter we will give an overview of the definition of the aerodynamic forces on a generic bridge section and we will present the sign conventions followed in our numerical procedure. We will show also the flutter derivatives and admittance functions notations adopted inside the code. A series of theoretical expression used during code validation will be here presented. As 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 suspension bridges. As an example, Figure 1.1 shows the Storebaelt Bridge that with a 1624 m long central span represents the longest suspension bridge in Europe.
Figure 1.1: Storebaelt Bridge, Denmark 1998 [1].
1.1.1
Main aspects of wind-bridge interaction
In order to understand the different problems related to wind action on bridges, it is necessary to define the aerodynamic forces. When a 3D structure is exposed to an air flow, three force components and three moment components can be generally considered. However since bridges are extended only in one direction, the primary concern regards its behavior when wind comes perpendicular to its longitudinal axis (Fig.1) and so only a total number of three actions needs to be considered: lift force FL, drag force FD and pitching moment M.
Figure 1.2: Aerodynamic forces on a generic deck section.
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 to the wind action is mainly governed by the deck aerodynamics. The aerodynamic forces applied to the generic bridge section, reported in Fig.1.2, can be expressed as:
FD = 1 2ρSV 2 CD(α, V∗) (1.1) FL = 1 2ρSV 2C L(α, V∗) (1.2) M = 1 2ρSBV 2C M(α, V∗) (1.3)
where ρ is the air density, V is the mean wind velocity, S is a reference body area, B is a reference body dimension, CD(α), CL(α), CM(α) are respectively the Drag,
Lift and Moment aerodynamic coefficients.
More precisely, aerodynamic coefficients are functions of the wind angle of attack α and of the reduced velocity. The angle of attack is defined with respect to a
1.1 Introduction 7 reference axis (see Fig.1.4) and considering small displacements and velocities can be formulated as:
α = θ + ψ = θ + w − B1 ˙ θ − ˙z
V + u − ˙y (1.4)
Reduced velocity is instead defined as: V∗ = V
f B (1.5)
It reproduces 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. 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 (1.6)
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 the bridge motion (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 design and the 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 linearise the aerodynamic forces. In practice, what really matters is the static rotation of the bridge θ0 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 (see Fig.1.2-1.3). Equation 1.4 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.
Figure 1.3: Wind reference system.
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 introduc-ing 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.1.4. Body motion has there-fore 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: one degree of freedom instability in the vertical or torsional mode, 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 larger.
− 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
1.1 Introduction 9 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.
Figure 1.4: Aerodynamic forces: sign convention.
1.2.1
Quasi-Steady Theory
The Quasi-Steady Theory (QST) is the most suitable approach to understand the physics of the problem [13, 14]. The QST is based on the assumption that the aeroelastic forces, acting on the bridge deck, are not influenced by the frequency of the deck motion. In that sense, the aeroelastic forces are the same static forces that are measured in the wind tunnel on still sectional models and their expressions are:
FD = 1 2ρSV 2 relCD(α) (1.7) FL= 1 2ρSV 2 relCL(α) (1.8) M = 1 2ρSBV 2 relCM(α) (1.9)
Due to its assumptions, the QST provides refined results when the reduced ve-locity V∗ assumes values greater than 15 ÷ 20. In the previous definition, V
rel is the
relative velocity of the fluid, f is the vibration frequency and B is the deck width. 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. Under this condition, the aerodynamic forces are not influenced by the motion frequency.
1.2 Description of the aerodynamic forces 11 The square of the relative velocity can be expressed as (see Fig.1.4):
Vrel2 = (V + u − ˙y)2+ (w − ˙z − B1θ)˙ 2 (1.10)
while the angle of attack α is still provided by Eq.1.4.
Making reference to the physical coordinates y, z and θ, the equations of motion of the system are:
myy + r¨ yy + k˙ yy = FDcos ψ − FLsin ψ = Fy mzz + r¨ z˙z + kzz = FDsin ψ + FLsin ψ = Fz Iθθ + r¨ θθ + k˙ θθ = M = Fθ (1.11) where my,z and Iθ 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 stiffness.
Figure 1.5: Elastically suspended deck section.
By substituting Eq.1.10 into Eq.1.7-1.8-1.9, the equations of motion Eq.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¨ zz + k˙ zz = 1 2ρS (V + u − ˙y)2+ (w − ˙z − B1zθ)˙ 2 (CD(α) sin ψ − CL(α) cos ψ) (1.13) Iθθ + r¨ θθ + k˙ θθ = 1 2ρBS (V + u − ˙y)2+ (w − ˙z − B1θθ)˙ 2 CM(α) (1.14)
This formulation is actually referred to as Corrected Quasi-Static Theory (QSTC, see Section 1.2.2), since the values taken for the reference body dimensions B1y,z,θ
are chosen after taking into consideration the flutter derivatives (see Section 1.2.2). By considering the displacement vector:
x = y z θ
Equations 1.12-1.13-1.14 can be rewritten in a matrix formulation:
[MS] ¨X + [RS] ˙X + [KS]X = Faero(X, ˙X, V, b) (1.15)
This equation represent 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).
Linearised equation of motion
The study of the aeroelastic problem can be simplified by considering the linear formulation of the aerodynamic forces, acting on a generic section of a bridge deck. As previously stated, the aerodynamic forces are in general a non-linear function of the wind speed, of the turbulent wind velocity components and of the bridge motion: Faero= Faero(X, ˙X, Vm, b) (1.16)
where Faero = {Fy, Fz, Fθ}T, X = {y, z, θ}T, ˙X = { ˙y, ˙z, ˙θ}T, b = {u, w}T.
The linearisation is performed around the static equilibrium configuration under the hypothesis of small displacements (X), small velocities ( ˙X) and small turbulent wind fluctuations (b).
The static equilibrium configuration, reached under mean wind speed conditions, is defined by the vectors:
X0 = y0 z0 θ0 ; X˙0 = 0 0 0 ; b0 = 0 0 0
and so the linearised expression of the aerodynamic forces acting on the bridge around can be formulated as:
Faero(X, ˙X, b) = Faero(X0, ˙X0, b0) + ∂Faero ∂X 0 (X− X0) + ∂Faero ∂ ˙X 0 ˙ X + ∂Faero ∂ ˙b 0 b
Making reference to the QSTC formulation of the aerodynamic forces: Fy =
1 2ρSV
2
rel(CD(α) cos ψ + CL(α) sin ψ) (1.17)
Fz =
1 2ρSV
2
rel(CD(α) sin ψ − CL(α) cos ψ) (1.18)
Fθ = 1 2ρBSV 2 relCM(α) (1.19) with α, ψ and V2
1.2 Description of the aerodynamic forces 13 Performing this change of coordinates:
¯ X0 = ¯ y = y − y0 ¯ z = z − z0 ¯ θ = θ − θ0
and defining the aerodynamic coefficients as linear functions of the angle of attack: CD(α) = CD0 + ∂CD ∂α 0 α = CD0 + KDα (1.20) CL(α) = CL0 + ∂CL ∂α 0 α = CL0 + KLα (1.21) CM(α) = CM0 + ∂CM ∂α 0 α = CM0 + KMα (1.22)
the linearised expressions of the aerodynamic forces are: Fy = 1 2ρS V 2− 2V ˙y + 2V u ((C D0 + KDα) − (CL0 + KLα)ψ) (1.23) Fz = 1 2ρS V 2− 2V ˙y + 2V u ((C D0 + KDα)ψ + (CL0 + KLα)) (1.24) Fθ = 1 2ρSB V 2− 2V ˙y + 2V u (C M0 + KMα) (1.25)
Developing the expressions while neglecting second order terms one obtains:
Fy= 1 2ρSV 2C D0+ 1 2ρSV 2 " KD θ +¯ w − ˙¯z − B1yθ˙¯ V ! − CL0 w − ˙¯z − B1yθ˙¯ V !# +1 2ρSV 2CD0(u − ˙¯y) (1.26) Fz= 1 2ρSV 2C L0+ 1 2ρSV 2 " KL θ +¯ w − ˙¯z − B1zθ˙¯ V ! − CD0 w − ˙¯z − B1zθ˙¯ V !# +1 2ρSV 2CL0(u − ˙¯y) (1.27) Fθ= 1 2ρSBV 2C M0+ 1 2ρSBV 2 " KM θ +¯ w − ˙¯z − B1θθ˙¯ V !# +1 2ρSBV 2CM0(u − ˙¯y) (1.28)
The static and dynamic parts of the aerodynamic forces can be subdivided and rewritten in a matrix formulation:
Faero,st = 1 2ρSV 2 CD0 CL0 BCM0 (1.29) Faero,dyn = 1 2ρSV 2 0 0 KD 0 0 KL 0 0 BKM X¯ −1 2ρSV 2CD0 (KD − CL0) B1y(KD− CL0) 2CL0 (KL+ CD0) B1z(KL+ CD0) 2CM0B BKM B1θBKM X˙¯ + 1 2ρSV 2CD0 (KD− CL0) 2CL0 (KL+ CD0) 2CM0B BKM b (1.30)
The formulation of Faero,dyn can then be compacted in:
Faero,dyn = −[Kaero] ¯X − [Raero] ˙¯X + [Am]b (1.31)
Finally, by substituting Eq.1.31 into Eq.1.15, one obtains the linearised equations of motion:
[MS] ¨X + [R¯ S+ Raero] ˙¯X + [KS+ Kaero] ¯X = [Am]b (1.32)
where [MS], [RS] and [KS] are the structural diagonal matrices for the section:
[MS] = my mz Iθ ; [RS] = ry rz rθ ; [KS] = ky kz kθ
[Raero] and [Kaero] are the equivalent damping and stiffness full matrices due to the
linearised aerodynamic forces: [Raero] = 1 2ρSV 2CD0 (KD − CL0) B1y(KD − CL0) 2CL0 (KL+ CD0) B1z(KL+ CD0) 2CM0B BKM B1θBKM (1.33) [Kaero] = − 1 2ρSV 2 0 0 KD 0 0 KL 0 0 BKM (1.34)
[Am]is the aerodynamic admittance matrix:
[Am] = 1 2ρSV 2CD0 (KD − CL0) 2CL0 (KL+ CD0) 2CM0B BKM (1.35)
Should be noted that [Kaero], [Raero] and [Am] matrices coefficients provided by
the QSTC 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
tur-bulence. 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
ab-sence 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 depend 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. A detailed overview of the aeroelastic problems is given in Appendix A.
1.2 Description of the aerodynamic forces 15
1.2.2
Forces identification by means of wind tunnel tests
In common practice, the definition of the aerodynamic coefficients of the equivalent damping/stiffness and admittance matrices is done trough wind tunnel tests. These coefficients, respectively referred to as flutter derivatives and admittance functions, can be defined on a wide range of reduced velocity V∗. The conclusions that have
been worked out making reference to the QST are still valid. In the following, we present also some theoretical expressions that have been used in the first step of the code validation process. They are continuous functions that have been used to reduce the degree of complexity of the procedure by-passing problems related to numerical interpolation, as it will be better explained in Section 5.2.
Flutter derivatives
Although in the bridge design field the flutter derivatives have become a generally accepted tool for comparing the aerodynamic characteristics of different deck sec-tions, various conventions are still in use [15]. Adopting the "PoliMi notation", the self-excited forces can be expressed as:
Fyse= 1 2ρV 2 BL(−p∗1 ˙z V − p ∗ 2 B ˙θ V + p ∗ 3θ + p ∗ 4 π 2V∗ ω 2 z B − p ∗ 5 ˙ y V + p ∗ 6 π 2V∗ ω 2 y B) (1.36) Fzse= 1 2ρV 2BL(−h∗ 1 ˙z V − h ∗ 2 B ˙θ V + h ∗ 3θ + h ∗ 4 π 2V∗ ω 2 z B − h ∗ 5 ˙ y V + h ∗ 6 π 2V∗ ω 2 y B) (1.37) Mse = 1 2ρV 2B2L(−a∗ 1 ˙z V − a ∗ 2 B ˙θ V + a ∗ 3θ + a ∗ 4 π 2V∗ ω 2 z B − a ∗ 5 ˙ y V + a ∗ 6 π 2V∗ ω 2 y B) (1.38) where: − Vω∗ = V∗/2π. − h∗
i: Flutter derivatives for lift force (i = 1, .., 6).
− a∗
i: Flutter derivatives for pitching moment (i = 1, .., 6).
− p∗
i: Flutter derivatives for drag force (i = 1, .., 6).
With matrix notation, the 3 × 3 [Kaero] and [Raero] matrices containing the
flutter derivatives coefficients, can be written as follow: [Kaero] = − 1 2ρBLV 2 p∗62Vπ∗ ω2 1 B p ∗ 4 π 2V∗ ω2 1 B p ∗ 3 h∗62Vπ∗ ω2 1 B h ∗ 4 π 2V∗ ω2 1 B h ∗ 3 a∗62Vπ∗ ω2 1 BB a ∗ 4 π 2V∗ ω2 1 BB a ∗ 3B (1.39)
[Raero] = 1 2ρBLV 2 p∗5V1 p∗1V1 p∗2BV h∗5V1 h∗1V1 h∗2BV a∗5V1B a∗1V1B a∗2BVB (1.40)
It is important to compare the [Kaero] and [Raero] matrix coefficients to those
evaluated through the QST. At high values of V∗, the [K
aero] and [Raero] matrix
coefficients tend to converge to the expression of the [Kaero] and [Raero] values of
the QST. 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. More precisely: B1y = p∗2 p∗ 1 ; B1z = h∗2 h∗ 1 ; B1θ = a∗2 a∗ 1 (1.41) 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 hold: − flutter derivatives p∗ 1−6: p∗1 = KD− CL; p∗2 = (KD − CL) B1y B ; p ∗ 3 = KD; (1.42) p∗4 = 0; p∗5 = 2CD; p∗6 = 0. − flutter derivatives h∗ 1−6: h∗1 = KL+ CD; h∗2 = (KL+ CD) B1z B ; h ∗ 3 = KL; (1.43) h∗4 = 0; h∗5 = 2CL; h∗6 = 0. − flutter derivatives a∗ 1−6: a∗1 = KM; a∗2 = KM B1θ B ; a ∗ 3 = KM; (1.44) a∗4 = 0; a∗5 = 2CM; a∗6 = 0.
Theodorsen (1935) defined the theoretical lift and moment acting on a harmon-ically oscillating airfoil moving in a laminar flow with mean speed V. According to Theodorsen’s theory [16], the lift force and pitching moment are given by:
Fzse = πρV2B " B ˙θ 4V − B ¨z 4V2 ! + C(f∗) θ − ˙z V + B ˙θ 4V !# (1.45) Mse= −πρB 2 4 −B 4z +¨ BV 2 ˙ θ + B 2 32 ¨ θ + B 4F se z (1.46)
1.2 Description of the aerodynamic forces 17 where the function C(f∗) is the Theodorsen circulatory function. The complex
function C(f∗) = F (f∗) + iG(f∗) is shown in Fig.1.6.
Figure 1.6: Theodorsen circulatory function C(f∗), real and imaginary parts.
F (f∗)and G(f∗)are the real and imaginary parts of the Theodorsen circulatory function C(f∗). F (f∗) and G(f∗)are shown in Fig.1.7 are given by:
F (f∗) = J1(J1+ Y0) + Y1(Y1− J0) (J1+ Y0)2+ (Y1− J0)2 (1.47) G(f∗) = − J1J0 + Y1Y0 (J1+ Y0)2+ (Y1− J0)2 (1.48) where Ji and Yi are Bessel functions of the first and second kind, respectively, of
order i.
Considering the "PoliMi notation", neglecting the drag equation Fse
y and
ne-glecting the flutter derivatives h∗
5−6 and a ∗
5−6, the following equations hold:
Fzse = 1 2ρV 2BL(−h∗ 1 ˙z V − h ∗ 2 B ˙θ V + h ∗ 3θ + h ∗ 4 π 2V∗ ω 2 z B) (1.49) Mse = 1 2ρV 2B2L(−a∗ 1 ˙z V − a ∗ 2 B ˙θ V + a ∗ 3θ + a ∗ 4 π 2V∗ ω 2 z B) (1.50)
Considering the theoretical lift and moment defined by Theodorsen acting on an airfoil and the previous expressions, the following flutter derivatives can be calcu-lated for flat plates:
Figure 1.7: Components of Theodorsen circulatory function versus reduced frequency f∗ and reduced velocity V∗.
− flutter derivatives h∗ 1−4: h∗1 = 2πF ; h∗2 = −2π 1 4+ F 4 + 1 2πV ∗ G ; (1.51) h∗3 = 2πF − π 2V∗G ; h∗4 = 1 + 2 πV ∗ G . − flutter derivatives a∗ 1−4: a∗1 = π 2F ; a ∗ 2 = π 2 1 4 − F 4 − 1 2πV ∗ G ; (1.52) a∗3 = π 2 F − π 2V∗G ; a∗4 = V ∗ 2πG.
where F and G are function of V∗. Figures 1.8-1.9 show the aerodynamic
deriva-tives for the flat plate in "PoliMi formulation".
In Figures 1.8-1.9 the coefficients coming from the Quasi Steady theory are also shown. If the QTS is used, KL and KM values must be taken equal to KL= 2π and
1.2 Description of the aerodynamic forces 19
Figure 1.8: Flutter derivatives h∗
1−4, theoretical expression for a flat plate
(PoliMi notation).
Figure 1.9: Flutter derivatives a∗
1−4, theoretical expression for a flat plate
Admittance functions
In the early sixties, Davenport (1962) proposed a model for calculating fluctuating wind load on a structure located in the atmospheric boundary layer ([17],[18]). The turbulence characteristics are combined with aerodynamic functions, called admit-tance functions, that convert the air flow properties into wind load on the structure. Following this approach, buffeting forces acting on a deck section of length L can be defined as: Fbuf f = Fy Fz M buf f = 1 2ρV 2BL χ∗yu χ∗yw χ∗zu χ∗zw Bχ∗θu Bχ∗θw u V w V (1.53) where χ∗(f∗)are the complex admittance functions (where f∗ = 1/V∗ is the reduced
frequency). For example, if we consider only the wind speed fluctuation w the three components of the forces can be expressed as:
Fy = 1 2ρV 2BLRe(χ∗ yw)Vw + iIm(χ ∗ yw)wV (1.54) Fz = 1 2ρV 2 BLRe(χ∗zw)Vw + iIm(χ∗zw)Vw (1.55) M = 1 2ρV 2 B2LRe(χ∗θw)Vw + iIm(χ∗θw)wV (1.56) 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 (Sears, Irwin..). One of the most commonly used expres-sion is the Davenport admittance:
χ∗yu = 2CDA(f∗); χ∗zu = 2CLA(f∗); χ∗θu= 2CMA(f∗) (1.57)
χ∗yw= (KD− CL)A(f∗); χ∗zw = (KL+ CD)A(f∗); χ∗θw = KMA(f∗) (1.58)
where A(f∗) is a real function that weighs in frequency the quasi-steady values of
buffeting forces:
A(f∗) = 2 (7f∗)2(7f
∗− 1 + e−7f∗
) (1.59)
The value of A(f∗) as a function of the reduced frequency/velocity is reported in
Fig.1.10-1.11.
The admittance matrix, containing all the admittance functions, can be written as follow: [Am] = 1 2ρV 2 BL χ∗yu χ∗yw χ∗zu χ∗zw Bχ∗θu Bχ∗θw (1.60)
1.2 Description of the aerodynamic forces 21
Figure 1.10: A(f∗) versus reduced frequency f∗.
1.3
Response in the frequency domain
Previously in Section 1.2, we have shown respectively how to define the aerodynamic equivalent matrices [Kaero]and [Raero]for the deck section and buffeting forces Fbuf f
and how to write the system of motion equations for the generic deck section. Al-though at first the formulations of the aerodynamic problem are derived in the time domain, they can be conveniently linearised and solved in the frequency domain. As it will be better explained in Chapter 2, wind turbulence is a random phenomenon and it must be studied using a stochastic approach. Solving the problem in the time domain, an infinite number of wind time histories would be necessary to obtain a significant statistical representation of the problem. Furthermore, for computational reasons, wind time histories have finite length and they are not able to capture low frequency variations of wind velocity. The advantages of the frequency domain are numerous:
− The complete statistics of the problem can be described with a single simula-tion;
− Low frequency turbulence components are taken into account;
− Turbulence characteristics and deck transfer functions are better described in the frequency domain;
− The response computation is made much more easier and does not involve numerical integration;
− The frequency domain is well suited for the prediction of extreme responses. Solving the problem in the frequency domain, the solution is computed on a certain range of frequencies. Since [Kaero], [Raero] and [Am] are functions of the reduced
velocity/frequency, this means that at each frequency step ∆f the aerodynamic ma-trices must be updated, always with reference to the deck static rotation θ0. The
definition of the flutter derivatives and admittance coefficients is therefore executed with a bilinear interpolation on the reduced velocity and the deck rotation.
1.3.1
Frequency Response Function
In order to define the analytical form of the aerodynamic problem in the frequency domain, it is convenient to consider a steady state harmonic excitation:
[MS] ¨X + [RS + Raero] ˙X + [KS + Kaero]X = F eiωt (1.61)
Let’s seek a solution in the same form of the excitation (steady state solution): X(t) =H(ω) F eiωt (1.62)
1.3 Response in the frequency domain 23 By substituting Eq.1.62 into Eq.1.61 one obtains:
−ω2[MS]H(ω) F eiωt+iω[RS+Raero]H(ω) F eiωt+[KS+Kaero]H(ω) F eiωt= F eiωt (1.63)
By simplifying the above expression, the complex transfer function, also known as Frequency Response Function(FRF), between the deck response ˜X(f )and buffeting force ˜Fbuf f(f ) is given by:
H(ω) = (−ω2M
S + iω RS+ Raero(V∗) + KS+ Kaero(V∗))−1 (1.64)
So finally the response X(f) can be computed as: ˜
X(f ) =H(f ) · ˜Fbuf f(f ) (1.65) By solving equation Eq.1.65 over the frequency range of interest, it is possible to describe completely the response of the deck. The matrix inversions involved in the definition of the FRF must be performed at each frequency step ∆f which usually demands a high computational effort, especially when the number of considered modes increases.
Chapter 2
Wind Numerical Generation
2.1
Introduction
Long-span bridges are typically immersed in the Atmospheric Boundary Layer (ABL), that typically extends up to 500 ÷ 1000m and is characterized with turbulent winds due to terrain friction. Figure 2.1 shows the example of the Yavuz Sultan Selim Bridge. Due to its randomness, a deterministic description of the wind in the ABL is impossible and so stochastic methods must be used. In the wind engineering con-text, it is important to acquire and elaborate data to describe the wind characteristic of the construction site in order to define the wind loads on the structure.
Figure 2.1: Yavuz Sultan Selim Bridge: even world’s tallest bridges are fully immersed in the ABL.
The great dimensions characterizing long-span bridges make the representation of turbulence coherence a crucial point in the analysis. In this Chapter we will present several empirical expressions describing wind turbulence and we will go over the concepts behind the innovative method that we developed for wind numeri-cal generation. It is a frequency domain re-elaboration of an existing method by
Shinozuka-Deodatis [11, 12] that makes use of FFT and was specifically designed to represent point and spatial coherence of wind turbulence.
2.2
Wind description
The analysis of large databases of records, performed by anemometric stations spread all over the globe for many years, is at the base of the knowledge of wind character-istics in the atmospheric boundary layer that we have today. Due to the turbulent nature of the ABL, the measurements provided by the stations are extremely irreg-ular. Nevertheless there are characteristics that are common among a long term set of measurements in the boundary layer, suggesting that we can develop empirical relationships for the variables of interest. Some of the recurring features are:
− the mean value of the velocity present typical changes over the day duration; − the wind speed has a limited range over the mean value, which is measurable
and characteristic of the turbulence;
− fluctuations cover a wide range of frequencies (time scales);
− the average wind velocity increases with the height from the ground;
− the patterns of gusts present some similarities at all heights, that is indicative of a spatial correlation of the wind turbulence.
Further considerations can be made by considering the typical wind spectrum, first estimated by Van der Hoven in 1957 (see Fig.2.2, [19]). The frequency of the wind speed variation can be associated to a characteristic size of eddies. Small eddies are characterized by shorter time periods, while large eddies have typical longer periods. By looking at the wind spectrum, three main peaks in the wind energy can be distinguished:
− the leftmost peak with a period of about 100h (4 days) corresponds to wind speed variations associated with the transit of fully developed weather systems that is named macro-meteorological peak;
− the peak at 24h shows the diurnal increase of wind speed during day and decrease at night;
− the rightmost peak with periods comprised between 10 minutes and less than 3 seconds is associated with turbulence of the ABL and is referred to as micro-meteorological peak.
The micro-meteorological peak is the one of interest for the wind engineering field. While it is possible to treat slow wind speed fluctuations as variations in the mean
2.2 Wind description 27 wind velocity producing only static effects on the structures, the high frequency fluctuations produce dynamic effects on the structures since the typical structural frequencies come in this range. Moreover, the lower is the structure frequency the higher is the wind energy at the same frequency.
The lack of wind speed variations having periods between 10 minutes and 1 hour is referred to as spectral gap. This gap represents a separation between the scales associated with wind climate and the ones associated with turbulence. Motions to the left of the gap are associated with the mean flow and motions to the right con-stitute the turbulence.
In order to model the design wind at a site, it is important to define its main characteristics that basically provide informations on the average wind, on the tur-bulence and on the gust patterns (correlation).
2.2.1
Mean wind velocity profile
The mean wind velocity U is defined as:
U (z, t) = U (z) + u(z, t) (2.1)
U (z) = 1 T
Z T
0
U (z, t)dt with T = 10min ÷ 1hour (2.2) The variation of the average wind speed over the height describes a mean wind velocity profile. Codes provide different formulas to define the mean wind velocity profile based on different analytic expressions. Two very well known examples are the logarithmic profile (Eurocode 1, [20]) and the power-law profile (North America, e.g. [21]): U (z) = u ∗ κ ln z z0 (2.3) U (z) = Uref z zref α (2.4) By considering one of the above expressions, the numerical procedure developed in this thesis is capable of generating wind inputs in any point of the bridge (deck, towers, cables). For simplicity, however, we have focused our analysis on the deck response only as this mainly governs bridge behaviour.
2.2 Wind description 29
2.2.2
Wind turbulence
Turbulence is a three dimensional phenomenon (see Fig.1.3). The variations appear in longitudinal, lateral and vertical direction. The longitudinal direction is defined as the mean wind direction so that the turbulence components have a zero mean value. In such a way, wind turbulence can be studied as a stochastic process. Turbulence intensity
Turbulence intensity is a dimensionless parameter defined for the three velocity components as: Iu(z) = σu(z) U (z) (2.5) Iv(z) = σv(z) U (z) ≈ 0.75Iu (2.6) Iw(z) = σw(z) U (z) ≈ 0.50Iu (2.7) where σu(z), σv(z), σw(z) are the standard deviations of the turbulence component
u, v, w and U(z) is the mean wind velocity, both at height z.
Expressions to define the turbulence intensity are provided by various design codes. As an example, Eurocode 1 proposes the following formulation [20]:
Iu(z) = kl co(z)·ln z z0
for zmin ≤ z ≤ zmax
Iu(z) = Iu(zmin) for z < zmin
(2.8) where kl is a turbulence factor and co is an orographic factor.
Integral length scales
Integral length scales are parameters representing the average size of gusts in a given direction. It is formally defined through the cross correlation function (reference is made to the integral length scale for turbulence component u measured in the longitudinal direction x):
Lxu = Z ∞
0
Ru(rx)drx (2.9)
where Ru(rx)is the cross correlation function between two different points separated
longitudinally (along the mean wind speed direction) by a distance rx and measured
simultaneously.
Having three turbulence components and three directions in which to measure the vortices, nine integral length scales can be defined:
Lx u Lyu Lzu Lx v Lyv Lzv Lxw Lyw Lzw (2.10)
The integral length scales depend on the height from the ground and the ter-rain roughness. Empirical formulations are suggested by many design codes. For example, Eurocode 1 proposes [20]:
Lxu = 300 z 200 α (2.11) Lx v ≈ 0.25Lxu Lxw ≈ 0.1Lxu Lyu ≈ 0.3Lx u Lzu ≈ 0.2Lxu (2.12) where α = 0.67 + 0.05 ln(z0).
Figure 2.4: Integral length scales of the longitudinal turbulence component u.
Power spectral density
The definition of the wind spectrum is at the base of the analysis. In literature, many formulations aiming at describing quantitatively the wind spectrum have been proposed. The Von Karman spectrum is one of them [22]:
f Su(f ) σ2 u = 4f Lxu U 1 + 70.8f Lxu U 2 5 6 (2.13)
The equation is in a non-dimensional form: Su(f ) is the power spectral density
of the velocity component u, f∗ = f Lx u
U is a non-dimensional frequency calculated
using the integral length scale Lx u.
2.2 Wind description 31 Von Karman proposed also equations for the lateral and vertical turbulence com-ponents [22]: f Si(f ) σ2 i = 4f Lxi U 1 + 755.2f Lxi U 2 1 + 283.2f Lxi U 2 11 6 i = v, w (2.14)
Eurocode 1 proposes a similar expression: f Su(f ) σ2 u = 6.8f Lxu U h 1 + 10f Lxu U i53 (2.15)
The dimensional form of the wind spectra shows clearly that the wind energy is concentrated at lower frequencies. That proves dynamic amplifications in the wind induced response to be more likely to happen if the structure frequency is low.
10-2 10-1 100 101 102 f* 10-3 10-2 10-1 100 Von Karman Eurocode 1
Figure 2.5: Non-dimensional power spectral density according to Von Karman and Eurocode 1.
Figure 2.6: Spectra of the three components of wind turbulence compared with the structural frequencies of some slender structures.
2.2.3
Point coherence of wind turbulence
The cross-correlation of different turbulence components in the same point of the space is quantified by the point coherence function.
Dealing with two generic random processes x(t) and y(t), the coherence function is defined as:
Λxy(f ) =
Sxy(f )
Sx(f ) · Sy(f )
(2.16) where Sx and Sy are the power spectral density functions (PSD) of the single
pro-cesses and Sxy is their cross power spectral density function (CPSD).
The CPSD can be defined in analogy with the power spectral density PSD: Sxy(f ) = 2
Z +∞
−∞
cxye−i2πf τdτ (2.17)
where cxy is the cross-correlation function describing the dependency of the two
processes.
Differently from the PSD, the CPSD is generally a complex function:
Sxy(f ) = Cxy(f ) + iQxy(f ) (2.18)
where the real part Cxy is referred to as co-spectral density (in phase) and the
imaginary part Qxy is called quad-spectral density (in quadrature).
For the particular case of wind turbulence, the CPSD of the longitudinal and vertical turbulence components in the same point is usually approximated as real and negative:
