Nikitas N-wind-Induced Dynamic Instabilities of Flexible Bridges (Phd Thesis)

Wind-Induced Dynamic Instabilities

of Flexible Bridges

Nikolaos Nikitas

Department of Civil Engineering

University of Bristol

A dissertation submitted to the University of Bristol in

accordance with the requirements for the degree of

Doctor of Philosophy in the Faculty of Engineering.

April 2011

Abstract

The wind-induced vibrations of flexible bridges and their components have long been

a major concern. Although a great level of sophistication has been reached in wind-

resistant design, there is still a significant threat from the wind. Most intriguingly, often

when one problem is solved a new one seems to arise in its place. This study examines

a selection of such peculiar aerodynamic issues involved in the routine ‘life’ of a bridge.

All of them share in common the need for further explanations to address previous

modelling omissions and weaknesses and offer new understanding of the underlying

phenomena.

Starting with flutter, an inverse scheme was employed to identify the flutter deriva-

tive description of aeroelastic loading using actual response measurements of a full-scale

suspension bridge. As expected for ambient data, the identification produced noisy es-

timates of the parameter values, yet clear trends could be distinguished. Encouragingly

the trends were in reasonably good agreement with results from wind tunnel tests on

similar cross-sections. Evidence of aeroelastic coupling between vertical and torsional

vibrations was identified from the recorded bridge data and the flutter wind speed for

the single-degree-of-freedom torsional instability case was estimated. It is believed this

is the first time this has been achieved based solely on full-scale data. The study shows

the viability of the method to identify the flutter derivatives from full-scale data, which

has rarely been attempted previously and never with such clarity of the results. This is

potentially useful for identifying the actual aeroelastic behaviour and safety of bridges

in service, particularly as uncertainties of wind tunnel tests, such as Reynolds number

dependence are overcome.

Next, quasi-steady galloping theory was revisited aiming to address previous incon-

sistencies and clarify the correct generic equations and their implications. The case of a

section free to vibrate in two orthogonal directions was considered, subject to flow at an

arbitrary angle of attack to the principal structural axes. Putting forward the correct

galloping criterion for this non-classical case, it was possible to quantify differences from

previous incorrect analysis of the galloping condition and structural damping demand.

The effects of the structural parameters on galloping thresholds were addressed, again

overcoming former shortcomings.

Finally there was an attempt to elucidate the influence of critical Reynolds number

on the apparent galloping instability of dry circular cables. This instability per se

cannot fully fit the earlier classical galloping description. Through a series of wind

tunnel tests flow conditions responsible for excessive motion were identified. These

conditions were more perplexing than the single-sided laminar separation bubble and

2

the associated steady lift that is generally believed to dominate the critical Reynolds

number range. For cylinders normal to the flow, discontinuities in the aerodynamic

loading appear to act like a quenching intermittency, effectively inhibiting response. For

other cable inclinations the unusual flow structures that emerge seem to be related to

the observed dynamic instability of the cable. According to this thesis, the transitional

behaviour of the boundary layer is entirely responsible for the large amplitude vibration

events of bridge cables in dry conditions attributed to so-called dry galloping.

Acknowledgements

I owe a great debt of gratitude to my supervisor John Macdonald. His support, his

trust in me, his advice and his critical views have been more than inspiring for this

work. I am sure that our endless, always unscheduled, discussions can be heard on

every page of this thesis. He introduced me to the world of bridge monitoring and he

made me piece of the legacy of the Clifton Suspension Bridge (CSB). Without him this

thesis would have never been this thesis.

I would also like to thank Jasna Jakobsen for all background information and support

provided when performing the identification analysis for the CSB. It was a pleasure to

work with her and learn from her.

Further all the ‘team’ that made possible the wind tunnel tests at the National Research

Council (NRC) of Canada deserve a special thank. Terje Andersen, Mike Savage, Brian

McAuliffe, Guy Larose, along with the technical staff at NRC have been more than

helpful partners.

I gratefully acknowledge the support of The Clifton Suspension Bridge Trust during the

CSB site tests and the financial support from EPSRC during my PhD course (under

John Macdonald’s Advanced Research Fellowship).

Last but not least, I want to thank my family in Drama for their understanding and love

throughout the period of my research at Bristol. Especially my father, a real modern

Ulysses in my mind, has a unique contribution to my work for he has never stopped

advising me on how to defeat my strongest enemy; myself.

From being to becoming

Author’s Declaration

I declare that the work in this dissertation was carried out in accordance with the

regulations of the University of Bristol. The work is original except where indicated by

special reference in the text and no part of the dissertation has been submitted for any

other degree.

Any views expressed in the dissertation are those of the author and in no way

represent those of the University of Bristol.

The dissertation has not been presented to any other University for examination

either in the United Kingdom or overseas.

Signed:

Dated:

Contents

1 Introduction 1

2 The Aeroelasticity Framework 5

2.1 Wind-induced structural loading . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Static loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Wind buffeting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.3 Vortex shedding . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.4 Galloping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.5 Flutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.6 Wake-induced loading . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1.7 Rain-wind Instabilities . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Circular galloping: myth or true? . . . . . . . . . . . . . . . . . . . . . . 25

2.2.1 Reynolds number effects . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.2 Inclination effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2.3 Instability mechanisms . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3 State of the art in bridge wind design . . . . . . . . . . . . . . . . . . . 35

2.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3 Identification of flutter derivatives from full-scale data 43

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 The case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3 Wind characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4 Response and modal parameters . . . . . . . . . . . . . . . . . . . . . . 50

3.4.1 Response Characteristics . . . . . . . . . . . . . . . . . . . . . . 50

3.4.2 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.5 Flutter derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.5.1 Flutter Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.5.2 Identification Method . . . . . . . . . . . . . . . . . . . . . . . . 55

3.5.3 Application to the Clifton Suspension Bridge . . . . . . . . . . . 57


i

ii CONTENTS

4 Quasi-steady galloping analysis revisited 67

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2 Quasi-steady derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2.1 Relevance to uniform continuous systems . . . . . . . . . . . . . 74

4.3 Application: quantifying differences . . . . . . . . . . . . . . . . . . . . . 75

4.4 The detuning effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82


5 Experiments on galloping vibration of a circular cylinder 89

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.2 Wind tunnel tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2.2 Setup details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.3.1 Overview and large responses . . . . . . . . . . . . . . . . . . . . 99

5.3.2 Pressure data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.4.1 Symmetry considerations . . . . . . . . . . . . . . . . . . . . . . 106

5.4.2 Mechanism implications . . . . . . . . . . . . . . . . . . . . . . . 111

5.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6 Conclusion and outlook 115

Publications 121

References 123

List of Tables

5.1 Orientation angles for studied cases. . . . . . . . . . . . . . . . . . . . . 93

5.2 Position details for the model. For rings and lowest cable end ‘distance

from floor’ refers to stagnation points, while for cobra probes ‘distance

from model’ refers to along-wind distance. . . . . . . . . . . . . . . . . . 98

iii

List of Figures

2.1 Mapping of Strouhal number against Reynolds number in the subcritical

range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Free vibration tests for a circular cylinder. . . . . . . . . . . . . . . . . . 10

2.3 Circular cylinder vibration phenomena past the lock-in range U=5m/s.

Top: vertical response record with fc=9Hz. Bottom: surface pressure at

transverse tap, fv=13Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Storebælt Bridge vortex-induced vertical motion at max (left) and min

(right) of the amplitude cycle. Encircled is a parked van with its view

distorted due to motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5 Typical galloping response curve for a rectangular prism. Inset the clas-

sical galloping mechanism illustrated. . . . . . . . . . . . . . . . . . . . . 15

2.6 Response of a rectangular prism with side ratio 2/1 against reduced

velocity for varying critical structural damping ratio. The prism has

Sr=0.081 that sets the relevant Ur for vortex resonance at ≈12.34. . . . 16

2.7 (a) Displacements and aeroelastic forces on a thin airfoil; (b) Displace-

ments and aeroelastic forces for a bridge section. . . . . . . . . . . . . . 17

2.8 (a) Quantitative difference of response characteristics for a full bridge

under different flow conditions. (b) A∗

2 from wind tunnel tests for a tor-

sionally unstable bridge section under laminar and turbulent flow condi-

tions. Changes appear minimal to sustain any substantial modification

in the flutter behaviour. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.9 (a) Rivulet formation on the circular cable section; (b) Inclination ge-

ometry of the inclined and yawed to the flow cable. . . . . . . . . . . . . 22

2.10 Upper water rivulet mean angular position during rain-wind vibrations

and lift force, displacement time series for a different large response

configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.11 Resume of wind tunnel test results on the proposed cables in the Higashi-

Kobe Bridge. Improvement in aerodynamic performance is apparent for

the solution with protuberances under all conditions. . . . . . . . . . . . 25

v

vi LIST OF FIGURES

2.12 Mean drag coefficient versus Reynolds number. On top, transitions (Tr)

from laminar (L) to turbulent (T) flow are presented in relation to sep-

aration points (S) and boundary layers (BL). . . . . . . . . . . . . . . . 28

2.13 (a) The axial flow, evidenced by light flags positioned inside the wake,

act towards inhibiting communication between shear layers and promot-

ing a secondary circulatory flow. The function described, simulates the

galloping of a circular cylinder equipped with a long splitter plate. (b)

Enhanced vortices are produced when axial vortices from the inclined

cable, mix and interact with ordinary Karman vortices. . . . . . . . . . 31

2.14 Dry cable instability design criteria together with real-bridge unstable

records. Dotted lines are due to the uncertainty in defining structural

damping values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.1 Clifton Suspension Bridge (CSB) elevation showing monitoring instru-

ment locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Sketch of the CSB cross-section . . . . . . . . . . . . . . . . . . . . . . . 46

3.3 Vertical and torsional modes of CSB . . . . . . . . . . . . . . . . . . . . 47

3.4 (a) Histogram of wind speeds during the 2003-04 recording period. (b)

Polar plots of 1h mean wind velocities from both anemometers. . . . . . 49

3.5 (a) 1h average wind speed over the monitoring period. (b) 1h RMS

vertical accelerations at the reference location over the monitoring period. 51

3.6 (a) RMS vertical accelerations σv, in relation to wind speed for all 1h

records. (b) Same as (a) for 1h records dominated by wind loading,

with RMS vertical accelerations now divided by the vertical turbulence

intensity. The power-law approximating the obtained trend is also plotted. 52

3.7 PSDs for different loading conditions for (a) vertical (b) torsional and

(c) lateral accelerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.8 (a) PSDs of filtered data for first vertical and torsional modes for the

maximum wind speed record. (b) The evolution of the coupling action

is evident in the vertical PSD for wind speeds above 11m/s. . . . . . . . 53

3.9 Decolouring process. In the 1DOF system, filter application will produce

corrected spectra, see dashed line, simulating white noise loading. For

the 2DOF system, filtering with FL will erroneously modify the true

aeroelastic coupling, see dashed line versus greyed area. . . . . . . . . . 57

3.10 Example covariance functions (scaled with variance) for the combined

two degrees-of-freedom plotted against time lag. . . . . . . . . . . . . . . 58

3.11 Flutter derivatives of Clifton Suspension Bridge from full-scale data,

compared with wind tunnel extracted flutter derivatives for various cross-

sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

LIST OF FIGURES vii

3.12 Identified H∗

1 flutter derivative from each 15-minute record. . . . . . . . 61

3.13 Flutter derivative A∗

2 with additional points from A∗

4. . . . . . . . . . . . 63

3.14 H∗

1 for different H-sections and a possible Reynolds number based ex-

planation for the H∗

1 inversion. Crossovers can initiate when Reynolds

number changes alter the multiplier of Ur in Eq.(3.6). . . . . . . . . . . 64

4.1 Geometry of a bluff section indicating lift and drag forces (L, D), relative

angle of attack (α) and principal structural axes (x, y). (a) The general

case with α0 6= 0 and the 2DOF motion potential. (b) The special case

for 1DOF across-wind oscillations. . . . . . . . . . . . . . . . . . . . . . 70

4.2 Sections used in the galloping analysis. . . . . . . . . . . . . . . . . . . . 76

4.3 Non-dimensional aerodynamic damping coefficients (SDH, min(Sxx, Syy),

S2D) as a function of angle of attack. Negative values indicate unstable

behaviour (in the absence of structural damping). . . . . . . . . . . . . . 78

4.3 (continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4 Comparison between the erroneous Ssc and the correct value for the 1D

rotated y-axis case, Syy, for (a) the square in Fig.4.2(d) and (b) the

triangle in Fig.4.2(f). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.5 Evolution of the aerodynamic damping solution for different values of

detuning, κ, for (a) the section in Fig.4.2(j) and α = 123◦ and (b) the

section in Fig.4.2(k) and α = 30◦. The lower branch is the important

one. In (a) for perfect tuning the solution is unstable (negative aerody-

namic damping) and for detuning above about 7% it approaches the 1D

solution, which in this case is stable. In (b), on the other hand, for per-

fect tuning the solution is stable and for detuning above 1% it becomes

unstable while moving towards the 1D solution. . . . . . . . . . . . . . . 83

4.6 Modal trajectories corresponding to Fig.4.5(a) for (a) κ = 1, (b) κ =

1.005, (c) κ = 1.05 and (d) κ = 1.1. The applicable Sdetuned value is also

indicated. Unstable modes are plotted with solid lines while stable ones

are dotted. Note for comparison that Sxx = 0.45, Syy = 0.06. For all

plots the structural damping value was c = 0. . . . . . . . . . . . . . . . 85

5.1 Transformation from real cable to wind tunnel model. . . . . . . . . . . 92

5.2 View of the NRC wind tunnel facility and its test section with the model

in place. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.3 Turbulence intensity and mean velocity across the wind tunnel section. . 95

5.4 Elevation of cable model showing instrumentation arrangement. . . . . . 97

5.5 Typical frequency response curves for three pressure taps. . . . . . . . . 98

viii LIST OF FIGURES

5.6 Three examples of motion traces during records of instability; Setup 2A

and 2C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.7 Proportion of total variance from 20 POMs (from all pressure tap data)

against Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.8 First Proper Orthogonal modeshapes for a set of dynamic and static tests103

5.9 Spectra of the lift coefficient on setup 2A, averaged over all four pressure

rings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.10 Correlation functions of lift coefficients between pressure rings. . . . . . 105

5.11 Mean pressure coefficient distribution around cylinder for a large re-

sponse case. Model setup 2A. . . . . . . . . . . . . . . . . . . . . . . . . 107

5.12 Drag evolution of the half-section with Reynolds number. Model setup

ϕ=60◦. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.13 Drag evolution of the half-section with Reynolds number. Model setup

ϕ=90◦. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.14 Ring 1 CL and CD transitional avalanche-like behaviour. Setup ϕ=90◦. 110

5.15 Pressure distributions during a transitional state succession. Model

setup ϕ=90◦. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Nomenclature

The precise interpretation of notation and abbreviations must be obtained from the

local context in which it is used and in which it will be explained. As a further guide,

the following is a non-exhaustive list of commonly used terms

Notation

A∗

i , H∗

i , P∗

i Flutter derivatives (i=1-6)

a1, γ, b1 Wake oscillator fitted parameters

B Representative sectional dimension (chord length)

b Half chord length

C(k) Theodorsen circulation function

CD,L,M Static force coefficients for D,L,M

CD1/2Static drag coefficient from half the cylinder section

Cij Covariance estimate between i, j series

Cp Static pressure coefficient

c Structural damping

D Static drag force

Db Buffeting drag force

Dse Self-excited drag force

d Across-wind dimension

FL(f) Filter function for lift force

Fx,y Force along direction x or y

f Frequency

fc Structural frequency

fv Vortex shedding frequency

Gij Modal integral of modes i, j

Hj(f) Frequency Response Function of mode j

h(s, t), p(s, t), α(s, t) Vertical, lateral and torsional displacement

hi(s), pi(s), αi(s) Vertical, lateral and torsional ith mode shape

Ii Turbulence intensity along component i

ix

x NOMENCLATURE

Ii Generalised inertia of mode i

ı Imaginary unit

|Jj(f)|2 Joint acceptance function of mode j

K Reduced cyclic frequency based on the full chord length

k Reduced cyclic frequency based on the half chord length

L Static lift force

Lse Self-excited lift force

Lb Buffeting lift forcex,y,zLi Turbulence length scale parameters

l Characteristic length

ℓ Maximum lags for covariance estimates

M Static overturning moment

Mb Buffeting overturning moment

Mse Self-excited overturning moment

m Mass

N Number of samples

n Number of lags in correlation function

Qi Generalised aerodynamic loading of mode i

qi, qi, qi, Generalised displacement, velocity and acceleration of mode i

Rij Correlation function

Re Reynolds number

rg Radius of gyration

S Power Spectral Density

Sc Scruton number

S2D Generalised two tuned degree-of-freedom galloping criterion

SDH Den Hartog galloping criterion

Ssc Non-generalised galloping criterion

Sij Dimensionless aerodynamic damping along i due to motion along j

Sr Strouhal number

s Space variable

U Wind velocity

Un Normal wind component

Ur Reduced wind velocity

Urel Relative wind velocity

u, v, w Longitudinal, transverse and vertical directions

x, y, z Directional motion variables

α Angle of attack

β Orientation angle

β∗ Wind-cable related angle

NOMENCLATURE xi

δij Kronecker delta

ε Roughness characteristic diameter

δij Kronecker delta

Θ(f) Sears function

θ Torsional motion

ϑ Cable inclination angle

θr Rivulet angular position

ζ Critical damping ratio

κ Detuning ratio

ν Kinematic viscosity

ρ Density of air

σi Standar deviation of i

τ Non-dimensional time variable

φ Mode shape function

ϕ Cable-wind angle

χ2(f) Aerodynamic admittance function

ω Cyclic frequency

Abbreviations

1DOF One Degree of Freedom

2D Two-dimensional

2DOF Two Degree of Freedom

3D Three-dimensional

3DOF Three Degree of Freedom

CEV Complex Eigenvalues Analysis

CHBM Covariance Block Hankel Matrix

CSB Clifton Suspension Bridge

DOF Degree of Freedom

ERA Eigenvalue Realization Algorithm

FHWA Federal Highway Administration

FRF Frequency Response Function

HDPE High Density Poly-Ethylene

IRF Impulse Response Function

IWCM Iterative Windowed Curve fitting Method

MIV Motion Induced Vortices

NRC National Research Council

xii NOMENCLATURE

PSD Power Spectral Density

POM Proper Orthogonal Mode

RMS Root Mean Square

SDOF Single Degree of Freedom

SVD Singular Value Decomposition

WIV Wake Induced Vibrations

Chapter 1

Introduction

Structural disasters have always served engineering as a trigger for moving forward. The

image of the Tacoma Narrows Bridge (TNB) collapsing back in 1940, under the action

of only moderate winds, has been one of the most striking failures ever recorded and

markedly influenced design practice. Since then bridge spans have expanded greatly

with no catastrophic event similar to the Tacoma Narrows failure recurring. This should

be attributed to the great wealth of knowledge that was developed in addressing this

notorious incident. Yet there are still remaining doubts about the exact mechanism

that instigated the dynamic instability. Most wind-structure interaction phenomena,

especially where bluff bodies are concerned, possess an intriguing duality. Explaining

them in a certain way does not always rule out alternate interpretations. Cooperative

phenomena seem to form and it is exactly when such hybrid cases are encountered

that our abilities of prediction break down. In any case the design framework currently

in use for aerodynamic effects on bridges has generally proved successful, combining

theory and wind tunnel experimentation. The latter acts as an effective ‘safety net’

that can warn us of potential aerodynamic problems. But apart from that, many in-

novative solutions have been conceived in wind tunnels. The form of many modern

bridges are actually owed to them. Through the wind tunnel testing for the Severn

Bridge the streamlined bridge section was invented, which allowed more efficient aero-

dynamically stable bridges. Nevertheless in a wind tunnel not all parameters of the

full-scale conditions can be faithfully reproduced, while size-effects could yield any links

between prototype and model inaccurate. Ultimately definitive information about the

wind resistance of a bridge can only be taken from the actual bridge. In the current

work a combination of full-scale monitoring, theory and wind tunnel tests is utilised in

investigating dynamic instabilities that threaten a bridge’s structural integrity.

The research begins by considering the old question of ‘flutter’ (flutter being the

widely accepted dynamic instability blamed for the TNB collapse) on an existing long

1

2 Chapter 1. Introduction

suspension bridge - the historic Clifton Suspension Bridge (CSB). As a matter of fact

the CSB belongs to the era of the first European steps in building suspension bridges.

Many examples of the time (e.g. Menai Straits Bridge, Brighton Chain Pier Bridge,

and Roche-Benard Bridge) had severe wind-induced vibrations leading to destruction

or serious damage. Still the issue addressed is not only the specific CSB application but

a more generic approach to answering challenges such as ‘Can we predict instabilities

for bridges that were not designed against them? ’ and ‘Can we predict the true safety

margins to compare with our design idealisations? ’. Field monitoring of dynamic re-

sponses combined with identification methods on the ambient data were performed in

order to estimate the vulnerability of the bridge to wind action. Exploring the potential

to identify actual full-scale behaviour rather than relying on wind tunnel tests is the

the main target of this section of the thesis.

Subsequently the interest shifts to another bridge-related issue; the dynamic in-

stability of bridge cables and the explanation of their excessive motion first witnessed

and reported on Japanese bridges in the early 1980’s. To justify this research further,

the TNB collapse is reconsidered. A recent explanation of the collapse says that just

before switching into an uncontrollable twisting motion, a pair of the middle cross-tie

cables on the bridge had snapped. Such an explanation could well mean that even

local parameters can have a major effect on the overall response, so before addressing

the overall design of the bridge the response characteristics of the discrete bridge parts

should be fully recognised first.

It has long been thought that a circular cable, due to perfect symmetry, cannot

gallop (as in classical Den Hartog galloping of iced transmission lines). This view

has recently been challenged and this study aims to uncover the details characterising

the instability generation mechanism known as dry galloping. The specific objective

adopted is to analyse the Reynolds number effects, which were previously overlooked.

For this purpose a series of large-scale wind tunnel tests was carried out with an aeroe-

lastic cable model inclined at different angles and equipped with pressure measuring

taps. There have been very few similar large-scale tests in the past and none of them

had equipment to measure the aerodynamic forces on a moving model.

The layout of the thesis is as follows:

• Chapter 2 introduces the aeroelasticity framework necessary to follow all concepts

discussed in this work. The literature is critically reviewed while information on

both loading characteristics and modelling practices are provided. The descrip-

tion of conventional well-known self-excitation phenomena such as galloping and

flutter is followed by a review of the more sparse research on dry galloping vibra-

3

tions. A series of controversial points are highlighted and additional justification

is given for the main study that follows.

• Chapter 3 presents the flutter derivative identification scheme performed on the

CSB full-scale ambient vibration data . The chapter commences by introduc-

ing background information on the monitoring procedure and the wind condi-

tions encountered on the CSB. Subsequently the stochastic subspace identifica-

tion method used is explained. Flutter derivative results are compared with other

similar sections and eventually an estimate for the flutter critical speed is made.

• Chapter 4 is devoted to addressing shortcomings of the current literature regard-

ing galloping, quantifying the differences in the results that can be observed with

different conditions or assumptions. The correct two-degree-of-freedom quasi-

steady galloping analysis is put forward for the first time, identifying the true

effect of the structural parameters on the dynamic instability. The analysis is

useful in understanding the characteristics of galloping and in interpreting the

observed response traces in the following chapter.

• Chapter 5 addresses the dry-galloping vibrations of circular cables that differ from

the instability considered in the previous chapter. It starts with describing the

experimental setup used and the large responses encountered. Next, a discussion

of the results regarding the influence of Reynolds number on the dry galloping

mechanism is made. The inclination angle appears to have key role in the oc-

curence of unstable motion. Discontinuities in the aerodynamic forces originating

from intermittent jumps between different flow states along with unusual flow

structures are unique features found due to the transitional behaviour.

• Chapter 6 concludes the thesis summarising the main findings and contributions

of the present research. It also includes suggestions for further work to advance

the present investigation.

Chapter 2

The Aeroelasticity Framework

Long-span bridges are exposed to severe wind action and the main target of this chapter,

before getting into the detailed analysis of this work, is to present the main attributes

of wind-structure interaction that are essential for the aerodynamic bridge design and

highlight yet unresolved issues that are later confronted. Background information at-

tempts to present both the loading characteristics as well as modelling practices that

have been employed in constructing wind-resistant structures. The interest centres on

self-excitation phenomena, which inherently introduce a feedback effect since the actual

aerodynamic forcing becomes a function of the flow-induced motion. Such phenomena,

where cause and effect are interlinked, naturally pose additional hurdles in their treat-

ment. Formulations and theoretical knowledge used in the later chapters are mostly

provided here.

A key feature is to also present the influence of Reynolds number on flows past cir-

cular cylinders, which later is proved to be an essential piece in the puzzle of excessive

bridge cable vibrations. For cable vibrations specifically, a number of inconsisten-

cies underlying the proposed explanation mechanisms are illustrated for the first time.

Throughout the chapter, real bridge examples are given in relation to the instability

issues discussed to further justify the needs for and benefits of the current research.

2.1 Wind-induced structural loading

In general, the wind approaching a structure can be decomposed into a steady and

a fluctuating vector component. In flexible bridges the gust loading falls well in the

range of their many low natural frequencies and on top of that even the uniform flow is

capable of imposing unsteady pressures, which can dynamically interact with structural

motion causing either a detrimental or a beneficial effect. The two mechanisms interact

5

6 Chapter 2. The Aeroelasticity Framework

with the net effect being in some cases a nonlinear one, which cannot be fully perceived

in the realm of simple linear superposition. There is also the static effect from the

flow, which is capable too of leading to failures, as in the case of torsional divergence.

What follows should be treated as a synopsis of the basic aerodynamic and aeroelastic

theory as included in the classical textbooks of Blevins [1], Simiu and Scanlan [2],

Zdravkovich [3] and Dowel et al. [4].

2.1.1 Static loads

The mean wind around a structure engenders static lift (L) and drag (D) forces together

with an overturning moment (M). The L, D, M expressions per unit length are given

by

L =ρBU2

2CL ,

D =ρBU2

2CD , (2.1)

M =ρB2U2

2CM ,

where ρ is the air density, B is a representative sectional dimension, U is the mean

wind velocity and CL,D,M are the static lift, drag and moment coefficients respectively.

CL,D,M are in probably all shapes functions of the angle of attack α. The scenario of

torsional divergence earlier referenced is brought about when CM has the tendency to

increase with increasing α. Thus for a structure with finite torsional stiffness, above a

critical value, wind will cause an ever-increasing α ultimately leading to destruction.

For dynamic loading a quasi-steady formulation considers that Eqs.(2.1) are still valid

with U replaced by the relative wind velocity Urel, which accounts for the influence

of either fluctuating wind components or structural motion. Complications with this

idealisation do exist, since evidently the introduction of rotational velocity waives the

uniqueness of such a Urel.

2.1.2 Wind buffeting

The unsteady loading imposed by wind turbulence is termed wind buffeting. Wind

turbulence, although chaotic and complex in its nature, is briefly characterised by a

number of parameters. These are:

• The turbulence intensity Ii = σi/U where σi is the Root Mean Square (RMS)

of the fluctuating wind component along i = u, v, w the longitudinal, transverse

and vertical directions relative to the wind.

2.1. Wind-induced structural loading 7

• The relevant auto-Power Spectral Densities (PSDs) Si(f) and cross-PSDs Si1i2(f),

i1 6= i2, where f is the frequency variable and it holds σ2i =

∫

∞

0 Si(f)df . Various

PSDs have been proposed in the literature for design purposes [5–7], wherein most

cases an efficient description of the fluctuations of strong winds is accomplished

through the von Karman model of isotropic turbulence [8].

• The length scale parameters x,y,zLi, nine in total, which designate a directional

(x, y, z) average size of turbulent eddies in each wind component i.

The classical treatise for buffeting loading, utilising the statistical concepts from ran-

dom vibration theory was first applied by Liepmann [9] to aircraft wings and later by

Davenport to long-span bridges [10, 11]. The formulation, expressing the force com-

ponents in quasi-steady terms, attempts to first establish the sectional loading PSDs

from the sectional (or point) directional wind PSDs. Further the generalised loading of

each mode is recovered by integration along the length, utilising the modal shape and

wind correlation information. In the frequency domain the PSD expression for vertical

response in each mode j is given by [11]

jSz(f) =(ρBU)2

4

[

(

CD +dCL

dα

)2

Sw(f) + 4C2LSu(f)

]

χ2(f)|Hj(f)|2|Jj(f)|2 , (2.2)

where Hj(f) is the Frequency Response Function (FRF) of mode j, |Jj(f)|2 is the

joint acceptance of mode j and χ2(f) is the admittance function. The joint acceptance

function is the means of translating the point-like load to a span-wise load, while the

admittance function should be deemed to be a correction factor to compensate for

the frequency dependence of the instantaneous aerodynamic load [12, 13] and for its

sectional variation. The first to theoretically evaluate such a function was Sears [14].

For a vertical gust w that is a sinusoidal time function of the form w = w0eı2πft (note ı

here is the imaginary unit), Sears derived the corresponding oscillatory lift on an airfoil

as

L(t) =ρBU

22πw0Θ(f)eı2πft , (2.3)

where Θ(f) is called the Sears function. Restating Eq.(2.3) in the frequency domain it

becomes

SL = (ρBUπ)2χ2(f)Sw , (2.4)

yielding quite easily that |Θ(f)|2 = χ2(f). Many other empirical options exist for

envisaging χ2(f), and there remains an ambiguity over which is the most appropriate

to use. In studies with long bridges with very low fundamental natural frequencies

the contribution from aerodynamic admittance is usually conservatively neglected [15],

though in some cases it is essential for the reliable response estimation [16]. As presented


in Hay’s [17] analysis for the Wye and Erskine Bridges, during narrow band vertical

response for the vertical displacement RMS values σz, the buffeting action should set

σz ∝ IU2.83 .1 (2.5)

The recovered relations from full-scale monitoring of the actual bridge responses were

actually containing a somewhat lower exponent than 2.83. In any case Eq.(2.5) shows

that response is proportional to turbulence intensity and is only asymptotically di-

verging. A final factor that is worth mentioning is the effect of signature turbulence,

which consists of wind fluctuations imposed not by the approach natural flow but by

the actual submerged structure or elements ahead of it. Any shape, unless very well

streamlined, will produce signature turbulence; however its effect in standard buffeting

analysis is generally ignored.

2.1.3 Vortex shedding

Vortices trailing behind bluff bodies are a very common picture in nature. Of course

the picture is not alone but is accompanied by forces with well defined dynamic char-

acteristics. Strouhal [18] was the first to observe and suggest that the frequency of the

wake oscillations follow

Sr =fvd

U, (2.6)

where Sr is the shape-dependent Strouhal constant, fv the vortex shedding frequency

and d the across-wind dimension. An across-wind force with frequency fv is exerted on

the submerged bluff body and as logically expected if the body is allowed to move across-

wind with a structural frequency fc, then the two frequencies’ coalescence will result in

resonance. Actually the classical conception of the wake structure (two counter-rotating

vortices shed alternately from each side during one cycle) implies that an along-wind

force also exists with twice the shedding frequency and is able too of causing resonance.

Unfortunately there are slight inconsistencies in this idealisation. The example of the

circular cylinder will allow to effectively present them. For a circular cylinder Sr is

strongly a function of Reynolds number (Re)

Re =Ud

ν, (2.7)

where ν is the kinematic viscosity. For subcritical values (i.e Re<≈ 105) customarily Sr

is considered to have a value of 0.2. Still this is not strictly true, the actual behaviour

is illustrated in Fig. 2.1 and is governed by discontinuity intervals as well as a strange

1Hay [17] does not explicitly state which component the turbulence intensity I refers to.


inversion of the monotonicity. The critical range behaviour is of far greater interest

(and variation) in large-scale engineering applications and will be discussed in detail

later.

0.000.17

0.19

0.21

0.23Sr

0.02 0.04√Re1/

2E4 2E3 1000 500 300 Re

retardedtransition

parallelshedding

5000 1300

240

230

180

360

0.06

Figure 2.1. Mapping of Strouhal number against Reynolds number in the subcritical range.

Adapted from Fey et al. [19].

A great number of experiments have revealed unique features of the aeroelastic char-

acter of the vortex shedding loading and response. The tests by Feng [20] were among

the first to demonstrate a series of intriguing nonlinear features, such as the capture of

fv from the structural vibration frequency in an extended range near resonance during

the phenomenon called ‘lock-in’. Seminal reviews on vortex phenomena refer to this

study [21–24]. Feng performed his wind tunnel tests with freely oscillating (restrictively

across-wind) cylinders for varying structural damping values in the subcritical range.

He measured the vibration amplitude Y (scaled with cylinder’s diameter d), fv, fc, the

phase difference η of the two throughout the lock-in region and for limited runs the

across-wind force coefficient dynamic amplitude CY . Part of his results are presented

in Fig. 2.2, where all recorded variables are plotted as functions of the reduced velocity

Ur=U/fc0d, with fc0 being the still air structural frequency. As shown in the figure

the lock-in behaviour establishes itself when approximately reaching the Strouhal res-

onance indicated earlier. From then on, fv remains equal to fc until the point where it

jumps back to fv ≈ 1.4fc following naturally Eq.(2.6). For other shapes the lock-in re-

gion is double-sided extending also to frequencies <fc. It is noteworthy that hysteresis

exists in the self-limiting amplitude response of Fig(2.2), with magnitude depending on

whether the wind velocity is increasing or decreasing. The upper branch attained for

increasing wind cannot be reached from rest displaying the interesting feature of depen-

dence on the ‘loading’ history. For the few runs that CY is recovered it is obvious that


its value is strongly amplified. Although not shown here, similar amplification holds

also for the along-wind dynamic forcing. It should be pointed out that the extreme re-

sponse and forcing do not match the initiation of the lock-in zone but they are situated

close to its middle. Finally the phase difference values along with their discontinuous

jump right in the heart of the synchronised regime, have become a great matter of

controversy and added trouble in the modelling task. In any case phase values indicate

both in phase and in quadrature forcing components. The detailed reasoning on phase

jumps was provided by Williamson and Roshko [22], who identified different modes

of vortex shedding and abrupt transitions between them in the amplitude-structural

frequency-wind velocity parameter space. Later work of Williamson with Jauvtis [25]

and Morse [26] completed, for the time being, the shedding mode characterisation.

Sr =0.198

Y-

U

5 6 7 84

Y-

Ur=U/fc0d

0.5

1.0

1.5

0.4

0.2

0.6

CY

1

2

0

0˚

50˚

100˚

η

fv

fcη

Y-

Y-from rest

-

d

CY-

fv,c /fc0

Figure 2.2. Free vibration tests for a circular cylinder. Adapted from Feng [20].


Outside the 1:1 (fc:fv) synchronisation range there is also a multitude of interesting

phenomena. Williamson and Roshko [22] identified a region of 1:3 subharmonic reso-

nance and reasoned that wake stability considerations prevent 1:2 similar occurrences.

As a matter of fact they point that subharmonic resonance should be possible for any

1:n ratio, with n being an odd number. Experiments from Durgin et al. [27] partly con-

firm this view by finding large response of free vibrating cylinders in such 1:3 regimes

but not in 1:2 or elsewhere. Still it is worth noting that such phenomena are very rare

and according to classical synchronisation theory should readily vanish when n>3 [28].

A final unresolved feature picked for this synopsis consists in the force-displacement

relation outside lock-in. As presented in Fig. 2.3 the displacement, top signal, is virtu-

ally a pure sinusoid (at fc=9Hz) of very stable magnitude. On the other hand the force

causing this displacement, illustrated in terms of the representative transverse pres-

sure tap trace in the lower signal, shows a strong modulation and a different frequency

(i.e. fv=13Hz). Thus a question surfaces on how these two signals can combine in a

cause-effect relation. According to Minorsky [29] this seems to be a classical example

of asynchronous excitation. A system possessing a stable focus point (i.e. initial equi-

librium point) followed by two adjacent limit-cycles (such can be a system expressed

by a polynomial of at least fifth order) could bifurcate and rest to the outer limit-cycle

when a suitably sized periodic action of random frequency is applied.

Figure 2.3. Circular cylinder vibration phenomena past the lock-in range after Ferguson and

Parkinson [30]. U=5m/s. Top: vertical response record with fc=9Hz. Bottom: surface pressure

at transverse tap, fv=13Hz.

Modelling of vortex wake phenomena is far from complete. Leaving aside the purely

computational treatises of discrete vortex potential flow models and the numerically


solved Navier Stokes equations, what remains to be the most efficient attacking tool

for our descriptive low-dimensional studies are the so-called wake-oscillator models.

Their conception belongs to Bishop and Hashan [31, 32] who were the first to suggest

that a cylinder’s wake behaves like a mechanical oscillator. Taking into account the

non-linear behaviour proven above, and particularly the lock-in and limit-cycle (the

stationary cylinder vortex shedding is what is idealised as limit-cycle) attributes, a

‘wise’ compatible modelling choice from the world of mechanics would be a Van der

Poll oscillator [33,34]. Such a model was first implemented by Hartlen and Currie [35]

and acquires the form

d2Y

dτ2+ 2ζ

dY

dτ+ Y =

ρd2ω20

8π2Sr2mCY ,

d2CY

dτ2− a1ω0

dCY

dτ+

γ

ω20

(dCY

dτ)3 + ω2

0CY = b1dY

dτ, (2.8)

where Y is the across-wind motion scaled with d, CY is the instantaneous across-

wind force coefficient, τ=2πfct is the non-dimensional time variable, ζ is the critical

structural damping ratio in still air, m is the mass per unit length, ω0 is the ratio of

shedding to structural frequency fv/fc and a1, γ, b1 are empirical constants to be fitted

from experiments. This archetypal form of the model assumes nonlinear phenomena

originating from the fluidic Van der Poll oscillator and being driven, in the case of

motion, by a linear coupling motion-dependent term. Hartlen and Currie employed

the first approximation solution of Kryloff and Bogoliuboff [33] and recovered parts

of the behaviour in Feng’s [20] experiments. Amplitude or phase hysteresis was not

apparent in analytical results but this is only due to the solution method employed.

Later variants of the model altered the fluid stiffness term, the fluid damping term, the

structural damping term or the form of the coupling-forcing term in Eqs.(2.8) improving

each time the match to experiments,(for a review see [36–39]). The same model was

also used on a less phenomenological basis, being derived from first principles, having

though CY substituted by a ‘hidden-flow’ variable [1]. The sophistication of the model

although extensive has been little concerned with a great branch of phenomena, such

as chaotic and quasi-periodic oscillations.

The preceding discussion tacitly assumed smooth flow conditions. The introduction

of turbulence would make vortices lose coherence along the body length and resonant

peaks in the CY spectrum degenerate, broaden, and evidently waive their efficacy in

setting up motion. Somewhat similar effects can be brought upon by surface protu-

berances, surface roughness, long splitter plates or more complex additions such as

wavy separation lines, or spirally arranged bumps [40, 41]. Earlier it was seen that

the ‘enhancement’ of vortex shedding during large response leads to amplification of


the dynamic loading. Similarly when the vortices shed by a body start losing their

strength, the mean pressure drag CD is expected to decrease. The most prominent

example of this rule is the circular cylinder and its behaviour along the Re range.

For bridges, although the British ‘revolution’ of streamlining sections was thought

to adequately handle vortex phenomena [42], this was not actually the case. Vortex-

induced vibrations have been quite systematic in bridges. The Wye Bridge had such

occurrences, but only of small amplitude, while the very similar Erskine Bridge did

not [17]. The discrepancy was reasoned in view of the higher turbulence intensities

measured at site for the latter, which as referenced earlier could disorganise the vortex

formation and propagation processes2. The Kessock Bridge [44] had similar observa-

tions with winds of only quite low turbulence intensities exciting moderate amplitude

bending oscillations. The extremely wind-prone Deer Isle Bridge [45,46] sustained vor-

tex oscillations in probably all of its configurations (being retrofitted or repaired in

many instances). For the Storebælt Bridge Larsen et al. [47] presented the excessive

form the phenomenon acquired, with Fig. 2.4 being indicative of amplitudes observed.

The Shanghai Lupu Bridge [48] is a unique reference, being an arch bridge with vortex

issues while on operation.

Figure 2.4. Storebælt Bridge vortex-induced vertical motion at max (left) and min (right) of

the amplitude cycle, after Larsen et al. [47]. Encircled is a parked van with its view distorted

due to motion.

The presentation in this section was mainly based on the circular cylinder paradigm

and one should expect that flexible bridges possessing cables, that can most of the times

fall into the circular cylinder category (disregarding only for the time being any incli-

nation), should have issues regarding them. Fortunately as presented by Virlogeux [49]

2This is not to be confused with the case of turbulence strengthening vortices as Matsumoto et

al. [43] observed on some bridge section types. Their analysis also includes non-classical Motion-Induced-Vortices (MIV) making up for the seeming inconsistency.


it can be easily shown, by considering typical values, that large amplitude vortex shed-

ding should not be a concern for bridge cables, since lower mode excitation is restricted

to quite low wind speeds of consequently limited energy content. Nevertheless, higher-

modes would receive large vortex forcing but due to their higher damping would not

produce excess motion. But the stresses exerted in the cables apart from being func-

tions of amplitude, they are also depending on curvature, which evidently increases

in higher modes. Thus even for low-amplitudes but higher mode persistent response,

large stress loading cycles will result, raising fatigue concerns. The record of a specific

cable in the Saint-Nazaire Bridge that initially sustained fatigue damage, was replaced

and later was found to be driven in large higher mode vortex oscillations, should be

deemed to be an indicative example.

2.1.4 Galloping

Classical galloping refers to across-wind motion arising due to the so-called ‘incidence

effect’, which translates to a wind forcing contribution originating from variations in

the effective flow incidence angle. It is considered equivalent to the condition

dCL

dα+ CD < 0 . (2.9)

Rotational asymmetry is evidently a basic requirement in the operating mechanism. A

simplistic view of galloping is provided in the inset sketch of Fig. 2.5, where a body

in a flow of velocity U is moving downward with velocity y, thus altering according to

definition, the effective flow incidence. The shear layer on the lower side moves closer to

the body, therefore getting more curved, while the upper side shear layer moves away

from the body and becomes less curved. As a result a net downward pressure force is

acted across the side faces (in contrast to an upward force that a streamlined airfoil

would experience) further assisting motion.

It will be shown in Chapter 4, Eqs.(4.7&4.8), that in this classical scenario the

threshold reduced wind velocity U0 for setting off galloping, calculated by means of

linear theory, is proportional to structural damping. Thereof a diverging response

should result. In reality a limiting mechanism operates to set response into a steady

state. Parkinson et al. [50,51] were among the first to implement nonlinear concepts to

evaluate steady galloping amplitudes that well match experimental observations. Their

analysis recovers that galloping increases roughly proportional to reduced velocity Ur,

as shown in Fig. 2.5, with also hysteresis effects and jump behaviour emerging in

the range noted by a dashed line and bounded between arrows. Later Novak [52–54]

proposed the notion of ‘universal response curves’ after rescaling axes in Fig. 2.5 with


Figure 2.5. Typical galloping response curve for a rectangular prism. Inset the classical

galloping mechanism illustrated.

Sc=ζm/ρd2. This dimensionless factor, named after Scruton, was first proposed by

Scruton [55] as the influencing parameter against most aeroelastic instabilities. Novak

extended on the distinction between ‘hard’ and ‘soft’ galloping oscillators, soft being

the ones able to gallop from rest while hard the ones in need of an initial hard ‘push’

to get into motion, presenting also a number of special cases where stable sections can

turn into weak hard ones.

Galloping forces in general terms should be considered as the product of flow-

afterbody interaction. Parameters altering any of the two constituents inevitably will

affect the instability characteristics. Introduction of turbulence for instance can turn

an unstable section to stable or the opposite, with any changes being strongly shape-

dependent. Similarly insertion of a splitter plate in the near wake can be quite dramatic

concerning the separation process [56,57] and induce galloping in cases where it would

not be expected, as in the examples of a rectangular section with along to across-

wind dimension ratio higher than 3 and the perfectly symmetric circular cylinder. The

splitter plate influence on galloping is the opposite of the influence earlier quoted on

vortex shedding, thus a question is raised on the link between galloping and vortex

shedding. In most practical applications in wind, the regions where the two phenomena

become dominant are well separated, with vortex shedding being confined in relatively

low reduced velocities and galloping appearing later only for much higher Ur values. A

typical example of aeroelastic response would be expected in the form of Fig. 2.6, with

vortex shedding giving a response ‘hump’ near Ur=5 and after a quiescent transition

period galloping taking over.


0 5 10 15 20

0.1

0.2

Ur

Damping ratio ζ:

0.37 %

0.76 %

1.40 %

2.12 %

4.40 %

U1

Y-

Y-

2

Figure 2.6. Response of a rectangular prism with side ratio 2/1 against reduced velocity for

varying critical structural damping ratio. The prism has Sr=0.081 that sets the relevant Ur

for vortex resonance at ≈12.34. Adapted from [53].

Actually the specific example of Fig. 2.6 although explained in this expected way

even in well respected textbooks, see [1], it contains some paradoxical features. Con-

trary to the aforesaid descriptions what seems like galloping here starts abruptly at

Ur ≈11 for all different values of structural damping, while in the ζ=4.40% case it

attains a rather unexpected decay. Taking into account that the rectangular prism

in hand has Sr=0.081, setting the relevant Ur for vortex resonance at ≈12.34, this

paradigm is probably a good indication of the rather complicated form that hybrid

vortex-galloping oscillations may obtain. For ζ=0.37% the galloping threshold is eval-

uated at U0=5.2 [53] but oddly is inhibited up to where vortex shedding is regularly

located. On the other hand for ζ=4.40% it follows U0=61, which explains the non-

increasing response character. The interaction range includes many more possibilities

to be pursued in later chapters. In the context of the present work the term galloping

will be used to characterise any motion-triggered translational aeroelastic instabilities

regardless of orientation and extending to also cover the combined participation of

orthogonal motion.

Galloping tends to be considered of less interest in cable-supported bridges, still

there were a number of instances in design or construction where it did show up. Ac-

cording to Virlogeux [58], the hexagonal bunches of strands to be installed on the

Normandy Bridge were abandoned due to galloping concerns that arose while in wind

tunnel testing. Instead a circular strand distribution was promoted, which was also cov-

ered by a high density polyethylene (HDPE) sheath with helical fillets. Wardlaw [59]


reports the wind tunnel investigation of the single tower of the Aratsu-Ohashi cable-

stayed Bridge where an apparent galloping susceptibility was documented throughout

the construction stages. A more controversial appearance should be the one docu-

mented in Deer Isle Bridge with deck motion amplitudes reaching up to 6100mm [46].

Although also other loading mechanisms could be held responsible for this incident the

characteristics of the bridge as presented by Cai et al. [60] readily support the galloping

scenario. Galloping events and the countermeasures for their treatment, for a series of

arch and truss bridges e.g. Burton Bridge, Bras d’Or Bridge and Commodore Bary

Bridge are presented by Wardlaw [59, 61]. Finally the case with excessive vibrations

recorded on the circular iced hangers of Storebælt Bridge [62], reminiscent of iced trans-

mission line galloping, is another phenomenon worthy of engineering attention along

with other less evident similar instances.

2.1.5 Flutter

Flutter is the aeroelastic instability that naturally follows galloping. It is nominally

of divergent character, rapidly building up, and has provided engineering history with

some of the most spectacular failure pictures. In different forms it is thought to have

been met first in slender bridges, dating as long as two centuries ago, and much later on

the torsionally weak wings (and tails) of World War I fighter aircraft, while nowadays

it is still one of the most serious concerns of aerodynamic design. Although many

classifications have been used by different authors, here the broadest categorisation

into coupled (or classical) flutter and Single-Degree-of-Freedom (SDOF) flutter will be

introduced.

Figure 2.7. (a) Displacements and aeroelastic forces on a thin airfoil; (b) Displacements and

aeroelastic forces for a bridge section

Coupled flutter was initially used to name the combined torsional-bending insta-

bility of airfoils. The phenomenon necessitates for a torsional and a bending mode

to oscillate at the same frequency but with a decisive phase difference between them,

that allows their cooperation to extract energy from the wind. Theodorsen [63] laid


the basis for ensuing flutter analysis by estimating the self-excited lift force (Lse) and

pitching moment (Mse) of a thin airfoil section immersed in an incompressible fluid

flow, while performing small amplitude harmonic vibrations of cyclic frequency ω. The

model problem is illustrated in Fig. 2.7(a), where typically the drag force and displace-

ment are unimportant and additionally it is assumed that the shear centre and chord

centre coincide. Theodorsen’s derivation established Lse, Mse as linear functions of

translation h, rotation α and of their first and second time derivatives, h, h, α, α,

where each overdot denotes one differentiation with respect to time. His well-known

solution is given as follows [64, 65]

Lse = πρb2[Uα+ h]− 2πρbUC(k)[Uα+ h+b

2α] ,

Mse = −πρb2[Ub

2α+

b2

8α] + πρb2UC(k)[Uα+ h+

b

2α] , (2.10)

where k=bω/U is the reduced cyclic frequency, based on the half chord length b, and

C(k) the complex Theodorsen circulation function that is assigned to time delays,

and is analytically expressible in terms of Bessel functions. C(k) becomes 1 for static

conditions, i.e. k →0, reverting to the quasi-steady formulation. It was early realised

[66–68] that direct application of Eqs.(2.10) to the analysis of bridges, owing to their

generally bluff sections, will yield inaccurate results. Hence the rationale of expressing

aeroelastic forces as linear k-dependent motion functions was preserved but C(k) was

replaced by a more reliable empirically determined substitute. Many options [69–71]

toward defining a reliable such substitute were formulated but the one originating from

Scanlan and co-workers [72, 73] became dominant. The Scanlan approach in its latest

amended form [74], expresses the self-excited force and moment components for the

bridge section in Fig. 2.7(b) as

Lse =1

2ρUB

[

KH∗

1

h

U+KH∗

2

Bα

U+K2H∗

3α+K2H∗

4

h

B+KH∗

5

p

U+K2H∗

6

p

B

]

,

Dse =1

2ρUB

[

KP ∗

1

p

U+KP ∗

2

Bα

U+K2P ∗

3α+K2P ∗

4

p

B+KP ∗

5

h

U+K2P ∗

6

h

B

]

,

Mse =1

2ρUB2

[

KA∗

1

h

U+KA∗

2

Bα

U+K2A∗

3α+K2A∗

4

h

B+KA∗

5

p

U+K2A∗

6

p

B

]

,

(2.11)

where K=Bω/U is the reduced cyclic frequency, based on the full deck length B this

time, and H∗

1−6, A∗

1−6, P∗

1−6 are the so-called flutter derivatives, which are functions

of K and are derived by means of sectional free or forced wind tunnel tests. No

inertial contributions are considered, since for heavy bridge decks in air they should


be minimal. P ∗ derivatives linked with Dse were a later addition to the formulation,

which actually proved critical for explaining the aeroelastic behaviour of the Akashi

Kaikyo full bridge model [75]. The linearisation concept utilised in deriving Eqs.(2.11)

should be applicable for small structural deflections, corresponding to only incipient

instability action, and in the absence of concerted vortex shedding with its subsequent

strong nonlinear characteristics.

The main attribute enabled by coupled flutter is having motions (one of them

being necessarily torsional) which when autonomously considered would behave stably,

i.e. have positive effective damping, but in common operation would exchange energy

between them through coupling terms H∗

2,3,5,6, A∗

1,4,5,6, P ∗

2,3,5,6 with a total positive

net energy effect for their system. Well streamlined bridges such as Severn, Lillebælt,

Burrard Inlet, Humber and Bosporus should exhibit catastrophic coupled flutter at very

high wind speeds, when interestingly part of their direct aeroelastic forces (velocity

products with H∗

1 , A∗

2, P ∗

1 ) contribute positive aerodynamic damping that adds to

the structural damping [76]. Note that these outcomes were derived only in wind

tunnel tests, and design wind speeds nowadays by far exceed the recorded operational

envelopes of modern real long span bridges, making the coupled flutter phenomenon a

quite improbable event. Still, complying with the strict flutter design guidelines has

imposed shape modifications on major modern bridge prototypes, referring in short

to the slot and stabiliser additions in the Akashi Kaikyo Bridge, the slot addition in

Zhejiang Xihoumen and Tsing Ma bridges [77], and the fairing addition to the Ting

Kau Bridge [78].

SDOF flutter should more accurately for the onomatology adopted in this work

be termed as torsional flutter, since the galloping convention introduced earlier should

cover any exclusively translational self-excited response. The mechanism operation re-

lies, as in incipient galloping, on an aerodynamic negative damping effect which reduces

structural damping and ultimately turns their sum’s sign to negative. Apparently from

Eqs.(2.11) A∗

2 becoming positive serves this goal. This function which is a typical

characteristic of many bluff sections previously used in bridges (e.g. H-sections [73]),

constitutes the most striking difference between airfoil and bridge behaviour. Thus

historically SDOF flutter was realised as a separated-flow phenomenon. Model stud-

ies have shown that there could be cases where a strongly positive A∗

2 coexists with a

tendency toward torsional-bending coupling. Then the intrinsic proclivity for torsional

flutter may drive the participation of vertical motion too, establishing optical (but not

functional) resemblance with coupled flutter.

Further on bridge flutter, its interaction with turbulence remains mystifying. Name-

ly a retardation of divergence of the full bridge response was early witnessed [12,76] as

qualitatively illustrated in Fig. 2.8(a). The phenomenon was initially accredited to the

20 Chapter 2. The Aeroelasticity FrameworkR

MS d

ispla

cem

ent

Wind velocity U

Laminar flow

Turbulent flow

(a)

0.2

-0.2

0 2 4

Turbulent flow

Laminar flow

fU/B

A2*

(b)

Figure 2.8. (a) Quantitative difference of response characteristics for a full bridge under

different flow conditions. Adapted from [59,76]; (b) A∗

2 from wind tunnel tests for a torsionally

unstable bridge section under laminar and turbulent flow conditions. Changes appear minimal

to sustain any substantial modification in the flutter behaviour. Data after [79].

effect of turbulence on flutter derivatives themselves, but a series of later studies [74,79]

proved that any alterations were minimal. A specific example for a torsionally weak

bridge section exposed to both laminar and turbulent flow conditions, is presented in

Fig. 2.8(b). Shown changes in A∗

2 are negligible, with even a minutely earlier negative

crossing for the turbulent scenario that should contradict experience. Other forms

of reasoning the flutter-turbulence interaction consist of multi-modal behaviour that

‘spreads’ the fluttering mode’s energy into other modes with various damping and

frequency characteristics, essentially setting up a complex tuned damping system [12,

76]. Finally explanations were acquired by means of the distracting action of turbulence

on the span-wise coherence of aeroelastic loading [80], although Scanlan also pointed out

that there is a possibility for such inhomogeneity to act detrimentally if local extreme

excitation regions, e.g. of very positive A∗

2, are introduced.

2.1.6 Wake-induced loading

Wake-induced loading refers to wind forces exerted on a body when this is situated in

the unsteady wake of another upstream bluff body. Meeting a ‘structured’ disturbed

stream of diffused and convected vorticity, when not far downstream of the source,

seems quite different than facing directly a fully developed turbulent flow, which also

has unsteady components but any localised imprints in it have been smeared out.

Notwithstanding the intuitive discrepancy, the forcing mechanism has been historically

perceived as only a special case of the previously presented instabilities and have been

explained on the same grounds. The term wake galloping is customarily employed to


characterise any excessive response events where wake loading seems to occur, still Assi

et al. [81] recently, revisiting an old problem set forward by Zdravkovich [82], exempli-

fied the case where a tandem pair of circular cylinders, not far apart (distance/diameter,

l/d=4), performs vibrations inherently different from common galloping. Galloping

should be successfully attained by quasi-steady theory, which strongly relies on the

mean wind velocity distribution. In the Wake-Induced-Vibrations (WIV) of Assi et al.

the unsteadiness brought upon by the vortices of the upstream obstacle are the critical

driving parameter and not the mean wind velocity distribution, while on the other hand

the phenomenon is non-resonant and induces a wake stiffness (or frequency) attribute

that will not match vortex shedding as presented earlier. Such phenomena, and any

interaction phenomena in general, will not be pursued further, but it is of relevance for

this work to shortly present them.

A very interesting feature in modelling wake interference effects is that ordinary

modelling ‘tools’ can capture their non-conservative nature once endowed with a ‘mem-

ory’ effect, which assumes the form of an artificially induced time-lag between motion

and aeroelastic force [83, 84]. This is striking when considering the discontinuous, or

even bistable [85] character observed. Bridge cables in parallel arrangements and on the

lee side of bridge towers, for certain wind directions, could evidently fall in the realm

of wake-induced response. Preventing such events consists of spacing cables far apart.

A distance of more than five or six diameters is thought to be an adequate remedy ac-

cording to Matsumoto et al. [86] and Tokoro et al. [87]. Records of large responses that

should be possibly accredited to wake effects can be found in many modern bridges.

In the Akashi Kaikyo Bridge, during the construction stage, hangers spaced at l/d=9

vibrated violently [88]. In another case of inclined stays this time, on the Second Severn

Crossing with even greater inter-cable distances (approximately 4m), relatively large

cable vibrations were witnessed in February 1999. Wind was blowing almost in parallel

to the cable fans, exciting all cables into first mode large transverse motions. The

phenomenon looks similar to tube arrays response, as in ‘breathing’ [49] for instance,

though the large spacing discourage engineers of linking it to a wake-related source. On

the Øresundsbron Bridge where inclined twin cables of d=250mm spaced 670mm apart

were used as stays, many extreme cable vibrations occurred [89]. The small spacing,

determined through a series of scaled wind tunnel tests though, is thought to be of some

connection to the events. Yet it is quite difficult to conclude, when even crude details

(e.g. of the vortex shedding process) due to the individual moving cable’s inclination

remain in ambiguity.


2.1.7 Rain-wind Instabilities

The cooperative action of rain and wind can produce forcing, which exceeds the previous

categorisation. The vibrations of yawed cable-stays in the Meiko-Nishi Bridge during

erection, reported by Hikami and Shiraishi [90], were the first to be attributed the

designation rain-wind instabilities. The name derives from the initial observation that

as soon as either rain or wind ceases, forcing dies down. Actually the contribution of

rain was even earlier postulated to be strongly influential in aeroelastic loading owing to

the monitoring work of Hardy and Bourdon [91] on transmission lines. Nowadays most

of the vibration recordings in many cable-stayed bridges concern the cables and are

classified as such phenomena, so evidently they deserve dedicated space in this study.

Figure 2.9. (a) Rivulet formation on the circular cable section; (b) Inclination geometry of

the inclined and yawed to the flow cable.

What seems to accompany the instability is the formation of water rivulets along

the cable length, as in Fig. 2.9. The formation is a process governed by many param-

eters such as the rainfall intensity, the cable inclination, the wind yaw, the treatment

and material of the cable surface and naturally the wind velocity and motion frequency.

There still remains some abstruseness on the subtle characteristics of these vibrations,

while in some cases contradicting results have been brought forward. Hikami and Shi-

raishi [90] performing wind tunnel tests on cable geometries with α=45◦, β=±45◦,

according to Fig. 2.9(b), observed that it is the upper rivulet motion that contributes

the excitation force with a lower rivulet only adding positive damping. Similarly Fla-

mand [92] for α=25◦, β=20◦–50◦ attained large vibrations when an upper rivulet (with

no lower one though) formed and oscillated in the circumferential direction. Further

attaching false fixed rivulets at the place where the real ones were previously seen, no


instability was found, thus yielding that it is the synchronised motion of the rivulet

that causes vibrations and not simply its appearance. Oppositely Bosdogiani and Oli-

vari [93] adopted the view that the motion of the liquid rivulets is not indispensable,

and by attaching rigid bars, of realistic size and shape, where rivulets would normally

form proved that classical galloping could emerge. On the same grounds Matsumoto

et al. [94] using a horizontal (α=0◦) circular cylinder, generically yawed (β=0◦–45◦) to

the flow, performed an extensive parametric study varying the angular position of an

artificial glued upper rivulet, in order to assess the effectiveness of the rivulet place-

ment. It is interesting to note that they obtained an unstable response even for the

typical non-yawed, non-inclined cross-flow scenario. Their results do not agree with the

rain-wind tests of Cosentino et al. [95] on a geometrically similar set-up. Still the impli-

cation of rain-wind vibrations on non-inclined cables was also made by Verwiebe [96],

who quotes the examples of hangers in two arch bridges that sustained such events just

before being commissioned. Verwiebe for cable orientations with α=30◦, β=0◦–90◦

presents three different vibration mechanisms underlying the rain-wind phenomenon.

In two of them the participation of both upper and lower rivulets is mandatory, with

resulting trajectories being planar across or along wind, depending on the symmetry

of the rivulets’ movement. For his remaining third mechanism only the lower rivulet

drives response resulting in an elliptic motion close to across-wind that can ultimately

cross over to purely across-wind when an upper rivulet forms.

Notwithstanding the discrepancies there are features unanimously accepted. Inci-

dents show up in a velocity restricted range visually similar to the vortex shedding

response of Fig. 2.2. Yet unlike vortex shedding the bounds of this range are inde-

pendent of the motion frequency [90,96] becoming functions of simply the wind speed.

Close to this observation lies that Reynolds numbers are of the order of 104–105, which

is nominally subcritical or very early critical. Amplitudes can reach up to several

meters i.e. >10d. There is some consistent repeatability of vibrations for a range of

effective β∗ (see Fig. 2.9 for the angle definition) between approximately 20◦ and 35◦,

but separately α and β are also relevant due to the gravity force influence on the water

rivulet motion. Flamand and co-workers [95, 97] recorded the water thickness and lift

force characteristics during large rain-wind induced excitation, as presented in Fig. 2.10.

Both plots are noisy and have intermittent ‘firing’ intervals, raising questions as to how

synchronisation can be feasible in such erratic waveforms.

The self-exciting character of rain-wind vibrations renders them similar to galloping

and alike formulations have been used for their modelling. Yamaguchi [98] employing

quasi-steady theory, showed that a sliding upper rivulet can oscillate along the cir-

cumference with a frequency originating from aerodynamic stiffness. As this frequency

measure varies with increasing wind speed, it approaches the cable frequency with their


12

7

2

-3

1 2 3 4

Lift force

Displacement

time (s)

cm, N/m

10 12 14 16 18 20 22 24 26 28 30

17

19

21

23

time (s)

position (°)

rivulet

U=11.5m/s, α=25°, β=50°α=25°, β=30°

Figure 2.10. Upper water rivulet mean angular position during rain-wind vibrations and

lift force, displacement time series for a different large response configuration. Adapted from

Flamand et al. [97].

coalescence setting off instability. This model along with its later successors does not

produce the random-like features shown in Fig. 2.10, however it has been deployed with

relative success in modern cable-stays (e.g. [99]) for treating a problem that is not yet

resolved in field. A long list of large-scale real events can be presented. The examples

of Puente Real Bridge, Veteran’s Memorial Bridge, Fred Hartman Bridge, Erasmus

Bridge, Dubrovnik Bridge, Farø and Higashi-Kobe Bridge are only indicative of how

widespread the phenomenon is. The actual case of Higashi-Kobe Bridge is of particular

interest. Fig. 2.11 from Kitazawa et al. [100] presents a summary of the preliminary

cable wind tunnel tests that were performed in order to mitigate the rain-wind response

of the proposed bridge design. Assuming that regular spaced protuberances along the

cable circumference will inhibit formation and motion of rivulets, it was found that in-

deed they are very effective in restraining large motions, keeping response low relative

to the plain smooth cylinder option. This was the verdict for all the tested wind speed

range and for both rainy and dry conditions. Thus such an aerodynamic measure was

applied to the newly erected bridge for first time ever.

Still there is another feature worthy of note in Fig. 2.11. For the cable encased in a

smooth circular duct, large vibrations were attained not only under rain but also in dry

conditions. The frequencies (at around 1Hz) are far off to reason direct Karman vortex

shedding resonance, and no water on the cable surface exists to form aerodynamically

unstable rivulets. Thus the origin of this new type of response sets forward a new puzzle

to later consider. Closing up this section it should be referred that large vibrations on

the protuberance-equipped cables of Higashi-Kobe Bridge did occur for an extreme

velocity outside the tested range of the preliminary tests (i.e. around 40m/s) [101].

It is the empiricism in the current state of knowledge that clearly necessitates for

additional testing and understanding of underlying mechanisms in order to get into a

state of successfully predicting problems.

2.2. Circular galloping: myth or true? 25

Figure 2.11. Resume of wind tunnel test results on the proposed cables in the Higashi-

Kobe Bridge. Improvement in aerodynamic performance is apparent for the solution with

protuberances under all conditions. Adapted from Kitazawa et al. [100].

2.2 Circular galloping: myth or true?

The unheralded diverging response illustrated for the plain cable in Fig. 2.11, seems

to resemble the classical galloping of Fig. 2.5. However a nominally perfect circular

body cannot fit galloping per se. Due to perfect symmetry any incidence effect is

expected to cancel out. As Parkinson argues in his cogent galloping review [23] for

such shapes “their afterbodies do not interfere with the separated shear layers and the

subsequent vortex formation, so that only vortex-induced vibration from rest will occur

for elastically-mounted cylinders, and galloping is not an issue”. This statement puts

forward a fundamental challenge: Is galloping possible for a circular cylinder? and if

not what is this new phenomenon captured by Fig. 2.11?

Answering these questions is far from obvious and entails first providing a short

background on aerodynamics specific to the circular cylinder. Actually Parkinson’s

quote is stripped from influences brought upon by Reynolds number transitions and

three-dimensionality of the flow, so it is natural to pursue discrepancies over these

parameters.

2.2.1 Reynolds number effects

For circular cylinders aerodynamic characteristics were early found to be decisively al-

tered by Reynolds number. Further to the subcritical Sr changes illustrated in Fig. 2.1,

the later critical and post-critical regions embody many interesting features (not only


in terms of Sr variation) that could well be held responsible for complex dynamic

behaviour of structures.

Bearman [102] was the first to systematically map the evolution of all main aero-

dynamic features in the Re range 105 to 7.5×105. He discovered that a discontinuous

jump in time-averaged base pressure, and concomitantly in time-averaged pressure

drag3 force, takes place at Re ≈3.4×105 suggesting that this is brought forward by

the establishment of a laminar separation bubble only on one side over the complete

length of the cylinder. Interestingly the bubble formed consistently on the same side,

despite the very smooth cylinder finish and the absence of apparent asymmetries in flow

conditions. A large steady lift force (i.e. CL ≈1.3) also resulted due to this one-sided

bubble. At the same time the frequency characteristics for lift, acquired through wake

velocity measurements, changed in an intermittent manner. For Re=3.55×105, when

the bubble was thought to be unstably bursting, Sr was transiting between two well

defined values at 0.23 and 0.32. Subsequently for a small Re increase the bubble sta-

bilised and acquired a single Sr ≈0.32. At larger Re a bubble similar to the first formed

on the other side of the cylinder bringing an end to the asymmetry-induced lift and

the so-called critical Reynolds region. Sr after this was about 0.48 with intensity more

than an order of magnitude lower than in any previous state. Bearman finally noted

that ‘contamination’ of the surface, e.g. by a dust particle, would trip the flow, alter the

flow uniformity over a considerable length, and transform the periodic vortex shedding

regardless of Sr to a wide-band process. An argument along the same lines, concerning

the high sensitivity while in near-critical Re, was earlier suggested by Humphreys [103],

who found that the addition of few light silk threads on the stagnation line can promote

regular 3D flow-patterning.

A substantial contribution to this insightful schema would come more than ten years

later. Kamiya et al. [104] performing similar experiments to Bearman, measured the

pressure distribution at a near-middle section and acquired plots translating meticu-

lously the laminar separation bubble notion to pressure profiles. In addition to before,

they obtained one-bubble states in alternate sides, waiving any uncertainty left that the

lift appearance may be an artefact caused only by geometric imperfections. They also

captured a hysteresis effect on all their recorded transitions (i.e. zero to one bubble,

one to two bubbles, and vice versa). This designates that the history of the flow is

crucial in the critical Re region. The actual transitions illustrated a transient period,

where steady lift coefficients would not jump directly to a large (or zero) lift value but

first gradually increase (or decrease) and this way reduce the amplitude of the ensuing

discontinuity. In terms of the formation of bubbles this could probably be seen as a

3Reference to drag throughout this study concerns the pressure part of drag, which is dominant formoderate to high Re numbers.


gradual growing with increasing growth rate. This conceptual view was also shared

by Almosnino and McAlister [105] who described the phenomenon as a supercritical

bifurcation.

Conversely, Schewe [106] in his seminal work, where he first proposed the similar-

ity of flow transitions to bifurcations he recorded sudden abrupt flow-state jumps and

consequently termed the bifurcations subcritical. In any case the bifurcations implied

by both Schewe and Almosnino and McAlister require higher order polynomials (at

least fifth, see [107]) for their accurate description. Schewe also suggested a number

of innovating ideas. Among them, he quotes that the observed transition phenomenon

is a hydrodynamic instability that should be treated in the framework of phase (or

critical) transitions. One particular contribution to the initial Bearman description

includes the recovery of the so-called critical fluctuations. Following Schewe, before

any discontinuous drag or lift jump takes place, the periodic shedding moves away

from its well defined Sr value, and acquires low frequency components that designate

the so-called critical slowing inherent in the proximity of any critical point. Another

interesting feature Schewe finds is that in the one-bubble regime, lift and drag PSDs

show the same pronounced frequency (Sr=0.33), which does not match the classical

shedding mode, where they should have a 1:2 relation. He estimates that this origi-

nates from vortices being strong only from the one side where a bubble has not yet

formed. Further he completes the characterisation of flow for Re up to 7.1×106, in

what should be deemed as the supercritical and transcritical Reynolds range. In the

course of increasing Re, a series of new overlapping transitions are postulated, where

bubbles progressively disappear towards reaching the fully turbulent state. In this final

turbulent state ordinary shedding seems to have revived at Sr ≈0.27, very close to the

subcritical value, a finding which has been known for many decades due to the work

of Roshko [108]. Roshko also quoted that the obtained mean pressure distributions for

these high Reynolds numbers are insensitive to the addition of a splitter plate in the

wake.

A concise schematic description of all the above was devised by Zdravkovich [109],

and is illustrated in Fig. 2.12. As shown, the time-averaged drag coefficient CD varies

as the position of the laminar-turbulent transition travels from the wake toward the

stagnation point. The one and two bubble states earlier defined, are mapped to the

designated TrBL1 and TrBL2-3 regimes. Most importantly Zdravkovich notes that

the whole classification should be valid only for a disturbance-free flow. The effects of

freestream turbulence or surface roughness could drastically alter the image. Although

most of the time both these influences are modelled as a simple shift to the left of the CD

curve in Fig. 2.12, the actual modifications are more subtle. The changing operations

concern exclusively the states where turbulence intrudes the shear and boundary layers,


Figure 2.12. Mean drag coefficient versus Reynolds number. On top, transitions (Tr) from

laminar (L) to turbulent (T) flow are presented in relation to separation points (S) and boundary

layers (BL). Adapted from Zravkovich [109].

termed TrSL and TrBL respectively. Large enough roughness values can for instance

completely preclude separation bubbles and thus sweep away the greatest part of TrBL,

whereas freestream turbulence would also very efficiently relocate and contract TrSL2.

All this description is exclusive to a static cylinder. Motion similarly perturbs the flow

by bringing vortex formation nearer to the cylinder surface, altering many details of

the resultant aerodynamic force. According to Humphries [110], who performed water-

tunnel tests on a large-scale flexible cylinder, motion can preserve vortex shedding

unaltered (i.e. without Sr transitions) throughout the critical Reynolds number range.

2.2.2 Inclination effects

Imagine a circular cylinder generically inclined to the flow. In this case, there is not only

the normal to the body wind component but also an axial wind contribution running

along the span-wise direction. This heuristic approach, where the flow is decomposed

into two independent orthogonal parts (i.e. independence principal), although appeal-

ing on its simplicity is not always accurate. The actual flow is far more complicated

and 3D features emerging from the complex fluid-structure interaction can render such

a two-component flow partition invalid.

Bursnall and Loftin [111] performed one of the very few test studies on yawed cylin-

ders inside the critical Reynolds number range. Their results suggest that the wind

component normal to the cylinder Un, is not enough to adequately characterise the

mean drag evolution. For their tested cases with cylinders inclined at 90◦, 75◦, 60◦, 45◦


and 30◦ to the freestream, they produced maps of Un versus the normal to the cylinder-

axis mean drag force, and obtained five different curves. They found that the lower

the inclination the lower the Un at which the critical drag-drop occurs. Namely for

the vertical cylinder the drag begins declining at 3.7×105, whereas rotating 45◦ or 60◦

(transiting to more shallow configurations) this value becomes 2.3×105 and 1.06×105

respectively. The situation is also preserved if the total wind is used instead of just

the normal wind component. Additionally there are vast discrepancies in the final

supercritical mean drag coefficients from different inclination set-ups. However, these

differences would much reduce when the coefficients are estimated based on the actual

wind speed. This sets a fundamental question on which would be the most appropriate

wind measure to use in our descriptions. Further, acquired pressure distributions re-

vealed that increasingly deviating from the vertical case, bubbles became less apparent

and less stable.

Ramberg [112] when treating inclined cylinders, focused on vortex shedding and

much lower Reynolds numbers. He found that for Strouhal number calculations, the

use of the normal wind component is a convincing approximation for a wide range of

inclinations. He also presented the increased geometric sensitivity that is inherent in

the inclined cylinder flow, by producing variously slanted, multistable wakes for only

minimal boundary alterations. These results were also corroborated by Shiraishi et

al. [113], who set off to test the validity of considering the yawed circular section as

analogous to an elliptical one. They disproved this view and with flow visualisations

illustrated that indeed streaklines bend and cross the static inclined cylinder at almost

right angles, which should readily support the use of simply the normal wind com-

ponent. Following Ramberg, they found that the addition of end-plates would alter

the shedding strength and uniformity. Resonant peaks in lift PSDs do not have the

sharpness seen in the non-inclined scenario. Yet, Ruscheweyh [114] testing a range of

inclined cantilevered (where probably also tip vortices contribute in loading) cylinders,

showed that the vortex shedding response would reduce due to inclination only for

higher Scruton numbers. Depending on the end-plate spacing, lift PSDs would also

attain a low frequency content, probably similar to what was previously referred as

critical transitional fluctuations. Actually Ramberg captured different wake modes co-

existing during his tests. Hence, critical-like fluctuations could result by transitions,

not on the boundary layers this time, but between the referenced wake modes. On an-

other aspect, exploiting the along-length inhomogeneity, Hayashi and Kawamura [115]

uncovered a pressure gradient on the lee side of their cylinder, regardless of boundary

conditions. Its sign promotes flow directed from downstream toward upstream, which

noticeably is opposite to the axial wind component. Detailed characteristics of this

combined axial flow are largely undefined.


2.2.3 Instability mechanisms

So is the circular cylinder response exemplified in Fig. 2.11 galloping? In terms of defi-

nitions it was actually named ‘dry galloping’, mainly due to its seeming divergence with

increasing wind speed, but whether it truly holds any resemblance to classical galloping

is a view seriously disputed. A series of recent studies [116–118] assign the phenomenon

to ordinary vortex shedding origin, and refute the galloping characterisation. In any

case, the study of such phenomena has lately received a lot of research interest. Large

vibrations of bridge cable-stays that were initially suspected to be due to rain, are now

clarified to have occurred under dry conditions. Although it is clear that to mitigate

the previously presented rain-wind vibrations one should inhibit the rivulet formation

and motion, the required countermeasures for dry wind vibrations are a disturbing

mystery. To answer it, it is essential to uncover the mechanisms stoking the instability.

Matsumoto and his co-workers with a series of seminal papers [94, 119, 120], were the

first to attempt an explanation of such unexpected wind behaviour. Initially it was sug-

gested that three different types of response can occur. A galloping type, which could

be either diverging or velocity restricted, a vortex shedding type with long period, and

their mixed type. It was postulated that rain-wind phenomena would also fall into this

broad framework. Rivulets when formed would only amplify these identical instability

types. Amplification is usually envisaged in a quasi-steady manner, simply imposing

an added geometric asymmetry.

Galloping type response

The galloping type response emerges due to the axial flow that runs in the lee side of

the inclined circular body. The early suggestion [94, 119] was that this axial flow is a

non-vortex flow, close in value to the approaching wind axial component. Acting as an

air-curtain it would simulate a long-rigid splitter plate that as indicated earlier it will

induce classical galloping on a plain circular section. The intensity of the axial flow,

is the critical parameter that decides the vibration occurrence. Evidently, according

to this rule, non-yawed cables are not susceptible to dry galloping. As a matter of

fact, Matsumoto added an external artificial uniform flow in the wake of a horizontal

cylinder, and observed large response similar to the yawed equivalent. This was consid-

ered a convincing proof of the suggested theory. The schematic representation of this

archetypal idea is given in Fig. 2.13(a). A splitter plate in principle has a dual oper-

ation. It reduces the mean drag force and inhibits vortex formation in the near wake.

A more accurate wording of the latter effect is that vortex formation is postponed to

far downstream, where it becomes ineffective in feeding back substantial forcing [121].

It was experimentally observed by Matsumoto [101], that when dry galloping occurs,


Figure 2.13. (a) The axial flow, evi-

denced by light flags positioned inside the

wake, act towards inhibiting communica-

tion between shear layers and promoting

a secondary circulatory flow. The func-

tion described, simulates the galloping of

a circular cylinder equipped with a long

splitter plate. (b) Enhanced vortices are

produced when axial vortices from the in-

clined cable, mix and interact with ordi-

nary Karman vortices. Adapted from Mat-

sumoto et al. [94, 119,123].

classical vortex shedding has become weak and intermittent. As a matter of fact it was

also stated that Karman vortices are capable of suppressing large low frequency mo-

tion, but no supporting explanation was given on this. In the latest theory amendment,

Matsumoto et al. [122] consider vortex mitigation to be an indication of the axial flow

intensity. When axial flow is strong then it faithfully resembles a long rigid splitter

plate that will completely inhibit the vortex forcing and readily cause galloping. On

the other hand the vortex appearance would mean that the axial flow is closer to a

less efficient perforated splitter plate, which could even become unable to set off the

instability. Subsequently unstable and diverging galloping are distinguished in terms

of the efficacy of the splitter plate analogue.

Vortex type response

This type of response is more ambiguous and probably controversial. Its original con-

ception was based on the observation that for both wind tunnel tests and field record-

ings, large events seemed to cluster at discrete reduced velocity ranges, at around

20, 40, 80, 120 etc. These figures look like following a certain pattern of multiples

of the reduced velocity that corresponds to ordinary shedding (i.e. 5). Still, no ev-

ident reason existed for this connection. The work of Bearman and Tombazis [124]

around the same period, provided a plausible explanation. They introduced a mild

three-dimensionality in the wake of an ellipse with a blunt trailing edge and acquired


wake velocity PSDs with span-wise distinct frequency peaks. The spatial transitions

between alternate shedding frequencies were accommodated by so-called vortex dis-

locations (or splittings). These dislocations were then associated a characteristic low

frequency of switching-states. Matsumoto et al. [120] suspected that a similar source

for low frequency loading could exist behind cables. Evidently the axial flow has to

become a vortex flow in this scenario. Acquiring wake velocity PSDs, Matsumoto et al.

found that even for a non-yawed cylinder there is a slight shedding variation along the

length. This among other justification, encouraged the belief that even normal to the

flow cylinders, latently have the ability of producing large response. In this explanation

attempt however, the frequency forcing characteristics in all cases could not be proved

to follow as submultiples of ordinary vortex shedding. Bearman and Tombazis with

their imposed three-dimensionality, could accurately control their dislocations’ posi-

tions but in Matsumoto’s et al. cables, dislocations, if any, are randomly distributed.

Thus although promising, this mechanism was abandoned for not fitting the specific

details. Yet the idea of a vortex type response was not invalidated.

Matsumoto et al. [123] came back soon afterwards claiming that the mechanism

could be founded on subharmonic resonance. As earlier noted Durgin et al. [27] pre-

viously observed strong 1:3 resonant vortex shedding response for a vertical cylinder.

Shirakashi et al. [125] argued that this was only an end effect, but Matsumoto et al.

married the two views and showed their applicability on inclined cable test results.

Flow visualisations presented an axial vortex originating from the upstream end and

propagating towards the cable’s middle. Once shed it interacted with ordinary Karman

vortices for enhancing every third produced vortex. Taking into account that for an

inclined cable Sr ≈0.15, the quoted timeliness results Ur=20, giving a well match to

the earlier said reduced velocity ranges. An illustration of the above description is given

in Fig. 2.13(b). Subsequently this mechanism successfully entered design guidelines for

bridge cables [49]. Two marked features found, are that turbulence can enhance this

instability and that axial flow is again the regulating parameter. When the latter is

increased diverging galloping will emerge. This makes disputable the actual distinction

between unstable galloping and the vortex-response type, which seem to overlap (at

least for inclined sections). As a matter of fact in his later published work Matsumoto

et al. [122] seem to view this mechanism as redundant. Thus all comes down to the

simple rule that complete mitigation of vortex shedding designates diverging galloping

while intermittent and partial mitigation unstable galloping. Even responses found

inside the critical Re region are considered to be due to the Karman vortex weakening

that takes place. Intriguingly, bluntly turning this rationale to prevention measures,

it could mean that vortex shedding suppression devices are galloping inducers. This

norm is obviously served by ordinary splitter plates.


Yeo and Jones [116, 117] and Zuo and Jones [118] are still ardent advocates of

the vortex type response. According to them the phenomenon is exclusive to inclined

cylinders and will be controlled by reduced wind velocity. Actually suggesting control

from reduced wind velocity, is another way of stating that any motions occurring, are

primarily forced vibrations. Yeo and Jones employed a hybrid numerical turbulence

modelling scheme to simulate the flow past a skewed horizontal circular cylinder at

Re=1.4×105 (calculated on freestream velocity), and recover the aerodynamic force

functioning. Increasing skew angle up to a critical value (with the flow thereafter fun-

damentally changing), Karman shedding is gradually suppressed and the axial velocity

component rises. The weakened shedding interacts with the axial flow and develops so-

called swirling structures, advancing along the length in organised patterns. Shedding

among others, would also have to serve as a signal carrier. The resultant travelling

forces, when considered sectionally, have PSDs with low frequency content, and look

both frequency and amplitude modulated. With added skewness, the lowest acquired

peak in the across-wind PSDs, draws away from the ordinary Karman value. It is pos-

tulated that such forcing component starts up a long period vortex resonance, which

could further get amplified due to motion. Eventually it is suggested that an efficient

plan for counteracting the whole process is to fully eliminate Karman vortex shed-

ding. This will deprive axial flow from the potential to nucleate swirling structures.

Awkwardly this is exactly the opposite from what Matsumoto et al. [122] advise.

Critical Reynolds number and galloping

In the previous section, critical Reynolds number was assigned a secondary role in cable

instabilities influencing them only due to contributing to vortex shedding mitigation.

Yet historically there have been cable-like examples which ideally fit reasoning exclu-

sively on Reynolds number. The high Re, along-current vibrations of an oil jetty at

Immingham [126], could convincingly be captured if the instantaneous drag force act-

ing on it, steeply decreases with Re, exactly as shown for the mean CD in Fig. 2.12.

Steam generator tubes in a number of nuclear power plants had identical issues [127].

Martin et al. [128] modelled in a quasi-steady way the flow-structure interaction and

obtained what looks like the benchmark mass-on-moving-belt problem [129]. Imagine a

circular cylinder oscillating while sweeping a smooth critical drag drop region. During

the part-cycle that it accelerates against the flow, Re becomes larger due to the relative

velocity increasing. Therefore, the drag force acting on the cylinder has progressively

smaller values, establishing a drag differential that points in the direction of the cylin-

der motion. Likewise, when the cylinder velocity reverses, the new difference in drag is

again in the direction of motion. This process will continuously pump energy into the


system until a stable limit-cycle is reached. Being in need of negative dCD/dRe, such

oscillations are evidently feasible only in the critical Reynolds number range.

The unusual feature with the similar vibrations of stranded power conductors over

the River Severn, was that motion could also be close to the vertical plane [130].

Richards [131] showed that Re related, shape-induced lift in skew winds and its subse-

quent changes (i.e. derivatives) would have a crucial function in the aeroelastic forcing.

Covering the strands with tape to form a smooth finish the instability disappeared.

Prophetically probably, Richards also warned about the actual non-zero lift measured

on the modified smooth cables. Macdonald et al. [132] applied a newly proposed gen-

eralised galloping theory [133–135], which quasi-steadily accounts for Reynolds number

effects, and analytically proved the source of the stranded Severn conductor problem

to lie on excessively negative dCD/dRe and CLdCL/dRe terms. Note that charging

the instability on Re derivatives could not work for purely across-wind vibrations.

Re would then remain unchanged, thus any Reynolds effect in a first approximation

negates. On a plain circular section steep gradients in both CD and CL exist at Re

nominally around 1-3×105, where laminar separation bubbles start forming. Accord-

ingly excessive force derivatives with respect to Re can potentially cause the same type

of galloping instability found in stranded cables.

However, there is much hesitation in accepting Re effects as a major piece in the

dry galloping puzzle. One of the main arguments against this is that Reynolds numbers

in most recorded events fall short for being characterised critical. Typically they are

of the order of 104, which with some (but not absolute) certainty classifies them as

subcritical. A second more insightful reason has to do with a fundamental objection.

When a system approaches a critical transition point, as this can be envisaged the case

here, it will get increasingly slower in recovering from small perturbations [136]. At any

instant the dynamic state carries the additional influence of adjacent (or even far apart)

previous states, establishing a memory effect. Quasi-steady theory, which idealises the

instantaneous force to be only a function of the instantaneous relative velocity, cannot

naturally cope with this requirement. Put into simpler words, Carassale et al. [137]

used the hysteresis evidence of Kamiya et al. [104] and Schewe [106] to question the

inclusion of Re derivatives in quasi-steady formulations. Such derivatives could be

non-unique or even indeterminate. However, they also utilised purely geometric quasi-

steady theory inside the critical Re region, where in principle this could be equally

inapplicable. Promoting a simple Reynolds number explanation for dry galloping, is

thought by many equivalent to proving the quasi-steady theory validity throughout

the critical range. Concluding this part, a paradoxical feature should be noted. The

strands-induced instability of the Severn conductor was cured by surface smoothening.

In modern cable-stayed bridges the instability-control tactics are to effectively roughen

2.3. State of the art in bridge wind design 35

the already smooth cables. More intriguingly the roughening measures could even seem

to resemble the old conductor’s shape, making up a clear inconsistency as commented

by Tanaka [138].

This extensive review of proposed mechanisms is a sine qua non for the scope of the

current thesis. The many contradicting elements presented, not adequately stressed

in many points, are the best indication that phenomena for which theory is only now

getting shaped are addressed. The diversity shown in explanations, dominates also

in field-observations. Records spread over a wide range of wind conditions, geometric

configurations, and structural characteristics perplexing any analysis. At the Iroise

Bridge, a long monitoring campaign indicated large, unforeseen cable vibrations in

what categorically was said to be the critical Re range [139]. On the other hand Zuo

and Jones, reporting on the monitoring of the Fred Hartman and Veterans Memorial

Bridges [140], identify all sorts of large cable vibrations but none in the nominal critical

Re regime. Interestingly they record that a particular non-ordinary-Karman event type

occurs for similar wind conditions, with similar modal features, for both dry and rainy

conditions. A connection is made with Ur, which seems to consistently fall near 40,

and subsequently the scenario that rain-wind and dry-wind vibrations are the same

forced phenomenon is advanced. Finally, Matsumoto et al. [101,122,141] summarise a

number of large-scale cable incidents in a selection of Japanese bridges. Most of their

events have Re>105, concentrate in a narrow defined cable-wind angle domain (i.e.

near ≈60◦), refer to single low mode motion and occur at Ur>100. Ordinary such high

reduced wind velocities would designate indubitable prevalence of quasi-steady theory,

but in this instance there also seem to be complications from complex unsteady effects.

2.3 State of the art in bridge wind design

After a long section focusing on aerodynamic issues only recently being explored, it is

time to illustrate how today’s bridge engineering design deals with them as well as with

the earlier presented older and more rigorously studied wind problems. The biggest

concern in bridge aerodynamic performance is flutter. Its interaction with gust loading

is inevitable in any real-case consideration. There are yet more interaction phenomena,

and fortunately many of them have reached an elaborate state of treatment. To this

end only the basic framework for combined flutter-buffeting evaluations, essential for

the following chapter, will be given.

In §2.1.5, flutter was discussed in view of sectional aeroelastic forces, but no connec-

tion was made to the estimation of stability limits when a full-bridge, with subsequent

multi-modal behaviour is the application in hand. Following the succinct derivation of


Jain et al. [142], for the most general case of a geometrically complex bridge with mixed

modes, analysis proceeds as follows. The deflection components, shown in Fig. 2.7(b),

are written by use of the dimensionless mode shapes hi(s), αi(s), pi(s) as

h(s, t) =∑

i

hi(s)Bqi(t) , α(s, t) =∑

i

αi(s)Bqi(t) , p(s, t) =∑

i

pi(s)Bqi(t) ,

(2.12)

where s is the distance along the deck span and qi is the ith mode generalised displace-

ment. The equation of motion becomes

Ii[

qi + 2ζiωiqi + ω2i qi]

= Qi , (2.13)

where ωi is the modal cyclic frequency, Ii is the generalised inertia and Qi is the

generalised aerodynamic force. The latter two are given by

Ii =l∫

0

(

m(s)h2iB2 + I(s)α2

i +m(s)p2iB2)

ds ,

Qi =

l∫

0

(

Lhi(s)B +Mαi(s) +Dpi(s)B)

ds , (2.14)

m(s) and I(s) being the mass and mass moment of inertia (about the section’s centre

of gravity) respectively and l is the deck span length. Let the lift, moment and drag per

unit span be linearly decomposed into the sum of self-excited and buffeting components

L = Lse + Lb , M = Mse +Mb , D = Dse +Db . (2.15)

In general there is also self-induced buffeting action, which is herein discarded. For

the self-excited parts, Eqs.(2.11) are employed additionally assuming that all flutter

derivatives are constant along l. For the modal integrals it is written

l∫

0

hi(s)aj(s)ds

l= Ghiαi , (2.16)

together with the rest five obvious hi(s), αi(s), pi(s) permutations. When Eqs.(2.13)

are Fourier-transformed into the reduced frequency (K) domain they can be expressed

in matrix form as

Eq = Qb , (2.17)

2.3. State of the art in bridge wind design 37

where ˚ denotes the Fourier transformation. The components of E and Qb are given

by

Eij = −K2δij + ıKAij(K) +Bij(K) ,

Aij(K) = 2ζiKiδij −ρB4lK

2Ii[

H∗

1Ghihj +H∗

2Ghiαj +H∗

5Ghipj +A∗

1Gαihj

+A∗

2Gαiαj +A∗

5Gαipj + P ∗

1Gpipj + P ∗

2Gpiαj + P ∗

5Gpihj

]

,

Bij(K) = K2i δij −

ρB4lK2

2Ii[

H∗

4Ghihj +H∗

3Ghiαj +H∗

6Ghipj +A∗

4Gαihj

+A∗

3Gαiαj +A∗

6Gαipj + P ∗

4Gpipj + P ∗

3Gpiαj + P ∗

6Gpihj

]

,

Qbi =ρB3

U2Ii

l∫

0

(

Lbhi(s) + Mbαi(s) + Dbpi(s))

ds , (2.18)

with δij=1 for i = j, δij=0 for i 6= j.

The multi-modal flutter critical condition is determined by solving the homogeneous

equivalent of Eq.(2.17). The evaluation consists of varying K and deriving different

sets of ω. The values of K and ω for which both the real and imaginary parts of

the determinant of matrix E become simultaneously zero, are the effective negative-

damping thresholds. The minimum wind speed calculated by these pairs, define the

flutter velocity. Thereon expressing Eq.(2.17) in PSDs form and utilising standard

concepts of random vibrations, the characteristics of the combined buffeting response

can be estimated. As seen by considering only the homogeneous solution, the flutter

condition does not include explicitly the buffeting influence. A complimentary analysis

accounting also for buffeting was presented by Scanlan [13]. He considers the classical

case of two interacting modes and calculates the average rate of change in their total

energy. This turning to positive may be taken as a sign of instability. Additionally a

slight variation regarding the generalised self-excited forces was proposed. According

to it, the flutter derivative products of Aij(K) and Bij(K) in Eqs.(2.18) are no more

functions of a single K. Instead they are evaluated at the aerodynamically modified Kj

corresponding to the relevant participating mode j (see also [12]). This latter analysis

variant will be used in an inverse way for estimating the values of flutter derivatives

for the CSB in the next chapter. The described flutter framework found application in

the latest and most important bridge designs, in-short referring to the examples of the

Storebælt Bridge [143], the Akashi Kaikyo Bridge [15] and the Messina Strait planned

bridge [144].

A great deal of specialised analysis also exists for the rest individual elements of a

bridge, and particularly for the versatile cable-stays. Treatments are mostly based on

quasi-steady aerodynamic theory with any unsteady concerns being confined to ordi-


nary Karman vortex shedding. As illustrated in §2.1.7 and §2.2.3, for typical circular

cables, many alarming phenomena occur beyond the reach of classical galloping and

classical vortex shedding. Subsequently they would be unaccounted in any typical de-

sign. Not having as yet hard evidence on the forcing attributes of such instabilities, the

only true solution for engineering practice is to recur to experiments. This obviously

serves both current and future design, since an empirical basis is constructed which

could then lead to the development of analytical tools. The core knowledge in prac-

tically dealing with dry cable galloping is expressed by Fig. 2.14, where curves in the

Sc–Ur parameter space bound the regions of safety. The initial dynamic tests of Saito

et al. [145], on an inclined bridge-cable replica, designate that the necessary Sc (mainly

seen as a structural damping requirement) for restraining vibrations is approximated

by 35√Sc = Ur. Such a connection implies that cables will always become unstable for

sufficiently high Ur. The unrealistically high damping values imposed for most cases by

the Saito et al. relation, led to a new test campaign sponsored by the Federal Highway

Administration (FHWA) [146]. FHWA results, indicate that the threat should partly

be released since for Sc>3 no large events were recorded. The latter is reminiscent of

the actual design guideline against rain-wind phenomena. There Sc>10 [138], similarly

on purely empirical grounds.

Figure 2.14. Dry cable instability design criteria together with real-bridge unstable records.

Dotted lines are due to the uncertainty in defining structural damping values. Adapted from

Matsumoto et al. [122,141] and Kumarasena et al. [146]

Thus the current state of understanding comprises two far apart instability criteria

without clear indication of which is the most appropriate. Large-scale records from

real bridges [122, 141], due to uncertainty over the exact structural damping values,

deceivingly seem to agree with both. Recently Matsumoto supported that the two cri-

2.4. Concluding Remarks 39

teria should be equally valid [122]; Saito’s et al. criterion designates unstable galloping

(cf.§2.2.3), while the FHWA criterion diverging galloping. FHWA data near Sc=4.5 in

Fig. 2.14 do not seem to agree with this assertion. It should be mentioned that Fig. 2.14

crudely groups points corresponding to different configurations, and most importantly

that some of these (especially unstable ones) may be non-repeatable.

In §2.1.3 it was noted in passing that alternative numerical methods can be used for

dealing with the wind-structure interaction modelling. The two alternatives quoted (i.e.

discrete vortex models and solution of the Navier Stokes equations) have many variants

depending on their computational implementations. In any case, for bridge structures

where the combination of wind conditions and size results in very high Reynolds num-

bers, there are noticeable difficulties with the operation of such schemes. Discrete

vortex methods suffer from inconsistencies due to the poor knowledge of how to ad-

just the vortex arrangement to accurately describe the turbulent flow. On the other

hand Navier Stokes solvers need to resolve a wide range of turbulence scales, which

numerically becomes extremely tedious. Often simplifications are put forward such as

time or spatial averaging of the equations, which necessitate explicit turbulence mod-

els. Tuning turbulence models for large separation problems remains ambiguous and

introduces non-robustness into calculations. A detailed review of functioning charac-

teristics and specific attributes of the methods is outside the scope of the current thesis.

Yet it is informative to briefly refer to a number of bridge applications that illustrate

the use and value of these analytical tools. For the flutter analysis earlier presented

the flutter derivatives, which are yet to be determined, are possible to calculate with

any of the two methods. Larsen et al. [143, 147] and Starossek et al. [148] numerically

established sensible agreement with experimental flutter derivatives for a large number

of both streamlined and bluff cross sections. For the circular cylinder flow the near

critical Reynolds regime still remains rather challenging to simulate. Recently Yeo and

Jones [116, 117] have treated the high Re inclined cylinder flow and their results are

of specific interest to the current study. A general conclusive point to be drawn out

is that although the trust in such computational methods has seriously increased over

the last two decades, they are still in need of validation. The simplifications and as-

sumptions made in order to reduce computational costs create uncertainties that can

only be waived through comparisons with experimental results.

2.4 Concluding Remarks

Apart from laying the ground for the analysis to follow, this chapter additionally tried to

establish questions. All of them should naturally fall into the broadest themes of: ‘How


efficient and reliable is the current state of knowledge? ’ and ‘Are there new elements

to complement this knowledge? ’. This thesis will attempt to touch upon both.

As presented, the best available defence that bridge engineering has against wind

is multi-modal flutter analysis. It aims primarily at keeping sound the deck, which

is probably the heart of a bridge. Although the method has reached a great level

of sophistication, concerns will always exist as to whether results are representative

and realistic. This is not surprising for a method that has developed inside wind

tunnels, based on scaled models. Jones et al. [149] and Katsuchi et al. [150], quote the

significance that full-scale ambient recorded data should have in a verification scheme.

Actually modern bridges cope quite well with flutter. Any size-effect or inconsistency

of modelling, if is there, does not seem to show up due to the structure operating really

far from the conditions where the phenomenon was estimated to unfold. Unfortunately

the best available real-scale example that could, when well instrumented, give us a

great lesson on flutter and expose potential flaws, lies 80m below water in the bottom

of Puget Sound (i.e. Tacoma Narrows). In any case, all monitoring attempts should

be seen as a chance of putting more reality into bridge modelling. This is exactly the

objective sought in the following chapter.

Further, it is also essential to understand the limitations and shortcomings of other

analytical tools currently in hand. Simple details (e.g. the exact geometric arrange-

ment), when correctly accounted for, can bring straightforward explanations on what

might have looked as an unforeseen aerodynamic event. And this is by no means an

easy task. In the context of this research, the quasi-steady galloping framework is re-

visited in order to produce an original contribution that recognises the true limits of

the galloping analysis and points out all past omissions and defects. Still there are also

truly new wind-structure interaction phenomena that cannot be cast into the existing

knowledge. For a latest F/A-18 fighter jet, unexpected unsteady events rose from the

newly shaped wings [151]. Similarly in long-span bridges new events of alarming am-

plitude came through their smooth cables, which were paradoxically shaped like that

to prevent old well-known dynamic instabilities. A number of proposed mechanisms

were devised to explain these large cable vibrations, hoping to become precursors of

a better analytical framework. Intriguingly, very few common grounds can be found

between different approaches. As illustrated, they seem to disagree even on basic prin-

ciples, often leading to exactly opposite results. The wind tunnel tests presented and

analysed in a later chapter, attempt to offer a different interpretation of this unique

phenomenon. The critical Reynolds number range by means of the associated com-

plex transitional behaviour is carefully examined to reveal all the attributes compatible

with the instability. Stepping towards a holistic approach of the aerodynamic bridge


design it is essential that the unique parts that compose it are adequately clarified and

understood. And this is exactly the notion that this thesis attempts to serve.

Chapter 3

Identification of flutter derivatives from

full-scale data

The estimated response of large flexible bridges to severe wind loads is prone to mod-

elling uncertainties that can only ultimately be assessed by full-scale testing. To this

end, ambient vibration data from full-scale monitoring of the historic Clifton Suspen-

sion Bridge (CSB) have been analysed in order to capture elements of true wind-bridge

interaction. The multi-modal flutter framework earlier presented, is herein employed in

an inverse investigation. Flutter derivative identification, which has rarely previously

been attempted on full-scale data, was performed to seize any trends towards aerody-

namic instability. The chapter does not intend to be a meticulous dynamic description

of CSB. Instead, a number of useful notes for today’s aeroelasticity will be drawn out,

while there is a clear potential for the old outdated CSB to become the test bed for

future advances.

3.1 Introduction

For large-scale structures the most rational way to proceed with predictions on the

reliability and operational safety, includes identification methods from response only

measurements. Especially for existing bridges, the risk of flutter can substantially be

verified in this way. A bluff bridge cross-section, unlike a flat plate or an airfoil, has

no analytical expression for the fluid forces exerted on it while in motion. Identifying

the critical wind speed for instability inevitably has to adopt some experimental, semi-

empirical or numerical foundation. Most commonly wind tunnel tests of scale models

are used for reproducing the flutter phenomenon leaving the question of the effects of

scaling. It is well established that minor details such as deck railings or roadway grills

and vents can strongly alter the aerodynamic performance (see Scanlan and Tomko [73],

43

44 Chapter 3. Identification of flutter derivatives from full-scale data

Jones et al. [152] and Matsumoto et al. [153]). Hence, analysis of the response of the real

bridge can clarify the validity of wind tunnel tests and even reveal aspects, which either

due to modelling assumptions or to loading irregularities, were previously concealed.

Aeroelastic parameters have rarely been obtained from full-scale bridge data. Okau-

chi et al. [154] were the first to attempt something relevant. Building a bridge section

model (at a large scale, roughly 1/10) and setting it on-site against real wind conditions,

they compared results with smaller wind tunnel equivalents. They suggested that, al-

though for turbulent flow the relative differences in the turbulence details can impose

inconsistencies, in general wind tunnel models produce representative results. Jakobsen

and Larose [155] addressed the problem on the Hoga Kusten Bridge and presented a

comparative analysis with wind tunnel results using a subspace identification technique

for extraction of flutter derivatives. Costa and Borri [156] essentially applied the same

identification routine, both on numerically simulated responses and on measured data

from the Iroise Bridge, quoting good performance of the method in each case. For all

of these bridge studies, the identification routine itself was found to be reliable, when

tested using simulated data produced with added variously coloured noise. Compar-

isons between full-scale and wind tunnel results were not unreasonable, but since the

full-scale bridges were far from flutter, the trends in flutter derivatives were not clear.

Another approach to the problem of identifying the aerodynamic effects on full-scale

bridge vibration characteristics was used by Macdonald [157] on the Second Severn

Crossing. Variations of effective damping ratios and natural frequencies with wind

speed were found, and some indications of aeroelastic modal coupling were identified on

the partially constructed bridge. In other full-scale studies, Littler [158] and Brownjohn

[159] on the Humber Bridge, Bietry et al. [16] on the Saint-Nazaire Bridge, Jensen et

al. [160] on the Great Belt Bridge, Ge and Tanaka [161] on the Hoga Kusten Bridge

during construction and Nagayama et al. [162] on the Hakucho Bridge all found some

trends of effective aerodynamic damping with wind speed, but coupling between modes

and flutter derivatives were not pursued.

The limited number of full-scale studies from which aeroelastic parameters have

been found, makes any new cases useful for extending knowledge on the viability of

system identification from site data and for interpreting actual bridge behaviour.

3.2 The case study

As part of this work, analysis is performed on full-scale vibration measurements from

the historic CSB, shown in Fig. 3.1. The CSB spans the Avon Gorge in Bristol, UK

and was designed by I.K. Brunel in 1830, although it was not completed until 1864

3.2. The case study 45

(Barlow [163]). It was one of the longest suspension bridges of that time, with a main

span of 214m. Wrought iron chains provide the suspension system, being the common

practice for such early long-span bridges. In the light of modern understanding of

bridge aerodynamics, the bridge cross-section (Fig. 3.2) and its light weight make it

potentially susceptible to wind-induced vibrations. Indeed, on a few occasions in its

lifespan large amplitude vibrations in strong winds have been reported.

FIGURES

LeighWoods

CliftonMaintenance craddle

Accelerometer reference cross-section

Accelerometer cross-section

Anemometer

214m

80.2m 107m

46m

Figure 3.1. Bridge elevation showing instrument locations. Based on figure after Barlow [163],

with permission from Thomas Telford Publishing.

On Christmas Day 1990 there was evidence of vertical motion at the bridge ends

of the order of 250mm, which translates to even larger amplitudes within the bridge

span. Both vertical and torsional deck motions were evident on a video recording of

the bridge towards the end of a storm. A similar large vibration event was reported

on 3 December 2006. Although no wind recordings exist from the bridge site itself on

these occasions, data from the nearest weather stations imply that the maximum 1h

mean wind speeds could have been around 20m/s. For recordings on site with wind

speeds up to 16m/s, coupling action between the first vertical and torsional modes

seemed to occur and the maximum vertical displacement at the ends of the bridge was

35mm (maximum measured elsewhere 92mm). The coupling action between modes

and the rapid growth of vibration amplitudes for a modest increase in wind speed,

indicate strong aeroelastic effects and make the bridge behaviour rather interesting.

Such characteristics are reminiscent of features observed, in a more severe form, on the

Tacoma Narrows (Farquharson et al. [164]) and Deer Isle (Kumarasena et al. [165,166])

bridges.

It is worth noting that ten suspension bridges from the same era as the CSB failed

due to wind between 1818 and 1889, including the Menai Straits Bridge, the Wheeling

Bridge and a span of the Brighton Chain Pier (Farquharson et al. [164]). In contrast

the CSB has survived virtually unscathed for over 140 years. According to empirical


Figure 3.2. Sketch of the bridge cross-section.

estimates, similar in most bridge design rules, it is potentially susceptible to flutter with

an estimated critical wind speed of not more than 20m/s. Therefore adverse aeroelastic

effects are expected to become significant in moderately strong winds. For the current

study, the wind conditions and bridge response recordings over two winter periods, from

November 2003 to March 2004 [167] and from December 2007 to February 2008, are

used. The site monitoring was conducted by Macdonald initially under a contract from

the Clifton Suspension Bridge Trust. Later the monitoring was continued for research

purposes with the assistance of the author. The data include several occasions with

moderately strong winds, and reasonable ranges of wind speeds and directions, thus

enabling a meaningful assessment of the wind effects on the bridge dynamics.

Two ultrasonic anemometers were mounted 61m either side of midspan, more than

5m above road level. Nine accelerometers were positioned at a series of cross-sections

along the bridge during an earlier analysis of the CSB dynamic response, enabling mode

shapes to be identified (Macdonald [168]). The first three vertical and torsional modes

are shown in Fig. 3.3. For the records considered here, two sets of three accelerome-

ters were positioned at midspan and at a cross-section slightly off centre (26.8m from

midspan) as illustrated in Fig. 3.1. This location was chosen as the reference cross-

section since all significant vibration modes could be measured there. All motion related

measurements below, refer to this location. Signals from all instruments were passed

through anti-aliasing filters with a cut-off frequency of 4Hz and were recorded at a

sampling rate of 12.5Hz. The primary aim here is to uncover the underlying mech-

anism of large amplitude response that the bridge was found to produce for certain

wind conditions, and explore the flutter potential. To do this the variation of modal

characteristics with wind velocity has to be determined.

Modal parameter estimates from a frequency based curve fitting technique (devised

by Macdonald [157]) are used here, together with a subspace stochastic identification

formulation especially modified to extract flutter derivatives (Jakobsen [169]). Due

3.3. Wind characteristics 47

fv1=0.293Hz,

fv2=0.424Hz,

fv3=0.657Hz,

ζv1=3.31%

ζv2=1.99%

ζv3=2.12%

ft1=0.356Hz,

ft2=0.498Hz,

ft3=0.759Hz,

ζt1=2.60%

ζt2=3.44%

ζt3=2.16%

Vertical Torsional

Figure 3.3. Experimentally obtained vertical and torsional modes for CSB, adapted from

Macdonald [168].

to lack of wind tunnel data from a scale model, flutter derivatives of other typical

bridge cross-sections, as presented by Scanlan and Tomko [73], are used for comparative

assessments. Cross-sections employed for this purpose are chosen to represent both the

low structural depth and the high parapets (perforated on the CSB), of the section in

hand.

In the following section, first a short background of the acquired wind measure-

ments is given. Typical wind speeds and wind turbulence conditions are described and

the local terrain effects are discussed. Subsequently attention is moved to the bridge

response details. All necessary modal background is provided, and the wind parameter

on them is distinguished. The last part containing the flutter derivative identification

scheme, starts by shortly commenting on the employed Covariance Block Hankel Ma-

trix (CBHM) formulation. Eventually a flutter velocity estimate is sought to compare

with theoretical predictions.

3.3 Wind characteristics

The topography around the CSB has a considerable effect on the local wind attributes.

As shown in the polar plots from both anemometers in Fig. 3.4, the wind speeds follow

certain trends with orientations. (True North is at 31◦ relative to the axes shown).

These trends differ markedly from the general wind pattern in the region away from the

local influences. The strongest winds in the absence of topographic effects are typically

from the south-west direction (at about 250◦ on Fig. 3.4 axes). The on-bridge measured

stronger winds, are aligned along the gorge and can be probably explained by funnelling

of the flow in these orientations and sheltering due to the high ground near the bridge

ends. Evidently strong winds from the south-west are greatly attenuated at the bridge

site, and virtually no wind from the north-east quadrant is experienced. It was also


observed that the correlation of the wind directions and wind speeds measured by the

two anemometers, was strongly influenced by the wind orientation. In particular, for

wind directions close to along the gorge even a small change in wind direction, results

a large but consistent variation in the ratio between the two individual anemometer

wind speed values.

A typical assumption is made that that wind loading is approximately stationary for

records up to one hour [2]. The maximum wind speed, averaged over one hour (unless

stated all wind information hereafter refers to 1h means), did not exceed 16m/s at the

bridge site, although higher speeds were measured at the nearest weather stations, for

winds from the south-west. A histogram of 1h average wind speeds at the bridge is

given in Fig. 3.4. Maximum 1s gusts were of the order of 26m/s for both anemometers.

In addition, the wind turbulence and angle of attack parameters were considered.

For wind turbulence there was a strong dependence on wind direction and a weaker

one on wind speed. High levels of turbulence (up to 60% longitudinal turbulence

intensity, Iu) were experienced, particularly for wind not along the gorge and for lower

wind speeds. In winds over 8m/s, which only occurred along the gorge, approximately

normal to the bridge, the mean longitudinal turbulence intensity was 21% and the mean

vertical turbulence intensity Iw, 10%. The vertical and across-wind (Iv, in the deck-

wind plane) turbulence intensities followed very similar patterns to the longitudinal

turbulence. For longitudinal turbulence up to 40%, the across-wind turbulence was

approximately equal to it and the vertical turbulence intensity approximately half the

value. These are typical relationships between the three components of turbulence. For

higher turbulence intensities measured, generally in lower wind speeds, the vertical and

across-wind turbulence intensities were relatively larger. For the vertical angle of attack

there was also strong dependence on the wind direction, and there were noticeable

differences in the measurements from the two anemometers. The presence of the bridge

itself is likely to have affected these measurements, as well as the topography of the

gorge, since the anemometers were relatively close to the deck. High vertical angles of

attack occurred, up to approximately ±10◦. It should be reminded that these values are

averaged over one hour periods. There was no significant difference in vertical angles

of attack for different wind speeds.

A wind aspect significant for the subsequent analysis, refers to the frequency content

of wind buffeting. Although the traffic loading seems to be reasonably well captured by

a white noise approximation (predominantly from step loading as vehicles drive onto

or off the suspended span), the same does not hold for wind. By comparing spectral

estimates deduced for various combinations of wind and traffic, it was found that above

3.3. Wind characteristics 49

Figure 3.4. (a) Histogram of wind speeds during the 2003-04 recording period. (b) Polar plots

of 1h mean wind velocities from both anemometers.


approximately 0.25Hz the wind loading spectra tailed off as f−ǫ with ǫ around -8/3,

producing a general loading spectrum of the form

Sload(f) = Swf−8/3 + St , (3.1)

where Sw is a constant for a given record, being a function of the wind parameters,

and St is the magnitude of white noise traffic loading specific for each record. The

frequency power exponent of -8/3 when compared with the -5/3 value corresponding

to isotropic Karman turbulence (referring to both Iu and Iw) customarily employed in

design, it implies that the product of the aerodynamic admittance and joint acceptance

functions in Eq.(2.2) should be inversely proportional to f .

3.4 Response and modal parameters

3.4.1 Response Characteristics

Fig. 3.5 shows the 1h average wind speeds over the whole of the first monitoring pe-

riod, and the corresponding Root Mean Square (RMS) vertical accelerations at the

reference cross-section. The RMS amplitudes normally show a clear daily cycle with

the varying traffic load, with a maximum vertical response of approximately 0.02m/s2.

By comparison it can be seen that only in wind speeds exceeding approximately 8m/s

does the response noticeably exceed the maximum traffic-induced response. The maxi-

mum wind-induced acceleration measured was approximately four times the maximum

traffic-induced acceleration. The torsional and lateral acceleration responses at the

reference cross-section followed very similar patterns to the vertical response over the

monitoring period, although the magnitudes of the responses were lower. The maxi-

mum instantaneous value of each component was found to be approximately six times

the 1h RMS value.

Dynamic displacements were calculated from the measured accelerations by double

integration and it was noticed that the response is dominated by low frequency modes.

The dominance of the low frequency modes is caused by the relatively higher wind

loading at low frequencies and the effect of the integration, which exaggerates low fre-

quency components. Whereas the maximum RMS acceleration due to wind loading was

approximately four times the maximum due to traffic loading, in terms of displacement

the maximum RMS response to wind was approximately 10 times that for traffic.

The influence of wind loading on the measured vertical accelerations is shown in

Fig. 3.6. Similar figures were obtained for the lateral and torsional accelerations. The

3.4. Response and modal parameters 51

10 17 24 1 8 15 22 29 5 12 19 26 2 9 16 23 1 8 15 220

2

4

6

8

10

12

14

16

November December January February MarchDate (2003-04)

1hr

aver

age

win

dsp

eed

(m/s)

(a)

10 17 24 1 8 15 22 29 5 12 19 26 2 9 16 23 1 8 15 220

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

November December January February MarchDate (2003-04)

RM

Sve

rtic

alacc

eler

ation

(m/s2

)

(b)

Figure 3.5. (a) 1h average wind speed over the monitoring period. (b) 1h RMS vertical

accelerations at the reference location over the monitoring period.

scatter of results, particularly at low wind speeds, is largely due to the varying traf-

fic contribution. The varying wind turbulence intensity also proved, as expected, to

have an effect. Excluding records dominated by traffic, and normalising by the verti-

cal turbulence intensity, gives a much clearer relationship with wind speed as shown

in Fig. 3.6(b). Vertical turbulence intensity is considered more appropriate for such

scaling, when bridge sections with relatively low mean lift coefficient CL (cf. Eq.(2.2))

are considered. Normalised RMS responses become a power law functions of the wind

speed as shown in Eq.(2.5). The power exponent is very close to the theoretical 2.83

buffeting value, which could imply that response is solely due to buffeting. Yet as shown

in Fig. 2.8 for turbulent conditions, a full bridge experiences the combined buffeting-

flutter action with a power law too. Also it is notable that no sharp peaks, that could

signify vortex-induced response, exist in Fig. 3.6(b).

3.4.2 Modal Analysis

Modal parameters were previously calculated from curve fitting acceleration Power

Spectral Densities (PSDs), using the Iterative Windowed Curve fitting Method (IWCM)

of Macdonald [157]. IWCM was specifically developed for the analysis of ambient

vibration data with in general non-white loading spectrum. The method iteratively

curve fits in the frequency domain a series of idealised single-degree-of-freedom (SDOF)

transfer functions, taking into account the shape of the loading spectrum, the effect of

windowing on the spectra (both from the measured data and from the idealised transfer

functions) and the contributions of multiple modes (in a linear sense).

Measurements were only taken on the suspended bridge deck, but all modes in-

evitably involve vibrations of other parts of the structure, particularly the chains.


0 2 4 6 8 10 12 14 160

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Wind speed (m/s)

RM

Sve

rtic

alacc

eler

ation,σ

v(m

/s2

)

(a)0 2 4 6 8 10 12 14 16

0

0.2

0.4

0.6

0.8

1

1.2

Wind speed (m/s)

Norm

alise

dR

MS

vert

icalacc

eler

ation,σ

v/I w

(m/s2

)

12:30am - 6:30am

Wind speed >8m/s

No vehicles

No vehicles or pedestrians

∝ (Wind speed)2.8

(b)

Figure 3.6. (a) RMS vertical accelerations σv, in relation to wind speed for all 1h records. (b)

Same as (a) for 1h records dominated by wind loading, with RMS vertical accelerations now

divided by the vertical turbulence intensity. The power-law approximating the obtained trend

is also plotted.

Analysis was performed for frequencies up to 3Hz with twelve vertical, eleven tor-

sional and four lateral modes being identified in this range, based on measurements

in low wind speeds (Macdonald [168]). Typical PSDs for three different loading sce-

narios for vertical, torsional and lateral accelerations are given in Fig. 3.7 to present

the effect of wind loading on the bridge response. An important detail is the proxim-

ity of the first vertical and torsional modes, with natural frequencies of 0.293Hz and

0.356Hz respectively (ratio 0.82). These are the first antisymmetric modes of each type

as depicted in Fig. (3.3). It appears that in the stronger winds they start to couple

in a possibly incipient flutter motion as evidenced by the small hump at 0.35Hz in the

vertical spectrum seen in Fig. 3.7(a) and more clearly in Fig. 3.8.

For further analysis it was desirable to reduce the actual multi-mode response to

a simpler two-degree-of-freedom equivalent, concerning only the relevant first vertical

and torsional modes. Low-pass filtering could not fully remove the contributions of the

second mode of each type, since they were very closely located. However, using the

measured responses at both accelerometer cross-sections (see Fig. 3.1) together with the

known mode shapes, the following operation was performed. With the second section

in the midspan, the first pair of modes being anti-symmetric is showing as zero. The

second pair of modes on the other hand, containing symmetric modes that maximise

there, could be readily identified. Transferring these modal displacement values to the

reference cross section and subtracting them from the total signal, results in almost

pure first mode motions.

Fig. 3.8(a) hence shows responses of the first pair of modes (z vertical and θ tor-

sional) for the highest recorded 1h wind speed (15.3m/s). The peak in the vertical PSD

3.4. Response and modal parameters 53

0 0.5 1 1.5 2 2.5 310

−7

10−6

10−5

10−4

10−3

10−2

10−1

Frequency (Hz)

Ver

tica

lacc

eler

ation

PSD

((m

/s2

)2/H

z)

Maximum wind, with trafficMinimal wind, Rush hour trafficModerate wind, only

(a)0 0.5 1 1.5 2 2.5 3

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

Frequency (Hz)

Tors

ionalacc

eler

ation

PSD

((ra

d/s2

)2/H

z)


(b)

Figure 3.7. PSDs for different loading condi-

tions for (a) vertical (b) torsional and (c) lat-

eral accelerations.

0 0.5 1 1.5 2 2.5 310

−7

10−6

10−5

10−4

10−3

10−2

10−1

Frequency (Hz)

Lat

eral

acce

lera

tion

PSD

((m

/s2)2

/Hz)


(c)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

2

4

6x 10

−3

f (Hz)

PSD

θ(r

ad/s

2)2

/H

z

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.03

0.06

0.09

f (Hz)

PSD

z(m

/s2)2

/Hz

(a)0 0.1 0.2 0.3 0.4 0.5 0.6

10−4

10−3

10−2

10−1

Frequency (Hz)

Ver

tica

lac

cele

ration

PSD

((m

/s2)2

/H

z)

15.3m/s

13.8m/s

11.9m/s

9.8m/s

7.9m/s

6.1m/s

1st Torsional mode

(b)

Figure 3.8. (a) PSDs of filtered data for first vertical and torsional modes for the maximum

wind speed record. (b) The evolution of the coupling action is evident in the vertical PSD for

wind speeds above 11m/s.


at the torsional frequency is strong evidence of coupling action. The coupling was not

evident for low winds and became stronger only for higher winds (Fig. 3.8(b)), setting

it most probably to be an aeroelastic effect. The coherence between the vertical and

torsional accelerations showed similar evidence, with the value around 0.35Hz rising to

approximately 0.6 in high winds compared with typical values below 0.4 in low winds

and at other frequencies. It is also worth noting that the second pair of modes showed

a similar tendency for coupling action in strong winds, due to their also similar shapes

(see Fig. 3.3) and close natural frequencies (ratio 0.424/0.498≈0.85). The next section

discusses in more detail the identification of the CSB flutter derivatives, so as to be

able to quantify the observed coupling signs and the tendency towards flutter.

3.5 Flutter derivatives

3.5.1 Flutter Analysis

According to the semi-empirical Selberg [170] equation for bridge sections resembling

flat plates, an estimate for the flutter speed is given by

Uf

fθ0B= C

√

rgm

ρB3[1− (

fz0fθ0

)2] , (3.2)

where Uf is the flutter speed, B the deck width, rg the radius of gyration, m the total

mass per unit length, C a constant depending on the section’s behavioural resemblance

to a flat plate, ρ the air density and fz0 and fθ0 the still air vertical and torsional

natural frequencies. For the first pair of natural frequencies, described above, substi-

tuting ρ=1.2kg/m3, B=9.46m, C=2, m=5370kg/m, rg=4m, the flutter onset speed is

estimated at approximately 18m/s, or 14m/s using the more conservative variant of

Eq.(3.2) in the British design rules [171]. These very low values are due to the low

torsional natural frequency, the close neighbourhood of vertical and torsional modes

and the low mass per unit length. However, the bridge cross-section is not a flat plate

and the uncertainty in the value of C (containing an experimental correction factor)

means Eq.(3.2) is not very reliable. Still, it gives a rough first approximation for the

flutter speed and this is within the range of wind speeds encountered at the site.

For evaluating the flutter behaviour, the classical multi-modal formulation presented

in §2.3 is employed for expressing the aeroelastic forces. Further it is assumed that the

considered modes have no considerable mixed components. Substituting z and θ to

the generalised displacements h1, α2 in Eqs.(2.18) (whereas α1=p1=h2=p2=0), the

3.5. Flutter derivatives 55

motion-dependent lift and moment, Lse and Mse become

Lse =1

2ρU2B

[

KH∗

1Gzzz

U+KH∗

2GzθBθ

U+K2H∗

3Gzθθ +K2H∗

4Gzzz

B

]

Mse =1

2ρU2B2

[

KA∗

1Gθzz

U+KA∗

2GθθBθ

U+K2A∗

3Gθθθ +K2A∗

4Gθzz

B

]

, (3.3)

where following the convention of Scanlan [12,13], the reduced frequenciesK(=2πfB/U)

and all flutter derivatives H∗

i , A∗

i are calculated based on the frequency of the mode

they refer to. Any amplitude or structural damping dependence of flutter derivatives

was not accounted for. In this basic 2D formulation, the motion-induced drag force,

Dse, and the effects of the along-wind motion are ignored, meaning that P ∗

i deriva-

tives are disregarded. Actually the standard linear decomposition of Dse was recently

questioned by Starossek et al. [148]. According to them for a bridge section symmetric

along the centreline, both positive and negative rotations will result the same exposed

area to the wind. Thus it is more natural that drag forcing with double the motion

frequency will result. Their assertion was corroborated by numerical simulations. In

any case along-wind displacements were quite low for all records in CSB.

3.5.2 Identification Method

A state space formulation of the dynamic problem was assembled using the Covariance

Block Hankel Matrix Method (CBHM method), which is founded on the widely used

Eigenvalue Realization Algorithm (ERA) described by Juang and Pappa [172]. The

formulation in CBHM is identical to ERA with the exception that instead of the Markov

parameters containing Impulse Response Functions (IRFs), covariance estimates of

output measured random data are employed. Jakobsen [169] first applied CHBM in the

estimation of flutter derivatives from wind tunnel tests and the method has since found

extensive use in aerodynamic applications and testing. Peeters and De Roeck [173],

Qin and Gu [174] and Siringoringo and Fujino [175] all describe the matrix derivation

in detail. Brownjohn et al. [176] found a relative advantage of the method over other

operational modal analysis approaches. The method is based on the Singular Value

Decomposition (SVD) and appropriate factorisation of a Hankel matrix built up by

covariance estimates of the output time series (i.e. displacements or accelerations in

this application). If y stands for the displacement matrix with z and θ in rows, then the

unbiased sample cross-covariance matrix to be used in the Hankel block construction

is given by

Cyy(i+ n, i) = Cyy(n) =1

N − n

∑

N−n

y(i+ n)yT (i), n = 0, 1, .., ℓ . (3.4)


where n is the number of sampling intervals for the discrete time delay n∆t, N is the

number of samples in the time series, ℓ is the maximum number of lags considered and

i is a counting index. The biased estimate, which only differs in the use of the denomi-

nator N instead of N − n, can be used instead with negligible differences for long time

records. The method exclusively handles white noise loading but here it was attempted

to also account for the actual shape of the loading spectra given in Eq.(3.1). Imagine

the SDOF system of Fig. 3.9 with frequency response function Hi(f). When loaded

by coloured noise, indicated by the non-flat PSD Sload, then the produced response

PSD will not preserve the shape of Hi(f) as the identification method necessitates. A

remedy would be to apply a filter function on the response PSD with magnitude equal

to the inverse square root of the loading PSD and zero phase lag. Thus the corrected

PSD, dashed line in Fig. 3.9, would look as if produced by a fictitious white noise load.

For the two-degree-of-freedom system similarly, ordinary filtering on the response data

in the frequency domain could account for the coloured lift and moment spectra. A new

attribute that has to be taken care of though, is preserving after filtering the exact cou-

pling between modes, that was artificially altered. The lift filter function FL in Fig. 3.9

for instance, will introduce increased coupling. This is not realistic because buffeting

action for each mode is assumed uncorrelated to the coupling action between modes

due to self-excited forcing. The correction for this deficiency, was to modify the final

extracted coupled derivatives with the ratio of the filter values in order to try reverting

to the initial coupling. Thus, the obtained H∗

2 and H∗

3 in the illustrated example, are

adjusted by dividing them with the ratio of FL(fθ)/FL(fz). Additional information on

the performance of identifications with coloured noise and specific attributes on the

relative sensitivity of each derivative were described by Jakobsen [169]. H∗

2 , H∗

4 and

A∗

4 were found to be affected the most from such a correction procedure. Interestingly

H∗

4 is not a coupling derivative. In practice, for the specific problem in hand the effect

of the actual wind spectra on the flutter derivatives was found to be insignificant.

The decomposition of the Hankel matrix recovers simultaneously all parameters of

the discrete time realisation. Knowing the modal stiffness and damping matrices for

the in wind and still air cases (pure structural stiffness and damping contributions)

allow one to separate the flutter derivatives. The whole method (having here the

dimensionality for the problem already decided as trivially two degrees-of-freedom)

relies on the choice of two parameters; the length of the individual time record N and

the number of time delays ℓ for which the covariance matrix is evaluated and stored

in the block Hankel matrix. The choice of both is investigated through a sensitivity

analysis together with inspection of the time evolution of the auto and cross-covariance

functions.


Sload(f)

f

f

f

|Hi(f)|2

Sresp(f)

fz fθ

FL(f)

f

Sz(f)

fz fθ f

FL(fθ)

FL(fz)

1DOF 2DOF

Figure 3.9. Decolouring process. In the 1DOF system, filter application will produce corrected

spectra, see dashed line, simulating white noise loading. For the 2DOF system, filtering with

FL will erroneously modify the true aeroelastic coupling, see dashed line versus greyed area.

Adapted from Jakobsen [169].

3.5.3 Application to the Clifton Suspension Bridge

The proximity of the fundamental vertical and torsional modes seems to encourage some

coupling action, which could potentially be the initial sign of classical flutter. The PSDs

in Fig. 3.8 indicate some non-negligible values of the H∗

2 or H∗

3 flutter derivatives, since

coupling occurs in the vertical PSD at the torsional motion frequency. The flutter

derivative identification was performed in one case with recorded acceleration data and

in another with displacements evaluated by double integration of the accelerations in

hand. Both cases produced identical results.

For the N and ℓ identification parameters, time records from 10 minutes to 1 hour

and lag ranges between 10 and 40 seconds were used, preserving analogies with sim-

ilar previous treatments. Example covariance functions, for moderately strong wind,

are plotted against time lag in Fig. 3.10. As previously demonstrated by Jakobsen

et al. [177], the suitable number of maximum time lags is strongly influenced by the

response character at different wind speeds. Higher wind speeds usually allow only a

shorter meaningful portion of the covariance function for accurately reproducing the

two-degree-of-freedom interaction, e.g. due to high aerodynamic damping of the pure

vertical response in the case of streamlined box-girders. For the example in hand an

optimum set of values, producing representative results, was found to be the combi-

nation of 15 minute records (N=11×210) with time delays up to 20 seconds (ℓ=250).

To justify the choice, such a lag value is slightly higher than the beating period of the


two frequencies in hand, while as seen in Fig. 3.10, Czθ peaks at approximately 10s.

Sensitivity of the identification for the ℓ range quoted, was only weak

0 10 20 30 40−1

−0.5

0

0.5

1

Czz

0 10 20 30 40−1

−0.5

0

0.5

1

Czθ

0 10 20 30 40−1

−0.5

0

0.5

1

Cθz

lag (s)0 10 20 30 40

−1

−0.5

0

0.5

1C

θθ

lag (s)

Figure 3.10. Example covariance functions (scaled with variance) for the combined two

degrees-of-freedom plotted against time lag.

Results for the CSB flutter derivatives are given in Fig. 3.11. No wind tunnel tests

have been undertaken on the CSB deck section, but where possible the site data are

compared with available wind tunnel results of other deck cross-sections. Sign conven-

tions for aerodynamic forces are as in Fig. 2.7, i.e. lift force and vertical displacement

pointing downwards and overturning moment with rotation positive for the windward

side of the bridge girder moving upwards. A sensitivity analysis on the measured wind

characteristics, such as the turbulence and the angle of attack, proved not to be able

to reproduce a clear picture of their effect. The identified trends of flutter derivatives

remained unaltered, but data were insufficient to quantify a systematic impact of the

investigated parameters.

Some of the derivatives in Fig. 3.11 appear to have an offset for still air wind

conditions. This has also been encountered in previous treatises both in wind tunnel

[169], and on site [155, 156]. Here it can mostly be attributed to effects such as the

distortion from traffic, influencing both the loading and the mass distribution, as well as

uncertainties in the modal masses and inaccuracies in the still air structural matrices.

The non inclusion of P ∗

i derivatives may also be of some influence. For off-diagonal

still air values there is no direct control on this offset, which can be purely a noise

artefact even in the absence of gyroscopic terms [178]. For diagonal values there was


Figure 3.11. Flutter derivatives of Clifton Suspension Bridge from full-scale data, compared

with wind tunnel extracted flutter derivatives for various cross-sections (after Scanlan and

Tomko [73] and adapted to Eqs.(3.3)). A∗

1 and A∗

3 for section G5 are negligible. H∗

4 and A∗

4

were not measured in the wind tunnel tests. Identified values correspond to binned and averaged

identified values.


an attempt to minimise the offset since for damping especially, the absolute values are

of great importance.

Although the identified flutter derivatives are noisy, unsurprisingly for full-scale am-

bient data, some trends are apparent. Consistent with the observed bridge behaviour,

the results indicate that, within the range of wind speeds recorded (maximum 15.3m/s),

the bridge is not susceptible to torsional flutter (so called ‘damping-driven flutter’ as

presented by Matsumoto et al. [179]), which was the reason for the famous Tacoma

Narrows Bridge collapse. Neither galloping1 seems an issue. This is due to having

close to negative A∗

2 and H∗

1 (direct damping derivatives), which will probably need a

further increase in reduced wind speed to initiate such alarming behaviour, if indeed it

does occur. However, H∗

1 apparently shows a steep positive gradient near the highest

wind speed recorded, suggesting it could become positive for higher wind speeds, pos-

sibly leading to instability. This trend persists regardless of the selected identification

parameters (N and ℓ), indicating it is not due to numerical errors, although the last

few points in the figure are from averages over few records, so their accuracy may be

limited. Fig. 3.12 shows the H∗

1 flutter derivative estimates from each 15-minute record,

before the averaging used for Fig. 3.11. Although only the last few points show the

apparent positive gradient, these points depart significantly from the trend at lower

reduced velocities and the differences are greater than the general scatter of points,

implying this is most probably a real effect. If this is confirmed, the effect of possible

positive H∗

1 (i.e. negative aerodynamic damping of vertical motion) could provide a

feasible explanation for the occasional observations of large vibrations of the bridge in

strong winds.

Actually such reversing H∗

1 behaviour, tending to self-excited oscillations, was first

met for the Tacoma Narrows Bridge section by Scanlan [73]. It was postulated then,

that the phenomenon is vortex shedding related. Yet oddly in the same study, an H-

section, with depth to width ratio equal to the Tacoma Narrows Bridge did not produce

any pronounced H∗

1 turn. Results from later wind tunnel tests by Neuhaus et al. [180],

plotted in Fig. 3.12, show an almost ever decreasing H∗

1 . Additionally the expected

reduced wind velocity for vortex oscillations on Tacoma should be Ur = U/fB≈0.55−1

according to [143]. This sets vortex shedding quite far apart (Fig. 3.12) to be capable

of causing lock-in and imposing a strong effect on H∗

1 . Analogously for the CSB, if the

captured phenomenon is assumed the same, the high turbulence intensities on-site make

quite improbable the explanation on a vortex-shedding basis. Thus, this monotonicity

inversion or in some cases simply multi-valuedness, seems an interesting unresolved

1Purely translational instabilities, according to the definition introduced in Chapter 2, are termedgalloping. However, the instability described here may elsewhere be found with the name SDOFtranslational flutter.


0 1 2 3 4 5 6 7−15

−10

−5

0

5

Urz(= 2π/K = UfzB

)

H∗ 1

identified

Tacoma [73]

Tacoma [180]

Tacoma nominal S−1

r

Figure 3.12. Identified H∗

1 flutter derivative from each 15-minute record. H∗

1 for the Tacoma

Narrows Bridge from Scanlan and Tomko [73] and from Neuhaus et al. [180] are given for

comparison.

issue. It is worth noting that the numerical derivation of H∗

1 for the Tacoma Narrows

section, practised by Larsen [143], did not reproduce the experimental convexity.

Dimensionally assessing the possibility of unstable motion in the pure vertical re-

sponse, a value of H∗

1≈6 needs to be reached. This estimate is based on the low

amplitude structural damping ratio of ζ=3.3% for the vertical mode [168] and on the as-

sumption that no beneficial amplitude-dependent increase in structural damping takes

place. Structural damping is believed to be so high, compared with modern suspen-

sion bridges, because of the many joints in the structure, particularly the wrought

iron suspension chains. The possibility that bias errors could have lead to erroneous

overestimation should be ruled out, since IWCM handles efficiently most artefacts.

The H∗

2 and H∗

3 derivatives, which control the coupling from torsional to vertical

motion, have small values. However at the higher wind speeds there is a noticeable

growing negative trend in H∗

3 , in line with the curves for other bridge profiles, which

potentially explains the previously illustrated coupled spectra in Fig. 3.8. The relative

influence of H∗

3 for the most coupled response record, translates to a vertical force

approximately 1/10 of the peak restoring elastic force for the mode.

The evolution of H∗

4 (aeroelastic direct vertical stiffness) reflects a reduction of

vertical natural frequency with increasing wind speed, although this could alternatively

be due to an amplitude dependence rather than the wind. Similarly A∗

3 (aeroelastic

direct torsional stiffness) illustrates a reduction in the torsional natural frequency with


increasing wind speed. The highest identified values of A∗

3 andH∗

4 translate to an actual

frequency drop of less than 7% in each case (cf. variation of up to 4% from traffic [168]).

Such values are higher than the 1%-3% found in [73], which could imply that the

recorded frequency shifts in CSB are actually due to a combinatory cause. In any

case variations are quite low. This reinforces previous observations that unlike airfoil

flutter, for bridges with bluff sections aeroelasticity influences more the damping than

the frequencies of the modes (see Scanlan and Tomko [73] and Billah and Scanlan [181]),

although in this study coupling could also become apparent.

In any case, from the sections in hand, a qualitative similarity was found with the

bluff section of Tacoma Narrows. For a proper estimation of the critical flutter wind

speed, through Complex Eigenvalue analysis (CEV) described in §2.3, data inclusive of

higher wind speeds are needed to extend the plots of Fig. 3.11. However, considering

that the Tacoma Narrows bridge failed under pure torsional flutter, due to A∗

2 (negative

torsional damping), focus is also put on this flutter derivative. Actually an estimate of

its value at higher wind speeds, can be obtained by utilising the relationships between

flutter derivatives initially proposed by Matsumoto [179] and here written as in Scanlan

et al. [182]

H∗

1 = KH∗

3 , H∗

4 = −KH∗

2 , A∗

1 = KA∗

3, A∗

4 = −KA∗

2 . (3.5)

These suggested relationships are based on the assertion that twisting θ and the

apparent angle of attack associated with the bridge girder vertical velocity generate

similar motion dependent forces. Scanlan et al. [182] present a simple elegant proof

based on the classical Theodorsen flutter treatise [63]. The relationships should be

mostly applicable for streamlined sections but were found to yield also accurate match

for many bluff cross sections too [179]. In [182] the streamlined section of the Tsurumi

Bridge was found to comply well with Eqs.(3.5), while the bluff section from Golden

Gate showed a much worse fit. Eqs.(3.5) are here employed to investigate the possible

extension of estimates of A∗

2 to higher reduced wind speeds through the measured values

of A∗

4. This can be achieved since the scaling of the reduced wind speed for A∗

4 uses the

lower vertical frequency giving higher reduced wind speeds than A∗

2, which is expressed

in relation to the torsional frequency. Consequently it is possible to better review the

possibility of single-degree-of-freedom torsional flutter on the CSB. Fig. 3.13 shows the

A∗

2 initial data with the additional points. Fitting the polynomial A∗

2 =0.12Urθ(0.3Urθ

– 1) provides an estimate for pure torsional flutter at Urθ≈6.3 i.e. at U≈21m/s. This

estimate is based on the low amplitude structural damping estimate for the torsional

mode where ζ=2.6%. No allowance for a potential beneficial increase of structural

damping with amplitude was considered. This, combined with the uncertainty in the

estimation of small aerodynamic forces represented by A∗

4 and with the uncertainty in


the relationships in Eqs.(3.5), expand the error margins of the estimation. Thus the

actual flutter wind speed could be higher, although results of this magnitude are still

significantly below today’s standards. For comparison, the modern Ting Kau Bridge

has an expected flutter velocity of more than 60m/s [78], while the latest Sutong and

Stonecutters bridges raise this value to an impressive 88m/s and 140m/s respectively

[77].

0 1 2 3 4 5 6 7−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Urθ(= 2π/K = UfθB

)

A∗ 2

A∗

2

A∗

4/(−K) (Matsumoto)

0.12Urθ(0.3Urθ − 1)

Tacoma

Figure 3.13. Flutter derivative A∗

2 with additional points from A∗

4 as suggested by Matsumoto

et al. [179]. The fitted polynomial, indicated by the broken line, was used for estimation of a

pure torsional flutter wind speed.


A similar flutter derivative identification scheme has been attempted also for the Ting

Kau Bridge, which benefits from a state of the art health monitoring system. Its flutter

derivatives Hi,Ai for i=1,2,3,4 were previously measured during scaled sectional wind

tunnel tests, thus providing the analysis a reference to compare with. Additionally,

data records are inclusive of the period that the bridge was hit by Typhoon York.

Evidently it should have been expected that the Ting Kau Bridge case study is far

more interesting and didactic than CSB. As a matter of fact it is not. The maximum

wind speed acquired at deck level on the Ting Kau Bridge was only 24.1m/s, too low in

comparison with the modelled flutter capacity of around 60m/s. Any flutter signs are

too weak and probably get masked by other influences, which contaminate the signal

parts assigned to flutter derivatives. Further all structural modes are three-dimensional

with their mixed character seriously complicating any analysis.


On the other hand the CSB is the simplification anyone would try to have for the

full-scale flutter problem. Modal complexities and unknowns are reduced, and on top

of that the operational wind velocities span an extensive regime closing to what could

be an instability threshold. It was proved that the historic bridge showed some similar

trends with the Tacoma Narrows Bridge, in terms of both the direct damping flutter

derivatives H∗

1 and A∗

2. Although it is now well accepted by the engineering society that

single-degree-of-freedom torsional flutter led the Tacoma Narrows Bridge to destruction,

there have also been some different views. Karman’s initial diagnosis was that vortex

shedding was the reason for the catastrophe, while Plaut [183] employing nonlinear

logic argued that initial small damage to a cross-tie introduced an asymmetry without

which the bridge would never go into unstable torsional motion, whatever many wind

tunnel tests unanimously support. Such ideas attempt to refute the whole bridge flutter

analysis in use today. CSB being a unique large scale example seemingly close to flutter,

can be used in exploring how the phenomenon truly unfolds (progress or die down by

leaking energy to other modes) and answer many remaining questions.

Ur

H∗

1

(

dCL

dα+ CD

)

Re2

< 0

(

dCL

dα+ CD

)

Re1

= c1 > 0

(

dCL

dα+ CD

)

Re0

= c0 > c1

Figure 3.14. H∗

1 for different H-sections [73] and a possible Reynolds number based expla-

nation for the H∗

1 inversion. Crossovers can initiate when Reynolds number changes alter the

multiplier of Ur in Eq.(3.6).

With the opportunity of the CSB identification another aerodynamic feature was

highlighted. Positive A∗

2 is the signature sign of H-sections. Yet, although often ne-

glected, H∗

1 for many H-profiles seems to change its gradient sign, also tending to a

positive unstable value, see the left of Fig. 3.14. Such behaviour was explained by

Scanlan as lock-in from vortex shedding. This would be quite unlikely for the inversion

in CSB that has not been seen to experience vortex shedding in any case. An alternate

explanation never suggested before could be founded on Reynolds number effects as


follows. H∗

1 could be quasi-steadily estimated by [184]

H∗

1 (Ur) = −(

dCL

dα+ CD

)∣

∣

∣

∣

α0

Ur

2π, (3.6)

where CD is the mean drag coefficient, α the angle of attack and α0 the equilibrium

angle of attack to which all flutter derivatives refer to. This linear estimate was found

to be very close to the true H∗

1 value for a number of bridge sections [169,184]. Suppose

that the critical Reynolds number Re1 is reached. Then CD will reduce and Eq.(3.6)

will result a less steep approximation. This is shown in Fig. 3.14. Potentially at Re1 a

crossover could occur that replicates the H-section behaviour. This could even lead to

a sign change for H∗

1 , when the Reynolds number change induces the classical galloping

condition, cf. Eq.(2.9). A CD drop with concomitant Strouhal number increase was

previously recorded in wind tunnel tests of the Storebælt bridge section by Schewe [185].

His attempt was to reason the 20% higher Strouhal number that was experienced for

the bridge on-site.

In general, this work attempted to contribute to the literature on the analysis of

aeroelastic effects from ambient vibration data on full-scale bridges, being one of very

few similar studies. On the specific issue of CSB’s flutter potential, the identified A∗

2

was extrapolated at higher reduced velocities by use of the A∗

4 results, and an apparent

trend was recovered. On this basis, a wind speed estimation for pure torsional flutter

has been made. Although uncertain, it is believed this is the first time this has been

achieved based solely on the actual full-scale dynamic performance of a bridge.

Concluding this chapter a future perspective can be given. It was early seen that

small shape modifications could enforce dramatic changes over flutter derivatives, at

least on scale models. Scanlan and Tomko [73] presented results from tests on two truss-

stiffened bluff sections for which A∗

2 on the addition of railroad tracks, or a short middle

traffic barrier became positive in where they were negative. This was considered a wind

tunnel artefact. For the CSB application recorded data after restoration work has taken

place on the bridge, indicate that there have been changes in flutter derivatives. Note

that none of the works induced any explicit shape alteration to the deck. The process of

understanding if such changes are truly of aerodynamic source is still ongoing. In any

case this could hold the key for increasing the safety limit of a true bridge monument.

Chapter 4

Quasi-steady galloping analysis revisited

The flutter derivative based definition of aeroelastic loads, shown in the previous chap-

ter, entails a series of laborious dynamic tests. Fortunately such descriptions become

redundant in the case of galloping vibrations, where simpler static tests of the bluff

body in hand together with quasi-steady theory can accurately capture the ensuing

dynamic behaviour. Specifically the condition Ur ≫ Sr−1 is considered sufficient for

enabling this simplification. Yet classical galloping analysis deals mostly with cases of

across-wind vibrations, leaving aside the more general situation where the wind and

motion may not be normal. This can arise in many circumstances, such as the motion

of a power transmission cable about its equilibrium configuration, which is swayed from

the vertical plane due to the mean wind, or for a tall slender structure in a skewed wind.

Furthermore the generalisation to such situations, when this had been made, has not

always been performed correctly.

This chapter aims at elucidating such shortcomings, thus naturally it is separated

from the introductory Chapter 2. Initially the correct equations for the quasi-steady

aerodynamic damping coefficients for the rotated system or wind are re-derived, and

differences from other variants are highlighted. Motion in two orthogonal structural

planes is considered, potentially giving coupled translational galloping, for which pre-

vious analysis has often been limited or has even arrived at erroneous conclusions. For

the two-degree-of-freedom case, the behaviour is dependent on the structural as well

as aerodynamic parameters, in particular the relative natural frequencies in the two

planes. Differences in the aerodynamic damping and zones of galloping instability are

quantified, between solutions from the correct perfectly tuned, well detuned and clas-

sical Den Hartog equations (and also an incorrect generalisation of it), for a variety of

typical cross-sectional shapes. The presentation intends to introduce concepts essential

for the subsequent parts of this thesis.

67

68 Chapter 4. Quasi-steady galloping analysis revisited

4.1 Introduction

Quasi-steady theory allows aerodynamic problems to be simplified vastly by replacing

the actual unsteady condition in hand to an equivalent static one, where only the

relative flow velocity is considered for capturing the relevant aerodynamic forcing. Its

most famous application is probably the galloping criterion put forward by Den Hartog

[186, 187] setting the condition for dynamically unstable behaviour of a single degree-

of-freedom (DOF) oscillator as:

F ′ = sinα(

−L+D′)

+ cosα(

L′ +D)

< 0 (4.1)

which is often (e.g. see Holmes [188, p117], Hemon and Santi [189, p856]) expressed in

terms of static force coefficients as:

Ssc = sinα(

−CL + C ′

D

)

+ cosα(

C ′

L + CD

)

< 0 (4.2)

where α is the angle of attack and L, D, CL, CD are the static lift and drag forces

and static lift and drag coefficients respectively, assumed for nominally 2D flow to be

functions only of α, and the prime indicates the derivative with respect to α. The

criterion is only a statement to avoid an undamped oscillator becoming negatively

damped due to aerodynamic action. Thus the whole problem reduces to determin-

ing the aerodynamic damping contribution and setting it equal and opposite to the

available structural damping. Eq.(4.2) presented as such is strictly not valid since its

trigonometric terms only apply for α = 0, corresponding to 1D across-wind oscillations,

which is tacitly ignored. In the general case (i.e. α 6= 0), the principal structural axes

may not be aligned along the flow direction and normal to it, for example for a vertical

section in skewed wind (or for a horizontal section in inclined wind), or considering the

static sway of a catenary due to the mean wind force. Then Eq.(4.2) fails to accurately

account for the effect of aerodynamic damping, as in the across-wind galloping scenario,

and even if employed in its correct Den Hartog stated form then it fails to describe the

true condition. The correct treatise, although partly presented in the literature already

(Richardson and Martuccelli [190], Blevins [1]) is generally not followed in practice

and the resulting errors in calculations rising from the mishandling have not previously

been quantified. To this end a number of benchmark cross-sections are considered to

illustrate the differences emerging in defining instability bounds.

In addition, the full extension of a generalised translational galloping criterion of

sections with principal axes arbitrarily inclined to the flow, has to cover motion in

both axes, including their coupling, which is especially important when they have close

natural frequencies. Such an analysis has previously been performed by a number of

4.1. Introduction 69

authors with specific scopes and sometimes with erroneous conclusions. In particular,

Jones [191] addressed coupled motion in two planes in some detail, but only for the

special case of α = 0 with identical natural frequencies in the vertical and horizon-

tal directions, and she concludes that no vertical galloping can occur when horizontal

motion is restrained. Although suggesting that this may be a reason for experimental

behaviour observed, it results only from mistreating boundary conditions and neglecting

the effect of the envisaged restraining force in the analysis. Liang et al. [192] and Li et

al. [193] used the formulation in terms of body co-ordinates, following Davenport [194],

and covered the seemingly more general case of α 6= 0 2D perfectly tuned coupled

motion. The fact that the frequencies are restrained to be tuned renders α arbitrary,

thus the results should only be equivalent to Jones’ [191] case and not a generalisa-

tion of it. By using force coefficients defined in body co-ordinates rather than wind

co-ordinates, the connections of their work to the Den Hartog criterion are unclear. In

the analysis a special behavioural subcase is missed and the inaccurate quote is put

forward that 2D coupled galloping oscillations may occur only when the fundamental

natural frequencies of a structure in the two orthogonal principal axial directions equal

each other. Macdonald and Larose [134, 135], focusing on the dry inclined galloping

of circular cables, accurately provided the full 2D aerodynamic damping contribution,

including also terms due to Reynolds number and 3D geometric effects. Also, both

resonant and non-resonant conditions between vibrations in the two planes were taken

into account. A similar treatise, though waiving (and questioning) the use of Reynolds

number dependent terms, was presented by Carassale et al. [137], who utilised Kro-

necker products and matrix calculus to derive a full aerodynamic damping matrix. In

both these research works the interest focused on circular cylinders and the objective

was beyond deriving a simple Den Hartog analogue for motion in two orthogonal planes

and testing it against the 1D requirement as is pursued here. Especially in Carassale

et al. [137], due to concentrating on circular sections, derivatives with respect to α

were not included in the analysis, rendering the formulation inapplicable to other clas-

sical galloping cases. It is worth referring also to Luongo and Piccardo [195], who

use bifurcation analysis to capture the limit cycle behaviour of detuned configurations,

again limited to Jones’ [191] schematic case. These references so far, alongside broader

3DOF treatises1, with different perspectives and not focusing on subtle translational

interaction details (e.g. [197, 198]), roughly cover the full range of available literature

on modelling two-degree-of-freedom (2DOF) galloping vibrations.

Other previous analysis of explicit 2DOF coupled quasi-steady instabilities has pri-

marily been concerned with the combination of translational and rotational motions

e.g. [1,199–201]. Still, there is an inherent incompatibility of the quasi-steady formula-

1Flutter galloping according to the terminology of Chabart and Lilien [196].


tion with rotations. There would always be some arbitrariness on selecting the point to

which a unique effective wind vector refers to. On the other hand, in many cases there

are similar structural conditions for translational motions in the two orthogonal planes

normal to the cylinder axis, rotational motion may not occur simultaneously, and there

is no complication in extending the classical 1DOF galloping treatment. These render

natural the analysis of 2DOF galloping. The work should be very relevant to the later

studied bridge cable motions. In what follows the correctly modified version of the

Den Hartog criterion for an arbitrary angle of attack for 1D motion and a solution

for the more generic motion in two orthogonal planes are re-derived. For the analysis

the full aerodynamic damping matrix is formed and the scenario of coupled galloping

oscillations is considered, which is a function of the structural parameters as well as

the aerodynamic ones and can lead to elliptical trajectories.

4.2 Quasi-steady derivations

The novel contribution of this chapter is not to propose new galloping criteria but, em-

ploying the current state-of-the-art [134, 135, 190], to quantify the difference between

the generalised 2DOF galloping scenario shown in Fig.4.1(a) and the normally con-

sidered special case in Fig.4.1(b) for pure across-wind motion. For completeness, the

quasi-steady aerodynamic damping derivations, yielding galloping criteria, are briefly

repeated hereafter with an added intention of highlighting the errors in Eq.(4.2). For

succinctness the variation of the force coefficients with Reynolds number is neglected,

but its incorporation is straightforward.

Figure 4.1. Geometry of a bluff section indicating lift and drag forces (L, D), relative angle

of attack (α) and principal structural axes (x, y). (a) The general case with α0 6= 0 and the

2DOF motion potential. (b) The special case for 1DOF across-wind oscillations.

The classical Den Hartog derivation starts typically by writing the mean aerody-

namic force, per unit length, along the y-axis in Fig.4.1, as a function of the mean drag

4.2. Quasi-steady derivations 71

and lift forces:

Fy = L cosα+D sinα . (4.3)

where L = 12ρU

2relBCL, D = 1

2ρU2relBCD, ρ is the fluid density, B is a reference di-

mension of the section and Urel is the relative velocity. For motion limited to the y

direction, expanding Fy around y=0 the standard damped equation of motion becomes

my + cy +mω2yy = Fy = Fy|y=0 + y ·dFy

dy

∣

∣

∣

∣

y=0

, (4.4)

where m is the cylinder mass per unit length, ωy is the circular natural frequency

(in the absence of wind), c is the structural damping coefficient, and dots represent

differentiation with respect to time. Noting that exclusively when the free-stream wind

velocity U and the motion velocity y are orthogonal and thus α = arctan(−y/U)

(Fig.4.1(b)),dFy

dy

∣

∣

∣

∣

y=0

= − 1

UF ′

y

∣

∣

α=0. (4.5)

From Eq.(4.3), the derivative of Fy with respect to α, which can be used for Taylor

expanding around any initial inclination α0, is

F ′

y = L′ cosα− L sinα+D′ sinα+D cosα . (4.6)

In Eq.(4.4), the aeroelastic force (the last term on the right hand side), is equivalent

to a linear viscous damping force. The condition for dynamic instability is simply that

the total effective damping coefficient is negative. Hence from Eqs.(4.5&4.6), noting

that for α = 0, U ′

rel = 0, it is easily shown that the galloping criterion is

−dFy

dy

∣

∣

∣

∣

y=0

=ρUB

2

(

C ′

L + CD

)

< −c , (4.7)

where the threshold U is proportional to c as noted in §2.1.4. For zero structural

damping, this reduces to the classical Den Hartog criterion presented earlier in Eq.2.9

SDH =(

C ′

L + CD

)

< 0 . (4.8)

which agrees with Eq.(4.2) for α = 0.

If the motion is not normal to the wind direction, Eq.(4.7) does not hold and there

are two problems with Eqs.(4.1&4.2). Firstly α 6= arctan(−y/U) so Eq.(4.5) is not

valid, which affects both Eqs.(4.1&4.2). Secondly in finding L′ and D′, U ′

rel 6= 0, so

extra terms are introduced. For the general case and for extending the analysis to

cover two orthogonal translational DOFs (see Fig.4.1(a)), which potentially can lead


to coupled response, the derivation follows.

Eq.(4.3) still holds and also

Fx = −L sinα+D cosα. (4.9)

Expanding Fy and Fx around zero motion, to find equivalent viscous damping terms,

similar to before,

Fy = Fy|x=y=0 + x ·dFy

dx

∣

∣

∣

∣

x=y=0

+ y ·dFy

dy

∣

∣

∣

∣

x=y=0

,

Fx = Fx|x=y=0 + x ·dFx

dx

∣

∣

∣

∣

x=y=0

+ y ·dFx

dy

∣

∣

∣

∣

x=y=0

. (4.10)

For evaluating the derivatives the chain rule is employed

d()

dx=

∂()

∂Urel· dUrel

dx+

∂()

∂α· dαdx

, (4.11)

and similarly for y.

Keeping in mind the relations

Urel =√

(Uy − y)2 + (Ux − x)2 , tanα =Uy − y

Ux − x, tanα0 =

Uy

Ux, (4.12)

finally the unit length full 2×2 aerodynamic damping matrix of a bluff section is ob-

tained:

Caero =

cxxa cxya

cyxa cyya

=

−dFx

dx−dFx

dy

−dFy

dx−dFy

dy

x=y=0

=ρBU

2

Sxx Sxy

Syx Syy

,

(4.13)

with

Sxx = CD(1 + cos2 α0)− (CL + C ′

D) sinα0 cosα0 + C ′

L sin2 α0 , (4.14a)

Sxy = −CL(1 + sin2 α0) + (CD − C ′

L) sinα0 cosα0 + C ′

D cos2 α0 , (4.14b)

Syx = CL(1 + cos2 α0) + (CD − C ′

L) sinα0 cosα0 − C ′

D sin2 α0 , (4.14c)

Syy = CD(1 + sin2 α0) + (CL + C ′

D) sinα0 cosα0 + C ′

L cos2 α0 . (4.14d)

The derivation is also valid for wind skewed to the cylinder axis, by employing the

wind component normal to the cylinder and the force coefficients with respect to that

component, as long as the independence principle is a viable approximation.

4.2. Quasi-steady derivations 73

The 1DOF instability thresholds for galloping in the x or y planes are simply when

the diagonal terms of Caero become negative (or more generally equal to minus the

structural damping coefficient). Evidently the non-dimensional aerodynamic damping

coefficients, Sxx and Syy in Eqs.(4.14a&d), differ from Ssc in Eq.(4.2), confirming that

it is incorrect. For α0 = 0, Syy in Eq.(4.14d) reduces to SDH in Eq.(4.8) (as does Sxx

in Eq.(4.14a) for α0 = ±90◦), as expected.

For the instability condition of the coupled response, an eigenvalue analysis has to

be performed for the 2DOF system, which is a function of the structural, as well as

the aerodynamic, parameters. In general a numerical solution is required, but for the

special case of equal mass (m), structural damping coefficient (c) and natural frequency

(ωx = ωy = ωn) for both DOFs, a closed form result is derived as below. The equations

of motion

mx+ (c+ cxxa)x+mω2xx = −cxyay ,

my + (c+ cyya)y +mω2yy = −cyxax , (4.15)

are assumed to possess a solution of the form

x = X exp(λt) , y = Y exp(λt) , (4.16)

where the eigenvalues λ and the amplitudes X, Y are in general complex valued.

Eqs.(4.15&4.16) yield

Y

X= −λ2 + cxx

m λ+ ω2x

λcxyam

= − λcyxam

λ2 +cyym λ+ ω2

y

, (4.17)

λ4 +

(

cxx + cyym

)

λ3 +

(

cxxcyy − cxyacyxam2

+ ω2x + ω2

y

)

λ2

+

(

cxxω2y + cyyω

2x

m

)

λ+ ω2xω

2y = 0 , (4.18)

where cxx = c + cxxa and cyy = c + cyya. For ωx = ωy = ωn, solving the biquadratic

Eq.(4.18) for the stability boundary (i.e. purely imaginary eigenvalues, λ=iω) results

in

ω = ωn , together with cxxcyy − cxyacyxa = 0 , (4.19)

or cxx + cyy = 0 (with ω not restricted to equal ωn) . (4.20)


Eq.(4.19) translates, by analogy with Eqs.(4.2, 4.8&4.14a&d), to the criterion for cou-

pled galloping (for no structural damping):

S2D =1

2

[

3CD + C ′

L ±√

(

CD − C ′

L

)2+ 8CL

(

C ′

D − CL

)

]

< 0 , (4.21)

where S2D denotes the non-dimensional effective aerodynamic damping coefficient of

the coupled motion (equivalent to Sxx and Syy for uncoupled motion) and the nega-

tive square root obviously gives the critical case. Here Y/X is real, indicating planar

trajectories. As expected Eq.(4.21) does not explicitly include α0.

The solution in Eq.(4.20) corresponds to the so-called complex response [134, 191],

which arises when the term under the square root in Eq.(4.21) is negative. Then the

criterion for coupled galloping becomes

S2D =1

2

(

3CD + C ′

L

)

< 0, (4.22)

which coincides with the real part in Eq.(4.21), but in addition, the frequencies of

the resulting two in-wind modes are released from being equal. This solution is often

missed (as in [192,193]) by constraining X and Y to be real. From Eq.(4.17), for λ = iω

with ω 6= ωn, Y/X is complex, indicating elliptical trajectories. Since the two modal

responses occur simultaneously at different frequencies, a 2D beating-type behaviour

occurs, as described in [134,191,195].

More generally, in the presence of structural damping (the same in both planes),

the right hand side of Eqs.(4.21&4.22) becomes −2c/ρBU (equivalent to Eq.(4.7)).

Also of interest is the case where the initial natural frequencies in the two DOFs

(ωx and ωy) are not equal. Then for the stability boundary, similar to Eqs.(4.17&4.18),

is given by [135]:

(cxxcyy − cxyacyxa)(cxx + cyy)(κ2cxx + cyy) + cxxcyy(1− κ2)2m2ω2

x = 0, (4.23)

where κ = ωy/ωx. This can generally only be solved numerically. For all detuned cases,

the trajectories become elliptical, similarly to the complex response for the perfectly

tuned system and to actual occurrences of galloping in the field.

4.2.1 Relevance to uniform continuous systems

It is worth noting that all the derived aerodynamic damping estimates (and hence the

galloping criteria), although referring explicitly to a unit length section, are often also

4.3. Application: quantifying differences 75

applicable for a uniform continuous system allowing motion in two orthogonal planes,

in a uniform flow. This can be easily proved by applying in Eqs.(4.15) the standard

separation of time and space variables,

x(s, t) =∑

n

φxn(s)qxn(t) , y(s, t) =∑

n

φyn(s)qyn(t), (4.24)

where s is the distance along the continuous system, φxn(s) and φyn(s) are the nth

undamped mode shapes in the x, y planes and qxn(t), qyn(t) the corresponding gener-

alised displacements. Employing apart from the standard orthogonality conditions for

same plane modes

∫

φxn1(s)φxn2

(s)ds = 0 for n1 6= n2 ,

∫

φyn1(s)φyn2

(s)ds = 0 for n1 6= n2 , (4.25)

the condition that the mode shapes in the two planes are the same (i.e. φxn(s) =

φyn(s) = φn(s)), for any pair of modes in the two planes, Eqs.(4.15) transforms to

∫

mφ(s)φ(s)qx(t)ds+

∫

cxxφ(s)φ(s)qx(t)ds+ ω2x

∫

mφ(s)φ(s)qx(t)ds

= −∫

cxyaφ(s)φ(s)qy(t)ds ,∫

mφ(s)φ(s)qy(t)ds+

∫

cyyφ(s)φ(s)qy(t)ds+ ω2y

∫

mφ(s)φ(s)qy(t)ds

= −∫

cyxaφ(s)φ(s)qx(t)ds . (4.26)

Since for a uniform section in a uniform wind, the generalised coordinates, mass per unit

length and damping coefficients are not functions of s, the integral term∫

φ(s)φ(s)ds

cancels out, yielding back again Eqs.(4.15) and thus rendering the deduced instability

thresholds in Eqs.(4.14a&d, 4.21&4.22) still valid. It is noted that when the aero-

dynamic damping coefficients cannot be deemed to be constants over the structural

length, as for instance for high rise bridge towers where the wind velocity profile is

significant, or for a varying section, then the integrals in Eq.(4.26) should be calculated

explicitly. However, in many cases the simplifying approach of uniform wind velocity

and section is adequate, and in the present situation it allows comparison between the

relatively simple different criteria presented above.

4.3 Application: quantifying differences

The differences in the results arising from the different galloping criteria are quantified

by utilising data of aerodynamic force coefficients for a variety of cross-sectional shapes.


Fortunately such data are available in the literature for many shapes (e.g. see [190,191,

202–207]), although they have almost exclusively been used in the Den Hartog criterion

only, which, as previously stressed, is not always the case in hand. For the current study

a number of representative shapes, as illustrated in Fig.4.2, were chosen to span a whole

range of possible relative differences between the different galloping criteria. The last

three iced cable shapes (Figs.4.2(j,k,&l)), although only specific examples of the infinite

number of possible iced geometries, were chosen for direct comparison with the work

of Jones [191], since, although she attempted to define the worst case for 1DOF or

perfectly tuned coupled galloping, she chose a presentation method that did not make

the actual differences in the results clear.

[190] [204] [204] [203] [205] [206]

[203] [190] [207] [191] [191] [191]

Figure 4.2. Sections with aerodynamic coefficients provided in the literature [190, 191, 203–

207], used in the galloping analysis herein.

Fig.4.3, presents the non-dimensional aerodynamic damping coefficients for each

galloping criterion, for each section, for the whole angle range of angles of attack that are

available. Negative values identify aerodynamically unstable regions, where galloping

would occur in the absence of structural damping. More generally, with structural

damping, galloping occurs when the non-dimensional aerodynamic damping coefficient

is below −2c/ρBU . For each shape, three lines are plotted, corresponding to the

aerodynamic damping contributions from: i) the classical Den Hartog summation, SDH,

in Eq.(4.8); ii) the more adverse of the two rotated 1DOF cases, Sxx or Syy, as given

in Eqs.(4.14a&d); and iii) the perfectly tuned 2DOF coupled motion case, S2D, in

Eqs.(4.21&4.22). It is pointed out that these correspond to three conceptually different

motion scenarios: i) applies to different aerodynamic angles of attack of the cross-

section, α, but with the motion always constrained to be purely across-wind; ii) applies

to the instance where the principal structural axes and cross-sectional shape are fixed to

each other and rotate together with respect to the wind (or the wind rotates relative to

the structural axes and shape) as in Fig.4.1(a), with α0 becoming the variable; and iii)

applies to combined 2D motion with perfect tuning of the structural natural frequencies

in the two planes, in which case the orientation of the structural axes is arbitrary,


reflected by Eqs.(4.21&4.22) being independent of α0, and only the orientation of the

cross-sectional shape with respect to the wind direction is then relevant. The sub-case

of the 2DOF complex response is distinguished in Fig.4.3 by plotting open circles. It

is notable that there is no case of instability linked to this scenario (i.e. S2D from

Eq.(4.22), when it applies, is never negative). This is in keeping with the suggestion by

Macdonald et al. [132], for galloping of a skewed stranded cable in the critical Reynolds

number range, that the combination of parameters required for galloping of a complex

response is unlikely to occur in practice. In addition, comparing Eqs.(4.8&4.22), since

CD is always positive, the condition for 2DOF complex galloping is less onerous than the

condition for pure across-wind galloping. Hence, in contrast to Jones’ [191] suggestion

that observations of elliptical galloping trajectories in the field can be attributed to

complex galloping, here it seems most probable that this is not the actual case.

The results presented here are for the cases of the wind and motion direction fixed at

right angles to each other or for the principal structural axes and cross-sectional shape

fixed to each other (or for perfect tuning in 2DOF). The full generalisation allows the

wind direction, principal structural axes and the orientation of the cross-sectional shape

to all be independent. This could occur, for example, for a transmission line, where

the wind is close to horizontal, the structural axes are given by the inclination of the

cable catenary in the mean wind, and the cross-sectional shape can rotate due to the

influence of gravity on any accreted ice. Such a situation is still covered by Eqs.(4.13-

4.15), where in Eq.(4.14) α0 is the angle between the wind direction and the structural

x-axis (Fig.4.1(a)), but CD, CL and their derivatives should be evaluated at the angle

of attack between the wind direction and the reference direction of the cross-sectional

shape (not necessarily equal to α0 due to the cross-section rotating).

Commenting on the individual plots in Fig.4.3, the first impression is that in most

cases all the values follow roughly similar trends and predict close instability zones with

respect to the angle of attack. Especially for sections being or resembling rectangles, in-

cluding the square in Fig.4.3(d), the rectangle with side ratio 3:1 in Fig.4.3(g), and the

rectangle with rounded ends in Fig.4.3(h), the instability zones from the three different

criteria are almost indistinguishable, showing some insensitivity of the susceptibility to

galloping for the different cases. The square and rectangle were actually chosen for this

study for exhibiting different characteristics, with the square galloping for zero angle

of attack and the 3:1 rectangular not (for a classical treatise on the instabilities of

rectangular sections with different side ratios see Nakamura and Hirata [57]), although

the most severe zone of instability is for an angle of attack near 10◦ in both cases. This

is the case for the section in Fig.4.3(h) also. A similar connection exists between the

rectangle in Fig.4.3(g) and the ellipsoid in Fig.4.3(c) with a strong instability near 70◦

for both, showing that sections very close to circular can still exhibit negative aerody-


Angle of attack α[◦]

SDH,m

in(S

xx,S

yy),S2D

(a)

0 20 40 60 80 100-3

-2

-1

0

1

2

3

4

Angle of attack α[◦]SDH,m

in(S

xx,S

yy),S2D

(b)

0 20 40 60 8010 30 50 70 90-3

-2

-1

0

1

2

3

4

5


SDH,m

in(S

xx,S

yy),S2D

(c)

0 20 40 60 8010 30 50 70 90-20

-15

-10

-5

0

5


SDH,m

in(S

xx,S

yy),S2D

(d)

0 20 40 60 80 9010 30 50 70-8

-6

-4

-2

0

2

4

6

8

10


SDH,m

in(S

xx,S

yy),S2D

(e)

0 10 20 30 40 50 60-4

-2

0

2

4

6

8

10


SDH,m

in(S

xx,S

yy),S2D

(f)

0 20 40 60 80 100 120 140 160 180-2

-1

0

1

2

3

4

5

6

Figure 4.3. Non-dimensional aerodynamic damping coefficients (SDH, min(Sxx, Syy), S2D)

as a function of angle of attack for the cross-sectional shapes given in Fig.4.2 (inset letters

link the two figures). Negative values indicate unstable behaviour (in the absence of structural

damping).



SDH,m

in(S

xx,S

yy),S2D

(g)

0 20 40 60 8010 30 50 70 90-5

0

5

10

15

20

25

30

35


SDH,m

in(S

xx,S

yy),S2D

(h)

0 20 40 60 80 100 120 140 160 180-25

-20

-15

-10

-5

0

5

10

15


SDH,m

in(S

xx,S

yy),S2D

(i)

0 20 40 60 80 100 120 140 160 180-4

-2

0

2

4

6

8


SDH,m

in(S

xx,S

yy),S2D

(j)

100 120 140 160 180 200 220 240 260-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3


SDH,m

in(S

xx,S

yy),S2D

(k)

0 20 40 60 80 100 120 140 160 180-8

-6

-4

-2

0

2

4

6

8


SDH,m

in(S

xx,S

yy),S2D

(l)

0 50 100 150 200 250 300 350-0.5

0

0.5

1

1.5

2

2.5

Figure 4.3 (continued)


namic damping values and consequently unstable behaviour, in this case because, after

separation at the sharp corner, the flow then does not reattach, causing a rapid drop

in lift. In any case, for all the figures referenced so far, the differences are sufficiently

small to consider that any of the above instability bounds works quite well in any ac-

tual case. Indeed the previous lack of a study to quantify the differences arising from

the use of different criteria can probably be linked to the fact that benchmark studies

of galloping analysis have often been performed on rectangles or rectangle-like shapes,

where the differences are unimportant.

On the other hand, another section, equally widely studied in wind tunnel tests and

historically connected with galloping, the D-section in Fig.4.3(a), shows more diverse

behaviour. For zero angle of attack, the Den Hartog summation (SDH) predicts zero

aerodynamic damping (the D-section is a hard oscillator [208] that, for motion exceeding

a certain amplitude, will gallop even for this angle of attack, but this is beyond the

scope of the current presentation). Evidently the lesser of Sxx and Syy (in this case Syy)

from Eqs.(4.14a&d) (referred to as the 1D rotated case hereafter) becomes the same as

SDH when α0 = 0, and slightly more unexpectedly the 2DOF solution (S2D) also falls

on the same value giving a common start for all three. As the angle of attack increases,

SDH departs from the other two, which have a negative peak near 40◦, representing

almost twice the negative aerodynamic damping as for SDH. This is a significant

difference and it clearly demonstrates that the appropriate galloping criterion should

always be applied carefully to the actual the problem in hand. Increasing the angle

of attack further, a smaller instability zone is expected near 100◦ where now the most

severe condition is for 2D motion and the Den Hartog summation estimates slightly

more negative aerodynamic damping than for the 1D rotated case. Near 80◦ it is seen

that SDH gives extremely positive aerodynamic damping. Looking more broadly it is

seen that actually for nearly all sections the Den Hartog summation gives the highest

positive aerodynamic damping value. Such extreme values are often very close to the

ones coming from the alternate 1D rotated case (the greater of Sxx or Syy), which

is not shown in Fig.4.3 that presents only the worse case. This also explains why in

Figs.4.3(f,g&h) the Den Hartog summation does not match the 1D rotated case for

zero angle of attack - the aerodynamic damping of along-wind vibrations is lower than

for across-wind, although both are positive.

Other sections considered also show notable differences between the outputs for

the different cases. The results for the triangle with a vertex angle 30◦ as shown in

Fig.4.3(f) show that near 20◦ the 1D rotated case is close to being stable while the

other two cases are clearly unstable, and around 30◦ the 2D case is unstable whereas

the other two are not. The same section near 180◦ (presenting a flat face to wind)

on the other hand shows all the three lines in Fig.4.3(f) coinciding. Similarly the



Ssc,S

yy

Ssc, Eq.(4.2)Syy, 1D rotated, Eq.(4.14d)

(a)

0 20 40 60 80 9010 30 50 70-8

-6

-4

-2

0

2

4

6

8

10


Ssc,S

yy

Ssc, Eq.(4.2)Syy, 1D rotated, Eq.(4.14d)

(b)

0 20 40 60 80 100 120 140 160 180-4

-2

0

2

4

6

Figure 4.4. Comparison between the erroneous Ssc (Eq.(4.2)) and the correct value for the

1D rotated y-axis case, Syy (Eq.(4.14d)), for (a) the square in Fig.4.2(d) and (b) the triangle

in Fig.4.2(f).

equilateral triangle (Fig.4.3(e)) exhibits significant differences, although the 1D and

2D rotated cases generally give close results. In addition it is of interest to note that

the two triangles behave quite differently (Fig.4.3(e&f)) when presenting their flat faces

to the wind although only a small vertex angle change has occured. Differences are also

apparent in Figs.4.3(b,j&k), with the 1D rotated case giving the least unstable results,

thus rendering the simple Den Hartog calculation to be unnecessarily conservative if

the structural axes rotate with the section. Conversely for the L-section in Fig.4.3(i),

in the most critical region near 60◦, the Den Hartog summation is unconservative.

Drawing some general conclusions, although in most cases the broad picture from

the three criteria is similar, the absolute aerodynamic damping values at certain angles

can be quite different. There are many instances where a section stable according to

one criterion can be unstable according to another, and there is no set sign for the

relative differences, with changes being possible even for the same shape in a different

range of angle of attack. Still, in almost all the examples interestingly the worst case

occurs for the 2DOF criterion.

It should be noted that the accuracy of the results is of course limited by the

accuracy of the available data. But additionally there is the need to convert the discrete

point measurements of static force coefficients into a continuous or piecewise continuous

function in order to determine their derivatives. Many options were pursued towards

this goal, including polynomial and harmonic curve fits of different orders. In any case

there is a need for a very high order for any function to accurately fit the measured

points, which was recognised early by Blevins [199]. In Blevins’ analysis the nonlinear


terms enter the equation of motion and subsequently solutions are sought to yield the

steady state amplitudes, thus it is detrimental that the introduction of different non-

linearity, from different fitted functions, strongly influences the results. However for the

purposes of the present analysis, different choices make little change to the results of the

comparison (except making the plots smoother). For this reason the simplest possible

piecewise linear assumption was picked for estimation of the derivatives in Fig.4.3.

One of the main initiatives of this section was to correct Eq.(4.2), but it is equally

important to show the error arising from its use. At first sight it is clear that for 180◦

rotation it is the opposite of the Den Hartog criterion, while for 90◦ any correlation with

the Den Hartog criterion should be deemed as fortuitous. Judging from the general

similarity that was earlier found between the Den Hartog case and the 1D rotated case,

it is expected that great differences from Eq.(4.2) can emerge. Actually it should be

noted that the correct direct equivalent of Eq.(4.2) is not the worse of Eqs.(4.14a&d),

as considered earlier, but only Eq.(4.14d), for motion in the y direction. As can be seen

in Fig.4.4(a), for plotted results for the square section, although Eq.(4.2) inaccurately

estimates a weak instability zone near 80◦ it otherwise predicts instabilities in agreement

with the correct result. This is only because a square’s critical zone occurs for small

angles of attack, where Eq.(4.2) should be close to the Den Hartog criterion. On the

other hand, for the triangular section in Fig.4.2(f) the errors are alarming (Fig.4.4(b)).

Apart from the initial coalescence of the two curves, Eq.(4.2) is consistently unable to

capture not only the value of the aerodynamic damping but even its correct sign. This

is true for most of the section shapes considered (Fig.4.2).

4.4 The detuning effect

The 2DOF solution presented earlier is restricted to perfectly tuned natural frequencies

in the two motion planes. It is evident that in a great number of situations different

natural frequencies exist for the different directions of motion. Such a case is found for

instance on cables, where, due to sag, the frequencies of odd in-plane modes are higher

than for the corresponding out-of-plane ones. For any detuning the coupled Eqs.(4.15)

still apply, but for increasing detuning the coupling terms on the right hand side move

further from the relevant natural frequency and hence have a reduced effect on the

behaviour. Eventually, for greatly detuned systems the coupling becomes irrelevant

and the system behaves like two uncoupled 1DOF systems in the orthogonal planes.

This poses the interesting questions of: i) what happens for close but not equal natural

frequencies and ii) what will the behaviour be for quite large detuning values.

Utilising Eq.(4.23) for all the different sections in hand it is found, as intuitively

expected, that the two 1DOF and the tuned 2DOF solutions define an envelope within

4.4. The detuning effect 83

Frequency ratio, κ = ωy

ωx

Sdetuned,S

xx,S

yy,S

2D

Sdetuned, 2D detuned solution, Eq.(4.23)

Sxx, Syy, 1D solutions, Eqs.(4.14a&d)

S2D, 2D tuned solutions, Eq.(4.21) (a)

1D y

1D x

0.8 0.9 1 1.1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Frequency ratio, κ = ωy

ωx

Sdetuned,S

xx,S

yy,S

2D

Sdetuned, 2D detuned solution, Eq.(4.23)

Sxx, Syy, 1D solutions, Eqs.(4.14a&d)

S2D, 2D tuned solutions, Eq.(4.21)

1D y

1D x

(b)

0.8 0.9 1 1.1-1

-0.5

0

0.5

1

1.5

2

Figure 4.5. Evolution of the aerodynamic damping solution for different values of detuning, κ,

for (a) the section in Fig.4.2(j) and α = 123◦ and (b) the section in Fig.4.2(k) and α = 30◦. The

lower branch is the important one. In (a) for perfect tuning the solution is unstable (negative

aerodynamic damping) and for detuning above about 7% it approaches the 1D solution, which

in this case is stable. In (b), on the other hand, for perfect tuning the solution is stable and for

detuning above 1% it becomes unstable while moving towards the 1D solution.

which the detuned coupled solutions always fall. The actual evolution with detuning

consists of starting from the tuned 2DOF solutions (using ± in Eq.(4.21)) and progres-

sively converging towards the 1DOF ones, as presented, for example, in Fig.4.5(a) for

the iced cable section of Fig.4.2(j) for an angle of attack of 123◦ (other parameters (m,

ωx, ρ, B, U) were taken from Jones [191]). The alternate path is given in Fig.4.5(b) for

the iced cable section of Fig.4.2(k) for an angle of attack of 30◦, where the 1D solution

is more onerous than the 2D tuned one. The actual rate of convergence is found to be

strongly dependent on the force coefficients and hence the angle of attack. The smaller

the initial distance between the coupled and uncoupled solutions in Fig.4.3, the slower

the rate of convergence seems to become. As clearly shown in Fig.4.5(a), for detuning

as low as 7% the 2D solution, which is unstable when tuned, becomes stable and ap-

proaches the 1D solution (see the lower critical branch in the figure and also Fig.4.3(j)

at 123◦).

Incidentally, this particular example gives the opportunity to correct an erroneous

conclusion reached by Jones [191]. Results from tests on this section by Nigol and

Buchan [209], where no galloping occurred, were taken as support for an assertion that

in general when along-wind motion is restrained then no instability can ever occur.

However, the reasoning was only a result of forgetting the associated external restraining

force in the balance of forces in the equation of motion. As seen in Fig.4.3(j) at 123◦,

picked as being the most critical orientation, the tuned 2D solutions and Den Hartog


criteria produce indistinguishable values. Thus in this case, with allowance for across-

wind motion, presence or absence of along-wind motion, whatever the frequency ratio

between the two, hardly changes the galloping threshold. The true reason for lack of

observed galloping in this case, which is theoretically only slightly unstable, is likely

related to the level of structural damping and/or slight inaccuracies in quasi-steady

theory, as discussed by Bearman et al. [202] or Luo et al. [210]. Bearman et al. ,

while studying the square sections’ aerodynamic behaviour, noted interaction effects

between galloping and vortex shedding that could delay galloping. Quite unexpectedly

the structural damping values during limit cycle oscillations were at least 13% higher

than the wind-off measured values. This was not a mere amplitude effect, and showed

strong interplay with turbulence conditions. Further the transition to high Ur, where

quasi-steady theory assumes wind forcing in quadrature with motion was not recorded.

Even for the highest attained Ur the phase advance of the aerodynamic force was

approximatelly 80◦ ahead of the motion. Luo et al. had a similar finding with unstable

galloping sections only slowly reaching the theoretical phase advance of 90◦.

Closing this short parenthesis there should be a note on the response trajectories,

which as already mentioned can range from planar to elliptical. As discussed above

and shown in Fig.4.5, the detuned solutions quickly approach the aerodynamic damping

values that correspond to the 1D solutions. However, they still remain qualitatively

different from them in terms of the trajectories, which are described by Eq.(4.17).

Such differences are presented for the case of Fig.4.5(a) in Fig.4.6. When perfectly

tuned (Fig.4.6(a)) two planar modes occur (not necessarily in orthogonal planes). For

small detuning values, the planar motion of the modes turns into ellipses with growing

magnitudes of their minor axes as the detuning increases. This is shown for κ = 1.005

in Fig.4.6(b). For larger detuning the axes of the elliptical modes also rotate as in

Fig.4.6(c) where κ = 1.05. This rotation continues until the principal structural axes

are reached, and when that is virtually accomplished (for detuning of the order of 10%

as in Fig.4.6(d)) the width of the trajectories reduces as they converge on the uncoupled

planar solutions. The detuning values required to produce essentially planar responses

are in fact much larger (of the order of 200% in this case). The relevant aerodynamic

damping estimates are also shown in Fig.4.6 to indicate instabilities and to establish

the link to Fig.4.5(a). It is a noteworthy conclusion that the elliptical galloping paths

observed in the field (see discussion in [191]), are almost certainly due to a detuning

effect between the structural axes.

4.4. The detuning effect 85

x

y

S2D = −1.24

S2D = 1.76

(a)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

x

y

Sdetuned = −1.22

Sdetuned = 1.74

(b)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

x

y

(c)

Sdetuned = −0.13

Sdetuned = 0.65

-1 -0.5 0 0.5 1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

x

y

(d)

Sdetuned = 0.48

Sdetuned = 0.04

-1 -0.5 0 0.5 1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Figure 4.6. Modal trajectories corresponding to Fig.4.5(a) for (a) κ = 1, (b) κ = 1.005, (c)

κ = 1.05 and (d) κ = 1.1. The applicable Sdetuned value is also indicated. Unstable modes

are plotted with solid lines while stable ones are dotted. Note for comparison that Sxx = 0.45,

Syy = 0.06. For all plots the structural damping value was c = 0.



Quasi-steady aerodynamic theory, which lies at the core of the current analysis, has

known limitations and it has long been recognised that in certain operational regimes

(only approximately identifiable) it breaks down. Still, for a broad range of conditions

for low frequency modes, quasi-steady analysis has a proven ability to predict aerody-

namic damping and galloping behaviour. The current chapter identifies cases where

limitations are introduced only because of shortcomings in the actual application of the

method.

In particular, the case sketched in Fig.4.1(a), with the structural axes inclined to

the wind direction, has hardly been properly considered before and the differences in

the behaviour from classical across-wind galloping had not been quantified. The Den

Hartog criterion is correct for pure across-wind vibrations, but not otherwise. Its ap-

plication, or even worse a faulted extension of it presented in Eq.(4.2), for the rotated

system or wind, can yield solutions that can range from close to even the opposite of the

correct ones. Although the Den Hartog summation often gives reasonable estimates of

the aerodynamic damping, it can in some circumstances give negative estimates of only

around half the magnitude of the real values, which is potentially unsafe. Conversely

in some other cases it can be unnecessarily conservative. Furthermore, the dynamic

stability of a section can be determined not only by its shape, the aerodynamic ori-

entation and the orientation of the principal structural axes (which may or may not

follow the aerodynamic orientation), but also by the proximity of the structural natural

frequencies in the two planes.

The correct equations for the non-dimensional aerodynamic damping coefficients,

and hence the instability criteria, have been devised for arbitrary relative orientations

of the system with respect to wind, and for perfectly tuned and well detuned natu-

ral frequencies. Such a presentation is essential in realising the way galloping theory

embodies geometric arrangement details and detuning. In particular detuning is a

parameter almost entirely neglected in the existing literature. Here, detuned results,

numerically obtained, always fall between the solutions for the 2D perfectly tuned case

and the more adverse of the two uncoupled 1D cases, so the more critical of the 1D

or 2D cases can be used conservatively. The equations provided are almost as easy to

apply as the classical Den Hartog equation, yet they avoid potential errors and give

accurate estimations of the aerodynamic damping and the propensity of a cylinder

to gallop. But it is important to use the particular equation relevant to the specific

problem being addressed.

A main application of the above design tool in bridges are the bridge cables. Being

of relatively small sectional dimension, having low motion frequencies due to their long


lengths, and moving both in and out of the bridge-cable plane they seem an excellent

practice field for the above analysis. An interesting illustration of a case where the

above description becomes relevant is the lightning protection system of a cable-stayed

bridge. This typically consists of ordinary stranded cables as in transmission lines. The

Rion Antirion Bridge, which suffered a cable stay failure due to a lightning strike, had

several issues with its lightning protection cables [211]. A simple galloping analysis as

above would be able to efficiently mitigate the problem. On the other hand, in view

of Eqs(4.14a,d&4.21) a perfectly circular cable with CL=C ′

L=C ′

D=0, CD>0 should not

gallop. The reasons why bridge stays do seem to gallop in reality are subsequently

pursued.

Chapter 5

Experiments on galloping vibration of a

circular cylinder

This chapter addresses large galloping-like vibrations of bridge cables, generically in-

clined and yawed to the flow. To this goal, wind tunnel experiments were performed for

various geometric arrangements of a rigid circular cylinder covered with a smooth high

density polyethylene (HDPE) duct as in real bridge stays. Both static and dynamic

configurations of the cylinder model were tested, while Reynolds numbers spanned the

estimated critical range 105–4.5×105. In what follows, initially the experimental appa-

ratus is presented and subsequently the flow features and excessive motions observed

are discussed. For motion frequencies far from Karman vortex resonance, unsteady

pressure measurements are utilised in order to uncover the aeroelastic forcing func-

tion. It is shown that a fundamental difference between the inclined and non-inclined

cylinder aerodynamics exists producing different pressure distributions and resulting in

alternate dynamic behaviour. Reynolds number-induced transitional behaviour seems

to be crucial in the recorded instability phenomenon in a way that was never suggested

before in the literature. Intermittent ‘jumps’ in overall sectional forces, wake discon-

tinuities between cell structures along the cylinder span-wise direction and axial flow

are conjectured to be key elements towards realising the complex mechanism of dry

galloping.

5.1 Introduction

As it was shown in §2.2, there are many conflicting theories on the explanation of dry

galloping. Most of them are qualitative and schematically demonstrate complex flow

structure interaction mechanisms that could instigate response. However, they are un-

able to seize the magnitude of the involved aerodynamic actions that is essential in

89

90 Chapter 5. Experiments on galloping vibration of a circular cylinder

designing mitigation measures. One exception to this rule, that allows for quantifying

damping thresholds, analogously to Chapter 4, is the Reynolds number based gener-

alised quasi-steady model initially proposed by Macdonald and Larose [133]. Circular

cylinders on different inclination-yaw configurations, could have a varying Reynolds

number dependence of their mean static force coefficients. This renders the existence

of dedicated sectional asymmetry terms, a sufficient but not necessary condition for

galloping.

The first wind tunnel tests, specifically designed to cover the range where Reynolds

number effects on a dry circular cylinder’s mean forcing become dominant (i.e. critical

Reynolds numbers), took place in 2001. Previous tests on moving cylinders in high

Reynolds numbers are extremely limited, mostly refer to marine applications and focus

on ordinary self-limited Karman vortex shedding motions e.g. [110, 212]. Most impor-

tantly they do not assess the influence of a cylinder’s inclination. The series of the 2001

tests, hereafter referred as Phase 1, was conducted on a 6.687m long inclined rigid ca-

ble model, spring mounted (in two perpendicular directions) in the National Research

Council Canada (NRC) open circuit propulsion wind tunnel. The tests produced data of

significant vibrations, including a record that was quoted being of diverging character,

see Cheng et al. [213,214]. This case, which appeared to be dry galloping, proved to be

non-reproducible in a single repeated run. The main reason for the non-repeatability,

was suspected to lie on ambient weather conditions change. The only explicit infor-

mation reporting on this is that during the unstable test it was raining outside while

the unsuccessful run arose on a dry sunny day. Yet it could have also been a system-

atic behaviour of a complex coupled system that even under identical conditions can

produce alternate results. The Phase 1 tests applied a range of parametric scenarios,

varying the cable inclination angle, critical structural damping ratio, support spring

orientation and surface roughness, with wind speeds covering the full nominal critical

Reynolds number range. Unfortunately there is no direct evidence of the actual flow

regimes encountered. Various attempts have been made to explain the significant vi-

brations observed. To complement these unique Phase 1 dynamic tests, a new test

series of static tests, referred to as Phase 2, was undertaken in the NRC 2m×3m closed

circuit wind tunnel. The cylinder this time was much smaller and testing parameters

such as end conditions, flow smoothness, aspect ratio and Mach numbers also differed.

The aim was to gain further insight into the cable forcing by testing a similarly inclined

static cylinder equipped with pressure measurement taps [215,216].

The mean static force coefficients subsequently deduced were used by Macdonald

and Larose [133–135] in their earlier referenced galloping framework to identify insta-

bility regions. The largest response incident from Phase 1 was adequately predicted but

obviously not also its non-repeatability. Other Phase 1 large response cases, of smaller

5.2. Wind tunnel tests 91

amplitude though, could not be traced by the galloping analysis and were thus waived

the galloping characterisation. Another very similar variant of quasi-steady analysis

was independently put forward by Carassale et al. [137]. In contrary to Macdonald and

Larose they consider the Reynolds number to have a ‘slow’ effect on the aerodynamic

forcing. This translates in disregarding all derivatives with respect to Re when form-

ing aerodynamic damping expressions (a process as in Eqs.(4.10&4.11)). The method

was still found capable of reproducing the observed dry galloping incident. Cheng et

al. [217] attempted to formulate a simplified Den Hartog-like criterion, different from

the above two, for studying the instability attributes. As a matter of fact the plausible

expression acquired is only product of mishandling derivatives’ estimates [217, p2272

Eqs.(A.11&A.12)]. When the correct derivation is performed results revert to the orig-

inal Macdonald and Larose, Carassale relations. In a distinct approach, Jakobsen et

al. [218] utilised the unsteady force components contained in the static measurements

to show that they could also cause large responses. And this is without employing any

force lock-in or amplitude dependent action. The ultimate decisive test for all these

treatises would be a comparison with the real dynamic forcing, which has never been

measured before.

In this chapter a newer series of wind tunnel tests, Phase 3, carried out under a

collaborative project between NRC, Canada, the University of Stavanger, Norway and

the University of Bristol, UK is described. It used the same large-scale aeroelastic model

as in Phase 1, but additionally instrumented with pressure taps and covering a different

range of parameters, strategically selected for large response to emerge. The main

objectives pursued herein are to identify the local characteristics of the aerodynamic

forcing causing the vibrations, and to highlight the true effects of Reynolds number.

5.2 Wind tunnel tests

5.2.1 Preliminary

Imagine the case of a real inclined bridge cable lying on a vertical plane subject to

horizontal arbitrary yawed wind as in Fig. 5.1. The cable orientation against the wind

can be uniquely described by the set of cable inclination angle ϑ and yaw angle β. When

modelling cable wind induced vibrations three directions are of major importance: the

mean wind direction, the cable orientation and the motion direction. Preserving the

way the wind sees the cable should leave unaltered the relative angles between them. In

wind tunnels, space restrictions often put limitations in the range of angle sets ϑ, β that

can be reproduced. This is also the current case. Since for the aerodynamic problem in

hand gravity is not relevant, as in rain-wind induced vibrations for instance, a rotational


Figure 5.1. Transformation from real cable to wind tunnel model.


transformation that will best fit the testing facility is performed, while keeping relative

angles constant. Thus ϑ and β convert to ϕ, which is the angle between the wind and

cable axis vectors, and α, which is the angle between the out-of plane motion direction

~x1 and the wind-cable plane normal ~x2. It is straightforward to prove that the mapping

operation yields the relations

cosφ = sinβ cos θ ,

tanα = cotβ/ sin θ . (5.1)

In the setup of both Phase 1&3 series, ϕ was implemented as the vertical angle inclina-

tion i.e. its horizontal projection is parallel to the wind. This corresponds to rotating

the real cable in Fig. 5.1 around the AB axis until the cable-wind plane coincides with

the vertical plane. Evidently α will be realised as the angle between the spring support

axes and the across-wind direction. Every geometric configuration ever tested with its

analogy to prototype set of angles and its naming convention as introduced in [146] is

given in Table 5.1.

Table 5.1. Orientation angles for studied cases.

Phase1

Setupϑ β ϕ α

Phase3

Setupϑ β ϕ α

(◦) (◦) (◦) (◦) (◦) (◦) (◦) (◦)

1B 45 90 45 0 2A 60 90 60 0

1C 30 54.7 45 54.7 2C 45 45 60 54.7

2A 60 90 60 0 4A 90 90 90 0

2C 45 45 60 54.7 4B 90 60 90 30

3A 35 90 35 0 5B 75.1 61 77 30

3B 20 60.6 35 58.7 5A 77 90 77 0

The (NRC) open circuit propulsion wind tunnel is an open loop wind tunnel blowing

air from the outside environment. Air is sucked via a conical intake from an electrically

powered fan with diameter 7.9m. The screens available for the inlet entry were not in

place during the time of Phase 3 tests. Before entering into the settling chamber the

flow passes additional screens, various stators and straightening vanes to counteract the

fan-induced swirl. Following, the air goes through a 1:6 contraction prior to entering

the test section. The test section is 12.2m long, 6.1m high and 3.1m wide. The model

was positioned within the second third of the tunnel after the end of the contraction.

A view of the facility and its inner section with the model in place is given in Fig. 5.2.

Right after completion of Phase 3 a calibration operation was performed by NRC staff,

acquiring local flow characteristics at 105 stations across the tested section area. Results

from data acquired, are illustrated in Fig.5.3. Turbulence intensities, referring to the


Figure 5.2. View of the NRC wind tunnel facility and its test section with the model in place.

total wind vector, were of the order of 0.5%. As it can be seen a vertical asymmetry

exists in the tunnel. This was quoted to be mainly due to the removal of the lower

fillets.

5.2.2 Setup details

In an attempt to reduce any potential scaling artefacts, the cable model had typical

properties of a full-scale bridge cable. The actual cylinder was a rigid steel pipe with

diameter d=140mm and thickness 16mm. It was covered with a smooth close fit HDPE

pipe of the same type that is used for covering cable strands on site. The duct had a

nominal outside diameter of 159mm. The final model was measured to have d=161.7mm

(though quoted as 160mm in all later discussion) and a total mass of 60.3kg/m. For

simply supported ends, these numbers give a natural bending frequency estimate of 7Hz.

The aspect ratio (l/d) was approaching 40, which according to West and Apelt [219]

is sufficient to minimise contamination of pressure data due to end effects. Still the

tested models of West and Apelt were horizontal and fitted with end plates, which is

not the case here. The wind tunnel blockage ratio is calculated at 5.3%. In line with

Blackburn [220], this value is sufficiently low not to cause added artificial span-wise

coherence. The wind speed (U) ranged for most tests between 10-40m/s, corresponding

roughly to Reynolds numbers of 105 to 4.5×105, fully covering the expected critical

range. The maximum Mach number tested was less than 0.12.


Figure 5.3. Turbulence intensity (Urms/U) and mean wind velocity variation with height

across the wind tunnel section.

The cable assembly was mounted on spring supports at both top and bottom,

allowing motion in the ‘sway’ and ‘heave’ directions normal to the cable axis as shown

in Fig. 5.4. A cable support was additionally provided at the top to carry the axial

component of the cable weight. The natural frequency for translational motions (f) was

set at approximately 1.4Hz. An end to end rotation also existed with a frequency around

2.21Hz. For practical reasons, the lower (down-wind) end of the cable support was

positioned inside the wind tunnel, whereas the upper one was outside the section (see

Fig. 5.4). The cable passed through a rectangular opening in the roof (approximately

0.43m along-wind × 0.42m across-wind) to allow motion. The orthogonal sets of springs

at both ends were connected to the cable body with spacer plates that indented to

inhibit any rotational motion. Structural damping ratios were attempted to be very low

to allow for aeroelastic effects to be distinguished. Values ranged between ζ=0.06-0.33%

for low amplitude response. It should be noted that the quoted numbers significantly

increased during high levels of response. For identical natural frequencies in the two

vibration planes, the orientation of the principal axes is arbitrary. However, real cables

have slight detuning of their natural frequencies in the vertical and horizontal directions

due to the cable sag. This was modelled with slightly different spring stiffness in the

two perpendicular directions. With detuning, the orientation of the principal axes

becomes significant. Former experience from Phase 1, indicated the importance of the


detuning parameter. The range of detuning between the natural frequencies in the two

planes was varied up to approximately 3%. To cover a variety of possible geometric

arrangements, the cable was tested at three different inclination angles attempting to

expand the previous Phase 1 range. As presented in Table 5.1, values of ϕ were 60◦, 77◦

and 90◦, with 60◦ being tested again to cover the previously quoted divergent response

record [214]. The support spring rotation angles were chosen as 0◦ and 54.7◦ for ϕ=

60◦, and 0◦ and 30◦ for the other inclinations. The rationale behind angle selections

is mostly founded on analysis from previous Phase 2 results [135, 137]. Further the

vertical cylinder scenario is a benchmark case that can allow for comparisons with

existing literature. It was identified that the principal axes in cases with α=54.7◦ did

not coincide with the rotated spring axes, indicating the existence of structural coupling.

For each of the inclination angles, a set of tests was also taken with the model fixed

in position, to measure the pressures on the stationary cylinder and compare with

the corresponding dynamic cases. The static tests for the inclinations of 60◦ and 90◦

were conducted with the hole around the top end of the cable both open and sealed

in order to assess the influence of the top end condition. Judging from the attained

mean pressure profiles, very small differences between the two alternatives established

that the top end effects were not significant, at least locally in the vicinity where the

pressures were measured.

The pressure measurement instrumentation was arranged in four rings of 32 pressure

taps each, more densely spaced on the leeward side, as illustrated in Fig. 5.4. Additional

lines of pressure taps were positioned close to the expected separation and back pressure

points at 100◦ and 150◦ from stagnation (measurements for when α=0) at one diameter

span-wise intervals. Taps were connected to four electronic pressure scanners embedded

in the model via 1mm diameter urethane tubing of varying length. All data had to be

corrected for the tubing frequency response. In short, the frequency response function

of each tap was measured prior to the experiments. Typical examples of them shown in

Fig. 5.5, look very similar to previous results obtained by Irwin et al. [221]. Using these

transfer functions the recorded measurements were corrected for magnitude and phase

distortions in the frequency domain. Details about the pressure measuring technique

on a similar application are given by Gatto et al. [222].

Both the displacements and accelerations of each end of the model were measured

in the two spring directions (termed heave and sway throughout, defined in Fig. 5.4).

Data sampling was performed at two different frequencies, 500Hz and 1250Hz, to assess

the significance of very high frequency components. It was found that the higher

sampling frequency was redundant. Being interested in fine details of the flow, the

turbulence intensity was measured with two dynamic three-component cobra probe

instruments, upstream and downstream of the cable model. Most of the upstream


cobra measurements were later found to be corrupted by noise and were discarded. An

additional effect that could influence the studied flow transitions was ‘dirt’ accumulation

on the cable surface particularly on the windward face. This was due to the open

return design of the wind tunnel, driving in air from outside. The size of these random

roughness anomalies (up to 1mm high) could be large enough (according to Achenbach

and Heinecke [223] and Shih et al. [224]) to alter the flow transition behaviour. To

minimise this effect the cable was regularly cleaned. The pattern established was that

an unclean cable would not present large motion. All the tests described below are

believed to not have been exposed to the influence of this uncontrolled parameter. On

a clean cable, surface roughness corresponded to a roughness-to-diameter ratio (ε/d)

of 6×10−6.

Figure 5.4. Elevation of cable model showing instrumentation arrangement. Some distances

are not in scale. For accurate positioning details consult Table 5.2.


Table 5.2. Position details for the model. For rings and lowest cable end ‘distance from floor’

refers to stagnation points, while for cobra probes ‘distance from model’ refers to along-wind

distance.

ϕ (◦) Distance from floor (m) Distance from model (m)

target/actual Ring1 Ring2 Ring3 Ring4 Low end Cobra1 Cobra2 Cobra1 Cobra2

60 / 59.4 3.87 3.594 3.043 2.63 0.859 4.34 3.19 3.018 0.133

77 / 76.7 4.231 3.919 3.296 2.829 0.827 4.34 3.24 2.381 0.305

90 / 90.9 4.152 3.832 3.192 2.712 0.655 4.34 3.19 1.702 0.304

Figure 5.5. Typical frequency response curves for three pressure taps.

5.3. Results 99

5.3 Results

5.3.1 Overview and large responses

Large responses of primarily across-wind character were observed, only for the incli-

nation of ϕ=60◦, for both cases of spring rotations examined, α= 0◦ and 54.7◦, and

for various combinations of spring tunings in the two vibration planes. All events refer

to the lowest translational modes with end to end motion always yielding much less

significant amplitudes. Results are consistent with previous findings by Matsumoto et

al. [120], who, for a horizontal cylinder restricted to planar across-wind motion, identi-

fied the range of yaw angles for which large cable vibrations occurred to be β ∈[22.5◦,45◦]. Note that cable orientations relatively close to parallel to the wind direction were

not considered and that the test wind speeds were intended to be in the subcritical

Reynolds number range. In the current study, the large responses for a cable-wind

angle of ϕ=60◦ fell in the unstable region, as previously identified in Phase 1, while

inclinations of ϕ=77◦, 90◦ did not produce any similarly large responses.

All large vibrations in the Phase 1&3 tests only occurred within a limited range of

wind speeds. This feature is reminiscent of typical Karman vortex shedding, although

the frequency content was very far from Karman vortex resonance (cf. Karman vortex

shedding corresponds to a reduced velocity Ur=U/fd ≈5 or lower, while in the tests

exhibiting large vibrations it was over 100). The feature also distinguishes the observed

response from classical Den Hartog galloping, which occurs for all wind speeds exceeding

a certain threshold (see Fig. 2.5). However such behaviour does not necessarily mean

that the term galloping is inappropriate. Galloping response of such a transient type,

though of different attributes, has previously been presented in Fig.2.6. Conjecturing

that instability may be triggered by some sustained boundary layer asymmetry in the

critical Reynolds number range, the excitation mechanism may be similar to normal

galloping but the restoration of symmetry in the supercritical Reynolds number range

for increased wind speeds may bring it to an end. This is exactly the underlying

mechanism implication of the quasi-steady analysis proposed by Macdonald and Larose

[133–135]. Their predicted unstable vibrations are limited to the critical Reynolds

number range, as observed, in contrast to classical galloping and to other proposed

mechanisms that are not limited to a specific wind speed range. However it should be

pointed out that the vibrations expected from the quasi-steady analysis for the setup

2A (see Table 5.1) were more along-wind, rather than across-wind as observed in Phase

3. The quasi-steady implementation of Carassale et al. [137] cannot predict any large

amplitude response for this case.

The maximum amplitude of vibrations observed was around 0.75d, which reached

the maximum available clearance of the hole through the wind tunnel roof and corre-


sponded to significantly increased structural damping values. The aerodynamic limit

state, if one exists, would occur at higher amplitude. In every case the trajectories of

the cable for large vibration events were elliptical, generally with the major axis at an

angle to the spring directions, suggesting coupling action induced by aeroelastic forces.

The motion trajectories evolution with detuning shown in Fig. 5.6 is qualitatively in

excellent agreement with theoretical quasi-steady analysis of coupled translational gal-

loping oscillations as presented earlier in Chapter 4. Relatively large detuning values

(in this case of only 2%) produce an ellipse aligned with one of the uncoupled degrees

of freedom (the more excited one) as in Fig. 5.6a. Closer tuning of the system (in this

case roughly 1% detuning) rotates the ellipse as in Fig. 5.6b&c, and perfect tuning

would lead to planar motion in the direction of the divergent coupled mode. The width

of the ellipse is controlled by the coupling action induced by the wind. Since in all

large amplitude cases the motion was predominantly across-wind, and in some cases

almost exclusively so (e.g. Fig. 5.6a), it seems that the dominant aerodynamic forcing

is across-wind and it is likely that the along-wind component arises as a secondary ef-

fect from coupling, giving a significant response in that direction for very close tuning.

5.3.2 Pressure data

When correlating large responses to Reynolds number it is observed that major events

were grouped in two distinct regions at approximately Re=2.5×105 and 3.5×105, falling

inside the boundaries of the drag crisis. Proper Orthogonal Decomposition (POD) was

performed on the pressure tap data and interestingly it was found that some consis-

tency of loading exists in the aforementioned regions. As seen in Fig. 5.7, which presents

the cumulative variance explained by a relatively small number of Proper Orthogonal

Modes (POMs), two peaks emerge, approximately coinciding with the Reynolds num-

bers for large responses. The number of POMs selected was such that the cumulative

variance could reach a value of around 90%, which is routinely selected when POD is

employed for filtering purposes. The trends presented remained even with alternate

choices of POM numbers. Very noticeably, the behaviour is similar for both the static

and dynamic tests indicating organisation of the loading even for the static cylinder,

although the actual POMs themselves were found to differ in the two cases. An actual

set of relevant modeshapes in both the static and dynamic tests is illustrated in Fig. 5.8.

The POMs selected are the most energetic ones and correspond to the points indicated

by arrows in Fig. 5.7. Interestingly modeshape coordinates for Ring 3 are quite similar

in both dynamic and static cases. The greatest variance contributions are located at

what seems to be the separation region on the left side. Actually for the dynamic tests

5.3. Results 101

−0.25 0 0.25

−1

0

1

Acr

oss

win

ddispla

cem

ent

(/d)

−0.25 0 0.25

−1

0

1

−0.25 0 0.25

−1

0

1

heave

sway

a

heave

sway

b

heave

sway

c

Along wind displacement (/d)

U=30m/sRe=3.1×105

U=32m/sRe=3.4×105

U=35m/sRe=3.7×105

Figure 5.6. Motion traces for cases a) ϕ=60◦, α=0◦, with frequency ratio in heave/sway

fh/fs=0.979 and structural damping ratio in heave/sway ζh/ζs=1.91; b) ϕ=60◦, α=0◦,

fh/fs=1.007, ζh/ζs=1.18 and c) ϕ=60◦, α=54.7◦, fh/fs=1.014, ζh/ζs = 0.94. The heave and

sway spring axes are also presented. ‘Along-wind’ really means normal to the cable in the

cable-wind plane rather than along the wind itself.


Figure 5.7. Proportion of total variance from 20 POMs (from all pressure tap data) against

Reynolds number. Model setup: ϕ=60◦ and, for dynamics tests, α=0◦.

all the Rings peak at around this location. On the right side there is a much broader

contribution, which seems to attain its maximum near the across-wind pressure tap.

The POD analysis indicated the existence of both localised and widespread energetic

modes to explain data variances. There were cases where time coefficients of certain

POMs from the static tests were primarily harmonic with a frequency close (but not

equal) to the structural frequency, raising the question as to whether this component

has the ability to lock in with motion to cause large response in the later dynamic tests.

The subcritical Reynolds number behaviour is dominated by periodic vortex shedding

and accordingly the possibility for fewer periodic modes to relatively accurately describe

the pressure tap measurements. Increasing Reynolds number into the critical range it

seems to destroy coherent structures and cause an increase in the dimensionality of the

underlying dynamics. Still, there are two breaks in the expected monotonic decrease in

the variance, which provide the system with energetic mechanisms that most probably

accommodate the large responses. The differences presented in Fig. 5.7 may seem small

but they are consistent and moreover seem to be amplified under the influence of large

scale motion, thus giving an indication of a possible lock-in action.

Power Spectral Densities (PSDs) of the fluctuating lift coefficient (CL), averaged

over the four pressure rings, for dynamic tests in different wind speeds are presented

in Fig. 5.9. The PSDs are normalised by multiplying by U/d, the inverse of the nor-

malisation of the frequency axis, to preserve the variance magnitude. The plot shows

the evolution of the wind forcing in four characteristic behavioural cases for different

5.3. Results 103

Figure 5.8. First Proper Orthogonal modeshapes for a set of dynamic and static tests. Tests

correspond to the cases indicated by arrows in Fig. 5.7. Interestingly the greatest variance

contributions originate from the near separation regions.

Figure 5.9. Spectra of the lift coefficient, averaged over all four pressure rings. Star-marked

points, 32m/s case, show the resonant peaks that correspond to the structural frequency of

1.4Hz and twice this value. Model setup 2A.


Reynolds numbers. For the subcritical range, represented by Re=1.2×105 (U=11m/s),

clear vortex shedding can be identified at a Strouhal (Sr) number of 0.17 corresponding

to 12Hz (cf. the natural frequency of 1.4Hz). When applying the independence prin-

cipal using the component of wind normal to the cable (Usinϕ) the estimate matches

the expected value for a static cylinder normal to the wind, Sr=0.2. Also the cable

inclination broadens the frequency range of the forcing, relative to the normal wind

case, which looks very similar to the effect of added turbulence [225]. Increasing the

Reynolds number, e.g. Re=2.5×105 (U=23m/s), leads to the vortices becoming inco-

herent, thus reducing the spectrum in the Strouhal reduced frequency region. At the

same time some very low frequency components emerge fd/U <0.05) and the spec-

trum becomes flat for a range of reduced frequencies from 0.1 to 0.2. This regime was

reached for much higher Reynolds numbers for the cable normal to the flow. For the

record for which there was a large cable response, at Re=3.4×105 (U=32m/s), there

are large sharp peaks in the spectrum at reduced frequencies corresponding to the mo-

tion frequency and twice this value, indicated by stars in Fig. 5.9. It seems that the

energy in the broad low frequency band locks in to the structural frequency and large

motion builds up, while some broader band low frequency forcing, probably related to

weak vortex shapes, still survives for fd/U <0.1. In the supercritical Reynolds number

regime, e.g. Re=4.3×105 (U= 40m/s), the low frequency components have vanished

and only low-level broadband excitation remains for fd/U <0.3 not having consistent

uniform periodic components. On the contrary, as expected according to Roshko’s

(normal cylinder) tests [108], the static equivalents did show up a narrow band process

near fd/U=0.2. It is important to note that the lift coefficients above were calculated

as the average from all four rings, whist there were marked differences between the

rings, particularly for the ϕ=60◦ inclination case. It is believed that this should not be

a consequence of end-effects. As a matter of fact there was some consistency between

rings for the two different sets of end conditions, which possibly implies that this be-

haviour is mainly sourced by the way the 3D flow pattern establishes itself regardless

of boundaries.

To investigate the relationship between the aerodynamic forces at the different rings,

lift and drag cross-correlation functions were estimated. Fig. 5.10 gives an illustration

of such functions showing the systematic force delays along the cable, indicated by the

asymmetry between the top right and bottom left of the figure. Such delays denote

some propagation in the axial direction only along with the flow. When the time lags

of the peak absolute values of the cross-correlation functions are transformed into a

propagation velocity, they compare quite well with the axial component of the free-

stream wind, although they are not constant along the length between different rings

but show a variation. This result is in excellent agreement with the numerical findings

5.3. Results 105

Figure 5.10. Correlation functions (R) of lift coefficients (CLiCLj) between rings (i, j) for the

subcritical dynamic test case of U=13m/s, Re=1.4×105. Propagation is evidently one-sided.

Model setup, ϕ=60◦ and α=0◦.

of Yeo and Jones [117]. Indications of axial propagation were clear in both the static

and dynamic tests for the cable inclined at ϕ=60◦ and, as possibly expected, ceased (or

almost ceased) for higher inclination values. Fig. 5.10 intentionally displays a subcritical

case with apparent vortex shedding, and was picked in order to match the Reynolds

number of the numerical simulations by Yeo and Jones [116, 117]. To continue on the

inhomogeneity point raised earlier, it is indicative from the auto-correlation functions,

along the diagonal of Fig. 5.10, that the state of shedding for each ring is very different.

For Rings 1-3 vortex shedding is highly damped in contrast to the strong periodic

phenomenon that is well known to cause large responses of cylinders normal to the

flow. The periodicity translates typically to Sr=0.17 which is around 20% lower than

the value recovered by Yeo and Jones. At Ring 4, most interestingly, vortex shedding

is almost entirely masked (but not vanished) by a low-frequency broadband process.

The case is reminiscent of so-called swirling structures and their associated pressure

distributions [117]. In any case, signature indications of some axial correlation persisted

throughout the wind speed range along the whole cable. Another possible effect of this

apparent secondary flow can be conjectured in view of static pressure (Cp) profiles such

as the ones given in Fig. 5.11. Ring 3, presents near 120◦ what looks as a sustained

attached flow or an axial-vortex type disturbance. Its pressure contribution is reaching

only up to the base pressure point (180◦). There the behaviour changes abruptly, which


appears to be consistent with the suggestion by Matsumoto et al. [120] that axial flow

on the lee side of the cable acts as a splitter plate. Still it should be highlighted that

only Ring 3 receives locally a splitting action if truly this is the case.

5.4 Discussion

5.4.1 Symmetry considerations

The force measuring technique of pressure taps employed (integrating pressures over

tributary areas) has the feature of allowing the assessment of the contribution of par-

ticular segments of the cross section to the total drag 1 and lift forces. Dealing with

an instability that is fundamentally fed by asymmetries, it would be useful to identify

where and how these arise. As indicated in §2.2, which meticulously described the

evolution of the flow characteristics for smooth flow past a normal circular cylinder

over the whole Reynolds number range, asymmetries can occur during the initiation of

critical transition. There the drag suddenly drops and considerable mean lift appears,

since half the section contains a laminar separation bubble while the other half does

not. The modified treatise, also allowing for the effects of turbulence and roughness

on flow ranges, is much more complex. For the current case, to assess the transitional

flow symmetry, the section illustrated in Fig.5.4 (for α=0◦) is split into a ‘right’ part

containing taps 4-3-2. . . -20 (with only half the 4 and 20 contributions) and a ‘left’ part

containing taps 4-5-6. . . -20 (again with half the 4 and 20 contributions).

There is some conceptual difference in the behaviour of individual rings while vary-

ing the inclination angle, ϕ. As presented in Fig.5.12, for the inclination of 60◦, the

drag crisis zone during static tests has very distinct features for different rings. For

Ring 2 the half-perimeter drag contributions (CD1/2) from the ‘right’ and ‘left’ parts

almost coincide while for Ring 3 there is consistent spacing during the drag drops.

These are signs of two bubbles forming simultaneously or sequentially for increasing

Reynolds number. Thus for a given Reynolds number inside the drag crisis region, two

states exist together along the cylinder length (Rings 2 and 3 are at a spacing of 4

diameters). But according to previous studies on smooth cylinders in smooth flow, the

simultaneous formation of two bubbles, as for Ring 2, does not occur in the critical

Reynolds number range but only in the supercritical or transcritical range.

A similar situation to the above observation was described by Zdravkovich [109],

who found that roughness of the order of only ε/d ∼=0.003, or alternatively turbulence,

not only shifts the drag crisis to lower Reynolds numbers but actually obliterates the

1Drag is here taken to be normal to the cable in the cable-wind plane

5.4. Discussion 107

Figure 5.11. Mean pressure coefficient distribution around cylinder for large response case of

U=32m/s, Re=3.4×105 (Fig.5.6b). Model setup ϕ=60◦, α=0◦.

critical state and causes a transition directly from the subcritical to the supercritical

state. More strikingly, in the presented results the two conditions are found to co-

exist stably (i.e. pressure profiles locally did not change state) along the cylinder

length. Neighbouring sections, with one and two bubbles respectively, will shed wake

vortices at different frequencies, thus becoming the source of vortex dislocations similar

to the ones identified by Bearman and Owen [40] in the wake of rectangular sections

with sinusoidal variation of the along width dimension (i.e. by introduction of wavy

front additions). Furthermore, the observation that a beyond-critical state emerges

very early is consistent with the acquired forcing spectra. As shown by Schewe [106],

the unstable supercritical state (supercritical to transcritical transition specifically) is

characterised by lift fluctuations at reduced frequencies in a broad band around 0.2,

along with stronger low frequency peaks, similar to the spectra found here (Fig. 5.9).

Note that differences between the two bubbles (or even inconsistency in the timing of

their existence) during the two bubble state can still give the opportunity for mean lift

to arise. Finally it should be stressed that what would appear as a second bubble can

yet be a non-conventional flow structure as earlier conjectured.

Considering the 77◦ and 90◦ inclination angles, different behaviour is found. All

rings seem to exhibit very similar behaviour between them for the mean drag force

coefficients. This is consistent with a previous finding of closer agreement for the

higher inclinations, between the mean coefficients from the four pressure tap rings

and those back-calculated from mean static displacements during the dynamic tests,


1 1.5 2 2.5 3 3.5 4 4.5

x 105

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Re

CD

1/2

Ring 2− rightRing 2− left

1 1.5 2 2.5 3 3.5 4 4.5

x 105

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Re

CD

1/2


Figure 5.12. Drag evolution acquired for ‘right’ (taps 4-3-. . . 20) and ‘left’ (taps 4-5-. . . 20)

parts of Rings 2 and 3, during static tests. Model setup ϕ=60◦. Double points correspond to

increasing and decreasing wind speed runs.

5.4. Discussion 109

0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 105

0

0.1

0.2

0.3

0.4

0.5

0.6

Re

CD

1/2


Figure 5.13. Drag evolution acquired for ‘right’ (taps 4-3-..20) and ‘left’ (taps 4-5-..20) parts

of Ring 3, during static tests. Model setup, ϕ=90◦.

which give a measure of the mean force over the whole cylinder length [226]. This

could well be due to the conjecture of increased inclination angles having a consistent

characteristic behaviour for the transitions of individual cylinder sections. Thus a

simple non-weighted averaging can be performed that closely matches the global forcing

coefficients. On the other hand, for ϕ=60◦, since different states exist along the cylinder

length, averaging over the four pressure tap rings with predefined weightings does not

so accurately represent the global forcing.

The drag crisis for Ring 3 for ϕ=90◦ is illustrated in Fig. 5.13. Similar one-bubble

formations exist on all rings in a (closely spaced) narrow defined range of Reynolds

number. It is significant that the Reynolds number corresponding to minimum drag is

increased relative to the previous inclined cable case (Fig. 5.12) as e.g. also in [111].

The difference is amplified further when the modification of calculating Re based on the

wind component normal to the cable, as successfully performed earlier for Sr, is used.

Logically some additional mechanism should be operating for the ‘early’ drag reduction

for the inclined cable. Apart from the evident curvature change of streamlines on the

cable surface, which translates to a higher effective Reynolds number, this could also

be due to the action of the quoted vortex dislocations, which when steadily located

along the length can act effectively towards drag reduction [40]. An interesting case of

enhanced drag reduction connected with stable flow patterns in the critical Reynolds

number range was previously given by Humphreys [103] when he performed static tests

tying fine silk threads at a cylinder’s stagnation points. One last feature not previously

explicitly reported was that of sudden avalanche-like (intermittency for ‘quiescent’ flow


Figure 5.14. Ring 1 CL and CD transitional avalanche-like behaviour. Setup ϕ=90◦,

Re=3.2×105, U=32m/s. Pressure distributions for time instants a, b, c, d are provided in

Fig.5.15.

conditions lends them this definition) jumps in the lift (and drag) coefficient, denoting

alternating transitions between symmetric and asymmetric states. Such a case is shown

in Fig.5.14, where Re=3.2×105, U=32m/s. These jumps, reminiscent of driven chaotic

vibrations (cc. the magnetoelastic problem [227]), could occur independently on all

rings, during both static and dynamic tests. Interestingly they were not recorded

as such for ϕ=60◦. Clearly due to this finding, when high-angle arrangements are

concerned, averaging of force coefficients should be performed cautiously. Analysis of

the pressure distributions around the cylinder confirmed that they were due to real

changes in the flow state. To illustrate this, four points (a,b,c,d) were selected in the

record of Fig.5.14 and their instantaneous pressures were drawn in Fig.5.15. As it can

be evidenced, all possible lift signs were realised. Time steps a and d do not only

produce opposite CL, but also quite distinct CD. This could well be due to a structural

asymmetry of the cylinder, however the fact that the lift flips sides during steady flow

seems as an indication that an unbiased circular behaviour prevails. It could thus be

a case where an ensemble of differently sized laminar separation bubbles (see §2.2)are possible. The sequence b→c→d was picked to surround a sudden drop in CL.

Noticeably during the prior to the jump instant b, a small one-sided ‘bump’ develops

in the back pressure region. When the intermittent process is midway through, at c,

two bumps can be seen in the same region in both sides. Strikingly this yields an

instantaneous Cp profile exactly as in Ring 3 in Fig.5.11.

5.4. Discussion 111

Figure 5.15. Pressure distributions during the state ‘vibration’ indicated in Fig.5.14. Steps

b, c, d describe the actual dynamic changes during a state jump. In the plots the radius

corresponds to Cp=1.

5.4.2 Mechanism implications

With all these observations in hand it is possible to speculate on possible flow mecha-

nisms underlying the dynamic excitation and on the influence of the cable inclination

angle. Some secondary flow that was identified in the case of ϕ=60◦ is evidently propa-

gating disturbances axially along the cable. Taking into account that no extreme state

jump, as in Fig. 5.14, was observed for the numerous recorded cases it seems reasonable

to suggest that, for this inclination, some reasonably ‘stable’ dynamic balance builds

up between sections along the cylinder. Still, the dynamic balance is built from cell

components that conceptually differ from each other thus becoming sources for vortex

dislocations. Each section is prone to the effects of turbulence and surface roughness,

as demonstrated by the frequent omission of the critical state in the transition from

subcritical to transcritical. Interaction between zones along the cylinder could possi-

bly allow for stable local formations of high lift to develop in the system (being an

indication of a structured asymmetry).

On the other hand, for ϕ=90◦ and 77◦ the sections appear to be less influenced

by the similar turbulence intensity and surface roughness conditions and all transition

states can clearly be distinguished. The existence of ‘delicate’ single bubble states

is probably proof of the good flow conditions during the whole series of tests. It is

suggested that the lack of significant secondary flow, providing the means for coupling,

inhibits the formation of a stable dynamic balance along the cylinder, allowing each

section to behave independently from its not too distant neighbours. The fact that


there seems to be some characteristic behaviour for the drag evolution, although data

are only available from a few sections, is not inconsistent with this conjecture. Although

considerable lift could occur locally, if the sign of lift is random, the expected total lift

on the cable could be zero, and in any case have unstable features. Vortex dislocations

if occurring (due to intermittent response) would also be unstable. Of course non-zero

total mean lift can stably occur due to the finite length of the cylinder or some exogenous

‘driving’ (e.g. blockage) or evidently if the lift sign statistics are not Gaussian. Yet

the above scenario could possibly explain the lack of large vibration incidents for high

inclination values in the Phase 3 tests. For lower cable-wind angles, the suggested

coherent dynamic behaviour over the length of the cable is more likely to result in

stable structured asymmetry (e.g. non-zero lift), which could be a significant factor in

a galloping mechanism.

5.5 Concluding remarks

Large amplitude vibrations of dry inclined cables are still an unravelled mystery with

ongoing research trying to establish connections and reasoning about the different pa-

rameters that may trigger instabilities. This chapter has attempted to add to the

previous observations of the behaviour, using the latest findings from a series of static

and dynamic tests on a realistic cable model equipped with pressure measurement taps

and identifying features not existing in the current literature.

An interesting finding was the identification of behaviour indicating two separation

bubbles existing very close to the subcritical Reynolds number range. This was only

found for the lowest cable-wind angle of ϕ=60◦ and ceased for ϕ=77◦ and 90◦. It

was also established that the cable at ϕ=60◦ could retain different flow states along

its length in a very stable way, while at the greater cable-wind angles the flow often

intermittently jumped between states, altering the lift sign and/or value. Moreover,

the Reynolds number value corresponding to minimum drag, nominally designating the

end of the critical state, was found to be significantly lower for the lowest cable-wind

angle. Accepting the limitation that only a few measurement sections were used, it

is suggested that some stable dynamic balance of different states builds up along the

cable for some inclination angles, probably connected with a specific range of along-

cylinder wind component values. Introduction of structures such as vortex dislocations,

resulting from a sustained asymmetry (e.g. as a distribution of pressure profiles similar

to Ring 3 in Fig. 5.11) along the cable, could be an important factor in the excitation

mechanism. On the other hand, it is thought that weak coupling, as appears to occur for

flow normal to the cylinder and more importantly avalache-like disrupting behaviour,

may tend to inhibit galloping.

5.5. Concluding remarks 113

Simultaneous existence of different flow states was suggested to be a case of different

sensitivity of inclination angles to (even low) turbulence intensity and roughness, but it

could alternatively be an effect of the inclination angle itself in smooth flow. The mean

features identified are indicative of the flow characteristics. In any case the critical

Reynolds number regime has been shown able to provide flow structures that could

well be responsible for dynamic instabilities of inclined cables. Ongoing work is aiming

to shed light on the links between these structures and the actual instability mechanism.

Chapter 6

Conclusion and outlook

The present work is a collection of aerodynamic studies concerning different parts of

flexible long-span bridges. In all cases an attempt was made to elucidate and understand

a number of distinct features that arise due to the complex structural interplay with

the wind. Specifically threatening self-excitation phenomena that affect bridges were

addressed. This concluding chapter aims to summarise the main findings and suggest

areas for further research.

Initially the flutter potential of the Clifton Suspension Bridge (CSB) was considered.

It is well accepted that plate girder sections as on the CSB are extremely vulnerable to

aerodynamic effects, yet in a modernised variant they are still in use. Utilising ambient

vibration measurements from a long-term monitoring campaign, it was possible to per-

form the conventional flutter analysis of Scanlan [12,13] in an inverse way and deduce

the flutter derivative description of the experienced aeroelastic loading. Records in-

cluding a quite wide variety of wind conditions gave a good range of reduced velocities

making possible further aeroelastic assessments. The results, obtained under uncon-

trolled conditions, seem to reliably follow values from wind tunnel sectional models of

similar sections. This is despite the fact that scaled tests are performed under homo-

geneous flow without the variations of natural wind. Interestingly it was found that

there is some similarity to the Tacoma Narrows Bridge, with single-degree-of-freedom

torsional instability being a possibility for the bridge for wind speeds not far beyond

the range experienced. For the highest obtained reduced velocities, flutter derivative

A∗

2 becomes positive (contributing negative torsional aerodynamic damping), reducing

the total available torsional damping. The actual A∗

2 function may be increasing even

more rapidly than estimated, since the possible amplitude-dependent increase of struc-

tural damping was not accounted for. Notably the analysis was successful in finding

an estimate of the critical flutter speed, based solely on full-scale measurements, for

the first time. The estimated value of approximately 21m/s is only slightly above the

115

116 Chapter 6. Conclusion and outlook

maximum recorded wind speed on the bridge. The findings also raised an interest-

ing question regarding the H∗

1 flutter derivative (responsible for vertical aerodynamic

damping). On the Tacoma Narrows Bridge the sign reversal of H∗

1 , similar to A∗

2,

was assigned to vortex shedding. However, it occurred far from the expected resonant

vortex shedding condition, which raises doubts about this explanation. On the CSB

there was no recorded vortex-induced response throughout the testing period, but it

shows a similar trend for H∗

1 (also at wind speeds far from the expected critical speed

for vortex shedding), suggesting that indeed it is due to some other cause. Aeroelastic

coupling action was also found but it was not a serious concern, at least for the range of

reduced velocities considered. This study essentially adds to the experience of aeroelas-

tic identification of bridges, which is very limited for full-scale structures, and provides

a practical example for any similar future study. In any case the main achievement

is that the best aeroelastic data of a real-scale bridge near critical behaviour to date

were clearly identified. The obvious means of complementing the analysis would be to

perform scaled wind tunnel tests or numerical simulations for comparison. There has

been no previous full-scale validation of bridge flutter analysis. This could substantiate

the empirical flutter framework currently in use and reassure engineering practice that

the safety margins that modern bridges are designed for are realistic. An added benefit

that could surface from the current analysis in the near future is putting forward a

convincing response to the long-standing question of how small localised changes can

improve the overall aerodynamic performance. This could be beneficial to assess the

wind risk of many existing bridges that were not designed using the recently devised

flutter-resisting framework.

Next a topic that has often been mishandled in the existing literature was consid-

ered;the generalisation of galloping in two dimensions. When the structural axes are

inclined to the wind direction, the original Den Hartog derivation for galloping motion

is not valid. The root of the error in some previous treatments (e.g. [189]) was iden-

tified and succinct expressions for generalised galloping modelling were devised. The

study was extended to cover two-degree-of-freedom motion with allowance for arbitrary

detuning between vibrations along the two principal axes. Although the foundations

for generalised 2DOF galloping were laid by Richardson and Martuccelli [190] in 1965

it has not previously been correctly and clearly presented and the implications quan-

tified. The presentation clarifies the way quasi-steady theory incorporates geometric

and structural details. Turning the derived force descriptions into instability criteria,

three benchmark galloping scenarios were considered: single-degree-of-freedom motion

normal to the wind and inclined to the wind and perfectly tuned two-degree-of-freedom

motion. These cover the range of possible instability boundaries. Employing published

data of static force coefficients it was possible to quantify the differences in the effective

117

aerodynamic damping (positive or negative) in the different cases. It was apparent that

large differences occur. Furthermore the influence of detuning on the evolution of the in-

stability has rarely been considered before. However it was shown that it is an essential

parameter in defining the true stability boundary. The investigation was able to refute

the suggestion that the introduction of an along-wind degree-of-freedom [137,191] will

necessarily stabilise a purely across-wind unstable motion. As presented, the potential

for the opposite behaviour also exists. A number of similar shortcomings in previous

analyses were also addressed, making the suggested updated galloping framework a

potentially valuable tool for wind studies of slender elements such as cables and bridge

towers.

For both the preceding sections, although Reynolds number (Re) was accepted as

a potentially influential variable, it was excluded from the analysis per se. The final

piece of this thesis attempted to tackle the controversial aerodynamic problem of dry

galloping of circular sections, in particular of inclined cables on cable-stayed bridges,

where the development of Reynolds-induced lift has been suspected of having a key role.

Therefore the study of Re effects was the focus of the final part of the investigation. An

experimental approach was adopted, testing both dynamic and static models of cables

with sectional dimensions as on real cable-stayed bridges. Observed large amplitude

responses were primarily of across-wind character and occurred close to the critical Re

and nowhere else. Quasi-steady theory, even in its most elaborate form [133], is unable

to fully interpret the details of most of the responses recorded. The results indicate

that only cylinders inclined at a limited range of angles exhibited large amplitude vi-

brations, while cylinders close to normal to the flow only experienced limited responses.

Strikingly it was recorded that for the near normal cable, aerodynamic forces vary in a

discontinuous non-stationary way, jumping between different laminar separation bubble

flow states and resulting in intermittent abrupt steps in the lift and drag time series.

This so-called avalanche-like behaviour was apparent during both static and dynamic

tests. The observed intermittent state jumps could not be identified on the cable in-

clined at angles that produced galloping-like responses, so they were conjectured to

have the function of a quenching disorder that effectively mitigates vibrations. In his

seminal work addressing the transitional Re behaviour of a smooth circular cylinder

normal to the flow, Schewe [228] was in search of period doubling phenomena to ac-

company the turbulent transitions on the cylinder’s surface. In the context of the new

findings it seems no surprise that he did not recover any. The current results confirm

that the route to the chaotic turbulent state in the boundary layers follow the alterna-

tive path of intermittency. Interestingly Schewe also found individual state jumps, but

these were global and were not in an ‘oscillating’ mode (i.e. once a jump occurred it

did not reverse) as in the present study. It would be most intriguing to discover how

118 Chapter 6. Conclusion and outlook

and why locally erratic behaviour with numerous asynchronous jumps can combine

over the cable length to result in the global spatio-temporal stable structure recorded

by Schewe. Due to a lack of global force measuring equipment in the experiments this

is not feasible from the current study. Intriguingly in §2.1.3 something similar was il-

lustrated for classical vortex shedding past the lock-in region(Fig. 2.3). Local pressure

fluctuations and global motion did not agree in terms of frequency content.

During large response events non-conventional flow structures near the back pres-

sure region that induce asymmetries were identified. Similarities were established be-

tween the current tests and the numerical simulations of Yeo and Jones [116, 117],

though no significant dynamic response occurred in the experiments near their simu-

lation region as they anticipated. Finally it was summarised how correlation changes

along the length of the cable, for cables inclined at various angles, influence the global

dynamics. It was surmised that the dry galloping region overlaps with the critical Re

regime not only because of an expected appearance of lift but due to the multitude of

complex flow features that emerge. If such features are possible to appear in other flow

regimes, which seems highly unlikely, then these would also be potential regions for dry

galloping.

In a study with similar aims to this one, Symes [229] conducted static wind tunnel

tests for a smooth circular cylinder normal to the flow, in the critical Re range. He

concluded that although nominally circular cylinders are normally treated as perfectly

symmetric, they could have a consistent rotational asymmetry that may lead to a clas-

sical galloping response similar to the one addressed in Chapter 4. It is uncertain how

such a behaviour would combine with the more complex picture captured here to form a

mixed origin instability. Additionally the range of Reynolds numbers where the critical

regime was found was unusually low, raising questions as to whether nominally precrit-

ical wind speeds on bridges could actually be critical. Future wind tunnel tests should

also verify the circularity of the model cylinder, which would enable consideration of

the role of the different possible galloping-like mechanisms involved.

The intermittent and noisy aerodynamic force fluctuations with the distinct abrupt

flow state jumps had a disturbance role since they did not allow motion-induced load-

ing to set in. Yet is is questionable whether this is always true. Potentially the self-

excitation mechanism, if existing, can carry on despite the appearance of noise or even

use the energy content in the noise spectrum to develop. Actually non-harmonic forces

with sharp noisy peaks have been seen during large rain-wind vibrations [97] (Chap-

ter 2, Fig. 2.10). Preliminary results from circular cylinders forced to vibrate with

large amplitudes indicate that there may well be such a different function even under

dry conditions. It should be pointed out that as Parkinson comments [23] (based on

the unique studies of Staubli [230] and Bearman [202]), in regions such as the critical

119

Re one, the equivalence between forced and free vibration tests could be invalid. Thus

it would be useful to complement the series of free vibration tests studied here with

equivalent forced ones and compare the relevant aerodynamic force characteristics. In

the spirit of Bishop and Hassan [32], who first suggested modelling the fluid-structure

interaction during vortex-shedding with a nonlinear Van der Poll oscillator, this study

wishes to conclude by proposing an alternative modelling for dry galloping vibrations.

Combining the actual ‘noisy’ intermittent state jumps with the suspected contribution

when ‘driving’ the cylinder during forced vibration tests, it is conceivable that a formu-

lation based on stochastic resonance could reproduce the dry galloping phenomenon.

The existence of a second harmonic in the motion-induced lift presented in Fig. 5.9, is

an additional detail that can also justify this choice.

Publications

Author’s publications related to the present thesis.

Journal papers

Nikitas N, Macdonald JHG, and Jakobsen JB. Wind induced Vibrations of the Clifton

Suspension Bridge. Wind Struct., 14(3):221-238, 2011.

Nikitas N, Macdonald JHG, and Jakobsen JB, Andersen TL. Critical Reynolds number

and galloping instabilities – Experiments on circular cylinders. Accepted in Exp. Fluids.

Nikitas N, and Macdonald JHG. Misconceptions and generalisations of the Den Hartog

galloping criterion. Submitted for publication in J. Eng. Mech.-ASCE.

Conference Proceedings

Nikitas N, Macdonald JHG, and Jakobsen JB. Full Scale Identification of Modal and

Aeroelastic Parameters of the Clifton Suspension Bridge. In 6th International Col-

loquium on Bluff Bodies Aerodynamics & Applications (BBAA VI), pages 135–138,

Milano, Italy, 2008.

Macdonald JHG, Nikitas N, Symes JA, Jakobsen JB, Andersen TL, Savage MG, and

McAuliffe BR. Large-scale wind tunnel tests of inclined cable vibrations- Preliminary

findings. In 8th UK Conference on Wind Engineering, Guildford, UK, 2008.

Nikitas N, Macdonald JHG, and Jakobsen JB., Andersen TL, Savage MG, McAuliffe

BR. Wind Tunnel testing of an inclined cable model-Pressure and motion characteris-

tics, Part I. In 5th European & African Conference on Wind Engineering (EACWE5),

Florence, Italy, 2009.

Jakobsen JB., Andersen TL, Macdonald JHG, Nikitas N, Savage MG, and McAuliffe

BR. Wind Tunnel testing of an inclined cable model-Pressure and motion characteris-

tics, Part II. In 5th European & African Conference on Wind Engineering (EACWE5),

Florence, Italy, 2009.

121

122 PUBLICATIONS

Andersen TL, Jakobsen JB, Macdonald JHG, Nikitas N, Larose GL, Savage MG,and

McAuliffe BR. Drag-crisis response of elastic cable-model. In 8th International Sym-

posium on Cable Dynamics, Paris, France, 2009.

Nikitas N, and Macdonald JHG. The Den Hartog galloping criterion revisited: a non-

classical case. In 9th UK Conference on Wind Engineering (WES-2010), Bristol, UK,

2010.

Zhang J, Au FTK, Li J, Nikitas N, Macdonald JHG, and Jacobsen JB. Identifying

bridge aeroelastic parameters from full-scale ambient vibration data. In 9th UK Con-

ference on Wind Engineering (WES-2010), Bristol, UK, 2010.

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