Theory of Bridge Aerodynamics (eBook)

PREFACE 7
CONTENTS 9
NOTATION 11
1 INTRODUCTION 17
1.1 General considerations 17
1.2 Random variables and stochastic processes 20
1.3 Basic flow and structural axis definitions 22
2 SOME BASIC STATISTICAL CONCEPTS IN WIND ENGINEERING 29
2.1 Parent probability distributions, mean value and variance 29
2.2 Time domain and ensemble statistics 31
2.3 Threshold crossing and peaks 43
2.4 Extreme values 46
2.5 Auto spectral density 49
2.6 Cross-spectral density 54
2.7 The connection between spectra and covariance 57
2.8 Coherence function and normalized co-spectrum 59
2.9 The spectral density of derivatives of processes 60
2.10 Spatial averaging in structural response calculations 61
3 STOCHASTIC DESCRIPTION OF TURBULENT WIND 69
3.1 Mean wind velocity 69
3.2 Single point statistics of wind turbulence 74
3.3 The spatial properties of wind turbulence 79
4 BASIC THEORY OF STOCHASTIC DYNAMIC RESPONSE CALCULATIONS 85
4.1 Modal analysis and dynamic equilibrium equations 85
4.2 Single mode single component response calculations 92
4.3 Single mode component response calculations 97
4.4 General multi-mode response calculations 100
5 WIND AND MOTION INDUCED LOADS 107
5.1 The buffeting theory 107
5.2 Aerodynamic derivatives 113
6 WIND INDUCED STATIC AND DYNAMIC RESPONSE CALCULATIONS 125
6.1 Introdction 125
6.2 The mean value of the response 129
6.3 Buffeting response 132
6.4 Vortex shedding 158
7 DETERMINATION OF CROSS SECTIONAL FORCES 173
7.1 Introduction 173
7.2 The mean value 179
7.3 The background quasi–static part 179
7.4 The resonant part 198
8 MOTION INDUCED INSTABILITIES 211
8.1 Introduction 211
8.2 Static divergence 215
8.3 Galloping 216
8.4 Dynamic stability limit in torsion 217
8.5 Flutter 219
Appendix A TIME DOMAIN SIMULATIONS 225
A.1 Introduction 225
A.2 Simulation of single point time series 226
A.3 Simulation of spatially non–coherent time series 229
A.4 The Cholesky decomposition 237
Appendix B DETERMINATION OF THE JOINT ACCEPTANCE FUNCTION 239
B.1 Closed form solutions 239
B.2 Numerical solutions 239
Appendix C AERODYNAMIC DERIVATIVES FROM SECTION MODEL DECAYS 243
REFERENCES 249
INDEX 251

Chapter 1 INTRODUCTION (p. 1)

1.1 General considerations

This text book focuses exclusively on the prediction of wind induced static and dynamic response of slender line-like civil engineering structures. Throughout the main part of the book it is taken for granted that the structure is horizontal, i.e. a bridge, but the theory is generally applicable to any line–like type of structure, and thus, it is equally applicable to e.g. a vertical tower.

It is a general assumption that structural behaviour is linear elastic and that any non-linear part of the relationship between load and structural displacement may be disregarded. It is also taken for granted that the main flow direction throughout the entire span of the structure is perpendicular to the axis in the direction of its span.

The wind velocity vector is split into three fluctuating orthogonal components, U in the main flow along–wind direction, and v and w in the across wind horizontal and vertical directions. For a relevant structural design situation it is assumed that U may be split into a mean value V that only varies with height above ground level and a fluctuating part u, i.e. U =V+ u& . V is the commonly known mean wind velocity, and u, v and w are the zero mean turbulence components, created by friction between the terrain and the flow of the main weather system.

It is taken for granted that the instantaneous wind velocity pressure is given by Bernoulli’s equation If an air flow is met by the obstacle of a more or less solid line-like body, the flow/structure interaction will give raise to forces acting on the body. Unless the body is extremely streamlined and the speed of the flow is very low and smooth, these forces will fluctuate. Firstly, the oncoming flow in which the body is submerged contains turbulence, i.e. it is itself fluctuating in time and space.

Secondly, on the surface of the body additional flow turbulence and vortices are created due to friction, and if the body has sharp edges the flow will separate on these edges and the flow passing the body is unstable in the sense that a variable part of it will alternate from one side to the other, causing vortices to be shed in the wake of the body.

And finally, if the body is flexible the fluctuating forces may cause the body to oscillate, and the alternating flow and the oscillating body may interact and generate further forces. Thus, the nature of wind forces may stem from pressure fluctuations (turbulence) in the oncoming flow, vortices shed on the surface and into the wake of the body, and from the interaction between the flow and the oscillating body itself.

The first of these effects is known as buffeting, the second as vortex shedding, and the third is usually labelled motion induced forces. In literature, the corresponding response calculations are usually treated separately.

The reason for this is that for most civil engineering structures they occur at their strongest in fairly separate wind velocity regions, i.e. vortex shedding is at its strongest at fairly low wind velocities, buffeting occur at stronger wind velocities, while motion induced forces are primarily associated with the highest wind velocities.