Theory of Bridge Aerodynamics

250 Pages·2006·7.86 MB·English

by  Einar Strømmen

TheoryofBridgeAerodynamics
Einar N. Strømmen
Theory of Bridge
Aerodynamics
ABC
ProfessorDr.EinarN.Strømmen
DepartmentofStructuralEngineering
NorwegianUniversity
ofScienceandTechnology
7491Trondheim,Norway
E-mail:[email protected]
LibraryofCongressControlNumber:2005936355
ISBN-10 3-540-30603-XSpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-30603-0SpringerBerlinHeidelbergNewYork
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To
Mary, Hannah, Kristian and Sigrid
PREFACE

This text book is intended for studies in wind engineering, with focus on the stochastic
theory of wind induced dynamic response calculations for slender bridges or other
line−like civil engineering type of structures. It contains the background assumptions
and hypothesis as well as the development of the computational theory that is necessary
for the prediction of wind induced fluctuating displacements and cross sectional forces.
The simple cases of static and quasi-static structural response calculations are for the
sake of completeness also included.
The text is at an advanced level in the sense that it requires a fairly comprehensive
knowledge of basic structural dynamics, particularly of solution procedures in a modal
format. None of the theory related to the determination of eigen−values and the
corresponding eigen−modes are included in this book, i.e. it is taken for granted that the
reader is familiar with this part of the theory of structural dynamics. Otherwise, the
reader will find the necessary subjects covered by e.g. Clough & Penzien [2] and
Meirovitch [3]. It is also advantageous that the reader has some knowledge of the theory
of statistical properties of stationary time series. However, while the theory of structural
dynamics is covered in a good number of text books, the theory of time series is not, and
therefore, the book contains most of the necessary treatment of stationary time series
(chapter 2).
The book does not cover special subjects such as rain-wind induced cable vibrations.
Nor does it cover all the various available theories for the description of vortex shedding,
as only one particular approach has been chosen. The same applies to the presentation of
time domain simulation procedures. Also, the book does not contain a large data base for
this particular field of engineering. For such a data base the reader should turn to e.g.
Engineering Science Data Unit (ESDU) [7] as well as the relevant standards in wind and
structural engineering.
The writing of this book would not have been possible had I not had the fortune of
working for nearly fifteen years together with Professor Erik Hjorth–Hansen on a
considerable number of wind engineering projects.
The drawings have been prepared by Anne Gaarden. Thanks to her and all others who
have contributed to the writing of this book.

Trondheim EinarN.Strømmen
August, 2005
CONTENTS

Preface vii
Notation x i

1 INTRODUCTION 1
1.1 General considerations 1
1.2 Random variables and stochastic processes 4
1.3 Basic flow and structural axis definitions 6
1.4 Structural design quantities 10

2 SOME BASIC STATISTICAL CONCEPTS
IN WIND ENGINEERING 13
2.1 Parent probability distributions, mean value and variance 13
2.2 Time domain and ensemble statistics 15
2.3 Threshold crossing and peaks 27
2.4 Extreme values 30
2.5 Auto spectral density 33
2.6 Cross spectral density 38
2.7 The connection between spectra and covariance 41
2.8 Coherence function and normalized co–spectrum 43
2.9 The spectral density of derivatives of processes 44
2.10 Spatial averaging in structural response calculations 45

3 STOCHASTIC DESCRIPTION OF
TURBULENT WIND 53
3.1 Mean wind velocity 53
3.2 Single point statistics of wind turbulence 58
3.3 The spatial properties of wind turbulence 63

4 BASIC THEORY OF STOCHASTIC DYNAMIC
RESPONSE CALCULATIONS 69
4.1 Modal analysis and dynamic equilibrium equations 69
4.2 Single mode single component response calculations 76
4.3 Single mode three component response calculations 81
4.4 General multi–mode response calculations 84

5 WIND AND MOTION INDUCED LOADS 91
5.1 The buffeting theory 91
5.2 Aerodynamic derivatives 97
5.3 Vortex shedding 102
CONT ENTS x

6 WIND INDUCED STATIC AND DYNAMIC
RESPONSE CALCULATIONS 109
6.1 Introduction 109
6.2 The mean value of the response 113
6.3 Buffeting response 116
6.4 Vortex shedding 142

7 DETERMINATION OF CROSS SECTIONAL
FORCES 157
7.1 Introduction 157
7.2 The mean value 163
7.3 The background quasi static part 163
7.4 The resonant part 182

8 MOTION INDUCED INSTABILITIES 195
8.1 Introduction 195
8.2 Static divergence 199
8.3 Galloping 200
8.4 Dynamic stability limit in torsion 201
8.5 Flutter 203

Appendix A: TIME DOMAIN SIMULATIONS 209
A.1 Introduction 209
A.2 Simulation of single point time series 210
A.3 Simulation of spatially non–coherent time series 213
A.4 The Cholesky decomposition 221

Appendix B: DETERMINATION OF THE JOINT
ACCEPTANCE FUNCTION 223
B.1 Closed form solutions 223
B.2 Numerical solutions 223

Appendix C: AERODYNAMIC DERIVATIVES FROM
SECTION MODEL DECAYS 227

References 233
Index 235
NOTATION

Matrices and vectors:
Matrices are in general bold upper case Latin or Greek letters, e.g. Q or Φ.
Vectors are in general bold lower case Latin or Greek letters, e.g. q or φ.
diag[⋅] is a diagonal matrix whose content is written within the brackets.
det(⋅) is the determinant of the matrix within the brackets.

Statistics:
E[⋅] is the average value of the variable within the brackets.
Pr[⋅] is the probability of the event given within the bracket.
P(x) is the cumulative probability function, P(x)=Pr[X ≤x].
p(x) is the probability density function of variable x.
Var(⋅) is the variance of the variable within the brackets.
Cov(⋅) is the covariance of the variable within the brackets.
Coh(⋅)is the coherence function of the content within the brackets.
R(⋅) is the auto- or cross-correlation function.
R is short for return period.
p
ρ(⋅) is the covariance (or correlation) coefficient of content within brackets.
ρ is a cross covariance or correlation matrix between a set of variables.
σ,σ2is the standard deviation, variance.
µ is a quantified small probability.

Imaginary quantities:
i is the imaginary unit (i.e. i = −1).
Re[⋅] is the real part of the variable within the brackets.
Im[⋅] is the imaginary part of the variable within the brackets.

Superscripts and bars above symbols:
Super-script T indicates the transposed of a vector or a matrix.
Super-script * indicates the complex conjugate of a quantity.
Dots above symbols (e.g. r(cid:5), (cid:5)r(cid:5)) indicates time derivatives, i.e. d/dt, d2/dt2.
xii NOTATION
A prime on a variable (e.g.C′ or φ′) indicates its derivative with respect to a relevant
L
variable (except t), e.g. C′ =dC dα and φ′=dφdx . Two primes is then the second
L L
derivative (e.g. φ′′=d2φ dx2 ) and so on.
Line ( − ) above a variable (e.g. C ) indicates its average value.
D
(cid:4)
A tilde ( ∼ ) above a symbol (e.g. M ) indicates a modal quantity.
i
A hat ( ∧ ) above a symbol (e.g. Bˆ ) indicates a normalised quantity.

The use of indexes:
Index x,y or z refers to the corresponding structural axis.
x ,y or z refers to the corresponding flow axis.
f f f
u,v or w refers to flow components.
i and j are mode shape numbers.
m refers to y,z or θ directions, n refers to u,v or w flow components.
p and k are in general used as node numbers.
F represents a cross sectional force component.
D,L,M refers to drag, lift and moment.
tot,B,R indicate total, background or resonant.
ae is short for aerodynamic, i.e. it indicates a flow induced quantity.
cr is short for critical.
max,min are short for maximum and minimum.
pv is short for peak value.
r is short for response.
s indicates quantities associated with vortex shedding.

Abbreviations:
CC and SC are short for cross-sectional neutral axis centre and shear centre.
FFT is short for Fast Fourier Transform. Sym. is short for symmetry.

∫ means integration over the wind exposed part of the structure.
Lexp
∫ means integration over the entire length of the structure.
L
NOTATION xiii
Latin letters

A Aerodynamic admittance functions (m = y, z or θ, n = u or w)
mn
A* −A* Aerodynamic derivatives associated with the motion in torsion
1 6
a Constant or Fourier coefficient
B Cross sectional width
B or Bˆ Buffeting dynamic load coefficient matrix
q q
b Constant, coefficient, band-width parameter
b or bˆ Mean wind load coefficient vector
q q
C or C, Damping coefficient or matrix containing damping coefficient
C Force coefficients at mean angle of incidence
C′ Slope of load coefficient curves at mean angle of incidence
c Constant, coefficient, Fourier amplitude
D Cross sectional depth
d Constant or coefficient
d Beam element displacement vector
E Modulus of elasticity
Eˆ,Eˆ Impedance, impedance matrix
e Eccentricity, distance between shear centre and cetroid
F,F Force vector, force at (beam) element level
f,f Frequency [Hz], eigen–frequency associated with mode i
i
f (⋅) Function of variable within brackets
G Modulus of elasticity in shear
G or G Influence function or matrix (F =V ,V ,M ,M or M )
F F y z x y z
g(⋅) Function of variable within brackets
H* −H* Aerodynamic derivatives associated with the across-wind motion
1 6
H or H Frequency response function or matrix
I Centroidal polar moment of inertia
p
I ,I St Venant torsion and warping constants
t w
I ,I ,I Turbulence intensity of flow components u, v or w
u v w
I , I Moment of inertia with respect to y or z axis
y z
I Identity matrix
I Turbulence matrix (I =diag[I I ] or I =diag[I I I ])
v v u w v u v w
i The imaginary unit (i.e. i = −1) or index variable