Theory of Bridge Aerodynamics

316 Pages·2010·6.1 MB·English

by  Einar Strømmen

TheoryofBridgeAerodynamics
Einar N. Strømmen
Theory of Bridge
Aerodynamics
ABC
Prof.Dr.EinarN.Strømmen
NorwegianUniversityofScienceandTechnology
DepartmentofStructuralEngineering
7491Trondheim
Norway
E-mail:[email protected]
LibraryofCongressControlNumber:2010928591
ISBN-13978-3-642-13659-7SpringerBerlinHeidelbergNewYork
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To
Mary, Hannah, Kristian and Sigrid
PREFACE TO 2ND EDITION
In this second edition a new chapter has been added covering the buffeting theory in a
finite element format. The motivation for this has been that a finite element format is
becoming more and more dominant in all areas of structural mechanics. It is streamlined
for computer programming, and it facilitates the use of general purpose routines that are
applicable in several types of structural engineering problems. In this book the finite
element formulation of the problem of dynamic response calculations follows the
general principle of virtual work, a general principle which may be found in many other
text books. While the buffeting wind load itself has with no trouble been included in a
finite element format, the main challenge has been to obtain a consistent formulation that
includes all the relevant motion induced forces. This has been important, because, while
many structures (e.g. long-span suspension bridges) may suffer greatly and become
unstable at high wind velocities, the same structures may also benefit from these effects
at the design wind velocity. It is well known that motion induced forces will change the
stiffness and damping properties of the combined structure and flow system. If
calculations are performed for a suitably close set of increasing mean wind velocities
and the changing mechanical properties (stiffness and damping) are updated from one
velocity to the next, then the response of the system may be followed up to wind
velocities close to the stability limit, i.e. up to response values that are perceived as
unduly large. Finite element calculations may be performed in time domain, in frequency
domain or converted into a modal format. All these options have been included.
Pursuing a time domain solution strategy requires the use of the so-called indicial
functions. The theory behind such a formulation is also covered, and the determination
of these functions from aerodynamic derivatives has been included in a separate
appendix.
A comment regarding the use of aerodynamic derivatives obtained from aeroelastic
wind tunnel experiments to predict structural response has been included.
It goes without saying that typing errors and calculation errors in examples that I so far
have come across in the first edition have been corrected.
Trondheim, January 2010 Einar N. Strömmen
[email protected]
PREFACE TO 1ST EDITION
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, August 2005 Einar N. Strömmen
[email protected]
CONTENTS
1 INTRODUCTION………………………………………………………………...1
1.1 General considerations………………………………………………………...1
1.2 Random variables and stochastic processes…………………………………...3
1.3 Basic flow and structural axis definitions……………………………..............6
1.4 Structural design quantities……………………………....................................9
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………………………………………………………………..29
2.5 Auto spectral density…………………………………………………………33
2.6 Cross-spectral density..……………………………………………………….38
2.7 The connection between spectra and covariance……………………………..40
2.8 Coherence function and normalized co-spectrum……………………………42
2.9 The spectral density of derivatives of processes……………………………...43
2.10 Spatial averaging in structural response calculations………………………...44
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………………………...80
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
XII CONTENTS
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…………………………………………...202
8.5 Flutter………………………………………………………………………..203
9 THE BUFFETING THEORY IN A FINITE ELEMENT FORMAT…….....209
9.1 Introduction………………………………………………………………….209
9.2 The element mechanical properties…………………………………………212
9.3 The wind load……………………………………………………………….220
9.4 The global analysis………………………………………………………….230
9.5 The time invariant static solution……………………………………………234
9.6 The quasi-static solution…………………………………………………….234
9.7 Dynamic response calculations in frequency domain………………………238
9.8 Frequency domain response calculations in modal coordinates…………….246
9.9 Dynamic response calculations in time domain…………………………….251
Appendix A: TIME DOMAIN SIMULATIONS………………………………….263
A.1 Introduction…………………………………………………………………263
A.2 Simulation of single point time series………………………………………264
A.3 Simulation of spatially non–coherent time series…………………………..267
A.4 The Cholesky decomposition……………………………………………….275
Appendix B: DETERMINATION OF THE JOINT ACCEPTANCE
FUNCTION…………………………………………………………..277
B.1 Closed form solutions……………………………………………………………..277
B.2 Numerical solutions……………………………………………………………….278
Appendix C: AERODYNAMIC DERIVATIVES FROM SECTION MODEL
DECAYS……………………………………………………………...281
CONTENTS XIII
Appendix D: DETERMINATION OF INDICIAL FUNCTIONS FROM
AERODYNAMIC DERIVATIVES………………………………...289
REFERENCES………….…………………………………………………………………….297
INDEX………….………………………………………………………………………………299